function d_is_int ( r ) !******************************************************************************* ! !! D_IS_INT determines if a double precision number represents an integer value. ! ! Modified: ! ! 07 September 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R, the number to be checked. ! ! Output, logical D_IS_INT, is TRUE if R is an integer value. ! implicit none ! integer, parameter :: i = 1 real ( kind = 8 ) r logical d_is_int ! if ( real ( huge ( i ), kind = 8 ) < r ) then d_is_int = .false. else if ( r < - real ( huge ( i ), kind = 8 ) ) then d_is_int = .false. else if ( r == real ( int ( r ), kind = 8 ) ) then d_is_int = .true. else d_is_int = .false. end if return end SUBROUTINE DBDSDC( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q, IQ, & WORK, IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! December 1, 1999 ! ! .. Scalar Arguments .. CHARACTER COMPQ, UPLO INTEGER INFO, LDU, LDVT, N ! .. ! .. Array Arguments .. INTEGER IQ( * ), IWORK( * ) DOUBLE PRECISION D( * ), E( * ), Q( * ), U( LDU, * ), & VT( LDVT, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DBDSDC computes the singular value decomposition (SVD) of a real ! N-by-N (upper or lower) bidiagonal matrix B: B = U * S * VT, ! using a divide and conquer method, where S is a diagonal matrix ! with non-negative diagonal elements (the singular values of B), and ! U and VT are orthogonal matrices of left and right singular vectors, ! respectively. DBDSDC can be used to compute all singular values, ! and optionally, singular vectors or singular vectors in compact form. ! ! This code makes very mild assumptions about floating point ! arithmetic. It will work on machines with a guard digit in ! add/subtract, or on those binary machines without guard digits ! which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. ! It could conceivably fail on hexadecimal or decimal machines ! without guard digits, but we know of none. See DLASD3 for details. ! ! The code currently call DLASDQ if singular values only are desired. ! However, it can be slightly modified to compute singular values ! using the divide and conquer method. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': B is upper bidiagonal. ! = 'L': B is lower bidiagonal. ! ! COMPQ (input) CHARACTER*1 ! Specifies whether singular vectors are to be computed ! as follows: ! = 'N': Compute singular values only; ! = 'P': Compute singular values and compute singular ! vectors in compact form; ! = 'I': Compute singular values and singular vectors. ! ! N (input) INTEGER ! The order of the matrix B. N >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the n diagonal elements of the bidiagonal matrix B. ! On exit, if INFO=0, the singular values of B. ! ! E (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the elements of E contain the offdiagonal ! elements of the bidiagonal matrix whose SVD is desired. ! On exit, E has been destroyed. ! ! U (output) DOUBLE PRECISION array, dimension (LDU,N) ! If COMPQ = 'I', then: ! On exit, if INFO = 0, U contains the left singular vectors ! of the bidiagonal matrix. ! For other values of COMPQ, U is not referenced. ! ! LDU (input) INTEGER ! The leading dimension of the array U. LDU >= 1. ! If singular vectors are desired, then LDU >= max( 1, N ). ! ! VT (output) DOUBLE PRECISION array, dimension (LDVT,N) ! If COMPQ = 'I', then: ! On exit, if INFO = 0, VT' contains the right singular ! vectors of the bidiagonal matrix. ! For other values of COMPQ, VT is not referenced. ! ! LDVT (input) INTEGER ! The leading dimension of the array VT. LDVT >= 1. ! If singular vectors are desired, then LDVT >= max( 1, N ). ! ! Q (output) DOUBLE PRECISION array, dimension (LDQ) ! If COMPQ = 'P', then: ! On exit, if INFO = 0, Q and IQ contain the left ! and right singular vectors in a compact form, ! requiring O(N log N) space instead of 2*N**2. ! In particular, Q contains all the DOUBLE PRECISION data in ! LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1)))) ! words of memory, where SMLSIZ is returned by ILAENV and ! is equal to the maximum size of the subproblems at the ! bottom of the computation tree (usually about 25). ! For other values of COMPQ, Q is not referenced. ! ! IQ (output) INTEGER array, dimension (LDIQ) ! If COMPQ = 'P', then: ! On exit, if INFO = 0, Q and IQ contain the left ! and right singular vectors in a compact form, ! requiring O(N log N) space instead of 2*N**2. ! In particular, IQ contains all INTEGER data in ! LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1)))) ! words of memory, where SMLSIZ is returned by ILAENV and ! is equal to the maximum size of the subproblems at the ! bottom of the computation tree (usually about 25). ! For other values of COMPQ, IQ is not referenced. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (LWORK) ! If COMPQ = 'N' then LWORK >= (4 * N). ! If COMPQ = 'P' then LWORK >= (6 * N). ! If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N). ! ! IWORK (workspace) INTEGER array, dimension (8*N) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: The algorithm failed to compute an singular value. ! The update process of divide and conquer failed. ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Huan Ren, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 ) ! .. ! .. Local Scalars .. INTEGER DIFL, DIFR, GIVCOL, GIVNUM, GIVPTR, I, IC, & ICOMPQ, IERR, II, IS, IU, IUPLO, IVT, J, K, KK, & MLVL, NM1, NSIZE, PERM, POLES, QSTART, SMLSIZ, & SMLSZP, SQRE, START, WSTART, Z DOUBLE PRECISION CS, EPS, ORGNRM, P, R, SN ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANST EXTERNAL LSAME, ILAENV, DLAMCH, DLANST ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLARTG, DLASCL, DLASD0, DLASDA, DLASDQ, & DLASET, DLASR, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, INT, LOG, SIGN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IUPLO = 0 IF( LSAME( UPLO, 'U' ) ) & IUPLO = 1 IF( LSAME( UPLO, 'L' ) ) & IUPLO = 2 IF( LSAME( COMPQ, 'N' ) ) THEN ICOMPQ = 0 ELSE IF( LSAME( COMPQ, 'P' ) ) THEN ICOMPQ = 1 ELSE IF( LSAME( COMPQ, 'I' ) ) THEN ICOMPQ = 2 ELSE ICOMPQ = -1 END IF IF( IUPLO.EQ.0 ) THEN INFO = -1 ELSE IF( ICOMPQ.LT.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( ( LDU.LT.1 ) .OR. ( ( ICOMPQ.EQ.2 ) .AND. ( LDU.LT. & N ) ) ) THEN INFO = -7 ELSE IF( ( LDVT.LT.1 ) .OR. ( ( ICOMPQ.EQ.2 ) .AND. ( LDVT.LT. & N ) ) ) THEN INFO = -9 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DBDSDC', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN SMLSIZ = ILAENV( 9, 'DBDSDC', ' ', 0, 0, 0, 0 ) IF( N.EQ.1 ) THEN IF( ICOMPQ.EQ.1 ) THEN Q( 1 ) = SIGN( ONE, D( 1 ) ) Q( 1+SMLSIZ*N ) = ONE ELSE IF( ICOMPQ.EQ.2 ) THEN U( 1, 1 ) = SIGN( ONE, D( 1 ) ) VT( 1, 1 ) = ONE END IF D( 1 ) = ABS( D( 1 ) ) RETURN END IF NM1 = N - 1 ! ! If matrix lower bidiagonal, rotate to be upper bidiagonal ! by applying Givens rotations on the left ! WSTART = 1 QSTART = 3 IF( ICOMPQ.EQ.1 ) THEN CALL DCOPY( N, D, 1, Q( 1 ), 1 ) CALL DCOPY( N-1, E, 1, Q( N+1 ), 1 ) END IF IF( IUPLO.EQ.2 ) THEN QSTART = 5 WSTART = 2*N - 1 DO I = 1, N - 1 CALL DLARTG( D( I ), E( I ), CS, SN, R ) D( I ) = R E( I ) = SN*D( I+1 ) D( I+1 ) = CS*D( I+1 ) IF( ICOMPQ.EQ.1 ) THEN Q( I+2*N ) = CS Q( I+3*N ) = SN ELSE IF( ICOMPQ.EQ.2 ) THEN WORK( I ) = CS WORK( NM1+I ) = -SN END IF end do END IF ! ! If ICOMPQ = 0, use DLASDQ to compute the singular values. ! IF( ICOMPQ.EQ.0 ) THEN CALL DLASDQ( 'U', 0, N, 0, 0, 0, D, E, VT, LDVT, U, LDU, U, & LDU, WORK( WSTART ), INFO ) GO TO 40 END IF ! ! If N is smaller than the minimum divide size SMLSIZ, then solve ! the problem with another solver. ! IF( N.LE.SMLSIZ ) THEN IF( ICOMPQ.EQ.2 ) THEN CALL DLASET( 'A', N, N, ZERO, ONE, U, LDU ) CALL DLASET( 'A', N, N, ZERO, ONE, VT, LDVT ) CALL DLASDQ( 'U', 0, N, N, N, 0, D, E, VT, LDVT, U, LDU, U, & LDU, WORK( WSTART ), INFO ) ELSE IF( ICOMPQ.EQ.1 ) THEN IU = 1 IVT = IU + N CALL DLASET( 'A', N, N, ZERO, ONE, Q( IU+( QSTART-1 )*N ), & N ) CALL DLASET( 'A', N, N, ZERO, ONE, Q( IVT+( QSTART-1 )*N ), & N ) CALL DLASDQ( 'U', 0, N, N, N, 0, D, E, & Q( IVT+( QSTART-1 )*N ), N, & Q( IU+( QSTART-1 )*N ), N, & Q( IU+( QSTART-1 )*N ), N, WORK( WSTART ), & INFO ) END IF GO TO 40 END IF ! IF( ICOMPQ.EQ.2 ) THEN CALL DLASET( 'A', N, N, ZERO, ONE, U, LDU ) CALL DLASET( 'A', N, N, ZERO, ONE, VT, LDVT ) END IF ! ! Scale. ! ORGNRM = DLANST( 'M', N, D, E ) IF( ORGNRM.EQ.ZERO ) & RETURN CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, IERR ) CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, NM1, 1, E, NM1, IERR ) ! EPS = DLAMCH( 'Epsilon' ) ! MLVL = INT( LOG( DBLE( N ) / DBLE( SMLSIZ+1 ) ) / LOG( TWO ) ) + 1 SMLSZP = SMLSIZ + 1 ! IF( ICOMPQ.EQ.1 ) THEN IU = 1 IVT = 1 + SMLSIZ DIFL = IVT + SMLSZP DIFR = DIFL + MLVL Z = DIFR + MLVL*2 IC = Z + MLVL IS = IC + 1 POLES = IS + 1 GIVNUM = POLES + 2*MLVL ! K = 1 GIVPTR = 2 PERM = 3 GIVCOL = PERM + MLVL END IF ! DO 20 I = 1, N IF( ABS( D( I ) ).LT.EPS ) THEN D( I ) = SIGN( EPS, D( I ) ) END IF 20 CONTINUE ! START = 1 SQRE = 0 ! DO 30 I = 1, NM1 IF( ( ABS( E( I ) ).LT.EPS ) .OR. ( I.EQ.NM1 ) ) THEN ! ! Subproblem found. First determine its size and then ! apply divide and conquer on it. ! IF( I.LT.NM1 ) THEN ! ! A subproblem with E(I) small for I < NM1. ! NSIZE = I - START + 1 ELSE IF( ABS( E( I ) ).GE.EPS ) THEN ! ! A subproblem with E(NM1) not too small but I = NM1. ! NSIZE = N - START + 1 ELSE ! ! A subproblem with E(NM1) small. This implies an ! 1-by-1 subproblem at D(N). Solve this 1-by-1 problem ! first. ! NSIZE = I - START + 1 IF( ICOMPQ.EQ.2 ) THEN U( N, N ) = SIGN( ONE, D( N ) ) VT( N, N ) = ONE ELSE IF( ICOMPQ.EQ.1 ) THEN Q( N+( QSTART-1 )*N ) = SIGN( ONE, D( N ) ) Q( N+( SMLSIZ+QSTART-1 )*N ) = ONE END IF D( N ) = ABS( D( N ) ) END IF IF( ICOMPQ.EQ.2 ) THEN CALL DLASD0( NSIZE, SQRE, D( START ), E( START ), & U( START, START ), LDU, VT( START, START ), & LDVT, SMLSIZ, IWORK, WORK( WSTART ), INFO ) ELSE CALL DLASDA( ICOMPQ, SMLSIZ, NSIZE, SQRE, D( START ), & E( START ), Q( START+( IU+QSTART-2 )*N ), N, & Q( START+( IVT+QSTART-2 )*N ), & IQ( START+K*N ), Q( START+( DIFL+QSTART-2 )* & N ), Q( START+( DIFR+QSTART-2 )*N ), & Q( START+( Z+QSTART-2 )*N ), & Q( START+( POLES+QSTART-2 )*N ), & IQ( START+GIVPTR*N ), IQ( START+GIVCOL*N ), & N, IQ( START+PERM*N ), & Q( START+( GIVNUM+QSTART-2 )*N ), & Q( START+( IC+QSTART-2 )*N ), & Q( START+( IS+QSTART-2 )*N ), & WORK( WSTART ), IWORK, INFO ) IF( INFO.NE.0 ) THEN RETURN END IF END IF START = I + 1 END IF 30 CONTINUE ! ! Unscale ! CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, IERR ) 40 CONTINUE ! ! Use Selection Sort to minimize swaps of singular vectors ! DO 60 II = 2, N I = II - 1 KK = I P = D( I ) DO 50 J = II, N IF( D( J ).GT.P ) THEN KK = J P = D( J ) END IF 50 CONTINUE IF( KK.NE.I ) THEN D( KK ) = D( I ) D( I ) = P IF( ICOMPQ.EQ.1 ) THEN IQ( I ) = KK ELSE IF( ICOMPQ.EQ.2 ) THEN CALL DSWAP( N, U( 1, I ), 1, U( 1, KK ), 1 ) CALL DSWAP( N, VT( I, 1 ), LDVT, VT( KK, 1 ), LDVT ) END IF ELSE IF( ICOMPQ.EQ.1 ) THEN IQ( I ) = I END IF 60 CONTINUE ! ! If ICOMPQ = 1, use IQ(N,1) as the indicator for UPLO ! IF( ICOMPQ.EQ.1 ) THEN IF( IUPLO.EQ.1 ) THEN IQ( N ) = 1 ELSE IQ( N ) = 0 END IF END IF ! ! If B is lower bidiagonal, update U by those Givens rotations ! which rotated B to be upper bidiagonal ! IF( ( IUPLO.EQ.2 ) .AND. ( ICOMPQ.EQ.2 ) ) & CALL DLASR( 'L', 'V', 'B', N, N, WORK( 1 ), WORK( N ), U, LDU ) ! RETURN ! ! End of DBDSDC ! END SUBROUTINE DBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, & LDU, C, LDC, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU ! .. ! .. Array Arguments .. DOUBLE PRECISION C( LDC, * ), D( * ), E( * ), U( LDU, * ), & VT( LDVT, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DBDSQR computes the singular value decomposition (SVD) of a real ! N-by-N (upper or lower) bidiagonal matrix B: B = Q * S * P' (P' ! denotes the transpose of P), where S is a diagonal matrix with ! non-negative diagonal elements (the singular values of B), and Q ! and P are orthogonal matrices. ! ! The routine computes S, and optionally computes U * Q, P' * VT, ! or Q' * C, for given real input matrices U, VT, and C. ! ! See "Computing Small Singular Values of Bidiagonal Matrices With ! Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, ! LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, ! no. 5, pp. 873-912, Sept 1990) and ! "Accurate singular values and differential qd algorithms," by ! B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics ! Department, University of California at Berkeley, July 1992 ! for a detailed description of the algorithm. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': B is upper bidiagonal; ! = 'L': B is lower bidiagonal. ! ! N (input) INTEGER ! The order of the matrix B. N >= 0. ! ! NCVT (input) INTEGER ! The number of columns of the matrix VT. NCVT >= 0. ! ! NRU (input) INTEGER ! The number of rows of the matrix U. NRU >= 0. ! ! NCC (input) INTEGER ! The number of columns of the matrix C. NCC >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the n diagonal elements of the bidiagonal matrix B. ! On exit, if INFO=0, the singular values of B in decreasing ! order. ! ! E (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the elements of E contain the ! offdiagonal elements of the bidiagonal matrix whose SVD ! is desired. On normal exit (INFO = 0), E is destroyed. ! If the algorithm does not converge (INFO > 0), D and E ! will contain the diagonal and superdiagonal elements of a ! bidiagonal matrix orthogonally equivalent to the one given ! as input. E(N) is used for workspace. ! ! VT (input/output) DOUBLE PRECISION array, dimension (LDVT, NCVT) ! On entry, an N-by-NCVT matrix VT. ! On exit, VT is overwritten by P' * VT. ! VT is not referenced if NCVT = 0. ! ! LDVT (input) INTEGER ! The leading dimension of the array VT. ! LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0. ! ! U (input/output) DOUBLE PRECISION array, dimension (LDU, N) ! On entry, an NRU-by-N matrix U. ! On exit, U is overwritten by U * Q. ! U is not referenced if NRU = 0. ! ! LDU (input) INTEGER ! The leading dimension of the array U. LDU >= max(1,NRU). ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC, NCC) ! On entry, an N-by-NCC matrix C. ! On exit, C is overwritten by Q' * C. ! C is not referenced if NCC = 0. ! ! LDC (input) INTEGER ! The leading dimension of the array C. ! LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (4*N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: If INFO = -i, the i-th argument had an illegal value ! > 0: the algorithm did not converge; D and E contain the ! elements of a bidiagonal matrix which is orthogonally ! similar to the input matrix B; if INFO = i, i ! elements of E have not converged to zero. ! ! Internal Parameters ! =================== ! ! TOLMUL DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8))) ! TOLMUL controls the convergence criterion of the QR loop. ! If it is positive, TOLMUL*EPS is the desired relative ! precision in the computed singular values. ! If it is negative, abs(TOLMUL*EPS*sigma_max) is the ! desired absolute accuracy in the computed singular ! values (corresponds to relative accuracy ! abs(TOLMUL*EPS) in the largest singular value. ! abs(TOLMUL) should be between 1 and 1/EPS, and preferably ! between 10 (for fast convergence) and .1/EPS ! (for there to be some accuracy in the results). ! Default is to lose at either one eighth or 2 of the ! available decimal digits in each computed singular value ! (whichever is smaller). ! ! MAXITR INTEGER, default = 6 ! MAXITR controls the maximum number of passes of the ! algorithm through its inner loop. The algorithms stops ! (and so fails to converge) if the number of passes ! through the inner loop exceeds MAXITR*N**2. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D0 ) DOUBLE PRECISION NEGONE PARAMETER ( NEGONE = -1.0D0 ) DOUBLE PRECISION HNDRTH PARAMETER ( HNDRTH = 0.01D0 ) DOUBLE PRECISION TEN PARAMETER ( TEN = 10.0D0 ) DOUBLE PRECISION HNDRD PARAMETER ( HNDRD = 100.0D0 ) DOUBLE PRECISION MEIGTH PARAMETER ( MEIGTH = -0.125D0 ) INTEGER MAXITR PARAMETER ( MAXITR = 6 ) ! .. ! .. Local Scalars .. LOGICAL LOWER, ROTATE INTEGER I, IDIR, ISUB, ITER, J, LL, LLL, M, MAXIT, NM1, & NM12, NM13, OLDLL, OLDM DOUBLE PRECISION ABSE, ABSS, COSL, COSR, CS, EPS, F, G, H, MU, & OLDCS, OLDSN, R, SHIFT, SIGMN, SIGMX, SINL, & SINR, SLL, SMAX, SMIN, SMINL, SMINLO, SMINOA, & SN, THRESH, TOL, TOLMUL, UNFL ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DLARTG, DLAS2, DLASQ1, DLASR, DLASV2, DROT, & DSCAL, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX, MIN, SIGN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 LOWER = LSAME( UPLO, 'L' ) IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LOWER ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NCVT.LT.0 ) THEN INFO = -3 ELSE IF( NRU.LT.0 ) THEN INFO = -4 ELSE IF( NCC.LT.0 ) THEN INFO = -5 ELSE IF( ( NCVT.EQ.0 .AND. LDVT.LT.1 ) .OR. & ( NCVT.GT.0 .AND. LDVT.LT.MAX( 1, N ) ) ) THEN INFO = -9 ELSE IF( LDU.LT.MAX( 1, NRU ) ) THEN INFO = -11 ELSE IF( ( NCC.EQ.0 .AND. LDC.LT.1 ) .OR. & ( NCC.GT.0 .AND. LDC.LT.MAX( 1, N ) ) ) THEN INFO = -13 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DBDSQR', -INFO ) RETURN END IF IF( N.EQ.0 ) & RETURN IF( N.EQ.1 ) & GO TO 160 ! ! ROTATE is true if any singular vectors desired, false otherwise ! ROTATE = ( NCVT.GT.0 ) .OR. ( NRU.GT.0 ) .OR. ( NCC.GT.0 ) ! ! If no singular vectors desired, use qd algorithm ! IF( .NOT.ROTATE ) THEN CALL DLASQ1( N, D, E, WORK, INFO ) RETURN END IF ! NM1 = N - 1 NM12 = NM1 + NM1 NM13 = NM12 + NM1 IDIR = 0 ! ! Get machine constants ! EPS = DLAMCH( 'Epsilon' ) UNFL = DLAMCH( 'Safe minimum' ) ! ! If matrix lower bidiagonal, rotate to be upper bidiagonal ! by applying Givens rotations on the left ! IF( LOWER ) THEN DO I = 1, N - 1 CALL DLARTG( D( I ), E( I ), CS, SN, R ) D( I ) = R E( I ) = SN*D( I+1 ) D( I+1 ) = CS*D( I+1 ) WORK( I ) = CS WORK( NM1+I ) = SN end do ! ! Update singular vectors if desired ! IF( NRU.GT.0 ) & CALL DLASR( 'R', 'V', 'F', NRU, N, WORK( 1 ), WORK( N ), U, & LDU ) IF( NCC.GT.0 ) & CALL DLASR( 'L', 'V', 'F', N, NCC, WORK( 1 ), WORK( N ), C, & LDC ) END IF ! ! Compute singular values to relative accuracy TOL ! (By setting TOL to be negative, algorithm will compute ! singular values to absolute accuracy ABS(TOL)*norm(input matrix)) ! TOLMUL = MAX( TEN, MIN( HNDRD, EPS**MEIGTH ) ) TOL = TOLMUL*EPS ! ! Compute approximate maximum, minimum singular values ! SMAX = ZERO DO 20 I = 1, N SMAX = MAX( SMAX, ABS( D( I ) ) ) 20 CONTINUE DO 30 I = 1, N - 1 SMAX = MAX( SMAX, ABS( E( I ) ) ) 30 CONTINUE SMINL = ZERO IF( TOL.GE.ZERO ) THEN ! ! Relative accuracy desired ! SMINOA = ABS( D( 1 ) ) IF( SMINOA.EQ.ZERO ) & GO TO 50 MU = SMINOA DO 40 I = 2, N MU = ABS( D( I ) )*( MU / ( MU+ABS( E( I-1 ) ) ) ) SMINOA = MIN( SMINOA, MU ) IF( SMINOA.EQ.ZERO ) & GO TO 50 40 CONTINUE 50 CONTINUE SMINOA = SMINOA / SQRT( DBLE( N ) ) THRESH = MAX( TOL*SMINOA, MAXITR*N*N*UNFL ) ELSE ! ! Absolute accuracy desired ! THRESH = MAX( ABS( TOL )*SMAX, MAXITR*N*N*UNFL ) END IF ! ! Prepare for main iteration loop for the singular values ! (MAXIT is the maximum number of passes through the inner ! loop permitted before nonconvergence signalled.) ! MAXIT = MAXITR*N*N ITER = 0 OLDLL = -1 OLDM = -1 ! ! M points to last element of unconverged part of matrix ! M = N ! ! Begin main iteration loop ! 60 CONTINUE ! ! Check for convergence or exceeding iteration count ! IF( M.LE.1 ) & GO TO 160 IF( ITER.GT.MAXIT ) & GO TO 200 ! ! Find diagonal block of matrix to work on ! IF( TOL.LT.ZERO .AND. ABS( D( M ) ).LE.THRESH ) & D( M ) = ZERO SMAX = ABS( D( M ) ) SMIN = SMAX DO 70 LLL = 1, M - 1 LL = M - LLL ABSS = ABS( D( LL ) ) ABSE = ABS( E( LL ) ) IF( TOL.LT.ZERO .AND. ABSS.LE.THRESH ) & D( LL ) = ZERO IF( ABSE.LE.THRESH ) & GO TO 80 SMIN = MIN( SMIN, ABSS ) SMAX = MAX( SMAX, ABSS, ABSE ) 70 CONTINUE LL = 0 GO TO 90 80 CONTINUE E( LL ) = ZERO ! ! Matrix splits since E(LL) = 0 ! IF( LL.EQ.M-1 ) THEN ! ! Convergence of bottom singular value, return to top of loop ! M = M - 1 GO TO 60 END IF 90 CONTINUE LL = LL + 1 ! ! E(LL) through E(M-1) are nonzero, E(LL-1) is zero ! IF( LL.EQ.M-1 ) THEN ! ! 2 by 2 block, handle separately ! CALL DLASV2( D( M-1 ), E( M-1 ), D( M ), SIGMN, SIGMX, SINR, & COSR, SINL, COSL ) D( M-1 ) = SIGMX E( M-1 ) = ZERO D( M ) = SIGMN ! ! Compute singular vectors, if desired ! IF( NCVT.GT.0 ) & CALL DROT( NCVT, VT( M-1, 1 ), LDVT, VT( M, 1 ), LDVT, COSR, & SINR ) IF( NRU.GT.0 ) & CALL DROT( NRU, U( 1, M-1 ), 1, U( 1, M ), 1, COSL, SINL ) IF( NCC.GT.0 ) & CALL DROT( NCC, C( M-1, 1 ), LDC, C( M, 1 ), LDC, COSL, & SINL ) M = M - 2 GO TO 60 END IF ! ! If working on new submatrix, choose shift direction ! (from larger end diagonal element towards smaller) ! IF( LL.GT.OLDM .OR. M.LT.OLDLL ) THEN IF( ABS( D( LL ) ).GE.ABS( D( M ) ) ) THEN ! ! Chase bulge from top (big end) to bottom (small end) ! IDIR = 1 ELSE ! ! Chase bulge from bottom (big end) to top (small end) ! IDIR = 2 END IF END IF ! ! Apply convergence tests ! IF( IDIR.EQ.1 ) THEN ! ! Run convergence test in forward direction ! First apply standard test to bottom of matrix ! IF( ABS( E( M-1 ) ).LE.ABS( TOL )*ABS( D( M ) ) .OR. & ( TOL.LT.ZERO .AND. ABS( E( M-1 ) ).LE.THRESH ) ) THEN E( M-1 ) = ZERO GO TO 60 END IF IF( TOL.GE.ZERO ) THEN ! ! If relative accuracy desired, apply convergence criterion forward ! MU = ABS( D( LL ) ) SMINL = MU DO LLL = LL, M - 1 IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN E( LLL ) = ZERO GO TO 60 END IF SMINLO = SMINL MU = ABS( D( LLL+1 ) )*( MU / ( MU+ABS( E( LLL ) ) ) ) SMINL = MIN( SMINL, MU ) end do END IF ELSE ! ! Run convergence test in backward direction ! First apply standard test to top of matrix ! IF( ABS( E( LL ) ).LE.ABS( TOL )*ABS( D( LL ) ) .OR. & ( TOL.LT.ZERO .AND. ABS( E( LL ) ).LE.THRESH ) ) THEN E( LL ) = ZERO GO TO 60 END IF ! IF( TOL.GE.ZERO ) THEN ! ! If relative accuracy desired, ! apply convergence criterion backward ! MU = ABS( D( M ) ) SMINL = MU DO 110 LLL = M - 1, LL, -1 IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN E( LLL ) = ZERO GO TO 60 END IF SMINLO = SMINL MU = ABS( D( LLL ) )*( MU / ( MU+ABS( E( LLL ) ) ) ) SMINL = MIN( SMINL, MU ) 110 CONTINUE END IF END IF OLDLL = LL OLDM = M ! ! Compute shift. First, test if shifting would ruin relative ! accuracy, and if so set the shift to zero. ! IF( TOL.GE.ZERO .AND. N*TOL*( SMINL / SMAX ).LE. & MAX( EPS, HNDRTH*TOL ) ) THEN ! ! Use a zero shift to avoid loss of relative accuracy ! SHIFT = ZERO ELSE ! ! Compute the shift from 2-by-2 block at end of matrix ! IF( IDIR.EQ.1 ) THEN SLL = ABS( D( LL ) ) CALL DLAS2( D( M-1 ), E( M-1 ), D( M ), SHIFT, R ) ELSE SLL = ABS( D( M ) ) CALL DLAS2( D( LL ), E( LL ), D( LL+1 ), SHIFT, R ) END IF ! ! Test if shift negligible, and if so set to zero ! IF( SLL.GT.ZERO ) THEN IF( ( SHIFT / SLL )**2.LT.EPS ) & SHIFT = ZERO END IF END IF ! ! Increment iteration count ! ITER = ITER + M - LL ! ! If SHIFT = 0, do simplified QR iteration ! IF( SHIFT.EQ.ZERO ) THEN IF( IDIR.EQ.1 ) THEN ! ! Chase bulge from top to bottom ! Save cosines and sines for later singular vector updates ! CS = ONE OLDCS = ONE DO 120 I = LL, M - 1 CALL DLARTG( D( I )*CS, E( I ), CS, SN, R ) IF( I.GT.LL ) & E( I-1 ) = OLDSN*R CALL DLARTG( OLDCS*R, D( I+1 )*SN, OLDCS, OLDSN, D( I ) ) WORK( I-LL+1 ) = CS WORK( I-LL+1+NM1 ) = SN WORK( I-LL+1+NM12 ) = OLDCS WORK( I-LL+1+NM13 ) = OLDSN 120 CONTINUE H = D( M )*CS D( M ) = H*OLDCS E( M-1 ) = H*OLDSN ! ! Update singular vectors ! IF( NCVT.GT.0 ) & CALL DLASR( 'L', 'V', 'F', M-LL+1, NCVT, WORK( 1 ), & WORK( N ), VT( LL, 1 ), LDVT ) IF( NRU.GT.0 ) & CALL DLASR( 'R', 'V', 'F', NRU, M-LL+1, WORK( NM12+1 ), & WORK( NM13+1 ), U( 1, LL ), LDU ) IF( NCC.GT.0 ) & CALL DLASR( 'L', 'V', 'F', M-LL+1, NCC, WORK( NM12+1 ), & WORK( NM13+1 ), C( LL, 1 ), LDC ) ! ! Test convergence ! IF( ABS( E( M-1 ) ).LE.THRESH ) & E( M-1 ) = ZERO ! ELSE ! ! Chase bulge from bottom to top ! Save cosines and sines for later singular vector updates ! CS = ONE OLDCS = ONE DO 130 I = M, LL + 1, -1 CALL DLARTG( D( I )*CS, E( I-1 ), CS, SN, R ) IF( I.LT.M ) & E( I ) = OLDSN*R CALL DLARTG( OLDCS*R, D( I-1 )*SN, OLDCS, OLDSN, D( I ) ) WORK( I-LL ) = CS WORK( I-LL+NM1 ) = -SN WORK( I-LL+NM12 ) = OLDCS WORK( I-LL+NM13 ) = -OLDSN 130 CONTINUE H = D( LL )*CS D( LL ) = H*OLDCS E( LL ) = H*OLDSN ! ! Update singular vectors ! IF( NCVT.GT.0 ) & CALL DLASR( 'L', 'V', 'B', M-LL+1, NCVT, WORK( NM12+1 ), & WORK( NM13+1 ), VT( LL, 1 ), LDVT ) IF( NRU.GT.0 ) & CALL DLASR( 'R', 'V', 'B', NRU, M-LL+1, WORK( 1 ), & WORK( N ), U( 1, LL ), LDU ) IF( NCC.GT.0 ) & CALL DLASR( 'L', 'V', 'B', M-LL+1, NCC, WORK( 1 ), & WORK( N ), C( LL, 1 ), LDC ) ! ! Test convergence ! IF( ABS( E( LL ) ).LE.THRESH ) & E( LL ) = ZERO END IF ELSE ! ! Use nonzero shift ! IF( IDIR.EQ.1 ) THEN ! ! Chase bulge from top to bottom ! Save cosines and sines for later singular vector updates ! F = ( ABS( D( LL ) )-SHIFT )* & ( SIGN( ONE, D( LL ) )+SHIFT / D( LL ) ) G = E( LL ) DO 140 I = LL, M - 1 CALL DLARTG( F, G, COSR, SINR, R ) IF( I.GT.LL ) & E( I-1 ) = R F = COSR*D( I ) + SINR*E( I ) E( I ) = COSR*E( I ) - SINR*D( I ) G = SINR*D( I+1 ) D( I+1 ) = COSR*D( I+1 ) CALL DLARTG( F, G, COSL, SINL, R ) D( I ) = R F = COSL*E( I ) + SINL*D( I+1 ) D( I+1 ) = COSL*D( I+1 ) - SINL*E( I ) IF( I.LT.M-1 ) THEN G = SINL*E( I+1 ) E( I+1 ) = COSL*E( I+1 ) END IF WORK( I-LL+1 ) = COSR WORK( I-LL+1+NM1 ) = SINR WORK( I-LL+1+NM12 ) = COSL WORK( I-LL+1+NM13 ) = SINL 140 CONTINUE E( M-1 ) = F ! ! Update singular vectors ! IF( NCVT.GT.0 ) & CALL DLASR( 'L', 'V', 'F', M-LL+1, NCVT, WORK( 1 ), & WORK( N ), VT( LL, 1 ), LDVT ) IF( NRU.GT.0 ) & CALL DLASR( 'R', 'V', 'F', NRU, M-LL+1, WORK( NM12+1 ), & WORK( NM13+1 ), U( 1, LL ), LDU ) IF( NCC.GT.0 ) & CALL DLASR( 'L', 'V', 'F', M-LL+1, NCC, WORK( NM12+1 ), & WORK( NM13+1 ), C( LL, 1 ), LDC ) ! ! Test convergence ! IF( ABS( E( M-1 ) ).LE.THRESH ) & E( M-1 ) = ZERO ! ELSE ! ! Chase bulge from bottom to top ! Save cosines and sines for later singular vector updates ! F = ( ABS( D( M ) )-SHIFT )*( SIGN( ONE, D( M ) )+SHIFT / & D( M ) ) G = E( M-1 ) DO 150 I = M, LL + 1, -1 CALL DLARTG( F, G, COSR, SINR, R ) IF( I.LT.M ) & E( I ) = R F = COSR*D( I ) + SINR*E( I-1 ) E( I-1 ) = COSR*E( I-1 ) - SINR*D( I ) G = SINR*D( I-1 ) D( I-1 ) = COSR*D( I-1 ) CALL DLARTG( F, G, COSL, SINL, R ) D( I ) = R F = COSL*E( I-1 ) + SINL*D( I-1 ) D( I-1 ) = COSL*D( I-1 ) - SINL*E( I-1 ) IF( I.GT.LL+1 ) THEN G = SINL*E( I-2 ) E( I-2 ) = COSL*E( I-2 ) END IF WORK( I-LL ) = COSR WORK( I-LL+NM1 ) = -SINR WORK( I-LL+NM12 ) = COSL WORK( I-LL+NM13 ) = -SINL 150 CONTINUE E( LL ) = F ! ! Test convergence ! IF( ABS( E( LL ) ).LE.THRESH ) & E( LL ) = ZERO ! ! Update singular vectors if desired ! IF( NCVT.GT.0 ) & CALL DLASR( 'L', 'V', 'B', M-LL+1, NCVT, WORK( NM12+1 ), & WORK( NM13+1 ), VT( LL, 1 ), LDVT ) IF( NRU.GT.0 ) & CALL DLASR( 'R', 'V', 'B', NRU, M-LL+1, WORK( 1 ), & WORK( N ), U( 1, LL ), LDU ) IF( NCC.GT.0 ) & CALL DLASR( 'L', 'V', 'B', M-LL+1, NCC, WORK( 1 ), & WORK( N ), C( LL, 1 ), LDC ) END IF END IF ! ! QR iteration finished, go back and check convergence ! GO TO 60 ! ! All singular values converged, so make them positive ! 160 CONTINUE DO 170 I = 1, N IF( D( I ).LT.ZERO ) THEN D( I ) = -D( I ) ! ! Change sign of singular vectors, if desired ! IF( NCVT.GT.0 ) & CALL DSCAL( NCVT, NEGONE, VT( I, 1 ), LDVT ) END IF 170 CONTINUE ! ! Sort the singular values into decreasing order (insertion sort on ! singular values, but only one transposition per singular vector) ! DO 190 I = 1, N - 1 ! ! Scan for smallest D(I) ! ISUB = 1 SMIN = D( 1 ) DO 180 J = 2, N + 1 - I IF( D( J ).LE.SMIN ) THEN ISUB = J SMIN = D( J ) END IF 180 CONTINUE IF( ISUB.NE.N+1-I ) THEN ! ! Swap singular values and vectors ! D( ISUB ) = D( N+1-I ) D( N+1-I ) = SMIN IF( NCVT.GT.0 ) & CALL DSWAP( NCVT, VT( ISUB, 1 ), LDVT, VT( N+1-I, 1 ), & LDVT ) IF( NRU.GT.0 ) & CALL DSWAP( NRU, U( 1, ISUB ), 1, U( 1, N+1-I ), 1 ) IF( NCC.GT.0 ) & CALL DSWAP( NCC, C( ISUB, 1 ), LDC, C( N+1-I, 1 ), LDC ) END IF 190 CONTINUE GO TO 220 ! ! Maximum number of iterations exceeded, failure to converge ! 200 CONTINUE INFO = 0 DO 210 I = 1, N - 1 IF( E( I ).NE.ZERO ) & INFO = INFO + 1 210 CONTINUE 220 CONTINUE RETURN ! ! End of DBDSQR ! END SUBROUTINE DDISNA( JOB, M, N, D, SEP, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER JOB INTEGER INFO, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ), SEP( * ) ! .. ! ! Purpose ! ======= ! ! DDISNA computes the reciprocal condition numbers for the eigenvectors ! of a real symmetric or complex Hermitian matrix or for the left or ! right singular vectors of a general m-by-n matrix. The reciprocal ! condition number is the 'gap' between the corresponding eigenvalue or ! singular value and the nearest other one. ! ! The bound on the error, measured by angle in radians, in the I-th ! computed vector is given by ! ! DLAMCH( 'E' ) * ( ANORM / SEP( I ) ) ! ! where ANORM = 2-norm(A) = max( abs( D(j) ) ). SEP(I) is not allowed ! to be smaller than DLAMCH( 'E' )*ANORM in order to limit the size of ! the error bound. ! ! DDISNA may also be used to compute error bounds for eigenvectors of ! the generalized symmetric definite eigenproblem. ! ! Arguments ! ========= ! ! JOB (input) CHARACTER*1 ! Specifies for which problem the reciprocal condition numbers ! should be computed: ! = 'E': the eigenvectors of a symmetric/Hermitian matrix; ! = 'L': the left singular vectors of a general matrix; ! = 'R': the right singular vectors of a general matrix. ! ! M (input) INTEGER ! The number of rows of the matrix. M >= 0. ! ! N (input) INTEGER ! If JOB = 'L' or 'R', the number of columns of the matrix, ! in which case N >= 0. Ignored if JOB = 'E'. ! ! D (input) DOUBLE PRECISION array, dimension (M) if JOB = 'E' ! dimension (min(M,N)) if JOB = 'L' or 'R' ! The eigenvalues (if JOB = 'E') or singular values (if JOB = ! 'L' or 'R') of the matrix, in either increasing or decreasing ! order. If singular values, they must be non-negative. ! ! SEP (output) DOUBLE PRECISION array, dimension (M) if JOB = 'E' ! dimension (min(M,N)) if JOB = 'L' or 'R' ! The reciprocal condition numbers of the vectors. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL DECR, EIGEN, INCR, LEFT, RIGHT, SING INTEGER I, K DOUBLE PRECISION ANORM, EPS, NEWGAP, OLDGAP, SAFMIN, THRESH ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 EIGEN = LSAME( JOB, 'E' ) LEFT = LSAME( JOB, 'L' ) RIGHT = LSAME( JOB, 'R' ) SING = LEFT .OR. RIGHT IF( EIGEN ) THEN K = M ELSE IF( SING ) THEN K = MIN( M, N ) END IF IF( .NOT.EIGEN .AND. .NOT.SING ) THEN INFO = -1 ELSE IF( M.LT.0 ) THEN INFO = -2 ELSE IF( K.LT.0 ) THEN INFO = -3 ELSE INCR = .TRUE. DECR = .TRUE. DO I = 1, K - 1 IF( INCR ) & INCR = INCR .AND. D( I ).LE.D( I+1 ) IF( DECR ) & DECR = DECR .AND. D( I ).GE.D( I+1 ) end do IF( SING .AND. K.GT.0 ) THEN IF( INCR ) & INCR = INCR .AND. ZERO.LE.D( 1 ) IF( DECR ) & DECR = DECR .AND. D( K ).GE.ZERO END IF IF( .NOT.( INCR .OR. DECR ) ) & INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DDISNA', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( K.EQ.0 ) & RETURN ! ! Compute reciprocal condition numbers ! IF( K.EQ.1 ) THEN SEP( 1 ) = DLAMCH( 'O' ) ELSE OLDGAP = ABS( D( 2 )-D( 1 ) ) SEP( 1 ) = OLDGAP DO 20 I = 2, K - 1 NEWGAP = ABS( D( I+1 )-D( I ) ) SEP( I ) = MIN( OLDGAP, NEWGAP ) OLDGAP = NEWGAP 20 CONTINUE SEP( K ) = OLDGAP END IF IF( SING ) THEN IF( ( LEFT .AND. M.GT.N ) .OR. ( RIGHT .AND. M.LT.N ) ) THEN IF( INCR ) & SEP( 1 ) = MIN( SEP( 1 ), D( 1 ) ) IF( DECR ) & SEP( K ) = MIN( SEP( K ), D( K ) ) END IF END IF ! ! Ensure that reciprocal condition numbers are not less than ! threshold, in order to limit the size of the error bound ! EPS = DLAMCH( 'E' ) SAFMIN = DLAMCH( 'S' ) ANORM = MAX( ABS( D( 1 ) ), ABS( D( K ) ) ) IF( ANORM.EQ.ZERO ) THEN THRESH = EPS ELSE THRESH = MAX( EPS*ANORM, SAFMIN ) END IF DO 30 I = 1, K SEP( I ) = MAX( SEP( I ), THRESH ) 30 CONTINUE ! RETURN ! ! End of DDISNA ! END SUBROUTINE DGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q, & LDQ, PT, LDPT, C, LDC, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER VECT INTEGER INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), C( LDC, * ), D( * ), E( * ), & PT( LDPT, * ), Q( LDQ, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGBBRD reduces a real general m-by-n band matrix A to upper ! bidiagonal form B by an orthogonal transformation: Q' * A * P = B. ! ! The routine computes B, and optionally forms Q or P', or computes ! Q'*C for a given matrix C. ! ! Arguments ! ========= ! ! VECT (input) CHARACTER*1 ! Specifies whether or not the matrices Q and P' are to be ! formed. ! = 'N': do not form Q or P'; ! = 'Q': form Q only; ! = 'P': form P' only; ! = 'B': form both. ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! NCC (input) INTEGER ! The number of columns of the matrix C. NCC >= 0. ! ! KL (input) INTEGER ! The number of subdiagonals of the matrix A. KL >= 0. ! ! KU (input) INTEGER ! The number of superdiagonals of the matrix A. KU >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) ! On entry, the m-by-n band matrix A, stored in rows 1 to ! KL+KU+1. The j-th column of A is stored in the j-th column of ! the array AB as follows: ! AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl). ! On exit, A is overwritten by values generated during the ! reduction. ! ! LDAB (input) INTEGER ! The leading dimension of the array A. LDAB >= KL+KU+1. ! ! D (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The diagonal elements of the bidiagonal matrix B. ! ! E (output) DOUBLE PRECISION array, dimension (min(M,N)-1) ! The superdiagonal elements of the bidiagonal matrix B. ! ! Q (output) DOUBLE PRECISION array, dimension (LDQ,M) ! If VECT = 'Q' or 'B', the m-by-m orthogonal matrix Q. ! If VECT = 'N' or 'P', the array Q is not referenced. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. ! LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise. ! ! PT (output) DOUBLE PRECISION array, dimension (LDPT,N) ! If VECT = 'P' or 'B', the n-by-n orthogonal matrix P'. ! If VECT = 'N' or 'Q', the array PT is not referenced. ! ! LDPT (input) INTEGER ! The leading dimension of the array PT. ! LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,NCC) ! On entry, an m-by-ncc matrix C. ! On exit, C is overwritten by Q'*C. ! C is not referenced if NCC = 0. ! ! LDC (input) INTEGER ! The leading dimension of the array C. ! LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (2*max(M,N)) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL WANTB, WANTC, WANTPT, WANTQ INTEGER I, INCA, J, J1, J2, KB, KB1, KK, KLM, KLU1, & KUN, L, MINMN, ML, ML0, MN, MU, MU0, NR, NRT DOUBLE PRECISION RA, RB, RC, RS ! .. ! .. External Subroutines .. EXTERNAL DLARGV, DLARTG, DLARTV, DLASET, DROT, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Executable Statements .. ! ! Test the input parameters ! WANTB = LSAME( VECT, 'B' ) WANTQ = LSAME( VECT, 'Q' ) .OR. WANTB WANTPT = LSAME( VECT, 'P' ) .OR. WANTB WANTC = NCC.GT.0 KLU1 = KL + KU + 1 INFO = 0 IF( .NOT.WANTQ .AND. .NOT.WANTPT .AND. .NOT.LSAME( VECT, 'N' ) ) & THEN INFO = -1 ELSE IF( M.LT.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( NCC.LT.0 ) THEN INFO = -4 ELSE IF( KL.LT.0 ) THEN INFO = -5 ELSE IF( KU.LT.0 ) THEN INFO = -6 ELSE IF( LDAB.LT.KLU1 ) THEN INFO = -8 ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. LDQ.LT.MAX( 1, M ) ) THEN INFO = -12 ELSE IF( LDPT.LT.1 .OR. WANTPT .AND. LDPT.LT.MAX( 1, N ) ) THEN INFO = -14 ELSE IF( LDC.LT.1 .OR. WANTC .AND. LDC.LT.MAX( 1, M ) ) THEN INFO = -16 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGBBRD', -INFO ) RETURN END IF ! ! Initialize Q and P' to the unit matrix, if needed ! IF( WANTQ ) & CALL DLASET( 'Full', M, M, ZERO, ONE, Q, LDQ ) IF( WANTPT ) & CALL DLASET( 'Full', N, N, ZERO, ONE, PT, LDPT ) ! ! Quick return if possible. ! IF( M.EQ.0 .OR. N.EQ.0 ) & RETURN ! MINMN = MIN( M, N ) ! IF( KL+KU.GT.1 ) THEN ! ! Reduce to upper bidiagonal form if KU > 0; if KU = 0, reduce ! first to lower bidiagonal form and then transform to upper ! bidiagonal ! IF( KU.GT.0 ) THEN ML0 = 1 MU0 = 2 ELSE ML0 = 2 MU0 = 1 END IF ! ! Wherever possible, plane rotations are generated and applied in ! vector operations of length NR over the index set J1:J2:KLU1. ! ! The sines of the plane rotations are stored in WORK(1:max(m,n)) ! and the cosines in WORK(max(m,n)+1:2*max(m,n)). ! MN = MAX( M, N ) KLM = MIN( M-1, KL ) KUN = MIN( N-1, KU ) KB = KLM + KUN KB1 = KB + 1 INCA = KB1*LDAB NR = 0 J1 = KLM + 2 J2 = 1 - KUN DO 90 I = 1, MINMN ! ! Reduce i-th column and i-th row of matrix to bidiagonal form ! ML = KLM + 1 MU = KUN + 1 DO 80 KK = 1, KB J1 = J1 + KB J2 = J2 + KB ! ! generate plane rotations to annihilate nonzero elements ! which have been created below the band ! IF( NR.GT.0 ) & CALL DLARGV( NR, AB( KLU1, J1-KLM-1 ), INCA, & WORK( J1 ), KB1, WORK( MN+J1 ), KB1 ) ! ! apply plane rotations from the left ! DO L = 1, KB IF( J2-KLM+L-1.GT.N ) THEN NRT = NR - 1 ELSE NRT = NR END IF IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( KLU1-L, J1-KLM+L-1 ), INCA, & AB( KLU1-L+1, J1-KLM+L-1 ), INCA, & WORK( MN+J1 ), WORK( J1 ), KB1 ) end do IF( ML.GT.ML0 ) THEN IF( ML.LE.M-I+1 ) THEN ! ! generate plane rotation to annihilate a(i+ml-1,i) ! within the band, and apply rotation from the left ! CALL DLARTG( AB( KU+ML-1, I ), AB( KU+ML, I ), & WORK( MN+I+ML-1 ), WORK( I+ML-1 ), & RA ) AB( KU+ML-1, I ) = RA IF( I.LT.N ) & CALL DROT( MIN( KU+ML-2, N-I ), & AB( KU+ML-2, I+1 ), LDAB-1, & AB( KU+ML-1, I+1 ), LDAB-1, & WORK( MN+I+ML-1 ), WORK( I+ML-1 ) ) END IF NR = NR + 1 J1 = J1 - KB1 END IF ! IF( WANTQ ) THEN ! ! accumulate product of plane rotations in Q ! DO 20 J = J1, J2, KB1 CALL DROT( M, Q( 1, J-1 ), 1, Q( 1, J ), 1, & WORK( MN+J ), WORK( J ) ) 20 CONTINUE END IF ! IF( WANTC ) THEN ! ! apply plane rotations to C ! DO 30 J = J1, J2, KB1 CALL DROT( NCC, C( J-1, 1 ), LDC, C( J, 1 ), LDC, & WORK( MN+J ), WORK( J ) ) 30 CONTINUE END IF ! IF( J2+KUN.GT.N ) THEN ! ! adjust J2 to keep within the bounds of the matrix ! NR = NR - 1 J2 = J2 - KB1 END IF ! DO 40 J = J1, J2, KB1 ! ! create nonzero element a(j-1,j+ku) above the band ! and store it in WORK(n+1:2*n) ! WORK( J+KUN ) = WORK( J )*AB( 1, J+KUN ) AB( 1, J+KUN ) = WORK( MN+J )*AB( 1, J+KUN ) 40 CONTINUE ! ! generate plane rotations to annihilate nonzero elements ! which have been generated above the band ! IF( NR.GT.0 ) & CALL DLARGV( NR, AB( 1, J1+KUN-1 ), INCA, & WORK( J1+KUN ), KB1, WORK( MN+J1+KUN ), & KB1 ) ! ! apply plane rotations from the right ! DO 50 L = 1, KB IF( J2+L-1.GT.M ) THEN NRT = NR - 1 ELSE NRT = NR END IF IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( L+1, J1+KUN-1 ), INCA, & AB( L, J1+KUN ), INCA, & WORK( MN+J1+KUN ), WORK( J1+KUN ), & KB1 ) 50 CONTINUE ! IF( ML.EQ.ML0 .AND. MU.GT.MU0 ) THEN IF( MU.LE.N-I+1 ) THEN ! ! generate plane rotation to annihilate a(i,i+mu-1) ! within the band, and apply rotation from the right ! CALL DLARTG( AB( KU-MU+3, I+MU-2 ), & AB( KU-MU+2, I+MU-1 ), & WORK( MN+I+MU-1 ), WORK( I+MU-1 ), & RA ) AB( KU-MU+3, I+MU-2 ) = RA CALL DROT( MIN( KL+MU-2, M-I ), & AB( KU-MU+4, I+MU-2 ), 1, & AB( KU-MU+3, I+MU-1 ), 1, & WORK( MN+I+MU-1 ), WORK( I+MU-1 ) ) END IF NR = NR + 1 J1 = J1 - KB1 END IF ! IF( WANTPT ) THEN ! ! accumulate product of plane rotations in P' ! DO 60 J = J1, J2, KB1 CALL DROT( N, PT( J+KUN-1, 1 ), LDPT, & PT( J+KUN, 1 ), LDPT, WORK( MN+J+KUN ), & WORK( J+KUN ) ) 60 CONTINUE END IF ! IF( J2+KB.GT.M ) THEN ! ! adjust J2 to keep within the bounds of the matrix ! NR = NR - 1 J2 = J2 - KB1 END IF ! DO 70 J = J1, J2, KB1 ! ! create nonzero element a(j+kl+ku,j+ku-1) below the ! band and store it in WORK(1:n) ! WORK( J+KB ) = WORK( J+KUN )*AB( KLU1, J+KUN ) AB( KLU1, J+KUN ) = WORK( MN+J+KUN )*AB( KLU1, J+KUN ) 70 CONTINUE ! IF( ML.GT.ML0 ) THEN ML = ML - 1 ELSE MU = MU - 1 END IF 80 CONTINUE 90 CONTINUE END IF IF( KU.EQ.0 .AND. KL.GT.0 ) THEN ! ! A has been reduced to lower bidiagonal form ! ! Transform lower bidiagonal form to upper bidiagonal by applying ! plane rotations from the left, storing diagonal elements in D ! and off-diagonal elements in E ! DO I = 1, MIN( M-1, N ) CALL DLARTG( AB( 1, I ), AB( 2, I ), RC, RS, RA ) D( I ) = RA IF( I.LT.N ) THEN E( I ) = RS*AB( 1, I+1 ) AB( 1, I+1 ) = RC*AB( 1, I+1 ) END IF IF( WANTQ ) & CALL DROT( M, Q( 1, I ), 1, Q( 1, I+1 ), 1, RC, RS ) IF( WANTC ) & CALL DROT( NCC, C( I, 1 ), LDC, C( I+1, 1 ), LDC, RC, & RS ) end do IF( M.LE.N ) & D( M ) = AB( 1, M ) ELSE IF( KU.GT.0 ) THEN ! ! A has been reduced to upper bidiagonal form ! IF( M.LT.N ) THEN ! ! Annihilate a(m,m+1) by applying plane rotations from the ! right, storing diagonal elements in D and off-diagonal ! elements in E ! RB = AB( KU, M+1 ) DO 110 I = M, 1, -1 CALL DLARTG( AB( KU+1, I ), RB, RC, RS, RA ) D( I ) = RA IF( I.GT.1 ) THEN RB = -RS*AB( KU, I ) E( I-1 ) = RC*AB( KU, I ) END IF IF( WANTPT ) & CALL DROT( N, PT( I, 1 ), LDPT, PT( M+1, 1 ), LDPT, & RC, RS ) 110 CONTINUE ELSE ! ! Copy off-diagonal elements to E and diagonal elements to D ! DO 120 I = 1, MINMN - 1 E( I ) = AB( KU, I+1 ) 120 CONTINUE DO 130 I = 1, MINMN D( I ) = AB( KU+1, I ) 130 CONTINUE END IF ELSE ! ! A is diagonal. Set elements of E to zero and copy diagonal ! elements to D. ! DO 140 I = 1, MINMN - 1 E( I ) = ZERO 140 CONTINUE DO 150 I = 1, MINMN D( I ) = AB( 1, I ) 150 CONTINUE END IF RETURN ! ! End of DGBBRD ! END SUBROUTINE DGBCON( NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND, & WORK, IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER NORM INTEGER INFO, KL, KU, LDAB, N DOUBLE PRECISION ANORM, RCOND ! .. ! .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION AB( LDAB, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGBCON estimates the reciprocal of the condition number of a real ! general band matrix A, in either the 1-norm or the infinity-norm, ! using the LU factorization computed by DGBTRF. ! ! An estimate is obtained for norm(inv(A)), and the reciprocal of the ! condition number is computed as ! RCOND = 1 / ( norm(A) * norm(inv(A)) ). ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies whether the 1-norm condition number or the ! infinity-norm condition number is required: ! = '1' or 'O': 1-norm; ! = 'I': Infinity-norm. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KL (input) INTEGER ! The number of subdiagonals within the band of A. KL >= 0. ! ! KU (input) INTEGER ! The number of superdiagonals within the band of A. KU >= 0. ! ! AB (input) DOUBLE PRECISION array, dimension (LDAB,N) ! Details of the LU factorization of the band matrix A, as ! computed by DGBTRF. U is stored as an upper triangular band ! matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and ! the multipliers used during the factorization are stored in ! rows KL+KU+2 to 2*KL+KU+1. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= 2*KL+KU+1. ! ! IPIV (input) INTEGER array, dimension (N) ! The pivot indices; for 1 <= i <= N, row i of the matrix was ! interchanged with row IPIV(i). ! ! ANORM (input) DOUBLE PRECISION ! If NORM = '1' or 'O', the 1-norm of the original matrix A. ! If NORM = 'I', the infinity-norm of the original matrix A. ! ! RCOND (output) DOUBLE PRECISION ! The reciprocal of the condition number of the matrix A, ! computed as RCOND = 1/(norm(A) * norm(inv(A))). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LNOTI, ONENRM CHARACTER NORMIN INTEGER IX, J, JP, KASE, KASE1, KD, LM DOUBLE PRECISION AINVNM, SCALE, SMLNUM, T ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DDOT, DLAMCH EXTERNAL LSAME, IDAMAX, DDOT, DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DLACON, DLATBS, DRSCL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' ) IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KL.LT.0 ) THEN INFO = -3 ELSE IF( KU.LT.0 ) THEN INFO = -4 ELSE IF( LDAB.LT.2*KL+KU+1 ) THEN INFO = -6 ELSE IF( ANORM.LT.ZERO ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGBCON', -INFO ) RETURN END IF ! ! Quick return if possible ! RCOND = ZERO IF( N.EQ.0 ) THEN RCOND = ONE RETURN ELSE IF( ANORM.EQ.ZERO ) THEN RETURN END IF ! SMLNUM = DLAMCH( 'Safe minimum' ) ! ! Estimate the norm of inv(A). ! AINVNM = ZERO NORMIN = 'N' IF( ONENRM ) THEN KASE1 = 1 ELSE KASE1 = 2 END IF KD = KL + KU + 1 LNOTI = KL.GT.0 KASE = 0 10 CONTINUE CALL DLACON( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.KASE1 ) THEN ! ! Multiply by inv(L). ! IF( LNOTI ) THEN DO 20 J = 1, N - 1 LM = MIN( KL, N-J ) JP = IPIV( J ) T = WORK( JP ) IF( JP.NE.J ) THEN WORK( JP ) = WORK( J ) WORK( J ) = T END IF CALL DAXPY( LM, -T, AB( KD+1, J ), 1, WORK( J+1 ), 1 ) 20 CONTINUE END IF ! ! Multiply by inv(U). ! CALL DLATBS( 'Upper', 'No transpose', 'Non-unit', NORMIN, N, & KL+KU, AB, LDAB, WORK, SCALE, WORK( 2*N+1 ), & INFO ) ELSE ! ! Multiply by inv(U'). ! CALL DLATBS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N, & KL+KU, AB, LDAB, WORK, SCALE, WORK( 2*N+1 ), & INFO ) ! ! Multiply by inv(L'). ! IF( LNOTI ) THEN DO 30 J = N - 1, 1, -1 LM = MIN( KL, N-J ) WORK( J ) = WORK( J ) - DDOT( LM, AB( KD+1, J ), 1, & WORK( J+1 ), 1 ) JP = IPIV( J ) IF( JP.NE.J ) THEN T = WORK( JP ) WORK( JP ) = WORK( J ) WORK( J ) = T END IF 30 CONTINUE END IF END IF ! ! Divide X by 1/SCALE if doing so will not cause overflow. ! NORMIN = 'Y' IF( SCALE.NE.ONE ) THEN IX = IDAMAX( N, WORK, 1 ) IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO ) & GO TO 40 CALL DRSCL( N, SCALE, WORK, 1 ) END IF GO TO 10 END IF ! ! Compute the estimate of the reciprocal condition number. ! IF( AINVNM.NE.ZERO ) & RCOND = ( ONE / AINVNM ) / ANORM ! 40 CONTINUE RETURN ! ! End of DGBCON ! END SUBROUTINE DGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, & AMAX, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. INTEGER INFO, KL, KU, LDAB, M, N DOUBLE PRECISION AMAX, COLCND, ROWCND ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), C( * ), R( * ) ! .. ! ! Purpose ! ======= ! ! DGBEQU computes row and column scalings intended to equilibrate an ! M-by-N band matrix A and reduce its condition number. R returns the ! row scale factors and C the column scale factors, chosen to try to ! make the largest element in each row and column of the matrix B with ! elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1. ! ! R(i) and C(j) are restricted to be between SMLNUM = smallest safe ! number and BIGNUM = largest safe number. Use of these scaling ! factors is not guaranteed to reduce the condition number of A but ! works well in practice. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! KL (input) INTEGER ! The number of subdiagonals within the band of A. KL >= 0. ! ! KU (input) INTEGER ! The number of superdiagonals within the band of A. KU >= 0. ! ! AB (input) DOUBLE PRECISION array, dimension (LDAB,N) ! The band matrix A, stored in rows 1 to KL+KU+1. The j-th ! column of A is stored in the j-th column of the array AB as ! follows: ! AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl). ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KL+KU+1. ! ! R (output) DOUBLE PRECISION array, dimension (M) ! If INFO = 0, or INFO > M, R contains the row scale factors ! for A. ! ! C (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, C contains the column scale factors for A. ! ! ROWCND (output) DOUBLE PRECISION ! If INFO = 0 or INFO > M, ROWCND contains the ratio of the ! smallest R(i) to the largest R(i). If ROWCND >= 0.1 and ! AMAX is neither too large nor too small, it is not worth ! scaling by R. ! ! COLCND (output) DOUBLE PRECISION ! If INFO = 0, COLCND contains the ratio of the smallest ! C(i) to the largest C(i). If COLCND >= 0.1, it is not ! worth scaling by C. ! ! AMAX (output) DOUBLE PRECISION ! Absolute value of largest matrix element. If AMAX is very ! close to overflow or very close to underflow, the matrix ! should be scaled. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, and i is ! <= M: the i-th row of A is exactly zero ! > M: the (i-M)-th column of A is exactly zero ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J, KD DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KL.LT.0 ) THEN INFO = -3 ELSE IF( KU.LT.0 ) THEN INFO = -4 ELSE IF( LDAB.LT.KL+KU+1 ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGBEQU', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 ) THEN ROWCND = ONE COLCND = ONE AMAX = ZERO RETURN END IF ! ! Get machine constants. ! SMLNUM = DLAMCH( 'S' ) BIGNUM = ONE / SMLNUM ! ! Compute row scale factors. ! R(1:m) = ZERO ! ! Find the maximum element in each row. ! KD = KU + 1 DO 30 J = 1, N DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M ) R( I ) = MAX( R( I ), ABS( AB( KD+I-J, J ) ) ) 20 CONTINUE 30 CONTINUE ! ! Find the maximum and minimum scale factors. ! RCMIN = BIGNUM RCMAX = ZERO DO 40 I = 1, M RCMAX = MAX( RCMAX, R( I ) ) RCMIN = MIN( RCMIN, R( I ) ) 40 CONTINUE AMAX = RCMAX ! IF( RCMIN.EQ.ZERO ) THEN ! ! Find the first zero scale factor and return an error code. ! DO 50 I = 1, M IF( R( I ).EQ.ZERO ) THEN INFO = I RETURN END IF 50 CONTINUE ELSE ! ! Invert the scale factors. ! DO 60 I = 1, M R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM ) 60 CONTINUE ! ! Compute ROWCND = min(R(I)) / max(R(I)) ! ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) END IF ! ! Compute column scale factors ! DO 70 J = 1, N C( J ) = ZERO 70 CONTINUE ! ! Find the maximum element in each column, ! assuming the row scaling computed above. ! KD = KU + 1 DO 90 J = 1, N DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M ) C( J ) = MAX( C( J ), ABS( AB( KD+I-J, J ) )*R( I ) ) 80 CONTINUE 90 CONTINUE ! ! Find the maximum and minimum scale factors. ! RCMIN = BIGNUM RCMAX = ZERO DO J = 1, N RCMIN = MIN( RCMIN, C( J ) ) RCMAX = MAX( RCMAX, C( J ) ) end do IF( RCMIN.EQ.ZERO ) THEN ! ! Find the first zero scale factor and return an error code. ! DO 110 J = 1, N IF( C( J ).EQ.ZERO ) THEN INFO = M + J RETURN END IF 110 CONTINUE ELSE ! ! Invert the scale factors. ! DO 120 J = 1, N C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM ) 120 CONTINUE ! ! Compute COLCND = min(C(J)) / max(C(J)) ! COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) END IF ! RETURN ! ! End of DGBEQU ! END SUBROUTINE DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, & IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER TRANS INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS ! .. ! .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), & BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DGBRFS improves the computed solution to a system of linear ! equations when the coefficient matrix is banded, and provides ! error bounds and backward error estimates for the solution. ! ! Arguments ! ========= ! ! TRANS (input) CHARACTER*1 ! Specifies the form of the system of equations: ! = 'N': A * X = B (No transpose) ! = 'T': A**T * X = B (Transpose) ! = 'C': A**H * X = B (Conjugate transpose = Transpose) ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KL (input) INTEGER ! The number of subdiagonals within the band of A. KL >= 0. ! ! KU (input) INTEGER ! The number of superdiagonals within the band of A. KU >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! AB (input) DOUBLE PRECISION array, dimension (LDAB,N) ! The original band matrix A, stored in rows 1 to KL+KU+1. ! The j-th column of A is stored in the j-th column of the ! array AB as follows: ! AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KL+KU+1. ! ! AFB (input) DOUBLE PRECISION array, dimension (LDAFB,N) ! Details of the LU factorization of the band matrix A, as ! computed by DGBTRF. U is stored as an upper triangular band ! matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and ! the multipliers used during the factorization are stored in ! rows KL+KU+2 to 2*KL+KU+1. ! ! LDAFB (input) INTEGER ! The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. ! ! IPIV (input) INTEGER array, dimension (N) ! The pivot indices from DGBTRF; for 1<=i<=N, row i of the ! matrix was interchanged with row IPIV(i). ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) ! The right hand side matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! On entry, the solution matrix X, as computed by DGBTRS. ! On exit, the improved solution matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Internal Parameters ! =================== ! ! ITMAX is the maximum number of steps of iterative refinement. ! ! ===================================================================== ! ! .. Parameters .. INTEGER ITMAX PARAMETER ( ITMAX = 5 ) DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D+0 ) DOUBLE PRECISION THREE PARAMETER ( THREE = 3.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOTRAN CHARACTER TRANST INTEGER COUNT, I, J, K, KASE, KK, NZ DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DGBMV, DGBTRS, DLACON, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 NOTRAN = LSAME( TRANS, 'N' ) IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. & LSAME( TRANS, 'C' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KL.LT.0 ) THEN INFO = -3 ELSE IF( KU.LT.0 ) THEN INFO = -4 ELSE IF( NRHS.LT.0 ) THEN INFO = -5 ELSE IF( LDAB.LT.KL+KU+1 ) THEN INFO = -7 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN INFO = -9 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -12 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -14 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGBRFS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN FERR(1:nrhs) = ZERO BERR(1:nrhs) = ZERO RETURN END IF ! IF( NOTRAN ) THEN TRANST = 'T' ELSE TRANST = 'N' END IF ! ! NZ = maximum number of nonzero elements in each row of A, plus 1 ! NZ = MIN( KL+KU+2, N+1 ) EPS = DLAMCH( 'Epsilon' ) SAFMIN = DLAMCH( 'Safe minimum' ) SAFE1 = NZ*SAFMIN SAFE2 = SAFE1 / EPS ! ! Do for each right hand side ! DO 140 J = 1, NRHS ! COUNT = 1 LSTRES = THREE 20 CONTINUE ! ! Loop until stopping criterion is satisfied. ! ! Compute residual R = B - op(A) * X, ! where op(A) = A, A**T, or A**H, depending on TRANS. ! CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) CALL DGBMV( TRANS, N, N, KL, KU, -ONE, AB, LDAB, X( 1, J ), 1, & ONE, WORK( N+1 ), 1 ) ! ! Compute componentwise relative backward error from formula ! ! max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) ! ! where abs(Z) is the componentwise absolute value of the matrix ! or vector Z. If the i-th component of the denominator is less ! than SAFE2, then SAFE1 is added to the i-th components of the ! numerator and denominator before dividing. ! DO 30 I = 1, N WORK( I ) = ABS( B( I, J ) ) 30 CONTINUE ! ! Compute abs(op(A))*abs(X) + abs(B). ! IF( NOTRAN ) THEN DO 50 K = 1, N KK = KU + 1 - K XK = ABS( X( K, J ) ) DO 40 I = MAX( 1, K-KU ), MIN( N, K+KL ) WORK( I ) = WORK( I ) + ABS( AB( KK+I, K ) )*XK 40 CONTINUE 50 CONTINUE ELSE DO 70 K = 1, N S = ZERO KK = KU + 1 - K DO 60 I = MAX( 1, K-KU ), MIN( N, K+KL ) S = S + ABS( AB( KK+I, K ) )*ABS( X( I, J ) ) 60 CONTINUE WORK( K ) = WORK( K ) + S 70 CONTINUE END IF S = ZERO DO 80 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) ELSE S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / & ( WORK( I )+SAFE1 ) ) END IF 80 CONTINUE BERR( J ) = S ! ! Test stopping criterion. Continue iterating if ! 1) The residual BERR(J) is larger than machine epsilon, and ! 2) BERR(J) decreased by at least a factor of 2 during the ! last iteration, and ! 3) At most ITMAX iterations tried. ! IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. & COUNT.LE.ITMAX ) THEN ! ! Update solution and try again. ! CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV, & WORK( N+1 ), N, INFO ) CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) LSTRES = BERR( J ) COUNT = COUNT + 1 GO TO 20 END IF ! ! Bound error from formula ! ! norm(X - XTRUE) / norm(X) .le. FERR = ! norm( abs(inv(op(A)))* ! ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) ! ! where ! norm(Z) is the magnitude of the largest component of Z ! inv(op(A)) is the inverse of op(A) ! abs(Z) is the componentwise absolute value of the matrix or ! vector Z ! NZ is the maximum number of nonzeros in any row of A, plus 1 ! EPS is machine epsilon ! ! The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) ! is incremented by SAFE1 if the i-th component of ! abs(op(A))*abs(X) + abs(B) is less than SAFE2. ! ! Use DLACON to estimate the infinity-norm of the matrix ! inv(op(A)) * diag(W), ! where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) ! DO 90 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) ELSE WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 END IF 90 CONTINUE ! KASE = 0 100 CONTINUE CALL DLACON( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), & KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN ! ! Multiply by diag(W)*inv(op(A)**T). ! CALL DGBTRS( TRANST, N, KL, KU, 1, AFB, LDAFB, IPIV, & WORK( N+1 ), N, INFO ) DO 110 I = 1, N WORK( N+I ) = WORK( N+I )*WORK( I ) 110 CONTINUE ELSE ! ! Multiply by inv(op(A))*diag(W). ! DO 120 I = 1, N WORK( N+I ) = WORK( N+I )*WORK( I ) 120 CONTINUE CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV, & WORK( N+1 ), N, INFO ) END IF GO TO 100 END IF ! ! Normalize error. ! LSTRES = ZERO DO 130 I = 1, N LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 130 CONTINUE IF( LSTRES.NE.ZERO ) & FERR( J ) = FERR( J ) / LSTRES ! 140 CONTINUE ! RETURN ! ! End of DGBRFS ! END SUBROUTINE DGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. INTEGER INFO, KL, KU, LDAB, LDB, N, NRHS ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DGBSV computes the solution to a real system of linear equations ! A * X = B, where A is a band matrix of order N with KL subdiagonals ! and KU superdiagonals, and X and B are N-by-NRHS matrices. ! ! The LU decomposition with partial pivoting and row interchanges is ! used to factor A as A = L * U, where L is a product of permutation ! and unit lower triangular matrices with KL subdiagonals, and U is ! upper triangular with KL+KU superdiagonals. The factored form of A ! is then used to solve the system of equations A * X = B. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The number of linear equations, i.e., the order of the ! matrix A. N >= 0. ! ! KL (input) INTEGER ! The number of subdiagonals within the band of A. KL >= 0. ! ! KU (input) INTEGER ! The number of superdiagonals within the band of A. KU >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) ! On entry, the matrix A in band storage, in rows KL+1 to ! 2*KL+KU+1; rows 1 to KL of the array need not be set. ! The j-th column of A is stored in the j-th column of the ! array AB as follows: ! AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL) ! On exit, details of the factorization: U is stored as an ! upper triangular band matrix with KL+KU superdiagonals in ! rows 1 to KL+KU+1, and the multipliers used during the ! factorization are stored in rows KL+KU+2 to 2*KL+KU+1. ! See below for further details. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= 2*KL+KU+1. ! ! IPIV (output) INTEGER array, dimension (N) ! The pivot indices that define the permutation matrix P; ! row i of the matrix was interchanged with row IPIV(i). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the N-by-NRHS right hand side matrix B. ! On exit, if INFO = 0, the N-by-NRHS solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, U(i,i) is exactly zero. The factorization ! has been completed, but the factor U is exactly ! singular, and the solution has not been computed. ! ! Further Details ! =============== ! ! The band storage scheme is illustrated by the following example, when ! M = N = 6, KL = 2, KU = 1: ! ! On entry: On exit: ! ! * * * + + + * * * u14 u25 u36 ! * * + + + + * * u13 u24 u35 u46 ! * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 ! a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 ! a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 * ! a31 a42 a53 a64 * * m31 m42 m53 m64 * * ! ! Array elements marked * are not used by the routine; elements marked ! + need not be set on entry, but are required by the routine to store ! elements of U because of fill-in resulting from the row interchanges. ! ! ===================================================================== ! ! .. External Subroutines .. EXTERNAL DGBTRF, DGBTRS, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF( N.LT.0 ) THEN INFO = -1 ELSE IF( KL.LT.0 ) THEN INFO = -2 ELSE IF( KU.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.0 ) THEN INFO = -4 ELSE IF( LDAB.LT.2*KL+KU+1 ) THEN INFO = -6 ELSE IF( LDB.LT.MAX( N, 1 ) ) THEN INFO = -9 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGBSV ', -INFO ) RETURN END IF ! ! Compute the LU factorization of the band matrix A. ! CALL DGBTRF( N, N, KL, KU, AB, LDAB, IPIV, INFO ) IF( INFO.EQ.0 ) THEN ! ! Solve the system A*X = B, overwriting B with X. ! CALL DGBTRS( 'No transpose', N, KL, KU, NRHS, AB, LDAB, IPIV, & B, LDB, INFO ) END IF RETURN ! ! End of DGBSV ! END SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, & LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, & RCOND, FERR, BERR, WORK, IWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER EQUED, FACT, TRANS INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), & BERR( * ), C( * ), FERR( * ), R( * ), & WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DGBSVX uses the LU factorization to compute the solution to a real ! system of linear equations A * X = B, A**T * X = B, or A**H * X = B, ! where A is a band matrix of order N with KL subdiagonals and KU ! superdiagonals, and X and B are N-by-NRHS matrices. ! ! Error bounds on the solution and a condition estimate are also ! provided. ! ! Description ! =========== ! ! The following steps are performed by this subroutine: ! ! 1. If FACT = 'E', real scaling factors are computed to equilibrate ! the system: ! TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B ! TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B ! TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B ! Whether or not the system will be equilibrated depends on the ! scaling of the matrix A, but if equilibration is used, A is ! overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') ! or diag(C)*B (if TRANS = 'T' or 'C'). ! ! 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the ! matrix A (after equilibration if FACT = 'E') as ! A = L * U, ! where L is a product of permutation and unit lower triangular ! matrices with KL subdiagonals, and U is upper triangular with ! KL+KU superdiagonals. ! ! 3. If some U(i,i)=0, so that U is exactly singular, then the routine ! returns with INFO = i. Otherwise, the factored form of A is used ! to estimate the condition number of the matrix A. If the ! reciprocal of the condition number is less than machine precision, ! INFO = N+1 is returned as a warning, but the routine still goes on ! to solve for X and compute error bounds as described below. ! ! 4. The system of equations is solved for X using the factored form ! of A. ! ! 5. Iterative refinement is applied to improve the computed solution ! matrix and calculate error bounds and backward error estimates ! for it. ! ! 6. If equilibration was used, the matrix X is premultiplied by ! diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so ! that it solves the original system before equilibration. ! ! Arguments ! ========= ! ! FACT (input) CHARACTER*1 ! Specifies whether or not the factored form of the matrix A is ! supplied on entry, and if not, whether the matrix A should be ! equilibrated before it is factored. ! = 'F': On entry, AFB and IPIV contain the factored form of ! A. If EQUED is not 'N', the matrix A has been ! equilibrated with scaling factors given by R and C. ! AB, AFB, and IPIV are not modified. ! = 'N': The matrix A will be copied to AFB and factored. ! = 'E': The matrix A will be equilibrated if necessary, then ! copied to AFB and factored. ! ! TRANS (input) CHARACTER*1 ! Specifies the form of the system of equations. ! = 'N': A * X = B (No transpose) ! = 'T': A**T * X = B (Transpose) ! = 'C': A**H * X = B (Transpose) ! ! N (input) INTEGER ! The number of linear equations, i.e., the order of the ! matrix A. N >= 0. ! ! KL (input) INTEGER ! The number of subdiagonals within the band of A. KL >= 0. ! ! KU (input) INTEGER ! The number of superdiagonals within the band of A. KU >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) ! On entry, the matrix A in band storage, in rows 1 to KL+KU+1. ! The j-th column of A is stored in the j-th column of the ! array AB as follows: ! AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl) ! ! If FACT = 'F' and EQUED is not 'N', then A must have been ! equilibrated by the scaling factors in R and/or C. AB is not ! modified if FACT = 'F' or 'N', or if FACT = 'E' and ! EQUED = 'N' on exit. ! ! On exit, if EQUED .ne. 'N', A is scaled as follows: ! EQUED = 'R': A := diag(R) * A ! EQUED = 'C': A := A * diag(C) ! EQUED = 'B': A := diag(R) * A * diag(C). ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KL+KU+1. ! ! AFB (input or output) DOUBLE PRECISION array, dimension (LDAFB,N) ! If FACT = 'F', then AFB is an input argument and on entry ! contains details of the LU factorization of the band matrix ! A, as computed by DGBTRF. U is stored as an upper triangular ! band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, ! and the multipliers used during the factorization are stored ! in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is ! the factored form of the equilibrated matrix A. ! ! If FACT = 'N', then AFB is an output argument and on exit ! returns details of the LU factorization of A. ! ! If FACT = 'E', then AFB is an output argument and on exit ! returns details of the LU factorization of the equilibrated ! matrix A (see the description of AB for the form of the ! equilibrated matrix). ! ! LDAFB (input) INTEGER ! The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. ! ! IPIV (input or output) INTEGER array, dimension (N) ! If FACT = 'F', then IPIV is an input argument and on entry ! contains the pivot indices from the factorization A = L*U ! as computed by DGBTRF; row i of the matrix was interchanged ! with row IPIV(i). ! ! If FACT = 'N', then IPIV is an output argument and on exit ! contains the pivot indices from the factorization A = L*U ! of the original matrix A. ! ! If FACT = 'E', then IPIV is an output argument and on exit ! contains the pivot indices from the factorization A = L*U ! of the equilibrated matrix A. ! ! EQUED (input or output) CHARACTER*1 ! Specifies the form of equilibration that was done. ! = 'N': No equilibration (always true if FACT = 'N'). ! = 'R': Row equilibration, i.e., A has been premultiplied by ! diag(R). ! = 'C': Column equilibration, i.e., A has been postmultiplied ! by diag(C). ! = 'B': Both row and column equilibration, i.e., A has been ! replaced by diag(R) * A * diag(C). ! EQUED is an input argument if FACT = 'F'; otherwise, it is an ! output argument. ! ! R (input or output) DOUBLE PRECISION array, dimension (N) ! The row scale factors for A. If EQUED = 'R' or 'B', A is ! multiplied on the left by diag(R); if EQUED = 'N' or 'C', R ! is not accessed. R is an input argument if FACT = 'F'; ! otherwise, R is an output argument. If FACT = 'F' and ! EQUED = 'R' or 'B', each element of R must be positive. ! ! C (input or output) DOUBLE PRECISION array, dimension (N) ! The column scale factors for A. If EQUED = 'C' or 'B', A is ! multiplied on the right by diag(C); if EQUED = 'N' or 'R', C ! is not accessed. C is an input argument if FACT = 'F'; ! otherwise, C is an output argument. If FACT = 'F' and ! EQUED = 'C' or 'B', each element of C must be positive. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the right hand side matrix B. ! On exit, ! if EQUED = 'N', B is not modified; ! if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by ! diag(R)*B; ! if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is ! overwritten by diag(C)*B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X ! to the original system of equations. Note that A and B are ! modified on exit if EQUED .ne. 'N', and the solution to the ! equilibrated system is inv(diag(C))*X if TRANS = 'N' and ! EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' ! and EQUED = 'R' or 'B'. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! RCOND (output) DOUBLE PRECISION ! The estimate of the reciprocal condition number of the matrix ! A after equilibration (if done). If RCOND is less than the ! machine precision (in particular, if RCOND = 0), the matrix ! is singular to working precision. This condition is ! indicated by a return code of INFO > 0. ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (3*N) ! On exit, WORK(1) contains the reciprocal pivot growth ! factor norm(A)/norm(U). The "max absolute element" norm is ! used. If WORK(1) is much less than 1, then the stability ! of the LU factorization of the (equilibrated) matrix A ! could be poor. This also means that the solution X, condition ! estimator RCOND, and forward error bound FERR could be ! unreliable. If factorization fails with 0 0: if INFO = i, and i is ! <= N: U(i,i) is exactly zero. The factorization ! has been completed, but the factor U is exactly ! singular, so the solution and error bounds ! could not be computed. RCOND = 0 is returned. ! = N+1: U is nonsingular, but RCOND is less than machine ! precision, meaning that the matrix is singular ! to working precision. Nevertheless, the ! solution and error bounds are computed because ! there are a number of situations where the ! computed solution can be more accurate than the ! value of RCOND would suggest. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU CHARACTER NORM INTEGER I, INFEQU, J, J1, J2 DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN, & ROWCND, RPVGRW, SMLNUM ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANGB, DLANTB EXTERNAL LSAME, DLAMCH, DLANGB, DLANTB ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGBCON, DGBEQU, DGBRFS, DGBTRF, DGBTRS, & DLACPY, DLAQGB, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Executable Statements .. ! INFO = 0 NOFACT = LSAME( FACT, 'N' ) EQUIL = LSAME( FACT, 'E' ) NOTRAN = LSAME( TRANS, 'N' ) IF( NOFACT .OR. EQUIL ) THEN EQUED = 'N' ROWEQU = .FALSE. COLEQU = .FALSE. ELSE ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) SMLNUM = DLAMCH( 'Safe minimum' ) BIGNUM = ONE / SMLNUM END IF ! ! Test the input parameters. ! IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) & THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. & LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KL.LT.0 ) THEN INFO = -4 ELSE IF( KU.LT.0 ) THEN INFO = -5 ELSE IF( NRHS.LT.0 ) THEN INFO = -6 ELSE IF( LDAB.LT.KL+KU+1 ) THEN INFO = -8 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN INFO = -10 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. & ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN INFO = -12 ELSE IF( ROWEQU ) THEN RCMIN = BIGNUM RCMAX = ZERO DO J = 1, N RCMIN = MIN( RCMIN, R( J ) ) RCMAX = MAX( RCMAX, R( J ) ) end do IF( RCMIN.LE.ZERO ) THEN INFO = -13 ELSE IF( N.GT.0 ) THEN ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) ELSE ROWCND = ONE END IF END IF IF( COLEQU .AND. INFO.EQ.0 ) THEN RCMIN = BIGNUM RCMAX = ZERO DO 20 J = 1, N RCMIN = MIN( RCMIN, C( J ) ) RCMAX = MAX( RCMAX, C( J ) ) 20 CONTINUE IF( RCMIN.LE.ZERO ) THEN INFO = -14 ELSE IF( N.GT.0 ) THEN COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) ELSE COLCND = ONE END IF END IF IF( INFO.EQ.0 ) THEN IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -16 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -18 END IF END IF END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGBSVX', -INFO ) RETURN END IF ! IF( EQUIL ) THEN ! ! Compute row and column scalings to equilibrate the matrix A. ! CALL DGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, & AMAX, INFEQU ) IF( INFEQU.EQ.0 ) THEN ! ! Equilibrate the matrix. ! CALL DLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, & AMAX, EQUED ) ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) END IF END IF ! ! Scale the right hand side. ! IF( NOTRAN ) THEN IF( ROWEQU ) THEN DO 40 J = 1, NRHS DO 30 I = 1, N B( I, J ) = R( I )*B( I, J ) 30 CONTINUE 40 CONTINUE END IF ELSE IF( COLEQU ) THEN DO 60 J = 1, NRHS DO 50 I = 1, N B( I, J ) = C( I )*B( I, J ) 50 CONTINUE 60 CONTINUE END IF ! IF( NOFACT .OR. EQUIL ) THEN ! ! Compute the LU factorization of the band matrix A. ! DO 70 J = 1, N J1 = MAX( J-KU, 1 ) J2 = MIN( J+KL, N ) CALL DCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1, & AFB( KL+KU+1-J+J1, J ), 1 ) 70 CONTINUE ! CALL DGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO ) ! ! Return if INFO is non-zero. ! IF( INFO.NE.0 ) THEN IF( INFO.GT.0 ) THEN ! ! Compute the reciprocal pivot growth factor of the ! leading rank-deficient INFO columns of A. ! ANORM = ZERO DO 90 J = 1, INFO DO 80 I = MAX( KU+2-J, 1 ), & MIN( N+KU+1-J, KL+KU+1 ) ANORM = MAX( ANORM, ABS( AB( I, J ) ) ) 80 CONTINUE 90 CONTINUE RPVGRW = DLANTB( 'M', 'U', 'N', INFO, & MIN( INFO-1, KL+KU ), AFB( MAX( 1, & KL+KU+2-INFO ), 1 ), LDAFB, WORK ) IF( RPVGRW.EQ.ZERO ) THEN RPVGRW = ONE ELSE RPVGRW = ANORM / RPVGRW END IF WORK( 1 ) = RPVGRW RCOND = ZERO END IF RETURN END IF END IF ! ! Compute the norm of the matrix A and the ! reciprocal pivot growth factor RPVGRW. ! IF( NOTRAN ) THEN NORM = '1' ELSE NORM = 'I' END IF ANORM = DLANGB( NORM, N, KL, KU, AB, LDAB, WORK ) RPVGRW = DLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, WORK ) IF( RPVGRW.EQ.ZERO ) THEN RPVGRW = ONE ELSE RPVGRW = DLANGB( 'M', N, KL, KU, AB, LDAB, WORK ) / RPVGRW END IF ! ! Compute the reciprocal of the condition number of A. ! CALL DGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND, & WORK, IWORK, INFO ) ! ! Set INFO = N+1 if the matrix is singular to working precision. ! IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) & INFO = N + 1 ! ! Compute the solution matrix X. ! CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL DGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX, & INFO ) ! ! Use iterative refinement to improve the computed solution and ! compute error bounds and backward error estimates for it. ! CALL DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, & B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO ) ! ! Transform the solution matrix X to a solution of the original ! system. ! IF( NOTRAN ) THEN IF( COLEQU ) THEN DO 110 J = 1, NRHS DO I = 1, N X( I, J ) = C( I )*X( I, J ) end do 110 CONTINUE DO 120 J = 1, NRHS FERR( J ) = FERR( J ) / COLCND 120 CONTINUE END IF ELSE IF( ROWEQU ) THEN DO 140 J = 1, NRHS DO 130 I = 1, N X( I, J ) = R( I )*X( I, J ) 130 CONTINUE 140 CONTINUE DO 150 J = 1, NRHS FERR( J ) = FERR( J ) / ROWCND 150 CONTINUE END IF ! WORK( 1 ) = RPVGRW RETURN ! ! End of DGBSVX ! END SUBROUTINE DGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. INTEGER INFO, KL, KU, LDAB, M, N ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION AB( LDAB, * ) ! .. ! ! Purpose ! ======= ! ! DGBTF2 computes an LU factorization of a real m-by-n band matrix A ! using partial pivoting with row interchanges. ! ! This is the unblocked version of the algorithm, calling Level 2 BLAS. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! KL (input) INTEGER ! The number of subdiagonals within the band of A. KL >= 0. ! ! KU (input) INTEGER ! The number of superdiagonals within the band of A. KU >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) ! On entry, the matrix A in band storage, in rows KL+1 to ! 2*KL+KU+1; rows 1 to KL of the array need not be set. ! The j-th column of A is stored in the j-th column of the ! array AB as follows: ! AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl) ! ! On exit, details of the factorization: U is stored as an ! upper triangular band matrix with KL+KU superdiagonals in ! rows 1 to KL+KU+1, and the multipliers used during the ! factorization are stored in rows KL+KU+2 to 2*KL+KU+1. ! See below for further details. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= 2*KL+KU+1. ! ! IPIV (output) INTEGER array, dimension (min(M,N)) ! The pivot indices; for 1 <= i <= min(M,N), row i of the ! matrix was interchanged with row IPIV(i). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = +i, U(i,i) is exactly zero. The factorization ! has been completed, but the factor U is exactly ! singular, and division by zero will occur if it is used ! to solve a system of equations. ! ! Further Details ! =============== ! ! The band storage scheme is illustrated by the following example, when ! M = N = 6, KL = 2, KU = 1: ! ! On entry: On exit: ! ! * * * + + + * * * u14 u25 u36 ! * * + + + + * * u13 u24 u35 u46 ! * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 ! a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 ! a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 * ! a31 a42 a53 a64 * * m31 m42 m53 m64 * * ! ! Array elements marked * are not used by the routine; elements marked ! + need not be set on entry, but are required by the routine to store ! elements of U, because of fill-in resulting from the row ! interchanges. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J, JP, JU, KM, KV ! .. ! .. External Functions .. INTEGER IDAMAX EXTERNAL IDAMAX ! .. ! .. External Subroutines .. EXTERNAL DGER, DSCAL, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! KV is the number of superdiagonals in the factor U, allowing for ! fill-in. ! KV = KU + KL ! ! Test the input parameters. ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KL.LT.0 ) THEN INFO = -3 ELSE IF( KU.LT.0 ) THEN INFO = -4 ELSE IF( LDAB.LT.KL+KV+1 ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGBTF2', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 ) & RETURN ! ! Gaussian elimination with partial pivoting ! ! Set fill-in elements in columns KU+2 to KV to zero. ! DO 20 J = KU + 2, MIN( KV, N ) DO I = KV - J + 2, KL AB( I, J ) = ZERO end do 20 CONTINUE ! ! JU is the index of the last column affected by the current stage ! of the factorization. ! JU = 1 ! DO 40 J = 1, MIN( M, N ) ! ! Set fill-in elements in column J+KV to zero. ! IF( J+KV.LE.N ) THEN DO 30 I = 1, KL AB( I, J+KV ) = ZERO 30 CONTINUE END IF ! ! Find pivot and test for singularity. KM is the number of ! subdiagonal elements in the current column. ! KM = MIN( KL, M-J ) JP = IDAMAX( KM+1, AB( KV+1, J ), 1 ) IPIV( J ) = JP + J - 1 IF( AB( KV+JP, J ).NE.ZERO ) THEN JU = MAX( JU, MIN( J+KU+JP-1, N ) ) ! ! Apply interchange to columns J to JU. ! IF( JP.NE.1 ) & CALL DSWAP( JU-J+1, AB( KV+JP, J ), LDAB-1, & AB( KV+1, J ), LDAB-1 ) ! IF( KM.GT.0 ) THEN ! ! Compute multipliers. ! CALL DSCAL( KM, ONE / AB( KV+1, J ), AB( KV+2, J ), 1 ) ! ! Update trailing submatrix within the band. ! IF( JU.GT.J ) & CALL DGER( KM, JU-J, -ONE, AB( KV+2, J ), 1, & AB( KV, J+1 ), LDAB-1, AB( KV+1, J+1 ), & LDAB-1 ) END IF ELSE ! ! If pivot is zero, set INFO to the index of the pivot ! unless a zero pivot has already been found. ! IF( INFO.EQ.0 ) & INFO = J END IF 40 CONTINUE RETURN ! ! End of DGBTF2 ! END SUBROUTINE DGBTRF( M, N, KL, KU, AB, LDAB, IPIV, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. INTEGER INFO, KL, KU, LDAB, M, N ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION AB( LDAB, * ) ! .. ! ! Purpose ! ======= ! ! DGBTRF computes an LU factorization of a real m-by-n band matrix A ! using partial pivoting with row interchanges. ! ! This is the blocked version of the algorithm, calling Level 3 BLAS. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! KL (input) INTEGER ! The number of subdiagonals within the band of A. KL >= 0. ! ! KU (input) INTEGER ! The number of superdiagonals within the band of A. KU >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) ! On entry, the matrix A in band storage, in rows KL+1 to ! 2*KL+KU+1; rows 1 to KL of the array need not be set. ! The j-th column of A is stored in the j-th column of the ! array AB as follows: ! AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl) ! ! On exit, details of the factorization: U is stored as an ! upper triangular band matrix with KL+KU superdiagonals in ! rows 1 to KL+KU+1, and the multipliers used during the ! factorization are stored in rows KL+KU+2 to 2*KL+KU+1. ! See below for further details. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= 2*KL+KU+1. ! ! IPIV (output) INTEGER array, dimension (min(M,N)) ! The pivot indices; for 1 <= i <= min(M,N), row i of the ! matrix was interchanged with row IPIV(i). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = +i, U(i,i) is exactly zero. The factorization ! has been completed, but the factor U is exactly ! singular, and division by zero will occur if it is used ! to solve a system of equations. ! ! Further Details ! =============== ! ! The band storage scheme is illustrated by the following example, when ! M = N = 6, KL = 2, KU = 1: ! ! On entry: On exit: ! ! * * * + + + * * * u14 u25 u36 ! * * + + + + * * u13 u24 u35 u46 ! * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 ! a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 ! a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 * ! a31 a42 a53 a64 * * m31 m42 m53 m64 * * ! ! Array elements marked * are not used by the routine; elements marked ! + need not be set on entry, but are required by the routine to store ! elements of U because of fill-in resulting from the row interchanges. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) INTEGER NBMAX, LDWORK PARAMETER ( NBMAX = 64, LDWORK = NBMAX+1 ) ! .. ! .. Local Scalars .. INTEGER I, I2, I3, II, IP, J, J2, J3, JB, JJ, JM, JP, & JU, K2, KM, KV, NB, NW DOUBLE PRECISION TEMP ! .. ! .. Local Arrays .. DOUBLE PRECISION WORK13( LDWORK, NBMAX ), & WORK31( LDWORK, NBMAX ) ! .. ! .. External Functions .. INTEGER IDAMAX, ILAENV EXTERNAL IDAMAX, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGBTF2, DGEMM, DGER, DLASWP, DSCAL, & DSWAP, DTRSM, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! KV is the number of superdiagonals in the factor U, allowing for ! fill-in ! KV = KU + KL ! ! Test the input parameters. ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KL.LT.0 ) THEN INFO = -3 ELSE IF( KU.LT.0 ) THEN INFO = -4 ELSE IF( LDAB.LT.KL+KV+1 ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGBTRF', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 ) & RETURN ! ! Determine the block size for this environment ! NB = ILAENV( 1, 'DGBTRF', ' ', M, N, KL, KU ) ! ! The block size must not exceed the limit set by the size of the ! local arrays WORK13 and WORK31. ! NB = MIN( NB, NBMAX ) ! IF( NB.LE.1 .OR. NB.GT.KL ) THEN ! ! Use unblocked code ! CALL DGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO ) ELSE ! ! Use blocked code ! ! Zero the superdiagonal elements of the work array WORK13 ! DO 20 J = 1, NB WORK13(1:j-1, J ) = ZERO 20 CONTINUE ! ! Zero the subdiagonal elements of the work array WORK31 ! DO 40 J = 1, NB DO 30 I = J + 1, NB WORK31( I, J ) = ZERO 30 CONTINUE 40 CONTINUE ! ! Gaussian elimination with partial pivoting ! ! Set fill-in elements in columns KU+2 to KV to zero ! DO 60 J = KU + 2, MIN( KV, N ) DO 50 I = KV - J + 2, KL AB( I, J ) = ZERO 50 CONTINUE 60 CONTINUE ! ! JU is the index of the last column affected by the current ! stage of the factorization ! JU = 1 ! DO 180 J = 1, MIN( M, N ), NB JB = MIN( NB, MIN( M, N )-J+1 ) ! ! The active part of the matrix is partitioned ! ! A11 A12 A13 ! A21 A22 A23 ! A31 A32 A33 ! ! Here A11, A21 and A31 denote the current block of JB columns ! which is about to be factorized. The number of rows in the ! partitioning are JB, I2, I3 respectively, and the numbers ! of columns are JB, J2, J3. The superdiagonal elements of A13 ! and the subdiagonal elements of A31 lie outside the band. ! I2 = MIN( KL-JB, M-J-JB+1 ) I3 = MIN( JB, M-J-KL+1 ) ! ! J2 and J3 are computed after JU has been updated. ! ! Factorize the current block of JB columns ! DO 80 JJ = J, J + JB - 1 ! ! Set fill-in elements in column JJ+KV to zero ! IF( JJ+KV.LE.N ) THEN DO 70 I = 1, KL AB( I, JJ+KV ) = ZERO 70 CONTINUE END IF ! ! Find pivot and test for singularity. KM is the number of ! subdiagonal elements in the current column. ! KM = MIN( KL, M-JJ ) JP = IDAMAX( KM+1, AB( KV+1, JJ ), 1 ) IPIV( JJ ) = JP + JJ - J IF( AB( KV+JP, JJ ).NE.ZERO ) THEN JU = MAX( JU, MIN( JJ+KU+JP-1, N ) ) IF( JP.NE.1 ) THEN ! ! Apply interchange to columns J to J+JB-1 ! IF( JP+JJ-1.LT.J+KL ) THEN ! CALL DSWAP( JB, AB( KV+1+JJ-J, J ), LDAB-1, & AB( KV+JP+JJ-J, J ), LDAB-1 ) ELSE ! ! The interchange affects columns J to JJ-1 of A31 ! which are stored in the work array WORK31 ! CALL DSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1, & WORK31( JP+JJ-J-KL, 1 ), LDWORK ) CALL DSWAP( J+JB-JJ, AB( KV+1, JJ ), LDAB-1, & AB( KV+JP, JJ ), LDAB-1 ) END IF END IF ! ! Compute multipliers ! CALL DSCAL( KM, ONE / AB( KV+1, JJ ), AB( KV+2, JJ ), & 1 ) ! ! Update trailing submatrix within the band and within ! the current block. JM is the index of the last column ! which needs to be updated. ! JM = MIN( JU, J+JB-1 ) IF( JM.GT.JJ ) & CALL DGER( KM, JM-JJ, -ONE, AB( KV+2, JJ ), 1, & AB( KV, JJ+1 ), LDAB-1, & AB( KV+1, JJ+1 ), LDAB-1 ) ELSE ! ! If pivot is zero, set INFO to the index of the pivot ! unless a zero pivot has already been found. ! IF( INFO.EQ.0 ) & INFO = JJ END IF ! ! Copy current column of A31 into the work array WORK31 ! NW = MIN( JJ-J+1, I3 ) IF( NW.GT.0 ) & CALL DCOPY( NW, AB( KV+KL+1-JJ+J, JJ ), 1, & WORK31( 1, JJ-J+1 ), 1 ) 80 CONTINUE IF( J+JB.LE.N ) THEN ! ! Apply the row interchanges to the other blocks. ! J2 = MIN( JU-J+1, KV ) - JB J3 = MAX( 0, JU-J-KV+1 ) ! ! Use DLASWP to apply the row interchanges to A12, A22, and ! A32. ! CALL DLASWP( J2, AB( KV+1-JB, J+JB ), LDAB-1, 1, JB, & IPIV( J ), 1 ) ! ! Adjust the pivot indices. ! DO 90 I = J, J + JB - 1 IPIV( I ) = IPIV( I ) + J - 1 90 CONTINUE ! ! Apply the row interchanges to A13, A23, and A33 ! columnwise. ! K2 = J - 1 + JB + J2 DO 110 I = 1, J3 JJ = K2 + I DO II = J + I - 1, J + JB - 1 IP = IPIV( II ) IF( IP.NE.II ) THEN TEMP = AB( KV+1+II-JJ, JJ ) AB( KV+1+II-JJ, JJ ) = AB( KV+1+IP-JJ, JJ ) AB( KV+1+IP-JJ, JJ ) = TEMP END IF end do 110 CONTINUE ! ! Update the relevant part of the trailing submatrix ! IF( J2.GT.0 ) THEN ! ! Update A12 ! CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Unit', & JB, J2, ONE, AB( KV+1, J ), LDAB-1, & AB( KV+1-JB, J+JB ), LDAB-1 ) ! IF( I2.GT.0 ) THEN ! ! Update A22 ! CALL DGEMM( 'No transpose', 'No transpose', I2, J2, & JB, -ONE, AB( KV+1+JB, J ), LDAB-1, & AB( KV+1-JB, J+JB ), LDAB-1, ONE, & AB( KV+1, J+JB ), LDAB-1 ) END IF ! IF( I3.GT.0 ) THEN ! ! Update A32 ! CALL DGEMM( 'No transpose', 'No transpose', I3, J2, & JB, -ONE, WORK31, LDWORK, & AB( KV+1-JB, J+JB ), LDAB-1, ONE, & AB( KV+KL+1-JB, J+JB ), LDAB-1 ) END IF END IF ! IF( J3.GT.0 ) THEN ! ! Copy the lower triangle of A13 into the work array ! WORK13 ! DO 130 JJ = 1, J3 DO 120 II = JJ, JB WORK13( II, JJ ) = AB( II-JJ+1, JJ+J+KV-1 ) 120 CONTINUE 130 CONTINUE ! ! Update A13 in the work array ! CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Unit', & JB, J3, ONE, AB( KV+1, J ), LDAB-1, & WORK13, LDWORK ) ! IF( I2.GT.0 ) THEN ! ! Update A23 ! CALL DGEMM( 'No transpose', 'No transpose', I2, J3, & JB, -ONE, AB( KV+1+JB, J ), LDAB-1, & WORK13, LDWORK, ONE, AB( 1+JB, J+KV ), & LDAB-1 ) END IF ! IF( I3.GT.0 ) THEN ! ! Update A33 ! CALL DGEMM( 'No transpose', 'No transpose', I3, J3, & JB, -ONE, WORK31, LDWORK, WORK13, & LDWORK, ONE, AB( 1+KL, J+KV ), LDAB-1 ) END IF ! ! Copy the lower triangle of A13 back into place ! DO 150 JJ = 1, J3 DO 140 II = JJ, JB AB( II-JJ+1, JJ+J+KV-1 ) = WORK13( II, JJ ) 140 CONTINUE 150 CONTINUE END IF ELSE ! ! Adjust the pivot indices. ! DO 160 I = J, J + JB - 1 IPIV( I ) = IPIV( I ) + J - 1 160 CONTINUE END IF ! ! Partially undo the interchanges in the current block to ! restore the upper triangular form of A31 and copy the upper ! triangle of A31 back into place ! DO 170 JJ = J + JB - 1, J, -1 JP = IPIV( JJ ) - JJ + 1 IF( JP.NE.1 ) THEN ! ! Apply interchange to columns J to JJ-1 ! IF( JP+JJ-1.LT.J+KL ) THEN ! ! The interchange does not affect A31 ! CALL DSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1, & AB( KV+JP+JJ-J, J ), LDAB-1 ) ELSE ! ! The interchange does affect A31 ! CALL DSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1, & WORK31( JP+JJ-J-KL, 1 ), LDWORK ) END IF END IF ! ! Copy the current column of A31 back into place ! NW = MIN( I3, JJ-J+1 ) IF( NW.GT.0 ) & CALL DCOPY( NW, WORK31( 1, JJ-J+1 ), 1, & AB( KV+KL+1-JJ+J, JJ ), 1 ) 170 CONTINUE 180 CONTINUE END IF ! RETURN ! ! End of DGBTRF ! END SUBROUTINE DGBTRS( TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER TRANS INTEGER INFO, KL, KU, LDAB, LDB, N, NRHS ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DGBTRS solves a system of linear equations ! A * X = B or A' * X = B ! with a general band matrix A using the LU factorization computed ! by DGBTRF. ! ! Arguments ! ========= ! ! TRANS (input) CHARACTER*1 ! Specifies the form of the system of equations. ! = 'N': A * X = B (No transpose) ! = 'T': A'* X = B (Transpose) ! = 'C': A'* X = B (Conjugate transpose = Transpose) ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KL (input) INTEGER ! The number of subdiagonals within the band of A. KL >= 0. ! ! KU (input) INTEGER ! The number of superdiagonals within the band of A. KU >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! AB (input) DOUBLE PRECISION array, dimension (LDAB,N) ! Details of the LU factorization of the band matrix A, as ! computed by DGBTRF. U is stored as an upper triangular band ! matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and ! the multipliers used during the factorization are stored in ! rows KL+KU+2 to 2*KL+KU+1. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= 2*KL+KU+1. ! ! IPIV (input) INTEGER array, dimension (N) ! The pivot indices; for 1 <= i <= N, row i of the matrix was ! interchanged with row IPIV(i). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the right hand side matrix B. ! On exit, the solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LNOTI, NOTRAN INTEGER I, J, KD, L, LM ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DGEMV, DGER, DSWAP, DTBSV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 NOTRAN = LSAME( TRANS, 'N' ) IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. & LSAME( TRANS, 'C' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KL.LT.0 ) THEN INFO = -3 ELSE IF( KU.LT.0 ) THEN INFO = -4 ELSE IF( NRHS.LT.0 ) THEN INFO = -5 ELSE IF( LDAB.LT.( 2*KL+KU+1 ) ) THEN INFO = -7 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -10 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGBTRS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) & RETURN ! KD = KU + KL + 1 LNOTI = KL.GT.0 ! IF( NOTRAN ) THEN ! ! Solve A*X = B. ! ! Solve L*X = B, overwriting B with X. ! ! L is represented as a product of permutations and unit lower ! triangular matrices L = P(1) * L(1) * ... * P(n-1) * L(n-1), ! where each transformation L(i) is a rank-one modification of ! the identity matrix. ! IF( LNOTI ) THEN DO J = 1, N - 1 LM = MIN( KL, N-J ) L = IPIV( J ) IF( L.NE.J ) & CALL DSWAP( NRHS, B( L, 1 ), LDB, B( J, 1 ), LDB ) CALL DGER( LM, NRHS, -ONE, AB( KD+1, J ), 1, B( J, 1 ), & LDB, B( J+1, 1 ), LDB ) end do END IF DO 20 I = 1, NRHS ! ! Solve U*X = B, overwriting B with X. ! CALL DTBSV( 'Upper', 'No transpose', 'Non-unit', N, KL+KU, & AB, LDAB, B( 1, I ), 1 ) 20 CONTINUE ! ELSE ! ! Solve A'*X = B. ! DO 30 I = 1, NRHS ! ! Solve U'*X = B, overwriting B with X. ! CALL DTBSV( 'Upper', 'Transpose', 'Non-unit', N, KL+KU, AB, & LDAB, B( 1, I ), 1 ) 30 CONTINUE ! ! Solve L'*X = B, overwriting B with X. ! IF( LNOTI ) THEN DO 40 J = N - 1, 1, -1 LM = MIN( KL, N-J ) CALL DGEMV( 'Transpose', LM, NRHS, -ONE, B( J+1, 1 ), & LDB, AB( KD+1, J ), 1, ONE, B( J, 1 ), LDB ) L = IPIV( J ) IF( L.NE.J ) & CALL DSWAP( NRHS, B( L, 1 ), LDB, B( J, 1 ), LDB ) 40 CONTINUE END IF END IF RETURN ! ! End of DGBTRS ! END SUBROUTINE DGEBAK( JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER JOB, SIDE INTEGER IHI, ILO, INFO, LDV, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION SCALE( * ), V( LDV, * ) ! .. ! ! Purpose ! ======= ! ! DGEBAK forms the right or left eigenvectors of a real general matrix ! by backward transformation on the computed eigenvectors of the ! balanced matrix output by DGEBAL. ! ! Arguments ! ========= ! ! JOB (input) CHARACTER*1 ! Specifies the type of backward transformation required: ! = 'N', do nothing, return immediately; ! = 'P', do backward transformation for permutation only; ! = 'S', do backward transformation for scaling only; ! = 'B', do backward transformations for both permutation and ! scaling. ! JOB must be the same as the argument JOB supplied to DGEBAL. ! ! SIDE (input) CHARACTER*1 ! = 'R': V contains right eigenvectors; ! = 'L': V contains left eigenvectors. ! ! N (input) INTEGER ! The number of rows of the matrix V. N >= 0. ! ! ILO (input) INTEGER ! IHI (input) INTEGER ! The integers ILO and IHI determined by DGEBAL. ! 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. ! ! SCALE (input) DOUBLE PRECISION array, dimension (N) ! Details of the permutation and scaling factors, as returned ! by DGEBAL. ! ! M (input) INTEGER ! The number of columns of the matrix V. M >= 0. ! ! V (input/output) DOUBLE PRECISION array, dimension (LDV,M) ! On entry, the matrix of right or left eigenvectors to be ! transformed, as returned by DHSEIN or DTREVC. ! On exit, V is overwritten by the transformed eigenvectors. ! ! LDV (input) INTEGER ! The leading dimension of the array V. LDV >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LEFTV, RIGHTV INTEGER I, II, K DOUBLE PRECISION S ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DSCAL, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Decode and Test the input parameters ! RIGHTV = LSAME( SIDE, 'R' ) LEFTV = LSAME( SIDE, 'L' ) ! INFO = 0 IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND. & .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN INFO = -1 ELSE IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN INFO = -5 ELSE IF( M.LT.0 ) THEN INFO = -7 ELSE IF( LDV.LT.MAX( 1, N ) ) THEN INFO = -9 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEBAK', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN IF( M.EQ.0 ) & RETURN IF( LSAME( JOB, 'N' ) ) & RETURN ! IF( ILO.EQ.IHI ) & GO TO 30 ! ! Backward balance ! IF( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'B' ) ) THEN ! IF( RIGHTV ) THEN DO I = ILO, IHI S = SCALE( I ) CALL DSCAL( M, S, V( I, 1 ), LDV ) end do END IF IF( LEFTV ) THEN DO 20 I = ILO, IHI S = ONE / SCALE( I ) CALL DSCAL( M, S, V( I, 1 ), LDV ) 20 CONTINUE END IF ! END IF ! ! Backward permutation ! ! For I = ILO-1 step -1 until 1, ! IHI+1 step 1 until N do -- ! 30 CONTINUE IF( LSAME( JOB, 'P' ) .OR. LSAME( JOB, 'B' ) ) THEN IF( RIGHTV ) THEN DO 40 II = 1, N I = II IF( I.GE.ILO .AND. I.LE.IHI ) & GO TO 40 IF( I.LT.ILO ) & I = ILO - II K = SCALE( I ) IF( K.EQ.I ) & GO TO 40 CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV ) 40 CONTINUE END IF ! IF( LEFTV ) THEN DO 50 II = 1, N I = II IF( I.GE.ILO .AND. I.LE.IHI ) & GO TO 50 IF( I.LT.ILO ) & I = ILO - II K = SCALE( I ) IF( K.EQ.I ) & GO TO 50 CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV ) 50 CONTINUE END IF END IF ! RETURN ! ! End of DGEBAK ! END SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOB INTEGER IHI, ILO, INFO, LDA, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), SCALE( * ) ! .. ! ! Purpose ! ======= ! ! DGEBAL balances a general real matrix A. This involves, first, ! permuting A by a similarity transformation to isolate eigenvalues ! in the first 1 to ILO-1 and last IHI+1 to N elements on the ! diagonal; and second, applying a diagonal similarity transformation ! to rows and columns ILO to IHI to make the rows and columns as ! close in norm as possible. Both steps are optional. ! ! Balancing may reduce the 1-norm of the matrix, and improve the ! accuracy of the computed eigenvalues and/or eigenvectors. ! ! Arguments ! ========= ! ! JOB (input) CHARACTER*1 ! Specifies the operations to be performed on A: ! = 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0 ! for i = 1,...,N; ! = 'P': permute only; ! = 'S': scale only; ! = 'B': both permute and scale. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the input matrix A. ! On exit, A is overwritten by the balanced matrix. ! If JOB = 'N', A is not referenced. ! See Further Details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! ILO (output) INTEGER ! IHI (output) INTEGER ! ILO and IHI are set to integers such that on exit ! A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N. ! If JOB = 'N' or 'S', ILO = 1 and IHI = N. ! ! SCALE (output) DOUBLE PRECISION array, dimension (N) ! Details of the permutations and scaling factors applied to ! A. If P(j) is the index of the row and column interchanged ! with row and column j and D(j) is the scaling factor ! applied to row and column j, then ! SCALE(j) = P(j) for j = 1,...,ILO-1 ! = D(j) for j = ILO,...,IHI ! = P(j) for j = IHI+1,...,N. ! The order in which the interchanges are made is N to IHI+1, ! then 1 to ILO-1. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! The permutations consist of row and column interchanges which put ! the matrix in the form ! ! ( T1 X Y ) ! P A P = ( 0 B Z ) ! ( 0 0 T2 ) ! ! where T1 and T2 are upper triangular matrices whose eigenvalues lie ! along the diagonal. The column indices ILO and IHI mark the starting ! and ending columns of the submatrix B. Balancing consists of applying ! a diagonal similarity transformation inv(D) * B * D to make the ! 1-norms of each row of B and its corresponding column nearly equal. ! The output matrix is ! ! ( T1 X*D Y ) ! ( 0 inv(D)*B*D inv(D)*Z ). ! ( 0 0 T2 ) ! ! Information about the permutations P and the diagonal matrix D is ! returned in the vector SCALE. ! ! This subroutine is based on the EISPACK routine BALANC. ! ! Modified by Tzu-Yi Chen, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) DOUBLE PRECISION SCLFAC PARAMETER ( SCLFAC = 0.8D+1 ) DOUBLE PRECISION FACTOR PARAMETER ( FACTOR = 0.95D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOCONV INTEGER I, ICA, IEXC, IRA, J, K, L, M DOUBLE PRECISION C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1, & SFMIN2 ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DLAMCH EXTERNAL LSAME, IDAMAX, DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DSCAL, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND. & .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEBAL', -INFO ) RETURN END IF ! K = 1 L = N ! IF( N.EQ.0 ) & GO TO 210 ! IF( LSAME( JOB, 'N' ) ) THEN SCALE(1:n) = ONE GO TO 210 END IF ! IF( LSAME( JOB, 'S' ) ) & GO TO 120 ! ! Permutation to isolate eigenvalues if possible ! GO TO 50 ! ! Row and column exchange. ! 20 CONTINUE SCALE( M ) = J IF( J.EQ.M ) & GO TO 30 ! CALL DSWAP( L, A( 1, J ), 1, A( 1, M ), 1 ) CALL DSWAP( N-K+1, A( J, K ), LDA, A( M, K ), LDA ) ! 30 CONTINUE GO TO ( 40, 80 )IEXC ! ! Search for rows isolating an eigenvalue and push them down. ! 40 CONTINUE IF( L.EQ.1 ) & GO TO 210 L = L - 1 ! 50 CONTINUE DO 70 J = L, 1, -1 ! DO 60 I = 1, L IF( I.EQ.J ) & GO TO 60 IF( A( J, I ).NE.ZERO ) & GO TO 70 60 CONTINUE ! M = L IEXC = 1 GO TO 20 70 CONTINUE ! GO TO 90 ! ! Search for columns isolating an eigenvalue and push them left. ! 80 CONTINUE K = K + 1 ! 90 CONTINUE DO 110 J = K, L ! DO 100 I = K, L IF( I.EQ.J ) & GO TO 100 IF( A( I, J ).NE.ZERO ) & GO TO 110 100 CONTINUE ! M = K IEXC = 2 GO TO 20 110 CONTINUE ! 120 CONTINUE DO 130 I = K, L SCALE( I ) = ONE 130 CONTINUE ! IF( LSAME( JOB, 'P' ) ) & GO TO 210 ! ! Balance the submatrix in rows K to L. ! ! Iterative loop for norm reduction ! SFMIN1 = DLAMCH( 'S' ) / DLAMCH( 'P' ) SFMAX1 = ONE / SFMIN1 SFMIN2 = SFMIN1*SCLFAC SFMAX2 = ONE / SFMIN2 140 CONTINUE NOCONV = .FALSE. ! DO 200 I = K, L C = ZERO R = ZERO ! DO 150 J = K, L IF( J.EQ.I ) & GO TO 150 C = C + ABS( A( J, I ) ) R = R + ABS( A( I, J ) ) 150 CONTINUE ICA = IDAMAX( L, A( 1, I ), 1 ) CA = ABS( A( ICA, I ) ) IRA = IDAMAX( N-K+1, A( I, K ), LDA ) RA = ABS( A( I, IRA+K-1 ) ) ! ! Guard against zero C or R due to underflow. ! IF( C.EQ.ZERO .OR. R.EQ.ZERO ) & GO TO 200 G = R / SCLFAC F = ONE S = C + R 160 CONTINUE IF( C.GE.G .OR. MAX( F, C, CA ).GE.SFMAX2 .OR. & MIN( R, G, RA ).LE.SFMIN2 )GO TO 170 F = F*SCLFAC C = C*SCLFAC CA = CA*SCLFAC R = R / SCLFAC G = G / SCLFAC RA = RA / SCLFAC GO TO 160 ! 170 CONTINUE G = C / SCLFAC 180 CONTINUE IF( G.LT.R .OR. MAX( R, RA ).GE.SFMAX2 .OR. & MIN( F, C, G, CA ).LE.SFMIN2 )GO TO 190 F = F / SCLFAC C = C / SCLFAC G = G / SCLFAC CA = CA / SCLFAC R = R*SCLFAC RA = RA*SCLFAC GO TO 180 ! ! Now balance. ! 190 CONTINUE IF( ( C+R ).GE.FACTOR*S ) & GO TO 200 IF( F.LT.ONE .AND. SCALE( I ).LT.ONE ) THEN IF( F*SCALE( I ).LE.SFMIN1 ) & GO TO 200 END IF IF( F.GT.ONE .AND. SCALE( I ).GT.ONE ) THEN IF( SCALE( I ).GE.SFMAX1 / F ) & GO TO 200 END IF G = ONE / F SCALE( I ) = SCALE( I )*F NOCONV = .TRUE. ! CALL DSCAL( N-K+1, G, A( I, K ), LDA ) CALL DSCAL( L, F, A( 1, I ), 1 ) ! 200 CONTINUE ! IF( NOCONV ) & GO TO 140 ! 210 CONTINUE ILO = K IHI = L ! RETURN ! ! End of DGEBAL ! END SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), & TAUQ( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGEBD2 reduces a real general m by n matrix A to upper or lower ! bidiagonal form B by an orthogonal transformation: Q' * A * P = B. ! ! If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows in the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns in the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the m by n general matrix to be reduced. ! On exit, ! if m >= n, the diagonal and the first superdiagonal are ! overwritten with the upper bidiagonal matrix B; the ! elements below the diagonal, with the array TAUQ, represent ! the orthogonal matrix Q as a product of elementary ! reflectors, and the elements above the first superdiagonal, ! with the array TAUP, represent the orthogonal matrix P as ! a product of elementary reflectors; ! if m < n, the diagonal and the first subdiagonal are ! overwritten with the lower bidiagonal matrix B; the ! elements below the first subdiagonal, with the array TAUQ, ! represent the orthogonal matrix Q as a product of ! elementary reflectors, and the elements above the diagonal, ! with the array TAUP, represent the orthogonal matrix P as ! a product of elementary reflectors. ! See Further Details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! D (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The diagonal elements of the bidiagonal matrix B: ! D(i) = A(i,i). ! ! E (output) DOUBLE PRECISION array, dimension (min(M,N)-1) ! The off-diagonal elements of the bidiagonal matrix B: ! if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; ! if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. ! ! TAUQ (output) DOUBLE PRECISION array dimension (min(M,N)) ! The scalar factors of the elementary reflectors which ! represent the orthogonal matrix Q. See Further Details. ! ! TAUP (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The scalar factors of the elementary reflectors which ! represent the orthogonal matrix P. See Further Details. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (max(M,N)) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! The matrices Q and P are represented as products of elementary ! reflectors: ! ! If m >= n, ! ! Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) ! ! Each H(i) and G(i) has the form: ! ! H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' ! ! where tauq and taup are real scalars, and v and u are real vectors; ! v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); ! u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); ! tauq is stored in TAUQ(i) and taup in TAUP(i). ! ! If m < n, ! ! Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) ! ! Each H(i) and G(i) has the form: ! ! H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' ! ! where tauq and taup are real scalars, and v and u are real vectors; ! v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); ! u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); ! tauq is stored in TAUQ(i) and taup in TAUP(i). ! ! The contents of A on exit are illustrated by the following examples: ! ! m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ! ! ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) ! ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) ! ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) ! ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) ! ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) ! ( v1 v2 v3 v4 v5 ) ! ! where d and e denote diagonal and off-diagonal elements of B, vi ! denotes an element of the vector defining H(i), and ui an element of ! the vector defining G(i). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I ! .. ! .. External Subroutines .. EXTERNAL DLARF, DLARFG, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 END IF IF( INFO.LT.0 ) THEN CALL XERBLA( 'DGEBD2', -INFO ) RETURN END IF ! IF( M.GE.N ) THEN ! ! Reduce to upper bidiagonal form ! DO 10 I = 1, N ! ! Generate elementary reflector H(i) to annihilate A(i+1:m,i) ! CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1, & TAUQ( I ) ) D( I ) = A( I, I ) A( I, I ) = ONE ! ! Apply H(i) to A(i:m,i+1:n) from the left ! CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAUQ( I ), & A( I, I+1 ), LDA, WORK ) A( I, I ) = D( I ) ! IF( I.LT.N ) THEN ! ! Generate elementary reflector G(i) to annihilate ! A(i,i+2:n) ! CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ), & LDA, TAUP( I ) ) E( I ) = A( I, I+1 ) A( I, I+1 ) = ONE ! ! Apply G(i) to A(i+1:m,i+1:n) from the right ! CALL DLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA, & TAUP( I ), A( I+1, I+1 ), LDA, WORK ) A( I, I+1 ) = E( I ) ELSE TAUP( I ) = ZERO END IF 10 CONTINUE ELSE ! ! Reduce to lower bidiagonal form ! DO 20 I = 1, M ! ! Generate elementary reflector G(i) to annihilate A(i,i+1:n) ! CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA, & TAUP( I ) ) D( I ) = A( I, I ) A( I, I ) = ONE ! ! Apply G(i) to A(i+1:m,i:n) from the right ! CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA, TAUP( I ), & A( MIN( I+1, M ), I ), LDA, WORK ) A( I, I ) = D( I ) ! IF( I.LT.M ) THEN ! ! Generate elementary reflector H(i) to annihilate ! A(i+2:m,i) ! CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1, & TAUQ( I ) ) E( I ) = A( I+1, I ) A( I+1, I ) = ONE ! ! Apply H(i) to A(i+1:m,i+1:n) from the left ! CALL DLARF( 'Left', M-I, N-I, A( I+1, I ), 1, TAUQ( I ), & A( I+1, I+1 ), LDA, WORK ) A( I+1, I ) = E( I ) ELSE TAUQ( I ) = ZERO END IF 20 CONTINUE END IF RETURN ! ! End of DGEBD2 ! END SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), & TAUQ( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGEBRD reduces a general real M-by-N matrix A to upper or lower ! bidiagonal form B by an orthogonal transformation: Q**T * A * P = B. ! ! If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows in the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns in the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N general matrix to be reduced. ! On exit, ! if m >= n, the diagonal and the first superdiagonal are ! overwritten with the upper bidiagonal matrix B; the ! elements below the diagonal, with the array TAUQ, represent ! the orthogonal matrix Q as a product of elementary ! reflectors, and the elements above the first superdiagonal, ! with the array TAUP, represent the orthogonal matrix P as ! a product of elementary reflectors; ! if m < n, the diagonal and the first subdiagonal are ! overwritten with the lower bidiagonal matrix B; the ! elements below the first subdiagonal, with the array TAUQ, ! represent the orthogonal matrix Q as a product of ! elementary reflectors, and the elements above the diagonal, ! with the array TAUP, represent the orthogonal matrix P as ! a product of elementary reflectors. ! See Further Details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! D (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The diagonal elements of the bidiagonal matrix B: ! D(i) = A(i,i). ! ! E (output) DOUBLE PRECISION array, dimension (min(M,N)-1) ! The off-diagonal elements of the bidiagonal matrix B: ! if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; ! if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. ! ! TAUQ (output) DOUBLE PRECISION array dimension (min(M,N)) ! The scalar factors of the elementary reflectors which ! represent the orthogonal matrix Q. See Further Details. ! ! TAUP (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The scalar factors of the elementary reflectors which ! represent the orthogonal matrix P. See Further Details. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The length of the array WORK. LWORK >= max(1,M,N). ! For optimum performance LWORK >= (M+N)*NB, where NB ! is the optimal blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! The matrices Q and P are represented as products of elementary ! reflectors: ! ! If m >= n, ! ! Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) ! ! Each H(i) and G(i) has the form: ! ! H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' ! ! where tauq and taup are real scalars, and v and u are real vectors; ! v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); ! u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); ! tauq is stored in TAUQ(i) and taup in TAUP(i). ! ! If m < n, ! ! Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) ! ! Each H(i) and G(i) has the form: ! ! H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' ! ! where tauq and taup are real scalars, and v and u are real vectors; ! v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); ! u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); ! tauq is stored in TAUQ(i) and taup in TAUP(i). ! ! The contents of A on exit are illustrated by the following examples: ! ! m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ! ! ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) ! ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) ! ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) ! ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) ! ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) ! ( v1 v2 v3 v4 v5 ) ! ! where d and e denote diagonal and off-diagonal elements of B, vi ! denotes an element of the vector defining H(i), and ui an element of ! the vector defining G(i). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB, & NBMIN, NX DOUBLE PRECISION WS ! .. ! .. External Subroutines .. EXTERNAL DGEBD2, DGEMM, DLABRD, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC DBLE, MAX, MIN ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 NB = MAX( 1, ILAENV( 1, 'DGEBRD', ' ', M, N, -1, -1 ) ) LWKOPT = ( M+N )*NB WORK( 1 ) = DBLE( LWKOPT ) LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 ELSE IF( LWORK.LT.MAX( 1, M, N ) .AND. .NOT.LQUERY ) THEN INFO = -10 END IF IF( INFO.LT.0 ) THEN CALL XERBLA( 'DGEBRD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! MINMN = MIN( M, N ) IF( MINMN.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! WS = MAX( M, N ) LDWRKX = M LDWRKY = N ! IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN ! ! Set the crossover point NX. ! NX = MAX( NB, ILAENV( 3, 'DGEBRD', ' ', M, N, -1, -1 ) ) ! ! Determine when to switch from blocked to unblocked code. ! IF( NX.LT.MINMN ) THEN WS = ( M+N )*NB IF( LWORK.LT.WS ) THEN ! ! Not enough work space for the optimal NB, consider using ! a smaller block size. ! NBMIN = ILAENV( 2, 'DGEBRD', ' ', M, N, -1, -1 ) IF( LWORK.GE.( M+N )*NBMIN ) THEN NB = LWORK / ( M+N ) ELSE NB = 1 NX = MINMN END IF END IF END IF ELSE NX = MINMN END IF ! DO 30 I = 1, MINMN - NX, NB ! ! Reduce rows and columns i:i+nb-1 to bidiagonal form and return ! the matrices X and Y which are needed to update the unreduced ! part of the matrix ! CALL DLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ), E( I ), & TAUQ( I ), TAUP( I ), WORK, LDWRKX, & WORK( LDWRKX*NB+1 ), LDWRKY ) ! ! Update the trailing submatrix A(i+nb:m,i+nb:n), using an update ! of the form A := A - V*Y' - X*U' ! CALL DGEMM( 'No transpose', 'Transpose', M-I-NB+1, N-I-NB+1, & NB, -ONE, A( I+NB, I ), LDA, & WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE, & A( I+NB, I+NB ), LDA ) CALL DGEMM( 'No transpose', 'No transpose', M-I-NB+1, N-I-NB+1, & NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA, & ONE, A( I+NB, I+NB ), LDA ) ! ! Copy diagonal and off-diagonal elements of B back into A ! IF( M.GE.N ) THEN DO J = I, I + NB - 1 A( J, J ) = D( J ) A( J, J+1 ) = E( J ) end do ELSE DO 20 J = I, I + NB - 1 A( J, J ) = D( J ) A( J+1, J ) = E( J ) 20 CONTINUE END IF 30 CONTINUE ! ! Use unblocked code to reduce the remainder of the matrix ! CALL DGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ), & TAUQ( I ), TAUP( I ), WORK, IINFO ) WORK( 1 ) = WS RETURN ! ! End of DGEBRD ! END SUBROUTINE DGECON( NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER NORM INTEGER INFO, LDA, N DOUBLE PRECISION ANORM, RCOND ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGECON estimates the reciprocal of the condition number of a general ! real matrix A, in either the 1-norm or the infinity-norm, using ! the LU factorization computed by DGETRF. ! ! An estimate is obtained for norm(inv(A)), and the reciprocal of the ! condition number is computed as ! RCOND = 1 / ( norm(A) * norm(inv(A)) ). ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies whether the 1-norm condition number or the ! infinity-norm condition number is required: ! = '1' or 'O': 1-norm; ! = 'I': Infinity-norm. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The factors L and U from the factorization A = P*L*U ! as computed by DGETRF. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! ANORM (input) DOUBLE PRECISION ! If NORM = '1' or 'O', the 1-norm of the original matrix A. ! If NORM = 'I', the infinity-norm of the original matrix A. ! ! RCOND (output) DOUBLE PRECISION ! The reciprocal of the condition number of the matrix A, ! computed as RCOND = 1/(norm(A) * norm(inv(A))). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (4*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL ONENRM CHARACTER NORMIN INTEGER IX, KASE, KASE1 DOUBLE PRECISION AINVNM, SCALE, SL, SMLNUM, SU ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DLAMCH EXTERNAL LSAME, IDAMAX, DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DLACON, DLATRS, DRSCL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' ) IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( ANORM.LT.ZERO ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGECON', -INFO ) RETURN END IF ! ! Quick return if possible ! RCOND = ZERO IF( N.EQ.0 ) THEN RCOND = ONE RETURN ELSE IF( ANORM.EQ.ZERO ) THEN RETURN END IF ! SMLNUM = DLAMCH( 'Safe minimum' ) ! ! Estimate the norm of inv(A). ! AINVNM = ZERO NORMIN = 'N' IF( ONENRM ) THEN KASE1 = 1 ELSE KASE1 = 2 END IF KASE = 0 10 CONTINUE CALL DLACON( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.KASE1 ) THEN ! ! Multiply by inv(L). ! CALL DLATRS( 'Lower', 'No transpose', 'Unit', NORMIN, N, A, & LDA, WORK, SL, WORK( 2*N+1 ), INFO ) ! ! Multiply by inv(U). ! CALL DLATRS( 'Upper', 'No transpose', 'Non-unit', NORMIN, N, & A, LDA, WORK, SU, WORK( 3*N+1 ), INFO ) ELSE ! ! Multiply by inv(U'). ! CALL DLATRS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N, A, & LDA, WORK, SU, WORK( 3*N+1 ), INFO ) ! ! Multiply by inv(L'). ! CALL DLATRS( 'Lower', 'Transpose', 'Unit', NORMIN, N, A, & LDA, WORK, SL, WORK( 2*N+1 ), INFO ) END IF ! ! Divide X by 1/(SL*SU) if doing so will not cause overflow. ! SCALE = SL*SU NORMIN = 'Y' IF( SCALE.NE.ONE ) THEN IX = IDAMAX( N, WORK, 1 ) IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO ) & GO TO 20 CALL DRSCL( N, SCALE, WORK, 1 ) END IF GO TO 10 END IF ! ! Compute the estimate of the reciprocal condition number. ! IF( AINVNM.NE.ZERO ) & RCOND = ( ONE / AINVNM ) / ANORM ! 20 CONTINUE RETURN ! ! End of DGECON ! END SUBROUTINE DGEEQU( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, M, N DOUBLE PRECISION AMAX, COLCND, ROWCND ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( * ), R( * ) ! .. ! ! Purpose ! ======= ! ! DGEEQU computes row and column scalings intended to equilibrate an ! M-by-N matrix A and reduce its condition number. R returns the row ! scale factors and C the column scale factors, chosen to try to make ! the largest element in each row and column of the matrix B with ! elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1. ! ! R(i) and C(j) are restricted to be between SMLNUM = smallest safe ! number and BIGNUM = largest safe number. Use of these scaling ! factors is not guaranteed to reduce the condition number of A but ! works well in practice. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The M-by-N matrix whose equilibration factors are ! to be computed. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! R (output) DOUBLE PRECISION array, dimension (M) ! If INFO = 0 or INFO > M, R contains the row scale factors ! for A. ! ! C (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, C contains the column scale factors for A. ! ! ROWCND (output) DOUBLE PRECISION ! If INFO = 0 or INFO > M, ROWCND contains the ratio of the ! smallest R(i) to the largest R(i). If ROWCND >= 0.1 and ! AMAX is neither too large nor too small, it is not worth ! scaling by R. ! ! COLCND (output) DOUBLE PRECISION ! If INFO = 0, COLCND contains the ratio of the smallest ! C(i) to the largest C(i). If COLCND >= 0.1, it is not ! worth scaling by C. ! ! AMAX (output) DOUBLE PRECISION ! Absolute value of largest matrix element. If AMAX is very ! close to overflow or very close to underflow, the matrix ! should be scaled. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, and i is ! <= M: the i-th row of A is exactly zero ! > M: the (i-M)-th column of A is exactly zero ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEEQU', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 ) THEN ROWCND = ONE COLCND = ONE AMAX = ZERO RETURN END IF ! ! Get machine constants. ! SMLNUM = DLAMCH( 'S' ) BIGNUM = ONE / SMLNUM ! ! Compute row scale factors. ! R(1:m) = ZERO ! ! Find the maximum element in each row. ! DO 30 J = 1, N DO 20 I = 1, M R( I ) = MAX( R( I ), ABS( A( I, J ) ) ) 20 CONTINUE 30 CONTINUE ! ! Find the maximum and minimum scale factors. ! RCMIN = BIGNUM RCMAX = ZERO DO 40 I = 1, M RCMAX = MAX( RCMAX, R( I ) ) RCMIN = MIN( RCMIN, R( I ) ) 40 CONTINUE AMAX = RCMAX ! IF( RCMIN.EQ.ZERO ) THEN ! ! Find the first zero scale factor and return an error code. ! DO 50 I = 1, M IF( R( I ).EQ.ZERO ) THEN INFO = I RETURN END IF 50 CONTINUE ELSE ! ! Invert the scale factors. ! DO 60 I = 1, M R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM ) 60 CONTINUE ! ! Compute ROWCND = min(R(I)) / max(R(I)) ! ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) END IF ! ! Compute column scale factors ! DO 70 J = 1, N C( J ) = ZERO 70 CONTINUE ! ! Find the maximum element in each column, ! assuming the row scaling computed above. ! DO 90 J = 1, N DO 80 I = 1, M C( J ) = MAX( C( J ), ABS( A( I, J ) )*R( I ) ) 80 CONTINUE 90 CONTINUE ! ! Find the maximum and minimum scale factors. ! RCMIN = BIGNUM RCMAX = ZERO DO J = 1, N RCMIN = MIN( RCMIN, C( J ) ) RCMAX = MAX( RCMAX, C( J ) ) end do IF( RCMIN.EQ.ZERO ) THEN ! ! Find the first zero scale factor and return an error code. ! DO 110 J = 1, N IF( C( J ).EQ.ZERO ) THEN INFO = M + J RETURN END IF 110 CONTINUE ELSE ! ! Invert the scale factors. ! DO 120 J = 1, N C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM ) 120 CONTINUE ! ! Compute COLCND = min(C(J)) / max(C(J)) ! COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) END IF ! RETURN ! ! End of DGEEQU ! END SUBROUTINE DGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI, & VS, LDVS, WORK, LWORK, BWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBVS, SORT INTEGER INFO, LDA, LDVS, LWORK, N, SDIM ! .. ! .. Array Arguments .. LOGICAL BWORK( * ) DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), & WR( * ) ! .. ! .. Function Arguments .. LOGICAL SELECT EXTERNAL SELECT ! .. ! ! Purpose ! ======= ! ! DGEES computes for an N-by-N real nonsymmetric matrix A, the ! eigenvalues, the real Schur form T, and, optionally, the matrix of ! Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T). ! ! Optionally, it also orders the eigenvalues on the diagonal of the ! real Schur form so that selected eigenvalues are at the top left. ! The leading columns of Z then form an orthonormal basis for the ! invariant subspace corresponding to the selected eigenvalues. ! ! A matrix is in real Schur form if it is upper quasi-triangular with ! 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the ! form ! [ a b ] ! [ c a ] ! ! where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc). ! ! Arguments ! ========= ! ! JOBVS (input) CHARACTER*1 ! = 'N': Schur vectors are not computed; ! = 'V': Schur vectors are computed. ! ! SORT (input) CHARACTER*1 ! Specifies whether or not to order the eigenvalues on the ! diagonal of the Schur form. ! = 'N': Eigenvalues are not ordered; ! = 'S': Eigenvalues are ordered (see SELECT). ! ! SELECT (input) LOGICAL FUNCTION of two DOUBLE PRECISION arguments ! SELECT must be declared EXTERNAL in the calling subroutine. ! If SORT = 'S', SELECT is used to select eigenvalues to sort ! to the top left of the Schur form. ! If SORT = 'N', SELECT is not referenced. ! An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if ! SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex ! conjugate pair of eigenvalues is selected, then both complex ! eigenvalues are selected. ! Note that a selected complex eigenvalue may no longer ! satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since ! ordering may change the value of complex eigenvalues ! (especially if the eigenvalue is ill-conditioned); in this ! case INFO is set to N+2 (see INFO below). ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the N-by-N matrix A. ! On exit, A has been overwritten by its real Schur form T. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! SDIM (output) INTEGER ! If SORT = 'N', SDIM = 0. ! If SORT = 'S', SDIM = number of eigenvalues (after sorting) ! for which SELECT is true. (Complex conjugate ! pairs for which SELECT is true for either ! eigenvalue count as 2.) ! ! WR (output) DOUBLE PRECISION array, dimension (N) ! WI (output) DOUBLE PRECISION array, dimension (N) ! WR and WI contain the real and imaginary parts, ! respectively, of the computed eigenvalues in the same order ! that they appear on the diagonal of the output Schur form T. ! Complex conjugate pairs of eigenvalues will appear ! consecutively with the eigenvalue having the positive ! imaginary part first. ! ! VS (output) DOUBLE PRECISION array, dimension (LDVS,N) ! If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur ! vectors. ! If JOBVS = 'N', VS is not referenced. ! ! LDVS (input) INTEGER ! The leading dimension of the array VS. LDVS >= 1; if ! JOBVS = 'V', LDVS >= N. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) contains the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,3*N). ! For good performance, LWORK must generally be larger. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! BWORK (workspace) LOGICAL array, dimension (N) ! Not referenced if SORT = 'N'. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = i, and i is ! <= N: the QR algorithm failed to compute all the ! eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI ! contain those eigenvalues which have converged; if ! JOBVS = 'V', VS contains the matrix which reduces A ! to its partially converged Schur form. ! = N+1: the eigenvalues could not be reordered because some ! eigenvalues were too close to separate (the problem ! is very ill-conditioned); ! = N+2: after reordering, roundoff changed values of some ! complex eigenvalues so that leading eigenvalues in ! the Schur form no longer satisfy SELECT=.TRUE. This ! could also be caused by underflow due to scaling. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL CURSL, LASTSL, LQUERY, LST2SL, SCALEA, WANTST, & WANTVS INTEGER HSWORK, I, I1, I2, IBAL, ICOND, IERR, IEVAL, & IHI, ILO, INXT, IP, ITAU, IWRK, K, MAXB, & MAXWRK, MINWRK DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, S, SEP, SMLNUM ! .. ! .. Local Arrays .. INTEGER IDUM( 1 ) DOUBLE PRECISION DUM( 1 ) ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLACPY, & DLASCL, DORGHR, DSWAP, DTRSEN, XERBLA ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LQUERY = ( LWORK.EQ.-1 ) WANTVS = LSAME( JOBVS, 'V' ) WANTST = LSAME( SORT, 'S' ) IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN INFO = -1 ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN INFO = -11 END IF ! ! Compute workspace ! (Note: Comments in the code beginning "Workspace:" describe the ! minimal amount of workspace needed at that point in the code, ! as well as the preferred amount for good performance. ! NB refers to the optimal block size for the immediately ! following subroutine, as returned by ILAENV. ! HSWORK refers to the workspace preferred by DHSEQR, as ! calculated below. HSWORK is computed assuming ILO=1 and IHI=N, ! the worst case.) ! MINWRK = 1 IF( INFO.EQ.0 .AND. ( LWORK.GE.1 .OR. LQUERY ) ) THEN MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 ) MINWRK = MAX( 1, 3*N ) IF( .NOT.WANTVS ) THEN MAXB = MAX( ILAENV( 8, 'DHSEQR', 'SN', N, 1, N, -1 ), 2 ) K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'SN', N, 1, & N, -1 ) ) ) HSWORK = MAX( K*( K+2 ), 2*N ) MAXWRK = MAX( MAXWRK, N+HSWORK, 1 ) ELSE MAXWRK = MAX( MAXWRK, 2*N+( N-1 )* & ILAENV( 1, 'DORGHR', ' ', N, 1, N, -1 ) ) MAXB = MAX( ILAENV( 8, 'DHSEQR', 'EN', N, 1, N, -1 ), 2 ) K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'EN', N, 1, & N, -1 ) ) ) HSWORK = MAX( K*( K+2 ), 2*N ) MAXWRK = MAX( MAXWRK, N+HSWORK, 1 ) END IF WORK( 1 ) = MAXWRK END IF IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -13 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEES ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) THEN SDIM = 0 RETURN END IF ! ! Get machine constants ! EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) SMLNUM = SQRT( SMLNUM ) / EPS BIGNUM = ONE / SMLNUM ! ! Scale A if max element outside range [SMLNUM,BIGNUM] ! ANRM = DLANGE( 'M', N, N, A, LDA, DUM ) SCALEA = .FALSE. IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN SCALEA = .TRUE. CSCALE = SMLNUM ELSE IF( ANRM.GT.BIGNUM ) THEN SCALEA = .TRUE. CSCALE = BIGNUM END IF IF( SCALEA ) & CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) ! ! Permute the matrix to make it more nearly triangular ! (Workspace: need N) ! IBAL = 1 CALL DGEBAL( 'P', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR ) ! ! Reduce to upper Hessenberg form ! (Workspace: need 3*N, prefer 2*N+N*NB) ! ITAU = N + IBAL IWRK = N + ITAU CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), & LWORK-IWRK+1, IERR ) ! IF( WANTVS ) THEN ! ! Copy Householder vectors to VS ! CALL DLACPY( 'L', N, N, A, LDA, VS, LDVS ) ! ! Generate orthogonal matrix in VS ! (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB) ! CALL DORGHR( N, ILO, IHI, VS, LDVS, WORK( ITAU ), WORK( IWRK ), & LWORK-IWRK+1, IERR ) END IF ! SDIM = 0 ! ! Perform QR iteration, accumulating Schur vectors in VS if desired ! (Workspace: need N+1, prefer N+HSWORK (see comments) ) ! IWRK = ITAU CALL DHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, WR, WI, VS, LDVS, & WORK( IWRK ), LWORK-IWRK+1, IEVAL ) IF( IEVAL.GT.0 ) & INFO = IEVAL ! ! Sort eigenvalues if desired ! IF( WANTST .AND. INFO.EQ.0 ) THEN IF( SCALEA ) THEN CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WR, N, IERR ) CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WI, N, IERR ) END IF DO I = 1, N BWORK( I ) = SELECT( WR( I ), WI( I ) ) end do ! ! Reorder eigenvalues and transform Schur vectors ! (Workspace: none needed) ! CALL DTRSEN( 'N', JOBVS, BWORK, N, A, LDA, VS, LDVS, WR, WI, & SDIM, S, SEP, WORK( IWRK ), LWORK-IWRK+1, IDUM, 1, & ICOND ) IF( ICOND.GT.0 ) & INFO = N + ICOND END IF ! IF( WANTVS ) THEN ! ! Undo balancing ! (Workspace: need N) ! CALL DGEBAK( 'P', 'R', N, ILO, IHI, WORK( IBAL ), N, VS, LDVS, & IERR ) END IF ! IF( SCALEA ) THEN ! ! Undo scaling for the Schur form of A ! CALL DLASCL( 'H', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR ) CALL DCOPY( N, A, LDA+1, WR, 1 ) IF( CSCALE.EQ.SMLNUM ) THEN ! ! If scaling back towards underflow, adjust WI if an ! offdiagonal element of a 2-by-2 block in the Schur form ! underflows. ! IF( IEVAL.GT.0 ) THEN I1 = IEVAL + 1 I2 = IHI - 1 CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, & MAX( ILO-1, 1 ), IERR ) ELSE IF( WANTST ) THEN I1 = 1 I2 = N - 1 ELSE I1 = ILO I2 = IHI - 1 END IF INXT = I1 - 1 DO 20 I = I1, I2 IF( I.LT.INXT ) & GO TO 20 IF( WI( I ).EQ.ZERO ) THEN INXT = I + 1 ELSE IF( A( I+1, I ).EQ.ZERO ) THEN WI( I ) = ZERO WI( I+1 ) = ZERO ELSE IF( A( I+1, I ).NE.ZERO .AND. A( I, I+1 ).EQ. & ZERO ) THEN WI( I ) = ZERO WI( I+1 ) = ZERO IF( I.GT.1 ) & CALL DSWAP( I-1, A( 1, I ), 1, A( 1, I+1 ), 1 ) IF( N.GT.I+1 ) & CALL DSWAP( N-I-1, A( I, I+2 ), LDA, & A( I+1, I+2 ), LDA ) CALL DSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ), 1 ) A( I, I+1 ) = A( I+1, I ) A( I+1, I ) = ZERO END IF INXT = I + 2 END IF 20 CONTINUE END IF ! ! Undo scaling for the imaginary part of the eigenvalues ! CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-IEVAL, 1, & WI( IEVAL+1 ), MAX( N-IEVAL, 1 ), IERR ) END IF ! IF( WANTST .AND. INFO.EQ.0 ) THEN ! ! Check if reordering successful ! LASTSL = .TRUE. LST2SL = .TRUE. SDIM = 0 IP = 0 DO 30 I = 1, N CURSL = SELECT( WR( I ), WI( I ) ) IF( WI( I ).EQ.ZERO ) THEN IF( CURSL ) & SDIM = SDIM + 1 IP = 0 IF( CURSL .AND. .NOT.LASTSL ) & INFO = N + 2 ELSE IF( IP.EQ.1 ) THEN ! ! Last eigenvalue of conjugate pair ! CURSL = CURSL .OR. LASTSL LASTSL = CURSL IF( CURSL ) & SDIM = SDIM + 2 IP = -1 IF( CURSL .AND. .NOT.LST2SL ) & INFO = N + 2 ELSE ! ! First eigenvalue of conjugate pair ! IP = 1 END IF END IF LST2SL = LASTSL LASTSL = CURSL 30 CONTINUE END IF ! WORK( 1 ) = MAXWRK RETURN ! ! End of DGEES ! END SUBROUTINE DGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, & WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK, & IWORK, LIWORK, BWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBVS, SENSE, SORT INTEGER INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM DOUBLE PRECISION RCONDE, RCONDV ! .. ! .. Array Arguments .. LOGICAL BWORK( * ) INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), & WR( * ) ! .. ! .. Function Arguments .. LOGICAL SELECT EXTERNAL SELECT ! .. ! ! Purpose ! ======= ! ! DGEESX computes for an N-by-N real nonsymmetric matrix A, the ! eigenvalues, the real Schur form T, and, optionally, the matrix of ! Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T). ! ! Optionally, it also orders the eigenvalues on the diagonal of the ! real Schur form so that selected eigenvalues are at the top left; ! computes a reciprocal condition number for the average of the ! selected eigenvalues (RCONDE); and computes a reciprocal condition ! number for the right invariant subspace corresponding to the ! selected eigenvalues (RCONDV). The leading columns of Z form an ! orthonormal basis for this invariant subspace. ! ! For further explanation of the reciprocal condition numbers RCONDE ! and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where ! these quantities are called s and sep respectively). ! ! A real matrix is in real Schur form if it is upper quasi-triangular ! with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in ! the form ! [ a b ] ! [ c a ] ! ! where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc). ! ! Arguments ! ========= ! ! JOBVS (input) CHARACTER*1 ! = 'N': Schur vectors are not computed; ! = 'V': Schur vectors are computed. ! ! SORT (input) CHARACTER*1 ! Specifies whether or not to order the eigenvalues on the ! diagonal of the Schur form. ! = 'N': Eigenvalues are not ordered; ! = 'S': Eigenvalues are ordered (see SELECT). ! ! SELECT (input) LOGICAL FUNCTION of two DOUBLE PRECISION arguments ! SELECT must be declared EXTERNAL in the calling subroutine. ! If SORT = 'S', SELECT is used to select eigenvalues to sort ! to the top left of the Schur form. ! If SORT = 'N', SELECT is not referenced. ! An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if ! SELECT(WR(j),WI(j)) is true; i.e., if either one of a ! complex conjugate pair of eigenvalues is selected, then both ! are. Note that a selected complex eigenvalue may no longer ! satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since ! ordering may change the value of complex eigenvalues ! (especially if the eigenvalue is ill-conditioned); in this ! case INFO may be set to N+3 (see INFO below). ! ! SENSE (input) CHARACTER*1 ! Determines which reciprocal condition numbers are computed. ! = 'N': None are computed; ! = 'E': Computed for average of selected eigenvalues only; ! = 'V': Computed for selected right invariant subspace only; ! = 'B': Computed for both. ! If SENSE = 'E', 'V' or 'B', SORT must equal 'S'. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the N-by-N matrix A. ! On exit, A is overwritten by its real Schur form T. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! SDIM (output) INTEGER ! If SORT = 'N', SDIM = 0. ! If SORT = 'S', SDIM = number of eigenvalues (after sorting) ! for which SELECT is true. (Complex conjugate ! pairs for which SELECT is true for either ! eigenvalue count as 2.) ! ! WR (output) DOUBLE PRECISION array, dimension (N) ! WI (output) DOUBLE PRECISION array, dimension (N) ! WR and WI contain the real and imaginary parts, respectively, ! of the computed eigenvalues, in the same order that they ! appear on the diagonal of the output Schur form T. Complex ! conjugate pairs of eigenvalues appear consecutively with the ! eigenvalue having the positive imaginary part first. ! ! VS (output) DOUBLE PRECISION array, dimension (LDVS,N) ! If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur ! vectors. ! If JOBVS = 'N', VS is not referenced. ! ! LDVS (input) INTEGER ! The leading dimension of the array VS. LDVS >= 1, and if ! JOBVS = 'V', LDVS >= N. ! ! RCONDE (output) DOUBLE PRECISION ! If SENSE = 'E' or 'B', RCONDE contains the reciprocal ! condition number for the average of the selected eigenvalues. ! Not referenced if SENSE = 'N' or 'V'. ! ! RCONDV (output) DOUBLE PRECISION ! If SENSE = 'V' or 'B', RCONDV contains the reciprocal ! condition number for the selected right invariant subspace. ! Not referenced if SENSE = 'N' or 'E'. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,3*N). ! Also, if SENSE = 'E' or 'V' or 'B', ! LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of ! selected eigenvalues computed by this routine. Note that ! N+2*SDIM*(N-SDIM) <= N+N*N/2. ! For good performance, LWORK must generally be larger. ! ! IWORK (workspace/output) INTEGER array, dimension (LIWORK) ! Not referenced if SENSE = 'N' or 'E'. ! On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. ! ! LIWORK (input) INTEGER ! The dimension of the array IWORK. ! LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM). ! ! BWORK (workspace) LOGICAL array, dimension (N) ! Not referenced if SORT = 'N'. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = i, and i is ! <= N: the QR algorithm failed to compute all the ! eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI ! contain those eigenvalues which have converged; if ! JOBVS = 'V', VS contains the transformation which ! reduces A to its partially converged Schur form. ! = N+1: the eigenvalues could not be reordered because some ! eigenvalues were too close to separate (the problem ! is very ill-conditioned); ! = N+2: after reordering, roundoff changed values of some ! complex eigenvalues so that leading eigenvalues in ! the Schur form no longer satisfy SELECT=.TRUE. This ! could also be caused by underflow due to scaling. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL CURSL, LASTSL, LST2SL, SCALEA, WANTSB, WANTSE, & WANTSN, WANTST, WANTSV, WANTVS INTEGER HSWORK, I, I1, I2, IBAL, ICOND, IERR, IEVAL, & IHI, ILO, INXT, IP, ITAU, IWRK, K, MAXB, & MAXWRK, MINWRK DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, SMLNUM ! .. ! .. Local Arrays .. DOUBLE PRECISION DUM( 1 ) ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLACPY, & DLASCL, DORGHR, DSWAP, DTRSEN, XERBLA ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 WANTVS = LSAME( JOBVS, 'V' ) WANTST = LSAME( SORT, 'S' ) WANTSN = LSAME( SENSE, 'N' ) WANTSE = LSAME( SENSE, 'E' ) WANTSV = LSAME( SENSE, 'V' ) WANTSB = LSAME( SENSE, 'B' ) IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN INFO = -1 ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR. & ( .NOT.WANTST .AND. .NOT.WANTSN ) ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN INFO = -12 END IF ! ! Compute workspace ! (Note: Comments in the code beginning "RWorkspace:" describe the ! minimal amount of real workspace needed at that point in the ! code, as well as the preferred amount for good performance. ! IWorkspace refers to integer workspace. ! NB refers to the optimal block size for the immediately ! following subroutine, as returned by ILAENV. ! HSWORK refers to the workspace preferred by DHSEQR, as ! calculated below. HSWORK is computed assuming ILO=1 and IHI=N, ! the worst case. ! If SENSE = 'E', 'V' or 'B', then the amount of workspace needed ! depends on SDIM, which is computed by the routine DTRSEN later ! in the code.) ! MINWRK = 1 IF( INFO.EQ.0 .AND. LWORK.GE.1 ) THEN MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 ) MINWRK = MAX( 1, 3*N ) IF( .NOT.WANTVS ) THEN MAXB = MAX( ILAENV( 8, 'DHSEQR', 'SN', N, 1, N, -1 ), 2 ) K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'SN', N, 1, & N, -1 ) ) ) HSWORK = MAX( K*( K+2 ), 2*N ) MAXWRK = MAX( MAXWRK, N+HSWORK, 1 ) ELSE MAXWRK = MAX( MAXWRK, 2*N+( N-1 )* & ILAENV( 1, 'DORGHR', ' ', N, 1, N, -1 ) ) MAXB = MAX( ILAENV( 8, 'DHSEQR', 'SV', N, 1, N, -1 ), 2 ) K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'SV', N, 1, & N, -1 ) ) ) HSWORK = MAX( K*( K+2 ), 2*N ) MAXWRK = MAX( MAXWRK, N+HSWORK, 1 ) END IF WORK( 1 ) = MAXWRK END IF IF( LWORK.LT.MINWRK ) THEN INFO = -16 END IF IF( LIWORK.LT.1 ) THEN INFO = -18 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEESX', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) THEN SDIM = 0 RETURN END IF ! ! Get machine constants ! EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) SMLNUM = SQRT( SMLNUM ) / EPS BIGNUM = ONE / SMLNUM ! ! Scale A if max element outside range [SMLNUM,BIGNUM] ! ANRM = DLANGE( 'M', N, N, A, LDA, DUM ) SCALEA = .FALSE. IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN SCALEA = .TRUE. CSCALE = SMLNUM ELSE IF( ANRM.GT.BIGNUM ) THEN SCALEA = .TRUE. CSCALE = BIGNUM END IF IF( SCALEA ) & CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) ! ! Permute the matrix to make it more nearly triangular ! (RWorkspace: need N) ! IBAL = 1 CALL DGEBAL( 'P', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR ) ! ! Reduce to upper Hessenberg form ! (RWorkspace: need 3*N, prefer 2*N+N*NB) ! ITAU = N + IBAL IWRK = N + ITAU CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), & LWORK-IWRK+1, IERR ) ! IF( WANTVS ) THEN ! ! Copy Householder vectors to VS ! CALL DLACPY( 'L', N, N, A, LDA, VS, LDVS ) ! ! Generate orthogonal matrix in VS ! (RWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB) ! CALL DORGHR( N, ILO, IHI, VS, LDVS, WORK( ITAU ), WORK( IWRK ), & LWORK-IWRK+1, IERR ) END IF ! SDIM = 0 ! ! Perform QR iteration, accumulating Schur vectors in VS if desired ! (RWorkspace: need N+1, prefer N+HSWORK (see comments) ) ! IWRK = ITAU CALL DHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, WR, WI, VS, LDVS, & WORK( IWRK ), LWORK-IWRK+1, IEVAL ) IF( IEVAL.GT.0 ) & INFO = IEVAL ! ! Sort eigenvalues if desired ! IF( WANTST .AND. INFO.EQ.0 ) THEN IF( SCALEA ) THEN CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WR, N, IERR ) CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WI, N, IERR ) END IF DO I = 1, N BWORK( I ) = SELECT( WR( I ), WI( I ) ) end do ! ! Reorder eigenvalues, transform Schur vectors, and compute ! reciprocal condition numbers ! (RWorkspace: if SENSE is not 'N', need N+2*SDIM*(N-SDIM) ! otherwise, need N ) ! (IWorkspace: if SENSE is 'V' or 'B', need SDIM*(N-SDIM) ! otherwise, need 0 ) ! CALL DTRSEN( SENSE, JOBVS, BWORK, N, A, LDA, VS, LDVS, WR, WI, & SDIM, RCONDE, RCONDV, WORK( IWRK ), LWORK-IWRK+1, & IWORK, LIWORK, ICOND ) IF( .NOT.WANTSN ) & MAXWRK = MAX( MAXWRK, N+2*SDIM*( N-SDIM ) ) IF( ICOND.EQ.-15 ) THEN ! ! Not enough real workspace ! INFO = -16 ELSE IF( ICOND.EQ.-17 ) THEN ! ! Not enough integer workspace ! INFO = -18 ELSE IF( ICOND.GT.0 ) THEN ! ! DTRSEN failed to reorder or to restore standard Schur form ! INFO = ICOND + N END IF END IF ! IF( WANTVS ) THEN ! ! Undo balancing ! (RWorkspace: need N) ! CALL DGEBAK( 'P', 'R', N, ILO, IHI, WORK( IBAL ), N, VS, LDVS, & IERR ) END IF ! IF( SCALEA ) THEN ! ! Undo scaling for the Schur form of A ! CALL DLASCL( 'H', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR ) CALL DCOPY( N, A, LDA+1, WR, 1 ) IF( ( WANTSV .OR. WANTSB ) .AND. INFO.EQ.0 ) THEN DUM( 1 ) = RCONDV CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR ) RCONDV = DUM( 1 ) END IF IF( CSCALE.EQ.SMLNUM ) THEN ! ! If scaling back towards underflow, adjust WI if an ! offdiagonal element of a 2-by-2 block in the Schur form ! underflows. ! IF( IEVAL.GT.0 ) THEN I1 = IEVAL + 1 I2 = IHI - 1 CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N, & IERR ) ELSE IF( WANTST ) THEN I1 = 1 I2 = N - 1 ELSE I1 = ILO I2 = IHI - 1 END IF INXT = I1 - 1 DO 20 I = I1, I2 IF( I.LT.INXT ) & GO TO 20 IF( WI( I ).EQ.ZERO ) THEN INXT = I + 1 ELSE IF( A( I+1, I ).EQ.ZERO ) THEN WI( I ) = ZERO WI( I+1 ) = ZERO ELSE IF( A( I+1, I ).NE.ZERO .AND. A( I, I+1 ).EQ. & ZERO ) THEN WI( I ) = ZERO WI( I+1 ) = ZERO IF( I.GT.1 ) & CALL DSWAP( I-1, A( 1, I ), 1, A( 1, I+1 ), 1 ) IF( N.GT.I+1 ) & CALL DSWAP( N-I-1, A( I, I+2 ), LDA, & A( I+1, I+2 ), LDA ) CALL DSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ), 1 ) A( I, I+1 ) = A( I+1, I ) A( I+1, I ) = ZERO END IF INXT = I + 2 END IF 20 CONTINUE END IF CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-IEVAL, 1, & WI( IEVAL+1 ), MAX( N-IEVAL, 1 ), IERR ) END IF ! IF( WANTST .AND. INFO.EQ.0 ) THEN ! ! Check if reordering successful ! LASTSL = .TRUE. LST2SL = .TRUE. SDIM = 0 IP = 0 DO 30 I = 1, N CURSL = SELECT( WR( I ), WI( I ) ) IF( WI( I ).EQ.ZERO ) THEN IF( CURSL ) & SDIM = SDIM + 1 IP = 0 IF( CURSL .AND. .NOT.LASTSL ) & INFO = N + 2 ELSE IF( IP.EQ.1 ) THEN ! ! Last eigenvalue of conjugate pair ! CURSL = CURSL .OR. LASTSL LASTSL = CURSL IF( CURSL ) & SDIM = SDIM + 2 IP = -1 IF( CURSL .AND. .NOT.LST2SL ) & INFO = N + 2 ELSE ! ! First eigenvalue of conjugate pair ! IP = 1 END IF END IF LST2SL = LASTSL LASTSL = CURSL 30 CONTINUE END IF ! WORK( 1 ) = MAXWRK IF( WANTSV .OR. WANTSB ) THEN IWORK( 1 ) = SDIM*( N-SDIM ) ELSE IWORK( 1 ) = 1 END IF ! RETURN ! ! End of DGEESX ! END SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, & LDVR, WORK, LWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! December 8, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBVL, JOBVR INTEGER INFO, LDA, LDVL, LDVR, LWORK, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), & WI( * ), WORK( * ), WR( * ) ! .. ! ! Purpose ! ======= ! ! DGEEV computes for an N-by-N real nonsymmetric matrix A, the ! eigenvalues and, optionally, the left and/or right eigenvectors. ! ! The right eigenvector v(j) of A satisfies ! A * v(j) = lambda(j) * v(j) ! where lambda(j) is its eigenvalue. ! The left eigenvector u(j) of A satisfies ! u(j)**H * A = lambda(j) * u(j)**H ! where u(j)**H denotes the conjugate transpose of u(j). ! ! The computed eigenvectors are normalized to have Euclidean norm ! equal to 1 and largest component real. ! ! Arguments ! ========= ! ! JOBVL (input) CHARACTER*1 ! = 'N': left eigenvectors of A are not computed; ! = 'V': left eigenvectors of A are computed. ! ! JOBVR (input) CHARACTER*1 ! = 'N': right eigenvectors of A are not computed; ! = 'V': right eigenvectors of A are computed. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the N-by-N matrix A. ! On exit, A has been overwritten. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! WR (output) DOUBLE PRECISION array, dimension (N) ! WI (output) DOUBLE PRECISION array, dimension (N) ! WR and WI contain the real and imaginary parts, ! respectively, of the computed eigenvalues. Complex ! conjugate pairs of eigenvalues appear consecutively ! with the eigenvalue having the positive imaginary part ! first. ! ! VL (output) DOUBLE PRECISION array, dimension (LDVL,N) ! If JOBVL = 'V', the left eigenvectors u(j) are stored one ! after another in the columns of VL, in the same order ! as their eigenvalues. ! If JOBVL = 'N', VL is not referenced. ! If the j-th eigenvalue is real, then u(j) = VL(:,j), ! the j-th column of VL. ! If the j-th and (j+1)-st eigenvalues form a complex ! conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and ! u(j+1) = VL(:,j) - i*VL(:,j+1). ! ! LDVL (input) INTEGER ! The leading dimension of the array VL. LDVL >= 1; if ! JOBVL = 'V', LDVL >= N. ! ! VR (output) DOUBLE PRECISION array, dimension (LDVR,N) ! If JOBVR = 'V', the right eigenvectors v(j) are stored one ! after another in the columns of VR, in the same order ! as their eigenvalues. ! If JOBVR = 'N', VR is not referenced. ! If the j-th eigenvalue is real, then v(j) = VR(:,j), ! the j-th column of VR. ! If the j-th and (j+1)-st eigenvalues form a complex ! conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and ! v(j+1) = VR(:,j) - i*VR(:,j+1). ! ! LDVR (input) INTEGER ! The leading dimension of the array VR. LDVR >= 1; if ! JOBVR = 'V', LDVR >= N. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,3*N), and ! if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N. For good ! performance, LWORK must generally be larger. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = i, the QR algorithm failed to compute all the ! eigenvalues, and no eigenvectors have been computed; ! elements i+1:N of WR and WI contain eigenvalues which ! have converged. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, SCALEA, WANTVL, WANTVR CHARACTER SIDE INTEGER HSWORK, I, IBAL, IERR, IHI, ILO, ITAU, IWRK, K, & MAXB, MAXWRK, MINWRK, NOUT DOUBLE PRECISION ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM, & SN ! .. ! .. Local Arrays .. LOGICAL SELECT( 1 ) DOUBLE PRECISION DUM( 1 ) ! .. ! .. External Subroutines .. EXTERNAL DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLACPY, DLARTG, & DLASCL, DORGHR, DROT, DSCAL, DTREVC, XERBLA ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX, ILAENV DOUBLE PRECISION DLAMCH, DLANGE, DLAPY2, DNRM2 EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DLANGE, DLAPY2, & DNRM2 ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LQUERY = ( LWORK.EQ.-1 ) WANTVL = LSAME( JOBVL, 'V' ) WANTVR = LSAME( JOBVR, 'V' ) IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN INFO = -1 ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN INFO = -9 ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN INFO = -11 END IF ! ! Compute workspace ! (Note: Comments in the code beginning "Workspace:" describe the ! minimal amount of workspace needed at that point in the code, ! as well as the preferred amount for good performance. ! NB refers to the optimal block size for the immediately ! following subroutine, as returned by ILAENV. ! HSWORK refers to the workspace preferred by DHSEQR, as ! calculated below. HSWORK is computed assuming ILO=1 and IHI=N, ! the worst case.) ! MINWRK = 1 IF( INFO.EQ.0 .AND. ( LWORK.GE.1 .OR. LQUERY ) ) THEN MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 ) IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN MINWRK = MAX( 1, 3*N ) MAXB = MAX( ILAENV( 8, 'DHSEQR', 'EN', N, 1, N, -1 ), 2 ) K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'EN', N, 1, & N, -1 ) ) ) HSWORK = MAX( K*( K+2 ), 2*N ) MAXWRK = MAX( MAXWRK, N+1, N+HSWORK ) ELSE MINWRK = MAX( 1, 4*N ) MAXWRK = MAX( MAXWRK, 2*N+( N-1 )* & ILAENV( 1, 'DORGHR', ' ', N, 1, N, -1 ) ) MAXB = MAX( ILAENV( 8, 'DHSEQR', 'SV', N, 1, N, -1 ), 2 ) K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'SV', N, 1, & N, -1 ) ) ) HSWORK = MAX( K*( K+2 ), 2*N ) MAXWRK = MAX( MAXWRK, N+1, N+HSWORK ) MAXWRK = MAX( MAXWRK, 4*N ) END IF WORK( 1 ) = MAXWRK END IF IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -13 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEEV ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Get machine constants ! EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) SMLNUM = SQRT( SMLNUM ) / EPS BIGNUM = ONE / SMLNUM ! ! Scale A if max element outside range [SMLNUM,BIGNUM] ! ANRM = DLANGE( 'M', N, N, A, LDA, DUM ) SCALEA = .FALSE. IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN SCALEA = .TRUE. CSCALE = SMLNUM ELSE IF( ANRM.GT.BIGNUM ) THEN SCALEA = .TRUE. CSCALE = BIGNUM END IF IF( SCALEA ) & CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) ! ! Balance the matrix ! (Workspace: need N) ! IBAL = 1 CALL DGEBAL( 'B', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR ) ! ! Reduce to upper Hessenberg form ! (Workspace: need 3*N, prefer 2*N+N*NB) ! ITAU = IBAL + N IWRK = ITAU + N CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), & LWORK-IWRK+1, IERR ) ! IF( WANTVL ) THEN ! ! Want left eigenvectors ! Copy Householder vectors to VL ! SIDE = 'L' CALL DLACPY( 'L', N, N, A, LDA, VL, LDVL ) ! ! Generate orthogonal matrix in VL ! (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB) ! CALL DORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ), & LWORK-IWRK+1, IERR ) ! ! Perform QR iteration, accumulating Schur vectors in VL ! (Workspace: need N+1, prefer N+HSWORK (see comments) ) ! IWRK = ITAU CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL, LDVL, & WORK( IWRK ), LWORK-IWRK+1, INFO ) ! IF( WANTVR ) THEN ! ! Want left and right eigenvectors ! Copy Schur vectors to VR ! SIDE = 'B' CALL DLACPY( 'F', N, N, VL, LDVL, VR, LDVR ) END IF ! ELSE IF( WANTVR ) THEN ! ! Want right eigenvectors ! Copy Householder vectors to VR ! SIDE = 'R' CALL DLACPY( 'L', N, N, A, LDA, VR, LDVR ) ! ! Generate orthogonal matrix in VR ! (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB) ! CALL DORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ), & LWORK-IWRK+1, IERR ) ! ! Perform QR iteration, accumulating Schur vectors in VR ! (Workspace: need N+1, prefer N+HSWORK (see comments) ) ! IWRK = ITAU CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR, & WORK( IWRK ), LWORK-IWRK+1, INFO ) ! ELSE ! ! Compute eigenvalues only ! (Workspace: need N+1, prefer N+HSWORK (see comments) ) ! IWRK = ITAU CALL DHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR, & WORK( IWRK ), LWORK-IWRK+1, INFO ) END IF ! ! If INFO > 0 from DHSEQR, then quit ! IF( INFO.GT.0 ) & GO TO 50 ! IF( WANTVL .OR. WANTVR ) THEN ! ! Compute left and/or right eigenvectors ! (Workspace: need 4*N) ! CALL DTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, & N, NOUT, WORK( IWRK ), IERR ) END IF ! IF( WANTVL ) THEN ! ! Undo balancing of left eigenvectors ! (Workspace: need N) ! CALL DGEBAK( 'B', 'L', N, ILO, IHI, WORK( IBAL ), N, VL, LDVL, & IERR ) ! ! Normalize left eigenvectors and make largest component real ! DO 20 I = 1, N IF( WI( I ).EQ.ZERO ) THEN SCL = ONE / DNRM2( N, VL( 1, I ), 1 ) CALL DSCAL( N, SCL, VL( 1, I ), 1 ) ELSE IF( WI( I ).GT.ZERO ) THEN SCL = ONE / DLAPY2( DNRM2( N, VL( 1, I ), 1 ), & DNRM2( N, VL( 1, I+1 ), 1 ) ) CALL DSCAL( N, SCL, VL( 1, I ), 1 ) CALL DSCAL( N, SCL, VL( 1, I+1 ), 1 ) DO K = 1, N WORK( IWRK+K-1 ) = VL( K, I )**2 + VL( K, I+1 )**2 end do K = IDAMAX( N, WORK( IWRK ), 1 ) CALL DLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R ) CALL DROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN ) VL( K, I+1 ) = ZERO END IF 20 CONTINUE END IF ! IF( WANTVR ) THEN ! ! Undo balancing of right eigenvectors ! (Workspace: need N) ! CALL DGEBAK( 'B', 'R', N, ILO, IHI, WORK( IBAL ), N, VR, LDVR, & IERR ) ! ! Normalize right eigenvectors and make largest component real ! DO 40 I = 1, N IF( WI( I ).EQ.ZERO ) THEN SCL = ONE / DNRM2( N, VR( 1, I ), 1 ) CALL DSCAL( N, SCL, VR( 1, I ), 1 ) ELSE IF( WI( I ).GT.ZERO ) THEN SCL = ONE / DLAPY2( DNRM2( N, VR( 1, I ), 1 ), & DNRM2( N, VR( 1, I+1 ), 1 ) ) CALL DSCAL( N, SCL, VR( 1, I ), 1 ) CALL DSCAL( N, SCL, VR( 1, I+1 ), 1 ) DO 30 K = 1, N WORK( IWRK+K-1 ) = VR( K, I )**2 + VR( K, I+1 )**2 30 CONTINUE K = IDAMAX( N, WORK( IWRK ), 1 ) CALL DLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R ) CALL DROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN ) VR( K, I+1 ) = ZERO END IF 40 CONTINUE END IF ! ! Undo scaling if necessary ! 50 CONTINUE IF( SCALEA ) THEN CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ), & MAX( N-INFO, 1 ), IERR ) CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WI( INFO+1 ), & MAX( N-INFO, 1 ), IERR ) IF( INFO.GT.0 ) THEN CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N, & IERR ) CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N, & IERR ) END IF END IF ! WORK( 1 ) = MAXWRK RETURN ! ! End of DGEEV ! END SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, & VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, & RCONDE, RCONDV, WORK, LWORK, IWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER BALANC, JOBVL, JOBVR, SENSE INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N DOUBLE PRECISION ABNRM ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), RCONDE( * ), RCONDV( * ), & SCALE( * ), VL( LDVL, * ), VR( LDVR, * ), & WI( * ), WORK( * ), WR( * ) ! .. ! ! Purpose ! ======= ! ! DGEEVX computes for an N-by-N real nonsymmetric matrix A, the ! eigenvalues and, optionally, the left and/or right eigenvectors. ! ! Optionally also, it computes a balancing transformation to improve ! the conditioning of the eigenvalues and eigenvectors (ILO, IHI, ! SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues ! (RCONDE), and reciprocal condition numbers for the right ! eigenvectors (RCONDV). ! ! The right eigenvector v(j) of A satisfies ! A * v(j) = lambda(j) * v(j) ! where lambda(j) is its eigenvalue. ! The left eigenvector u(j) of A satisfies ! u(j)**H * A = lambda(j) * u(j)**H ! where u(j)**H denotes the conjugate transpose of u(j). ! ! The computed eigenvectors are normalized to have Euclidean norm ! equal to 1 and largest component real. ! ! Balancing a matrix means permuting the rows and columns to make it ! more nearly upper triangular, and applying a diagonal similarity ! transformation D * A * D**(-1), where D is a diagonal matrix, to ! make its rows and columns closer in norm and the condition numbers ! of its eigenvalues and eigenvectors smaller. The computed ! reciprocal condition numbers correspond to the balanced matrix. ! Permuting rows and columns will not change the condition numbers ! (in exact arithmetic) but diagonal scaling will. For further ! explanation of balancing, see section 4.10.2 of the LAPACK ! Users' Guide. ! ! Arguments ! ========= ! ! BALANC (input) CHARACTER*1 ! Indicates how the input matrix should be diagonally scaled ! and/or permuted to improve the conditioning of its ! eigenvalues. ! = 'N': Do not diagonally scale or permute; ! = 'P': Perform permutations to make the matrix more nearly ! upper triangular. Do not diagonally scale; ! = 'S': Diagonally scale the matrix, i.e. replace A by ! D*A*D**(-1), where D is a diagonal matrix chosen ! to make the rows and columns of A more equal in ! norm. Do not permute; ! = 'B': Both diagonally scale and permute A. ! ! Computed reciprocal condition numbers will be for the matrix ! after balancing and/or permuting. Permuting does not change ! condition numbers (in exact arithmetic), but balancing does. ! ! JOBVL (input) CHARACTER*1 ! = 'N': left eigenvectors of A are not computed; ! = 'V': left eigenvectors of A are computed. ! If SENSE = 'E' or 'B', JOBVL must = 'V'. ! ! JOBVR (input) CHARACTER*1 ! = 'N': right eigenvectors of A are not computed; ! = 'V': right eigenvectors of A are computed. ! If SENSE = 'E' or 'B', JOBVR must = 'V'. ! ! SENSE (input) CHARACTER*1 ! Determines which reciprocal condition numbers are computed. ! = 'N': None are computed; ! = 'E': Computed for eigenvalues only; ! = 'V': Computed for right eigenvectors only; ! = 'B': Computed for eigenvalues and right eigenvectors. ! ! If SENSE = 'E' or 'B', both left and right eigenvectors ! must also be computed (JOBVL = 'V' and JOBVR = 'V'). ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the N-by-N matrix A. ! On exit, A has been overwritten. If JOBVL = 'V' or ! JOBVR = 'V', A contains the real Schur form of the balanced ! version of the input matrix A. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! WR (output) DOUBLE PRECISION array, dimension (N) ! WI (output) DOUBLE PRECISION array, dimension (N) ! WR and WI contain the real and imaginary parts, ! respectively, of the computed eigenvalues. Complex ! conjugate pairs of eigenvalues will appear consecutively ! with the eigenvalue having the positive imaginary part ! first. ! ! VL (output) DOUBLE PRECISION array, dimension (LDVL,N) ! If JOBVL = 'V', the left eigenvectors u(j) are stored one ! after another in the columns of VL, in the same order ! as their eigenvalues. ! If JOBVL = 'N', VL is not referenced. ! If the j-th eigenvalue is real, then u(j) = VL(:,j), ! the j-th column of VL. ! If the j-th and (j+1)-st eigenvalues form a complex ! conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and ! u(j+1) = VL(:,j) - i*VL(:,j+1). ! ! LDVL (input) INTEGER ! The leading dimension of the array VL. LDVL >= 1; if ! JOBVL = 'V', LDVL >= N. ! ! VR (output) DOUBLE PRECISION array, dimension (LDVR,N) ! If JOBVR = 'V', the right eigenvectors v(j) are stored one ! after another in the columns of VR, in the same order ! as their eigenvalues. ! If JOBVR = 'N', VR is not referenced. ! If the j-th eigenvalue is real, then v(j) = VR(:,j), ! the j-th column of VR. ! If the j-th and (j+1)-st eigenvalues form a complex ! conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and ! v(j+1) = VR(:,j) - i*VR(:,j+1). ! ! LDVR (input) INTEGER ! The leading dimension of the array VR. LDVR >= 1, and if ! JOBVR = 'V', LDVR >= N. ! ! ILO,IHI (output) INTEGER ! ILO and IHI are integer values determined when A was ! balanced. The balanced A(i,j) = 0 if I > J and ! J = 1,...,ILO-1 or I = IHI+1,...,N. ! ! SCALE (output) DOUBLE PRECISION array, dimension (N) ! Details of the permutations and scaling factors applied ! when balancing A. If P(j) is the index of the row and column ! interchanged with row and column j, and D(j) is the scaling ! factor applied to row and column j, then ! SCALE(J) = P(J), for J = 1,...,ILO-1 ! = D(J), for J = ILO,...,IHI ! = P(J) for J = IHI+1,...,N. ! The order in which the interchanges are made is N to IHI+1, ! then 1 to ILO-1. ! ! ABNRM (output) DOUBLE PRECISION ! The one-norm of the balanced matrix (the maximum ! of the sum of absolute values of elements of any column). ! ! RCONDE (output) DOUBLE PRECISION array, dimension (N) ! RCONDE(j) is the reciprocal condition number of the j-th ! eigenvalue. ! ! RCONDV (output) DOUBLE PRECISION array, dimension (N) ! RCONDV(j) is the reciprocal condition number of the j-th ! right eigenvector. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. If SENSE = 'N' or 'E', ! LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V', ! LWORK >= 3*N. If SENSE = 'V' or 'B', LWORK >= N*(N+6). ! For good performance, LWORK must generally be larger. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace) INTEGER array, dimension (2*N-2) ! If SENSE = 'N' or 'E', not referenced. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = i, the QR algorithm failed to compute all the ! eigenvalues, and no eigenvectors or condition numbers ! have been computed; elements 1:ILO-1 and i+1:N of WR ! and WI contain eigenvalues which have converged. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE, & WNTSNN, WNTSNV CHARACTER JOB, SIDE INTEGER HSWORK, I, ICOND, IERR, ITAU, IWRK, K, MAXB, & MAXWRK, MINWRK, NOUT DOUBLE PRECISION ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM, & SN ! .. ! .. Local Arrays .. LOGICAL SELECT( 1 ) DOUBLE PRECISION DUM( 1 ) ! .. ! .. External Subroutines .. EXTERNAL DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLACPY, DLARTG, & DLASCL, DORGHR, DROT, DSCAL, DTREVC, DTRSNA, & XERBLA ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX, ILAENV DOUBLE PRECISION DLAMCH, DLANGE, DLAPY2, DNRM2 EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DLANGE, DLAPY2, & DNRM2 ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LQUERY = ( LWORK.EQ.-1 ) WANTVL = LSAME( JOBVL, 'V' ) WANTVR = LSAME( JOBVR, 'V' ) WNTSNN = LSAME( SENSE, 'N' ) WNTSNE = LSAME( SENSE, 'E' ) WNTSNV = LSAME( SENSE, 'V' ) WNTSNB = LSAME( SENSE, 'B' ) IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, & 'S' ) .OR. LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) ) & THEN INFO = -1 ELSE IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN INFO = -2 ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN INFO = -3 ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR. & ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND. & WANTVR ) ) ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN INFO = -11 ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN INFO = -13 END IF ! ! Compute workspace ! (Note: Comments in the code beginning "Workspace:" describe the ! minimal amount of workspace needed at that point in the code, ! as well as the preferred amount for good performance. ! NB refers to the optimal block size for the immediately ! following subroutine, as returned by ILAENV. ! HSWORK refers to the workspace preferred by DHSEQR, as ! calculated below. HSWORK is computed assuming ILO=1 and IHI=N, ! the worst case.) ! MINWRK = 1 IF( INFO.EQ.0 .AND. ( LWORK.GE.1 .OR. LQUERY ) ) THEN MAXWRK = N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 ) IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN MINWRK = MAX( 1, 2*N ) IF( .NOT.WNTSNN ) & MINWRK = MAX( MINWRK, N*N+6*N ) MAXB = MAX( ILAENV( 8, 'DHSEQR', 'SN', N, 1, N, -1 ), 2 ) IF( WNTSNN ) THEN K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'EN', N, & 1, N, -1 ) ) ) ELSE K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'SN', N, & 1, N, -1 ) ) ) END IF HSWORK = MAX( K*( K+2 ), 2*N ) MAXWRK = MAX( MAXWRK, 1, HSWORK ) IF( .NOT.WNTSNN ) & MAXWRK = MAX( MAXWRK, N*N+6*N ) ELSE MINWRK = MAX( 1, 3*N ) IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) ) & MINWRK = MAX( MINWRK, N*N+6*N ) MAXB = MAX( ILAENV( 8, 'DHSEQR', 'SN', N, 1, N, -1 ), 2 ) K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'DHSEQR', 'EN', N, 1, & N, -1 ) ) ) HSWORK = MAX( K*( K+2 ), 2*N ) MAXWRK = MAX( MAXWRK, 1, HSWORK ) MAXWRK = MAX( MAXWRK, N+( N-1 )* & ILAENV( 1, 'DORGHR', ' ', N, 1, N, -1 ) ) IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) ) & MAXWRK = MAX( MAXWRK, N*N+6*N ) MAXWRK = MAX( MAXWRK, 3*N, 1 ) END IF WORK( 1 ) = MAXWRK END IF IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -21 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEEVX', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Get machine constants ! EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) SMLNUM = SQRT( SMLNUM ) / EPS BIGNUM = ONE / SMLNUM ! ! Scale A if max element outside range [SMLNUM,BIGNUM] ! ICOND = 0 ANRM = DLANGE( 'M', N, N, A, LDA, DUM ) SCALEA = .FALSE. IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN SCALEA = .TRUE. CSCALE = SMLNUM ELSE IF( ANRM.GT.BIGNUM ) THEN SCALEA = .TRUE. CSCALE = BIGNUM END IF IF( SCALEA ) & CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) ! ! Balance the matrix and compute ABNRM ! CALL DGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR ) ABNRM = DLANGE( '1', N, N, A, LDA, DUM ) IF( SCALEA ) THEN DUM( 1 ) = ABNRM CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR ) ABNRM = DUM( 1 ) END IF ! ! Reduce to upper Hessenberg form ! (Workspace: need 2*N, prefer N+N*NB) ! ITAU = 1 IWRK = ITAU + N CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), & LWORK-IWRK+1, IERR ) ! IF( WANTVL ) THEN ! ! Want left eigenvectors ! Copy Householder vectors to VL ! SIDE = 'L' CALL DLACPY( 'L', N, N, A, LDA, VL, LDVL ) ! ! Generate orthogonal matrix in VL ! (Workspace: need 2*N-1, prefer N+(N-1)*NB) ! CALL DORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ), & LWORK-IWRK+1, IERR ) ! ! Perform QR iteration, accumulating Schur vectors in VL ! (Workspace: need 1, prefer HSWORK (see comments) ) ! IWRK = ITAU CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL, LDVL, & WORK( IWRK ), LWORK-IWRK+1, INFO ) ! IF( WANTVR ) THEN ! ! Want left and right eigenvectors ! Copy Schur vectors to VR ! SIDE = 'B' CALL DLACPY( 'F', N, N, VL, LDVL, VR, LDVR ) END IF ! ELSE IF( WANTVR ) THEN ! ! Want right eigenvectors ! Copy Householder vectors to VR ! SIDE = 'R' CALL DLACPY( 'L', N, N, A, LDA, VR, LDVR ) ! ! Generate orthogonal matrix in VR ! (Workspace: need 2*N-1, prefer N+(N-1)*NB) ! CALL DORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ), & LWORK-IWRK+1, IERR ) ! ! Perform QR iteration, accumulating Schur vectors in VR ! (Workspace: need 1, prefer HSWORK (see comments) ) ! IWRK = ITAU CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR, & WORK( IWRK ), LWORK-IWRK+1, INFO ) ! ELSE ! ! Compute eigenvalues only ! If condition numbers desired, compute Schur form ! IF( WNTSNN ) THEN JOB = 'E' ELSE JOB = 'S' END IF ! ! (Workspace: need 1, prefer HSWORK (see comments) ) ! IWRK = ITAU CALL DHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR, & WORK( IWRK ), LWORK-IWRK+1, INFO ) END IF ! ! If INFO > 0 from DHSEQR, then quit ! IF( INFO.GT.0 ) & GO TO 50 ! IF( WANTVL .OR. WANTVR ) THEN ! ! Compute left and/or right eigenvectors ! (Workspace: need 3*N) ! CALL DTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, & N, NOUT, WORK( IWRK ), IERR ) END IF ! ! Compute condition numbers if desired ! (Workspace: need N*N+6*N unless SENSE = 'E') ! IF( .NOT.WNTSNN ) THEN CALL DTRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, & RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, IWORK, & ICOND ) END IF ! IF( WANTVL ) THEN ! ! Undo balancing of left eigenvectors ! CALL DGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, N, VL, LDVL, & IERR ) ! ! Normalize left eigenvectors and make largest component real ! DO 20 I = 1, N IF( WI( I ).EQ.ZERO ) THEN SCL = ONE / DNRM2( N, VL( 1, I ), 1 ) CALL DSCAL( N, SCL, VL( 1, I ), 1 ) ELSE IF( WI( I ).GT.ZERO ) THEN SCL = ONE / DLAPY2( DNRM2( N, VL( 1, I ), 1 ), & DNRM2( N, VL( 1, I+1 ), 1 ) ) CALL DSCAL( N, SCL, VL( 1, I ), 1 ) CALL DSCAL( N, SCL, VL( 1, I+1 ), 1 ) DO K = 1, N WORK( K ) = VL( K, I )**2 + VL( K, I+1 )**2 end do K = IDAMAX( N, WORK, 1 ) CALL DLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R ) CALL DROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN ) VL( K, I+1 ) = ZERO END IF 20 CONTINUE END IF ! IF( WANTVR ) THEN ! ! Undo balancing of right eigenvectors ! CALL DGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, N, VR, LDVR, & IERR ) ! ! Normalize right eigenvectors and make largest component real ! DO 40 I = 1, N IF( WI( I ).EQ.ZERO ) THEN SCL = ONE / DNRM2( N, VR( 1, I ), 1 ) CALL DSCAL( N, SCL, VR( 1, I ), 1 ) ELSE IF( WI( I ).GT.ZERO ) THEN SCL = ONE / DLAPY2( DNRM2( N, VR( 1, I ), 1 ), & DNRM2( N, VR( 1, I+1 ), 1 ) ) CALL DSCAL( N, SCL, VR( 1, I ), 1 ) CALL DSCAL( N, SCL, VR( 1, I+1 ), 1 ) DO 30 K = 1, N WORK( K ) = VR( K, I )**2 + VR( K, I+1 )**2 30 CONTINUE K = IDAMAX( N, WORK, 1 ) CALL DLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R ) CALL DROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN ) VR( K, I+1 ) = ZERO END IF 40 CONTINUE END IF ! ! Undo scaling if necessary ! 50 CONTINUE IF( SCALEA ) THEN CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ), & MAX( N-INFO, 1 ), IERR ) CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WI( INFO+1 ), & MAX( N-INFO, 1 ), IERR ) IF( INFO.EQ.0 ) THEN IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 ) & CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N, & IERR ) ELSE CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N, & IERR ) CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N, & IERR ) END IF END IF ! WORK( 1 ) = MAXWRK RETURN ! ! End of DGEEVX ! END SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, & ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, & LWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBVSL, JOBVSR INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), & B( LDB, * ), BETA( * ), VSL( LDVSL, * ), & VSR( LDVSR, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! This routine is deprecated and has been replaced by routine DGGES. ! ! DGEGS computes for a pair of N-by-N real nonsymmetric matrices A, B: ! the generalized eigenvalues (alphar +/- alphai*i, beta), the real ! Schur form (A, B), and optionally left and/or right Schur vectors ! (VSL and VSR). ! ! (If only the generalized eigenvalues are needed, use the driver DGEGV ! instead.) ! ! A generalized eigenvalue for a pair of matrices (A,B) is, roughly ! speaking, a scalar w or a ratio alpha/beta = w, such that A - w*B ! is singular. It is usually represented as the pair (alpha,beta), ! as there is a reasonable interpretation for beta=0, and even for ! both being zero. A good beginning reference is the book, "Matrix ! Computations", by G. Golub & C. van Loan (Johns Hopkins U. Press) ! ! The (generalized) Schur form of a pair of matrices is the result of ! multiplying both matrices on the left by one orthogonal matrix and ! both on the right by another orthogonal matrix, these two orthogonal ! matrices being chosen so as to bring the pair of matrices into ! (real) Schur form. ! ! A pair of matrices A, B is in generalized real Schur form if B is ! upper triangular with non-negative diagonal and A is block upper ! triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond ! to real generalized eigenvalues, while 2-by-2 blocks of A will be ! "standardized" by making the corresponding elements of B have the ! form: ! [ a 0 ] ! [ 0 b ] ! ! and the pair of corresponding 2-by-2 blocks in A and B will ! have a complex conjugate pair of generalized eigenvalues. ! ! The left and right Schur vectors are the columns of VSL and VSR, ! respectively, where VSL and VSR are the orthogonal matrices ! which reduce A and B to Schur form: ! ! Schur form of (A,B) = ( (VSL)**T A (VSR), (VSL)**T B (VSR) ) ! ! Arguments ! ========= ! ! JOBVSL (input) CHARACTER*1 ! = 'N': do not compute the left Schur vectors; ! = 'V': compute the left Schur vectors. ! ! JOBVSR (input) CHARACTER*1 ! = 'N': do not compute the right Schur vectors; ! = 'V': compute the right Schur vectors. ! ! N (input) INTEGER ! The order of the matrices A, B, VSL, and VSR. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the first of the pair of matrices whose generalized ! eigenvalues and (optionally) Schur vectors are to be ! computed. ! On exit, the generalized Schur form of A. ! Note: to avoid overflow, the Frobenius norm of the matrix ! A should be less than the overflow threshold. ! ! LDA (input) INTEGER ! The leading dimension of A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB, N) ! On entry, the second of the pair of matrices whose ! generalized eigenvalues and (optionally) Schur vectors are ! to be computed. ! On exit, the generalized Schur form of B. ! Note: to avoid overflow, the Frobenius norm of the matrix ! B should be less than the overflow threshold. ! ! LDB (input) INTEGER ! The leading dimension of B. LDB >= max(1,N). ! ! ALPHAR (output) DOUBLE PRECISION array, dimension (N) ! ALPHAI (output) DOUBLE PRECISION array, dimension (N) ! BETA (output) DOUBLE PRECISION array, dimension (N) ! On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will ! be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i, ! j=1,...,N and BETA(j),j=1,...,N are the diagonals of the ! complex Schur form (A,B) that would result if the 2-by-2 ! diagonal blocks of the real Schur form of (A,B) were further ! reduced to triangular form using 2-by-2 complex unitary ! transformations. If ALPHAI(j) is zero, then the j-th ! eigenvalue is real; if positive, then the j-th and (j+1)-st ! eigenvalues are a complex conjugate pair, with ALPHAI(j+1) ! negative. ! ! Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) ! may easily over- or underflow, and BETA(j) may even be zero. ! Thus, the user should avoid naively computing the ratio ! alpha/beta. However, ALPHAR and ALPHAI will be always less ! than and usually comparable with norm(A) in magnitude, and ! BETA always less than and usually comparable with norm(B). ! ! VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N) ! If JOBVSL = 'V', VSL will contain the left Schur vectors. ! (See "Purpose", above.) ! Not referenced if JOBVSL = 'N'. ! ! LDVSL (input) INTEGER ! The leading dimension of the matrix VSL. LDVSL >=1, and ! if JOBVSL = 'V', LDVSL >= N. ! ! VSR (output) DOUBLE PRECISION array, dimension (LDVSR,N) ! If JOBVSR = 'V', VSR will contain the right Schur vectors. ! (See "Purpose", above.) ! Not referenced if JOBVSR = 'N'. ! ! LDVSR (input) INTEGER ! The leading dimension of the matrix VSR. LDVSR >= 1, and ! if JOBVSR = 'V', LDVSR >= N. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,4*N). ! For good performance, LWORK must generally be larger. ! To compute the optimal value of LWORK, call ILAENV to get ! blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute: ! NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR ! The optimal LWORK is 2*N + N*(NB+1). ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! = 1,...,N: ! The QZ iteration failed. (A,B) are not in Schur ! form, but ALPHAR(j), ALPHAI(j), and BETA(j) should ! be correct for j=INFO+1,...,N. ! > N: errors that usually indicate LAPACK problems: ! =N+1: error return from DGGBAL ! =N+2: error return from DGEQRF ! =N+3: error return from DORMQR ! =N+4: error return from DORGQR ! =N+5: error return from DGGHRD ! =N+6: error return from DHGEQZ (other than failed ! iteration) ! =N+7: error return from DGGBAK (computing VSL) ! =N+8: error return from DGGBAK (computing VSR) ! =N+9: error return from DLASCL (various places) ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL ILASCL, ILBSCL, ILVSL, ILVSR, LQUERY INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO, & IRIGHT, IROWS, ITAU, IWORK, LOPT, LWKMIN, & LWKOPT, NB, NB1, NB2, NB3 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, & SAFMIN, SMLNUM ! .. ! .. External Subroutines .. EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY, & DLASCL, DLASET, DORGQR, DORMQR, XERBLA ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE ! .. ! .. Intrinsic Functions .. INTRINSIC INT, MAX ! .. ! .. Executable Statements .. ! ! Decode the input arguments ! IF( LSAME( JOBVSL, 'N' ) ) THEN IJOBVL = 1 ILVSL = .FALSE. ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN IJOBVL = 2 ILVSL = .TRUE. ELSE IJOBVL = -1 ILVSL = .FALSE. END IF ! IF( LSAME( JOBVSR, 'N' ) ) THEN IJOBVR = 1 ILVSR = .FALSE. ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN IJOBVR = 2 ILVSR = .TRUE. ELSE IJOBVR = -1 ILVSR = .FALSE. END IF ! ! Test the input arguments ! LWKMIN = MAX( 4*N, 1 ) LWKOPT = LWKMIN WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) INFO = 0 IF( IJOBVL.LE.0 ) THEN INFO = -1 ELSE IF( IJOBVR.LE.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN INFO = -12 ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN INFO = -14 ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN INFO = -16 END IF ! IF( INFO.EQ.0 ) THEN NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 ) NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 ) NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 ) NB = MAX( NB1, NB2, NB3 ) LOPT = 2*N + N*( NB+1 ) WORK( 1 ) = LOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEGS ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Get machine constants ! EPS = DLAMCH( 'E' )*DLAMCH( 'B' ) SAFMIN = DLAMCH( 'S' ) SMLNUM = N*SAFMIN / EPS BIGNUM = ONE / SMLNUM ! ! Scale A if max element outside range [SMLNUM,BIGNUM] ! ANRM = DLANGE( 'M', N, N, A, LDA, WORK ) ILASCL = .FALSE. IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN ANRMTO = SMLNUM ILASCL = .TRUE. ELSE IF( ANRM.GT.BIGNUM ) THEN ANRMTO = BIGNUM ILASCL = .TRUE. END IF ! IF( ILASCL ) THEN CALL DLASCL( 'G', -1, -1, ANRM, ANRMTO, N, N, A, LDA, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 9 RETURN END IF END IF ! ! Scale B if max element outside range [SMLNUM,BIGNUM] ! BNRM = DLANGE( 'M', N, N, B, LDB, WORK ) ILBSCL = .FALSE. IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN BNRMTO = SMLNUM ILBSCL = .TRUE. ELSE IF( BNRM.GT.BIGNUM ) THEN BNRMTO = BIGNUM ILBSCL = .TRUE. END IF ! IF( ILBSCL ) THEN CALL DLASCL( 'G', -1, -1, BNRM, BNRMTO, N, N, B, LDB, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 9 RETURN END IF END IF ! ! Permute the matrix to make it more nearly triangular ! Workspace layout: (2*N words -- "work..." not actually used) ! left_permutation, right_permutation, work... ! ILEFT = 1 IRIGHT = N + 1 IWORK = IRIGHT + N CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ), & WORK( IRIGHT ), WORK( IWORK ), IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 1 GO TO 10 END IF ! ! Reduce B to triangular form, and initialize VSL and/or VSR ! Workspace layout: ("work..." must have at least N words) ! left_permutation, right_permutation, tau, work... ! IROWS = IHI + 1 - ILO ICOLS = N + 1 - ILO ITAU = IWORK IWORK = ITAU + IROWS CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), & WORK( IWORK ), LWORK+1-IWORK, IINFO ) IF( IINFO.GE.0 ) & LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) IF( IINFO.NE.0 ) THEN INFO = N + 2 GO TO 10 END IF ! CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB, & WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ), & LWORK+1-IWORK, IINFO ) IF( IINFO.GE.0 ) & LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) IF( IINFO.NE.0 ) THEN INFO = N + 3 GO TO 10 END IF ! IF( ILVSL ) THEN CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL ) CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, & VSL( ILO+1, ILO ), LDVSL ) CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL, & WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK, & IINFO ) IF( IINFO.GE.0 ) & LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) IF( IINFO.NE.0 ) THEN INFO = N + 4 GO TO 10 END IF END IF ! IF( ILVSR ) & CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR ) ! ! Reduce to generalized Hessenberg form ! CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL, & LDVSL, VSR, LDVSR, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 5 GO TO 10 END IF ! ! Perform QZ algorithm, computing Schur vectors if desired ! Workspace layout: ("work..." must have at least 1 word) ! left_permutation, right_permutation, work... ! IWORK = ITAU CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, & ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, & WORK( IWORK ), LWORK+1-IWORK, IINFO ) IF( IINFO.GE.0 ) & LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) IF( IINFO.NE.0 ) THEN IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN INFO = IINFO ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN INFO = IINFO - N ELSE INFO = N + 6 END IF GO TO 10 END IF ! ! Apply permutation to VSL and VSR ! IF( ILVSL ) THEN CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ), & WORK( IRIGHT ), N, VSL, LDVSL, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 7 GO TO 10 END IF END IF IF( ILVSR ) THEN CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ), & WORK( IRIGHT ), N, VSR, LDVSR, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 8 GO TO 10 END IF END IF ! ! Undo scaling ! IF( ILASCL ) THEN CALL DLASCL( 'H', -1, -1, ANRMTO, ANRM, N, N, A, LDA, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 9 RETURN END IF CALL DLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAR, N, & IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 9 RETURN END IF CALL DLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAI, N, & IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 9 RETURN END IF END IF ! IF( ILBSCL ) THEN CALL DLASCL( 'U', -1, -1, BNRMTO, BNRM, N, N, B, LDB, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 9 RETURN END IF CALL DLASCL( 'G', -1, -1, BNRMTO, BNRM, N, 1, BETA, N, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 9 RETURN END IF END IF ! 10 CONTINUE WORK( 1 ) = LWKOPT ! RETURN ! ! End of DGEGS ! END SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, & BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBVL, JOBVR INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), & B( LDB, * ), BETA( * ), VL( LDVL, * ), & VR( LDVR, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! This routine is deprecated and has been replaced by routine DGGEV. ! ! DGEGV computes for a pair of n-by-n real nonsymmetric matrices A and ! B, the generalized eigenvalues (alphar +/- alphai*i, beta), and ! optionally, the left and/or right generalized eigenvectors (VL and ! VR). ! ! A generalized eigenvalue for a pair of matrices (A,B) is, roughly ! speaking, a scalar w or a ratio alpha/beta = w, such that A - w*B ! is singular. It is usually represented as the pair (alpha,beta), ! as there is a reasonable interpretation for beta=0, and even for ! both being zero. A good beginning reference is the book, "Matrix ! Computations", by G. Golub & C. van Loan (Johns Hopkins U. Press) ! ! A right generalized eigenvector corresponding to a generalized ! eigenvalue w for a pair of matrices (A,B) is a vector r such ! that (A - w B) r = 0 . A left generalized eigenvector is a vector ! l such that l**H * (A - w B) = 0, where l**H is the ! conjugate-transpose of l. ! ! Note: this routine performs "full balancing" on A and B -- see ! "Further Details", below. ! ! Arguments ! ========= ! ! JOBVL (input) CHARACTER*1 ! = 'N': do not compute the left generalized eigenvectors; ! = 'V': compute the left generalized eigenvectors. ! ! JOBVR (input) CHARACTER*1 ! = 'N': do not compute the right generalized eigenvectors; ! = 'V': compute the right generalized eigenvectors. ! ! N (input) INTEGER ! The order of the matrices A, B, VL, and VR. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the first of the pair of matrices whose ! generalized eigenvalues and (optionally) generalized ! eigenvectors are to be computed. ! On exit, the contents will have been destroyed. (For a ! description of the contents of A on exit, see "Further ! Details", below.) ! ! LDA (input) INTEGER ! The leading dimension of A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB, N) ! On entry, the second of the pair of matrices whose ! generalized eigenvalues and (optionally) generalized ! eigenvectors are to be computed. ! On exit, the contents will have been destroyed. (For a ! description of the contents of B on exit, see "Further ! Details", below.) ! ! LDB (input) INTEGER ! The leading dimension of B. LDB >= max(1,N). ! ! ALPHAR (output) DOUBLE PRECISION array, dimension (N) ! ALPHAI (output) DOUBLE PRECISION array, dimension (N) ! BETA (output) DOUBLE PRECISION array, dimension (N) ! On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will ! be the generalized eigenvalues. If ALPHAI(j) is zero, then ! the j-th eigenvalue is real; if positive, then the j-th and ! (j+1)-st eigenvalues are a complex conjugate pair, with ! ALPHAI(j+1) negative. ! ! Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) ! may easily over- or underflow, and BETA(j) may even be zero. ! Thus, the user should avoid naively computing the ratio ! alpha/beta. However, ALPHAR and ALPHAI will be always less ! than and usually comparable with norm(A) in magnitude, and ! BETA always less than and usually comparable with norm(B). ! ! VL (output) DOUBLE PRECISION array, dimension (LDVL,N) ! If JOBVL = 'V', the left generalized eigenvectors. (See ! "Purpose", above.) Real eigenvectors take one column, ! complex take two columns, the first for the real part and ! the second for the imaginary part. Complex eigenvectors ! correspond to an eigenvalue with positive imaginary part. ! Each eigenvector will be scaled so the largest component ! will have abs(real part) + abs(imag. part) = 1, *except* ! that for eigenvalues with alpha=beta=0, a zero vector will ! be returned as the corresponding eigenvector. ! Not referenced if JOBVL = 'N'. ! ! LDVL (input) INTEGER ! The leading dimension of the matrix VL. LDVL >= 1, and ! if JOBVL = 'V', LDVL >= N. ! ! VR (output) DOUBLE PRECISION array, dimension (LDVR,N) ! If JOBVR = 'V', the right generalized eigenvectors. (See ! "Purpose", above.) Real eigenvectors take one column, ! complex take two columns, the first for the real part and ! the second for the imaginary part. Complex eigenvectors ! correspond to an eigenvalue with positive imaginary part. ! Each eigenvector will be scaled so the largest component ! will have abs(real part) + abs(imag. part) = 1, *except* ! that for eigenvalues with alpha=beta=0, a zero vector will ! be returned as the corresponding eigenvector. ! Not referenced if JOBVR = 'N'. ! ! LDVR (input) INTEGER ! The leading dimension of the matrix VR. LDVR >= 1, and ! if JOBVR = 'V', LDVR >= N. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,8*N). ! For good performance, LWORK must generally be larger. ! To compute the optimal value of LWORK, call ILAENV to get ! blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute: ! NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR; ! The optimal LWORK is: ! 2*N + MAX( 6*N, N*(NB+1) ). ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! = 1,...,N: ! The QZ iteration failed. No eigenvectors have been ! calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) ! should be correct for j=INFO+1,...,N. ! > N: errors that usually indicate LAPACK problems: ! =N+1: error return from DGGBAL ! =N+2: error return from DGEQRF ! =N+3: error return from DORMQR ! =N+4: error return from DORGQR ! =N+5: error return from DGGHRD ! =N+6: error return from DHGEQZ (other than failed ! iteration) ! =N+7: error return from DTGEVC ! =N+8: error return from DGGBAK (computing VL) ! =N+9: error return from DGGBAK (computing VR) ! =N+10: error return from DLASCL (various calls) ! ! Further Details ! =============== ! ! Balancing ! --------- ! ! This driver calls DGGBAL to both permute and scale rows and columns ! of A and B. The permutations PL and PR are chosen so that PL*A*PR ! and PL*B*R will be upper triangular except for the diagonal blocks ! A(i:j,i:j) and B(i:j,i:j), with i and j as close together as ! possible. The diagonal scaling matrices DL and DR are chosen so ! that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to ! one (except for the elements that start out zero.) ! ! After the eigenvalues and eigenvectors of the balanced matrices ! have been computed, DGGBAK transforms the eigenvectors back to what ! they would have been (in perfect arithmetic) if they had not been ! balanced. ! ! Contents of A and B on Exit ! -------- -- - --- - -- ---- ! ! If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or ! both), then on exit the arrays A and B will contain the real Schur ! form[*] of the "balanced" versions of A and B. If no eigenvectors ! are computed, then only the diagonal blocks will be correct. ! ! [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations", ! by Golub & van Loan, pub. by Johns Hopkins U. Press. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY CHARACTER CHTEMP INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO, & IN, IRIGHT, IROWS, ITAU, IWORK, JC, JR, LOPT, & LWKMIN, LWKOPT, NB, NB1, NB2, NB3 DOUBLE PRECISION ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM, & BNRM1, BNRM2, EPS, ONEPLS, SAFMAX, SAFMIN, & SALFAI, SALFAR, SBETA, SCALE, TEMP ! .. ! .. Local Arrays .. LOGICAL LDUMMA( 1 ) ! .. ! .. External Subroutines .. EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY, & DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, XERBLA ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, INT, MAX ! .. ! .. Executable Statements .. ! ! Decode the input arguments ! IF( LSAME( JOBVL, 'N' ) ) THEN IJOBVL = 1 ILVL = .FALSE. ELSE IF( LSAME( JOBVL, 'V' ) ) THEN IJOBVL = 2 ILVL = .TRUE. ELSE IJOBVL = -1 ILVL = .FALSE. END IF ! IF( LSAME( JOBVR, 'N' ) ) THEN IJOBVR = 1 ILVR = .FALSE. ELSE IF( LSAME( JOBVR, 'V' ) ) THEN IJOBVR = 2 ILVR = .TRUE. ELSE IJOBVR = -1 ILVR = .FALSE. END IF ILV = ILVL .OR. ILVR ! ! Test the input arguments ! LWKMIN = MAX( 8*N, 1 ) LWKOPT = LWKMIN WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) INFO = 0 IF( IJOBVL.LE.0 ) THEN INFO = -1 ELSE IF( IJOBVR.LE.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN INFO = -12 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN INFO = -14 ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN INFO = -16 END IF ! IF( INFO.EQ.0 ) THEN NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 ) NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 ) NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 ) NB = MAX( NB1, NB2, NB3 ) LOPT = 2*N + MAX( 6*N, N*( NB+1 ) ) WORK( 1 ) = LOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEGV ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Get machine constants ! EPS = DLAMCH( 'E' )*DLAMCH( 'B' ) SAFMIN = DLAMCH( 'S' ) SAFMIN = SAFMIN + SAFMIN SAFMAX = ONE / SAFMIN ONEPLS = ONE + ( 4*EPS ) ! ! Scale A ! ANRM = DLANGE( 'M', N, N, A, LDA, WORK ) ANRM1 = ANRM ANRM2 = ONE IF( ANRM.LT.ONE ) THEN IF( SAFMAX*ANRM.LT.ONE ) THEN ANRM1 = SAFMIN ANRM2 = SAFMAX*ANRM END IF END IF ! IF( ANRM.GT.ZERO ) THEN CALL DLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 10 RETURN END IF END IF ! ! Scale B ! BNRM = DLANGE( 'M', N, N, B, LDB, WORK ) BNRM1 = BNRM BNRM2 = ONE IF( BNRM.LT.ONE ) THEN IF( SAFMAX*BNRM.LT.ONE ) THEN BNRM1 = SAFMIN BNRM2 = SAFMAX*BNRM END IF END IF ! IF( BNRM.GT.ZERO ) THEN CALL DLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 10 RETURN END IF END IF ! ! Permute the matrix to make it more nearly triangular ! Workspace layout: (8*N words -- "work" requires 6*N words) ! left_permutation, right_permutation, work... ! ILEFT = 1 IRIGHT = N + 1 IWORK = IRIGHT + N CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ), & WORK( IRIGHT ), WORK( IWORK ), IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 1 GO TO 120 END IF ! ! Reduce B to triangular form, and initialize VL and/or VR ! Workspace layout: ("work..." must have at least N words) ! left_permutation, right_permutation, tau, work... ! IROWS = IHI + 1 - ILO IF( ILV ) THEN ICOLS = N + 1 - ILO ELSE ICOLS = IROWS END IF ITAU = IWORK IWORK = ITAU + IROWS CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), & WORK( IWORK ), LWORK+1-IWORK, IINFO ) IF( IINFO.GE.0 ) & LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) IF( IINFO.NE.0 ) THEN INFO = N + 2 GO TO 120 END IF ! CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB, & WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ), & LWORK+1-IWORK, IINFO ) IF( IINFO.GE.0 ) & LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) IF( IINFO.NE.0 ) THEN INFO = N + 3 GO TO 120 END IF ! IF( ILVL ) THEN CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL ) CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, & VL( ILO+1, ILO ), LDVL ) CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL, & WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK, & IINFO ) IF( IINFO.GE.0 ) & LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) IF( IINFO.NE.0 ) THEN INFO = N + 4 GO TO 120 END IF END IF ! IF( ILVR ) & CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR ) ! ! Reduce to generalized Hessenberg form ! IF( ILV ) THEN ! ! Eigenvectors requested -- work on whole matrix. ! CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL, & LDVL, VR, LDVR, IINFO ) ELSE CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA, & B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO ) END IF IF( IINFO.NE.0 ) THEN INFO = N + 5 GO TO 120 END IF ! ! Perform QZ algorithm ! Workspace layout: ("work..." must have at least 1 word) ! left_permutation, right_permutation, work... ! IWORK = ITAU IF( ILV ) THEN CHTEMP = 'S' ELSE CHTEMP = 'E' END IF CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, & ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, & WORK( IWORK ), LWORK+1-IWORK, IINFO ) IF( IINFO.GE.0 ) & LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) IF( IINFO.NE.0 ) THEN IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN INFO = IINFO ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN INFO = IINFO - N ELSE INFO = N + 6 END IF GO TO 120 END IF ! IF( ILV ) THEN ! ! Compute Eigenvectors (DTGEVC requires 6*N words of workspace) ! IF( ILVL ) THEN IF( ILVR ) THEN CHTEMP = 'B' ELSE CHTEMP = 'L' END IF ELSE CHTEMP = 'R' END IF ! CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL, & VR, LDVR, N, IN, WORK( IWORK ), IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 7 GO TO 120 END IF ! ! Undo balancing on VL and VR, rescale ! IF( ILVL ) THEN CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ), & WORK( IRIGHT ), N, VL, LDVL, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 8 GO TO 120 END IF DO 50 JC = 1, N IF( ALPHAI( JC ).LT.ZERO ) & GO TO 50 TEMP = ZERO IF( ALPHAI( JC ).EQ.ZERO ) THEN DO JR = 1, N TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) ) end do ELSE DO 20 JR = 1, N TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+ & ABS( VL( JR, JC+1 ) ) ) 20 CONTINUE END IF IF( TEMP.LT.SAFMIN ) & GO TO 50 TEMP = ONE / TEMP IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 30 JR = 1, N VL( JR, JC ) = VL( JR, JC )*TEMP 30 CONTINUE ELSE DO 40 JR = 1, N VL( JR, JC ) = VL( JR, JC )*TEMP VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP 40 CONTINUE END IF 50 CONTINUE END IF IF( ILVR ) THEN CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ), & WORK( IRIGHT ), N, VR, LDVR, IINFO ) IF( IINFO.NE.0 ) THEN INFO = N + 9 GO TO 120 END IF DO 100 JC = 1, N IF( ALPHAI( JC ).LT.ZERO ) & GO TO 100 TEMP = ZERO IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 60 JR = 1, N TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) ) 60 CONTINUE ELSE DO 70 JR = 1, N TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+ & ABS( VR( JR, JC+1 ) ) ) 70 CONTINUE END IF IF( TEMP.LT.SAFMIN ) & GO TO 100 TEMP = ONE / TEMP IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 80 JR = 1, N VR( JR, JC ) = VR( JR, JC )*TEMP 80 CONTINUE ELSE DO 90 JR = 1, N VR( JR, JC ) = VR( JR, JC )*TEMP VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP 90 CONTINUE END IF 100 CONTINUE END IF ! ! End of eigenvector calculation ! END IF ! ! Undo scaling in alpha, beta ! ! Note: this does not give the alpha and beta for the unscaled ! problem. ! ! Un-scaling is limited to avoid underflow in alpha and beta ! if they are significant. ! DO 110 JC = 1, N ABSAR = ABS( ALPHAR( JC ) ) ABSAI = ABS( ALPHAI( JC ) ) ABSB = ABS( BETA( JC ) ) SALFAR = ANRM*ALPHAR( JC ) SALFAI = ANRM*ALPHAI( JC ) SBETA = BNRM*BETA( JC ) ILIMIT = .FALSE. SCALE = ONE ! ! Check for significant underflow in ALPHAI ! IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE. & MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN ILIMIT = .TRUE. SCALE = ( ONEPLS*SAFMIN / ANRM1 ) / & MAX( ONEPLS*SAFMIN, ANRM2*ABSAI ) ! ELSE IF( SALFAI.EQ.ZERO ) THEN ! ! If insignificant underflow in ALPHAI, then make the ! conjugate eigenvalue real. ! IF( ALPHAI( JC ).LT.ZERO .AND. JC.GT.1 ) THEN ALPHAI( JC-1 ) = ZERO ELSE IF( ALPHAI( JC ).GT.ZERO .AND. JC.LT.N ) THEN ALPHAI( JC+1 ) = ZERO END IF END IF ! ! Check for significant underflow in ALPHAR ! IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE. & MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN ILIMIT = .TRUE. SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / ANRM1 ) / & MAX( ONEPLS*SAFMIN, ANRM2*ABSAR ) ) END IF ! ! Check for significant underflow in BETA ! IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE. & MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN ILIMIT = .TRUE. SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / BNRM1 ) / & MAX( ONEPLS*SAFMIN, BNRM2*ABSB ) ) END IF ! ! Check for possible overflow when limiting scaling ! IF( ILIMIT ) THEN TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ), & ABS( SBETA ) ) IF( TEMP.GT.ONE ) & SCALE = SCALE / TEMP IF( SCALE.LT.ONE ) & ILIMIT = .FALSE. END IF ! ! Recompute un-scaled ALPHAR, ALPHAI, BETA if necessary. ! IF( ILIMIT ) THEN SALFAR = ( SCALE*ALPHAR( JC ) )*ANRM SALFAI = ( SCALE*ALPHAI( JC ) )*ANRM SBETA = ( SCALE*BETA( JC ) )*BNRM END IF ALPHAR( JC ) = SALFAR ALPHAI( JC ) = SALFAI BETA( JC ) = SBETA 110 CONTINUE ! 120 CONTINUE WORK( 1 ) = LWKOPT ! RETURN ! ! End of DGEGV ! END SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. INTEGER IHI, ILO, INFO, LDA, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGEHD2 reduces a real general matrix A to upper Hessenberg form H by ! an orthogonal similarity transformation: Q' * A * Q = H . ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! ILO (input) INTEGER ! IHI (input) INTEGER ! It is assumed that A is already upper triangular in rows ! and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally ! set by a previous call to DGEBAL; otherwise they should be ! set to 1 and N respectively. See Further Details. ! 1 <= ILO <= IHI <= max(1,N). ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the n by n general matrix to be reduced. ! On exit, the upper triangle and the first subdiagonal of A ! are overwritten with the upper Hessenberg matrix H, and the ! elements below the first subdiagonal, with the array TAU, ! represent the orthogonal matrix Q as a product of elementary ! reflectors. See Further Details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! TAU (output) DOUBLE PRECISION array, dimension (N-1) ! The scalar factors of the elementary reflectors (see Further ! Details). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of (ihi-ilo) elementary ! reflectors ! ! Q = H(ilo) H(ilo+1) . . . H(ihi-1). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on ! exit in A(i+2:ihi,i), and tau in TAU(i). ! ! The contents of A are illustrated by the following example, with ! n = 7, ilo = 2 and ihi = 6: ! ! on entry, on exit, ! ! ( a a a a a a a ) ( a a h h h h a ) ! ( a a a a a a ) ( a h h h h a ) ! ( a a a a a a ) ( h h h h h h ) ! ( a a a a a a ) ( v2 h h h h h ) ! ( a a a a a a ) ( v2 v3 h h h h ) ! ( a a a a a a ) ( v2 v3 v4 h h h ) ! ( a ) ( a ) ! ! where a denotes an element of the original matrix A, h denotes a ! modified element of the upper Hessenberg matrix H, and vi denotes an ! element of the vector defining H(i). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I DOUBLE PRECISION AII ! .. ! .. External Subroutines .. EXTERNAL DLARF, DLARFG, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 IF( N.LT.0 ) THEN INFO = -1 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN INFO = -2 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEHD2', -INFO ) RETURN END IF ! DO 10 I = ILO, IHI - 1 ! ! Compute elementary reflector H(i) to annihilate A(i+2:ihi,i) ! CALL DLARFG( IHI-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1, & TAU( I ) ) AII = A( I+1, I ) A( I+1, I ) = ONE ! ! Apply H(i) to A(1:ihi,i+1:ihi) from the right ! CALL DLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ), & A( 1, I+1 ), LDA, WORK ) ! ! Apply H(i) to A(i+1:ihi,i+1:n) from the left ! CALL DLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1, TAU( I ), & A( I+1, I+1 ), LDA, WORK ) ! A( I+1, I ) = AII 10 CONTINUE ! RETURN ! ! End of DGEHD2 ! END SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER IHI, ILO, INFO, LDA, LWORK, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGEHRD reduces a real general matrix A to upper Hessenberg form H by ! an orthogonal similarity transformation: Q' * A * Q = H . ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! ILO (input) INTEGER ! IHI (input) INTEGER ! It is assumed that A is already upper triangular in rows ! and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally ! set by a previous call to DGEBAL; otherwise they should be ! set to 1 and N respectively. See Further Details. ! 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the N-by-N general matrix to be reduced. ! On exit, the upper triangle and the first subdiagonal of A ! are overwritten with the upper Hessenberg matrix H, and the ! elements below the first subdiagonal, with the array TAU, ! represent the orthogonal matrix Q as a product of elementary ! reflectors. See Further Details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! TAU (output) DOUBLE PRECISION array, dimension (N-1) ! The scalar factors of the elementary reflectors (see Further ! Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to ! zero. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The length of the array WORK. LWORK >= max(1,N). ! For optimum performance LWORK >= N*NB, where NB is the ! optimal blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of (ihi-ilo) elementary ! reflectors ! ! Q = H(ilo) H(ilo+1) . . . H(ihi-1). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on ! exit in A(i+2:ihi,i), and tau in TAU(i). ! ! The contents of A are illustrated by the following example, with ! n = 7, ilo = 2 and ihi = 6: ! ! on entry, on exit, ! ! ( a a a a a a a ) ( a a h h h h a ) ! ( a a a a a a ) ( a h h h h a ) ! ( a a a a a a ) ( h h h h h h ) ! ( a a a a a a ) ( v2 h h h h h ) ! ( a a a a a a ) ( v2 v3 h h h h ) ! ( a a a a a a ) ( v2 v3 v4 h h h ) ! ( a ) ( a ) ! ! where a denotes an element of the original matrix A, h denotes a ! modified element of the upper Hessenberg matrix H, and vi denotes an ! element of the vector defining H(i). ! ! ===================================================================== ! ! .. Parameters .. INTEGER NBMAX, LDT PARAMETER ( NBMAX = 64, LDT = NBMAX+1 ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, IB, IINFO, IWS, LDWORK, LWKOPT, NB, NBMIN, & NH, NX DOUBLE PRECISION EI ! .. ! .. Local Arrays .. DOUBLE PRECISION T( LDT, NBMAX ) ! .. ! .. External Subroutines .. EXTERNAL DGEHD2, DGEMM, DLAHRD, DLARFB, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) ) LWKOPT = N*NB WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) IF( N.LT.0 ) THEN INFO = -1 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN INFO = -2 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEHRD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Set elements 1:ILO-1 and IHI:N-1 of TAU to zero ! TAU(1:ilo-1) = ZERO DO 20 I = MAX( 1, IHI ), N - 1 TAU( I ) = ZERO 20 CONTINUE ! ! Quick return if possible ! NH = IHI - ILO + 1 IF( NH.LE.1 ) THEN WORK( 1 ) = 1 RETURN END IF ! ! Determine the block size. ! NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) ) NBMIN = 2 IWS = 1 IF( NB.GT.1 .AND. NB.LT.NH ) THEN ! ! Determine when to cross over from blocked to unblocked code ! (last block is always handled by unblocked code). ! NX = MAX( NB, ILAENV( 3, 'DGEHRD', ' ', N, ILO, IHI, -1 ) ) IF( NX.LT.NH ) THEN ! ! Determine if workspace is large enough for blocked code. ! IWS = N*NB IF( LWORK.LT.IWS ) THEN ! ! Not enough workspace to use optimal NB: determine the ! minimum value of NB, and reduce NB or force use of ! unblocked code. ! NBMIN = MAX( 2, ILAENV( 2, 'DGEHRD', ' ', N, ILO, IHI, & -1 ) ) IF( LWORK.GE.N*NBMIN ) THEN NB = LWORK / N ELSE NB = 1 END IF END IF END IF END IF LDWORK = N ! IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN ! ! Use unblocked code below ! I = ILO ! ELSE ! ! Use blocked code ! DO 30 I = ILO, IHI - 1 - NX, NB IB = MIN( NB, IHI-I ) ! ! Reduce columns i:i+ib-1 to Hessenberg form, returning the ! matrices V and T of the block reflector H = I - V*T*V' ! which performs the reduction, and also the matrix Y = A*V*T ! CALL DLAHRD( IHI, I, IB, A( 1, I ), LDA, TAU( I ), T, LDT, & WORK, LDWORK ) ! ! Apply the block reflector H to A(1:ihi,i+ib:ihi) from the ! right, computing A := A - Y * V'. V(i+ib,ib-1) must be set ! to 1. ! EI = A( I+IB, I+IB-1 ) A( I+IB, I+IB-1 ) = ONE CALL DGEMM( 'No transpose', 'Transpose', IHI, IHI-I-IB+1, & IB, -ONE, WORK, LDWORK, A( I+IB, I ), LDA, ONE, & A( 1, I+IB ), LDA ) A( I+IB, I+IB-1 ) = EI ! ! Apply the block reflector H to A(i+1:ihi,i+ib:n) from the ! left ! CALL DLARFB( 'Left', 'Transpose', 'Forward', 'Columnwise', & IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA, T, LDT, & A( I+1, I+IB ), LDA, WORK, LDWORK ) 30 CONTINUE END IF ! ! Use unblocked code to reduce the rest of the matrix ! CALL DGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO ) WORK( 1 ) = IWS ! RETURN ! ! End of DGEHRD ! END SUBROUTINE DGELQ2( M, N, A, LDA, TAU, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGELQ2 computes an LQ factorization of a real m by n matrix A: ! A = L * Q. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the m by n matrix A. ! On exit, the elements on and below the diagonal of the array ! contain the m by min(m,n) lower trapezoidal matrix L (L is ! lower triangular if m <= n); the elements above the diagonal, ! with the array TAU, represent the orthogonal matrix Q as a ! product of elementary reflectors (see Further Details). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! TAU (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The scalar factors of the elementary reflectors (see Further ! Details). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (M) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of elementary reflectors ! ! Q = H(k) . . . H(2) H(1), where k = min(m,n). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n), ! and tau in TAU(i). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, K DOUBLE PRECISION AII ! .. ! .. External Subroutines .. EXTERNAL DLARF, DLARFG, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGELQ2', -INFO ) RETURN END IF ! K = MIN( M, N ) ! DO 10 I = 1, K ! ! Generate elementary reflector H(i) to annihilate A(i,i+1:n) ! CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA, & TAU( I ) ) IF( I.LT.M ) THEN ! ! Apply H(i) to A(i+1:m,i:n) from the right ! AII = A( I, I ) A( I, I ) = ONE CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA, TAU( I ), & A( I+1, I ), LDA, WORK ) A( I, I ) = AII END IF 10 CONTINUE RETURN ! ! End of DGELQ2 ! END SUBROUTINE DGELQF( M, N, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGELQF computes an LQ factorization of a real M-by-N matrix A: ! A = L * Q. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, the elements on and below the diagonal of the array ! contain the m-by-min(m,n) lower trapezoidal matrix L (L is ! lower triangular if m <= n); the elements above the diagonal, ! with the array TAU, represent the orthogonal matrix Q as a ! product of elementary reflectors (see Further Details). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! TAU (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The scalar factors of the elementary reflectors (see Further ! Details). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,M). ! For optimum performance LWORK >= M*NB, where NB is the ! optimal blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of elementary reflectors ! ! Q = H(k) . . . H(2) H(1), where k = min(m,n). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n), ! and tau in TAU(i). ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB, & NBMIN, NX ! .. ! .. External Subroutines .. EXTERNAL DGELQ2, DLARFB, DLARFT, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 NB = ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) LWKOPT = M*NB WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 ELSE IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGELQF', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! K = MIN( M, N ) IF( K.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! NBMIN = 2 NX = 0 IWS = M IF( NB.GT.1 .AND. NB.LT.K ) THEN ! ! Determine when to cross over from blocked to unblocked code. ! NX = MAX( 0, ILAENV( 3, 'DGELQF', ' ', M, N, -1, -1 ) ) IF( NX.LT.K ) THEN ! ! Determine if workspace is large enough for blocked code. ! LDWORK = M IWS = LDWORK*NB IF( LWORK.LT.IWS ) THEN ! ! Not enough workspace to use optimal NB: reduce NB and ! determine the minimum value of NB. ! NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'DGELQF', ' ', M, N, -1, & -1 ) ) END IF END IF END IF ! IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN ! ! Use blocked code initially ! DO 10 I = 1, K - NX, NB IB = MIN( K-I+1, NB ) ! ! Compute the LQ factorization of the current block ! A(i:i+ib-1,i:n) ! CALL DGELQ2( IB, N-I+1, A( I, I ), LDA, TAU( I ), WORK, & IINFO ) IF( I+IB.LE.M ) THEN ! ! Form the triangular factor of the block reflector ! H = H(i) H(i+1) . . . H(i+ib-1) ! CALL DLARFT( 'Forward', 'Rowwise', N-I+1, IB, A( I, I ), & LDA, TAU( I ), WORK, LDWORK ) ! ! Apply H to A(i+ib:m,i:n) from the right ! CALL DLARFB( 'Right', 'No transpose', 'Forward', & 'Rowwise', M-I-IB+1, N-I+1, IB, A( I, I ), & LDA, WORK, LDWORK, A( I+IB, I ), LDA, & WORK( IB+1 ), LDWORK ) END IF 10 CONTINUE ELSE I = 1 END IF ! ! Use unblocked code to factor the last or only block. ! IF( I.LE.K ) & CALL DGELQ2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK, & IINFO ) ! WORK( 1 ) = IWS RETURN ! ! End of DGELQF ! END SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, & INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER TRANS INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGELS solves overdetermined or underdetermined real linear systems ! involving an M-by-N matrix A, or its transpose, using a QR or LQ ! factorization of A. It is assumed that A has full rank. ! ! The following options are provided: ! ! 1. If TRANS = 'N' and m >= n: find the least squares solution of ! an overdetermined system, i.e., solve the least squares problem ! minimize || B - A*X ||. ! ! 2. If TRANS = 'N' and m < n: find the minimum norm solution of ! an underdetermined system A * X = B. ! ! 3. If TRANS = 'T' and m >= n: find the minimum norm solution of ! an undetermined system A**T * X = B. ! ! 4. If TRANS = 'T' and m < n: find the least squares solution of ! an overdetermined system, i.e., solve the least squares problem ! minimize || B - A**T * X ||. ! ! Several right hand side vectors b and solution vectors x can be ! handled in a single call; they are stored as the columns of the ! M-by-NRHS right hand side matrix B and the N-by-NRHS solution ! matrix X. ! ! Arguments ! ========= ! ! TRANS (input) CHARACTER ! = 'N': the linear system involves A; ! = 'T': the linear system involves A**T. ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of ! columns of the matrices B and X. NRHS >=0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, ! if M >= N, A is overwritten by details of its QR ! factorization as returned by DGEQRF; ! if M < N, A is overwritten by details of its LQ ! factorization as returned by DGELQF. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the matrix B of right hand side vectors, stored ! columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS ! if TRANS = 'T'. ! On exit, B is overwritten by the solution vectors, stored ! columnwise: ! if TRANS = 'N' and m >= n, rows 1 to n of B contain the least ! squares solution vectors; the residual sum of squares for the ! solution in each column is given by the sum of squares of ! elements N+1 to M in that column; ! if TRANS = 'N' and m < n, rows 1 to N of B contain the ! minimum norm solution vectors; ! if TRANS = 'T' and m >= n, rows 1 to M of B contain the ! minimum norm solution vectors; ! if TRANS = 'T' and m < n, rows 1 to M of B contain the ! least squares solution vectors; the residual sum of squares ! for the solution in each column is given by the sum of ! squares of elements M+1 to N in that column. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= MAX(1,M,N). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! LWORK >= max( 1, MN + max( MN, NRHS ) ). ! For optimal performance, ! LWORK >= max( 1, MN + max( MN, NRHS )*NB ). ! where MN = min(M,N) and NB is the optimum block size. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, TPSD INTEGER BROW, I, IASCL, IBSCL, J, MN, NB, SCLLEN, WSIZE DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMLNUM ! .. ! .. Local Arrays .. DOUBLE PRECISION RWORK( 1 ) ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE ! .. ! .. External Subroutines .. EXTERNAL DGELQF, DGEQRF, DLASCL, DLASET, DORMLQ, DORMQR, & DTRSM, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC DBLE, MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments. ! INFO = 0 MN = MIN( M, N ) LQUERY = ( LWORK.EQ.-1 ) IF( .NOT.( LSAME( TRANS, 'N' ) .OR. LSAME( TRANS, 'T' ) ) ) THEN INFO = -1 ELSE IF( M.LT.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -6 ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN INFO = -8 ELSE IF( LWORK.LT.MAX( 1, MN+MAX( MN, NRHS ) ) .AND. .NOT.LQUERY ) & THEN INFO = -10 END IF ! ! Figure out optimal block size ! IF( INFO.EQ.0 .OR. INFO.EQ.-10 ) THEN ! TPSD = .TRUE. IF( LSAME( TRANS, 'N' ) ) & TPSD = .FALSE. ! IF( M.GE.N ) THEN NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) IF( TPSD ) THEN NB = MAX( NB, ILAENV( 1, 'DORMQR', 'LN', M, NRHS, N, & -1 ) ) ELSE NB = MAX( NB, ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N, & -1 ) ) END IF ELSE NB = ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) IF( TPSD ) THEN NB = MAX( NB, ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M, & -1 ) ) ELSE NB = MAX( NB, ILAENV( 1, 'DORMLQ', 'LN', N, NRHS, M, & -1 ) ) END IF END IF ! WSIZE = MAX( 1, MN+MAX( MN, NRHS )*NB ) WORK( 1 ) = DBLE( WSIZE ) ! END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGELS ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( MIN( M, N, NRHS ).EQ.0 ) THEN CALL DLASET( 'Full', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) RETURN END IF ! ! Get machine parameters ! SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) ! ! Scale A, B if max element outside range [SMLNUM,BIGNUM] ! ANRM = DLANGE( 'M', M, N, A, LDA, RWORK ) IASCL = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN ! ! Scale matrix norm up to SMLNUM ! CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO ) IASCL = 1 ELSE IF( ANRM.GT.BIGNUM ) THEN ! ! Scale matrix norm down to BIGNUM ! CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO ) IASCL = 2 ELSE IF( ANRM.EQ.ZERO ) THEN ! ! Matrix all zero. Return zero solution. ! CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) GO TO 50 END IF ! BROW = M IF( TPSD ) & BROW = N BNRM = DLANGE( 'M', BROW, NRHS, B, LDB, RWORK ) IBSCL = 0 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN ! ! Scale matrix norm up to SMLNUM ! CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, BROW, NRHS, B, LDB, & INFO ) IBSCL = 1 ELSE IF( BNRM.GT.BIGNUM ) THEN ! ! Scale matrix norm down to BIGNUM ! CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, BROW, NRHS, B, LDB, & INFO ) IBSCL = 2 END IF ! IF( M.GE.N ) THEN ! ! compute QR factorization of A ! CALL DGEQRF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ), LWORK-MN, & INFO ) ! ! workspace at least N, optimally N*NB ! IF( .NOT.TPSD ) THEN ! ! Least-Squares Problem min || A * X - B || ! ! B(1:M,1:NRHS) := Q' * B(1:M,1:NRHS) ! CALL DORMQR( 'Left', 'Transpose', M, NRHS, N, A, LDA, & WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN, & INFO ) ! ! workspace at least NRHS, optimally NRHS*NB ! ! B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS) ! CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', N, & NRHS, ONE, A, LDA, B, LDB ) ! SCLLEN = N ! ELSE ! ! Overdetermined system of equations A' * X = B ! ! B(1:N,1:NRHS) := inv(R') * B(1:N,1:NRHS) ! CALL DTRSM( 'Left', 'Upper', 'Transpose', 'Non-unit', N, & NRHS, ONE, A, LDA, B, LDB ) ! ! B(N+1:M,1:NRHS) = ZERO ! DO 20 J = 1, NRHS DO 10 I = N + 1, M B( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ! ! B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS) ! CALL DORMQR( 'Left', 'No transpose', M, NRHS, N, A, LDA, & WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN, & INFO ) ! ! workspace at least NRHS, optimally NRHS*NB ! SCLLEN = M ! END IF ! ELSE ! ! Compute LQ factorization of A ! CALL DGELQF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ), LWORK-MN, & INFO ) ! ! workspace at least M, optimally M*NB. ! IF( .NOT.TPSD ) THEN ! ! underdetermined system of equations A * X = B ! ! B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS) ! CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Non-unit', M, & NRHS, ONE, A, LDA, B, LDB ) ! ! B(M+1:N,1:NRHS) = 0 ! DO 40 J = 1, NRHS DO 30 I = M + 1, N B( I, J ) = ZERO 30 CONTINUE 40 CONTINUE ! ! B(1:N,1:NRHS) := Q(1:N,:)' * B(1:M,1:NRHS) ! CALL DORMLQ( 'Left', 'Transpose', N, NRHS, M, A, LDA, & WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN, & INFO ) ! ! workspace at least NRHS, optimally NRHS*NB ! SCLLEN = N ! ELSE ! ! overdetermined system min || A' * X - B || ! ! B(1:N,1:NRHS) := Q * B(1:N,1:NRHS) ! CALL DORMLQ( 'Left', 'No transpose', N, NRHS, M, A, LDA, & WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN, & INFO ) ! ! workspace at least NRHS, optimally NRHS*NB ! ! B(1:M,1:NRHS) := inv(L') * B(1:M,1:NRHS) ! CALL DTRSM( 'Left', 'Lower', 'Transpose', 'Non-unit', M, & NRHS, ONE, A, LDA, B, LDB ) ! SCLLEN = M ! END IF ! END IF ! ! Undo scaling ! IF( IASCL.EQ.1 ) THEN CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, SCLLEN, NRHS, B, LDB, & INFO ) ELSE IF( IASCL.EQ.2 ) THEN CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, SCLLEN, NRHS, B, LDB, & INFO ) END IF IF( IBSCL.EQ.1 ) THEN CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, SCLLEN, NRHS, B, LDB, & INFO ) ELSE IF( IBSCL.EQ.2 ) THEN CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, SCLLEN, NRHS, B, LDB, & INFO ) END IF ! 50 CONTINUE WORK( 1 ) = DBLE( WSIZE ) ! RETURN ! ! End of DGELS ! END SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, & WORK, LWORK, IWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGELSD computes the minimum-norm solution to a real linear least ! squares problem: ! minimize 2-norm(| b - A*x |) ! using the singular value decomposition (SVD) of A. A is an M-by-N ! matrix which may be rank-deficient. ! ! Several right hand side vectors b and solution vectors x can be ! handled in a single call; they are stored as the columns of the ! M-by-NRHS right hand side matrix B and the N-by-NRHS solution ! matrix X. ! ! The problem is solved in three steps: ! (1) Reduce the coefficient matrix A to bidiagonal form with ! Householder transformations, reducing the original problem ! into a "bidiagonal least squares problem" (BLS) ! (2) Solve the BLS using a divide and conquer approach. ! (3) Apply back all the Householder tranformations to solve ! the original least squares problem. ! ! The effective rank of A is determined by treating as zero those ! singular values which are less than RCOND times the largest singular ! value. ! ! The divide and conquer algorithm makes very mild assumptions about ! floating point arithmetic. It will work on machines with a guard ! digit in add/subtract, or on those binary machines without guard ! digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or ! Cray-2. It could conceivably fail on hexadecimal or decimal machines ! without guard digits, but we know of none. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of A. M >= 0. ! ! N (input) INTEGER ! The number of columns of A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, A has been destroyed. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the M-by-NRHS right hand side matrix B. ! On exit, B is overwritten by the N-by-NRHS solution ! matrix X. If m >= n and RANK = n, the residual ! sum-of-squares for the solution in the i-th column is given ! by the sum of squares of elements n+1:m in that column. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,max(M,N)). ! ! S (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The singular values of A in decreasing order. ! The condition number of A in the 2-norm = S(1)/S(min(m,n)). ! ! RCOND (input) DOUBLE PRECISION ! RCOND is used to determine the effective rank of A. ! Singular values S(i) <= RCOND*S(1) are treated as zero. ! If RCOND < 0, machine precision is used instead. ! ! RANK (output) INTEGER ! The effective rank of A, i.e., the number of singular values ! which are greater than RCOND*S(1). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK must be at least 1. ! The exact minimum amount of workspace needed depends on M, ! N and NRHS. As long as LWORK is at least ! 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2, ! if M is greater than or equal to N or ! 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, ! if M is less than N, the code will execute correctly. ! SMLSIZ is returned by ILAENV and is equal to the maximum ! size of the subproblems at the bottom of the computation ! tree (usually about 25), and ! NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) ! For good performance, LWORK should generally be larger. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace) INTEGER array, dimension (LIWORK) ! LIWORK >= 3 * MINMN * NLVL + 11 * MINMN, ! where MINMN = MIN( M,N ). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: the algorithm for computing the SVD failed to converge; ! if INFO = i, i off-diagonal elements of an intermediate ! bidiagonal form did not converge to zero. ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Ren-Cang Li, Computer Science Division, University of ! California at Berkeley, USA ! Osni Marques, LBNL/NERSC, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ, & LDWORK, MAXMN, MAXWRK, MINMN, MINWRK, MM, & MNTHR, NLVL, NWORK, SMLSIZ, WLALSD DOUBLE PRECISION ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM ! .. ! .. External Subroutines .. EXTERNAL DGEBRD, DGELQF, DGEQRF, DLABAD, DLACPY, DLALSD, & DLASCL, DLASET, DORMBR, DORMLQ, DORMQR, XERBLA ! .. ! .. External Functions .. INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL ILAENV, DLAMCH, DLANGE ! .. ! .. Intrinsic Functions .. INTRINSIC DBLE, INT, LOG, MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments. ! INFO = 0 MINMN = MIN( M, N ) MAXMN = MAX( M, N ) MNTHR = ILAENV( 6, 'DGELSD', ' ', M, N, NRHS, -1 ) LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN INFO = -7 END IF ! SMLSIZ = ILAENV( 9, 'DGELSD', ' ', 0, 0, 0, 0 ) ! ! Compute workspace. ! (Note: Comments in the code beginning "Workspace:" describe the ! minimal amount of workspace needed at that point in the code, ! as well as the preferred amount for good performance. ! NB refers to the optimal block size for the immediately ! following subroutine, as returned by ILAENV.) ! MINWRK = 1 MINMN = MAX( 1, MINMN ) NLVL = MAX( INT( LOG( DBLE( MINMN ) / DBLE( SMLSIZ+1 ) ) / & LOG( TWO ) ) + 1, 0 ) ! IF( INFO.EQ.0 ) THEN MAXWRK = 0 MM = M IF( M.GE.N .AND. M.GE.MNTHR ) THEN ! ! Path 1a - overdetermined, with many more rows than columns. ! MM = N MAXWRK = MAX( MAXWRK, N+N*ILAENV( 1, 'DGEQRF', ' ', M, N, & -1, -1 ) ) MAXWRK = MAX( MAXWRK, N+NRHS* & ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N, -1 ) ) END IF IF( M.GE.N ) THEN ! ! Path 1 - overdetermined or exactly determined. ! MAXWRK = MAX( MAXWRK, 3*N+( MM+N )* & ILAENV( 1, 'DGEBRD', ' ', MM, N, -1, -1 ) ) MAXWRK = MAX( MAXWRK, 3*N+NRHS* & ILAENV( 1, 'DORMBR', 'QLT', MM, NRHS, N, -1 ) ) MAXWRK = MAX( MAXWRK, 3*N+( N-1 )* & ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, N, -1 ) ) WLALSD = 9*N+2*N*SMLSIZ+8*N*NLVL+N*NRHS+(SMLSIZ+1)**2 MAXWRK = MAX( MAXWRK, 3*N+WLALSD ) MINWRK = MAX( 3*N+MM, 3*N+NRHS, 3*N+WLALSD ) END IF IF( N.GT.M ) THEN WLALSD = 9*M+2*M*SMLSIZ+8*M*NLVL+M*NRHS+(SMLSIZ+1)**2 IF( N.GE.MNTHR ) THEN ! ! Path 2a - underdetermined, with many more columns ! than rows. ! MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) MAXWRK = MAX( MAXWRK, M*M+4*M+2*M* & ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS* & ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )* & ILAENV( 1, 'DORMBR', 'PLN', M, NRHS, M, -1 ) ) IF( NRHS.GT.1 ) THEN MAXWRK = MAX( MAXWRK, M*M+M+M*NRHS ) ELSE MAXWRK = MAX( MAXWRK, M*M+2*M ) END IF MAXWRK = MAX( MAXWRK, M+NRHS* & ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+WLALSD ) ELSE ! ! Path 2 - remaining underdetermined cases. ! MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'DGEBRD', ' ', M, N, & -1, -1 ) MAXWRK = MAX( MAXWRK, 3*M+NRHS* & ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, N, -1 ) ) MAXWRK = MAX( MAXWRK, 3*M+M* & ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, 3*M+WLALSD ) END IF MINWRK = MAX( 3*M+NRHS, 3*M+M, 3*M+WLALSD ) END IF MINWRK = MIN( MINWRK, MAXWRK ) WORK( 1 ) = MAXWRK IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -12 END IF END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGELSD', -INFO ) RETURN ELSE IF( LQUERY ) THEN GO TO 10 END IF ! ! Quick return if possible. ! IF( M.EQ.0 .OR. N.EQ.0 ) THEN RANK = 0 RETURN END IF ! ! Get machine parameters. ! EPS = DLAMCH( 'P' ) SFMIN = DLAMCH( 'S' ) SMLNUM = SFMIN / EPS BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) ! ! Scale A if max entry outside range [SMLNUM,BIGNUM]. ! ANRM = DLANGE( 'M', M, N, A, LDA, WORK ) IASCL = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN ! ! Scale matrix norm up to SMLNUM. ! CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO ) IASCL = 1 ELSE IF( ANRM.GT.BIGNUM ) THEN ! ! Scale matrix norm down to BIGNUM. ! CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO ) IASCL = 2 ELSE IF( ANRM.EQ.ZERO ) THEN ! ! Matrix all zero. Return zero solution. ! CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 ) RANK = 0 GO TO 10 END IF ! ! Scale B if max entry outside range [SMLNUM,BIGNUM]. ! BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK ) IBSCL = 0 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN ! ! Scale matrix norm up to SMLNUM. ! CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO ) IBSCL = 1 ELSE IF( BNRM.GT.BIGNUM ) THEN ! ! Scale matrix norm down to BIGNUM. ! CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO ) IBSCL = 2 END IF ! ! If M < N make sure certain entries of B are zero. ! IF( M.LT.N ) & CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB ) ! ! Overdetermined case. ! IF( M.GE.N ) THEN ! ! Path 1 - overdetermined or exactly determined. ! MM = M IF( M.GE.MNTHR ) THEN ! ! Path 1a - overdetermined, with many more rows than columns. ! MM = N ITAU = 1 NWORK = ITAU + N ! ! Compute A=Q*R. ! (Workspace: need 2*N, prefer N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), & LWORK-NWORK+1, INFO ) ! ! Multiply B by transpose(Q). ! (Workspace: need N+NRHS, prefer N+NRHS*NB) ! CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ), B, & LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) ! ! Zero out below R. ! IF( N.GT.1 ) THEN CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA ) END IF END IF ! IE = 1 ITAUQ = IE + N ITAUP = ITAUQ + N NWORK = ITAUP + N ! ! Bidiagonalize R in A. ! (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB) ! CALL DGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, & INFO ) ! ! Multiply B by transpose of left bidiagonalizing vectors of R. ! (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB) ! CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ), & B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) ! ! Solve the bidiagonal least squares problem. ! CALL DLALSD( 'U', SMLSIZ, N, NRHS, S, WORK( IE ), B, LDB, & RCOND, RANK, WORK( NWORK ), IWORK, INFO ) IF( INFO.NE.0 ) THEN GO TO 10 END IF ! ! Multiply B by right bidiagonalizing vectors of R. ! CALL DORMBR( 'P', 'L', 'N', N, NRHS, N, A, LDA, WORK( ITAUP ), & B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) ! ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+ & MAX( M, 2*M-4, NRHS, N-3*M ) ) THEN ! ! Path 2a - underdetermined, with many more columns than rows ! and sufficient workspace for an efficient algorithm. ! LDWORK = M IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ), & M*LDA+M+M*NRHS ) )LDWORK = LDA ITAU = 1 NWORK = M + 1 ! ! Compute A=L*Q. ! (Workspace: need 2*M, prefer M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), & LWORK-NWORK+1, INFO ) IL = NWORK ! ! Copy L to WORK(IL), zeroing out above its diagonal. ! CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ), & LDWORK ) IE = IL + LDWORK*M ITAUQ = IE + M ITAUP = ITAUQ + M NWORK = ITAUP + M ! ! Bidiagonalize L in WORK(IL). ! (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB) ! CALL DGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ), & LWORK-NWORK+1, INFO ) ! ! Multiply B by transpose of left bidiagonalizing vectors of L. ! (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB) ! CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK, & WORK( ITAUQ ), B, LDB, WORK( NWORK ), & LWORK-NWORK+1, INFO ) ! ! Solve the bidiagonal least squares problem. ! CALL DLALSD( 'U', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB, & RCOND, RANK, WORK( NWORK ), IWORK, INFO ) IF( INFO.NE.0 ) THEN GO TO 10 END IF ! ! Multiply B by right bidiagonalizing vectors of L. ! CALL DORMBR( 'P', 'L', 'N', M, NRHS, M, WORK( IL ), LDWORK, & WORK( ITAUP ), B, LDB, WORK( NWORK ), & LWORK-NWORK+1, INFO ) ! ! Zero out below first M rows of B. ! CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB ) NWORK = ITAU + M ! ! Multiply transpose(Q) by B. ! (Workspace: need M+NRHS, prefer M+NRHS*NB) ! CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B, & LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) ! ELSE ! ! Path 2 - remaining underdetermined cases. ! IE = 1 ITAUQ = IE + M ITAUP = ITAUQ + M NWORK = ITAUP + M ! ! Bidiagonalize A. ! (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB) ! CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, & INFO ) ! ! Multiply B by transpose of left bidiagonalizing vectors. ! (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB) ! CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ), & B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) ! ! Solve the bidiagonal least squares problem. ! CALL DLALSD( 'L', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB, & RCOND, RANK, WORK( NWORK ), IWORK, INFO ) IF( INFO.NE.0 ) THEN GO TO 10 END IF ! ! Multiply B by right bidiagonalizing vectors of A. ! CALL DORMBR( 'P', 'L', 'N', N, NRHS, M, A, LDA, WORK( ITAUP ), & B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) ! END IF ! ! Undo scaling. ! IF( IASCL.EQ.1 ) THEN CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO ) CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN, & INFO ) ELSE IF( IASCL.EQ.2 ) THEN CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO ) CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN, & INFO ) END IF IF( IBSCL.EQ.1 ) THEN CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO ) ELSE IF( IBSCL.EQ.2 ) THEN CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO ) END IF ! 10 CONTINUE WORK( 1 ) = MAXWRK RETURN ! ! End of DGELSD ! END SUBROUTINE DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, & WORK, LWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGELSS computes the minimum norm solution to a real linear least ! squares problem: ! ! Minimize 2-norm(| b - A*x |). ! ! using the singular value decomposition (SVD) of A. A is an M-by-N ! matrix which may be rank-deficient. ! ! Several right hand side vectors b and solution vectors x can be ! handled in a single call; they are stored as the columns of the ! M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix ! X. ! ! The effective rank of A is determined by treating as zero those ! singular values which are less than RCOND times the largest singular ! value. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, the first min(m,n) rows of A are overwritten with ! its right singular vectors, stored rowwise. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the M-by-NRHS right hand side matrix B. ! On exit, B is overwritten by the N-by-NRHS solution ! matrix X. If m >= n and RANK = n, the residual ! sum-of-squares for the solution in the i-th column is given ! by the sum of squares of elements n+1:m in that column. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,max(M,N)). ! ! S (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The singular values of A in decreasing order. ! The condition number of A in the 2-norm = S(1)/S(min(m,n)). ! ! RCOND (input) DOUBLE PRECISION ! RCOND is used to determine the effective rank of A. ! Singular values S(i) <= RCOND*S(1) are treated as zero. ! If RCOND < 0, machine precision is used instead. ! ! RANK (output) INTEGER ! The effective rank of A, i.e., the number of singular values ! which are greater than RCOND*S(1). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= 1, and also: ! LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS ) ! For good performance, LWORK should generally be larger. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: the algorithm for computing the SVD failed to converge; ! if INFO = i, i off-diagonal elements of an intermediate ! bidiagonal form did not converge to zero. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER BDSPAC, BL, CHUNK, I, IASCL, IBSCL, IE, IL, & ITAU, ITAUP, ITAUQ, IWORK, LDWORK, MAXMN, & MAXWRK, MINMN, MINWRK, MM, MNTHR DOUBLE PRECISION ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM, THR ! .. ! .. Local Arrays .. DOUBLE PRECISION VDUM( 1 ) ! .. ! .. External Subroutines .. EXTERNAL DBDSQR, DCOPY, DGEBRD, DGELQF, DGEMM, DGEMV, & DGEQRF, DLABAD, DLACPY, DLASCL, DLASET, DORGBR, & DORMBR, DORMLQ, DORMQR, DRSCL, XERBLA ! .. ! .. External Functions .. INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL ILAENV, DLAMCH, DLANGE ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 MINMN = MIN( M, N ) MAXMN = MAX( M, N ) MNTHR = ILAENV( 6, 'DGELSS', ' ', M, N, NRHS, -1 ) LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN INFO = -7 END IF ! ! Compute workspace ! (Note: Comments in the code beginning "Workspace:" describe the ! minimal amount of workspace needed at that point in the code, ! as well as the preferred amount for good performance. ! NB refers to the optimal block size for the immediately ! following subroutine, as returned by ILAENV.) ! MINWRK = 1 IF( INFO.EQ.0 .AND. ( LWORK.GE.1 .OR. LQUERY ) ) THEN MAXWRK = 0 MM = M IF( M.GE.N .AND. M.GE.MNTHR ) THEN ! ! Path 1a - overdetermined, with many more rows than columns ! MM = N MAXWRK = MAX( MAXWRK, N+N*ILAENV( 1, 'DGEQRF', ' ', M, N, & -1, -1 ) ) MAXWRK = MAX( MAXWRK, N+NRHS* & ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N, -1 ) ) END IF IF( M.GE.N ) THEN ! ! Path 1 - overdetermined or exactly determined ! ! Compute workspace needed for DBDSQR ! BDSPAC = MAX( 1, 5*N ) MAXWRK = MAX( MAXWRK, 3*N+( MM+N )* & ILAENV( 1, 'DGEBRD', ' ', MM, N, -1, -1 ) ) MAXWRK = MAX( MAXWRK, 3*N+NRHS* & ILAENV( 1, 'DORMBR', 'QLT', MM, NRHS, N, -1 ) ) MAXWRK = MAX( MAXWRK, 3*N+( N-1 )* & ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) ) MAXWRK = MAX( MAXWRK, BDSPAC ) MAXWRK = MAX( MAXWRK, N*NRHS ) MINWRK = MAX( 3*N+MM, 3*N+NRHS, BDSPAC ) MAXWRK = MAX( MINWRK, MAXWRK ) END IF IF( N.GT.M ) THEN ! ! Compute workspace needed for DBDSQR ! BDSPAC = MAX( 1, 5*M ) MINWRK = MAX( 3*M+NRHS, 3*M+N, BDSPAC ) IF( N.GE.MNTHR ) THEN ! ! Path 2a - underdetermined, with many more columns ! than rows ! MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) MAXWRK = MAX( MAXWRK, M*M+4*M+2*M* & ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS* & ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )* & ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+M+BDSPAC ) IF( NRHS.GT.1 ) THEN MAXWRK = MAX( MAXWRK, M*M+M+M*NRHS ) ELSE MAXWRK = MAX( MAXWRK, M*M+2*M ) END IF MAXWRK = MAX( MAXWRK, M+NRHS* & ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M, -1 ) ) ELSE ! ! Path 2 - underdetermined ! MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'DGEBRD', ' ', M, N, & -1, -1 ) MAXWRK = MAX( MAXWRK, 3*M+NRHS* & ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, 3*M+M* & ILAENV( 1, 'DORGBR', 'P', M, N, M, -1 ) ) MAXWRK = MAX( MAXWRK, BDSPAC ) MAXWRK = MAX( MAXWRK, N*NRHS ) END IF END IF MAXWRK = MAX( MINWRK, MAXWRK ) WORK( 1 ) = MAXWRK END IF ! MINWRK = MAX( MINWRK, 1 ) IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) & INFO = -12 IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGELSS', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 ) THEN RANK = 0 RETURN END IF ! ! Get machine parameters ! EPS = DLAMCH( 'P' ) SFMIN = DLAMCH( 'S' ) SMLNUM = SFMIN / EPS BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) ! ! Scale A if max element outside range [SMLNUM,BIGNUM] ! ANRM = DLANGE( 'M', M, N, A, LDA, WORK ) IASCL = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN ! ! Scale matrix norm up to SMLNUM ! CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO ) IASCL = 1 ELSE IF( ANRM.GT.BIGNUM ) THEN ! ! Scale matrix norm down to BIGNUM ! CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO ) IASCL = 2 ELSE IF( ANRM.EQ.ZERO ) THEN ! ! Matrix all zero. Return zero solution. ! CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 ) RANK = 0 GO TO 70 END IF ! ! Scale B if max element outside range [SMLNUM,BIGNUM] ! BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK ) IBSCL = 0 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN ! ! Scale matrix norm up to SMLNUM ! CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO ) IBSCL = 1 ELSE IF( BNRM.GT.BIGNUM ) THEN ! ! Scale matrix norm down to BIGNUM ! CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO ) IBSCL = 2 END IF ! ! Overdetermined case ! IF( M.GE.N ) THEN ! ! Path 1 - overdetermined or exactly determined ! MM = M IF( M.GE.MNTHR ) THEN ! ! Path 1a - overdetermined, with many more rows than columns ! MM = N ITAU = 1 IWORK = ITAU + N ! ! Compute A=Q*R ! (Workspace: need 2*N, prefer N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ), & LWORK-IWORK+1, INFO ) ! ! Multiply B by transpose(Q) ! (Workspace: need N+NRHS, prefer N+NRHS*NB) ! CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ), B, & LDB, WORK( IWORK ), LWORK-IWORK+1, INFO ) ! ! Zero out below R ! IF( N.GT.1 ) & CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA ) END IF ! IE = 1 ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Bidiagonalize R in A ! (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB) ! CALL DGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1, & INFO ) ! ! Multiply B by transpose of left bidiagonalizing vectors of R ! (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB) ! CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ), & B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO ) ! ! Generate right bidiagonalizing vectors of R in A ! (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB) ! CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, INFO ) IWORK = IE + N ! ! Perform bidiagonal QR iteration ! multiply B by transpose of left singular vectors ! compute right singular vectors in A ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', N, N, 0, NRHS, S, WORK( IE ), A, LDA, VDUM, & 1, B, LDB, WORK( IWORK ), INFO ) IF( INFO.NE.0 ) & GO TO 70 ! ! Multiply B by reciprocals of singular values ! THR = MAX( RCOND*S( 1 ), SFMIN ) IF( RCOND.LT.ZERO ) & THR = MAX( EPS*S( 1 ), SFMIN ) RANK = 0 DO 10 I = 1, N IF( S( I ).GT.THR ) THEN CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB ) RANK = RANK + 1 ELSE CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB ) END IF 10 CONTINUE ! ! Multiply B by right singular vectors ! (Workspace: need N, prefer N*NRHS) ! IF( LWORK.GE.LDB*NRHS .AND. NRHS.GT.1 ) THEN CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, A, LDA, B, LDB, ZERO, & WORK, LDB ) CALL DLACPY( 'G', N, NRHS, WORK, LDB, B, LDB ) ELSE IF( NRHS.GT.1 ) THEN CHUNK = LWORK / N DO 20 I = 1, NRHS, CHUNK BL = MIN( NRHS-I+1, CHUNK ) CALL DGEMM( 'T', 'N', N, BL, N, ONE, A, LDA, B( 1, I ), & LDB, ZERO, WORK, N ) CALL DLACPY( 'G', N, BL, WORK, N, B( 1, I ), LDB ) 20 CONTINUE ELSE CALL DGEMV( 'T', N, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 ) CALL DCOPY( N, WORK, 1, B, 1 ) END IF ! ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+ & MAX( M, 2*M-4, NRHS, N-3*M ) ) THEN ! ! Path 2a - underdetermined, with many more columns than rows ! and sufficient workspace for an efficient algorithm ! LDWORK = M IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ), & M*LDA+M+M*NRHS ) )LDWORK = LDA ITAU = 1 IWORK = M + 1 ! ! Compute A=L*Q ! (Workspace: need 2*M, prefer M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ), & LWORK-IWORK+1, INFO ) IL = IWORK ! ! Copy L to WORK(IL), zeroing out above it ! CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ), & LDWORK ) IE = IL + LDWORK*M ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Bidiagonalize L in WORK(IL) ! (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB) ! CALL DGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, INFO ) ! ! Multiply B by transpose of left bidiagonalizing vectors of L ! (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB) ! CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK, & WORK( ITAUQ ), B, LDB, WORK( IWORK ), & LWORK-IWORK+1, INFO ) ! ! Generate right bidiagonalizing vectors of R in WORK(IL) ! (Workspace: need M*M+5*M-1, prefer M*M+4*M+(M-1)*NB) ! CALL DORGBR( 'P', M, M, M, WORK( IL ), LDWORK, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, INFO ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, ! computing right singular vectors of L in WORK(IL) and ! multiplying B by transpose of left singular vectors ! (Workspace: need M*M+M+BDSPAC) ! CALL DBDSQR( 'U', M, M, 0, NRHS, S, WORK( IE ), WORK( IL ), & LDWORK, A, LDA, B, LDB, WORK( IWORK ), INFO ) IF( INFO.NE.0 ) & GO TO 70 ! ! Multiply B by reciprocals of singular values ! THR = MAX( RCOND*S( 1 ), SFMIN ) IF( RCOND.LT.ZERO ) & THR = MAX( EPS*S( 1 ), SFMIN ) RANK = 0 DO 30 I = 1, M IF( S( I ).GT.THR ) THEN CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB ) RANK = RANK + 1 ELSE CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB ) END IF 30 CONTINUE IWORK = IE ! ! Multiply B by right singular vectors of L in WORK(IL) ! (Workspace: need M*M+2*M, prefer M*M+M+M*NRHS) ! IF( LWORK.GE.LDB*NRHS+IWORK-1 .AND. NRHS.GT.1 ) THEN CALL DGEMM( 'T', 'N', M, NRHS, M, ONE, WORK( IL ), LDWORK, & B, LDB, ZERO, WORK( IWORK ), LDB ) CALL DLACPY( 'G', M, NRHS, WORK( IWORK ), LDB, B, LDB ) ELSE IF( NRHS.GT.1 ) THEN CHUNK = ( LWORK-IWORK+1 ) / M DO 40 I = 1, NRHS, CHUNK BL = MIN( NRHS-I+1, CHUNK ) CALL DGEMM( 'T', 'N', M, BL, M, ONE, WORK( IL ), LDWORK, & B( 1, I ), LDB, ZERO, WORK( IWORK ), N ) CALL DLACPY( 'G', M, BL, WORK( IWORK ), N, B( 1, I ), & LDB ) 40 CONTINUE ELSE CALL DGEMV( 'T', M, M, ONE, WORK( IL ), LDWORK, B( 1, 1 ), & 1, ZERO, WORK( IWORK ), 1 ) CALL DCOPY( M, WORK( IWORK ), 1, B( 1, 1 ), 1 ) END IF ! ! Zero out below first M rows of B ! CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB ) IWORK = ITAU + M ! ! Multiply transpose(Q) by B ! (Workspace: need M+NRHS, prefer M+NRHS*NB) ! CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B, & LDB, WORK( IWORK ), LWORK-IWORK+1, INFO ) ! ELSE ! ! Path 2 - remaining underdetermined cases ! IE = 1 ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Bidiagonalize A ! (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB) ! CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1, & INFO ) ! ! Multiply B by transpose of left bidiagonalizing vectors ! (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB) ! CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ), & B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO ) ! ! Generate right bidiagonalizing vectors in A ! (Workspace: need 4*M, prefer 3*M+M*NB) ! CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, INFO ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, ! computing right singular vectors of A in A and ! multiplying B by transpose of left singular vectors ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'L', M, N, 0, NRHS, S, WORK( IE ), A, LDA, VDUM, & 1, B, LDB, WORK( IWORK ), INFO ) IF( INFO.NE.0 ) & GO TO 70 ! ! Multiply B by reciprocals of singular values ! THR = MAX( RCOND*S( 1 ), SFMIN ) IF( RCOND.LT.ZERO ) & THR = MAX( EPS*S( 1 ), SFMIN ) RANK = 0 DO 50 I = 1, M IF( S( I ).GT.THR ) THEN CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB ) RANK = RANK + 1 ELSE CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB ) END IF 50 CONTINUE ! ! Multiply B by right singular vectors of A ! (Workspace: need N, prefer N*NRHS) ! IF( LWORK.GE.LDB*NRHS .AND. NRHS.GT.1 ) THEN CALL DGEMM( 'T', 'N', N, NRHS, M, ONE, A, LDA, B, LDB, ZERO, & WORK, LDB ) CALL DLACPY( 'F', N, NRHS, WORK, LDB, B, LDB ) ELSE IF( NRHS.GT.1 ) THEN CHUNK = LWORK / N DO 60 I = 1, NRHS, CHUNK BL = MIN( NRHS-I+1, CHUNK ) CALL DGEMM( 'T', 'N', N, BL, M, ONE, A, LDA, B( 1, I ), & LDB, ZERO, WORK, N ) CALL DLACPY( 'F', N, BL, WORK, N, B( 1, I ), LDB ) 60 CONTINUE ELSE CALL DGEMV( 'T', M, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 ) CALL DCOPY( N, WORK, 1, B, 1 ) END IF END IF ! ! Undo scaling ! IF( IASCL.EQ.1 ) THEN CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO ) CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN, & INFO ) ELSE IF( IASCL.EQ.2 ) THEN CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO ) CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN, & INFO ) END IF IF( IBSCL.EQ.1 ) THEN CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO ) ELSE IF( IBSCL.EQ.2 ) THEN CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO ) END IF ! 70 CONTINUE WORK( 1 ) = MAXWRK RETURN ! ! End of DGELSS ! END SUBROUTINE DGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, & WORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LDB, M, N, NRHS, RANK DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. INTEGER JPVT( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! This routine is deprecated and has been replaced by routine DGELSY. ! ! DGELSX computes the minimum-norm solution to a real linear least ! squares problem: ! minimize || A * X - B || ! using a complete orthogonal factorization of A. A is an M-by-N ! matrix which may be rank-deficient. ! ! Several right hand side vectors b and solution vectors x can be ! handled in a single call; they are stored as the columns of the ! M-by-NRHS right hand side matrix B and the N-by-NRHS solution ! matrix X. ! ! The routine first computes a QR factorization with column pivoting: ! A * P = Q * [ R11 R12 ] ! [ 0 R22 ] ! with R11 defined as the largest leading submatrix whose estimated ! condition number is less than 1/RCOND. The order of R11, RANK, ! is the effective rank of A. ! ! Then, R22 is considered to be negligible, and R12 is annihilated ! by orthogonal transformations from the right, arriving at the ! complete orthogonal factorization: ! A * P = Q * [ T11 0 ] * Z ! [ 0 0 ] ! The minimum-norm solution is then ! X = P * Z' [ inv(T11)*Q1'*B ] ! [ 0 ] ! where Q1 consists of the first RANK columns of Q. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of ! columns of matrices B and X. NRHS >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, A has been overwritten by details of its ! complete orthogonal factorization. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the M-by-NRHS right hand side matrix B. ! On exit, the N-by-NRHS solution matrix X. ! If m >= n and RANK = n, the residual sum-of-squares for ! the solution in the i-th column is given by the sum of ! squares of elements N+1:M in that column. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,M,N). ! ! JPVT (input/output) INTEGER array, dimension (N) ! On entry, if JPVT(i) .ne. 0, the i-th column of A is an ! initial column, otherwise it is a free column. Before ! the QR factorization of A, all initial columns are ! permuted to the leading positions; only the remaining ! free columns are moved as a result of column pivoting ! during the factorization. ! On exit, if JPVT(i) = k, then the i-th column of A*P ! was the k-th column of A. ! ! RCOND (input) DOUBLE PRECISION ! RCOND is used to determine the effective rank of A, which ! is defined as the order of the largest leading triangular ! submatrix R11 in the QR factorization with pivoting of A, ! whose estimated condition number < 1/RCOND. ! ! RANK (output) INTEGER ! The effective rank of A, i.e., the order of the submatrix ! R11. This is the same as the order of the submatrix T11 ! in the complete orthogonal factorization of A. ! ! WORK (workspace) DOUBLE PRECISION array, dimension ! (max( min(M,N)+3*N, 2*min(M,N)+NRHS )), ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. INTEGER IMAX, IMIN PARAMETER ( IMAX = 1, IMIN = 2 ) DOUBLE PRECISION ZERO, ONE, DONE, NTDONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, DONE = ZERO, & NTDONE = ONE ) ! .. ! .. Local Scalars .. INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, K, MN DOUBLE PRECISION ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX, & SMAXPR, SMIN, SMINPR, SMLNUM, T1, T2 ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL DLAMCH, DLANGE ! .. ! .. External Subroutines .. EXTERNAL DGEQPF, DLAIC1, DLASCL, DLASET, DLATZM, DORM2R, & DTRSM, DTZRQF, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Executable Statements .. ! MN = MIN( M, N ) ISMIN = MN + 1 ISMAX = 2*MN + 1 ! ! Test the input arguments. ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN INFO = -7 END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGELSX', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( MIN( M, N, NRHS ).EQ.0 ) THEN RANK = 0 RETURN END IF ! ! Get machine parameters ! SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) ! ! Scale A, B if max elements outside range [SMLNUM,BIGNUM] ! ANRM = DLANGE( 'M', M, N, A, LDA, WORK ) IASCL = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN ! ! Scale matrix norm up to SMLNUM ! CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO ) IASCL = 1 ELSE IF( ANRM.GT.BIGNUM ) THEN ! ! Scale matrix norm down to BIGNUM ! CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO ) IASCL = 2 ELSE IF( ANRM.EQ.ZERO ) THEN ! ! Matrix all zero. Return zero solution. ! CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) RANK = 0 GO TO 100 END IF ! BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK ) IBSCL = 0 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN ! ! Scale matrix norm up to SMLNUM ! CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO ) IBSCL = 1 ELSE IF( BNRM.GT.BIGNUM ) THEN ! ! Scale matrix norm down to BIGNUM ! CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO ) IBSCL = 2 END IF ! ! Compute QR factorization with column pivoting of A: ! A * P = Q * R ! CALL DGEQPF( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ), INFO ) ! ! workspace 3*N. Details of Householder rotations stored ! in WORK(1:MN). ! ! Determine RANK using incremental condition estimation ! WORK( ISMIN ) = ONE WORK( ISMAX ) = ONE SMAX = ABS( A( 1, 1 ) ) SMIN = SMAX IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN RANK = 0 CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) GO TO 100 ELSE RANK = 1 END IF ! 10 CONTINUE IF( RANK.LT.MN ) THEN I = RANK + 1 CALL DLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ), & A( I, I ), SMINPR, S1, C1 ) CALL DLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ), & A( I, I ), SMAXPR, S2, C2 ) ! IF( SMAXPR*RCOND.LE.SMINPR ) THEN DO 20 I = 1, RANK WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 ) WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 ) 20 CONTINUE WORK( ISMIN+RANK ) = C1 WORK( ISMAX+RANK ) = C2 SMIN = SMINPR SMAX = SMAXPR RANK = RANK + 1 GO TO 10 END IF END IF ! ! Logically partition R = [ R11 R12 ] ! [ 0 R22 ] ! where R11 = R(1:RANK,1:RANK) ! ! [R11,R12] = [ T11, 0 ] * Y ! IF( RANK.LT.N ) & CALL DTZRQF( RANK, N, A, LDA, WORK( MN+1 ), INFO ) ! ! Details of Householder rotations stored in WORK(MN+1:2*MN) ! ! B(1:M,1:NRHS) := Q' * B(1:M,1:NRHS) ! CALL DORM2R( 'Left', 'Transpose', M, NRHS, MN, A, LDA, WORK( 1 ), & B, LDB, WORK( 2*MN+1 ), INFO ) ! ! workspace NRHS ! ! B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS) ! CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK, & NRHS, ONE, A, LDA, B, LDB ) ! DO 40 I = RANK + 1, N DO 30 J = 1, NRHS B( I, J ) = ZERO 30 CONTINUE 40 CONTINUE ! ! B(1:N,1:NRHS) := Y' * B(1:N,1:NRHS) ! IF( RANK.LT.N ) THEN DO 50 I = 1, RANK CALL DLATZM( 'Left', N-RANK+1, NRHS, A( I, RANK+1 ), LDA, & WORK( MN+I ), B( I, 1 ), B( RANK+1, 1 ), LDB, & WORK( 2*MN+1 ) ) 50 CONTINUE END IF ! ! workspace NRHS ! ! B(1:N,1:NRHS) := P * B(1:N,1:NRHS) ! DO 90 J = 1, NRHS DO 60 I = 1, N WORK( 2*MN+I ) = NTDONE 60 CONTINUE DO 80 I = 1, N IF( WORK( 2*MN+I ).EQ.NTDONE ) THEN IF( JPVT( I ).NE.I ) THEN K = I T1 = B( K, J ) T2 = B( JPVT( K ), J ) 70 CONTINUE B( JPVT( K ), J ) = T1 WORK( 2*MN+K ) = DONE T1 = T2 K = JPVT( K ) T2 = B( JPVT( K ), J ) IF( JPVT( K ).NE.I ) & GO TO 70 B( I, J ) = T1 WORK( 2*MN+K ) = DONE END IF END IF 80 CONTINUE 90 CONTINUE ! ! Undo scaling ! IF( IASCL.EQ.1 ) THEN CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO ) CALL DLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA, & INFO ) ELSE IF( IASCL.EQ.2 ) THEN CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO ) CALL DLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA, & INFO ) END IF IF( IBSCL.EQ.1 ) THEN CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO ) ELSE IF( IBSCL.EQ.2 ) THEN CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO ) END IF ! 100 CONTINUE ! RETURN ! ! End of DGELSX ! END SUBROUTINE DGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, & WORK, LWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. INTEGER JPVT( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGELSY computes the minimum-norm solution to a real linear least ! squares problem: ! minimize || A * X - B || ! using a complete orthogonal factorization of A. A is an M-by-N ! matrix which may be rank-deficient. ! ! Several right hand side vectors b and solution vectors x can be ! handled in a single call; they are stored as the columns of the ! M-by-NRHS right hand side matrix B and the N-by-NRHS solution ! matrix X. ! ! The routine first computes a QR factorization with column pivoting: ! A * P = Q * [ R11 R12 ] ! [ 0 R22 ] ! with R11 defined as the largest leading submatrix whose estimated ! condition number is less than 1/RCOND. The order of R11, RANK, ! is the effective rank of A. ! ! Then, R22 is considered to be negligible, and R12 is annihilated ! by orthogonal transformations from the right, arriving at the ! complete orthogonal factorization: ! A * P = Q * [ T11 0 ] * Z ! [ 0 0 ] ! The minimum-norm solution is then ! X = P * Z' [ inv(T11)*Q1'*B ] ! [ 0 ] ! where Q1 consists of the first RANK columns of Q. ! ! This routine is basically identical to the original xGELSX except ! three differences: ! o The call to the subroutine xGEQPF has been substituted by the ! the call to the subroutine xGEQP3. This subroutine is a Blas-3 ! version of the QR factorization with column pivoting. ! o Matrix B (the right hand side) is updated with Blas-3. ! o The permutation of matrix B (the right hand side) is faster and ! more simple. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of ! columns of matrices B and X. NRHS >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, A has been overwritten by details of its ! complete orthogonal factorization. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the M-by-NRHS right hand side matrix B. ! On exit, the N-by-NRHS solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,M,N). ! ! JPVT (input/output) INTEGER array, dimension (N) ! On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted ! to the front of AP, otherwise column i is a free column. ! On exit, if JPVT(i) = k, then the i-th column of AP ! was the k-th column of A. ! ! RCOND (input) DOUBLE PRECISION ! RCOND is used to determine the effective rank of A, which ! is defined as the order of the largest leading triangular ! submatrix R11 in the QR factorization with pivoting of A, ! whose estimated condition number < 1/RCOND. ! ! RANK (output) INTEGER ! The effective rank of A, i.e., the order of the submatrix ! R11. This is the same as the order of the submatrix T11 ! in the complete orthogonal factorization of A. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! The unblocked strategy requires that: ! LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ), ! where MN = min( M, N ). ! The block algorithm requires that: ! LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ), ! where NB is an upper bound on the blocksize returned ! by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR, ! and DORMRZ. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: If INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! Based on contributions by ! A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA ! E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain ! G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain ! ! ===================================================================== ! ! .. Parameters .. INTEGER IMAX, IMIN PARAMETER ( IMAX = 1, IMIN = 2 ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, LWKOPT, MN, & NB, NB1, NB2, NB3, NB4 DOUBLE PRECISION ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX, & SMAXPR, SMIN, SMINPR, SMLNUM, WSIZE ! .. ! .. External Functions .. INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL ILAENV, DLAMCH, DLANGE ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEQP3, DLABAD, DLAIC1, DLASCL, DLASET, & DORMQR, DORMRZ, DTRSM, DTZRZF, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX, MIN ! .. ! .. Executable Statements .. ! MN = MIN( M, N ) ISMIN = MN + 1 ISMAX = 2*MN + 1 ! ! Test the input arguments. ! INFO = 0 NB1 = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) NB2 = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 ) NB3 = ILAENV( 1, 'DORMQR', ' ', M, N, NRHS, -1 ) NB4 = ILAENV( 1, 'DORMRQ', ' ', M, N, NRHS, -1 ) NB = MAX( NB1, NB2, NB3, NB4 ) LWKOPT = MAX( 1, MN+2*N+NB*( N+1 ), 2*MN+NB*NRHS ) WORK( 1 ) = DBLE( LWKOPT ) LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN INFO = -7 ELSE IF( LWORK.LT.MAX( 1, MN+3*N+1, 2*MN+NRHS ) .AND. .NOT. & LQUERY ) THEN INFO = -12 END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGELSY', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( MIN( M, N, NRHS ).EQ.0 ) THEN RANK = 0 RETURN END IF ! ! Get machine parameters ! SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) ! ! Scale A, B if max entries outside range [SMLNUM,BIGNUM] ! ANRM = DLANGE( 'M', M, N, A, LDA, WORK ) IASCL = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN ! ! Scale matrix norm up to SMLNUM ! CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO ) IASCL = 1 ELSE IF( ANRM.GT.BIGNUM ) THEN ! ! Scale matrix norm down to BIGNUM ! CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO ) IASCL = 2 ELSE IF( ANRM.EQ.ZERO ) THEN ! ! Matrix all zero. Return zero solution. ! CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) RANK = 0 GO TO 70 END IF ! BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK ) IBSCL = 0 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN ! ! Scale matrix norm up to SMLNUM ! CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO ) IBSCL = 1 ELSE IF( BNRM.GT.BIGNUM ) THEN ! ! Scale matrix norm down to BIGNUM ! CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO ) IBSCL = 2 END IF ! ! Compute QR factorization with column pivoting of A: ! A * P = Q * R ! CALL DGEQP3( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ), & LWORK-MN, INFO ) WSIZE = MN + WORK( MN+1 ) ! ! workspace: MN+2*N+NB*(N+1). ! Details of Householder rotations stored in WORK(1:MN). ! ! Determine RANK using incremental condition estimation ! WORK( ISMIN ) = ONE WORK( ISMAX ) = ONE SMAX = ABS( A( 1, 1 ) ) SMIN = SMAX IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN RANK = 0 CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) GO TO 70 ELSE RANK = 1 END IF ! 10 CONTINUE IF( RANK.LT.MN ) THEN I = RANK + 1 CALL DLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ), & A( I, I ), SMINPR, S1, C1 ) CALL DLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ), & A( I, I ), SMAXPR, S2, C2 ) ! IF( SMAXPR*RCOND.LE.SMINPR ) THEN DO 20 I = 1, RANK WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 ) WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 ) 20 CONTINUE WORK( ISMIN+RANK ) = C1 WORK( ISMAX+RANK ) = C2 SMIN = SMINPR SMAX = SMAXPR RANK = RANK + 1 GO TO 10 END IF END IF ! ! workspace: 3*MN. ! ! Logically partition R = [ R11 R12 ] ! [ 0 R22 ] ! where R11 = R(1:RANK,1:RANK) ! ! [R11,R12] = [ T11, 0 ] * Y ! IF( RANK.LT.N ) & CALL DTZRZF( RANK, N, A, LDA, WORK( MN+1 ), WORK( 2*MN+1 ), & LWORK-2*MN, INFO ) ! ! workspace: 2*MN. ! Details of Householder rotations stored in WORK(MN+1:2*MN) ! ! B(1:M,1:NRHS) := Q' * B(1:M,1:NRHS) ! CALL DORMQR( 'Left', 'Transpose', M, NRHS, MN, A, LDA, WORK( 1 ), & B, LDB, WORK( 2*MN+1 ), LWORK-2*MN, INFO ) WSIZE = MAX( WSIZE, 2*MN+WORK( 2*MN+1 ) ) ! ! workspace: 2*MN+NB*NRHS. ! ! B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS) ! CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK, & NRHS, ONE, A, LDA, B, LDB ) ! DO 40 J = 1, NRHS DO 30 I = RANK + 1, N B( I, J ) = ZERO 30 CONTINUE 40 CONTINUE ! ! B(1:N,1:NRHS) := Y' * B(1:N,1:NRHS) ! IF( RANK.LT.N ) THEN CALL DORMRZ( 'Left', 'Transpose', N, NRHS, RANK, N-RANK, A, & LDA, WORK( MN+1 ), B, LDB, WORK( 2*MN+1 ), & LWORK-2*MN, INFO ) END IF ! ! workspace: 2*MN+NRHS. ! ! B(1:N,1:NRHS) := P * B(1:N,1:NRHS) ! DO 60 J = 1, NRHS DO 50 I = 1, N WORK( JPVT( I ) ) = B( I, J ) 50 CONTINUE CALL DCOPY( N, WORK( 1 ), 1, B( 1, J ), 1 ) 60 CONTINUE ! ! workspace: N. ! ! Undo scaling ! IF( IASCL.EQ.1 ) THEN CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO ) CALL DLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA, & INFO ) ELSE IF( IASCL.EQ.2 ) THEN CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO ) CALL DLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA, & INFO ) END IF IF( IBSCL.EQ.1 ) THEN CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO ) ELSE IF( IBSCL.EQ.2 ) THEN CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO ) END IF ! 70 CONTINUE WORK( 1 ) = DBLE( LWKOPT ) ! RETURN ! ! End of DGELSY ! END SUBROUTINE DGEQL2( M, N, A, LDA, TAU, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGEQL2 computes a QL factorization of a real m by n matrix A: ! A = Q * L. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the m by n matrix A. ! On exit, if m >= n, the lower triangle of the subarray ! A(m-n+1:m,1:n) contains the n by n lower triangular matrix L; ! if m <= n, the elements on and below the (n-m)-th ! superdiagonal contain the m by n lower trapezoidal matrix L; ! the remaining elements, with the array TAU, represent the ! orthogonal matrix Q as a product of elementary reflectors ! (see Further Details). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! TAU (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The scalar factors of the elementary reflectors (see Further ! Details). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of elementary reflectors ! ! Q = H(k) . . . H(2) H(1), where k = min(m,n). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in ! A(1:m-k+i-1,n-k+i), and tau in TAU(i). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, K DOUBLE PRECISION AII ! .. ! .. External Subroutines .. EXTERNAL DLARF, DLARFG, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEQL2', -INFO ) RETURN END IF ! K = MIN( M, N ) ! DO 10 I = K, 1, -1 ! ! Generate elementary reflector H(i) to annihilate ! A(1:m-k+i-1,n-k+i) ! CALL DLARFG( M-K+I, A( M-K+I, N-K+I ), A( 1, N-K+I ), 1, & TAU( I ) ) ! ! Apply H(i) to A(1:m-k+i,1:n-k+i-1) from the left ! AII = A( M-K+I, N-K+I ) A( M-K+I, N-K+I ) = ONE CALL DLARF( 'Left', M-K+I, N-K+I-1, A( 1, N-K+I ), 1, TAU( I ), & A, LDA, WORK ) A( M-K+I, N-K+I ) = AII 10 CONTINUE RETURN ! ! End of DGEQL2 ! END SUBROUTINE DGEQLF( M, N, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGEQLF computes a QL factorization of a real M-by-N matrix A: ! A = Q * L. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, ! if m >= n, the lower triangle of the subarray ! A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L; ! if m <= n, the elements on and below the (n-m)-th ! superdiagonal contain the M-by-N lower trapezoidal matrix L; ! the remaining elements, with the array TAU, represent the ! orthogonal matrix Q as a product of elementary reflectors ! (see Further Details). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! TAU (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The scalar factors of the elementary reflectors (see Further ! Details). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,N). ! For optimum performance LWORK >= N*NB, where NB is the ! optimal blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of elementary reflectors ! ! Q = H(k) . . . H(2) H(1), where k = min(m,n). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in ! A(1:m-k+i-1,n-k+i), and tau in TAU(i). ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT, & MU, NB, NBMIN, NU, NX ! .. ! .. External Subroutines .. EXTERNAL DGEQL2, DLARFB, DLARFT, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 NB = ILAENV( 1, 'DGEQLF', ' ', M, N, -1, -1 ) LWKOPT = N*NB WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEQLF', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! K = MIN( M, N ) IF( K.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! NBMIN = 2 NX = 1 IWS = N IF( NB.GT.1 .AND. NB.LT.K ) THEN ! ! Determine when to cross over from blocked to unblocked code. ! NX = MAX( 0, ILAENV( 3, 'DGEQLF', ' ', M, N, -1, -1 ) ) IF( NX.LT.K ) THEN ! ! Determine if workspace is large enough for blocked code. ! LDWORK = N IWS = LDWORK*NB IF( LWORK.LT.IWS ) THEN ! ! Not enough workspace to use optimal NB: reduce NB and ! determine the minimum value of NB. ! NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'DGEQLF', ' ', M, N, -1, & -1 ) ) END IF END IF END IF ! IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN ! ! Use blocked code initially. ! The last kk columns are handled by the block method. ! KI = ( ( K-NX-1 ) / NB )*NB KK = MIN( K, KI+NB ) ! DO 10 I = K - KK + KI + 1, K - KK + 1, -NB IB = MIN( K-I+1, NB ) ! ! Compute the QL factorization of the current block ! A(1:m-k+i+ib-1,n-k+i:n-k+i+ib-1) ! CALL DGEQL2( M-K+I+IB-1, IB, A( 1, N-K+I ), LDA, TAU( I ), & WORK, IINFO ) IF( N-K+I.GT.1 ) THEN ! ! Form the triangular factor of the block reflector ! H = H(i+ib-1) . . . H(i+1) H(i) ! CALL DLARFT( 'Backward', 'Columnwise', M-K+I+IB-1, IB, & A( 1, N-K+I ), LDA, TAU( I ), WORK, LDWORK ) ! ! Apply H' to A(1:m-k+i+ib-1,1:n-k+i-1) from the left ! CALL DLARFB( 'Left', 'Transpose', 'Backward', & 'Columnwise', M-K+I+IB-1, N-K+I-1, IB, & A( 1, N-K+I ), LDA, WORK, LDWORK, A, LDA, & WORK( IB+1 ), LDWORK ) END IF 10 CONTINUE MU = M - K + I + NB - 1 NU = N - K + I + NB - 1 ELSE MU = M NU = N END IF ! ! Use unblocked code to factor the last or only block ! IF( MU.GT.0 .AND. NU.GT.0 ) & CALL DGEQL2( MU, NU, A, LDA, TAU, WORK, IINFO ) ! WORK( 1 ) = IWS RETURN ! ! End of DGEQLF ! END SUBROUTINE DGEQP3( M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LWORK, M, N ! .. ! .. Array Arguments .. INTEGER JPVT( * ) DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGEQP3 computes a QR factorization with column pivoting of a ! matrix A: A*P = Q*R using Level 3 BLAS. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, the upper triangle of the array contains the ! min(M,N)-by-N upper trapezoidal matrix R; the elements below ! the diagonal, together with the array TAU, represent the ! orthogonal matrix Q as a product of min(M,N) elementary ! reflectors. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! JPVT (input/output) INTEGER array, dimension (N) ! On entry, if JPVT(J).ne.0, the J-th column of A is permuted ! to the front of A*P (a leading column); if JPVT(J)=0, ! the J-th column of A is a free column. ! On exit, if JPVT(J)=K, then the J-th column of A*P was the ! the K-th column of A. ! ! TAU (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The scalar factors of the elementary reflectors. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO=0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= 3*N+1. ! For optimal performance LWORK >= 2*N+( N+1 )*NB, where NB ! is the optimal blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of elementary reflectors ! ! Q = H(1) H(2) . . . H(k), where k = min(m,n). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real/complex scalar, and v is a real/complex vector ! with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in ! A(i+1:m,i), and tau in TAU(i). ! ! Based on contributions by ! G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain ! X. Sun, Computer Science Dept., Duke University, USA ! ! ===================================================================== ! ! .. Parameters .. INTEGER INB, INBMIN, IXOVER PARAMETER ( INB = 1, INBMIN = 2, IXOVER = 3 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER FJB, IWS, J, JB, LWKOPT, MINMN, MINWS, NA, NB, & NBMIN, NFXD, NX, SM, SMINMN, SN, TOPBMN ! .. ! .. External Subroutines .. EXTERNAL DGEQRF, DLAQP2, DLAQPS, DORMQR, DSWAP, XERBLA ! .. ! .. External Functions .. INTEGER ILAENV DOUBLE PRECISION DNRM2 EXTERNAL ILAENV, DNRM2 ! .. ! .. Intrinsic Functions .. INTRINSIC INT, MAX, MIN ! .. ! .. Executable Statements .. ! IWS = 3*N + 1 MINMN = MIN( M, N ) ! ! Test input arguments ! ==================== ! INFO = 0 NB = ILAENV( INB, 'DGEQRF', ' ', M, N, -1, -1 ) LWKOPT = 2*N + ( N+1 )*NB WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 ELSE IF( ( LWORK.LT.IWS ) .AND. .NOT.LQUERY ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEQP3', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible. ! IF( MINMN.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! ! Move initial columns up front. ! NFXD = 1 DO 10 J = 1, N IF( JPVT( J ).NE.0 ) THEN IF( J.NE.NFXD ) THEN CALL DSWAP( M, A( 1, J ), 1, A( 1, NFXD ), 1 ) JPVT( J ) = JPVT( NFXD ) JPVT( NFXD ) = J ELSE JPVT( J ) = J END IF NFXD = NFXD + 1 ELSE JPVT( J ) = J END IF 10 CONTINUE NFXD = NFXD - 1 ! ! Factorize fixed columns ! ======================= ! ! Compute the QR factorization of fixed columns and update ! remaining columns. ! IF( NFXD.GT.0 ) THEN NA = MIN( M, NFXD ) !CC CALL DGEQR2( M, NA, A, LDA, TAU, WORK, INFO ) CALL DGEQRF( M, NA, A, LDA, TAU, WORK, LWORK, INFO ) IWS = MAX( IWS, INT( WORK( 1 ) ) ) IF( NA.LT.N ) THEN !CC CALL DORM2R( 'Left', 'Transpose', M, N-NA, NA, A, LDA, !CC $ TAU, A( 1, NA+1 ), LDA, WORK, INFO ) CALL DORMQR( 'Left', 'Transpose', M, N-NA, NA, A, LDA, TAU, & A( 1, NA+1 ), LDA, WORK, LWORK, INFO ) IWS = MAX( IWS, INT( WORK( 1 ) ) ) END IF END IF ! ! Factorize free columns ! ====================== ! IF( NFXD.LT.MINMN ) THEN ! SM = M - NFXD SN = N - NFXD SMINMN = MINMN - NFXD ! ! Determine the block size. ! NB = ILAENV( INB, 'DGEQRF', ' ', SM, SN, -1, -1 ) NBMIN = 2 NX = 0 ! IF( ( NB.GT.1 ) .AND. ( NB.LT.SMINMN ) ) THEN ! ! Determine when to cross over from blocked to unblocked code. ! NX = MAX( 0, ILAENV( IXOVER, 'DGEQRF', ' ', SM, SN, -1, & -1 ) ) ! ! IF( NX.LT.SMINMN ) THEN ! ! Determine if workspace is large enough for blocked code. ! MINWS = 2*SN + ( SN+1 )*NB IWS = MAX( IWS, MINWS ) IF( LWORK.LT.MINWS ) THEN ! ! Not enough workspace to use optimal NB: Reduce NB and ! determine the minimum value of NB. ! NB = ( LWORK-2*SN ) / ( SN+1 ) NBMIN = MAX( 2, ILAENV( INBMIN, 'DGEQRF', ' ', SM, SN, & -1, -1 ) ) ! ! END IF END IF END IF ! ! Initialize partial column norms. The first N elements of work ! store the exact column norms. ! DO 20 J = NFXD + 1, N WORK( J ) = DNRM2( SM, A( NFXD+1, J ), 1 ) WORK( N+J ) = WORK( J ) 20 CONTINUE ! IF( ( NB.GE.NBMIN ) .AND. ( NB.LT.SMINMN ) .AND. & ( NX.LT.SMINMN ) ) THEN ! ! Use blocked code initially. ! J = NFXD + 1 ! ! Compute factorization: while loop. ! ! TOPBMN = MINMN - NX 30 CONTINUE IF( J.LE.TOPBMN ) THEN JB = MIN( NB, TOPBMN-J+1 ) ! ! Factorize JB columns among columns J:N. ! CALL DLAQPS( M, N-J+1, J-1, JB, FJB, A( 1, J ), LDA, & JPVT( J ), TAU( J ), WORK( J ), WORK( N+J ), & WORK( 2*N+1 ), WORK( 2*N+JB+1 ), N-J+1 ) ! J = J + FJB GO TO 30 END IF ELSE J = NFXD + 1 END IF ! ! Use unblocked code to factor the last or only block. ! ! IF( J.LE.MINMN ) & CALL DLAQP2( M, N-J+1, J-1, A( 1, J ), LDA, JPVT( J ), & TAU( J ), WORK( J ), WORK( N+J ), & WORK( 2*N+1 ) ) ! END IF ! WORK( 1 ) = IWS RETURN ! ! End of DGEQP3 ! END SUBROUTINE DGEQPF( M, N, A, LDA, JPVT, TAU, WORK, INFO ) ! ! -- LAPACK test routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, M, N ! .. ! .. Array Arguments .. INTEGER JPVT( * ) DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! This routine is deprecated and has been replaced by routine DGEQP3. ! ! DGEQPF computes a QR factorization with column pivoting of a ! real M-by-N matrix A: A*P = Q*R. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0 ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, the upper triangle of the array contains the ! min(M,N)-by-N upper triangular matrix R; the elements ! below the diagonal, together with the array TAU, ! represent the orthogonal matrix Q as a product of ! min(m,n) elementary reflectors. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! JPVT (input/output) INTEGER array, dimension (N) ! On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted ! to the front of A*P (a leading column); if JPVT(i) = 0, ! the i-th column of A is a free column. ! On exit, if JPVT(i) = k, then the i-th column of A*P ! was the k-th column of A. ! ! TAU (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The scalar factors of the elementary reflectors. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of elementary reflectors ! ! Q = H(1) H(2) . . . H(n) ! ! Each H(i) has the form ! ! H = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i). ! ! The matrix P is represented in jpvt as follows: If ! jpvt(j) = i ! then the jth column of P is the ith canonical unit vector. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, ITEMP, J, MA, MN, PVT DOUBLE PRECISION AII, TEMP, TEMP2 ! .. ! .. External Subroutines .. EXTERNAL DGEQR2, DLARF, DLARFG, DORM2R, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT ! .. ! .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DNRM2 EXTERNAL IDAMAX, DNRM2 ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEQPF', -INFO ) RETURN END IF ! MN = MIN( M, N ) ! ! Move initial columns up front ! ITEMP = 1 DO 10 I = 1, N IF( JPVT( I ).NE.0 ) THEN IF( I.NE.ITEMP ) THEN CALL DSWAP( M, A( 1, I ), 1, A( 1, ITEMP ), 1 ) JPVT( I ) = JPVT( ITEMP ) JPVT( ITEMP ) = I ELSE JPVT( I ) = I END IF ITEMP = ITEMP + 1 ELSE JPVT( I ) = I END IF 10 CONTINUE ITEMP = ITEMP - 1 ! ! Compute the QR factorization and update remaining columns ! IF( ITEMP.GT.0 ) THEN MA = MIN( ITEMP, M ) CALL DGEQR2( M, MA, A, LDA, TAU, WORK, INFO ) IF( MA.LT.N ) THEN CALL DORM2R( 'Left', 'Transpose', M, N-MA, MA, A, LDA, TAU, & A( 1, MA+1 ), LDA, WORK, INFO ) END IF END IF ! IF( ITEMP.LT.MN ) THEN ! ! Initialize partial column norms. The first n elements of ! work store the exact column norms. ! DO 20 I = ITEMP + 1, N WORK( I ) = DNRM2( M-ITEMP, A( ITEMP+1, I ), 1 ) WORK( N+I ) = WORK( I ) 20 CONTINUE ! ! Compute factorization ! DO 40 I = ITEMP + 1, MN ! ! Determine ith pivot column and swap if necessary ! PVT = ( I-1 ) + IDAMAX( N-I+1, WORK( I ), 1 ) ! IF( PVT.NE.I ) THEN CALL DSWAP( M, A( 1, PVT ), 1, A( 1, I ), 1 ) ITEMP = JPVT( PVT ) JPVT( PVT ) = JPVT( I ) JPVT( I ) = ITEMP WORK( PVT ) = WORK( I ) WORK( N+PVT ) = WORK( N+I ) END IF ! ! Generate elementary reflector H(i) ! IF( I.LT.M ) THEN CALL DLARFG( M-I+1, A( I, I ), A( I+1, I ), 1, TAU( I ) ) ELSE CALL DLARFG( 1, A( M, M ), A( M, M ), 1, TAU( M ) ) END IF ! IF( I.LT.N ) THEN ! ! Apply H(i) to A(i:m,i+1:n) from the left ! AII = A( I, I ) A( I, I ) = ONE CALL DLARF( 'LEFT', M-I+1, N-I, A( I, I ), 1, TAU( I ), & A( I, I+1 ), LDA, WORK( 2*N+1 ) ) A( I, I ) = AII END IF ! ! Update partial column norms ! DO 30 J = I + 1, N IF( WORK( J ).NE.ZERO ) THEN TEMP = ONE - ( ABS( A( I, J ) ) / WORK( J ) )**2 TEMP = MAX( TEMP, ZERO ) TEMP2 = ONE + 0.05D0*TEMP* & ( WORK( J ) / WORK( N+J ) )**2 IF( TEMP2.EQ.ONE ) THEN IF( M-I.GT.0 ) THEN WORK( J ) = DNRM2( M-I, A( I+1, J ), 1 ) WORK( N+J ) = WORK( J ) ELSE WORK( J ) = ZERO WORK( N+J ) = ZERO END IF ELSE WORK( J ) = WORK( J )*SQRT( TEMP ) END IF END IF 30 CONTINUE ! 40 CONTINUE END IF RETURN ! ! End of DGEQPF ! END SUBROUTINE DGEQR2( M, N, A, LDA, TAU, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGEQR2 computes a QR factorization of a real m by n matrix A: ! A = Q * R. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the m by n matrix A. ! On exit, the elements on and above the diagonal of the array ! contain the min(m,n) by n upper trapezoidal matrix R (R is ! upper triangular if m >= n); the elements below the diagonal, ! with the array TAU, represent the orthogonal matrix Q as a ! product of elementary reflectors (see Further Details). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! TAU (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The scalar factors of the elementary reflectors (see Further ! Details). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of elementary reflectors ! ! Q = H(1) H(2) . . . H(k), where k = min(m,n). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), ! and tau in TAU(i). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, K DOUBLE PRECISION AII ! .. ! .. External Subroutines .. EXTERNAL DLARF, DLARFG, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEQR2', -INFO ) RETURN END IF ! K = MIN( M, N ) ! DO 10 I = 1, K ! ! Generate elementary reflector H(i) to annihilate A(i+1:m,i) ! CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1, & TAU( I ) ) IF( I.LT.N ) THEN ! ! Apply H(i) to A(i:m,i+1:n) from the left ! AII = A( I, I ) A( I, I ) = ONE CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAU( I ), & A( I, I+1 ), LDA, WORK ) A( I, I ) = AII END IF 10 CONTINUE RETURN ! ! End of DGEQR2 ! END SUBROUTINE DGEQRF( M, N, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGEQRF computes a QR factorization of a real M-by-N matrix A: ! A = Q * R. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, the elements on and above the diagonal of the array ! contain the min(M,N)-by-N upper trapezoidal matrix R (R is ! upper triangular if m >= n); the elements below the diagonal, ! with the array TAU, represent the orthogonal matrix Q as a ! product of min(m,n) elementary reflectors (see Further ! Details). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! TAU (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The scalar factors of the elementary reflectors (see Further ! Details). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,N). ! For optimum performance LWORK >= N*NB, where NB is ! the optimal blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of elementary reflectors ! ! Q = H(1) H(2) . . . H(k), where k = min(m,n). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), ! and tau in TAU(i). ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB, & NBMIN, NX ! .. ! .. External Subroutines .. EXTERNAL DGEQR2, DLARFB, DLARFT, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) LWKOPT = N*NB WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGEQRF', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! K = MIN( M, N ) IF( K.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! NBMIN = 2 NX = 0 IWS = N IF( NB.GT.1 .AND. NB.LT.K ) THEN ! ! Determine when to cross over from blocked to unblocked code. ! NX = MAX( 0, ILAENV( 3, 'DGEQRF', ' ', M, N, -1, -1 ) ) IF( NX.LT.K ) THEN ! ! Determine if workspace is large enough for blocked code. ! LDWORK = N IWS = LDWORK*NB IF( LWORK.LT.IWS ) THEN ! ! Not enough workspace to use optimal NB: reduce NB and ! determine the minimum value of NB. ! NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'DGEQRF', ' ', M, N, -1, & -1 ) ) END IF END IF END IF ! IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN ! ! Use blocked code initially ! DO 10 I = 1, K - NX, NB IB = MIN( K-I+1, NB ) ! ! Compute the QR factorization of the current block ! A(i:m,i:i+ib-1) ! CALL DGEQR2( M-I+1, IB, A( I, I ), LDA, TAU( I ), WORK, & IINFO ) IF( I+IB.LE.N ) THEN ! ! Form the triangular factor of the block reflector ! H = H(i) H(i+1) . . . H(i+ib-1) ! CALL DLARFT( 'Forward', 'Columnwise', M-I+1, IB, & A( I, I ), LDA, TAU( I ), WORK, LDWORK ) ! ! Apply H' to A(i:m,i+ib:n) from the left ! CALL DLARFB( 'Left', 'Transpose', 'Forward', & 'Columnwise', M-I+1, N-I-IB+1, IB, & A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ), & LDA, WORK( IB+1 ), LDWORK ) END IF 10 CONTINUE ELSE I = 1 END IF ! ! Use unblocked code to factor the last or only block. ! IF( I.LE.K ) & CALL DGEQR2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK, & IINFO ) ! WORK( 1 ) = IWS RETURN ! ! End of DGEQRF ! END SUBROUTINE DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, & X, LDX, FERR, BERR, WORK, IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER TRANS INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS ! .. ! .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), & BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DGERFS improves the computed solution to a system of linear ! equations and provides error bounds and backward error estimates for ! the solution. ! ! Arguments ! ========= ! ! TRANS (input) CHARACTER*1 ! Specifies the form of the system of equations: ! = 'N': A * X = B (No transpose) ! = 'T': A**T * X = B (Transpose) ! = 'C': A**H * X = B (Conjugate transpose = Transpose) ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The original N-by-N matrix A. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! AF (input) DOUBLE PRECISION array, dimension (LDAF,N) ! The factors L and U from the factorization A = P*L*U ! as computed by DGETRF. ! ! LDAF (input) INTEGER ! The leading dimension of the array AF. LDAF >= max(1,N). ! ! IPIV (input) INTEGER array, dimension (N) ! The pivot indices from DGETRF; for 1<=i<=N, row i of the ! matrix was interchanged with row IPIV(i). ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) ! The right hand side matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! On entry, the solution matrix X, as computed by DGETRS. ! On exit, the improved solution matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Internal Parameters ! =================== ! ! ITMAX is the maximum number of steps of iterative refinement. ! ! ===================================================================== ! ! .. Parameters .. INTEGER ITMAX PARAMETER ( ITMAX = 5 ) DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D+0 ) DOUBLE PRECISION THREE PARAMETER ( THREE = 3.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOTRAN CHARACTER TRANST INTEGER COUNT, I, J, K, KASE, NZ DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DGEMV, DGETRS, DLACON, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 NOTRAN = LSAME( TRANS, 'N' ) IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. & LSAME( TRANS, 'C' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -10 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -12 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGERFS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN DO 10 J = 1, NRHS FERR( J ) = ZERO BERR( J ) = ZERO 10 CONTINUE RETURN END IF ! IF( NOTRAN ) THEN TRANST = 'T' ELSE TRANST = 'N' END IF ! ! NZ = maximum number of nonzero elements in each row of A, plus 1 ! NZ = N + 1 EPS = DLAMCH( 'Epsilon' ) SAFMIN = DLAMCH( 'Safe minimum' ) SAFE1 = NZ*SAFMIN SAFE2 = SAFE1 / EPS ! ! Do for each right hand side ! DO 140 J = 1, NRHS ! COUNT = 1 LSTRES = THREE 20 CONTINUE ! ! Loop until stopping criterion is satisfied. ! ! Compute residual R = B - op(A) * X, ! where op(A) = A, A**T, or A**H, depending on TRANS. ! CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) CALL DGEMV( TRANS, N, N, -ONE, A, LDA, X( 1, J ), 1, ONE, & WORK( N+1 ), 1 ) ! ! Compute componentwise relative backward error from formula ! ! max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) ! ! where abs(Z) is the componentwise absolute value of the matrix ! or vector Z. If the i-th component of the denominator is less ! than SAFE2, then SAFE1 is added to the i-th components of the ! numerator and denominator before dividing. ! DO 30 I = 1, N WORK( I ) = ABS( B( I, J ) ) 30 CONTINUE ! ! Compute abs(op(A))*abs(X) + abs(B). ! IF( NOTRAN ) THEN DO 50 K = 1, N XK = ABS( X( K, J ) ) DO 40 I = 1, N WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 40 CONTINUE 50 CONTINUE ELSE DO 70 K = 1, N S = ZERO DO 60 I = 1, N S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 60 CONTINUE WORK( K ) = WORK( K ) + S 70 CONTINUE END IF S = ZERO DO 80 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) ELSE S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / & ( WORK( I )+SAFE1 ) ) END IF 80 CONTINUE BERR( J ) = S ! ! Test stopping criterion. Continue iterating if ! 1) The residual BERR(J) is larger than machine epsilon, and ! 2) BERR(J) decreased by at least a factor of 2 during the ! last iteration, and ! 3) At most ITMAX iterations tried. ! IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. & COUNT.LE.ITMAX ) THEN ! ! Update solution and try again. ! CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N, & INFO ) CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) LSTRES = BERR( J ) COUNT = COUNT + 1 GO TO 20 END IF ! ! Bound error from formula ! ! norm(X - XTRUE) / norm(X) .le. FERR = ! norm( abs(inv(op(A)))* ! ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) ! ! where ! norm(Z) is the magnitude of the largest component of Z ! inv(op(A)) is the inverse of op(A) ! abs(Z) is the componentwise absolute value of the matrix or ! vector Z ! NZ is the maximum number of nonzeros in any row of A, plus 1 ! EPS is machine epsilon ! ! The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) ! is incremented by SAFE1 if the i-th component of ! abs(op(A))*abs(X) + abs(B) is less than SAFE2. ! ! Use DLACON to estimate the infinity-norm of the matrix ! inv(op(A)) * diag(W), ! where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) ! DO 90 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) ELSE WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 END IF 90 CONTINUE ! KASE = 0 100 CONTINUE CALL DLACON( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), & KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN ! ! Multiply by diag(W)*inv(op(A)**T). ! CALL DGETRS( TRANST, N, 1, AF, LDAF, IPIV, WORK( N+1 ), & N, INFO ) DO 110 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 110 CONTINUE ELSE ! ! Multiply by inv(op(A))*diag(W). ! DO 120 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 120 CONTINUE CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N, & INFO ) END IF GO TO 100 END IF ! ! Normalize error. ! LSTRES = ZERO DO 130 I = 1, N LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 130 CONTINUE IF( LSTRES.NE.ZERO ) & FERR( J ) = FERR( J ) / LSTRES ! 140 CONTINUE ! RETURN ! ! End of DGERFS ! END SUBROUTINE DGERQ2( M, N, A, LDA, TAU, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGERQ2 computes an RQ factorization of a real m by n matrix A: ! A = R * Q. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the m by n matrix A. ! On exit, if m <= n, the upper triangle of the subarray ! A(1:m,n-m+1:n) contains the m by m upper triangular matrix R; ! if m >= n, the elements on and above the (m-n)-th subdiagonal ! contain the m by n upper trapezoidal matrix R; the remaining ! elements, with the array TAU, represent the orthogonal matrix ! Q as a product of elementary reflectors (see Further ! Details). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! TAU (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The scalar factors of the elementary reflectors (see Further ! Details). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (M) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of elementary reflectors ! ! Q = H(1) H(2) . . . H(k), where k = min(m,n). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in ! A(m-k+i,1:n-k+i-1), and tau in TAU(i). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, K DOUBLE PRECISION AII ! .. ! .. External Subroutines .. EXTERNAL DLARF, DLARFG, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGERQ2', -INFO ) RETURN END IF ! K = MIN( M, N ) ! DO 10 I = K, 1, -1 ! ! Generate elementary reflector H(i) to annihilate ! A(m-k+i,1:n-k+i-1) ! CALL DLARFG( N-K+I, A( M-K+I, N-K+I ), A( M-K+I, 1 ), LDA, & TAU( I ) ) ! ! Apply H(i) to A(1:m-k+i-1,1:n-k+i) from the right ! AII = A( M-K+I, N-K+I ) A( M-K+I, N-K+I ) = ONE CALL DLARF( 'Right', M-K+I-1, N-K+I, A( M-K+I, 1 ), LDA, & TAU( I ), A, LDA, WORK ) A( M-K+I, N-K+I ) = AII 10 CONTINUE RETURN ! ! End of DGERQ2 ! END SUBROUTINE DGERQF( M, N, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGERQF computes an RQ factorization of a real M-by-N matrix A: ! A = R * Q. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, ! if m <= n, the upper triangle of the subarray ! A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R; ! if m >= n, the elements on and above the (m-n)-th subdiagonal ! contain the M-by-N upper trapezoidal matrix R; ! the remaining elements, with the array TAU, represent the ! orthogonal matrix Q as a product of min(m,n) elementary ! reflectors (see Further Details). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! TAU (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The scalar factors of the elementary reflectors (see Further ! Details). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,M). ! For optimum performance LWORK >= M*NB, where NB is ! the optimal blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of elementary reflectors ! ! Q = H(1) H(2) . . . H(k), where k = min(m,n). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in ! A(m-k+i,1:n-k+i-1), and tau in TAU(i). ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT, & MU, NB, NBMIN, NU, NX ! .. ! .. External Subroutines .. EXTERNAL DGERQ2, DLARFB, DLARFT, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 NB = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 ) LWKOPT = M*NB WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 ELSE IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGERQF', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! K = MIN( M, N ) IF( K.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! NBMIN = 2 NX = 1 IWS = M IF( NB.GT.1 .AND. NB.LT.K ) THEN ! ! Determine when to cross over from blocked to unblocked code. ! NX = MAX( 0, ILAENV( 3, 'DGERQF', ' ', M, N, -1, -1 ) ) IF( NX.LT.K ) THEN ! ! Determine if workspace is large enough for blocked code. ! LDWORK = M IWS = LDWORK*NB IF( LWORK.LT.IWS ) THEN ! ! Not enough workspace to use optimal NB: reduce NB and ! determine the minimum value of NB. ! NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'DGERQF', ' ', M, N, -1, & -1 ) ) END IF END IF END IF ! IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN ! ! Use blocked code initially. ! The last kk rows are handled by the block method. ! KI = ( ( K-NX-1 ) / NB )*NB KK = MIN( K, KI+NB ) ! DO 10 I = K - KK + KI + 1, K - KK + 1, -NB IB = MIN( K-I+1, NB ) ! ! Compute the RQ factorization of the current block ! A(m-k+i:m-k+i+ib-1,1:n-k+i+ib-1) ! CALL DGERQ2( IB, N-K+I+IB-1, A( M-K+I, 1 ), LDA, TAU( I ), & WORK, IINFO ) IF( M-K+I.GT.1 ) THEN ! ! Form the triangular factor of the block reflector ! H = H(i+ib-1) . . . H(i+1) H(i) ! CALL DLARFT( 'Backward', 'Rowwise', N-K+I+IB-1, IB, & A( M-K+I, 1 ), LDA, TAU( I ), WORK, LDWORK ) ! ! Apply H to A(1:m-k+i-1,1:n-k+i+ib-1) from the right ! CALL DLARFB( 'Right', 'No transpose', 'Backward', & 'Rowwise', M-K+I-1, N-K+I+IB-1, IB, & A( M-K+I, 1 ), LDA, WORK, LDWORK, A, LDA, & WORK( IB+1 ), LDWORK ) END IF 10 CONTINUE MU = M - K + I + NB - 1 NU = N - K + I + NB - 1 ELSE MU = M NU = N END IF ! ! Use unblocked code to factor the last or only block ! IF( MU.GT.0 .AND. NU.GT.0 ) & CALL DGERQ2( MU, NU, A, LDA, TAU, WORK, IINFO ) ! WORK( 1 ) = IWS RETURN ! ! End of DGERQF ! END SUBROUTINE DGESC2( N, A, LDA, RHS, IPIV, JPIV, SCALE ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER LDA, N DOUBLE PRECISION SCALE ! .. ! .. Array Arguments .. INTEGER IPIV( * ), JPIV( * ) DOUBLE PRECISION A( LDA, * ), RHS( * ) ! .. ! ! Purpose ! ======= ! ! DGESC2 solves a system of linear equations ! ! A * X = scale* RHS ! ! with a general N-by-N matrix A using the LU factorization with ! complete pivoting computed by DGETC2. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix A. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the LU part of the factorization of the n-by-n ! matrix A computed by DGETC2: A = P * L * U * Q ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1, N). ! ! RHS (input/output) DOUBLE PRECISION array, dimension (N). ! On entry, the right hand side vector b. ! On exit, the solution vector X. ! ! IPIV (iput) INTEGER array, dimension (N). ! The pivot indices; for 1 <= i <= N, row i of the ! matrix has been interchanged with row IPIV(i). ! ! JPIV (iput) INTEGER array, dimension (N). ! The pivot indices; for 1 <= j <= N, column j of the ! matrix has been interchanged with column JPIV(j). ! ! SCALE (output) DOUBLE PRECISION ! On exit, SCALE contains the scale factor. SCALE is chosen ! 0 <= SCALE <= 1 to prevent owerflow in the solution. ! ! Further Details ! =============== ! ! Based on contributions by ! Bo Kagstrom and Peter Poromaa, Department of Computing Science, ! Umea University, S-901 87 Umea, Sweden. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, TWO PARAMETER ( ONE = 1.0D+0, TWO = 2.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J DOUBLE PRECISION BIGNUM, EPS, SMLNUM, TEMP ! .. ! .. External Subroutines .. EXTERNAL DLASWP, DSCAL ! .. ! .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DLAMCH EXTERNAL IDAMAX, DLAMCH ! .. ! .. Intrinsic Functions .. INTRINSIC ABS ! .. ! .. Executable Statements .. ! ! Set constant to control owerflow ! EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) / EPS BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) ! ! Apply permutations IPIV to RHS ! CALL DLASWP( 1, RHS, LDA, 1, N-1, IPIV, 1 ) ! ! Solve for L part ! DO 20 I = 1, N - 1 DO 10 J = I + 1, N RHS( J ) = RHS( J ) - A( J, I )*RHS( I ) 10 CONTINUE 20 CONTINUE ! ! Solve for U part ! SCALE = ONE ! ! Check for scaling ! I = IDAMAX( N, RHS, 1 ) IF( TWO*SMLNUM*ABS( RHS( I ) ).GT.ABS( A( N, N ) ) ) THEN TEMP = ( ONE / TWO ) / ABS( RHS( I ) ) CALL DSCAL( N, TEMP, RHS( 1 ), 1 ) SCALE = SCALE*TEMP END IF ! DO 40 I = N, 1, -1 TEMP = ONE / A( I, I ) RHS( I ) = RHS( I )*TEMP DO 30 J = I + 1, N RHS( I ) = RHS( I ) - RHS( J )*( A( I, J )*TEMP ) 30 CONTINUE 40 CONTINUE ! ! Apply permutations JPIV to the solution (RHS) ! CALL DLASWP( 1, RHS, LDA, 1, N-1, JPIV, -1 ) RETURN ! ! End of DGESC2 ! END SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, & LWORK, IWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), S( * ), U( LDU, * ), & VT( LDVT, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGESDD computes the singular value decomposition (SVD) of a real ! M-by-N matrix A, optionally computing the left and right singular ! vectors. If singular vectors are desired, it uses a ! divide-and-conquer algorithm. ! ! The SVD is written ! ! A = U * SIGMA * transpose(V) ! ! where SIGMA is an M-by-N matrix which is zero except for its ! min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and ! V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA ! are the singular values of A; they are real and non-negative, and ! are returned in descending order. The first min(m,n) columns of ! U and V are the left and right singular vectors of A. ! ! Note that the routine returns VT = V**T, not V. ! ! The divide and conquer algorithm makes very mild assumptions about ! floating point arithmetic. It will work on machines with a guard ! digit in add/subtract, or on those binary machines without guard ! digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or ! Cray-2. It could conceivably fail on hexadecimal or decimal machines ! without guard digits, but we know of none. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! Specifies options for computing all or part of the matrix U: ! = 'A': all M columns of U and all N rows of V**T are ! returned in the arrays U and VT; ! = 'S': the first min(M,N) columns of U and the first ! min(M,N) rows of V**T are returned in the arrays U ! and VT; ! = 'O': If M >= N, the first N columns of U are overwritten ! on the array A and all rows of V**T are returned in ! the array VT; ! otherwise, all columns of U are returned in the ! array U and the first M rows of V**T are overwritten ! in the array VT; ! = 'N': no columns of U or rows of V**T are computed. ! ! M (input) INTEGER ! The number of rows of the input matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the input matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, ! if JOBZ = 'O', A is overwritten with the first N columns ! of U (the left singular vectors, stored ! columnwise) if M >= N; ! A is overwritten with the first M rows ! of V**T (the right singular vectors, stored ! rowwise) otherwise. ! if JOBZ .ne. 'O', the contents of A are destroyed. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! S (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The singular values of A, sorted so that S(i) >= S(i+1). ! ! U (output) DOUBLE PRECISION array, dimension (LDU,UCOL) ! UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N; ! UCOL = min(M,N) if JOBZ = 'S'. ! If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M ! orthogonal matrix U; ! if JOBZ = 'S', U contains the first min(M,N) columns of U ! (the left singular vectors, stored columnwise); ! if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced. ! ! LDU (input) INTEGER ! The leading dimension of the array U. LDU >= 1; if ! JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M. ! ! VT (output) DOUBLE PRECISION array, dimension (LDVT,N) ! If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the ! N-by-N orthogonal matrix V**T; ! if JOBZ = 'S', VT contains the first min(M,N) rows of ! V**T (the right singular vectors, stored rowwise); ! if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced. ! ! LDVT (input) INTEGER ! The leading dimension of the array VT. LDVT >= 1; if ! JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N; ! if JOBZ = 'S', LDVT >= min(M,N). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK; ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= 1. ! If JOBZ = 'N', ! LWORK >= 3*min(M,N) + max(max(M,N),6*min(M,N)). ! If JOBZ = 'O', ! LWORK >= 3*min(M,N)*min(M,N) + ! max(max(M,N),5*min(M,N)*min(M,N)+4*min(M,N)). ! If JOBZ = 'S' or 'A' ! LWORK >= 3*min(M,N)*min(M,N) + ! max(max(M,N),4*min(M,N)*min(M,N)+4*min(M,N)). ! For good performance, LWORK should generally be larger. ! If LWORK < 0 but other input arguments are legal, WORK(1) ! returns the optimal LWORK. ! ! IWORK (workspace) INTEGER array, dimension (8*min(M,N)) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: DBDSDC did not converge, updating process failed. ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Huan Ren, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, WNTQA, WNTQAS, WNTQN, WNTQO, WNTQS INTEGER BDSPAC, BLK, CHUNK, I, IE, IERR, IL, & IR, ISCL, ITAU, ITAUP, ITAUQ, IU, IVT, LDWKVT, & LDWRKL, LDWRKR, LDWRKU, MAXWRK, MINMN, MINWRK, & MNTHR, NWORK, WRKBL DOUBLE PRECISION ANRM, BIGNUM, EPS, SMLNUM ! .. ! .. Local Arrays .. INTEGER IDUM( 1 ) DOUBLE PRECISION DUM( 1 ) ! .. ! .. External Subroutines .. EXTERNAL DBDSDC, DGEBRD, DGELQF, DGEMM, DGEQRF, DLACPY, & DLASCL, DLASET, DORGBR, DORGLQ, DORGQR, DORMBR, & XERBLA ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL DLAMCH, DLANGE, ILAENV, LSAME ! .. ! .. Intrinsic Functions .. INTRINSIC DBLE, INT, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 MINMN = MIN( M, N ) MNTHR = INT( MINMN*11.0D0 / 6.0D0 ) WNTQA = LSAME( JOBZ, 'A' ) WNTQS = LSAME( JOBZ, 'S' ) WNTQAS = WNTQA .OR. WNTQS WNTQO = LSAME( JOBZ, 'O' ) WNTQN = LSAME( JOBZ, 'N' ) MINWRK = 1 MAXWRK = 1 LQUERY = ( LWORK.EQ.-1 ) ! IF( .NOT.( WNTQA .OR. WNTQS .OR. WNTQO .OR. WNTQN ) ) THEN INFO = -1 ELSE IF( M.LT.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LDU.LT.1 .OR. ( WNTQAS .AND. LDU.LT.M ) .OR. & ( WNTQO .AND. M.LT.N .AND. LDU.LT.M ) ) THEN INFO = -8 ELSE IF( LDVT.LT.1 .OR. ( WNTQA .AND. LDVT.LT.N ) .OR. & ( WNTQS .AND. LDVT.LT.MINMN ) .OR. & ( WNTQO .AND. M.GE.N .AND. LDVT.LT.N ) ) THEN INFO = -10 END IF ! ! Compute workspace ! (Note: Comments in the code beginning "Workspace:" describe the ! minimal amount of workspace needed at that point in the code, ! as well as the preferred amount for good performance. ! NB refers to the optimal block size for the immediately ! following subroutine, as returned by ILAENV.) ! IF( INFO.EQ.0 .AND. M.GT.0 .AND. N.GT.0 ) THEN IF( M.GE.N ) THEN ! ! Compute space needed for DBDSDC ! IF( WNTQN ) THEN BDSPAC = 7*N ELSE BDSPAC = 3*N*N + 4*N END IF IF( M.GE.MNTHR ) THEN IF( WNTQN ) THEN ! ! Path 1 (M much larger than N, JOBZ='N') ! WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, & -1 ) WRKBL = MAX( WRKBL, 3*N+2*N* & ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) ) MAXWRK = MAX( WRKBL, BDSPAC+N ) MINWRK = BDSPAC + N ELSE IF( WNTQO ) THEN ! ! Path 2 (M much larger than N, JOBZ='O') ! WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M, & N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+2*N* & ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC+3*N ) MAXWRK = WRKBL + 2*N*N MINWRK = BDSPAC + 2*N*N + 3*N ELSE IF( WNTQS ) THEN ! ! Path 3 (M much larger than N, JOBZ='S') ! WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M, & N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+2*N* & ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC+3*N ) MAXWRK = WRKBL + N*N MINWRK = BDSPAC + N*N + 3*N ELSE IF( WNTQA ) THEN ! ! Path 4 (M much larger than N, JOBZ='A') ! WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'DORGQR', ' ', M, & M, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+2*N* & ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC+3*N ) MAXWRK = WRKBL + N*N MINWRK = BDSPAC + N*N + 3*N END IF ELSE ! ! Path 5 (M at least N, but not much larger) ! WRKBL = 3*N + ( M+N )*ILAENV( 1, 'DGEBRD', ' ', M, N, -1, & -1 ) IF( WNTQN ) THEN MAXWRK = MAX( WRKBL, BDSPAC+3*N ) MINWRK = 3*N + MAX( M, BDSPAC ) ELSE IF( WNTQO ) THEN WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORMBR', 'QLN', M, N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC+3*N ) MAXWRK = WRKBL + M*N MINWRK = 3*N + MAX( M, N*N+BDSPAC ) ELSE IF( WNTQS ) THEN WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORMBR', 'QLN', M, N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) ) MAXWRK = MAX( WRKBL, BDSPAC+3*N ) MINWRK = 3*N + MAX( M, BDSPAC ) ELSE IF( WNTQA ) THEN WRKBL = MAX( WRKBL, 3*N+M* & ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) ) MAXWRK = MAX( MAXWRK, BDSPAC+3*N ) MINWRK = 3*N + MAX( M, BDSPAC ) END IF END IF ELSE ! ! Compute space needed for DBDSDC ! IF( WNTQN ) THEN BDSPAC = 7*M ELSE BDSPAC = 3*M*M + 4*M END IF IF( N.GE.MNTHR ) THEN IF( WNTQN ) THEN ! ! Path 1t (N much larger than M, JOBZ='N') ! WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, & -1 ) WRKBL = MAX( WRKBL, 3*M+2*M* & ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) MAXWRK = MAX( WRKBL, BDSPAC+M ) MINWRK = BDSPAC + M ELSE IF( WNTQO ) THEN ! ! Path 2t (N much larger than M, JOBZ='O') ! WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M, & N, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+2*M* & ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORMBR', 'QLN', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORMBR', 'PRT', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC+3*M ) MAXWRK = WRKBL + 2*M*M MINWRK = BDSPAC + 2*M*M + 3*M ELSE IF( WNTQS ) THEN ! ! Path 3t (N much larger than M, JOBZ='S') ! WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M, & N, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+2*M* & ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORMBR', 'QLN', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORMBR', 'PRT', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC+3*M ) MAXWRK = WRKBL + M*M MINWRK = BDSPAC + M*M + 3*M ELSE IF( WNTQA ) THEN ! ! Path 4t (N much larger than M, JOBZ='A') ! WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'DORGLQ', ' ', N, & N, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+2*M* & ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORMBR', 'QLN', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORMBR', 'PRT', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC+3*M ) MAXWRK = WRKBL + M*M MINWRK = BDSPAC + M*M + 3*M END IF ELSE ! ! Path 5t (N greater than M, but not much larger) ! WRKBL = 3*M + ( M+N )*ILAENV( 1, 'DGEBRD', ' ', M, N, -1, & -1 ) IF( WNTQN ) THEN MAXWRK = MAX( WRKBL, BDSPAC+3*M ) MINWRK = 3*M + MAX( N, BDSPAC ) ELSE IF( WNTQO ) THEN WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) ) WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORMBR', 'PRT', M, N, M, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC+3*M ) MAXWRK = WRKBL + M*N MINWRK = 3*M + MAX( N, M*M+BDSPAC ) ELSE IF( WNTQS ) THEN WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) ) WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORMBR', 'PRT', M, N, M, -1 ) ) MAXWRK = MAX( WRKBL, BDSPAC+3*M ) MINWRK = 3*M + MAX( N, BDSPAC ) ELSE IF( WNTQA ) THEN WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) ) WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORMBR', 'PRT', N, N, M, -1 ) ) MAXWRK = MAX( WRKBL, BDSPAC+3*M ) MINWRK = 3*M + MAX( N, BDSPAC ) END IF END IF END IF WORK( 1 ) = MAXWRK END IF ! IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -12 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGESDD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 ) THEN IF( LWORK.GE.1 ) & WORK( 1 ) = ONE RETURN END IF ! ! Get machine constants ! EPS = DLAMCH( 'P' ) SMLNUM = SQRT( DLAMCH( 'S' ) ) / EPS BIGNUM = ONE / SMLNUM ! ! Scale A if max element outside range [SMLNUM,BIGNUM] ! ANRM = DLANGE( 'M', M, N, A, LDA, DUM ) ISCL = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN ISCL = 1 CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, IERR ) ELSE IF( ANRM.GT.BIGNUM ) THEN ISCL = 1 CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, IERR ) END IF ! IF( M.GE.N ) THEN ! ! A has at least as many rows as columns. If A has sufficiently ! more rows than columns, first reduce using the QR ! decomposition (if sufficient workspace available) ! IF( M.GE.MNTHR ) THEN ! IF( WNTQN ) THEN ! ! Path 1 (M much larger than N, JOBZ='N') ! No singular vectors to be computed ! ITAU = 1 NWORK = ITAU + N ! ! Compute A=Q*R ! (Workspace: need 2*N, prefer N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! ! Zero out below R ! CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA ) IE = 1 ITAUQ = IE + N ITAUP = ITAUQ + N NWORK = ITAUP + N ! ! Bidiagonalize R in A ! (Workspace: need 4*N, prefer 3*N+2*N*NB) ! CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, & IERR ) NWORK = IE + N ! ! Perform bidiagonal SVD, computing singular values only ! (Workspace: need N+BDSPAC) ! CALL DBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM, 1, & DUM, IDUM, WORK( NWORK ), IWORK, INFO ) ! ELSE IF( WNTQO ) THEN ! ! Path 2 (M much larger than N, JOBZ = 'O') ! N left singular vectors to be overwritten on A and ! N right singular vectors to be computed in VT ! IR = 1 ! ! WORK(IR) is LDWRKR by N ! IF( LWORK.GE.LDA*N+N*N+3*N+BDSPAC ) THEN LDWRKR = LDA ELSE LDWRKR = ( LWORK-N*N-3*N-BDSPAC ) / N END IF ITAU = IR + LDWRKR*N NWORK = ITAU + N ! ! Compute A=Q*R ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! ! Copy R to WORK(IR), zeroing out below it ! CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR ) CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, WORK( IR+1 ), & LDWRKR ) ! ! Generate Q in A ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ), & WORK( NWORK ), LWORK-NWORK+1, IERR ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N NWORK = ITAUP + N ! ! Bidiagonalize R in VT, copying result to WORK(IR) ! (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB) ! CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! ! WORK(IU) is N by N ! IU = NWORK NWORK = IU + N*N ! ! Perform bidiagonal SVD, computing left singular vectors ! of bidiagonal matrix in WORK(IU) and computing right ! singular vectors of bidiagonal matrix in VT ! (Workspace: need N+N*N+BDSPAC) ! CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ), N, & VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK, & INFO ) ! ! Overwrite WORK(IU) by left singular vectors of R ! and VT by right singular vectors of R ! (Workspace: need 2*N*N+3*N, prefer 2*N*N+2*N+N*NB) ! CALL DORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR, & WORK( ITAUQ ), WORK( IU ), N, WORK( NWORK ), & LWORK-NWORK+1, IERR ) CALL DORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ), LDWRKR, & WORK( ITAUP ), VT, LDVT, WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! ! Multiply Q in A by left singular vectors of R in ! WORK(IU), storing result in WORK(IR) and copying to A ! (Workspace: need 2*N*N, prefer N*N+M*N) ! DO 10 I = 1, M, LDWRKR CHUNK = MIN( M-I+1, LDWRKR ) CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ), & LDA, WORK( IU ), N, ZERO, WORK( IR ), & LDWRKR ) CALL DLACPY( 'F', CHUNK, N, WORK( IR ), LDWRKR, & A( I, 1 ), LDA ) 10 CONTINUE ! ELSE IF( WNTQS ) THEN ! ! Path 3 (M much larger than N, JOBZ='S') ! N left singular vectors to be computed in U and ! N right singular vectors to be computed in VT ! IR = 1 ! ! WORK(IR) is N by N ! LDWRKR = N ITAU = IR + LDWRKR*N NWORK = ITAU + N ! ! Compute A=Q*R ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! ! Copy R to WORK(IR), zeroing out below it ! CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR ) CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, WORK( IR+1 ), & LDWRKR ) ! ! Generate Q in A ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ), & WORK( NWORK ), LWORK-NWORK+1, IERR ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N NWORK = ITAUP + N ! ! Bidiagonalize R in WORK(IR) ! (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB) ! CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! ! Perform bidiagonal SVD, computing left singular vectors ! of bidiagoal matrix in U and computing right singular ! vectors of bidiagonal matrix in VT ! (Workspace: need N+BDSPAC) ! CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT, & LDVT, DUM, IDUM, WORK( NWORK ), IWORK, & INFO ) ! ! Overwrite U by left singular vectors of R and VT ! by right singular vectors of R ! (Workspace: need N*N+3*N, prefer N*N+2*N+N*NB) ! CALL DORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR, & WORK( ITAUQ ), U, LDU, WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! CALL DORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ), LDWRKR, & WORK( ITAUP ), VT, LDVT, WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! ! Multiply Q in A by left singular vectors of R in ! WORK(IR), storing result in U ! (Workspace: need N*N) ! CALL DLACPY( 'F', N, N, U, LDU, WORK( IR ), LDWRKR ) CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA, WORK( IR ), & LDWRKR, ZERO, U, LDU ) ! ELSE IF( WNTQA ) THEN ! ! Path 4 (M much larger than N, JOBZ='A') ! M left singular vectors to be computed in U and ! N right singular vectors to be computed in VT ! IU = 1 ! ! WORK(IU) is N by N ! LDWRKU = N ITAU = IU + LDWRKU*N NWORK = ITAU + N ! ! Compute A=Q*R, copying result to U ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), & LWORK-NWORK+1, IERR ) CALL DLACPY( 'L', M, N, A, LDA, U, LDU ) ! ! Generate Q in U ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ), & WORK( NWORK ), LWORK-NWORK+1, IERR ) ! ! Produce R in A, zeroing out other entries ! CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N NWORK = ITAUP + N ! ! Bidiagonalize R in A ! (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB) ! CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, & IERR ) ! ! Perform bidiagonal SVD, computing left singular vectors ! of bidiagonal matrix in WORK(IU) and computing right ! singular vectors of bidiagonal matrix in VT ! (Workspace: need N+N*N+BDSPAC) ! CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ), N, & VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK, & INFO ) ! ! Overwrite WORK(IU) by left singular vectors of R and VT ! by right singular vectors of R ! (Workspace: need N*N+3*N, prefer N*N+2*N+N*NB) ! CALL DORMBR( 'Q', 'L', 'N', N, N, N, A, LDA, & WORK( ITAUQ ), WORK( IU ), LDWRKU, & WORK( NWORK ), LWORK-NWORK+1, IERR ) CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA, & WORK( ITAUP ), VT, LDVT, WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! ! Multiply Q in U by left singular vectors of R in ! WORK(IU), storing result in A ! (Workspace: need N*N) ! CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU, WORK( IU ), & LDWRKU, ZERO, A, LDA ) ! ! Copy left singular vectors of A from A to U ! CALL DLACPY( 'F', M, N, A, LDA, U, LDU ) ! END IF ! ELSE ! ! M .LT. MNTHR ! ! Path 5 (M at least N, but not much larger) ! Reduce to bidiagonal form without QR decomposition ! IE = 1 ITAUQ = IE + N ITAUP = ITAUQ + N NWORK = ITAUP + N ! ! Bidiagonalize A ! (Workspace: need 3*N+M, prefer 3*N+(M+N)*NB) ! CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, & IERR ) IF( WNTQN ) THEN ! ! Perform bidiagonal SVD, only computing singular values ! (Workspace: need N+BDSPAC) ! CALL DBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM, 1, & DUM, IDUM, WORK( NWORK ), IWORK, INFO ) ELSE IF( WNTQO ) THEN IU = NWORK IF( LWORK.GE.M*N+3*N+BDSPAC ) THEN ! ! WORK( IU ) is M by N ! LDWRKU = M NWORK = IU + LDWRKU*N CALL DLASET( 'F', M, N, ZERO, ZERO, WORK( IU ), & LDWRKU ) ELSE ! ! WORK( IU ) is N by N ! LDWRKU = N NWORK = IU + LDWRKU*N ! ! WORK(IR) is LDWRKR by N ! IR = NWORK LDWRKR = ( LWORK-N*N-3*N ) / N END IF NWORK = IU + LDWRKU*N ! ! Perform bidiagonal SVD, computing left singular vectors ! of bidiagonal matrix in WORK(IU) and computing right ! singular vectors of bidiagonal matrix in VT ! (Workspace: need N+N*N+BDSPAC) ! CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ), & LDWRKU, VT, LDVT, DUM, IDUM, WORK( NWORK ), & IWORK, INFO ) ! ! Overwrite VT by right singular vectors of A ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA, & WORK( ITAUP ), VT, LDVT, WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! IF( LWORK.GE.M*N+3*N+BDSPAC ) THEN ! ! Overwrite WORK(IU) by left singular vectors of A ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DORMBR( 'Q', 'L', 'N', M, N, N, A, LDA, & WORK( ITAUQ ), WORK( IU ), LDWRKU, & WORK( NWORK ), LWORK-NWORK+1, IERR ) ! ! Copy left singular vectors of A from WORK(IU) to A ! CALL DLACPY( 'F', M, N, WORK( IU ), LDWRKU, A, LDA ) ELSE ! ! Generate Q in A ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ), & WORK( NWORK ), LWORK-NWORK+1, IERR ) ! ! Multiply Q in A by left singular vectors of ! bidiagonal matrix in WORK(IU), storing result in ! WORK(IR) and copying to A ! (Workspace: need 2*N*N, prefer N*N+M*N) ! DO 20 I = 1, M, LDWRKR CHUNK = MIN( M-I+1, LDWRKR ) CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ), & LDA, WORK( IU ), LDWRKU, ZERO, & WORK( IR ), LDWRKR ) CALL DLACPY( 'F', CHUNK, N, WORK( IR ), LDWRKR, & A( I, 1 ), LDA ) 20 CONTINUE END IF ! ELSE IF( WNTQS ) THEN ! ! Perform bidiagonal SVD, computing left singular vectors ! of bidiagonal matrix in U and computing right singular ! vectors of bidiagonal matrix in VT ! (Workspace: need N+BDSPAC) ! CALL DLASET( 'F', M, N, ZERO, ZERO, U, LDU ) CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT, & LDVT, DUM, IDUM, WORK( NWORK ), IWORK, & INFO ) ! ! Overwrite U by left singular vectors of A and VT ! by right singular vectors of A ! (Workspace: need 3*N, prefer 2*N+N*NB) ! CALL DORMBR( 'Q', 'L', 'N', M, N, N, A, LDA, & WORK( ITAUQ ), U, LDU, WORK( NWORK ), & LWORK-NWORK+1, IERR ) CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA, & WORK( ITAUP ), VT, LDVT, WORK( NWORK ), & LWORK-NWORK+1, IERR ) ELSE IF( WNTQA ) THEN ! ! Perform bidiagonal SVD, computing left singular vectors ! of bidiagonal matrix in U and computing right singular ! vectors of bidiagonal matrix in VT ! (Workspace: need N+BDSPAC) ! CALL DLASET( 'F', M, M, ZERO, ZERO, U, LDU ) CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT, & LDVT, DUM, IDUM, WORK( NWORK ), IWORK, & INFO ) ! ! Set the right corner of U to identity matrix ! CALL DLASET( 'F', M-N, M-N, ZERO, ONE, U( N+1, N+1 ), & LDU ) ! ! Overwrite U by left singular vectors of A and VT ! by right singular vectors of A ! (Workspace: need N*N+2*N+M, prefer N*N+2*N+M*NB) ! CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA, & WORK( ITAUQ ), U, LDU, WORK( NWORK ), & LWORK-NWORK+1, IERR ) CALL DORMBR( 'P', 'R', 'T', N, N, M, A, LDA, & WORK( ITAUP ), VT, LDVT, WORK( NWORK ), & LWORK-NWORK+1, IERR ) END IF ! END IF ! ELSE ! ! A has more columns than rows. If A has sufficiently more ! columns than rows, first reduce using the LQ decomposition (if ! sufficient workspace available) ! IF( N.GE.MNTHR ) THEN ! IF( WNTQN ) THEN ! ! Path 1t (N much larger than M, JOBZ='N') ! No singular vectors to be computed ! ITAU = 1 NWORK = ITAU + M ! ! Compute A=L*Q ! (Workspace: need 2*M, prefer M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! ! Zero out above L ! CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ), LDA ) IE = 1 ITAUQ = IE + M ITAUP = ITAUQ + M NWORK = ITAUP + M ! ! Bidiagonalize L in A ! (Workspace: need 4*M, prefer 3*M+2*M*NB) ! CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, & IERR ) NWORK = IE + M ! ! Perform bidiagonal SVD, computing singular values only ! (Workspace: need M+BDSPAC) ! CALL DBDSDC( 'U', 'N', M, S, WORK( IE ), DUM, 1, DUM, 1, & DUM, IDUM, WORK( NWORK ), IWORK, INFO ) ! ELSE IF( WNTQO ) THEN ! ! Path 2t (N much larger than M, JOBZ='O') ! M right singular vectors to be overwritten on A and ! M left singular vectors to be computed in U ! IVT = 1 ! ! IVT is M by M ! IL = IVT + M*M IF( LWORK.GE.M*N+M*M+3*M+BDSPAC ) THEN ! ! WORK(IL) is M by N ! LDWRKL = M CHUNK = N ELSE LDWRKL = M CHUNK = ( LWORK-M*M ) / M END IF ITAU = IL + LDWRKL*M NWORK = ITAU + M ! ! Compute A=L*Q ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! ! Copy L to WORK(IL), zeroing about above it ! CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, & WORK( IL+LDWRKL ), LDWRKL ) ! ! Generate Q in A ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ), & WORK( NWORK ), LWORK-NWORK+1, IERR ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M NWORK = ITAUP + M ! ! Bidiagonalize L in WORK(IL) ! (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB) ! CALL DGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! ! Perform bidiagonal SVD, computing left singular vectors ! of bidiagonal matrix in U, and computing right singular ! vectors of bidiagonal matrix in WORK(IVT) ! (Workspace: need M+M*M+BDSPAC) ! CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU, & WORK( IVT ), M, DUM, IDUM, WORK( NWORK ), & IWORK, INFO ) ! ! Overwrite U by left singular vectors of L and WORK(IVT) ! by right singular vectors of L ! (Workspace: need 2*M*M+3*M, prefer 2*M*M+2*M+M*NB) ! CALL DORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL, & WORK( ITAUQ ), U, LDU, WORK( NWORK ), & LWORK-NWORK+1, IERR ) CALL DORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ), LDWRKL, & WORK( ITAUP ), WORK( IVT ), M, & WORK( NWORK ), LWORK-NWORK+1, IERR ) ! ! Multiply right singular vectors of L in WORK(IVT) by Q ! in A, storing result in WORK(IL) and copying to A ! (Workspace: need 2*M*M, prefer M*M+M*N) ! DO 30 I = 1, N, CHUNK BLK = MIN( N-I+1, CHUNK ) CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IVT ), M, & A( 1, I ), LDA, ZERO, WORK( IL ), LDWRKL ) CALL DLACPY( 'F', M, BLK, WORK( IL ), LDWRKL, & A( 1, I ), LDA ) 30 CONTINUE ! ELSE IF( WNTQS ) THEN ! ! Path 3t (N much larger than M, JOBZ='S') ! M right singular vectors to be computed in VT and ! M left singular vectors to be computed in U ! IL = 1 ! ! WORK(IL) is M by M ! LDWRKL = M ITAU = IL + LDWRKL*M NWORK = ITAU + M ! ! Compute A=L*Q ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! ! Copy L to WORK(IL), zeroing out above it ! CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, & WORK( IL+LDWRKL ), LDWRKL ) ! ! Generate Q in A ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ), & WORK( NWORK ), LWORK-NWORK+1, IERR ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M NWORK = ITAUP + M ! ! Bidiagonalize L in WORK(IU), copying result to U ! (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB) ! CALL DGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! ! Perform bidiagonal SVD, computing left singular vectors ! of bidiagonal matrix in U and computing right singular ! vectors of bidiagonal matrix in VT ! (Workspace: need M+BDSPAC) ! CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU, VT, & LDVT, DUM, IDUM, WORK( NWORK ), IWORK, & INFO ) ! ! Overwrite U by left singular vectors of L and VT ! by right singular vectors of L ! (Workspace: need M*M+3*M, prefer M*M+2*M+M*NB) ! CALL DORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL, & WORK( ITAUQ ), U, LDU, WORK( NWORK ), & LWORK-NWORK+1, IERR ) CALL DORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ), LDWRKL, & WORK( ITAUP ), VT, LDVT, WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! ! Multiply right singular vectors of L in WORK(IL) by ! Q in A, storing result in VT ! (Workspace: need M*M) ! CALL DLACPY( 'F', M, M, VT, LDVT, WORK( IL ), LDWRKL ) CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IL ), LDWRKL, & A, LDA, ZERO, VT, LDVT ) ! ELSE IF( WNTQA ) THEN ! ! Path 4t (N much larger than M, JOBZ='A') ! N right singular vectors to be computed in VT and ! M left singular vectors to be computed in U ! IVT = 1 ! ! WORK(IVT) is M by M ! LDWKVT = M ITAU = IVT + LDWKVT*M NWORK = ITAU + M ! ! Compute A=L*Q, copying result to VT ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), & LWORK-NWORK+1, IERR ) CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT ) ! ! Generate Q in VT ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ), & WORK( NWORK ), LWORK-NWORK+1, IERR ) ! ! Produce L in A, zeroing out other entries ! CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ), LDA ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M NWORK = ITAUP + M ! ! Bidiagonalize L in A ! (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB) ! CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, & IERR ) ! ! Perform bidiagonal SVD, computing left singular vectors ! of bidiagonal matrix in U and computing right singular ! vectors of bidiagonal matrix in WORK(IVT) ! (Workspace: need M+M*M+BDSPAC) ! CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU, & WORK( IVT ), LDWKVT, DUM, IDUM, & WORK( NWORK ), IWORK, INFO ) ! ! Overwrite U by left singular vectors of L and WORK(IVT) ! by right singular vectors of L ! (Workspace: need M*M+3*M, prefer M*M+2*M+M*NB) ! CALL DORMBR( 'Q', 'L', 'N', M, M, M, A, LDA, & WORK( ITAUQ ), U, LDU, WORK( NWORK ), & LWORK-NWORK+1, IERR ) CALL DORMBR( 'P', 'R', 'T', M, M, M, A, LDA, & WORK( ITAUP ), WORK( IVT ), LDWKVT, & WORK( NWORK ), LWORK-NWORK+1, IERR ) ! ! Multiply right singular vectors of L in WORK(IVT) by ! Q in VT, storing result in A ! (Workspace: need M*M) ! CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IVT ), LDWKVT, & VT, LDVT, ZERO, A, LDA ) ! ! Copy right singular vectors of A from A to VT ! CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT ) ! END IF ! ELSE ! ! N .LT. MNTHR ! ! Path 5t (N greater than M, but not much larger) ! Reduce to bidiagonal form without LQ decomposition ! IE = 1 ITAUQ = IE + M ITAUP = ITAUQ + M NWORK = ITAUP + M ! ! Bidiagonalize A ! (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB) ! CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, & IERR ) IF( WNTQN ) THEN ! ! Perform bidiagonal SVD, only computing singular values ! (Workspace: need M+BDSPAC) ! CALL DBDSDC( 'L', 'N', M, S, WORK( IE ), DUM, 1, DUM, 1, & DUM, IDUM, WORK( NWORK ), IWORK, INFO ) ELSE IF( WNTQO ) THEN LDWKVT = M IVT = NWORK IF( LWORK.GE.M*N+3*M+BDSPAC ) THEN ! ! WORK( IVT ) is M by N ! CALL DLASET( 'F', M, N, ZERO, ZERO, WORK( IVT ), & LDWKVT ) NWORK = IVT + LDWKVT*N ELSE ! ! WORK( IVT ) is M by M ! NWORK = IVT + LDWKVT*M IL = NWORK ! ! WORK(IL) is M by CHUNK ! CHUNK = ( LWORK-M*M-3*M ) / M END IF ! ! Perform bidiagonal SVD, computing left singular vectors ! of bidiagonal matrix in U and computing right singular ! vectors of bidiagonal matrix in WORK(IVT) ! (Workspace: need M*M+BDSPAC) ! CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, & WORK( IVT ), LDWKVT, DUM, IDUM, & WORK( NWORK ), IWORK, INFO ) ! ! Overwrite U by left singular vectors of A ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA, & WORK( ITAUQ ), U, LDU, WORK( NWORK ), & LWORK-NWORK+1, IERR ) ! IF( LWORK.GE.M*N+3*M+BDSPAC ) THEN ! ! Overwrite WORK(IVT) by left singular vectors of A ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DORMBR( 'P', 'R', 'T', M, N, M, A, LDA, & WORK( ITAUP ), WORK( IVT ), LDWKVT, & WORK( NWORK ), LWORK-NWORK+1, IERR ) ! ! Copy right singular vectors of A from WORK(IVT) to A ! CALL DLACPY( 'F', M, N, WORK( IVT ), LDWKVT, A, LDA ) ELSE ! ! Generate P**T in A ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ), & WORK( NWORK ), LWORK-NWORK+1, IERR ) ! ! Multiply Q in A by right singular vectors of ! bidiagonal matrix in WORK(IVT), storing result in ! WORK(IL) and copying to A ! (Workspace: need 2*M*M, prefer M*M+M*N) ! DO 40 I = 1, N, CHUNK BLK = MIN( N-I+1, CHUNK ) CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IVT ), & LDWKVT, A( 1, I ), LDA, ZERO, & WORK( IL ), M ) CALL DLACPY( 'F', M, BLK, WORK( IL ), M, A( 1, I ), & LDA ) 40 CONTINUE END IF ELSE IF( WNTQS ) THEN ! ! Perform bidiagonal SVD, computing left singular vectors ! of bidiagonal matrix in U and computing right singular ! vectors of bidiagonal matrix in VT ! (Workspace: need M+BDSPAC) ! CALL DLASET( 'F', M, N, ZERO, ZERO, VT, LDVT ) CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT, & LDVT, DUM, IDUM, WORK( NWORK ), IWORK, & INFO ) ! ! Overwrite U by left singular vectors of A and VT ! by right singular vectors of A ! (Workspace: need 3*M, prefer 2*M+M*NB) ! CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA, & WORK( ITAUQ ), U, LDU, WORK( NWORK ), & LWORK-NWORK+1, IERR ) CALL DORMBR( 'P', 'R', 'T', M, N, M, A, LDA, & WORK( ITAUP ), VT, LDVT, WORK( NWORK ), & LWORK-NWORK+1, IERR ) ELSE IF( WNTQA ) THEN ! ! Perform bidiagonal SVD, computing left singular vectors ! of bidiagonal matrix in U and computing right singular ! vectors of bidiagonal matrix in VT ! (Workspace: need M+BDSPAC) ! CALL DLASET( 'F', N, N, ZERO, ZERO, VT, LDVT ) CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT, & LDVT, DUM, IDUM, WORK( NWORK ), IWORK, & INFO ) ! ! Set the right corner of VT to identity matrix ! CALL DLASET( 'F', N-M, N-M, ZERO, ONE, VT( M+1, M+1 ), & LDVT ) ! ! Overwrite U by left singular vectors of A and VT ! by right singular vectors of A ! (Workspace: need 2*M+N, prefer 2*M+N*NB) ! CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA, & WORK( ITAUQ ), U, LDU, WORK( NWORK ), & LWORK-NWORK+1, IERR ) CALL DORMBR( 'P', 'R', 'T', N, N, M, A, LDA, & WORK( ITAUP ), VT, LDVT, WORK( NWORK ), & LWORK-NWORK+1, IERR ) END IF ! END IF ! END IF ! ! Undo scaling if necessary ! IF( ISCL.EQ.1 ) THEN IF( ANRM.GT.BIGNUM ) & CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN, & IERR ) IF( ANRM.LT.SMLNUM ) & CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN, & IERR ) END IF ! ! Return optimal workspace in WORK(1) ! WORK( 1 ) = DBLE( MAXWRK ) ! RETURN ! ! End of DGESDD ! END SUBROUTINE DGESV( N, NRHS, A, LDA, IPIV, B, LDB, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LDB, N, NRHS ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DGESV computes the solution to a real system of linear equations ! A * X = B, ! where A is an N-by-N matrix and X and B are N-by-NRHS matrices. ! ! The LU decomposition with partial pivoting and row interchanges is ! used to factor A as ! A = P * L * U, ! where P is a permutation matrix, L is unit lower triangular, and U is ! upper triangular. The factored form of A is then used to solve the ! system of equations A * X = B. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The number of linear equations, i.e., the order of the ! matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the N-by-N coefficient matrix A. ! On exit, the factors L and U from the factorization ! A = P*L*U; the unit diagonal elements of L are not stored. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! IPIV (output) INTEGER array, dimension (N) ! The pivot indices that define the permutation matrix P; ! row i of the matrix was interchanged with row IPIV(i). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the N-by-NRHS matrix of right hand side matrix B. ! On exit, if INFO = 0, the N-by-NRHS solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, U(i,i) is exactly zero. The factorization ! has been completed, but the factor U is exactly ! singular, so the solution could not be computed. ! ! ===================================================================== ! ! .. External Subroutines .. EXTERNAL DGETRF, DGETRS, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF( N.LT.0 ) THEN INFO = -1 ELSE IF( NRHS.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGESV ', -INFO ) RETURN END IF ! ! Compute the LU factorization of A. ! CALL DGETRF( N, N, A, LDA, IPIV, INFO ) IF( INFO.EQ.0 ) THEN ! ! Solve the system A*X = B, overwriting B with X. ! CALL DGETRS( 'No transpose', N, NRHS, A, LDA, IPIV, B, LDB, & INFO ) END IF RETURN ! ! End of DGESV ! END SUBROUTINE DGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT, & WORK, LWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBU, JOBVT INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), S( * ), U( LDU, * ), & VT( LDVT, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGESVD computes the singular value decomposition (SVD) of a real ! M-by-N matrix A, optionally computing the left and/or right singular ! vectors. The SVD is written ! ! A = U * SIGMA * transpose(V) ! ! where SIGMA is an M-by-N matrix which is zero except for its ! min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and ! V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA ! are the singular values of A; they are real and non-negative, and ! are returned in descending order. The first min(m,n) columns of ! U and V are the left and right singular vectors of A. ! ! Note that the routine returns V**T, not V. ! ! Arguments ! ========= ! ! JOBU (input) CHARACTER*1 ! Specifies options for computing all or part of the matrix U: ! = 'A': all M columns of U are returned in array U: ! = 'S': the first min(m,n) columns of U (the left singular ! vectors) are returned in the array U; ! = 'O': the first min(m,n) columns of U (the left singular ! vectors) are overwritten on the array A; ! = 'N': no columns of U (no left singular vectors) are ! computed. ! ! JOBVT (input) CHARACTER*1 ! Specifies options for computing all or part of the matrix ! V**T: ! = 'A': all N rows of V**T are returned in the array VT; ! = 'S': the first min(m,n) rows of V**T (the right singular ! vectors) are returned in the array VT; ! = 'O': the first min(m,n) rows of V**T (the right singular ! vectors) are overwritten on the array A; ! = 'N': no rows of V**T (no right singular vectors) are ! computed. ! ! JOBVT and JOBU cannot both be 'O'. ! ! M (input) INTEGER ! The number of rows of the input matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the input matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, ! if JOBU = 'O', A is overwritten with the first min(m,n) ! columns of U (the left singular vectors, ! stored columnwise); ! if JOBVT = 'O', A is overwritten with the first min(m,n) ! rows of V**T (the right singular vectors, ! stored rowwise); ! if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A ! are destroyed. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! S (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The singular values of A, sorted so that S(i) >= S(i+1). ! ! U (output) DOUBLE PRECISION array, dimension (LDU,UCOL) ! (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'. ! If JOBU = 'A', U contains the M-by-M orthogonal matrix U; ! if JOBU = 'S', U contains the first min(m,n) columns of U ! (the left singular vectors, stored columnwise); ! if JOBU = 'N' or 'O', U is not referenced. ! ! LDU (input) INTEGER ! The leading dimension of the array U. LDU >= 1; if ! JOBU = 'S' or 'A', LDU >= M. ! ! VT (output) DOUBLE PRECISION array, dimension (LDVT,N) ! If JOBVT = 'A', VT contains the N-by-N orthogonal matrix ! V**T; ! if JOBVT = 'S', VT contains the first min(m,n) rows of ! V**T (the right singular vectors, stored rowwise); ! if JOBVT = 'N' or 'O', VT is not referenced. ! ! LDVT (input) INTEGER ! The leading dimension of the array VT. LDVT >= 1; if ! JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK; ! if INFO > 0, WORK(2:MIN(M,N)) contains the unconverged ! superdiagonal elements of an upper bidiagonal matrix B ! whose diagonal is in S (not necessarily sorted). B ! satisfies A = U * B * VT, so it has the same singular values ! as A, and singular vectors related by U and VT. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= 1. ! LWORK >= MAX(3*MIN(M,N)+MAX(M,N),5*MIN(M,N)). ! For good performance, LWORK should generally be larger. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if DBDSQR did not converge, INFO specifies how many ! superdiagonals of an intermediate bidiagonal form B ! did not converge to zero. See the description of WORK ! above for details. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, WNTUA, WNTUAS, WNTUN, WNTUO, WNTUS, & WNTVA, WNTVAS, WNTVN, WNTVO, WNTVS INTEGER BDSPAC, BLK, CHUNK, I, IE, IERR, IR, ISCL, & ITAU, ITAUP, ITAUQ, IU, IWORK, LDWRKR, LDWRKU, & MAXWRK, MINMN, MINWRK, MNTHR, NCU, NCVT, NRU, & NRVT, WRKBL DOUBLE PRECISION ANRM, BIGNUM, EPS, SMLNUM ! .. ! .. Local Arrays .. DOUBLE PRECISION DUM( 1 ) ! .. ! .. External Subroutines .. EXTERNAL DBDSQR, DGEBRD, DGELQF, DGEMM, DGEQRF, DLACPY, & DLASCL, DLASET, DORGBR, DORGLQ, DORGQR, DORMBR, & XERBLA ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 MINMN = MIN( M, N ) MNTHR = ILAENV( 6, 'DGESVD', JOBU // JOBVT, M, N, 0, 0 ) WNTUA = LSAME( JOBU, 'A' ) WNTUS = LSAME( JOBU, 'S' ) WNTUAS = WNTUA .OR. WNTUS WNTUO = LSAME( JOBU, 'O' ) WNTUN = LSAME( JOBU, 'N' ) WNTVA = LSAME( JOBVT, 'A' ) WNTVS = LSAME( JOBVT, 'S' ) WNTVAS = WNTVA .OR. WNTVS WNTVO = LSAME( JOBVT, 'O' ) WNTVN = LSAME( JOBVT, 'N' ) MINWRK = 1 LQUERY = ( LWORK.EQ.-1 ) ! IF( .NOT.( WNTUA .OR. WNTUS .OR. WNTUO .OR. WNTUN ) ) THEN INFO = -1 ELSE IF( .NOT.( WNTVA .OR. WNTVS .OR. WNTVO .OR. WNTVN ) .OR. & ( WNTVO .AND. WNTUO ) ) THEN INFO = -2 ELSE IF( M.LT.0 ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -6 ELSE IF( LDU.LT.1 .OR. ( WNTUAS .AND. LDU.LT.M ) ) THEN INFO = -9 ELSE IF( LDVT.LT.1 .OR. ( WNTVA .AND. LDVT.LT.N ) .OR. & ( WNTVS .AND. LDVT.LT.MINMN ) ) THEN INFO = -11 END IF ! ! Compute workspace ! (Note: Comments in the code beginning "Workspace:" describe the ! minimal amount of workspace needed at that point in the code, ! as well as the preferred amount for good performance. ! NB refers to the optimal block size for the immediately ! following subroutine, as returned by ILAENV.) ! IF( INFO.EQ.0 .AND. ( LWORK.GE.1 .OR. LQUERY ) .AND. M.GT.0 .AND. & N.GT.0 ) THEN IF( M.GE.N ) THEN ! ! Compute space needed for DBDSQR ! BDSPAC = 5*N IF( M.GE.MNTHR ) THEN IF( WNTUN ) THEN ! ! Path 1 (M much larger than N, JOBU='N') ! MAXWRK = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, & -1 ) MAXWRK = MAX( MAXWRK, 3*N+2*N* & ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) ) IF( WNTVO .OR. WNTVAS ) & MAXWRK = MAX( MAXWRK, 3*N+( N-1 )* & ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) ) MAXWRK = MAX( MAXWRK, BDSPAC ) MINWRK = MAX( 4*N, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) ELSE IF( WNTUO .AND. WNTVN ) THEN ! ! Path 2 (M much larger than N, JOBU='O', JOBVT='N') ! WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M, & N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+2*N* & ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC ) MAXWRK = MAX( N*N+WRKBL, N*N+M*N+N ) MINWRK = MAX( 3*N+M, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) ELSE IF( WNTUO .AND. WNTVAS ) THEN ! ! Path 3 (M much larger than N, JOBU='O', JOBVT='S' or ! 'A') ! WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M, & N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+2*N* & ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+( N-1 )* & ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC ) MAXWRK = MAX( N*N+WRKBL, N*N+M*N+N ) MINWRK = MAX( 3*N+M, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) ELSE IF( WNTUS .AND. WNTVN ) THEN ! ! Path 4 (M much larger than N, JOBU='S', JOBVT='N') ! WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M, & N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+2*N* & ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC ) MAXWRK = N*N + WRKBL MINWRK = MAX( 3*N+M, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) ELSE IF( WNTUS .AND. WNTVO ) THEN ! ! Path 5 (M much larger than N, JOBU='S', JOBVT='O') ! WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M, & N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+2*N* & ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+( N-1 )* & ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC ) MAXWRK = 2*N*N + WRKBL MINWRK = MAX( 3*N+M, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) ELSE IF( WNTUS .AND. WNTVAS ) THEN ! ! Path 6 (M much larger than N, JOBU='S', JOBVT='S' or ! 'A') ! WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M, & N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+2*N* & ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+( N-1 )* & ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC ) MAXWRK = N*N + WRKBL MINWRK = MAX( 3*N+M, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) ELSE IF( WNTUA .AND. WNTVN ) THEN ! ! Path 7 (M much larger than N, JOBU='A', JOBVT='N') ! WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'DORGQR', ' ', M, & M, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+2*N* & ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC ) MAXWRK = N*N + WRKBL MINWRK = MAX( 3*N+M, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) ELSE IF( WNTUA .AND. WNTVO ) THEN ! ! Path 8 (M much larger than N, JOBU='A', JOBVT='O') ! WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'DORGQR', ' ', M, & M, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+2*N* & ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+( N-1 )* & ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC ) MAXWRK = 2*N*N + WRKBL MINWRK = MAX( 3*N+M, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) ELSE IF( WNTUA .AND. WNTVAS ) THEN ! ! Path 9 (M much larger than N, JOBU='A', JOBVT='S' or ! 'A') ! WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'DORGQR', ' ', M, & M, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+2*N* & ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*N+N* & ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, 3*N+( N-1 )* & ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC ) MAXWRK = N*N + WRKBL MINWRK = MAX( 3*N+M, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) END IF ELSE ! ! Path 10 (M at least N, but not much larger) ! MAXWRK = 3*N + ( M+N )*ILAENV( 1, 'DGEBRD', ' ', M, N, & -1, -1 ) IF( WNTUS .OR. WNTUO ) & MAXWRK = MAX( MAXWRK, 3*N+N* & ILAENV( 1, 'DORGBR', 'Q', M, N, N, -1 ) ) IF( WNTUA ) & MAXWRK = MAX( MAXWRK, 3*N+M* & ILAENV( 1, 'DORGBR', 'Q', M, M, N, -1 ) ) IF( .NOT.WNTVN ) & MAXWRK = MAX( MAXWRK, 3*N+( N-1 )* & ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) ) MAXWRK = MAX( MAXWRK, BDSPAC ) MINWRK = MAX( 3*N+M, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) END IF ELSE ! ! Compute space needed for DBDSQR ! BDSPAC = 5*M IF( N.GE.MNTHR ) THEN IF( WNTVN ) THEN ! ! Path 1t(N much larger than M, JOBVT='N') ! MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, & -1 ) MAXWRK = MAX( MAXWRK, 3*M+2*M* & ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) IF( WNTUO .OR. WNTUAS ) & MAXWRK = MAX( MAXWRK, 3*M+M* & ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) ) MAXWRK = MAX( MAXWRK, BDSPAC ) MINWRK = MAX( 4*M, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) ELSE IF( WNTVO .AND. WNTUN ) THEN ! ! Path 2t(N much larger than M, JOBU='N', JOBVT='O') ! WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M, & N, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+2*M* & ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*M+( M-1 )* & ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC ) MAXWRK = MAX( M*M+WRKBL, M*M+M*N+M ) MINWRK = MAX( 3*M+N, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) ELSE IF( WNTVO .AND. WNTUAS ) THEN ! ! Path 3t(N much larger than M, JOBU='S' or 'A', ! JOBVT='O') ! WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M, & N, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+2*M* & ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*M+( M-1 )* & ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC ) MAXWRK = MAX( M*M+WRKBL, M*M+M*N+M ) MINWRK = MAX( 3*M+N, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) ELSE IF( WNTVS .AND. WNTUN ) THEN ! ! Path 4t(N much larger than M, JOBU='N', JOBVT='S') ! WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M, & N, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+2*M* & ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*M+( M-1 )* & ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC ) MAXWRK = M*M + WRKBL MINWRK = MAX( 3*M+N, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) ELSE IF( WNTVS .AND. WNTUO ) THEN ! ! Path 5t(N much larger than M, JOBU='O', JOBVT='S') ! WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M, & N, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+2*M* & ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*M+( M-1 )* & ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC ) MAXWRK = 2*M*M + WRKBL MINWRK = MAX( 3*M+N, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) ELSE IF( WNTVS .AND. WNTUAS ) THEN ! ! Path 6t(N much larger than M, JOBU='S' or 'A', ! JOBVT='S') ! WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M, & N, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+2*M* & ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*M+( M-1 )* & ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC ) MAXWRK = M*M + WRKBL MINWRK = MAX( 3*M+N, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) ELSE IF( WNTVA .AND. WNTUN ) THEN ! ! Path 7t(N much larger than M, JOBU='N', JOBVT='A') ! WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'DORGLQ', ' ', N, & N, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+2*M* & ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*M+( M-1 )* & ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC ) MAXWRK = M*M + WRKBL MINWRK = MAX( 3*M+N, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) ELSE IF( WNTVA .AND. WNTUO ) THEN ! ! Path 8t(N much larger than M, JOBU='O', JOBVT='A') ! WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'DORGLQ', ' ', N, & N, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+2*M* & ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*M+( M-1 )* & ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC ) MAXWRK = 2*M*M + WRKBL MINWRK = MAX( 3*M+N, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) ELSE IF( WNTVA .AND. WNTUAS ) THEN ! ! Path 9t(N much larger than M, JOBU='S' or 'A', ! JOBVT='A') ! WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 ) WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'DORGLQ', ' ', N, & N, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+2*M* & ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) ) WRKBL = MAX( WRKBL, 3*M+( M-1 )* & ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, 3*M+M* & ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) ) WRKBL = MAX( WRKBL, BDSPAC ) MAXWRK = M*M + WRKBL MINWRK = MAX( 3*M+N, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) END IF ELSE ! ! Path 10t(N greater than M, but not much larger) ! MAXWRK = 3*M + ( M+N )*ILAENV( 1, 'DGEBRD', ' ', M, N, & -1, -1 ) IF( WNTVS .OR. WNTVO ) & MAXWRK = MAX( MAXWRK, 3*M+M* & ILAENV( 1, 'DORGBR', 'P', M, N, M, -1 ) ) IF( WNTVA ) & MAXWRK = MAX( MAXWRK, 3*M+N* & ILAENV( 1, 'DORGBR', 'P', N, N, M, -1 ) ) IF( .NOT.WNTUN ) & MAXWRK = MAX( MAXWRK, 3*M+( M-1 )* & ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) ) MAXWRK = MAX( MAXWRK, BDSPAC ) MINWRK = MAX( 3*M+N, BDSPAC ) MAXWRK = MAX( MAXWRK, MINWRK ) END IF END IF WORK( 1 ) = MAXWRK END IF ! IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -13 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGESVD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 ) THEN IF( LWORK.GE.1 ) & WORK( 1 ) = ONE RETURN END IF ! ! Get machine constants ! EPS = DLAMCH( 'P' ) SMLNUM = SQRT( DLAMCH( 'S' ) ) / EPS BIGNUM = ONE / SMLNUM ! ! Scale A if max element outside range [SMLNUM,BIGNUM] ! ANRM = DLANGE( 'M', M, N, A, LDA, DUM ) ISCL = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN ISCL = 1 CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, IERR ) ELSE IF( ANRM.GT.BIGNUM ) THEN ISCL = 1 CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, IERR ) END IF ! IF( M.GE.N ) THEN ! ! A has at least as many rows as columns. If A has sufficiently ! more rows than columns, first reduce using the QR ! decomposition (if sufficient workspace available) ! IF( M.GE.MNTHR ) THEN ! IF( WNTUN ) THEN ! ! Path 1 (M much larger than N, JOBU='N') ! No left singular vectors to be computed ! ITAU = 1 IWORK = ITAU + N ! ! Compute A=Q*R ! (Workspace: need 2*N, prefer N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Zero out below R ! CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA ) IE = 1 ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Bidiagonalize R in A ! (Workspace: need 4*N, prefer 3*N+2*N*NB) ! CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1, & IERR ) NCVT = 0 IF( WNTVO .OR. WNTVAS ) THEN ! ! If right singular vectors desired, generate P'. ! (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB) ! CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) NCVT = N END IF IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing right ! singular vectors of A in A if desired ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', N, NCVT, 0, 0, S, WORK( IE ), A, LDA, & DUM, 1, DUM, 1, WORK( IWORK ), INFO ) ! ! If right singular vectors desired in VT, copy them there ! IF( WNTVAS ) & CALL DLACPY( 'F', N, N, A, LDA, VT, LDVT ) ! ELSE IF( WNTUO .AND. WNTVN ) THEN ! ! Path 2 (M much larger than N, JOBU='O', JOBVT='N') ! N left singular vectors to be overwritten on A and ! no right singular vectors to be computed ! IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN ! ! Sufficient workspace for a fast algorithm ! IR = 1 IF( LWORK.GE.MAX( WRKBL, LDA*N+N )+LDA*N ) THEN ! ! WORK(IU) is LDA by N, WORK(IR) is LDA by N ! LDWRKU = LDA LDWRKR = LDA ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N+N )+N*N ) THEN ! ! WORK(IU) is LDA by N, WORK(IR) is N by N ! LDWRKU = LDA LDWRKR = N ELSE ! ! WORK(IU) is LDWRKU by N, WORK(IR) is N by N ! LDWRKU = ( LWORK-N*N-N ) / N LDWRKR = N END IF ITAU = IR + LDWRKR*N IWORK = ITAU + N ! ! Compute A=Q*R ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy R to WORK(IR) and zero out below it ! CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR ) CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, WORK( IR+1 ), & LDWRKR ) ! ! Generate Q in A ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Bidiagonalize R in WORK(IR) ! (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB) ! CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Generate left vectors bidiagonalizing R ! (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB) ! CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR, & WORK( ITAUQ ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of R in WORK(IR) ! (Workspace: need N*N+BDSPAC) ! CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ), DUM, 1, & WORK( IR ), LDWRKR, DUM, 1, & WORK( IWORK ), INFO ) IU = IE + N ! ! Multiply Q in A by left singular vectors of R in ! WORK(IR), storing result in WORK(IU) and copying to A ! (Workspace: need N*N+2*N, prefer N*N+M*N+N) ! DO 10 I = 1, M, LDWRKU CHUNK = MIN( M-I+1, LDWRKU ) CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ), & LDA, WORK( IR ), LDWRKR, ZERO, & WORK( IU ), LDWRKU ) CALL DLACPY( 'F', CHUNK, N, WORK( IU ), LDWRKU, & A( I, 1 ), LDA ) 10 CONTINUE ! ELSE ! ! Insufficient workspace for a fast algorithm ! IE = 1 ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Bidiagonalize A ! (Workspace: need 3*N+M, prefer 3*N+(M+N)*NB) ! CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Generate left vectors bidiagonalizing A ! (Workspace: need 4*N, prefer 3*N+N*NB) ! CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of A in A ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', N, 0, M, 0, S, WORK( IE ), DUM, 1, & A, LDA, DUM, 1, WORK( IWORK ), INFO ) ! END IF ! ELSE IF( WNTUO .AND. WNTVAS ) THEN ! ! Path 3 (M much larger than N, JOBU='O', JOBVT='S' or 'A') ! N left singular vectors to be overwritten on A and ! N right singular vectors to be computed in VT ! IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN ! ! Sufficient workspace for a fast algorithm ! IR = 1 IF( LWORK.GE.MAX( WRKBL, LDA*N+N )+LDA*N ) THEN ! ! WORK(IU) is LDA by N and WORK(IR) is LDA by N ! LDWRKU = LDA LDWRKR = LDA ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N+N )+N*N ) THEN ! ! WORK(IU) is LDA by N and WORK(IR) is N by N ! LDWRKU = LDA LDWRKR = N ELSE ! ! WORK(IU) is LDWRKU by N and WORK(IR) is N by N ! LDWRKU = ( LWORK-N*N-N ) / N LDWRKR = N END IF ITAU = IR + LDWRKR*N IWORK = ITAU + N ! ! Compute A=Q*R ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy R to VT, zeroing out below it ! CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT ) CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, VT( 2, 1 ), & LDVT ) ! ! Generate Q in A ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Bidiagonalize R in VT, copying result to WORK(IR) ! (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB) ! CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'L', N, N, VT, LDVT, WORK( IR ), LDWRKR ) ! ! Generate left vectors bidiagonalizing R in WORK(IR) ! (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB) ! CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR, & WORK( ITAUQ ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate right vectors bidiagonalizing R in VT ! (Workspace: need N*N+4*N-1, prefer N*N+3*N+(N-1)*NB) ! CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of R in WORK(IR) and computing right ! singular vectors of R in VT ! (Workspace: need N*N+BDSPAC) ! CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT, LDVT, & WORK( IR ), LDWRKR, DUM, 1, & WORK( IWORK ), INFO ) IU = IE + N ! ! Multiply Q in A by left singular vectors of R in ! WORK(IR), storing result in WORK(IU) and copying to A ! (Workspace: need N*N+2*N, prefer N*N+M*N+N) ! DO 20 I = 1, M, LDWRKU CHUNK = MIN( M-I+1, LDWRKU ) CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ), & LDA, WORK( IR ), LDWRKR, ZERO, & WORK( IU ), LDWRKU ) CALL DLACPY( 'F', CHUNK, N, WORK( IU ), LDWRKU, & A( I, 1 ), LDA ) 20 CONTINUE ! ELSE ! ! Insufficient workspace for a fast algorithm ! ITAU = 1 IWORK = ITAU + N ! ! Compute A=Q*R ! (Workspace: need 2*N, prefer N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy R to VT, zeroing out below it ! CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT ) CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, VT( 2, 1 ), & LDVT ) ! ! Generate Q in A ! (Workspace: need 2*N, prefer N+N*NB) ! CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Bidiagonalize R in VT ! (Workspace: need 4*N, prefer 3*N+2*N*NB) ! CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Multiply Q in A by left vectors bidiagonalizing R ! (Workspace: need 3*N+M, prefer 3*N+M*NB) ! CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT, & WORK( ITAUQ ), A, LDA, WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate right vectors bidiagonalizing R in VT ! (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB) ! CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of A in A and computing right ! singular vectors of A in VT ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), VT, LDVT, & A, LDA, DUM, 1, WORK( IWORK ), INFO ) ! END IF ! ELSE IF( WNTUS ) THEN ! IF( WNTVN ) THEN ! ! Path 4 (M much larger than N, JOBU='S', JOBVT='N') ! N left singular vectors to be computed in U and ! no right singular vectors to be computed ! IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN ! ! Sufficient workspace for a fast algorithm ! IR = 1 IF( LWORK.GE.WRKBL+LDA*N ) THEN ! ! WORK(IR) is LDA by N ! LDWRKR = LDA ELSE ! ! WORK(IR) is N by N ! LDWRKR = N END IF ITAU = IR + LDWRKR*N IWORK = ITAU + N ! ! Compute A=Q*R ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy R to WORK(IR), zeroing out below it ! CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), & LDWRKR ) CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, & WORK( IR+1 ), LDWRKR ) ! ! Generate Q in A ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Bidiagonalize R in WORK(IR) ! (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB) ! CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, & WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate left vectors bidiagonalizing R in WORK(IR) ! (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB) ! CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR, & WORK( ITAUQ ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of R in WORK(IR) ! (Workspace: need N*N+BDSPAC) ! CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ), DUM, & 1, WORK( IR ), LDWRKR, DUM, 1, & WORK( IWORK ), INFO ) ! ! Multiply Q in A by left singular vectors of R in ! WORK(IR), storing result in U ! (Workspace: need N*N) ! CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA, & WORK( IR ), LDWRKR, ZERO, U, LDU ) ! ELSE ! ! Insufficient workspace for a fast algorithm ! ITAU = 1 IWORK = ITAU + N ! ! Compute A=Q*R, copying result to U ! (Workspace: need 2*N, prefer N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'L', M, N, A, LDA, U, LDU ) ! ! Generate Q in U ! (Workspace: need 2*N, prefer N+N*NB) ! CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Zero out below R in A ! CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), & LDA ) ! ! Bidiagonalize R in A ! (Workspace: need 4*N, prefer 3*N+2*N*NB) ! CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Multiply Q in U by left vectors bidiagonalizing R ! (Workspace: need 3*N+M, prefer 3*N+M*NB) ! CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA, & WORK( ITAUQ ), U, LDU, WORK( IWORK ), & LWORK-IWORK+1, IERR ) IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of A in U ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', N, 0, M, 0, S, WORK( IE ), DUM, & 1, U, LDU, DUM, 1, WORK( IWORK ), & INFO ) ! END IF ! ELSE IF( WNTVO ) THEN ! ! Path 5 (M much larger than N, JOBU='S', JOBVT='O') ! N left singular vectors to be computed in U and ! N right singular vectors to be overwritten on A ! IF( LWORK.GE.2*N*N+MAX( 4*N, BDSPAC ) ) THEN ! ! Sufficient workspace for a fast algorithm ! IU = 1 IF( LWORK.GE.WRKBL+2*LDA*N ) THEN ! ! WORK(IU) is LDA by N and WORK(IR) is LDA by N ! LDWRKU = LDA IR = IU + LDWRKU*N LDWRKR = LDA ELSE IF( LWORK.GE.WRKBL+( LDA+N )*N ) THEN ! ! WORK(IU) is LDA by N and WORK(IR) is N by N ! LDWRKU = LDA IR = IU + LDWRKU*N LDWRKR = N ELSE ! ! WORK(IU) is N by N and WORK(IR) is N by N ! LDWRKU = N IR = IU + LDWRKU*N LDWRKR = N END IF ITAU = IR + LDWRKR*N IWORK = ITAU + N ! ! Compute A=Q*R ! (Workspace: need 2*N*N+2*N, prefer 2*N*N+N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy R to WORK(IU), zeroing out below it ! CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ), & LDWRKU ) CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, & WORK( IU+1 ), LDWRKU ) ! ! Generate Q in A ! (Workspace: need 2*N*N+2*N, prefer 2*N*N+N+N*NB) ! CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Bidiagonalize R in WORK(IU), copying result to ! WORK(IR) ! (Workspace: need 2*N*N+4*N, ! prefer 2*N*N+3*N+2*N*NB) ! CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S, & WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU, & WORK( IR ), LDWRKR ) ! ! Generate left bidiagonalizing vectors in WORK(IU) ! (Workspace: need 2*N*N+4*N, prefer 2*N*N+3*N+N*NB) ! CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU, & WORK( ITAUQ ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate right bidiagonalizing vectors in WORK(IR) ! (Workspace: need 2*N*N+4*N-1, ! prefer 2*N*N+3*N+(N-1)*NB) ! CALL DORGBR( 'P', N, N, N, WORK( IR ), LDWRKR, & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of R in WORK(IU) and computing ! right singular vectors of R in WORK(IR) ! (Workspace: need 2*N*N+BDSPAC) ! CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), & WORK( IR ), LDWRKR, WORK( IU ), & LDWRKU, DUM, 1, WORK( IWORK ), INFO ) ! ! Multiply Q in A by left singular vectors of R in ! WORK(IU), storing result in U ! (Workspace: need N*N) ! CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA, & WORK( IU ), LDWRKU, ZERO, U, LDU ) ! ! Copy right singular vectors of R to A ! (Workspace: need N*N) ! CALL DLACPY( 'F', N, N, WORK( IR ), LDWRKR, A, & LDA ) ! ELSE ! ! Insufficient workspace for a fast algorithm ! ITAU = 1 IWORK = ITAU + N ! ! Compute A=Q*R, copying result to U ! (Workspace: need 2*N, prefer N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'L', M, N, A, LDA, U, LDU ) ! ! Generate Q in U ! (Workspace: need 2*N, prefer N+N*NB) ! CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Zero out below R in A ! CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), & LDA ) ! ! Bidiagonalize R in A ! (Workspace: need 4*N, prefer 3*N+2*N*NB) ! CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Multiply Q in U by left vectors bidiagonalizing R ! (Workspace: need 3*N+M, prefer 3*N+M*NB) ! CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA, & WORK( ITAUQ ), U, LDU, WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate right vectors bidiagonalizing R in A ! (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB) ! CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of A in U and computing right ! singular vectors of A in A ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), A, & LDA, U, LDU, DUM, 1, WORK( IWORK ), & INFO ) ! END IF ! ELSE IF( WNTVAS ) THEN ! ! Path 6 (M much larger than N, JOBU='S', JOBVT='S' ! or 'A') ! N left singular vectors to be computed in U and ! N right singular vectors to be computed in VT ! IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN ! ! Sufficient workspace for a fast algorithm ! IU = 1 IF( LWORK.GE.WRKBL+LDA*N ) THEN ! ! WORK(IU) is LDA by N ! LDWRKU = LDA ELSE ! ! WORK(IU) is N by N ! LDWRKU = N END IF ITAU = IU + LDWRKU*N IWORK = ITAU + N ! ! Compute A=Q*R ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy R to WORK(IU), zeroing out below it ! CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ), & LDWRKU ) CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, & WORK( IU+1 ), LDWRKU ) ! ! Generate Q in A ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Bidiagonalize R in WORK(IU), copying result to VT ! (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB) ! CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S, & WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU, VT, & LDVT ) ! ! Generate left bidiagonalizing vectors in WORK(IU) ! (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB) ! CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU, & WORK( ITAUQ ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate right bidiagonalizing vectors in VT ! (Workspace: need N*N+4*N-1, ! prefer N*N+3*N+(N-1)*NB) ! CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of R in WORK(IU) and computing ! right singular vectors of R in VT ! (Workspace: need N*N+BDSPAC) ! CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT, & LDVT, WORK( IU ), LDWRKU, DUM, 1, & WORK( IWORK ), INFO ) ! ! Multiply Q in A by left singular vectors of R in ! WORK(IU), storing result in U ! (Workspace: need N*N) ! CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA, & WORK( IU ), LDWRKU, ZERO, U, LDU ) ! ELSE ! ! Insufficient workspace for a fast algorithm ! ITAU = 1 IWORK = ITAU + N ! ! Compute A=Q*R, copying result to U ! (Workspace: need 2*N, prefer N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'L', M, N, A, LDA, U, LDU ) ! ! Generate Q in U ! (Workspace: need 2*N, prefer N+N*NB) ! CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy R to VT, zeroing out below it ! CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT ) CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, VT( 2, 1 ), & LDVT ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Bidiagonalize R in VT ! (Workspace: need 4*N, prefer 3*N+2*N*NB) ! CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Multiply Q in U by left bidiagonalizing vectors ! in VT ! (Workspace: need 3*N+M, prefer 3*N+M*NB) ! CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT, & WORK( ITAUQ ), U, LDU, WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate right bidiagonalizing vectors in VT ! (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB) ! CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of A in U and computing right ! singular vectors of A in VT ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), VT, & LDVT, U, LDU, DUM, 1, WORK( IWORK ), & INFO ) ! END IF ! END IF ! ELSE IF( WNTUA ) THEN ! IF( WNTVN ) THEN ! ! Path 7 (M much larger than N, JOBU='A', JOBVT='N') ! M left singular vectors to be computed in U and ! no right singular vectors to be computed ! IF( LWORK.GE.N*N+MAX( N+M, 4*N, BDSPAC ) ) THEN ! ! Sufficient workspace for a fast algorithm ! IR = 1 IF( LWORK.GE.WRKBL+LDA*N ) THEN ! ! WORK(IR) is LDA by N ! LDWRKR = LDA ELSE ! ! WORK(IR) is N by N ! LDWRKR = N END IF ITAU = IR + LDWRKR*N IWORK = ITAU + N ! ! Compute A=Q*R, copying result to U ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'L', M, N, A, LDA, U, LDU ) ! ! Copy R to WORK(IR), zeroing out below it ! CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), & LDWRKR ) CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, & WORK( IR+1 ), LDWRKR ) ! ! Generate Q in U ! (Workspace: need N*N+N+M, prefer N*N+N+M*NB) ! CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Bidiagonalize R in WORK(IR) ! (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB) ! CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, & WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate left bidiagonalizing vectors in WORK(IR) ! (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB) ! CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR, & WORK( ITAUQ ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of R in WORK(IR) ! (Workspace: need N*N+BDSPAC) ! CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ), DUM, & 1, WORK( IR ), LDWRKR, DUM, 1, & WORK( IWORK ), INFO ) ! ! Multiply Q in U by left singular vectors of R in ! WORK(IR), storing result in A ! (Workspace: need N*N) ! CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU, & WORK( IR ), LDWRKR, ZERO, A, LDA ) ! ! Copy left singular vectors of A from A to U ! CALL DLACPY( 'F', M, N, A, LDA, U, LDU ) ! ELSE ! ! Insufficient workspace for a fast algorithm ! ITAU = 1 IWORK = ITAU + N ! ! Compute A=Q*R, copying result to U ! (Workspace: need 2*N, prefer N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'L', M, N, A, LDA, U, LDU ) ! ! Generate Q in U ! (Workspace: need N+M, prefer N+M*NB) ! CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Zero out below R in A ! CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), & LDA ) ! ! Bidiagonalize R in A ! (Workspace: need 4*N, prefer 3*N+2*N*NB) ! CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Multiply Q in U by left bidiagonalizing vectors ! in A ! (Workspace: need 3*N+M, prefer 3*N+M*NB) ! CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA, & WORK( ITAUQ ), U, LDU, WORK( IWORK ), & LWORK-IWORK+1, IERR ) IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of A in U ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', N, 0, M, 0, S, WORK( IE ), DUM, & 1, U, LDU, DUM, 1, WORK( IWORK ), & INFO ) ! END IF ! ELSE IF( WNTVO ) THEN ! ! Path 8 (M much larger than N, JOBU='A', JOBVT='O') ! M left singular vectors to be computed in U and ! N right singular vectors to be overwritten on A ! IF( LWORK.GE.2*N*N+MAX( N+M, 4*N, BDSPAC ) ) THEN ! ! Sufficient workspace for a fast algorithm ! IU = 1 IF( LWORK.GE.WRKBL+2*LDA*N ) THEN ! ! WORK(IU) is LDA by N and WORK(IR) is LDA by N ! LDWRKU = LDA IR = IU + LDWRKU*N LDWRKR = LDA ELSE IF( LWORK.GE.WRKBL+( LDA+N )*N ) THEN ! ! WORK(IU) is LDA by N and WORK(IR) is N by N ! LDWRKU = LDA IR = IU + LDWRKU*N LDWRKR = N ELSE ! ! WORK(IU) is N by N and WORK(IR) is N by N ! LDWRKU = N IR = IU + LDWRKU*N LDWRKR = N END IF ITAU = IR + LDWRKR*N IWORK = ITAU + N ! ! Compute A=Q*R, copying result to U ! (Workspace: need 2*N*N+2*N, prefer 2*N*N+N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'L', M, N, A, LDA, U, LDU ) ! ! Generate Q in U ! (Workspace: need 2*N*N+N+M, prefer 2*N*N+N+M*NB) ! CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy R to WORK(IU), zeroing out below it ! CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ), & LDWRKU ) CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, & WORK( IU+1 ), LDWRKU ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Bidiagonalize R in WORK(IU), copying result to ! WORK(IR) ! (Workspace: need 2*N*N+4*N, ! prefer 2*N*N+3*N+2*N*NB) ! CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S, & WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU, & WORK( IR ), LDWRKR ) ! ! Generate left bidiagonalizing vectors in WORK(IU) ! (Workspace: need 2*N*N+4*N, prefer 2*N*N+3*N+N*NB) ! CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU, & WORK( ITAUQ ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate right bidiagonalizing vectors in WORK(IR) ! (Workspace: need 2*N*N+4*N-1, ! prefer 2*N*N+3*N+(N-1)*NB) ! CALL DORGBR( 'P', N, N, N, WORK( IR ), LDWRKR, & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of R in WORK(IU) and computing ! right singular vectors of R in WORK(IR) ! (Workspace: need 2*N*N+BDSPAC) ! CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), & WORK( IR ), LDWRKR, WORK( IU ), & LDWRKU, DUM, 1, WORK( IWORK ), INFO ) ! ! Multiply Q in U by left singular vectors of R in ! WORK(IU), storing result in A ! (Workspace: need N*N) ! CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU, & WORK( IU ), LDWRKU, ZERO, A, LDA ) ! ! Copy left singular vectors of A from A to U ! CALL DLACPY( 'F', M, N, A, LDA, U, LDU ) ! ! Copy right singular vectors of R from WORK(IR) to A ! CALL DLACPY( 'F', N, N, WORK( IR ), LDWRKR, A, & LDA ) ! ELSE ! ! Insufficient workspace for a fast algorithm ! ITAU = 1 IWORK = ITAU + N ! ! Compute A=Q*R, copying result to U ! (Workspace: need 2*N, prefer N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'L', M, N, A, LDA, U, LDU ) ! ! Generate Q in U ! (Workspace: need N+M, prefer N+M*NB) ! CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Zero out below R in A ! CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), & LDA ) ! ! Bidiagonalize R in A ! (Workspace: need 4*N, prefer 3*N+2*N*NB) ! CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Multiply Q in U by left bidiagonalizing vectors ! in A ! (Workspace: need 3*N+M, prefer 3*N+M*NB) ! CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA, & WORK( ITAUQ ), U, LDU, WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate right bidiagonalizing vectors in A ! (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB) ! CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of A in U and computing right ! singular vectors of A in A ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), A, & LDA, U, LDU, DUM, 1, WORK( IWORK ), & INFO ) ! END IF ! ELSE IF( WNTVAS ) THEN ! ! Path 9 (M much larger than N, JOBU='A', JOBVT='S' ! or 'A') ! M left singular vectors to be computed in U and ! N right singular vectors to be computed in VT ! IF( LWORK.GE.N*N+MAX( N+M, 4*N, BDSPAC ) ) THEN ! ! Sufficient workspace for a fast algorithm ! IU = 1 IF( LWORK.GE.WRKBL+LDA*N ) THEN ! ! WORK(IU) is LDA by N ! LDWRKU = LDA ELSE ! ! WORK(IU) is N by N ! LDWRKU = N END IF ITAU = IU + LDWRKU*N IWORK = ITAU + N ! ! Compute A=Q*R, copying result to U ! (Workspace: need N*N+2*N, prefer N*N+N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'L', M, N, A, LDA, U, LDU ) ! ! Generate Q in U ! (Workspace: need N*N+N+M, prefer N*N+N+M*NB) ! CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy R to WORK(IU), zeroing out below it ! CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ), & LDWRKU ) CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, & WORK( IU+1 ), LDWRKU ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Bidiagonalize R in WORK(IU), copying result to VT ! (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB) ! CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S, & WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU, VT, & LDVT ) ! ! Generate left bidiagonalizing vectors in WORK(IU) ! (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB) ! CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU, & WORK( ITAUQ ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate right bidiagonalizing vectors in VT ! (Workspace: need N*N+4*N-1, ! prefer N*N+3*N+(N-1)*NB) ! CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of R in WORK(IU) and computing ! right singular vectors of R in VT ! (Workspace: need N*N+BDSPAC) ! CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT, & LDVT, WORK( IU ), LDWRKU, DUM, 1, & WORK( IWORK ), INFO ) ! ! Multiply Q in U by left singular vectors of R in ! WORK(IU), storing result in A ! (Workspace: need N*N) ! CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU, & WORK( IU ), LDWRKU, ZERO, A, LDA ) ! ! Copy left singular vectors of A from A to U ! CALL DLACPY( 'F', M, N, A, LDA, U, LDU ) ! ELSE ! ! Insufficient workspace for a fast algorithm ! ITAU = 1 IWORK = ITAU + N ! ! Compute A=Q*R, copying result to U ! (Workspace: need 2*N, prefer N+N*NB) ! CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'L', M, N, A, LDA, U, LDU ) ! ! Generate Q in U ! (Workspace: need N+M, prefer N+M*NB) ! CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy R from A to VT, zeroing out below it ! CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT ) CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, VT( 2, 1 ), & LDVT ) IE = ITAU ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Bidiagonalize R in VT ! (Workspace: need 4*N, prefer 3*N+2*N*NB) ! CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Multiply Q in U by left bidiagonalizing vectors ! in VT ! (Workspace: need 3*N+M, prefer 3*N+M*NB) ! CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT, & WORK( ITAUQ ), U, LDU, WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate right bidiagonalizing vectors in VT ! (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB) ! CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + N ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of A in U and computing right ! singular vectors of A in VT ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), VT, & LDVT, U, LDU, DUM, 1, WORK( IWORK ), & INFO ) ! END IF ! END IF ! END IF ! ELSE ! ! M .LT. MNTHR ! ! Path 10 (M at least N, but not much larger) ! Reduce to bidiagonal form without QR decomposition ! IE = 1 ITAUQ = IE + N ITAUP = ITAUQ + N IWORK = ITAUP + N ! ! Bidiagonalize A ! (Workspace: need 3*N+M, prefer 3*N+(M+N)*NB) ! CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1, & IERR ) IF( WNTUAS ) THEN ! ! If left singular vectors desired in U, copy result to U ! and generate left bidiagonalizing vectors in U ! (Workspace: need 3*N+NCU, prefer 3*N+NCU*NB) ! CALL DLACPY( 'L', M, N, A, LDA, U, LDU ) IF( WNTUS ) & NCU = N IF( WNTUA ) & NCU = M CALL DORGBR( 'Q', M, NCU, N, U, LDU, WORK( ITAUQ ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) END IF IF( WNTVAS ) THEN ! ! If right singular vectors desired in VT, copy result to ! VT and generate right bidiagonalizing vectors in VT ! (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB) ! CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT ) CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) END IF IF( WNTUO ) THEN ! ! If left singular vectors desired in A, generate left ! bidiagonalizing vectors in A ! (Workspace: need 4*N, prefer 3*N+N*NB) ! CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) END IF IF( WNTVO ) THEN ! ! If right singular vectors desired in A, generate right ! bidiagonalizing vectors in A ! (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB) ! CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) END IF IWORK = IE + N IF( WNTUAS .OR. WNTUO ) & NRU = M IF( WNTUN ) & NRU = 0 IF( WNTVAS .OR. WNTVO ) & NCVT = N IF( WNTVN ) & NCVT = 0 IF( ( .NOT.WNTUO ) .AND. ( .NOT.WNTVO ) ) THEN ! ! Perform bidiagonal QR iteration, if desired, computing ! left singular vectors in U and computing right singular ! vectors in VT ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', N, NCVT, NRU, 0, S, WORK( IE ), VT, & LDVT, U, LDU, DUM, 1, WORK( IWORK ), INFO ) ELSE IF( ( .NOT.WNTUO ) .AND. WNTVO ) THEN ! ! Perform bidiagonal QR iteration, if desired, computing ! left singular vectors in U and computing right singular ! vectors in A ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', N, NCVT, NRU, 0, S, WORK( IE ), A, LDA, & U, LDU, DUM, 1, WORK( IWORK ), INFO ) ELSE ! ! Perform bidiagonal QR iteration, if desired, computing ! left singular vectors in A and computing right singular ! vectors in VT ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', N, NCVT, NRU, 0, S, WORK( IE ), VT, & LDVT, A, LDA, DUM, 1, WORK( IWORK ), INFO ) END IF ! END IF ! ELSE ! ! A has more columns than rows. If A has sufficiently more ! columns than rows, first reduce using the LQ decomposition (if ! sufficient workspace available) ! IF( N.GE.MNTHR ) THEN ! IF( WNTVN ) THEN ! ! Path 1t(N much larger than M, JOBVT='N') ! No right singular vectors to be computed ! ITAU = 1 IWORK = ITAU + M ! ! Compute A=L*Q ! (Workspace: need 2*M, prefer M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Zero out above L ! CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ), LDA ) IE = 1 ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Bidiagonalize L in A ! (Workspace: need 4*M, prefer 3*M+2*M*NB) ! CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1, & IERR ) IF( WNTUO .OR. WNTUAS ) THEN ! ! If left singular vectors desired, generate Q ! (Workspace: need 4*M, prefer 3*M+M*NB) ! CALL DORGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) END IF IWORK = IE + M NRU = 0 IF( WNTUO .OR. WNTUAS ) & NRU = M ! ! Perform bidiagonal QR iteration, computing left singular ! vectors of A in A if desired ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', M, 0, NRU, 0, S, WORK( IE ), DUM, 1, A, & LDA, DUM, 1, WORK( IWORK ), INFO ) ! ! If left singular vectors desired in U, copy them there ! IF( WNTUAS ) & CALL DLACPY( 'F', M, M, A, LDA, U, LDU ) ! ELSE IF( WNTVO .AND. WNTUN ) THEN ! ! Path 2t(N much larger than M, JOBU='N', JOBVT='O') ! M right singular vectors to be overwritten on A and ! no left singular vectors to be computed ! IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN ! ! Sufficient workspace for a fast algorithm ! IR = 1 IF( LWORK.GE.MAX( WRKBL, LDA*N+M )+LDA*M ) THEN ! ! WORK(IU) is LDA by N and WORK(IR) is LDA by M ! LDWRKU = LDA CHUNK = N LDWRKR = LDA ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N+M )+M*M ) THEN ! ! WORK(IU) is LDA by N and WORK(IR) is M by M ! LDWRKU = LDA CHUNK = N LDWRKR = M ELSE ! ! WORK(IU) is M by CHUNK and WORK(IR) is M by M ! LDWRKU = M CHUNK = ( LWORK-M*M-M ) / M LDWRKR = M END IF ITAU = IR + LDWRKR*M IWORK = ITAU + M ! ! Compute A=L*Q ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy L to WORK(IR) and zero out above it ! CALL DLACPY( 'L', M, M, A, LDA, WORK( IR ), LDWRKR ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, & WORK( IR+LDWRKR ), LDWRKR ) ! ! Generate Q in A ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Bidiagonalize L in WORK(IR) ! (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB) ! CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Generate right vectors bidiagonalizing L ! (Workspace: need M*M+4*M-1, prefer M*M+3*M+(M-1)*NB) ! CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR, & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, computing right ! singular vectors of L in WORK(IR) ! (Workspace: need M*M+BDSPAC) ! CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ), & WORK( IR ), LDWRKR, DUM, 1, DUM, 1, & WORK( IWORK ), INFO ) IU = IE + M ! ! Multiply right singular vectors of L in WORK(IR) by Q ! in A, storing result in WORK(IU) and copying to A ! (Workspace: need M*M+2*M, prefer M*M+M*N+M) ! DO 30 I = 1, N, CHUNK BLK = MIN( N-I+1, CHUNK ) CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IR ), & LDWRKR, A( 1, I ), LDA, ZERO, & WORK( IU ), LDWRKU ) CALL DLACPY( 'F', M, BLK, WORK( IU ), LDWRKU, & A( 1, I ), LDA ) 30 CONTINUE ! ELSE ! ! Insufficient workspace for a fast algorithm ! IE = 1 ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Bidiagonalize A ! (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB) ! CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Generate right vectors bidiagonalizing A ! (Workspace: need 4*M, prefer 3*M+M*NB) ! CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, computing right ! singular vectors of A in A ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'L', M, N, 0, 0, S, WORK( IE ), A, LDA, & DUM, 1, DUM, 1, WORK( IWORK ), INFO ) ! END IF ! ELSE IF( WNTVO .AND. WNTUAS ) THEN ! ! Path 3t(N much larger than M, JOBU='S' or 'A', JOBVT='O') ! M right singular vectors to be overwritten on A and ! M left singular vectors to be computed in U ! IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN ! ! Sufficient workspace for a fast algorithm ! IR = 1 IF( LWORK.GE.MAX( WRKBL, LDA*N+M )+LDA*M ) THEN ! ! WORK(IU) is LDA by N and WORK(IR) is LDA by M ! LDWRKU = LDA CHUNK = N LDWRKR = LDA ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N+M )+M*M ) THEN ! ! WORK(IU) is LDA by N and WORK(IR) is M by M ! LDWRKU = LDA CHUNK = N LDWRKR = M ELSE ! ! WORK(IU) is M by CHUNK and WORK(IR) is M by M ! LDWRKU = M CHUNK = ( LWORK-M*M-M ) / M LDWRKR = M END IF ITAU = IR + LDWRKR*M IWORK = ITAU + M ! ! Compute A=L*Q ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy L to U, zeroing about above it ! CALL DLACPY( 'L', M, M, A, LDA, U, LDU ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1, 2 ), & LDU ) ! ! Generate Q in A ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Bidiagonalize L in U, copying result to WORK(IR) ! (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB) ! CALL DGEBRD( M, M, U, LDU, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'U', M, M, U, LDU, WORK( IR ), LDWRKR ) ! ! Generate right vectors bidiagonalizing L in WORK(IR) ! (Workspace: need M*M+4*M-1, prefer M*M+3*M+(M-1)*NB) ! CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR, & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate left vectors bidiagonalizing L in U ! (Workspace: need M*M+4*M, prefer M*M+3*M+M*NB) ! CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of L in U, and computing right ! singular vectors of L in WORK(IR) ! (Workspace: need M*M+BDSPAC) ! CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ), & WORK( IR ), LDWRKR, U, LDU, DUM, 1, & WORK( IWORK ), INFO ) IU = IE + M ! ! Multiply right singular vectors of L in WORK(IR) by Q ! in A, storing result in WORK(IU) and copying to A ! (Workspace: need M*M+2*M, prefer M*M+M*N+M)) ! DO 40 I = 1, N, CHUNK BLK = MIN( N-I+1, CHUNK ) CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IR ), & LDWRKR, A( 1, I ), LDA, ZERO, & WORK( IU ), LDWRKU ) CALL DLACPY( 'F', M, BLK, WORK( IU ), LDWRKU, & A( 1, I ), LDA ) 40 CONTINUE ! ELSE ! ! Insufficient workspace for a fast algorithm ! ITAU = 1 IWORK = ITAU + M ! ! Compute A=L*Q ! (Workspace: need 2*M, prefer M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy L to U, zeroing out above it ! CALL DLACPY( 'L', M, M, A, LDA, U, LDU ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1, 2 ), & LDU ) ! ! Generate Q in A ! (Workspace: need 2*M, prefer M+M*NB) ! CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Bidiagonalize L in U ! (Workspace: need 4*M, prefer 3*M+2*M*NB) ! CALL DGEBRD( M, M, U, LDU, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Multiply right vectors bidiagonalizing L by Q in A ! (Workspace: need 3*M+N, prefer 3*M+N*NB) ! CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU, & WORK( ITAUP ), A, LDA, WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate left vectors bidiagonalizing L in U ! (Workspace: need 4*M, prefer 3*M+M*NB) ! CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of A in U and computing right ! singular vectors of A in A ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), A, LDA, & U, LDU, DUM, 1, WORK( IWORK ), INFO ) ! END IF ! ELSE IF( WNTVS ) THEN ! IF( WNTUN ) THEN ! ! Path 4t(N much larger than M, JOBU='N', JOBVT='S') ! M right singular vectors to be computed in VT and ! no left singular vectors to be computed ! IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN ! ! Sufficient workspace for a fast algorithm ! IR = 1 IF( LWORK.GE.WRKBL+LDA*M ) THEN ! ! WORK(IR) is LDA by M ! LDWRKR = LDA ELSE ! ! WORK(IR) is M by M ! LDWRKR = M END IF ITAU = IR + LDWRKR*M IWORK = ITAU + M ! ! Compute A=L*Q ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy L to WORK(IR), zeroing out above it ! CALL DLACPY( 'L', M, M, A, LDA, WORK( IR ), & LDWRKR ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, & WORK( IR+LDWRKR ), LDWRKR ) ! ! Generate Q in A ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Bidiagonalize L in WORK(IR) ! (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB) ! CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S, & WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate right vectors bidiagonalizing L in ! WORK(IR) ! (Workspace: need M*M+4*M, prefer M*M+3*M+(M-1)*NB) ! CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR, & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, computing right ! singular vectors of L in WORK(IR) ! (Workspace: need M*M+BDSPAC) ! CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ), & WORK( IR ), LDWRKR, DUM, 1, DUM, 1, & WORK( IWORK ), INFO ) ! ! Multiply right singular vectors of L in WORK(IR) by ! Q in A, storing result in VT ! (Workspace: need M*M) ! CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IR ), & LDWRKR, A, LDA, ZERO, VT, LDVT ) ! ELSE ! ! Insufficient workspace for a fast algorithm ! ITAU = 1 IWORK = ITAU + M ! ! Compute A=L*Q ! (Workspace: need 2*M, prefer M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy result to VT ! CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT ) ! ! Generate Q in VT ! (Workspace: need 2*M, prefer M+M*NB) ! CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Zero out above L in A ! CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ), & LDA ) ! ! Bidiagonalize L in A ! (Workspace: need 4*M, prefer 3*M+2*M*NB) ! CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Multiply right vectors bidiagonalizing L by Q in VT ! (Workspace: need 3*M+N, prefer 3*M+N*NB) ! CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA, & WORK( ITAUP ), VT, LDVT, & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, computing right ! singular vectors of A in VT ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', M, N, 0, 0, S, WORK( IE ), VT, & LDVT, DUM, 1, DUM, 1, WORK( IWORK ), & INFO ) ! END IF ! ELSE IF( WNTUO ) THEN ! ! Path 5t(N much larger than M, JOBU='O', JOBVT='S') ! M right singular vectors to be computed in VT and ! M left singular vectors to be overwritten on A ! IF( LWORK.GE.2*M*M+MAX( 4*M, BDSPAC ) ) THEN ! ! Sufficient workspace for a fast algorithm ! IU = 1 IF( LWORK.GE.WRKBL+2*LDA*M ) THEN ! ! WORK(IU) is LDA by M and WORK(IR) is LDA by M ! LDWRKU = LDA IR = IU + LDWRKU*M LDWRKR = LDA ELSE IF( LWORK.GE.WRKBL+( LDA+M )*M ) THEN ! ! WORK(IU) is LDA by M and WORK(IR) is M by M ! LDWRKU = LDA IR = IU + LDWRKU*M LDWRKR = M ELSE ! ! WORK(IU) is M by M and WORK(IR) is M by M ! LDWRKU = M IR = IU + LDWRKU*M LDWRKR = M END IF ITAU = IR + LDWRKR*M IWORK = ITAU + M ! ! Compute A=L*Q ! (Workspace: need 2*M*M+2*M, prefer 2*M*M+M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy L to WORK(IU), zeroing out below it ! CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ), & LDWRKU ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, & WORK( IU+LDWRKU ), LDWRKU ) ! ! Generate Q in A ! (Workspace: need 2*M*M+2*M, prefer 2*M*M+M+M*NB) ! CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Bidiagonalize L in WORK(IU), copying result to ! WORK(IR) ! (Workspace: need 2*M*M+4*M, ! prefer 2*M*M+3*M+2*M*NB) ! CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S, & WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU, & WORK( IR ), LDWRKR ) ! ! Generate right bidiagonalizing vectors in WORK(IU) ! (Workspace: need 2*M*M+4*M-1, ! prefer 2*M*M+3*M+(M-1)*NB) ! CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU, & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate left bidiagonalizing vectors in WORK(IR) ! (Workspace: need 2*M*M+4*M, prefer 2*M*M+3*M+M*NB) ! CALL DORGBR( 'Q', M, M, M, WORK( IR ), LDWRKR, & WORK( ITAUQ ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of L in WORK(IR) and computing ! right singular vectors of L in WORK(IU) ! (Workspace: need 2*M*M+BDSPAC) ! CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ), & WORK( IU ), LDWRKU, WORK( IR ), & LDWRKR, DUM, 1, WORK( IWORK ), INFO ) ! ! Multiply right singular vectors of L in WORK(IU) by ! Q in A, storing result in VT ! (Workspace: need M*M) ! CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ), & LDWRKU, A, LDA, ZERO, VT, LDVT ) ! ! Copy left singular vectors of L to A ! (Workspace: need M*M) ! CALL DLACPY( 'F', M, M, WORK( IR ), LDWRKR, A, & LDA ) ! ELSE ! ! Insufficient workspace for a fast algorithm ! ITAU = 1 IWORK = ITAU + M ! ! Compute A=L*Q, copying result to VT ! (Workspace: need 2*M, prefer M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT ) ! ! Generate Q in VT ! (Workspace: need 2*M, prefer M+M*NB) ! CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Zero out above L in A ! CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ), & LDA ) ! ! Bidiagonalize L in A ! (Workspace: need 4*M, prefer 3*M+2*M*NB) ! CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Multiply right vectors bidiagonalizing L by Q in VT ! (Workspace: need 3*M+N, prefer 3*M+N*NB) ! CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA, & WORK( ITAUP ), VT, LDVT, & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Generate left bidiagonalizing vectors of L in A ! (Workspace: need 4*M, prefer 3*M+M*NB) ! CALL DORGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, compute left ! singular vectors of A in A and compute right ! singular vectors of A in VT ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT, & LDVT, A, LDA, DUM, 1, WORK( IWORK ), & INFO ) ! END IF ! ELSE IF( WNTUAS ) THEN ! ! Path 6t(N much larger than M, JOBU='S' or 'A', ! JOBVT='S') ! M right singular vectors to be computed in VT and ! M left singular vectors to be computed in U ! IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN ! ! Sufficient workspace for a fast algorithm ! IU = 1 IF( LWORK.GE.WRKBL+LDA*M ) THEN ! ! WORK(IU) is LDA by N ! LDWRKU = LDA ELSE ! ! WORK(IU) is LDA by M ! LDWRKU = M END IF ITAU = IU + LDWRKU*M IWORK = ITAU + M ! ! Compute A=L*Q ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy L to WORK(IU), zeroing out above it ! CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ), & LDWRKU ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, & WORK( IU+LDWRKU ), LDWRKU ) ! ! Generate Q in A ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Bidiagonalize L in WORK(IU), copying result to U ! (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB) ! CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S, & WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU, U, & LDU ) ! ! Generate right bidiagonalizing vectors in WORK(IU) ! (Workspace: need M*M+4*M-1, ! prefer M*M+3*M+(M-1)*NB) ! CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU, & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate left bidiagonalizing vectors in U ! (Workspace: need M*M+4*M, prefer M*M+3*M+M*NB) ! CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of L in U and computing right ! singular vectors of L in WORK(IU) ! (Workspace: need M*M+BDSPAC) ! CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ), & WORK( IU ), LDWRKU, U, LDU, DUM, 1, & WORK( IWORK ), INFO ) ! ! Multiply right singular vectors of L in WORK(IU) by ! Q in A, storing result in VT ! (Workspace: need M*M) ! CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ), & LDWRKU, A, LDA, ZERO, VT, LDVT ) ! ELSE ! ! Insufficient workspace for a fast algorithm ! ITAU = 1 IWORK = ITAU + M ! ! Compute A=L*Q, copying result to VT ! (Workspace: need 2*M, prefer M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT ) ! ! Generate Q in VT ! (Workspace: need 2*M, prefer M+M*NB) ! CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy L to U, zeroing out above it ! CALL DLACPY( 'L', M, M, A, LDA, U, LDU ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1, 2 ), & LDU ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Bidiagonalize L in U ! (Workspace: need 4*M, prefer 3*M+2*M*NB) ! CALL DGEBRD( M, M, U, LDU, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Multiply right bidiagonalizing vectors in U by Q ! in VT ! (Workspace: need 3*M+N, prefer 3*M+N*NB) ! CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU, & WORK( ITAUP ), VT, LDVT, & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Generate left bidiagonalizing vectors in U ! (Workspace: need 4*M, prefer 3*M+M*NB) ! CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of A in U and computing right ! singular vectors of A in VT ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT, & LDVT, U, LDU, DUM, 1, WORK( IWORK ), & INFO ) ! END IF ! END IF ! ELSE IF( WNTVA ) THEN ! IF( WNTUN ) THEN ! ! Path 7t(N much larger than M, JOBU='N', JOBVT='A') ! N right singular vectors to be computed in VT and ! no left singular vectors to be computed ! IF( LWORK.GE.M*M+MAX( N+M, 4*M, BDSPAC ) ) THEN ! ! Sufficient workspace for a fast algorithm ! IR = 1 IF( LWORK.GE.WRKBL+LDA*M ) THEN ! ! WORK(IR) is LDA by M ! LDWRKR = LDA ELSE ! ! WORK(IR) is M by M ! LDWRKR = M END IF ITAU = IR + LDWRKR*M IWORK = ITAU + M ! ! Compute A=L*Q, copying result to VT ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT ) ! ! Copy L to WORK(IR), zeroing out above it ! CALL DLACPY( 'L', M, M, A, LDA, WORK( IR ), & LDWRKR ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, & WORK( IR+LDWRKR ), LDWRKR ) ! ! Generate Q in VT ! (Workspace: need M*M+M+N, prefer M*M+M+N*NB) ! CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Bidiagonalize L in WORK(IR) ! (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB) ! CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S, & WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate right bidiagonalizing vectors in WORK(IR) ! (Workspace: need M*M+4*M-1, ! prefer M*M+3*M+(M-1)*NB) ! CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR, & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, computing right ! singular vectors of L in WORK(IR) ! (Workspace: need M*M+BDSPAC) ! CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ), & WORK( IR ), LDWRKR, DUM, 1, DUM, 1, & WORK( IWORK ), INFO ) ! ! Multiply right singular vectors of L in WORK(IR) by ! Q in VT, storing result in A ! (Workspace: need M*M) ! CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IR ), & LDWRKR, VT, LDVT, ZERO, A, LDA ) ! ! Copy right singular vectors of A from A to VT ! CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT ) ! ELSE ! ! Insufficient workspace for a fast algorithm ! ITAU = 1 IWORK = ITAU + M ! ! Compute A=L*Q, copying result to VT ! (Workspace: need 2*M, prefer M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT ) ! ! Generate Q in VT ! (Workspace: need M+N, prefer M+N*NB) ! CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Zero out above L in A ! CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ), & LDA ) ! ! Bidiagonalize L in A ! (Workspace: need 4*M, prefer 3*M+2*M*NB) ! CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Multiply right bidiagonalizing vectors in A by Q ! in VT ! (Workspace: need 3*M+N, prefer 3*M+N*NB) ! CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA, & WORK( ITAUP ), VT, LDVT, & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, computing right ! singular vectors of A in VT ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', M, N, 0, 0, S, WORK( IE ), VT, & LDVT, DUM, 1, DUM, 1, WORK( IWORK ), & INFO ) ! END IF ! ELSE IF( WNTUO ) THEN ! ! Path 8t(N much larger than M, JOBU='O', JOBVT='A') ! N right singular vectors to be computed in VT and ! M left singular vectors to be overwritten on A ! IF( LWORK.GE.2*M*M+MAX( N+M, 4*M, BDSPAC ) ) THEN ! ! Sufficient workspace for a fast algorithm ! IU = 1 IF( LWORK.GE.WRKBL+2*LDA*M ) THEN ! ! WORK(IU) is LDA by M and WORK(IR) is LDA by M ! LDWRKU = LDA IR = IU + LDWRKU*M LDWRKR = LDA ELSE IF( LWORK.GE.WRKBL+( LDA+M )*M ) THEN ! ! WORK(IU) is LDA by M and WORK(IR) is M by M ! LDWRKU = LDA IR = IU + LDWRKU*M LDWRKR = M ELSE ! ! WORK(IU) is M by M and WORK(IR) is M by M ! LDWRKU = M IR = IU + LDWRKU*M LDWRKR = M END IF ITAU = IR + LDWRKR*M IWORK = ITAU + M ! ! Compute A=L*Q, copying result to VT ! (Workspace: need 2*M*M+2*M, prefer 2*M*M+M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT ) ! ! Generate Q in VT ! (Workspace: need 2*M*M+M+N, prefer 2*M*M+M+N*NB) ! CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy L to WORK(IU), zeroing out above it ! CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ), & LDWRKU ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, & WORK( IU+LDWRKU ), LDWRKU ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Bidiagonalize L in WORK(IU), copying result to ! WORK(IR) ! (Workspace: need 2*M*M+4*M, ! prefer 2*M*M+3*M+2*M*NB) ! CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S, & WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU, & WORK( IR ), LDWRKR ) ! ! Generate right bidiagonalizing vectors in WORK(IU) ! (Workspace: need 2*M*M+4*M-1, ! prefer 2*M*M+3*M+(M-1)*NB) ! CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU, & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate left bidiagonalizing vectors in WORK(IR) ! (Workspace: need 2*M*M+4*M, prefer 2*M*M+3*M+M*NB) ! CALL DORGBR( 'Q', M, M, M, WORK( IR ), LDWRKR, & WORK( ITAUQ ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of L in WORK(IR) and computing ! right singular vectors of L in WORK(IU) ! (Workspace: need 2*M*M+BDSPAC) ! CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ), & WORK( IU ), LDWRKU, WORK( IR ), & LDWRKR, DUM, 1, WORK( IWORK ), INFO ) ! ! Multiply right singular vectors of L in WORK(IU) by ! Q in VT, storing result in A ! (Workspace: need M*M) ! CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ), & LDWRKU, VT, LDVT, ZERO, A, LDA ) ! ! Copy right singular vectors of A from A to VT ! CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT ) ! ! Copy left singular vectors of A from WORK(IR) to A ! CALL DLACPY( 'F', M, M, WORK( IR ), LDWRKR, A, & LDA ) ! ELSE ! ! Insufficient workspace for a fast algorithm ! ITAU = 1 IWORK = ITAU + M ! ! Compute A=L*Q, copying result to VT ! (Workspace: need 2*M, prefer M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT ) ! ! Generate Q in VT ! (Workspace: need M+N, prefer M+N*NB) ! CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Zero out above L in A ! CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ), & LDA ) ! ! Bidiagonalize L in A ! (Workspace: need 4*M, prefer 3*M+2*M*NB) ! CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Multiply right bidiagonalizing vectors in A by Q ! in VT ! (Workspace: need 3*M+N, prefer 3*M+N*NB) ! CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA, & WORK( ITAUP ), VT, LDVT, & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Generate left bidiagonalizing vectors in A ! (Workspace: need 4*M, prefer 3*M+M*NB) ! CALL DORGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of A in A and computing right ! singular vectors of A in VT ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT, & LDVT, A, LDA, DUM, 1, WORK( IWORK ), & INFO ) ! END IF ! ELSE IF( WNTUAS ) THEN ! ! Path 9t(N much larger than M, JOBU='S' or 'A', ! JOBVT='A') ! N right singular vectors to be computed in VT and ! M left singular vectors to be computed in U ! IF( LWORK.GE.M*M+MAX( N+M, 4*M, BDSPAC ) ) THEN ! ! Sufficient workspace for a fast algorithm ! IU = 1 IF( LWORK.GE.WRKBL+LDA*M ) THEN ! ! WORK(IU) is LDA by M ! LDWRKU = LDA ELSE ! ! WORK(IU) is M by M ! LDWRKU = M END IF ITAU = IU + LDWRKU*M IWORK = ITAU + M ! ! Compute A=L*Q, copying result to VT ! (Workspace: need M*M+2*M, prefer M*M+M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT ) ! ! Generate Q in VT ! (Workspace: need M*M+M+N, prefer M*M+M+N*NB) ! CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy L to WORK(IU), zeroing out above it ! CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ), & LDWRKU ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, & WORK( IU+LDWRKU ), LDWRKU ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Bidiagonalize L in WORK(IU), copying result to U ! (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB) ! CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S, & WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU, U, & LDU ) ! ! Generate right bidiagonalizing vectors in WORK(IU) ! (Workspace: need M*M+4*M, prefer M*M+3*M+(M-1)*NB) ! CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU, & WORK( ITAUP ), WORK( IWORK ), & LWORK-IWORK+1, IERR ) ! ! Generate left bidiagonalizing vectors in U ! (Workspace: need M*M+4*M, prefer M*M+3*M+M*NB) ! CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of L in U and computing right ! singular vectors of L in WORK(IU) ! (Workspace: need M*M+BDSPAC) ! CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ), & WORK( IU ), LDWRKU, U, LDU, DUM, 1, & WORK( IWORK ), INFO ) ! ! Multiply right singular vectors of L in WORK(IU) by ! Q in VT, storing result in A ! (Workspace: need M*M) ! CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ), & LDWRKU, VT, LDVT, ZERO, A, LDA ) ! ! Copy right singular vectors of A from A to VT ! CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT ) ! ELSE ! ! Insufficient workspace for a fast algorithm ! ITAU = 1 IWORK = ITAU + M ! ! Compute A=L*Q, copying result to VT ! (Workspace: need 2*M, prefer M+M*NB) ! CALL DGELQF( M, N, A, LDA, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT ) ! ! Generate Q in VT ! (Workspace: need M+N, prefer M+N*NB) ! CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Copy L to U, zeroing out above it ! CALL DLACPY( 'L', M, M, A, LDA, U, LDU ) CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1, 2 ), & LDU ) IE = ITAU ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Bidiagonalize L in U ! (Workspace: need 4*M, prefer 3*M+2*M*NB) ! CALL DGEBRD( M, M, U, LDU, S, WORK( IE ), & WORK( ITAUQ ), WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Multiply right bidiagonalizing vectors in U by Q ! in VT ! (Workspace: need 3*M+N, prefer 3*M+N*NB) ! CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU, & WORK( ITAUP ), VT, LDVT, & WORK( IWORK ), LWORK-IWORK+1, IERR ) ! ! Generate left bidiagonalizing vectors in U ! (Workspace: need 4*M, prefer 3*M+M*NB) ! CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) IWORK = IE + M ! ! Perform bidiagonal QR iteration, computing left ! singular vectors of A in U and computing right ! singular vectors of A in VT ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT, & LDVT, U, LDU, DUM, 1, WORK( IWORK ), & INFO ) ! END IF ! END IF ! END IF ! ELSE ! ! N .LT. MNTHR ! ! Path 10t(N greater than M, but not much larger) ! Reduce to bidiagonal form without LQ decomposition ! IE = 1 ITAUQ = IE + M ITAUP = ITAUQ + M IWORK = ITAUP + M ! ! Bidiagonalize A ! (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB) ! CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), & WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1, & IERR ) IF( WNTUAS ) THEN ! ! If left singular vectors desired in U, copy result to U ! and generate left bidiagonalizing vectors in U ! (Workspace: need 4*M-1, prefer 3*M+(M-1)*NB) ! CALL DLACPY( 'L', M, M, A, LDA, U, LDU ) CALL DORGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) END IF IF( WNTVAS ) THEN ! ! If right singular vectors desired in VT, copy result to ! VT and generate right bidiagonalizing vectors in VT ! (Workspace: need 3*M+NRVT, prefer 3*M+NRVT*NB) ! CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT ) IF( WNTVA ) & NRVT = N IF( WNTVS ) & NRVT = M CALL DORGBR( 'P', NRVT, N, M, VT, LDVT, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) END IF IF( WNTUO ) THEN ! ! If left singular vectors desired in A, generate left ! bidiagonalizing vectors in A ! (Workspace: need 4*M-1, prefer 3*M+(M-1)*NB) ! CALL DORGBR( 'Q', M, M, N, A, LDA, WORK( ITAUQ ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) END IF IF( WNTVO ) THEN ! ! If right singular vectors desired in A, generate right ! bidiagonalizing vectors in A ! (Workspace: need 4*M, prefer 3*M+M*NB) ! CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ), & WORK( IWORK ), LWORK-IWORK+1, IERR ) END IF IWORK = IE + M IF( WNTUAS .OR. WNTUO ) & NRU = M IF( WNTUN ) & NRU = 0 IF( WNTVAS .OR. WNTVO ) & NCVT = N IF( WNTVN ) & NCVT = 0 IF( ( .NOT.WNTUO ) .AND. ( .NOT.WNTVO ) ) THEN ! ! Perform bidiagonal QR iteration, if desired, computing ! left singular vectors in U and computing right singular ! vectors in VT ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'L', M, NCVT, NRU, 0, S, WORK( IE ), VT, & LDVT, U, LDU, DUM, 1, WORK( IWORK ), INFO ) ELSE IF( ( .NOT.WNTUO ) .AND. WNTVO ) THEN ! ! Perform bidiagonal QR iteration, if desired, computing ! left singular vectors in U and computing right singular ! vectors in A ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'L', M, NCVT, NRU, 0, S, WORK( IE ), A, LDA, & U, LDU, DUM, 1, WORK( IWORK ), INFO ) ELSE ! ! Perform bidiagonal QR iteration, if desired, computing ! left singular vectors in A and computing right singular ! vectors in VT ! (Workspace: need BDSPAC) ! CALL DBDSQR( 'L', M, NCVT, NRU, 0, S, WORK( IE ), VT, & LDVT, A, LDA, DUM, 1, WORK( IWORK ), INFO ) END IF ! END IF ! END IF ! ! If DBDSQR failed to converge, copy unconverged superdiagonals ! to WORK( 2:MINMN ) ! IF( INFO.NE.0 ) THEN IF( IE.GT.2 ) THEN DO 50 I = 1, MINMN - 1 WORK( I+1 ) = WORK( I+IE-1 ) 50 CONTINUE END IF IF( IE.LT.2 ) THEN DO 60 I = MINMN - 1, 1, -1 WORK( I+1 ) = WORK( I+IE-1 ) 60 CONTINUE END IF END IF ! ! Undo scaling if necessary ! IF( ISCL.EQ.1 ) THEN IF( ANRM.GT.BIGNUM ) & CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN, & IERR ) IF( INFO.NE.0 .AND. ANRM.GT.BIGNUM ) & CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN-1, 1, WORK( 2 ), & MINMN, IERR ) IF( ANRM.LT.SMLNUM ) & CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN, & IERR ) IF( INFO.NE.0 .AND. ANRM.LT.SMLNUM ) & CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN-1, 1, WORK( 2 ), & MINMN, IERR ) END IF ! ! Return optimal workspace in WORK(1) ! WORK( 1 ) = MAXWRK ! RETURN ! ! End of DGESVD ! END SUBROUTINE DGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, & EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, & WORK, IWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER EQUED, FACT, TRANS INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), & BERR( * ), C( * ), FERR( * ), R( * ), & WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DGESVX uses the LU factorization to compute the solution to a real ! system of linear equations ! A * X = B, ! where A is an N-by-N matrix and X and B are N-by-NRHS matrices. ! ! Error bounds on the solution and a condition estimate are also ! provided. ! ! Description ! =========== ! ! The following steps are performed: ! ! 1. If FACT = 'E', real scaling factors are computed to equilibrate ! the system: ! TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B ! TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B ! TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B ! Whether or not the system will be equilibrated depends on the ! scaling of the matrix A, but if equilibration is used, A is ! overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') ! or diag(C)*B (if TRANS = 'T' or 'C'). ! ! 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the ! matrix A (after equilibration if FACT = 'E') as ! A = P * L * U, ! where P is a permutation matrix, L is a unit lower triangular ! matrix, and U is upper triangular. ! ! 3. If some U(i,i)=0, so that U is exactly singular, then the routine ! returns with INFO = i. Otherwise, the factored form of A is used ! to estimate the condition number of the matrix A. If the ! reciprocal of the condition number is less than machine precision, ! INFO = N+1 is returned as a warning, but the routine still goes on ! to solve for X and compute error bounds as described below. ! ! 4. The system of equations is solved for X using the factored form ! of A. ! ! 5. Iterative refinement is applied to improve the computed solution ! matrix and calculate error bounds and backward error estimates ! for it. ! ! 6. If equilibration was used, the matrix X is premultiplied by ! diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so ! that it solves the original system before equilibration. ! ! Arguments ! ========= ! ! FACT (input) CHARACTER*1 ! Specifies whether or not the factored form of the matrix A is ! supplied on entry, and if not, whether the matrix A should be ! equilibrated before it is factored. ! = 'F': On entry, AF and IPIV contain the factored form of A. ! If EQUED is not 'N', the matrix A has been ! equilibrated with scaling factors given by R and C. ! A, AF, and IPIV are not modified. ! = 'N': The matrix A will be copied to AF and factored. ! = 'E': The matrix A will be equilibrated if necessary, then ! copied to AF and factored. ! ! TRANS (input) CHARACTER*1 ! Specifies the form of the system of equations: ! = 'N': A * X = B (No transpose) ! = 'T': A**T * X = B (Transpose) ! = 'C': A**H * X = B (Transpose) ! ! N (input) INTEGER ! The number of linear equations, i.e., the order of the ! matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is ! not 'N', then A must have been equilibrated by the scaling ! factors in R and/or C. A is not modified if FACT = 'F' or ! 'N', or if FACT = 'E' and EQUED = 'N' on exit. ! ! On exit, if EQUED .ne. 'N', A is scaled as follows: ! EQUED = 'R': A := diag(R) * A ! EQUED = 'C': A := A * diag(C) ! EQUED = 'B': A := diag(R) * A * diag(C). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N) ! If FACT = 'F', then AF is an input argument and on entry ! contains the factors L and U from the factorization ! A = P*L*U as computed by DGETRF. If EQUED .ne. 'N', then ! AF is the factored form of the equilibrated matrix A. ! ! If FACT = 'N', then AF is an output argument and on exit ! returns the factors L and U from the factorization A = P*L*U ! of the original matrix A. ! ! If FACT = 'E', then AF is an output argument and on exit ! returns the factors L and U from the factorization A = P*L*U ! of the equilibrated matrix A (see the description of A for ! the form of the equilibrated matrix). ! ! LDAF (input) INTEGER ! The leading dimension of the array AF. LDAF >= max(1,N). ! ! IPIV (input or output) INTEGER array, dimension (N) ! If FACT = 'F', then IPIV is an input argument and on entry ! contains the pivot indices from the factorization A = P*L*U ! as computed by DGETRF; row i of the matrix was interchanged ! with row IPIV(i). ! ! If FACT = 'N', then IPIV is an output argument and on exit ! contains the pivot indices from the factorization A = P*L*U ! of the original matrix A. ! ! If FACT = 'E', then IPIV is an output argument and on exit ! contains the pivot indices from the factorization A = P*L*U ! of the equilibrated matrix A. ! ! EQUED (input or output) CHARACTER*1 ! Specifies the form of equilibration that was done. ! = 'N': No equilibration (always true if FACT = 'N'). ! = 'R': Row equilibration, i.e., A has been premultiplied by ! diag(R). ! = 'C': Column equilibration, i.e., A has been postmultiplied ! by diag(C). ! = 'B': Both row and column equilibration, i.e., A has been ! replaced by diag(R) * A * diag(C). ! EQUED is an input argument if FACT = 'F'; otherwise, it is an ! output argument. ! ! R (input or output) DOUBLE PRECISION array, dimension (N) ! The row scale factors for A. If EQUED = 'R' or 'B', A is ! multiplied on the left by diag(R); if EQUED = 'N' or 'C', R ! is not accessed. R is an input argument if FACT = 'F'; ! otherwise, R is an output argument. If FACT = 'F' and ! EQUED = 'R' or 'B', each element of R must be positive. ! ! C (input or output) DOUBLE PRECISION array, dimension (N) ! The column scale factors for A. If EQUED = 'C' or 'B', A is ! multiplied on the right by diag(C); if EQUED = 'N' or 'R', C ! is not accessed. C is an input argument if FACT = 'F'; ! otherwise, C is an output argument. If FACT = 'F' and ! EQUED = 'C' or 'B', each element of C must be positive. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the N-by-NRHS right hand side matrix B. ! On exit, ! if EQUED = 'N', B is not modified; ! if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by ! diag(R)*B; ! if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is ! overwritten by diag(C)*B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X ! to the original system of equations. Note that A and B are ! modified on exit if EQUED .ne. 'N', and the solution to the ! equilibrated system is inv(diag(C))*X if TRANS = 'N' and ! EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' ! and EQUED = 'R' or 'B'. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! RCOND (output) DOUBLE PRECISION ! The estimate of the reciprocal condition number of the matrix ! A after equilibration (if done). If RCOND is less than the ! machine precision (in particular, if RCOND = 0), the matrix ! is singular to working precision. This condition is ! indicated by a return code of INFO > 0. ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (4*N) ! On exit, WORK(1) contains the reciprocal pivot growth ! factor norm(A)/norm(U). The "max absolute element" norm is ! used. If WORK(1) is much less than 1, then the stability ! of the LU factorization of the (equilibrated) matrix A ! could be poor. This also means that the solution X, condition ! estimator RCOND, and forward error bound FERR could be ! unreliable. If factorization fails with 0 0: if INFO = i, and i is ! <= N: U(i,i) is exactly zero. The factorization has ! been completed, but the factor U is exactly ! singular, so the solution and error bounds ! could not be computed. RCOND = 0 is returned. ! = N+1: U is nonsingular, but RCOND is less than machine ! precision, meaning that the matrix is singular ! to working precision. Nevertheless, the ! solution and error bounds are computed because ! there are a number of situations where the ! computed solution can be more accurate than the ! value of RCOND would suggest. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU CHARACTER NORM INTEGER I, INFEQU, J DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN, & ROWCND, RPVGRW, SMLNUM ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANGE, DLANTR EXTERNAL LSAME, DLAMCH, DLANGE, DLANTR ! .. ! .. External Subroutines .. EXTERNAL DGECON, DGEEQU, DGERFS, DGETRF, DGETRS, DLACPY, & DLAQGE, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! INFO = 0 NOFACT = LSAME( FACT, 'N' ) EQUIL = LSAME( FACT, 'E' ) NOTRAN = LSAME( TRANS, 'N' ) IF( NOFACT .OR. EQUIL ) THEN EQUED = 'N' ROWEQU = .FALSE. COLEQU = .FALSE. ELSE ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) SMLNUM = DLAMCH( 'Safe minimum' ) BIGNUM = ONE / SMLNUM END IF ! ! Test the input parameters. ! IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) & THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. & LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. & ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN INFO = -10 ELSE IF( ROWEQU ) THEN RCMIN = BIGNUM RCMAX = ZERO DO 10 J = 1, N RCMIN = MIN( RCMIN, R( J ) ) RCMAX = MAX( RCMAX, R( J ) ) 10 CONTINUE IF( RCMIN.LE.ZERO ) THEN INFO = -11 ELSE IF( N.GT.0 ) THEN ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) ELSE ROWCND = ONE END IF END IF IF( COLEQU .AND. INFO.EQ.0 ) THEN RCMIN = BIGNUM RCMAX = ZERO DO 20 J = 1, N RCMIN = MIN( RCMIN, C( J ) ) RCMAX = MAX( RCMAX, C( J ) ) 20 CONTINUE IF( RCMIN.LE.ZERO ) THEN INFO = -12 ELSE IF( N.GT.0 ) THEN COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) ELSE COLCND = ONE END IF END IF IF( INFO.EQ.0 ) THEN IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -14 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -16 END IF END IF END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGESVX', -INFO ) RETURN END IF ! IF( EQUIL ) THEN ! ! Compute row and column scalings to equilibrate the matrix A. ! CALL DGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFEQU ) IF( INFEQU.EQ.0 ) THEN ! ! Equilibrate the matrix. ! CALL DLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, & EQUED ) ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) END IF END IF ! ! Scale the right hand side. ! IF( NOTRAN ) THEN IF( ROWEQU ) THEN DO 40 J = 1, NRHS DO 30 I = 1, N B( I, J ) = R( I )*B( I, J ) 30 CONTINUE 40 CONTINUE END IF ELSE IF( COLEQU ) THEN DO 60 J = 1, NRHS DO 50 I = 1, N B( I, J ) = C( I )*B( I, J ) 50 CONTINUE 60 CONTINUE END IF ! IF( NOFACT .OR. EQUIL ) THEN ! ! Compute the LU factorization of A. ! CALL DLACPY( 'Full', N, N, A, LDA, AF, LDAF ) CALL DGETRF( N, N, AF, LDAF, IPIV, INFO ) ! ! Return if INFO is non-zero. ! IF( INFO.NE.0 ) THEN IF( INFO.GT.0 ) THEN ! ! Compute the reciprocal pivot growth factor of the ! leading rank-deficient INFO columns of A. ! RPVGRW = DLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF, & WORK ) IF( RPVGRW.EQ.ZERO ) THEN RPVGRW = ONE ELSE RPVGRW = DLANGE( 'M', N, INFO, A, LDA, WORK ) / RPVGRW END IF WORK( 1 ) = RPVGRW RCOND = ZERO END IF RETURN END IF END IF ! ! Compute the norm of the matrix A and the ! reciprocal pivot growth factor RPVGRW. ! IF( NOTRAN ) THEN NORM = '1' ELSE NORM = 'I' END IF ANORM = DLANGE( NORM, N, N, A, LDA, WORK ) RPVGRW = DLANTR( 'M', 'U', 'N', N, N, AF, LDAF, WORK ) IF( RPVGRW.EQ.ZERO ) THEN RPVGRW = ONE ELSE RPVGRW = DLANGE( 'M', N, N, A, LDA, WORK ) / RPVGRW END IF ! ! Compute the reciprocal of the condition number of A. ! CALL DGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, IWORK, INFO ) ! ! Set INFO = N+1 if the matrix is singular to working precision. ! IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) & INFO = N + 1 ! ! Compute the solution matrix X. ! CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL DGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO ) ! ! Use iterative refinement to improve the computed solution and ! compute error bounds and backward error estimates for it. ! CALL DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, & LDX, FERR, BERR, WORK, IWORK, INFO ) ! ! Transform the solution matrix X to a solution of the original ! system. ! IF( NOTRAN ) THEN IF( COLEQU ) THEN DO 80 J = 1, NRHS DO 70 I = 1, N X( I, J ) = C( I )*X( I, J ) 70 CONTINUE 80 CONTINUE DO 90 J = 1, NRHS FERR( J ) = FERR( J ) / COLCND 90 CONTINUE END IF ELSE IF( ROWEQU ) THEN DO 110 J = 1, NRHS DO 100 I = 1, N X( I, J ) = R( I )*X( I, J ) 100 CONTINUE 110 CONTINUE DO 120 J = 1, NRHS FERR( J ) = FERR( J ) / ROWCND 120 CONTINUE END IF ! WORK( 1 ) = RPVGRW RETURN ! ! End of DGESVX ! END SUBROUTINE DGETC2( N, A, LDA, IPIV, JPIV, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, N ! .. ! .. Array Arguments .. INTEGER IPIV( * ), JPIV( * ) DOUBLE PRECISION A( LDA, * ) ! .. ! ! Purpose ! ======= ! ! DGETC2 computes an LU factorization with complete pivoting of the ! n-by-n matrix A. The factorization has the form A = P * L * U * Q, ! where P and Q are permutation matrices, L is lower triangular with ! unit diagonal elements and U is upper triangular. ! ! This is the Level 2 BLAS algorithm. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the n-by-n matrix A to be factored. ! On exit, the factors L and U from the factorization ! A = P*L*U*Q; the unit diagonal elements of L are not stored. ! If U(k, k) appears to be less than SMIN, U(k, k) is given the ! value of SMIN, i.e., giving a nonsingular perturbed system. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! IPIV (output) INTEGER array, dimension(N). ! The pivot indices; for 1 <= i <= N, row i of the ! matrix has been interchanged with row IPIV(i). ! ! JPIV (output) INTEGER array, dimension(N). ! The pivot indices; for 1 <= j <= N, column j of the ! matrix has been interchanged with column JPIV(j). ! ! INFO (output) INTEGER ! = 0: successful exit ! > 0: if INFO = k, U(k, k) is likely to produce owerflow if ! we try to solve for x in Ax = b. So U is perturbed to ! avoid the overflow. ! ! Further Details ! =============== ! ! Based on contributions by ! Bo Kagstrom and Peter Poromaa, Department of Computing Science, ! Umea University, S-901 87 Umea, Sweden. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, IP, IPV, J, JP, JPV DOUBLE PRECISION BIGNUM, EPS, SMIN, SMLNUM, XMAX ! .. ! .. External Subroutines .. EXTERNAL DGER, DSWAP ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. Executable Statements .. ! ! Set constants to control overflow ! INFO = 0 EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) / EPS BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) ! ! Factorize A using complete pivoting. ! Set pivots less than SMIN to SMIN. ! DO 40 I = 1, N - 1 ! ! Find max element in matrix A ! XMAX = ZERO DO 20 IP = I, N DO 10 JP = I, N IF( ABS( A( IP, JP ) ).GE.XMAX ) THEN XMAX = ABS( A( IP, JP ) ) IPV = IP JPV = JP END IF 10 CONTINUE 20 CONTINUE IF( I.EQ.1 ) & SMIN = MAX( EPS*XMAX, SMLNUM ) ! ! Swap rows ! IF( IPV.NE.I ) & CALL DSWAP( N, A( IPV, 1 ), LDA, A( I, 1 ), LDA ) IPIV( I ) = IPV ! ! Swap columns ! IF( JPV.NE.I ) & CALL DSWAP( N, A( 1, JPV ), 1, A( 1, I ), 1 ) JPIV( I ) = JPV ! ! Check for singularity ! IF( ABS( A( I, I ) ).LT.SMIN ) THEN INFO = I A( I, I ) = SMIN END IF DO 30 J = I + 1, N A( J, I ) = A( J, I ) / A( I, I ) 30 CONTINUE CALL DGER( N-I, N-I, -ONE, A( I+1, I ), 1, A( I, I+1 ), LDA, & A( I+1, I+1 ), LDA ) 40 CONTINUE ! IF( ABS( A( N, N ) ).LT.SMIN ) THEN INFO = N A( N, N ) = SMIN END IF ! RETURN ! ! End of DGETC2 ! END SUBROUTINE DGETF2( M, N, A, LDA, IPIV, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1992 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, M, N ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION A( LDA, * ) ! .. ! ! Purpose ! ======= ! ! DGETF2 computes an LU factorization of a general m-by-n matrix A ! using partial pivoting with row interchanges. ! ! The factorization has the form ! A = P * L * U ! where P is a permutation matrix, L is lower triangular with unit ! diagonal elements (lower trapezoidal if m > n), and U is upper ! triangular (upper trapezoidal if m < n). ! ! This is the right-looking Level 2 BLAS version of the algorithm. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the m by n matrix to be factored. ! On exit, the factors L and U from the factorization ! A = P*L*U; the unit diagonal elements of L are not stored. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! IPIV (output) INTEGER array, dimension (min(M,N)) ! The pivot indices; for 1 <= i <= min(M,N), row i of the ! matrix was interchanged with row IPIV(i). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -k, the k-th argument had an illegal value ! > 0: if INFO = k, U(k,k) is exactly zero. The factorization ! has been completed, but the factor U is exactly ! singular, and division by zero will occur if it is used ! to solve a system of equations. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER J, JP ! .. ! .. External Functions .. INTEGER IDAMAX EXTERNAL IDAMAX ! .. ! .. External Subroutines .. EXTERNAL DGER, DSCAL, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGETF2', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 ) & RETURN ! DO 10 J = 1, MIN( M, N ) ! ! Find pivot and test for singularity. ! JP = J - 1 + IDAMAX( M-J+1, A( J, J ), 1 ) IPIV( J ) = JP IF( A( JP, J ).NE.ZERO ) THEN ! ! Apply the interchange to columns 1:N. ! IF( JP.NE.J ) & CALL DSWAP( N, A( J, 1 ), LDA, A( JP, 1 ), LDA ) ! ! Compute elements J+1:M of J-th column. ! IF( J.LT.M ) & CALL DSCAL( M-J, ONE / A( J, J ), A( J+1, J ), 1 ) ! ELSE IF( INFO.EQ.0 ) THEN ! INFO = J END IF ! IF( J.LT.MIN( M, N ) ) THEN ! ! Update trailing submatrix. ! CALL DGER( M-J, N-J, -ONE, A( J+1, J ), 1, A( J, J+1 ), LDA, & A( J+1, J+1 ), LDA ) END IF 10 CONTINUE RETURN ! ! End of DGETF2 ! END SUBROUTINE DGETRF( M, N, A, LDA, IPIV, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, M, N ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION A( LDA, * ) ! .. ! ! Purpose ! ======= ! ! DGETRF computes an LU factorization of a general M-by-N matrix A ! using partial pivoting with row interchanges. ! ! The factorization has the form ! A = P * L * U ! where P is a permutation matrix, L is lower triangular with unit ! diagonal elements (lower trapezoidal if m > n), and U is upper ! triangular (upper trapezoidal if m < n). ! ! This is the right-looking Level 3 BLAS version of the algorithm. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix to be factored. ! On exit, the factors L and U from the factorization ! A = P*L*U; the unit diagonal elements of L are not stored. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! IPIV (output) INTEGER array, dimension (min(M,N)) ! The pivot indices; for 1 <= i <= min(M,N), row i of the ! matrix was interchanged with row IPIV(i). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, U(i,i) is exactly zero. The factorization ! has been completed, but the factor U is exactly ! singular, and division by zero will occur if it is used ! to solve a system of equations. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, IINFO, J, JB, NB ! .. ! .. External Subroutines .. EXTERNAL DGEMM, DGETF2, DLASWP, DTRSM, XERBLA ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGETRF', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 ) & RETURN ! ! Determine the block size for this environment. ! NB = ILAENV( 1, 'DGETRF', ' ', M, N, -1, -1 ) IF( NB.LE.1 .OR. NB.GE.MIN( M, N ) ) THEN ! ! Use unblocked code. ! CALL DGETF2( M, N, A, LDA, IPIV, INFO ) ELSE ! ! Use blocked code. ! DO 20 J = 1, MIN( M, N ), NB JB = MIN( MIN( M, N )-J+1, NB ) ! ! Factor diagonal and subdiagonal blocks and test for exact ! singularity. ! CALL DGETF2( M-J+1, JB, A( J, J ), LDA, IPIV( J ), IINFO ) ! ! Adjust INFO and the pivot indices. ! IF( INFO.EQ.0 .AND. IINFO.GT.0 ) & INFO = IINFO + J - 1 DO 10 I = J, MIN( M, J+JB-1 ) IPIV( I ) = J - 1 + IPIV( I ) 10 CONTINUE ! ! Apply interchanges to columns 1:J-1. ! CALL DLASWP( J-1, A, LDA, J, J+JB-1, IPIV, 1 ) ! IF( J+JB.LE.N ) THEN ! ! Apply interchanges to columns J+JB:N. ! CALL DLASWP( N-J-JB+1, A( 1, J+JB ), LDA, J, J+JB-1, & IPIV, 1 ) ! ! Compute block row of U. ! CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Unit', JB, & N-J-JB+1, ONE, A( J, J ), LDA, A( J, J+JB ), & LDA ) IF( J+JB.LE.M ) THEN ! ! Update trailing submatrix. ! CALL DGEMM( 'No transpose', 'No transpose', M-J-JB+1, & N-J-JB+1, JB, -ONE, A( J+JB, J ), LDA, & A( J, J+JB ), LDA, ONE, A( J+JB, J+JB ), & LDA ) END IF END IF 20 CONTINUE END IF RETURN ! ! End of DGETRF ! END SUBROUTINE DGETRI( N, A, LDA, IPIV, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LWORK, N ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION A( LDA, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGETRI computes the inverse of a matrix using the LU factorization ! computed by DGETRF. ! ! This method inverts U and then computes inv(A) by solving the system ! inv(A)*L = inv(U) for inv(A). ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the factors L and U from the factorization ! A = P*L*U as computed by DGETRF. ! On exit, if INFO = 0, the inverse of the original matrix A. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! IPIV (input) INTEGER array, dimension (N) ! The pivot indices from DGETRF; for 1<=i<=N, row i of the ! matrix was interchanged with row IPIV(i). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO=0, then WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,N). ! For optimal performance LWORK >= N*NB, where NB is ! the optimal blocksize returned by ILAENV. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, U(i,i) is exactly zero; the matrix is ! singular and its inverse could not be computed. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, IWS, J, JB, JJ, JP, LDWORK, LWKOPT, NB, & NBMIN, NN ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. External Subroutines .. EXTERNAL DGEMM, DGEMV, DSWAP, DTRSM, DTRTRI, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 NB = ILAENV( 1, 'DGETRI', ' ', N, -1, -1, -1 ) LWKOPT = N*NB WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) IF( N.LT.0 ) THEN INFO = -1 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -3 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGETRI', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Form inv(U). If INFO > 0 from DTRTRI, then U is singular, ! and the inverse is not computed. ! CALL DTRTRI( 'Upper', 'Non-unit', N, A, LDA, INFO ) IF( INFO.GT.0 ) & RETURN ! NBMIN = 2 LDWORK = N IF( NB.GT.1 .AND. NB.LT.N ) THEN IWS = MAX( LDWORK*NB, 1 ) IF( LWORK.LT.IWS ) THEN NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'DGETRI', ' ', N, -1, -1, -1 ) ) END IF ELSE IWS = N END IF ! ! Solve the equation inv(A)*L = inv(U) for inv(A). ! IF( NB.LT.NBMIN .OR. NB.GE.N ) THEN ! ! Use unblocked code. ! DO 20 J = N, 1, -1 ! ! Copy current column of L to WORK and replace with zeros. ! DO 10 I = J + 1, N WORK( I ) = A( I, J ) A( I, J ) = ZERO 10 CONTINUE ! ! Compute current column of inv(A). ! IF( J.LT.N ) & CALL DGEMV( 'No transpose', N, N-J, -ONE, A( 1, J+1 ), & LDA, WORK( J+1 ), 1, ONE, A( 1, J ), 1 ) 20 CONTINUE ELSE ! ! Use blocked code. ! NN = ( ( N-1 ) / NB )*NB + 1 DO 50 J = NN, 1, -NB JB = MIN( NB, N-J+1 ) ! ! Copy current block column of L to WORK and replace with ! zeros. ! DO 40 JJ = J, J + JB - 1 DO 30 I = JJ + 1, N WORK( I+( JJ-J )*LDWORK ) = A( I, JJ ) A( I, JJ ) = ZERO 30 CONTINUE 40 CONTINUE ! ! Compute current block column of inv(A). ! IF( J+JB.LE.N ) & CALL DGEMM( 'No transpose', 'No transpose', N, JB, & N-J-JB+1, -ONE, A( 1, J+JB ), LDA, & WORK( J+JB ), LDWORK, ONE, A( 1, J ), LDA ) CALL DTRSM( 'Right', 'Lower', 'No transpose', 'Unit', N, JB, & ONE, WORK( J ), LDWORK, A( 1, J ), LDA ) 50 CONTINUE END IF ! ! Apply column interchanges. ! DO 60 J = N - 1, 1, -1 JP = IPIV( J ) IF( JP.NE.J ) & CALL DSWAP( N, A( 1, J ), 1, A( 1, JP ), 1 ) 60 CONTINUE ! WORK( 1 ) = IWS RETURN ! ! End of DGETRI ! END SUBROUTINE DGETRS( TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER TRANS INTEGER INFO, LDA, LDB, N, NRHS ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DGETRS solves a system of linear equations ! A * X = B or A' * X = B ! with a general N-by-N matrix A using the LU factorization computed ! by DGETRF. ! ! Arguments ! ========= ! ! TRANS (input) CHARACTER*1 ! Specifies the form of the system of equations: ! = 'N': A * X = B (No transpose) ! = 'T': A'* X = B (Transpose) ! = 'C': A'* X = B (Conjugate transpose = Transpose) ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The factors L and U from the factorization A = P*L*U ! as computed by DGETRF. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! IPIV (input) INTEGER array, dimension (N) ! The pivot indices from DGETRF; for 1<=i<=N, row i of the ! matrix was interchanged with row IPIV(i). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the right hand side matrix B. ! On exit, the solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOTRAN ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DLASWP, DTRSM, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 NOTRAN = LSAME( TRANS, 'N' ) IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. & LSAME( TRANS, 'C' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGETRS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) & RETURN ! IF( NOTRAN ) THEN ! ! Solve A * X = B. ! ! Apply row interchanges to the right hand sides. ! CALL DLASWP( NRHS, B, LDB, 1, N, IPIV, 1 ) ! ! Solve L*X = B, overwriting B with X. ! CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Unit', N, NRHS, & ONE, A, LDA, B, LDB ) ! ! Solve U*X = B, overwriting B with X. ! CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', N, & NRHS, ONE, A, LDA, B, LDB ) ELSE ! ! Solve A' * X = B. ! ! Solve U'*X = B, overwriting B with X. ! CALL DTRSM( 'Left', 'Upper', 'Transpose', 'Non-unit', N, NRHS, & ONE, A, LDA, B, LDB ) ! ! Solve L'*X = B, overwriting B with X. ! CALL DTRSM( 'Left', 'Lower', 'Transpose', 'Unit', N, NRHS, ONE, & A, LDA, B, LDB ) ! ! Apply row interchanges to the solution vectors. ! CALL DLASWP( NRHS, B, LDB, 1, N, IPIV, -1 ) END IF ! RETURN ! ! End of DGETRS ! END SUBROUTINE DGGBAK( JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, & LDV, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER JOB, SIDE INTEGER IHI, ILO, INFO, LDV, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION LSCALE( * ), RSCALE( * ), V( LDV, * ) ! .. ! ! Purpose ! ======= ! ! DGGBAK forms the right or left eigenvectors of a real generalized ! eigenvalue problem A*x = lambda*B*x, by backward transformation on ! the computed eigenvectors of the balanced pair of matrices output by ! DGGBAL. ! ! Arguments ! ========= ! ! JOB (input) CHARACTER*1 ! Specifies the type of backward transformation required: ! = 'N': do nothing, return immediately; ! = 'P': do backward transformation for permutation only; ! = 'S': do backward transformation for scaling only; ! = 'B': do backward transformations for both permutation and ! scaling. ! JOB must be the same as the argument JOB supplied to DGGBAL. ! ! SIDE (input) CHARACTER*1 ! = 'R': V contains right eigenvectors; ! = 'L': V contains left eigenvectors. ! ! N (input) INTEGER ! The number of rows of the matrix V. N >= 0. ! ! ILO (input) INTEGER ! IHI (input) INTEGER ! The integers ILO and IHI determined by DGGBAL. ! 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. ! ! LSCALE (input) DOUBLE PRECISION array, dimension (N) ! Details of the permutations and/or scaling factors applied ! to the left side of A and B, as returned by DGGBAL. ! ! RSCALE (input) DOUBLE PRECISION array, dimension (N) ! Details of the permutations and/or scaling factors applied ! to the right side of A and B, as returned by DGGBAL. ! ! M (input) INTEGER ! The number of columns of the matrix V. M >= 0. ! ! V (input/output) DOUBLE PRECISION array, dimension (LDV,M) ! On entry, the matrix of right or left eigenvectors to be ! transformed, as returned by DTGEVC. ! On exit, V is overwritten by the transformed eigenvectors. ! ! LDV (input) INTEGER ! The leading dimension of the matrix V. LDV >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! See R.C. Ward, Balancing the generalized eigenvalue problem, ! SIAM J. Sci. Stat. Comp. 2 (1981), 141-152. ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL LEFTV, RIGHTV INTEGER I, K ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DSCAL, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters ! RIGHTV = LSAME( SIDE, 'R' ) LEFTV = LSAME( SIDE, 'L' ) ! INFO = 0 IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND. & .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN INFO = -1 ELSE IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( ILO.LT.1 ) THEN INFO = -4 ELSE IF( IHI.LT.ILO .OR. IHI.GT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( M.LT.0 ) THEN INFO = -6 ELSE IF( LDV.LT.MAX( 1, N ) ) THEN INFO = -10 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGGBAK', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN IF( M.EQ.0 ) & RETURN IF( LSAME( JOB, 'N' ) ) & RETURN ! IF( ILO.EQ.IHI ) & GO TO 30 ! ! Backward balance ! IF( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'B' ) ) THEN ! ! Backward transformation on right eigenvectors ! IF( RIGHTV ) THEN DO 10 I = ILO, IHI CALL DSCAL( M, RSCALE( I ), V( I, 1 ), LDV ) 10 CONTINUE END IF ! ! Backward transformation on left eigenvectors ! IF( LEFTV ) THEN DO 20 I = ILO, IHI CALL DSCAL( M, LSCALE( I ), V( I, 1 ), LDV ) 20 CONTINUE END IF END IF ! ! Backward permutation ! 30 CONTINUE IF( LSAME( JOB, 'P' ) .OR. LSAME( JOB, 'B' ) ) THEN ! ! Backward permutation on right eigenvectors ! IF( RIGHTV ) THEN IF( ILO.EQ.1 ) & GO TO 50 ! DO 40 I = ILO - 1, 1, -1 K = RSCALE( I ) IF( K.EQ.I ) & GO TO 40 CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV ) 40 CONTINUE ! 50 CONTINUE IF( IHI.EQ.N ) & GO TO 70 DO 60 I = IHI + 1, N K = RSCALE( I ) IF( K.EQ.I ) & GO TO 60 CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV ) 60 CONTINUE END IF ! ! Backward permutation on left eigenvectors ! 70 CONTINUE IF( LEFTV ) THEN IF( ILO.EQ.1 ) & GO TO 90 DO 80 I = ILO - 1, 1, -1 K = LSCALE( I ) IF( K.EQ.I ) & GO TO 80 CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV ) 80 CONTINUE ! 90 CONTINUE IF( IHI.EQ.N ) & GO TO 110 DO 100 I = IHI + 1, N K = LSCALE( I ) IF( K.EQ.I ) & GO TO 100 CALL DSWAP( M, V( I, 1 ), LDV, V( K, 1 ), LDV ) 100 CONTINUE END IF END IF ! 110 CONTINUE ! RETURN ! ! End of DGGBAK ! END SUBROUTINE DGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, & RSCALE, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER JOB INTEGER IHI, ILO, INFO, LDA, LDB, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), LSCALE( * ), & RSCALE( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGGBAL balances a pair of general real matrices (A,B). This ! involves, first, permuting A and B by similarity transformations to ! isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N ! elements on the diagonal; and second, applying a diagonal similarity ! transformation to rows and columns ILO to IHI to make the rows ! and columns as close in norm as possible. Both steps are optional. ! ! Balancing may reduce the 1-norm of the matrices, and improve the ! accuracy of the computed eigenvalues and/or eigenvectors in the ! generalized eigenvalue problem A*x = lambda*B*x. ! ! Arguments ! ========= ! ! JOB (input) CHARACTER*1 ! Specifies the operations to be performed on A and B: ! = 'N': none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0 ! and RSCALE(I) = 1.0 for i = 1,...,N. ! = 'P': permute only; ! = 'S': scale only; ! = 'B': both permute and scale. ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the input matrix A. ! On exit, A is overwritten by the balanced matrix. ! If JOB = 'N', A is not referenced. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,N) ! On entry, the input matrix B. ! On exit, B is overwritten by the balanced matrix. ! If JOB = 'N', B is not referenced. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! ILO (output) INTEGER ! IHI (output) INTEGER ! ILO and IHI are set to integers such that on exit ! A(i,j) = 0 and B(i,j) = 0 if i > j and ! j = 1,...,ILO-1 or i = IHI+1,...,N. ! If JOB = 'N' or 'S', ILO = 1 and IHI = N. ! ! LSCALE (output) DOUBLE PRECISION array, dimension (N) ! Details of the permutations and scaling factors applied ! to the left side of A and B. If P(j) is the index of the ! row interchanged with row j, and D(j) ! is the scaling factor applied to row j, then ! LSCALE(j) = P(j) for J = 1,...,ILO-1 ! = D(j) for J = ILO,...,IHI ! = P(j) for J = IHI+1,...,N. ! The order in which the interchanges are made is N to IHI+1, ! then 1 to ILO-1. ! ! RSCALE (output) DOUBLE PRECISION array, dimension (N) ! Details of the permutations and scaling factors applied ! to the right side of A and B. If P(j) is the index of the ! column interchanged with column j, and D(j) ! is the scaling factor applied to column j, then ! LSCALE(j) = P(j) for J = 1,...,ILO-1 ! = D(j) for J = ILO,...,IHI ! = P(j) for J = IHI+1,...,N. ! The order in which the interchanges are made is N to IHI+1, ! then 1 to ILO-1. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (6*N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! See R.C. WARD, Balancing the generalized eigenvalue problem, ! SIAM J. Sci. Stat. Comp. 2 (1981), 141-152. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, HALF, ONE PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 ) DOUBLE PRECISION THREE, SCLFAC PARAMETER ( THREE = 3.0D+0, SCLFAC = 1.0D+1 ) ! .. ! .. Local Scalars .. INTEGER I, ICAB, IFLOW, IP1, IR, IRAB, IT, J, JC, JP1, & K, KOUNT, L, LCAB, LM1, LRAB, LSFMAX, LSFMIN, & M, NR, NRP2 DOUBLE PRECISION ALPHA, BASL, BETA, CAB, CMAX, COEF, COEF2, & COEF5, COR, EW, EWC, GAMMA, PGAMMA, RAB, SFMAX, & SFMIN, SUM, T, TA, TB, TC ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DDOT, DLAMCH EXTERNAL LSAME, IDAMAX, DDOT, DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DSCAL, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, INT, LOG10, MAX, MIN, SIGN ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND. & .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGGBAL', -INFO ) RETURN END IF ! K = 1 L = N ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( LSAME( JOB, 'N' ) ) THEN ILO = 1 IHI = N DO 10 I = 1, N LSCALE( I ) = ONE RSCALE( I ) = ONE 10 CONTINUE RETURN END IF ! IF( K.EQ.L ) THEN ILO = 1 IHI = 1 LSCALE( 1 ) = ONE RSCALE( 1 ) = ONE RETURN END IF ! IF( LSAME( JOB, 'S' ) ) & GO TO 190 ! GO TO 30 ! ! Permute the matrices A and B to isolate the eigenvalues. ! ! Find row with one nonzero in columns 1 through L ! 20 CONTINUE L = LM1 IF( L.NE.1 ) & GO TO 30 ! RSCALE( 1 ) = 1 LSCALE( 1 ) = 1 GO TO 190 ! 30 CONTINUE LM1 = L - 1 DO 80 I = L, 1, -1 DO 40 J = 1, LM1 JP1 = J + 1 IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO ) & GO TO 50 40 CONTINUE J = L GO TO 70 ! 50 CONTINUE DO 60 J = JP1, L IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO ) & GO TO 80 60 CONTINUE J = JP1 - 1 ! 70 CONTINUE M = L IFLOW = 1 GO TO 160 80 CONTINUE GO TO 100 ! ! Find column with one nonzero in rows K through N ! 90 CONTINUE K = K + 1 ! 100 CONTINUE DO 150 J = K, L DO 110 I = K, LM1 IP1 = I + 1 IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO ) & GO TO 120 110 CONTINUE I = L GO TO 140 120 CONTINUE DO 130 I = IP1, L IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO ) & GO TO 150 130 CONTINUE I = IP1 - 1 140 CONTINUE M = K IFLOW = 2 GO TO 160 150 CONTINUE GO TO 190 ! ! Permute rows M and I ! 160 CONTINUE LSCALE( M ) = I IF( I.EQ.M ) & GO TO 170 CALL DSWAP( N-K+1, A( I, K ), LDA, A( M, K ), LDA ) CALL DSWAP( N-K+1, B( I, K ), LDB, B( M, K ), LDB ) ! ! Permute columns M and J ! 170 CONTINUE RSCALE( M ) = J IF( J.EQ.M ) & GO TO 180 CALL DSWAP( L, A( 1, J ), 1, A( 1, M ), 1 ) CALL DSWAP( L, B( 1, J ), 1, B( 1, M ), 1 ) ! 180 CONTINUE GO TO ( 20, 90 )IFLOW ! 190 CONTINUE ILO = K IHI = L ! IF( ILO.EQ.IHI ) & RETURN ! IF( LSAME( JOB, 'P' ) ) & RETURN ! ! Balance the submatrix in rows ILO to IHI. ! NR = IHI - ILO + 1 DO 200 I = ILO, IHI RSCALE( I ) = ZERO LSCALE( I ) = ZERO ! WORK( I ) = ZERO WORK( I+N ) = ZERO WORK( I+2*N ) = ZERO WORK( I+3*N ) = ZERO WORK( I+4*N ) = ZERO WORK( I+5*N ) = ZERO 200 CONTINUE ! ! Compute right side vector in resulting linear equations ! BASL = LOG10( SCLFAC ) DO 240 I = ILO, IHI DO 230 J = ILO, IHI TB = B( I, J ) TA = A( I, J ) IF( TA.EQ.ZERO ) & GO TO 210 TA = LOG10( ABS( TA ) ) / BASL 210 CONTINUE IF( TB.EQ.ZERO ) & GO TO 220 TB = LOG10( ABS( TB ) ) / BASL 220 CONTINUE WORK( I+4*N ) = WORK( I+4*N ) - TA - TB WORK( J+5*N ) = WORK( J+5*N ) - TA - TB 230 CONTINUE 240 CONTINUE ! COEF = ONE / DBLE( 2*NR ) COEF2 = COEF*COEF COEF5 = HALF*COEF2 NRP2 = NR + 2 BETA = ZERO IT = 1 ! ! Start generalized conjugate gradient iteration ! 250 CONTINUE ! GAMMA = DDOT( NR, WORK( ILO+4*N ), 1, WORK( ILO+4*N ), 1 ) + & DDOT( NR, WORK( ILO+5*N ), 1, WORK( ILO+5*N ), 1 ) ! EW = ZERO EWC = ZERO DO 260 I = ILO, IHI EW = EW + WORK( I+4*N ) EWC = EWC + WORK( I+5*N ) 260 CONTINUE ! GAMMA = COEF*GAMMA - COEF2*( EW**2+EWC**2 ) - COEF5*( EW-EWC )**2 IF( GAMMA.EQ.ZERO ) & GO TO 350 IF( IT.NE.1 ) & BETA = GAMMA / PGAMMA T = COEF5*( EWC-THREE*EW ) TC = COEF5*( EW-THREE*EWC ) ! CALL DSCAL( NR, BETA, WORK( ILO ), 1 ) CALL DSCAL( NR, BETA, WORK( ILO+N ), 1 ) ! CALL DAXPY( NR, COEF, WORK( ILO+4*N ), 1, WORK( ILO+N ), 1 ) CALL DAXPY( NR, COEF, WORK( ILO+5*N ), 1, WORK( ILO ), 1 ) ! DO 270 I = ILO, IHI WORK( I ) = WORK( I ) + TC WORK( I+N ) = WORK( I+N ) + T 270 CONTINUE ! ! Apply matrix to vector ! DO 300 I = ILO, IHI KOUNT = 0 SUM = ZERO DO 290 J = ILO, IHI IF( A( I, J ).EQ.ZERO ) & GO TO 280 KOUNT = KOUNT + 1 SUM = SUM + WORK( J ) 280 CONTINUE IF( B( I, J ).EQ.ZERO ) & GO TO 290 KOUNT = KOUNT + 1 SUM = SUM + WORK( J ) 290 CONTINUE WORK( I+2*N ) = DBLE( KOUNT )*WORK( I+N ) + SUM 300 CONTINUE ! DO 330 J = ILO, IHI KOUNT = 0 SUM = ZERO DO 320 I = ILO, IHI IF( A( I, J ).EQ.ZERO ) & GO TO 310 KOUNT = KOUNT + 1 SUM = SUM + WORK( I+N ) 310 CONTINUE IF( B( I, J ).EQ.ZERO ) & GO TO 320 KOUNT = KOUNT + 1 SUM = SUM + WORK( I+N ) 320 CONTINUE WORK( J+3*N ) = DBLE( KOUNT )*WORK( J ) + SUM 330 CONTINUE ! SUM = DDOT( NR, WORK( ILO+N ), 1, WORK( ILO+2*N ), 1 ) + & DDOT( NR, WORK( ILO ), 1, WORK( ILO+3*N ), 1 ) ALPHA = GAMMA / SUM ! ! Determine correction to current iteration ! CMAX = ZERO DO 340 I = ILO, IHI COR = ALPHA*WORK( I+N ) IF( ABS( COR ).GT.CMAX ) & CMAX = ABS( COR ) LSCALE( I ) = LSCALE( I ) + COR COR = ALPHA*WORK( I ) IF( ABS( COR ).GT.CMAX ) & CMAX = ABS( COR ) RSCALE( I ) = RSCALE( I ) + COR 340 CONTINUE IF( CMAX.LT.HALF ) & GO TO 350 ! CALL DAXPY( NR, -ALPHA, WORK( ILO+2*N ), 1, WORK( ILO+4*N ), 1 ) CALL DAXPY( NR, -ALPHA, WORK( ILO+3*N ), 1, WORK( ILO+5*N ), 1 ) ! PGAMMA = GAMMA IT = IT + 1 IF( IT.LE.NRP2 ) & GO TO 250 ! ! End generalized conjugate gradient iteration ! 350 CONTINUE SFMIN = DLAMCH( 'S' ) SFMAX = ONE / SFMIN LSFMIN = INT( LOG10( SFMIN ) / BASL+ONE ) LSFMAX = INT( LOG10( SFMAX ) / BASL ) DO 360 I = ILO, IHI IRAB = IDAMAX( N-ILO+1, A( I, ILO ), LDA ) RAB = ABS( A( I, IRAB+ILO-1 ) ) IRAB = IDAMAX( N-ILO+1, B( I, ILO ), LDA ) RAB = MAX( RAB, ABS( B( I, IRAB+ILO-1 ) ) ) LRAB = INT( LOG10( RAB+SFMIN ) / BASL+ONE ) IR = LSCALE( I ) + SIGN( HALF, LSCALE( I ) ) IR = MIN( MAX( IR, LSFMIN ), LSFMAX, LSFMAX-LRAB ) LSCALE( I ) = SCLFAC**IR ICAB = IDAMAX( IHI, A( 1, I ), 1 ) CAB = ABS( A( ICAB, I ) ) ICAB = IDAMAX( IHI, B( 1, I ), 1 ) CAB = MAX( CAB, ABS( B( ICAB, I ) ) ) LCAB = INT( LOG10( CAB+SFMIN ) / BASL+ONE ) JC = RSCALE( I ) + SIGN( HALF, RSCALE( I ) ) JC = MIN( MAX( JC, LSFMIN ), LSFMAX, LSFMAX-LCAB ) RSCALE( I ) = SCLFAC**JC 360 CONTINUE ! ! Row scaling of matrices A and B ! DO 370 I = ILO, IHI CALL DSCAL( N-ILO+1, LSCALE( I ), A( I, ILO ), LDA ) CALL DSCAL( N-ILO+1, LSCALE( I ), B( I, ILO ), LDB ) 370 CONTINUE ! ! Column scaling of matrices A and B ! DO 380 J = ILO, IHI CALL DSCAL( IHI, RSCALE( J ), A( 1, J ), 1 ) CALL DSCAL( IHI, RSCALE( J ), B( 1, J ), 1 ) 380 CONTINUE ! RETURN ! ! End of DGGBAL ! END SUBROUTINE DGGES( JOBVSL, JOBVSR, SORT, DELCTG, N, A, LDA, B, LDB, & SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, & LDVSR, WORK, LWORK, BWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBVSL, JOBVSR, SORT INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM ! .. ! .. Array Arguments .. LOGICAL BWORK( * ) DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), & B( LDB, * ), BETA( * ), VSL( LDVSL, * ), & VSR( LDVSR, * ), WORK( * ) ! .. ! .. Function Arguments .. LOGICAL DELCTG EXTERNAL DELCTG ! .. ! ! Purpose ! ======= ! ! DGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B), ! the generalized eigenvalues, the generalized real Schur form (S,T), ! optionally, the left and/or right matrices of Schur vectors (VSL and ! VSR). This gives the generalized Schur factorization ! ! (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T ) ! ! Optionally, it also orders the eigenvalues so that a selected cluster ! of eigenvalues appears in the leading diagonal blocks of the upper ! quasi-triangular matrix S and the upper triangular matrix T.The ! leading columns of VSL and VSR then form an orthonormal basis for the ! corresponding left and right eigenspaces (deflating subspaces). ! ! (If only the generalized eigenvalues are needed, use the driver ! DGGEV instead, which is faster.) ! ! A generalized eigenvalue for a pair of matrices (A,B) is a scalar w ! or a ratio alpha/beta = w, such that A - w*B is singular. It is ! usually represented as the pair (alpha,beta), as there is a ! reasonable interpretation for beta=0 or both being zero. ! ! A pair of matrices (S,T) is in generalized real Schur form if T is ! upper triangular with non-negative diagonal and S is block upper ! triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond ! to real generalized eigenvalues, while 2-by-2 blocks of S will be ! "standardized" by making the corresponding elements of T have the ! form: ! [ a 0 ] ! [ 0 b ] ! ! and the pair of corresponding 2-by-2 blocks in S and T will have a ! complex conjugate pair of generalized eigenvalues. ! ! ! Arguments ! ========= ! ! JOBVSL (input) CHARACTER*1 ! = 'N': do not compute the left Schur vectors; ! = 'V': compute the left Schur vectors. ! ! JOBVSR (input) CHARACTER*1 ! = 'N': do not compute the right Schur vectors; ! = 'V': compute the right Schur vectors. ! ! SORT (input) CHARACTER*1 ! Specifies whether or not to order the eigenvalues on the ! diagonal of the generalized Schur form. ! = 'N': Eigenvalues are not ordered; ! = 'S': Eigenvalues are ordered (see DELZTG); ! ! DELZTG (input) LOGICAL FUNCTION of three DOUBLE PRECISION arguments ! DELZTG must be declared EXTERNAL in the calling subroutine. ! If SORT = 'N', DELZTG is not referenced. ! If SORT = 'S', DELZTG is used to select eigenvalues to sort ! to the top left of the Schur form. ! An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if ! DELZTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either ! one of a complex conjugate pair of eigenvalues is selected, ! then both complex eigenvalues are selected. ! ! Note that in the ill-conditioned case, a selected complex ! eigenvalue may no longer satisfy DELZTG(ALPHAR(j),ALPHAI(j), ! BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2 ! in this case. ! ! N (input) INTEGER ! The order of the matrices A, B, VSL, and VSR. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the first of the pair of matrices. ! On exit, A has been overwritten by its generalized Schur ! form S. ! ! LDA (input) INTEGER ! The leading dimension of A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB, N) ! On entry, the second of the pair of matrices. ! On exit, B has been overwritten by its generalized Schur ! form T. ! ! LDB (input) INTEGER ! The leading dimension of B. LDB >= max(1,N). ! ! SDIM (output) INTEGER ! If SORT = 'N', SDIM = 0. ! If SORT = 'S', SDIM = number of eigenvalues (after sorting) ! for which DELZTG is true. (Complex conjugate pairs for which ! DELZTG is true for either eigenvalue count as 2.) ! ! ALPHAR (output) DOUBLE PRECISION array, dimension (N) ! ALPHAI (output) DOUBLE PRECISION array, dimension (N) ! BETA (output) DOUBLE PRECISION array, dimension (N) ! On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will ! be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i, ! and BETA(j),j=1,...,N are the diagonals of the complex Schur ! form (S,T) that would result if the 2-by-2 diagonal blocks of ! the real Schur form of (A,B) were further reduced to ! triangular form using 2-by-2 complex unitary transformations. ! If ALPHAI(j) is zero, then the j-th eigenvalue is real; if ! positive, then the j-th and (j+1)-st eigenvalues are a ! complex conjugate pair, with ALPHAI(j+1) negative. ! ! Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) ! may easily over- or underflow, and BETA(j) may even be zero. ! Thus, the user should avoid naively computing the ratio. ! However, ALPHAR and ALPHAI will be always less than and ! usually comparable with norm(A) in magnitude, and BETA always ! less than and usually comparable with norm(B). ! ! VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N) ! If JOBVSL = 'V', VSL will contain the left Schur vectors. ! Not referenced if JOBVSL = 'N'. ! ! LDVSL (input) INTEGER ! The leading dimension of the matrix VSL. LDVSL >=1, and ! if JOBVSL = 'V', LDVSL >= N. ! ! VSR (output) DOUBLE PRECISION array, dimension (LDVSR,N) ! If JOBVSR = 'V', VSR will contain the right Schur vectors. ! Not referenced if JOBVSR = 'N'. ! ! LDVSR (input) INTEGER ! The leading dimension of the matrix VSR. LDVSR >= 1, and ! if JOBVSR = 'V', LDVSR >= N. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= 8*N+16. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! BWORK (workspace) LOGICAL array, dimension (N) ! Not referenced if SORT = 'N'. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! = 1,...,N: ! The QZ iteration failed. (A,B) are not in Schur ! form, but ALPHAR(j), ALPHAI(j), and BETA(j) should ! be correct for j=INFO+1,...,N. ! > N: =N+1: other than QZ iteration failed in DHGEQZ. ! =N+2: after reordering, roundoff changed values of ! some complex eigenvalues so that leading ! eigenvalues in the Generalized Schur form no ! longer satisfy DELZTG=.TRUE. This could also ! be caused due to scaling. ! =N+3: reordering failed in DTGSEN. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL, & LQUERY, LST2SL, WANTST INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, & ILO, IP, IRIGHT, IROWS, ITAU, IWRK, MAXWRK, & MINWRK DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL, & PVSR, SAFMAX, SAFMIN, SMLNUM ! .. ! .. Local Arrays .. INTEGER IDUM( 1 ) DOUBLE PRECISION DIF( 2 ) ! .. ! .. External Subroutines .. EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD, & DLACPY, DLASCL, DLASET, DORGQR, DORMQR, DTGSEN, & XERBLA ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! ! Decode the input arguments ! IF( LSAME( JOBVSL, 'N' ) ) THEN IJOBVL = 1 ILVSL = .FALSE. ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN IJOBVL = 2 ILVSL = .TRUE. ELSE IJOBVL = -1 ILVSL = .FALSE. END IF ! IF( LSAME( JOBVSR, 'N' ) ) THEN IJOBVR = 1 ILVSR = .FALSE. ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN IJOBVR = 2 ILVSR = .TRUE. ELSE IJOBVR = -1 ILVSR = .FALSE. END IF ! WANTST = LSAME( SORT, 'S' ) ! ! Test the input arguments ! INFO = 0 LQUERY = ( LWORK.EQ.-1 ) IF( IJOBVL.LE.0 ) THEN INFO = -1 ELSE IF( IJOBVR.LE.0 ) THEN INFO = -2 ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN INFO = -15 ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN INFO = -17 END IF ! ! Compute workspace ! (Note: Comments in the code beginning "Workspace:" describe the ! minimal amount of workspace needed at that point in the code, ! as well as the preferred amount for good performance. ! NB refers to the optimal block size for the immediately ! following subroutine, as returned by ILAENV.) ! MINWRK = 1 IF( INFO.EQ.0 .AND. ( LWORK.GE.1 .OR. LQUERY ) ) THEN MINWRK = 7*( N+1 ) + 16 MAXWRK = 7*( N+1 ) + N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 ) + & 16 IF( ILVSL ) THEN MAXWRK = MAX( MAXWRK, 7*( N+1 )+N* & ILAENV( 1, 'DORGQR', ' ', N, 1, N, -1 ) ) END IF WORK( 1 ) = MAXWRK END IF ! IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) & INFO = -19 IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGGES ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) THEN SDIM = 0 RETURN END IF ! ! Get machine constants ! EPS = DLAMCH( 'P' ) SAFMIN = DLAMCH( 'S' ) SAFMAX = ONE / SAFMIN CALL DLABAD( SAFMIN, SAFMAX ) SMLNUM = SQRT( SAFMIN ) / EPS BIGNUM = ONE / SMLNUM ! ! Scale A if max element outside range [SMLNUM,BIGNUM] ! ANRM = DLANGE( 'M', N, N, A, LDA, WORK ) ILASCL = .FALSE. IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN ANRMTO = SMLNUM ILASCL = .TRUE. ELSE IF( ANRM.GT.BIGNUM ) THEN ANRMTO = BIGNUM ILASCL = .TRUE. END IF IF( ILASCL ) & CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR ) ! ! Scale B if max element outside range [SMLNUM,BIGNUM] ! BNRM = DLANGE( 'M', N, N, B, LDB, WORK ) ILBSCL = .FALSE. IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN BNRMTO = SMLNUM ILBSCL = .TRUE. ELSE IF( BNRM.GT.BIGNUM ) THEN BNRMTO = BIGNUM ILBSCL = .TRUE. END IF IF( ILBSCL ) & CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR ) ! ! Permute the matrix to make it more nearly triangular ! (Workspace: need 6*N + 2*N space for storing balancing factors) ! ILEFT = 1 IRIGHT = N + 1 IWRK = IRIGHT + N CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ), & WORK( IRIGHT ), WORK( IWRK ), IERR ) ! ! Reduce B to triangular form (QR decomposition of B) ! (Workspace: need N, prefer N*NB) ! IROWS = IHI + 1 - ILO ICOLS = N + 1 - ILO ITAU = IWRK IWRK = ITAU + IROWS CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), & WORK( IWRK ), LWORK+1-IWRK, IERR ) ! ! Apply the orthogonal transformation to matrix A ! (Workspace: need N, prefer N*NB) ! CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB, & WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ), & LWORK+1-IWRK, IERR ) ! ! Initialize VSL ! (Workspace: need N, prefer N*NB) ! IF( ILVSL ) THEN CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL ) CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, & VSL( ILO+1, ILO ), LDVSL ) CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL, & WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR ) END IF ! ! Initialize VSR ! IF( ILVSR ) & CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR ) ! ! Reduce to generalized Hessenberg form ! (Workspace: none needed) ! CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL, & LDVSL, VSR, LDVSR, IERR ) ! ! Perform QZ algorithm, computing Schur vectors if desired ! (Workspace: need N) ! IWRK = ITAU CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, & ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, & WORK( IWRK ), LWORK+1-IWRK, IERR ) IF( IERR.NE.0 ) THEN IF( IERR.GT.0 .AND. IERR.LE.N ) THEN INFO = IERR ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN INFO = IERR - N ELSE INFO = N + 1 END IF GO TO 50 END IF ! ! Sort eigenvalues ALPHA/BETA if desired ! (Workspace: need 4*N+16 ) ! SDIM = 0 IF( WANTST ) THEN ! ! Undo scaling on eigenvalues before DELZTGing ! IF( ILASCL ) THEN CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, & IERR ) CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, & IERR ) END IF IF( ILBSCL ) & CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) ! ! Select eigenvalues ! DO 10 I = 1, N BWORK( I ) = DELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) ) 10 CONTINUE ! CALL DTGSEN( 0, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB, ALPHAR, & ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, SDIM, PVSL, & PVSR, DIF, WORK( IWRK ), LWORK-IWRK+1, IDUM, 1, & IERR ) IF( IERR.EQ.1 ) & INFO = N + 3 ! END IF ! ! Apply back-permutation to VSL and VSR ! (Workspace: none needed) ! IF( ILVSL ) & CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ), & WORK( IRIGHT ), N, VSL, LDVSL, IERR ) ! IF( ILVSR ) & CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ), & WORK( IRIGHT ), N, VSR, LDVSR, IERR ) ! ! Check if unscaling would cause over/underflow, if so, rescale ! (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of ! B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I) ! IF( ILASCL ) THEN DO 20 I = 1, N IF( ALPHAI( I ).NE.ZERO ) THEN IF( ( ALPHAR( I ) / SAFMAX ).GT.( ANRMTO / ANRM ) .OR. & ( SAFMIN / ALPHAR( I ) ).GT.( ANRM / ANRMTO ) ) THEN WORK( 1 ) = ABS( A( I, I ) / ALPHAR( I ) ) BETA( I ) = BETA( I )*WORK( 1 ) ALPHAR( I ) = ALPHAR( I )*WORK( 1 ) ALPHAI( I ) = ALPHAI( I )*WORK( 1 ) ELSE IF( ( ALPHAI( I ) / SAFMAX ).GT. & ( ANRMTO / ANRM ) .OR. & ( SAFMIN / ALPHAI( I ) ).GT.( ANRM / ANRMTO ) ) & THEN WORK( 1 ) = ABS( A( I, I+1 ) / ALPHAI( I ) ) BETA( I ) = BETA( I )*WORK( 1 ) ALPHAR( I ) = ALPHAR( I )*WORK( 1 ) ALPHAI( I ) = ALPHAI( I )*WORK( 1 ) END IF END IF 20 CONTINUE END IF ! IF( ILBSCL ) THEN DO 30 I = 1, N IF( ALPHAI( I ).NE.ZERO ) THEN IF( ( BETA( I ) / SAFMAX ).GT.( BNRMTO / BNRM ) .OR. & ( SAFMIN / BETA( I ) ).GT.( BNRM / BNRMTO ) ) THEN WORK( 1 ) = ABS( B( I, I ) / BETA( I ) ) BETA( I ) = BETA( I )*WORK( 1 ) ALPHAR( I ) = ALPHAR( I )*WORK( 1 ) ALPHAI( I ) = ALPHAI( I )*WORK( 1 ) END IF END IF 30 CONTINUE END IF ! ! Undo scaling ! IF( ILASCL ) THEN CALL DLASCL( 'H', 0, 0, ANRMTO, ANRM, N, N, A, LDA, IERR ) CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR ) CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR ) END IF ! IF( ILBSCL ) THEN CALL DLASCL( 'U', 0, 0, BNRMTO, BNRM, N, N, B, LDB, IERR ) CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) END IF ! IF( WANTST ) THEN ! ! Check if reordering is correct ! LASTSL = .TRUE. LST2SL = .TRUE. SDIM = 0 IP = 0 DO 40 I = 1, N CURSL = DELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) ) IF( ALPHAI( I ).EQ.ZERO ) THEN IF( CURSL ) & SDIM = SDIM + 1 IP = 0 IF( CURSL .AND. .NOT.LASTSL ) & INFO = N + 2 ELSE IF( IP.EQ.1 ) THEN ! ! Last eigenvalue of conjugate pair ! CURSL = CURSL .OR. LASTSL LASTSL = CURSL IF( CURSL ) & SDIM = SDIM + 2 IP = -1 IF( CURSL .AND. .NOT.LST2SL ) & INFO = N + 2 ELSE ! ! First eigenvalue of conjugate pair ! IP = 1 END IF END IF LST2SL = LASTSL LASTSL = CURSL 40 CONTINUE ! END IF ! 50 CONTINUE ! WORK( 1 ) = MAXWRK ! RETURN ! ! End of DGGES ! END SUBROUTINE DGGESX( JOBVSL, JOBVSR, SORT, DELCTG, SENSE, N, A, LDA, & B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, & VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK, & LIWORK, BWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBVSL, JOBVSR, SENSE, SORT INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N, & SDIM ! .. ! .. Array Arguments .. LOGICAL BWORK( * ) INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), & B( LDB, * ), BETA( * ), RCONDE( 2 ), & RCONDV( 2 ), VSL( LDVSL, * ), VSR( LDVSR, * ), & WORK( * ) ! .. ! .. Function Arguments .. LOGICAL DELCTG EXTERNAL DELCTG ! .. ! ! Purpose ! ======= ! ! DGGESX computes for a pair of N-by-N real nonsymmetric matrices ! (A,B), the generalized eigenvalues, the real Schur form (S,T), and, ! optionally, the left and/or right matrices of Schur vectors (VSL and ! VSR). This gives the generalized Schur factorization ! ! (A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T ) ! ! Optionally, it also orders the eigenvalues so that a selected cluster ! of eigenvalues appears in the leading diagonal blocks of the upper ! quasi-triangular matrix S and the upper triangular matrix T; computes ! a reciprocal condition number for the average of the selected ! eigenvalues (RCONDE); and computes a reciprocal condition number for ! the right and left deflating subspaces corresponding to the selected ! eigenvalues (RCONDV). The leading columns of VSL and VSR then form ! an orthonormal basis for the corresponding left and right eigenspaces ! (deflating subspaces). ! ! A generalized eigenvalue for a pair of matrices (A,B) is a scalar w ! or a ratio alpha/beta = w, such that A - w*B is singular. It is ! usually represented as the pair (alpha,beta), as there is a ! reasonable interpretation for beta=0 or for both being zero. ! ! A pair of matrices (S,T) is in generalized real Schur form if T is ! upper triangular with non-negative diagonal and S is block upper ! triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond ! to real generalized eigenvalues, while 2-by-2 blocks of S will be ! "standardized" by making the corresponding elements of T have the ! form: ! [ a 0 ] ! [ 0 b ] ! ! and the pair of corresponding 2-by-2 blocks in S and T will have a ! complex conjugate pair of generalized eigenvalues. ! ! ! Arguments ! ========= ! ! JOBVSL (input) CHARACTER*1 ! = 'N': do not compute the left Schur vectors; ! = 'V': compute the left Schur vectors. ! ! JOBVSR (input) CHARACTER*1 ! = 'N': do not compute the right Schur vectors; ! = 'V': compute the right Schur vectors. ! ! SORT (input) CHARACTER*1 ! Specifies whether or not to order the eigenvalues on the ! diagonal of the generalized Schur form. ! = 'N': Eigenvalues are not ordered; ! = 'S': Eigenvalues are ordered (see DELZTG). ! ! DELZTG (input) LOGICAL FUNCTION of three DOUBLE PRECISION arguments ! DELZTG must be declared EXTERNAL in the calling subroutine. ! If SORT = 'N', DELZTG is not referenced. ! If SORT = 'S', DELZTG is used to select eigenvalues to sort ! to the top left of the Schur form. ! An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if ! DELZTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either ! one of a complex conjugate pair of eigenvalues is selected, ! then both complex eigenvalues are selected. ! Note that a selected complex eigenvalue may no longer satisfy ! DELZTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering, ! since ordering may change the value of complex eigenvalues ! (especially if the eigenvalue is ill-conditioned), in this ! case INFO is set to N+3. ! ! SENSE (input) CHARACTER ! Determines which reciprocal condition numbers are computed. ! = 'N' : None are computed; ! = 'E' : Computed for average of selected eigenvalues only; ! = 'V' : Computed for selected deflating subspaces only; ! = 'B' : Computed for both. ! If SENSE = 'E', 'V', or 'B', SORT must equal 'S'. ! ! N (input) INTEGER ! The order of the matrices A, B, VSL, and VSR. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the first of the pair of matrices. ! On exit, A has been overwritten by its generalized Schur ! form S. ! ! LDA (input) INTEGER ! The leading dimension of A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB, N) ! On entry, the second of the pair of matrices. ! On exit, B has been overwritten by its generalized Schur ! form T. ! ! LDB (input) INTEGER ! The leading dimension of B. LDB >= max(1,N). ! ! SDIM (output) INTEGER ! If SORT = 'N', SDIM = 0. ! If SORT = 'S', SDIM = number of eigenvalues (after sorting) ! for which DELZTG is true. (Complex conjugate pairs for which ! DELZTG is true for either eigenvalue count as 2.) ! ! ALPHAR (output) DOUBLE PRECISION array, dimension (N) ! ALPHAI (output) DOUBLE PRECISION array, dimension (N) ! BETA (output) DOUBLE PRECISION array, dimension (N) ! On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will ! be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i ! and BETA(j),j=1,...,N are the diagonals of the complex Schur ! form (S,T) that would result if the 2-by-2 diagonal blocks of ! the real Schur form of (A,B) were further reduced to ! triangular form using 2-by-2 complex unitary transformations. ! If ALPHAI(j) is zero, then the j-th eigenvalue is real; if ! positive, then the j-th and (j+1)-st eigenvalues are a ! complex conjugate pair, with ALPHAI(j+1) negative. ! ! Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) ! may easily over- or underflow, and BETA(j) may even be zero. ! Thus, the user should avoid naively computing the ratio. ! However, ALPHAR and ALPHAI will be always less than and ! usually comparable with norm(A) in magnitude, and BETA always ! less than and usually comparable with norm(B). ! ! VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N) ! If JOBVSL = 'V', VSL will contain the left Schur vectors. ! Not referenced if JOBVSL = 'N'. ! ! LDVSL (input) INTEGER ! The leading dimension of the matrix VSL. LDVSL >=1, and ! if JOBVSL = 'V', LDVSL >= N. ! ! VSR (output) DOUBLE PRECISION array, dimension (LDVSR,N) ! If JOBVSR = 'V', VSR will contain the right Schur vectors. ! Not referenced if JOBVSR = 'N'. ! ! LDVSR (input) INTEGER ! The leading dimension of the matrix VSR. LDVSR >= 1, and ! if JOBVSR = 'V', LDVSR >= N. ! ! RCONDE (output) DOUBLE PRECISION array, dimension ( 2 ) ! If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the ! reciprocal condition numbers for the average of the selected ! eigenvalues. ! Not referenced if SENSE = 'N' or 'V'. ! ! RCONDV (output) DOUBLE PRECISION array, dimension ( 2 ) ! If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the ! reciprocal condition numbers for the selected deflating ! subspaces. ! Not referenced if SENSE = 'N' or 'E'. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= 8*(N+1)+16. ! If SENSE = 'E', 'V', or 'B', ! LWORK >= MAX( 8*(N+1)+16, 2*SDIM*(N-SDIM) ). ! ! IWORK (workspace) INTEGER array, dimension (LIWORK) ! Not referenced if SENSE = 'N'. ! ! LIWORK (input) INTEGER ! The dimension of the array WORK. LIWORK >= N+6. ! ! BWORK (workspace) LOGICAL array, dimension (N) ! Not referenced if SORT = 'N'. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! = 1,...,N: ! The QZ iteration failed. (A,B) are not in Schur ! form, but ALPHAR(j), ALPHAI(j), and BETA(j) should ! be correct for j=INFO+1,...,N. ! > N: =N+1: other than QZ iteration failed in DHGEQZ ! =N+2: after reordering, roundoff changed values of ! some complex eigenvalues so that leading ! eigenvalues in the Generalized Schur form no ! longer satisfy DELZTG=.TRUE. This could also ! be caused due to scaling. ! =N+3: reordering failed in DTGSEN. ! ! Further details ! =============== ! ! An approximate (asymptotic) bound on the average absolute error of ! the selected eigenvalues is ! ! EPS * norm((A, B)) / RCONDE( 1 ). ! ! An approximate (asymptotic) bound on the maximum angular error in ! the computed deflating subspaces is ! ! EPS * norm((A, B)) / RCONDV( 2 ). ! ! See LAPACK User's Guide, section 4.11 for more information. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL, & LST2SL, WANTSB, WANTSE, WANTSN, WANTST, WANTSV INTEGER I, ICOLS, IERR, IHI, IJOB, IJOBVL, IJOBVR, & ILEFT, ILO, IP, IRIGHT, IROWS, ITAU, IWRK, & LIWMIN, MAXWRK, MINWRK DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PL, & PR, SAFMAX, SAFMIN, SMLNUM ! .. ! .. Local Arrays .. DOUBLE PRECISION DIF( 2 ) ! .. ! .. External Subroutines .. EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD, & DLACPY, DLASCL, DLASET, DORGQR, DORMQR, DTGSEN, & XERBLA ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! ! Decode the input arguments ! IF( LSAME( JOBVSL, 'N' ) ) THEN IJOBVL = 1 ILVSL = .FALSE. ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN IJOBVL = 2 ILVSL = .TRUE. ELSE IJOBVL = -1 ILVSL = .FALSE. END IF ! IF( LSAME( JOBVSR, 'N' ) ) THEN IJOBVR = 1 ILVSR = .FALSE. ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN IJOBVR = 2 ILVSR = .TRUE. ELSE IJOBVR = -1 ILVSR = .FALSE. END IF ! WANTST = LSAME( SORT, 'S' ) WANTSN = LSAME( SENSE, 'N' ) WANTSE = LSAME( SENSE, 'E' ) WANTSV = LSAME( SENSE, 'V' ) WANTSB = LSAME( SENSE, 'B' ) IF( WANTSN ) THEN IJOB = 0 IWORK( 1 ) = 1 ELSE IF( WANTSE ) THEN IJOB = 1 ELSE IF( WANTSV ) THEN IJOB = 2 ELSE IF( WANTSB ) THEN IJOB = 4 END IF ! ! Test the input arguments ! INFO = 0 IF( IJOBVL.LE.0 ) THEN INFO = -1 ELSE IF( IJOBVR.LE.0 ) THEN INFO = -2 ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN INFO = -3 ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR. & ( .NOT.WANTST .AND. .NOT.WANTSN ) ) THEN INFO = -5 ELSE IF( N.LT.0 ) THEN INFO = -6 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -10 ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN INFO = -16 ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN INFO = -18 END IF ! ! Compute workspace ! (Note: Comments in the code beginning "Workspace:" describe the ! minimal amount of workspace needed at that point in the code, ! as well as the preferred amount for good performance. ! NB refers to the optimal block size for the immediately ! following subroutine, as returned by ILAENV.) ! MINWRK = 1 IF( INFO.EQ.0 .AND. LWORK.GE.1 ) THEN MINWRK = 8*( N+1 ) + 16 MAXWRK = 7*( N+1 ) + N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 ) + & 16 IF( ILVSL ) THEN MAXWRK = MAX( MAXWRK, 8*( N+1 )+N* & ILAENV( 1, 'DORGQR', ' ', N, 1, N, -1 )+16 ) END IF WORK( 1 ) = MAXWRK END IF IF( .NOT.WANTSN ) THEN LIWMIN = 1 ELSE LIWMIN = N + 6 END IF IWORK( 1 ) = LIWMIN ! IF( INFO.EQ.0 .AND. LWORK.LT.MINWRK ) THEN INFO = -22 ELSE IF( INFO.EQ.0 .AND. IJOB.GE.1 ) THEN IF( LIWORK.LT.LIWMIN ) & INFO = -24 END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGGESX', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) THEN SDIM = 0 RETURN END IF ! ! Get machine constants ! EPS = DLAMCH( 'P' ) SAFMIN = DLAMCH( 'S' ) SAFMAX = ONE / SAFMIN CALL DLABAD( SAFMIN, SAFMAX ) SMLNUM = SQRT( SAFMIN ) / EPS BIGNUM = ONE / SMLNUM ! ! Scale A if max element outside range [SMLNUM,BIGNUM] ! ANRM = DLANGE( 'M', N, N, A, LDA, WORK ) ILASCL = .FALSE. IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN ANRMTO = SMLNUM ILASCL = .TRUE. ELSE IF( ANRM.GT.BIGNUM ) THEN ANRMTO = BIGNUM ILASCL = .TRUE. END IF IF( ILASCL ) & CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR ) ! ! Scale B if max element outside range [SMLNUM,BIGNUM] ! BNRM = DLANGE( 'M', N, N, B, LDB, WORK ) ILBSCL = .FALSE. IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN BNRMTO = SMLNUM ILBSCL = .TRUE. ELSE IF( BNRM.GT.BIGNUM ) THEN BNRMTO = BIGNUM ILBSCL = .TRUE. END IF IF( ILBSCL ) & CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR ) ! ! Permute the matrix to make it more nearly triangular ! (Workspace: need 6*N + 2*N for permutation parameters) ! ILEFT = 1 IRIGHT = N + 1 IWRK = IRIGHT + N CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ), & WORK( IRIGHT ), WORK( IWRK ), IERR ) ! ! Reduce B to triangular form (QR decomposition of B) ! (Workspace: need N, prefer N*NB) ! IROWS = IHI + 1 - ILO ICOLS = N + 1 - ILO ITAU = IWRK IWRK = ITAU + IROWS CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), & WORK( IWRK ), LWORK+1-IWRK, IERR ) ! ! Apply the orthogonal transformation to matrix A ! (Workspace: need N, prefer N*NB) ! CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB, & WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ), & LWORK+1-IWRK, IERR ) ! ! Initialize VSL ! (Workspace: need N, prefer N*NB) ! IF( ILVSL ) THEN CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL ) CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, & VSL( ILO+1, ILO ), LDVSL ) CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL, & WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR ) END IF ! ! Initialize VSR ! IF( ILVSR ) & CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR ) ! ! Reduce to generalized Hessenberg form ! (Workspace: none needed) ! CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL, & LDVSL, VSR, LDVSR, IERR ) ! SDIM = 0 ! ! Perform QZ algorithm, computing Schur vectors if desired ! (Workspace: need N) ! IWRK = ITAU CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, & ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, & WORK( IWRK ), LWORK+1-IWRK, IERR ) IF( IERR.NE.0 ) THEN IF( IERR.GT.0 .AND. IERR.LE.N ) THEN INFO = IERR ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN INFO = IERR - N ELSE INFO = N + 1 END IF GO TO 60 END IF ! ! Sort eigenvalues ALPHA/BETA and compute the reciprocal of ! condition number(s) ! (Workspace: If IJOB >= 1, need MAX( 8*(N+1), 2*SDIM*(N-SDIM) ) ! otherwise, need 8*(N+1) ) ! IF( WANTST ) THEN ! ! Undo scaling on eigenvalues before DELZTGing ! IF( ILASCL ) THEN CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, & IERR ) CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, & IERR ) END IF IF( ILBSCL ) & CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) ! ! Select eigenvalues ! DO 10 I = 1, N BWORK( I ) = DELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) ) 10 CONTINUE ! ! Reorder eigenvalues, transform Generalized Schur vectors, and ! compute reciprocal condition numbers ! CALL DTGSEN( IJOB, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB, & ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, & SDIM, PL, PR, DIF, WORK( IWRK ), LWORK-IWRK+1, & IWORK, LIWORK, IERR ) ! IF( IJOB.GE.1 ) & MAXWRK = MAX( MAXWRK, 2*SDIM*( N-SDIM ) ) IF( IERR.EQ.-22 ) THEN ! ! not enough real workspace ! INFO = -22 ELSE RCONDE( 1 ) = PL RCONDE( 2 ) = PR RCONDV( 1 ) = DIF( 1 ) RCONDV( 2 ) = DIF( 2 ) IF( IERR.EQ.1 ) & INFO = N + 3 END IF ! END IF ! ! Apply permutation to VSL and VSR ! (Workspace: none needed) ! IF( ILVSL ) & CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ), & WORK( IRIGHT ), N, VSL, LDVSL, IERR ) ! IF( ILVSR ) & CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ), & WORK( IRIGHT ), N, VSR, LDVSR, IERR ) ! ! Check if unscaling would cause over/underflow, if so, rescale ! (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of ! B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I) ! IF( ILASCL ) THEN DO 20 I = 1, N IF( ALPHAI( I ).NE.ZERO ) THEN IF( ( ALPHAR( I ) / SAFMAX ).GT.( ANRMTO / ANRM ) .OR. & ( SAFMIN / ALPHAR( I ) ).GT.( ANRM / ANRMTO ) ) THEN WORK( 1 ) = ABS( A( I, I ) / ALPHAR( I ) ) BETA( I ) = BETA( I )*WORK( 1 ) ALPHAR( I ) = ALPHAR( I )*WORK( 1 ) ALPHAI( I ) = ALPHAI( I )*WORK( 1 ) ELSE IF( ( ALPHAI( I ) / SAFMAX ).GT. & ( ANRMTO / ANRM ) .OR. & ( SAFMIN / ALPHAI( I ) ).GT.( ANRM / ANRMTO ) ) & THEN WORK( 1 ) = ABS( A( I, I+1 ) / ALPHAI( I ) ) BETA( I ) = BETA( I )*WORK( 1 ) ALPHAR( I ) = ALPHAR( I )*WORK( 1 ) ALPHAI( I ) = ALPHAI( I )*WORK( 1 ) END IF END IF 20 CONTINUE END IF ! IF( ILBSCL ) THEN DO 30 I = 1, N IF( ALPHAI( I ).NE.ZERO ) THEN IF( ( BETA( I ) / SAFMAX ).GT.( BNRMTO / BNRM ) .OR. & ( SAFMIN / BETA( I ) ).GT.( BNRM / BNRMTO ) ) THEN WORK( 1 ) = ABS( B( I, I ) / BETA( I ) ) BETA( I ) = BETA( I )*WORK( 1 ) ALPHAR( I ) = ALPHAR( I )*WORK( 1 ) ALPHAI( I ) = ALPHAI( I )*WORK( 1 ) END IF END IF 30 CONTINUE END IF ! ! Undo scaling ! IF( ILASCL ) THEN CALL DLASCL( 'H', 0, 0, ANRMTO, ANRM, N, N, A, LDA, IERR ) CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR ) CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR ) END IF ! IF( ILBSCL ) THEN CALL DLASCL( 'U', 0, 0, BNRMTO, BNRM, N, N, B, LDB, IERR ) CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) END IF ! 40 CONTINUE ! IF( WANTST ) THEN ! ! Check if reordering is correct ! LASTSL = .TRUE. LST2SL = .TRUE. SDIM = 0 IP = 0 DO 50 I = 1, N CURSL = DELCTG( ALPHAR( I ), ALPHAI( I ), BETA( I ) ) IF( ALPHAI( I ).EQ.ZERO ) THEN IF( CURSL ) & SDIM = SDIM + 1 IP = 0 IF( CURSL .AND. .NOT.LASTSL ) & INFO = N + 2 ELSE IF( IP.EQ.1 ) THEN ! ! Last eigenvalue of conjugate pair ! CURSL = CURSL .OR. LASTSL LASTSL = CURSL IF( CURSL ) & SDIM = SDIM + 2 IP = -1 IF( CURSL .AND. .NOT.LST2SL ) & INFO = N + 2 ELSE ! ! First eigenvalue of conjugate pair ! IP = 1 END IF END IF LST2SL = LASTSL LASTSL = CURSL 50 CONTINUE ! END IF ! 60 CONTINUE ! WORK( 1 ) = MAXWRK IWORK( 1 ) = LIWMIN ! RETURN ! ! End of DGGESX ! END SUBROUTINE DGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, & BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBVL, JOBVR INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), & B( LDB, * ), BETA( * ), VL( LDVL, * ), & VR( LDVR, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B) ! the generalized eigenvalues, and optionally, the left and/or right ! generalized eigenvectors. ! ! A generalized eigenvalue for a pair of matrices (A,B) is a scalar ! lambda or a ratio alpha/beta = lambda, such that A - lambda*B is ! singular. It is usually represented as the pair (alpha,beta), as ! there is a reasonable interpretation for beta=0, and even for both ! being zero. ! ! The right eigenvector v(j) corresponding to the eigenvalue lambda(j) ! of (A,B) satisfies ! ! A * v(j) = lambda(j) * B * v(j). ! ! The left eigenvector u(j) corresponding to the eigenvalue lambda(j) ! of (A,B) satisfies ! ! u(j)**H * A = lambda(j) * u(j)**H * B . ! ! where u(j)**H is the conjugate-transpose of u(j). ! ! ! Arguments ! ========= ! ! JOBVL (input) CHARACTER*1 ! = 'N': do not compute the left generalized eigenvectors; ! = 'V': compute the left generalized eigenvectors. ! ! JOBVR (input) CHARACTER*1 ! = 'N': do not compute the right generalized eigenvectors; ! = 'V': compute the right generalized eigenvectors. ! ! N (input) INTEGER ! The order of the matrices A, B, VL, and VR. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the matrix A in the pair (A,B). ! On exit, A has been overwritten. ! ! LDA (input) INTEGER ! The leading dimension of A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB, N) ! On entry, the matrix B in the pair (A,B). ! On exit, B has been overwritten. ! ! LDB (input) INTEGER ! The leading dimension of B. LDB >= max(1,N). ! ! ALPHAR (output) DOUBLE PRECISION array, dimension (N) ! ALPHAI (output) DOUBLE PRECISION array, dimension (N) ! BETA (output) DOUBLE PRECISION array, dimension (N) ! On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will ! be the generalized eigenvalues. If ALPHAI(j) is zero, then ! the j-th eigenvalue is real; if positive, then the j-th and ! (j+1)-st eigenvalues are a complex conjugate pair, with ! ALPHAI(j+1) negative. ! ! Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) ! may easily over- or underflow, and BETA(j) may even be zero. ! Thus, the user should avoid naively computing the ratio ! alpha/beta. However, ALPHAR and ALPHAI will be always less ! than and usually comparable with norm(A) in magnitude, and ! BETA always less than and usually comparable with norm(B). ! ! VL (output) DOUBLE PRECISION array, dimension (LDVL,N) ! If JOBVL = 'V', the left eigenvectors u(j) are stored one ! after another in the columns of VL, in the same order as ! their eigenvalues. If the j-th eigenvalue is real, then ! u(j) = VL(:,j), the j-th column of VL. If the j-th and ! (j+1)-th eigenvalues form a complex conjugate pair, then ! u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1). ! Each eigenvector will be scaled so the largest component have ! abs(real part)+abs(imag. part)=1. ! Not referenced if JOBVL = 'N'. ! ! LDVL (input) INTEGER ! The leading dimension of the matrix VL. LDVL >= 1, and ! if JOBVL = 'V', LDVL >= N. ! ! VR (output) DOUBLE PRECISION array, dimension (LDVR,N) ! If JOBVR = 'V', the right eigenvectors v(j) are stored one ! after another in the columns of VR, in the same order as ! their eigenvalues. If the j-th eigenvalue is real, then ! v(j) = VR(:,j), the j-th column of VR. If the j-th and ! (j+1)-th eigenvalues form a complex conjugate pair, then ! v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). ! Each eigenvector will be scaled so the largest component have ! abs(real part)+abs(imag. part)=1. ! Not referenced if JOBVR = 'N'. ! ! LDVR (input) INTEGER ! The leading dimension of the matrix VR. LDVR >= 1, and ! if JOBVR = 'V', LDVR >= N. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,8*N). ! For good performance, LWORK must generally be larger. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! = 1,...,N: ! The QZ iteration failed. No eigenvectors have been ! calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) ! should be correct for j=INFO+1,...,N. ! > N: =N+1: other than QZ iteration failed in DHGEQZ. ! =N+2: error return from DTGEVC. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY CHARACTER CHTEMP INTEGER ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO, & IN, IRIGHT, IROWS, ITAU, IWRK, JC, JR, MAXWRK, & MINWRK DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, & SMLNUM, TEMP ! .. ! .. Local Arrays .. LOGICAL LDUMMA( 1 ) ! .. ! .. External Subroutines .. EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY, & DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, XERBLA ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! ! Decode the input arguments ! IF( LSAME( JOBVL, 'N' ) ) THEN IJOBVL = 1 ILVL = .FALSE. ELSE IF( LSAME( JOBVL, 'V' ) ) THEN IJOBVL = 2 ILVL = .TRUE. ELSE IJOBVL = -1 ILVL = .FALSE. END IF ! IF( LSAME( JOBVR, 'N' ) ) THEN IJOBVR = 1 ILVR = .FALSE. ELSE IF( LSAME( JOBVR, 'V' ) ) THEN IJOBVR = 2 ILVR = .TRUE. ELSE IJOBVR = -1 ILVR = .FALSE. END IF ILV = ILVL .OR. ILVR ! ! Test the input arguments ! INFO = 0 LQUERY = ( LWORK.EQ.-1 ) IF( IJOBVL.LE.0 ) THEN INFO = -1 ELSE IF( IJOBVR.LE.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN INFO = -12 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN INFO = -14 END IF ! ! Compute workspace ! (Note: Comments in the code beginning "Workspace:" describe the ! minimal amount of workspace needed at that point in the code, ! as well as the preferred amount for good performance. ! NB refers to the optimal block size for the immediately ! following subroutine, as returned by ILAENV. The workspace is ! computed assuming ILO = 1 and IHI = N, the worst case.) ! MINWRK = 1 IF( INFO.EQ.0 .AND. ( LWORK.GE.1 .OR. LQUERY ) ) THEN MAXWRK = 7*N + N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 ) MINWRK = MAX( 1, 8*N ) WORK( 1 ) = MAXWRK END IF ! IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) & INFO = -16 ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGGEV ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Get machine constants ! EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) SMLNUM = SQRT( SMLNUM ) / EPS BIGNUM = ONE / SMLNUM ! ! Scale A if max element outside range [SMLNUM,BIGNUM] ! ANRM = DLANGE( 'M', N, N, A, LDA, WORK ) ILASCL = .FALSE. IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN ANRMTO = SMLNUM ILASCL = .TRUE. ELSE IF( ANRM.GT.BIGNUM ) THEN ANRMTO = BIGNUM ILASCL = .TRUE. END IF IF( ILASCL ) & CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR ) ! ! Scale B if max element outside range [SMLNUM,BIGNUM] ! BNRM = DLANGE( 'M', N, N, B, LDB, WORK ) ILBSCL = .FALSE. IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN BNRMTO = SMLNUM ILBSCL = .TRUE. ELSE IF( BNRM.GT.BIGNUM ) THEN BNRMTO = BIGNUM ILBSCL = .TRUE. END IF IF( ILBSCL ) & CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR ) ! ! Permute the matrices A, B to isolate eigenvalues if possible ! (Workspace: need 6*N) ! ILEFT = 1 IRIGHT = N + 1 IWRK = IRIGHT + N CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ), & WORK( IRIGHT ), WORK( IWRK ), IERR ) ! ! Reduce B to triangular form (QR decomposition of B) ! (Workspace: need N, prefer N*NB) ! IROWS = IHI + 1 - ILO IF( ILV ) THEN ICOLS = N + 1 - ILO ELSE ICOLS = IROWS END IF ITAU = IWRK IWRK = ITAU + IROWS CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), & WORK( IWRK ), LWORK+1-IWRK, IERR ) ! ! Apply the orthogonal transformation to matrix A ! (Workspace: need N, prefer N*NB) ! CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB, & WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ), & LWORK+1-IWRK, IERR ) ! ! Initialize VL ! (Workspace: need N, prefer N*NB) ! IF( ILVL ) THEN CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL ) CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, & VL( ILO+1, ILO ), LDVL ) CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL, & WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR ) END IF ! ! Initialize VR ! IF( ILVR ) & CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR ) ! ! Reduce to generalized Hessenberg form ! (Workspace: none needed) ! IF( ILV ) THEN ! ! Eigenvectors requested -- work on whole matrix. ! CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL, & LDVL, VR, LDVR, IERR ) ELSE CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA, & B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR ) END IF ! ! Perform QZ algorithm (Compute eigenvalues, and optionally, the ! Schur forms and Schur vectors) ! (Workspace: need N) ! IWRK = ITAU IF( ILV ) THEN CHTEMP = 'S' ELSE CHTEMP = 'E' END IF CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, & ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, & WORK( IWRK ), LWORK+1-IWRK, IERR ) IF( IERR.NE.0 ) THEN IF( IERR.GT.0 .AND. IERR.LE.N ) THEN INFO = IERR ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN INFO = IERR - N ELSE INFO = N + 1 END IF GO TO 110 END IF ! ! Compute Eigenvectors ! (Workspace: need 6*N) ! IF( ILV ) THEN IF( ILVL ) THEN IF( ILVR ) THEN CHTEMP = 'B' ELSE CHTEMP = 'L' END IF ELSE CHTEMP = 'R' END IF CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL, & VR, LDVR, N, IN, WORK( IWRK ), IERR ) IF( IERR.NE.0 ) THEN INFO = N + 2 GO TO 110 END IF ! ! Undo balancing on VL and VR and normalization ! (Workspace: none needed) ! IF( ILVL ) THEN CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ), & WORK( IRIGHT ), N, VL, LDVL, IERR ) DO 50 JC = 1, N IF( ALPHAI( JC ).LT.ZERO ) & GO TO 50 TEMP = ZERO IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 10 JR = 1, N TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) ) 10 CONTINUE ELSE DO 20 JR = 1, N TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+ & ABS( VL( JR, JC+1 ) ) ) 20 CONTINUE END IF IF( TEMP.LT.SMLNUM ) & GO TO 50 TEMP = ONE / TEMP IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 30 JR = 1, N VL( JR, JC ) = VL( JR, JC )*TEMP 30 CONTINUE ELSE DO 40 JR = 1, N VL( JR, JC ) = VL( JR, JC )*TEMP VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP 40 CONTINUE END IF 50 CONTINUE END IF IF( ILVR ) THEN CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ), & WORK( IRIGHT ), N, VR, LDVR, IERR ) DO 100 JC = 1, N IF( ALPHAI( JC ).LT.ZERO ) & GO TO 100 TEMP = ZERO IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 60 JR = 1, N TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) ) 60 CONTINUE ELSE DO 70 JR = 1, N TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+ & ABS( VR( JR, JC+1 ) ) ) 70 CONTINUE END IF IF( TEMP.LT.SMLNUM ) & GO TO 100 TEMP = ONE / TEMP IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 80 JR = 1, N VR( JR, JC ) = VR( JR, JC )*TEMP 80 CONTINUE ELSE DO 90 JR = 1, N VR( JR, JC ) = VR( JR, JC )*TEMP VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP 90 CONTINUE END IF 100 CONTINUE END IF ! ! End of eigenvector calculation ! END IF ! ! Undo scaling if necessary ! IF( ILASCL ) THEN CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR ) CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR ) END IF ! IF( ILBSCL ) THEN CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) END IF ! 110 CONTINUE ! WORK( 1 ) = MAXWRK ! RETURN ! ! End of DGGEV ! END SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, & ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO, & IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, & RCONDV, WORK, LWORK, IWORK, BWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER BALANC, JOBVL, JOBVR, SENSE INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N DOUBLE PRECISION ABNRM, BBNRM ! .. ! .. Array Arguments .. LOGICAL BWORK( * ) INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), & B( LDB, * ), BETA( * ), LSCALE( * ), & RCONDE( * ), RCONDV( * ), RSCALE( * ), & VL( LDVL, * ), VR( LDVR, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B) ! the generalized eigenvalues, and optionally, the left and/or right ! generalized eigenvectors. ! ! Optionally also, it computes a balancing transformation to improve ! the conditioning of the eigenvalues and eigenvectors (ILO, IHI, ! LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for ! the eigenvalues (RCONDE), and reciprocal condition numbers for the ! right eigenvectors (RCONDV). ! ! A generalized eigenvalue for a pair of matrices (A,B) is a scalar ! lambda or a ratio alpha/beta = lambda, such that A - lambda*B is ! singular. It is usually represented as the pair (alpha,beta), as ! there is a reasonable interpretation for beta=0, and even for both ! being zero. ! ! The right eigenvector v(j) corresponding to the eigenvalue lambda(j) ! of (A,B) satisfies ! ! A * v(j) = lambda(j) * B * v(j) . ! ! The left eigenvector u(j) corresponding to the eigenvalue lambda(j) ! of (A,B) satisfies ! ! u(j)**H * A = lambda(j) * u(j)**H * B. ! ! where u(j)**H is the conjugate-transpose of u(j). ! ! ! Arguments ! ========= ! ! BALANC (input) CHARACTER*1 ! Specifies the balance option to be performed. ! = 'N': do not diagonally scale or permute; ! = 'P': permute only; ! = 'S': scale only; ! = 'B': both permute and scale. ! Computed reciprocal condition numbers will be for the ! matrices after permuting and/or balancing. Permuting does ! not change condition numbers (in exact arithmetic), but ! balancing does. ! ! JOBVL (input) CHARACTER*1 ! = 'N': do not compute the left generalized eigenvectors; ! = 'V': compute the left generalized eigenvectors. ! ! JOBVR (input) CHARACTER*1 ! = 'N': do not compute the right generalized eigenvectors; ! = 'V': compute the right generalized eigenvectors. ! ! SENSE (input) CHARACTER*1 ! Determines which reciprocal condition numbers are computed. ! = 'N': none are computed; ! = 'E': computed for eigenvalues only; ! = 'V': computed for eigenvectors only; ! = 'B': computed for eigenvalues and eigenvectors. ! ! N (input) INTEGER ! The order of the matrices A, B, VL, and VR. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the matrix A in the pair (A,B). ! On exit, A has been overwritten. If JOBVL='V' or JOBVR='V' ! or both, then A contains the first part of the real Schur ! form of the "balanced" versions of the input A and B. ! ! LDA (input) INTEGER ! The leading dimension of A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB, N) ! On entry, the matrix B in the pair (A,B). ! On exit, B has been overwritten. If JOBVL='V' or JOBVR='V' ! or both, then B contains the second part of the real Schur ! form of the "balanced" versions of the input A and B. ! ! LDB (input) INTEGER ! The leading dimension of B. LDB >= max(1,N). ! ! ALPHAR (output) DOUBLE PRECISION array, dimension (N) ! ALPHAI (output) DOUBLE PRECISION array, dimension (N) ! BETA (output) DOUBLE PRECISION array, dimension (N) ! On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will ! be the generalized eigenvalues. If ALPHAI(j) is zero, then ! the j-th eigenvalue is real; if positive, then the j-th and ! (j+1)-st eigenvalues are a complex conjugate pair, with ! ALPHAI(j+1) negative. ! ! Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) ! may easily over- or underflow, and BETA(j) may even be zero. ! Thus, the user should avoid naively computing the ratio ! ALPHA/BETA. However, ALPHAR and ALPHAI will be always less ! than and usually comparable with norm(A) in magnitude, and ! BETA always less than and usually comparable with norm(B). ! ! VL (output) DOUBLE PRECISION array, dimension (LDVL,N) ! If JOBVL = 'V', the left eigenvectors u(j) are stored one ! after another in the columns of VL, in the same order as ! their eigenvalues. If the j-th eigenvalue is real, then ! u(j) = VL(:,j), the j-th column of VL. If the j-th and ! (j+1)-th eigenvalues form a complex conjugate pair, then ! u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1). ! Each eigenvector will be scaled so the largest component have ! abs(real part) + abs(imag. part) = 1. ! Not referenced if JOBVL = 'N'. ! ! LDVL (input) INTEGER ! The leading dimension of the matrix VL. LDVL >= 1, and ! if JOBVL = 'V', LDVL >= N. ! ! VR (output) DOUBLE PRECISION array, dimension (LDVR,N) ! If JOBVR = 'V', the right eigenvectors v(j) are stored one ! after another in the columns of VR, in the same order as ! their eigenvalues. If the j-th eigenvalue is real, then ! v(j) = VR(:,j), the j-th column of VR. If the j-th and ! (j+1)-th eigenvalues form a complex conjugate pair, then ! v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). ! Each eigenvector will be scaled so the largest component have ! abs(real part) + abs(imag. part) = 1. ! Not referenced if JOBVR = 'N'. ! ! LDVR (input) INTEGER ! The leading dimension of the matrix VR. LDVR >= 1, and ! if JOBVR = 'V', LDVR >= N. ! ! ILO,IHI (output) INTEGER ! ILO and IHI are integer values such that on exit ! A(i,j) = 0 and B(i,j) = 0 if i > j and ! j = 1,...,ILO-1 or i = IHI+1,...,N. ! If BALANC = 'N' or 'S', ILO = 1 and IHI = N. ! ! LSCALE (output) DOUBLE PRECISION array, dimension (N) ! Details of the permutations and scaling factors applied ! to the left side of A and B. If PL(j) is the index of the ! row interchanged with row j, and DL(j) is the scaling ! factor applied to row j, then ! LSCALE(j) = PL(j) for j = 1,...,ILO-1 ! = DL(j) for j = ILO,...,IHI ! = PL(j) for j = IHI+1,...,N. ! The order in which the interchanges are made is N to IHI+1, ! then 1 to ILO-1. ! ! RSCALE (output) DOUBLE PRECISION array, dimension (N) ! Details of the permutations and scaling factors applied ! to the right side of A and B. If PR(j) is the index of the ! column interchanged with column j, and DR(j) is the scaling ! factor applied to column j, then ! RSCALE(j) = PR(j) for j = 1,...,ILO-1 ! = DR(j) for j = ILO,...,IHI ! = PR(j) for j = IHI+1,...,N ! The order in which the interchanges are made is N to IHI+1, ! then 1 to ILO-1. ! ! ABNRM (output) DOUBLE PRECISION ! The one-norm of the balanced matrix A. ! ! BBNRM (output) DOUBLE PRECISION ! The one-norm of the balanced matrix B. ! ! RCONDE (output) DOUBLE PRECISION array, dimension (N) ! If SENSE = 'E' or 'B', the reciprocal condition numbers of ! the selected eigenvalues, stored in consecutive elements of ! the array. For a complex conjugate pair of eigenvalues two ! consecutive elements of RCONDE are set to the same value. ! Thus RCONDE(j), RCONDV(j), and the j-th columns of VL and VR ! all correspond to the same eigenpair (but not in general the ! j-th eigenpair, unless all eigenpairs are selected). ! If SENSE = 'V', RCONDE is not referenced. ! ! RCONDV (output) DOUBLE PRECISION array, dimension (N) ! If SENSE = 'V' or 'B', the estimated reciprocal condition ! numbers of the selected eigenvectors, stored in consecutive ! elements of the array. For a complex eigenvector two ! consecutive elements of RCONDV are set to the same value. If ! the eigenvalues cannot be reordered to compute RCONDV(j), ! RCONDV(j) is set to 0; this can only occur when the true ! value would be very small anyway. ! If SENSE = 'E', RCONDV is not referenced. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,6*N). ! If SENSE = 'E', LWORK >= 12*N. ! If SENSE = 'V' or 'B', LWORK >= 2*N*N+12*N+16. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace) INTEGER array, dimension (N+6) ! If SENSE = 'E', IWORK is not referenced. ! ! BWORK (workspace) LOGICAL array, dimension (N) ! If SENSE = 'N', BWORK is not referenced. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! = 1,...,N: ! The QZ iteration failed. No eigenvectors have been ! calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) ! should be correct for j=INFO+1,...,N. ! > N: =N+1: other than QZ iteration failed in DHGEQZ. ! =N+2: error return from DTGEVC. ! ! Further Details ! =============== ! ! Balancing a matrix pair (A,B) includes, first, permuting rows and ! columns to isolate eigenvalues, second, applying diagonal similarity ! transformation to the rows and columns to make the rows and columns ! as close in norm as possible. The computed reciprocal condition ! numbers correspond to the balanced matrix. Permuting rows and columns ! will not change the condition numbers (in exact arithmetic) but ! diagonal scaling will. For further explanation of balancing, see ! section 4.11.1.2 of LAPACK Users' Guide. ! ! An approximate error bound on the chordal distance between the i-th ! computed generalized eigenvalue w and the corresponding exact ! eigenvalue lambda is ! ! chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I) ! ! An approximate error bound for the angle between the i-th computed ! eigenvector VL(i) or VR(i) is given by ! ! EPS * norm(ABNRM, BBNRM) / DIF(i). ! ! For further explanation of the reciprocal condition numbers RCONDE ! and RCONDV, see section 4.11 of LAPACK User's Guide. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, PAIR, & WANTSB, WANTSE, WANTSN, WANTSV CHARACTER CHTEMP INTEGER I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS, & ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK, & MINWRK, MM DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, & SMLNUM, TEMP ! .. ! .. Local Arrays .. LOGICAL LDUMMA( 1 ) ! .. ! .. External Subroutines .. EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY, & DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, DTGSNA, & XERBLA ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! ! Decode the input arguments ! IF( LSAME( JOBVL, 'N' ) ) THEN IJOBVL = 1 ILVL = .FALSE. ELSE IF( LSAME( JOBVL, 'V' ) ) THEN IJOBVL = 2 ILVL = .TRUE. ELSE IJOBVL = -1 ILVL = .FALSE. END IF ! IF( LSAME( JOBVR, 'N' ) ) THEN IJOBVR = 1 ILVR = .FALSE. ELSE IF( LSAME( JOBVR, 'V' ) ) THEN IJOBVR = 2 ILVR = .TRUE. ELSE IJOBVR = -1 ILVR = .FALSE. END IF ILV = ILVL .OR. ILVR ! WANTSN = LSAME( SENSE, 'N' ) WANTSE = LSAME( SENSE, 'E' ) WANTSV = LSAME( SENSE, 'V' ) WANTSB = LSAME( SENSE, 'B' ) ! ! Test the input arguments ! INFO = 0 LQUERY = ( LWORK.EQ.-1 ) IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, & 'S' ) .OR. LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) ) & THEN INFO = -1 ELSE IF( IJOBVL.LE.0 ) THEN INFO = -2 ELSE IF( IJOBVR.LE.0 ) THEN INFO = -3 ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSB .OR. WANTSV ) ) & THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN INFO = -14 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN INFO = -16 END IF ! ! Compute workspace ! (Note: Comments in the code beginning "Workspace:" describe the ! minimal amount of workspace needed at that point in the code, ! as well as the preferred amount for good performance. ! NB refers to the optimal block size for the immediately ! following subroutine, as returned by ILAENV. The workspace is ! computed assuming ILO = 1 and IHI = N, the worst case.) ! MINWRK = 1 IF( INFO.EQ.0 .AND. ( LWORK.GE.1 .OR. LQUERY ) ) THEN MAXWRK = 5*N + N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 ) MINWRK = MAX( 1, 6*N ) IF( WANTSE ) THEN MINWRK = MAX( 1, 12*N ) ELSE IF( WANTSV .OR. WANTSB ) THEN MINWRK = 2*N*N + 12*N + 16 MAXWRK = MAX( MAXWRK, 2*N*N+12*N+16 ) END IF WORK( 1 ) = MAXWRK END IF ! IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -26 END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGGEVX', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! ! Get machine constants ! EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) SMLNUM = SQRT( SMLNUM ) / EPS BIGNUM = ONE / SMLNUM ! ! Scale A if max element outside range [SMLNUM,BIGNUM] ! ANRM = DLANGE( 'M', N, N, A, LDA, WORK ) ILASCL = .FALSE. IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN ANRMTO = SMLNUM ILASCL = .TRUE. ELSE IF( ANRM.GT.BIGNUM ) THEN ANRMTO = BIGNUM ILASCL = .TRUE. END IF IF( ILASCL ) & CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR ) ! ! Scale B if max element outside range [SMLNUM,BIGNUM] ! BNRM = DLANGE( 'M', N, N, B, LDB, WORK ) ILBSCL = .FALSE. IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN BNRMTO = SMLNUM ILBSCL = .TRUE. ELSE IF( BNRM.GT.BIGNUM ) THEN BNRMTO = BIGNUM ILBSCL = .TRUE. END IF IF( ILBSCL ) & CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR ) ! ! Permute and/or balance the matrix pair (A,B) ! (Workspace: need 6*N) ! CALL DGGBAL( BALANC, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, & WORK, IERR ) ! ! Compute ABNRM and BBNRM ! ABNRM = DLANGE( '1', N, N, A, LDA, WORK( 1 ) ) IF( ILASCL ) THEN WORK( 1 ) = ABNRM CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, 1, 1, WORK( 1 ), 1, & IERR ) ABNRM = WORK( 1 ) END IF ! BBNRM = DLANGE( '1', N, N, B, LDB, WORK( 1 ) ) IF( ILBSCL ) THEN WORK( 1 ) = BBNRM CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, 1, 1, WORK( 1 ), 1, & IERR ) BBNRM = WORK( 1 ) END IF ! ! Reduce B to triangular form (QR decomposition of B) ! (Workspace: need N, prefer N*NB ) ! IROWS = IHI + 1 - ILO IF( ILV .OR. .NOT.WANTSN ) THEN ICOLS = N + 1 - ILO ELSE ICOLS = IROWS END IF ITAU = 1 IWRK = ITAU + IROWS CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), & WORK( IWRK ), LWORK+1-IWRK, IERR ) ! ! Apply the orthogonal transformation to A ! (Workspace: need N, prefer N*NB) ! CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB, & WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ), & LWORK+1-IWRK, IERR ) ! ! Initialize VL and/or VR ! (Workspace: need N, prefer N*NB) ! IF( ILVL ) THEN CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL ) CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, & VL( ILO+1, ILO ), LDVL ) CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL, & WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR ) END IF ! IF( ILVR ) & CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR ) ! ! Reduce to generalized Hessenberg form ! (Workspace: none needed) ! IF( ILV .OR. .NOT.WANTSN ) THEN ! ! Eigenvectors requested -- work on whole matrix. ! CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL, & LDVL, VR, LDVR, IERR ) ELSE CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA, & B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR ) END IF ! ! Perform QZ algorithm (Compute eigenvalues, and optionally, the ! Schur forms and Schur vectors) ! (Workspace: need N) ! IF( ILV .OR. .NOT.WANTSN ) THEN CHTEMP = 'S' ELSE CHTEMP = 'E' END IF ! CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, & ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, & LWORK, IERR ) IF( IERR.NE.0 ) THEN IF( IERR.GT.0 .AND. IERR.LE.N ) THEN INFO = IERR ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN INFO = IERR - N ELSE INFO = N + 1 END IF GO TO 130 END IF ! ! Compute Eigenvectors and estimate condition numbers if desired ! (Workspace: DTGEVC: need 6*N ! DTGSNA: need 2*N*(N+2)+16 if SENSE = 'V' or 'B', ! need N otherwise ) ! IF( ILV .OR. .NOT.WANTSN ) THEN IF( ILV ) THEN IF( ILVL ) THEN IF( ILVR ) THEN CHTEMP = 'B' ELSE CHTEMP = 'L' END IF ELSE CHTEMP = 'R' END IF ! CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, & LDVL, VR, LDVR, N, IN, WORK, IERR ) IF( IERR.NE.0 ) THEN INFO = N + 2 GO TO 130 END IF END IF ! IF( .NOT.WANTSN ) THEN ! ! compute eigenvectors (DTGEVC) and estimate condition ! numbers (DTGSNA). Note that the definition of the condition ! number is not invariant under transformation (u,v) to ! (Q*u, Z*v), where (u,v) are eigenvectors of the generalized ! Schur form (S,T), Q and Z are orthogonal matrices. In order ! to avoid using extra 2*N*N workspace, we have to recalculate ! eigenvectors and estimate one condition numbers at a time. ! PAIR = .FALSE. DO 20 I = 1, N ! IF( PAIR ) THEN PAIR = .FALSE. GO TO 20 END IF MM = 1 IF( I.LT.N ) THEN IF( A( I+1, I ).NE.ZERO ) THEN PAIR = .TRUE. MM = 2 END IF END IF ! DO 10 J = 1, N BWORK( J ) = .FALSE. 10 CONTINUE IF( MM.EQ.1 ) THEN BWORK( I ) = .TRUE. ELSE IF( MM.EQ.2 ) THEN BWORK( I ) = .TRUE. BWORK( I+1 ) = .TRUE. END IF ! IWRK = MM*N + 1 IWRK1 = IWRK + MM*N ! ! Compute a pair of left and right eigenvectors. ! (compute workspace: need up to 4*N + 6*N) ! IF( WANTSE .OR. WANTSB ) THEN CALL DTGEVC( 'B', 'S', BWORK, N, A, LDA, B, LDB, & WORK( 1 ), N, WORK( IWRK ), N, MM, M, & WORK( IWRK1 ), IERR ) IF( IERR.NE.0 ) THEN INFO = N + 2 GO TO 130 END IF END IF ! CALL DTGSNA( SENSE, 'S', BWORK, N, A, LDA, B, LDB, & WORK( 1 ), N, WORK( IWRK ), N, RCONDE( I ), & RCONDV( I ), MM, M, WORK( IWRK1 ), & LWORK-IWRK1+1, IWORK, IERR ) ! 20 CONTINUE END IF END IF ! ! Undo balancing on VL and VR and normalization ! (Workspace: none needed) ! IF( ILVL ) THEN CALL DGGBAK( BALANC, 'L', N, ILO, IHI, LSCALE, RSCALE, N, VL, & LDVL, IERR ) ! DO 70 JC = 1, N IF( ALPHAI( JC ).LT.ZERO ) & GO TO 70 TEMP = ZERO IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 30 JR = 1, N TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) ) 30 CONTINUE ELSE DO 40 JR = 1, N TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+ & ABS( VL( JR, JC+1 ) ) ) 40 CONTINUE END IF IF( TEMP.LT.SMLNUM ) & GO TO 70 TEMP = ONE / TEMP IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 50 JR = 1, N VL( JR, JC ) = VL( JR, JC )*TEMP 50 CONTINUE ELSE DO 60 JR = 1, N VL( JR, JC ) = VL( JR, JC )*TEMP VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP 60 CONTINUE END IF 70 CONTINUE END IF IF( ILVR ) THEN CALL DGGBAK( BALANC, 'R', N, ILO, IHI, LSCALE, RSCALE, N, VR, & LDVR, IERR ) DO 120 JC = 1, N IF( ALPHAI( JC ).LT.ZERO ) & GO TO 120 TEMP = ZERO IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 80 JR = 1, N TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) ) 80 CONTINUE ELSE DO 90 JR = 1, N TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+ & ABS( VR( JR, JC+1 ) ) ) 90 CONTINUE END IF IF( TEMP.LT.SMLNUM ) & GO TO 120 TEMP = ONE / TEMP IF( ALPHAI( JC ).EQ.ZERO ) THEN DO 100 JR = 1, N VR( JR, JC ) = VR( JR, JC )*TEMP 100 CONTINUE ELSE DO 110 JR = 1, N VR( JR, JC ) = VR( JR, JC )*TEMP VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP 110 CONTINUE END IF 120 CONTINUE END IF ! ! Undo scaling if necessary ! IF( ILASCL ) THEN CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR ) CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR ) END IF ! IF( ILBSCL ) THEN CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) END IF ! 130 CONTINUE WORK( 1 ) = MAXWRK ! RETURN ! ! End of DGGEVX ! END SUBROUTINE DGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK, & INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LWORK, M, N, P ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), D( * ), WORK( * ), & X( * ), Y( * ) ! .. ! ! Purpose ! ======= ! ! DGGGLM solves a general Gauss-Markov linear model (GLM) problem: ! ! minimize || y ||_2 subject to d = A*x + B*y ! x ! ! where A is an N-by-M matrix, B is an N-by-P matrix, and d is a ! given N-vector. It is assumed that M <= N <= M+P, and ! ! rank(A) = M and rank( A B ) = N. ! ! Under these assumptions, the constrained equation is always ! consistent, and there is a unique solution x and a minimal 2-norm ! solution y, which is obtained using a generalized QR factorization ! of A and B. ! ! In particular, if matrix B is square nonsingular, then the problem ! GLM is equivalent to the following weighted linear least squares ! problem ! ! minimize || inv(B)*(d-A*x) ||_2 ! x ! ! where inv(B) denotes the inverse of B. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The number of rows of the matrices A and B. N >= 0. ! ! M (input) INTEGER ! The number of columns of the matrix A. 0 <= M <= N. ! ! P (input) INTEGER ! The number of columns of the matrix B. P >= N-M. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,M) ! On entry, the N-by-M matrix A. ! On exit, A is destroyed. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,P) ! On entry, the N-by-P matrix B. ! On exit, B is destroyed. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, D is the left hand side of the GLM equation. ! On exit, D is destroyed. ! ! X (output) DOUBLE PRECISION array, dimension (M) ! Y (output) DOUBLE PRECISION array, dimension (P) ! On exit, X and Y are the solutions of the GLM problem. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,N+M+P). ! For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB, ! where NB is an upper bound for the optimal blocksizes for ! DGEQRF, SGERQF, DORMQR and SORMRQ. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! =================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, LOPT, LWKOPT, NB, NB1, NB2, NB3, NB4, NP ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEMV, DGGQRF, DORMQR, DORMRQ, DTRSV, & XERBLA ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Intrinsic Functions .. INTRINSIC INT, MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 NP = MIN( N, P ) NB1 = ILAENV( 1, 'DGEQRF', ' ', N, M, -1, -1 ) NB2 = ILAENV( 1, 'DGERQF', ' ', N, M, -1, -1 ) NB3 = ILAENV( 1, 'DORMQR', ' ', N, M, P, -1 ) NB4 = ILAENV( 1, 'DORMRQ', ' ', N, M, P, -1 ) NB = MAX( NB1, NB2, NB3, NB4 ) LWKOPT = M + NP + MAX( N, P )*NB WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) IF( N.LT.0 ) THEN INFO = -1 ELSE IF( M.LT.0 .OR. M.GT.N ) THEN INFO = -2 ELSE IF( P.LT.0 .OR. P.LT.N-M ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LWORK.LT.MAX( 1, N+M+P ) .AND. .NOT.LQUERY ) THEN INFO = -12 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGGGLM', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Compute the GQR factorization of matrices A and B: ! ! Q'*A = ( R11 ) M, Q'*B*Z' = ( T11 T12 ) M ! ( 0 ) N-M ( 0 T22 ) N-M ! M M+P-N N-M ! ! where R11 and T22 are upper triangular, and Q and Z are ! orthogonal. ! CALL DGGQRF( N, M, P, A, LDA, WORK, B, LDB, WORK( M+1 ), & WORK( M+NP+1 ), LWORK-M-NP, INFO ) LOPT = WORK( M+NP+1 ) ! ! Update left-hand-side vector d = Q'*d = ( d1 ) M ! ( d2 ) N-M ! CALL DORMQR( 'Left', 'Transpose', N, 1, M, A, LDA, WORK, D, & MAX( 1, N ), WORK( M+NP+1 ), LWORK-M-NP, INFO ) LOPT = MAX( LOPT, INT( WORK( M+NP+1 ) ) ) ! ! Solve T22*y2 = d2 for y2 ! CALL DTRSV( 'Upper', 'No transpose', 'Non unit', N-M, & B( M+1, M+P-N+1 ), LDB, D( M+1 ), 1 ) CALL DCOPY( N-M, D( M+1 ), 1, Y( M+P-N+1 ), 1 ) ! ! Set y1 = 0 ! DO 10 I = 1, M + P - N Y( I ) = ZERO 10 CONTINUE ! ! Update d1 = d1 - T12*y2 ! CALL DGEMV( 'No transpose', M, N-M, -ONE, B( 1, M+P-N+1 ), LDB, & Y( M+P-N+1 ), 1, ONE, D, 1 ) ! ! Solve triangular system: R11*x = d1 ! CALL DTRSV( 'Upper', 'No Transpose', 'Non unit', M, A, LDA, D, 1 ) ! ! Copy D to X ! CALL DCOPY( M, D, 1, X, 1 ) ! ! Backward transformation y = Z'*y ! CALL DORMRQ( 'Left', 'Transpose', P, 1, NP, & B( MAX( 1, N-P+1 ), 1 ), LDB, WORK( M+1 ), Y, & MAX( 1, P ), WORK( M+NP+1 ), LWORK-M-NP, INFO ) WORK( 1 ) = M + NP + MAX( LOPT, INT( WORK( M+NP+1 ) ) ) ! RETURN ! ! End of DGGGLM ! END SUBROUTINE DGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, & LDQ, Z, LDZ, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER COMPQ, COMPZ INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), & Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DGGHRD reduces a pair of real matrices (A,B) to generalized upper ! Hessenberg form using orthogonal transformations, where A is a ! general matrix and B is upper triangular: Q' * A * Z = H and ! Q' * B * Z = T, where H is upper Hessenberg, T is upper triangular, ! and Q and Z are orthogonal, and ' means transpose. ! ! The orthogonal matrices Q and Z are determined as products of Givens ! rotations. They may either be formed explicitly, or they may be ! postmultiplied into input matrices Q1 and Z1, so that ! ! Q1 * A * Z1' = (Q1*Q) * H * (Z1*Z)' ! Q1 * B * Z1' = (Q1*Q) * T * (Z1*Z)' ! ! Arguments ! ========= ! ! COMPQ (input) CHARACTER*1 ! = 'N': do not compute Q; ! = 'I': Q is initialized to the unit matrix, and the ! orthogonal matrix Q is returned; ! = 'V': Q must contain an orthogonal matrix Q1 on entry, ! and the product Q1*Q is returned. ! ! COMPZ (input) CHARACTER*1 ! = 'N': do not compute Z; ! = 'I': Z is initialized to the unit matrix, and the ! orthogonal matrix Z is returned; ! = 'V': Z must contain an orthogonal matrix Z1 on entry, ! and the product Z1*Z is returned. ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! ILO (input) INTEGER ! IHI (input) INTEGER ! It is assumed that A is already upper triangular in rows and ! columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set ! by a previous call to DGGBAL; otherwise they should be set ! to 1 and N respectively. ! 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the N-by-N general matrix to be reduced. ! On exit, the upper triangle and the first subdiagonal of A ! are overwritten with the upper Hessenberg matrix H, and the ! rest is set to zero. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB, N) ! On entry, the N-by-N upper triangular matrix B. ! On exit, the upper triangular matrix T = Q' B Z. The ! elements below the diagonal are set to zero. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N) ! If COMPQ='N': Q is not referenced. ! If COMPQ='I': on entry, Q need not be set, and on exit it ! contains the orthogonal matrix Q, where Q' ! is the product of the Givens transformations ! which are applied to A and B on the left. ! If COMPQ='V': on entry, Q must contain an orthogonal matrix ! Q1, and on exit this is overwritten by Q1*Q. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. ! LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise. ! ! Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N) ! If COMPZ='N': Z is not referenced. ! If COMPZ='I': on entry, Z need not be set, and on exit it ! contains the orthogonal matrix Z, which is ! the product of the Givens transformations ! which are applied to A and B on the right. ! If COMPZ='V': on entry, Z must contain an orthogonal matrix ! Z1, and on exit this is overwritten by Z1*Z. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. ! LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! This routine reduces A to Hessenberg and B to triangular form by ! an unblocked reduction, as described in _Matrix_Computations_, ! by Golub and Van Loan (Johns Hopkins Press.) ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL ILQ, ILZ INTEGER ICOMPQ, ICOMPZ, JCOL, JROW DOUBLE PRECISION C, S, TEMP ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DLARTG, DLASET, DROT, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Decode COMPQ ! IF( LSAME( COMPQ, 'N' ) ) THEN ILQ = .FALSE. ICOMPQ = 1 ELSE IF( LSAME( COMPQ, 'V' ) ) THEN ILQ = .TRUE. ICOMPQ = 2 ELSE IF( LSAME( COMPQ, 'I' ) ) THEN ILQ = .TRUE. ICOMPQ = 3 ELSE ICOMPQ = 0 END IF ! ! Decode COMPZ ! IF( LSAME( COMPZ, 'N' ) ) THEN ILZ = .FALSE. ICOMPZ = 1 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN ILZ = .TRUE. ICOMPZ = 2 ELSE IF( LSAME( COMPZ, 'I' ) ) THEN ILZ = .TRUE. ICOMPZ = 3 ELSE ICOMPZ = 0 END IF ! ! Test the input parameters. ! INFO = 0 IF( ICOMPQ.LE.0 ) THEN INFO = -1 ELSE IF( ICOMPZ.LE.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( ILO.LT.1 ) THEN INFO = -4 ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( ( ILQ .AND. LDQ.LT.N ) .OR. LDQ.LT.1 ) THEN INFO = -11 ELSE IF( ( ILZ .AND. LDZ.LT.N ) .OR. LDZ.LT.1 ) THEN INFO = -13 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGGHRD', -INFO ) RETURN END IF ! ! Initialize Q and Z if desired. ! IF( ICOMPQ.EQ.3 ) & CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ ) IF( ICOMPZ.EQ.3 ) & CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ ) ! ! Quick return if possible ! IF( N.LE.1 ) & RETURN ! ! Zero out lower triangle of B ! DO 20 JCOL = 1, N - 1 DO 10 JROW = JCOL + 1, N B( JROW, JCOL ) = ZERO 10 CONTINUE 20 CONTINUE ! ! Reduce A and B ! DO 40 JCOL = ILO, IHI - 2 ! DO 30 JROW = IHI, JCOL + 2, -1 ! ! Step 1: rotate rows JROW-1, JROW to kill A(JROW,JCOL) ! TEMP = A( JROW-1, JCOL ) CALL DLARTG( TEMP, A( JROW, JCOL ), C, S, & A( JROW-1, JCOL ) ) A( JROW, JCOL ) = ZERO CALL DROT( N-JCOL, A( JROW-1, JCOL+1 ), LDA, & A( JROW, JCOL+1 ), LDA, C, S ) CALL DROT( N+2-JROW, B( JROW-1, JROW-1 ), LDB, & B( JROW, JROW-1 ), LDB, C, S ) IF( ILQ ) & CALL DROT( N, Q( 1, JROW-1 ), 1, Q( 1, JROW ), 1, C, S ) ! ! Step 2: rotate columns JROW, JROW-1 to kill B(JROW,JROW-1) ! TEMP = B( JROW, JROW ) CALL DLARTG( TEMP, B( JROW, JROW-1 ), C, S, & B( JROW, JROW ) ) B( JROW, JROW-1 ) = ZERO CALL DROT( IHI, A( 1, JROW ), 1, A( 1, JROW-1 ), 1, C, S ) CALL DROT( JROW-1, B( 1, JROW ), 1, B( 1, JROW-1 ), 1, C, & S ) IF( ILZ ) & CALL DROT( N, Z( 1, JROW ), 1, Z( 1, JROW-1 ), 1, C, S ) 30 CONTINUE 40 CONTINUE ! RETURN ! ! End of DGGHRD ! END SUBROUTINE DGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, & INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LWORK, M, N, P ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( * ), D( * ), & WORK( * ), X( * ) ! .. ! ! Purpose ! ======= ! ! DGGLSE solves the linear equality-constrained least squares (LSE) ! problem: ! ! minimize || c - A*x ||_2 subject to B*x = d ! ! where A is an M-by-N matrix, B is a P-by-N matrix, c is a given ! M-vector, and d is a given P-vector. It is assumed that ! P <= N <= M+P, and ! ! rank(B) = P and rank( ( A ) ) = N. ! ( ( B ) ) ! ! These conditions ensure that the LSE problem has a unique solution, ! which is obtained using a GRQ factorization of the matrices B and A. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrices A and B. N >= 0. ! ! P (input) INTEGER ! The number of rows of the matrix B. 0 <= P <= N <= M+P. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, A is destroyed. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,N) ! On entry, the P-by-N matrix B. ! On exit, B is destroyed. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,P). ! ! C (input/output) DOUBLE PRECISION array, dimension (M) ! On entry, C contains the right hand side vector for the ! least squares part of the LSE problem. ! On exit, the residual sum of squares for the solution ! is given by the sum of squares of elements N-P+1 to M of ! vector C. ! ! D (input/output) DOUBLE PRECISION array, dimension (P) ! On entry, D contains the right hand side vector for the ! constrained equation. ! On exit, D is destroyed. ! ! X (output) DOUBLE PRECISION array, dimension (N) ! On exit, X is the solution of the LSE problem. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,M+N+P). ! For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB, ! where NB is an upper bound for the optimal blocksizes for ! DGEQRF, SGERQF, DORMQR and SORMRQ. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER LOPT, LWKOPT, MN, NB, NB1, NB2, NB3, NB4, NR ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DGEMV, DGGRQF, DORMQR, DORMRQ, & DTRMV, DTRSV, XERBLA ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Intrinsic Functions .. INTRINSIC INT, MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 MN = MIN( M, N ) NB1 = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) NB2 = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 ) NB3 = ILAENV( 1, 'DORMQR', ' ', M, N, P, -1 ) NB4 = ILAENV( 1, 'DORMRQ', ' ', M, N, P, -1 ) NB = MAX( NB1, NB2, NB3, NB4 ) LWKOPT = P + MN + MAX( M, N )*NB WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( P.LT.0 .OR. P.GT.N .OR. P.LT.N-M ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN INFO = -7 ELSE IF( LWORK.LT.MAX( 1, M+N+P ) .AND. .NOT.LQUERY ) THEN INFO = -12 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGGLSE', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Compute the GRQ factorization of matrices B and A: ! ! B*Q' = ( 0 T12 ) P Z'*A*Q' = ( R11 R12 ) N-P ! N-P P ( 0 R22 ) M+P-N ! N-P P ! ! where T12 and R11 are upper triangular, and Q and Z are ! orthogonal. ! CALL DGGRQF( P, M, N, B, LDB, WORK, A, LDA, WORK( P+1 ), & WORK( P+MN+1 ), LWORK-P-MN, INFO ) LOPT = WORK( P+MN+1 ) ! ! Update c = Z'*c = ( c1 ) N-P ! ( c2 ) M+P-N ! CALL DORMQR( 'Left', 'Transpose', M, 1, MN, A, LDA, WORK( P+1 ), & C, MAX( 1, M ), WORK( P+MN+1 ), LWORK-P-MN, INFO ) LOPT = MAX( LOPT, INT( WORK( P+MN+1 ) ) ) ! ! Solve T12*x2 = d for x2 ! CALL DTRSV( 'Upper', 'No transpose', 'Non unit', P, B( 1, N-P+1 ), & LDB, D, 1 ) ! ! Update c1 ! CALL DGEMV( 'No transpose', N-P, P, -ONE, A( 1, N-P+1 ), LDA, D, & 1, ONE, C, 1 ) ! ! Sovle R11*x1 = c1 for x1 ! CALL DTRSV( 'Upper', 'No transpose', 'Non unit', N-P, A, LDA, C, & 1 ) ! ! Put the solutions in X ! CALL DCOPY( N-P, C, 1, X, 1 ) CALL DCOPY( P, D, 1, X( N-P+1 ), 1 ) ! ! Compute the residual vector: ! IF( M.LT.N ) THEN NR = M + P - N CALL DGEMV( 'No transpose', NR, N-M, -ONE, A( N-P+1, M+1 ), & LDA, D( NR+1 ), 1, ONE, C( N-P+1 ), 1 ) ELSE NR = P END IF CALL DTRMV( 'Upper', 'No transpose', 'Non unit', NR, & A( N-P+1, N-P+1 ), LDA, D, 1 ) CALL DAXPY( NR, -ONE, D, 1, C( N-P+1 ), 1 ) ! ! Backward transformation x = Q'*x ! CALL DORMRQ( 'Left', 'Transpose', N, 1, P, B, LDB, WORK( 1 ), X, & N, WORK( P+MN+1 ), LWORK-P-MN, INFO ) WORK( 1 ) = P + MN + MAX( LOPT, INT( WORK( P+MN+1 ) ) ) ! RETURN ! ! End of DGGLSE ! END SUBROUTINE DGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, & LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LWORK, M, N, P ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), & WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGGQRF computes a generalized QR factorization of an N-by-M matrix A ! and an N-by-P matrix B: ! ! A = Q*R, B = Q*T*Z, ! ! where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal ! matrix, and R and T assume one of the forms: ! ! if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N, ! ( 0 ) N-M N M-N ! M ! ! where R11 is upper triangular, and ! ! if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P, ! P-N N ( T21 ) P ! P ! ! where T12 or T21 is upper triangular. ! ! In particular, if B is square and nonsingular, the GQR factorization ! of A and B implicitly gives the QR factorization of inv(B)*A: ! ! inv(B)*A = Z'*(inv(T)*R) ! ! where inv(B) denotes the inverse of the matrix B, and Z' denotes the ! transpose of the matrix Z. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The number of rows of the matrices A and B. N >= 0. ! ! M (input) INTEGER ! The number of columns of the matrix A. M >= 0. ! ! P (input) INTEGER ! The number of columns of the matrix B. P >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,M) ! On entry, the N-by-M matrix A. ! On exit, the elements on and above the diagonal of the array ! contain the min(N,M)-by-M upper trapezoidal matrix R (R is ! upper triangular if N >= M); the elements below the diagonal, ! with the array TAUA, represent the orthogonal matrix Q as a ! product of min(N,M) elementary reflectors (see Further ! Details). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! TAUA (output) DOUBLE PRECISION array, dimension (min(N,M)) ! The scalar factors of the elementary reflectors which ! represent the orthogonal matrix Q (see Further Details). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,P) ! On entry, the N-by-P matrix B. ! On exit, if N <= P, the upper triangle of the subarray ! B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; ! if N > P, the elements on and above the (N-P)-th subdiagonal ! contain the N-by-P upper trapezoidal matrix T; the remaining ! elements, with the array TAUB, represent the orthogonal ! matrix Z as a product of elementary reflectors (see Further ! Details). ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! TAUB (output) DOUBLE PRECISION array, dimension (min(N,P)) ! The scalar factors of the elementary reflectors which ! represent the orthogonal matrix Z (see Further Details). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,N,M,P). ! For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3), ! where NB1 is the optimal blocksize for the QR factorization ! of an N-by-M matrix, NB2 is the optimal blocksize for the ! RQ factorization of an N-by-P matrix, and NB3 is the optimal ! blocksize for a call of DORMQR. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of elementary reflectors ! ! Q = H(1) H(2) . . . H(k), where k = min(n,m). ! ! Each H(i) has the form ! ! H(i) = I - taua * v * v' ! ! where taua is a real scalar, and v is a real vector with ! v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), ! and taua in TAUA(i). ! To form Q explicitly, use LAPACK subroutine DORGQR. ! To use Q to update another matrix, use LAPACK subroutine DORMQR. ! ! The matrix Z is represented as a product of elementary reflectors ! ! Z = H(1) H(2) . . . H(k), where k = min(n,p). ! ! Each H(i) has the form ! ! H(i) = I - taub * v * v' ! ! where taub is a real scalar, and v is a real vector with ! v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in ! B(n-k+i,1:p-k+i-1), and taub in TAUB(i). ! To form Z explicitly, use LAPACK subroutine DORGRQ. ! To use Z to update another matrix, use LAPACK subroutine DORMRQ. ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL LQUERY INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3 ! .. ! .. External Subroutines .. EXTERNAL DGEQRF, DGERQF, DORMQR, XERBLA ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Intrinsic Functions .. INTRINSIC INT, MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 NB1 = ILAENV( 1, 'DGEQRF', ' ', N, M, -1, -1 ) NB2 = ILAENV( 1, 'DGERQF', ' ', N, P, -1, -1 ) NB3 = ILAENV( 1, 'DORMQR', ' ', N, M, P, -1 ) NB = MAX( NB1, NB2, NB3 ) LWKOPT = MAX( N, M, P )*NB WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) IF( N.LT.0 ) THEN INFO = -1 ELSE IF( M.LT.0 ) THEN INFO = -2 ELSE IF( P.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LWORK.LT.MAX( 1, N, M, P ) .AND. .NOT.LQUERY ) THEN INFO = -11 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGGQRF', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! QR factorization of N-by-M matrix A: A = Q*R ! CALL DGEQRF( N, M, A, LDA, TAUA, WORK, LWORK, INFO ) LOPT = WORK( 1 ) ! ! Update B := Q'*B. ! CALL DORMQR( 'Left', 'Transpose', N, P, MIN( N, M ), A, LDA, TAUA, & B, LDB, WORK, LWORK, INFO ) LOPT = MAX( LOPT, INT( WORK( 1 ) ) ) ! ! RQ factorization of N-by-P matrix B: B = T*Z. ! CALL DGERQF( N, P, B, LDB, TAUB, WORK, LWORK, INFO ) WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) ) ! RETURN ! ! End of DGGQRF ! END SUBROUTINE DGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, & LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LWORK, M, N, P ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), & WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGGRQF computes a generalized RQ factorization of an M-by-N matrix A ! and a P-by-N matrix B: ! ! A = R*Q, B = Z*T*Q, ! ! where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal ! matrix, and R and T assume one of the forms: ! ! if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N, ! N-M M ( R21 ) N ! N ! ! where R12 or R21 is upper triangular, and ! ! if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P, ! ( 0 ) P-N P N-P ! N ! ! where T11 is upper triangular. ! ! In particular, if B is square and nonsingular, the GRQ factorization ! of A and B implicitly gives the RQ factorization of A*inv(B): ! ! A*inv(B) = (R*inv(T))*Z' ! ! where inv(B) denotes the inverse of the matrix B, and Z' denotes the ! transpose of the matrix Z. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! P (input) INTEGER ! The number of rows of the matrix B. P >= 0. ! ! N (input) INTEGER ! The number of columns of the matrices A and B. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, if M <= N, the upper triangle of the subarray ! A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R; ! if M > N, the elements on and above the (M-N)-th subdiagonal ! contain the M-by-N upper trapezoidal matrix R; the remaining ! elements, with the array TAUA, represent the orthogonal ! matrix Q as a product of elementary reflectors (see Further ! Details). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! TAUA (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The scalar factors of the elementary reflectors which ! represent the orthogonal matrix Q (see Further Details). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,N) ! On entry, the P-by-N matrix B. ! On exit, the elements on and above the diagonal of the array ! contain the min(P,N)-by-N upper trapezoidal matrix T (T is ! upper triangular if P >= N); the elements below the diagonal, ! with the array TAUB, represent the orthogonal matrix Z as a ! product of elementary reflectors (see Further Details). ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,P). ! ! TAUB (output) DOUBLE PRECISION array, dimension (min(P,N)) ! The scalar factors of the elementary reflectors which ! represent the orthogonal matrix Z (see Further Details). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,N,M,P). ! For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3), ! where NB1 is the optimal blocksize for the RQ factorization ! of an M-by-N matrix, NB2 is the optimal blocksize for the ! QR factorization of a P-by-N matrix, and NB3 is the optimal ! blocksize for a call of DORMRQ. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INF0= -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of elementary reflectors ! ! Q = H(1) H(2) . . . H(k), where k = min(m,n). ! ! Each H(i) has the form ! ! H(i) = I - taua * v * v' ! ! where taua is a real scalar, and v is a real vector with ! v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in ! A(m-k+i,1:n-k+i-1), and taua in TAUA(i). ! To form Q explicitly, use LAPACK subroutine DORGRQ. ! To use Q to update another matrix, use LAPACK subroutine DORMRQ. ! ! The matrix Z is represented as a product of elementary reflectors ! ! Z = H(1) H(2) . . . H(k), where k = min(p,n). ! ! Each H(i) has the form ! ! H(i) = I - taub * v * v' ! ! where taub is a real scalar, and v is a real vector with ! v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i), ! and taub in TAUB(i). ! To form Z explicitly, use LAPACK subroutine DORGQR. ! To use Z to update another matrix, use LAPACK subroutine DORMQR. ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL LQUERY INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3 ! .. ! .. External Subroutines .. EXTERNAL DGEQRF, DGERQF, DORMRQ, XERBLA ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Intrinsic Functions .. INTRINSIC INT, MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 NB1 = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 ) NB2 = ILAENV( 1, 'DGEQRF', ' ', P, N, -1, -1 ) NB3 = ILAENV( 1, 'DORMRQ', ' ', M, N, P, -1 ) NB = MAX( NB1, NB2, NB3 ) LWKOPT = MAX( N, M, P )*NB WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( P.LT.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN INFO = -8 ELSE IF( LWORK.LT.MAX( 1, M, P, N ) .AND. .NOT.LQUERY ) THEN INFO = -11 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGGRQF', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! RQ factorization of M-by-N matrix A: A = R*Q ! CALL DGERQF( M, N, A, LDA, TAUA, WORK, LWORK, INFO ) LOPT = WORK( 1 ) ! ! Update B := B*Q' ! CALL DORMRQ( 'Right', 'Transpose', P, N, MIN( M, N ), & A( MAX( 1, M-N+1 ), 1 ), LDA, TAUA, B, LDB, WORK, & LWORK, INFO ) LOPT = MAX( LOPT, INT( WORK( 1 ) ) ) ! ! QR factorization of P-by-N matrix B: B = Z*T ! CALL DGEQRF( P, N, B, LDB, TAUB, WORK, LWORK, INFO ) WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) ) ! RETURN ! ! End of DGGRQF ! END SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, & LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, & IWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBQ, JOBU, JOBV INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ), & BETA( * ), Q( LDQ, * ), U( LDU, * ), & V( LDV, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGGSVD computes the generalized singular value decomposition (GSVD) ! of an M-by-N real matrix A and P-by-N real matrix B: ! ! U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) ! ! where U, V and Q are orthogonal matrices, and Z' is the transpose ! of Z. Let K+L = the effective numerical rank of the matrix (A',B')', ! then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and ! D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the ! following structures, respectively: ! ! If M-K-L >= 0, ! ! K L ! D1 = K ( I 0 ) ! L ( 0 C ) ! M-K-L ( 0 0 ) ! ! K L ! D2 = L ( 0 S ) ! P-L ( 0 0 ) ! ! N-K-L K L ! ( 0 R ) = K ( 0 R11 R12 ) ! L ( 0 0 R22 ) ! ! where ! ! C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), ! S = diag( BETA(K+1), ... , BETA(K+L) ), ! C**2 + S**2 = I. ! ! R is stored in A(1:K+L,N-K-L+1:N) on exit. ! ! If M-K-L < 0, ! ! K M-K K+L-M ! D1 = K ( I 0 0 ) ! M-K ( 0 C 0 ) ! ! K M-K K+L-M ! D2 = M-K ( 0 S 0 ) ! K+L-M ( 0 0 I ) ! P-L ( 0 0 0 ) ! ! N-K-L K M-K K+L-M ! ( 0 R ) = K ( 0 R11 R12 R13 ) ! M-K ( 0 0 R22 R23 ) ! K+L-M ( 0 0 0 R33 ) ! ! where ! ! C = diag( ALPHA(K+1), ... , ALPHA(M) ), ! S = diag( BETA(K+1), ... , BETA(M) ), ! C**2 + S**2 = I. ! ! (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored ! ( 0 R22 R23 ) ! in B(M-K+1:L,N+M-K-L+1:N) on exit. ! ! The routine computes C, S, R, and optionally the orthogonal ! transformation matrices U, V and Q. ! ! In particular, if B is an N-by-N nonsingular matrix, then the GSVD of ! A and B implicitly gives the SVD of A*inv(B): ! A*inv(B) = U*(D1*inv(D2))*V'. ! If ( A',B')' has orthonormal columns, then the GSVD of A and B is ! also equal to the CS decomposition of A and B. Furthermore, the GSVD ! can be used to derive the solution of the eigenvalue problem: ! A'*A x = lambda* B'*B x. ! In some literature, the GSVD of A and B is presented in the form ! U'*A*X = ( 0 D1 ), V'*B*X = ( 0 D2 ) ! where U and V are orthogonal and X is nonsingular, D1 and D2 are ! ``diagonal''. The former GSVD form can be converted to the latter ! form by taking the nonsingular matrix X as ! ! X = Q*( I 0 ) ! ( 0 inv(R) ). ! ! Arguments ! ========= ! ! JOBU (input) CHARACTER*1 ! = 'U': Orthogonal matrix U is computed; ! = 'N': U is not computed. ! ! JOBV (input) CHARACTER*1 ! = 'V': Orthogonal matrix V is computed; ! = 'N': V is not computed. ! ! JOBQ (input) CHARACTER*1 ! = 'Q': Orthogonal matrix Q is computed; ! = 'N': Q is not computed. ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrices A and B. N >= 0. ! ! P (input) INTEGER ! The number of rows of the matrix B. P >= 0. ! ! K (output) INTEGER ! L (output) INTEGER ! On exit, K and L specify the dimension of the subblocks ! described in the Purpose section. ! K + L = effective numerical rank of (A',B')'. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, A contains the triangular matrix R, or part of R. ! See Purpose for details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,N) ! On entry, the P-by-N matrix B. ! On exit, B contains the triangular matrix R if M-K-L < 0. ! See Purpose for details. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDA >= max(1,P). ! ! ALPHA (output) DOUBLE PRECISION array, dimension (N) ! BETA (output) DOUBLE PRECISION array, dimension (N) ! On exit, ALPHA and BETA contain the generalized singular ! value pairs of A and B; ! ALPHA(1:K) = 1, ! BETA(1:K) = 0, ! and if M-K-L >= 0, ! ALPHA(K+1:K+L) = C, ! BETA(K+1:K+L) = S, ! or if M-K-L < 0, ! ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0 ! BETA(K+1:M) =S, BETA(M+1:K+L) =1 ! and ! ALPHA(K+L+1:N) = 0 ! BETA(K+L+1:N) = 0 ! ! U (output) DOUBLE PRECISION array, dimension (LDU,M) ! If JOBU = 'U', U contains the M-by-M orthogonal matrix U. ! If JOBU = 'N', U is not referenced. ! ! LDU (input) INTEGER ! The leading dimension of the array U. LDU >= max(1,M) if ! JOBU = 'U'; LDU >= 1 otherwise. ! ! V (output) DOUBLE PRECISION array, dimension (LDV,P) ! If JOBV = 'V', V contains the P-by-P orthogonal matrix V. ! If JOBV = 'N', V is not referenced. ! ! LDV (input) INTEGER ! The leading dimension of the array V. LDV >= max(1,P) if ! JOBV = 'V'; LDV >= 1 otherwise. ! ! Q (output) DOUBLE PRECISION array, dimension (LDQ,N) ! If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q. ! If JOBQ = 'N', Q is not referenced. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. LDQ >= max(1,N) if ! JOBQ = 'Q'; LDQ >= 1 otherwise. ! ! WORK (workspace) DOUBLE PRECISION array, ! dimension (max(3*N,M,P)+N) ! ! IWORK (workspace/output) INTEGER array, dimension (N) ! On exit, IWORK stores the sorting information. More ! precisely, the following loop will sort ALPHA ! for I = K+1, min(M,K+L) ! swap ALPHA(I) and ALPHA(IWORK(I)) ! endfor ! such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N). ! ! INFO (output)INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = 1, the Jacobi-type procedure failed to ! converge. For further details, see subroutine DTGSJA. ! ! Internal Parameters ! =================== ! ! TOLA DOUBLE PRECISION ! TOLB DOUBLE PRECISION ! TOLA and TOLB are the thresholds to determine the effective ! rank of (A',B')'. Generally, they are set to ! TOLA = MAX(M,N)*norm(A)*MAZHEPS, ! TOLB = MAX(P,N)*norm(B)*MAZHEPS. ! The size of TOLA and TOLB may affect the size of backward ! errors of the decomposition. ! ! Further Details ! =============== ! ! 2-96 Based on modifications by ! Ming Gu and Huan Ren, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL WANTQ, WANTU, WANTV INTEGER I, IBND, ISUB, J, NCYCLE DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL LSAME, DLAMCH, DLANGE ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGGSVP, DTGSJA, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters ! WANTU = LSAME( JOBU, 'U' ) WANTV = LSAME( JOBV, 'V' ) WANTQ = LSAME( JOBQ, 'Q' ) ! INFO = 0 IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN INFO = -3 ELSE IF( M.LT.0 ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( P.LT.0 ) THEN INFO = -6 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -10 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN INFO = -12 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN INFO = -16 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN INFO = -18 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN INFO = -20 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGGSVD', -INFO ) RETURN END IF ! ! Compute the Frobenius norm of matrices A and B ! ANORM = DLANGE( '1', M, N, A, LDA, WORK ) BNORM = DLANGE( '1', P, N, B, LDB, WORK ) ! ! Get machine precision and set up threshold for determining ! the effective numerical rank of the matrices A and B. ! ULP = DLAMCH( 'Precision' ) UNFL = DLAMCH( 'Safe Minimum' ) TOLA = MAX( M, N )*MAX( ANORM, UNFL )*ULP TOLB = MAX( P, N )*MAX( BNORM, UNFL )*ULP ! ! Preprocessing ! CALL DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, & TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, WORK, & WORK( N+1 ), INFO ) ! ! Compute the GSVD of two upper "triangular" matrices ! CALL DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, & TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, & WORK, NCYCLE, INFO ) ! ! Sort the singular values and store the pivot indices in IWORK ! Copy ALPHA to WORK, then sort ALPHA in WORK ! CALL DCOPY( N, ALPHA, 1, WORK, 1 ) IBND = MIN( L, M-K ) DO 20 I = 1, IBND ! ! Scan for largest ALPHA(K+I) ! ISUB = I SMAX = WORK( K+I ) DO 10 J = I + 1, IBND TEMP = WORK( K+J ) IF( TEMP.GT.SMAX ) THEN ISUB = J SMAX = TEMP END IF 10 CONTINUE IF( ISUB.NE.I ) THEN WORK( K+ISUB ) = WORK( K+I ) WORK( K+I ) = SMAX IWORK( K+I ) = K + ISUB ELSE IWORK( K+I ) = K + I END IF 20 CONTINUE ! RETURN ! ! End of DGGSVD ! END SUBROUTINE DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, & TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, & IWORK, TAU, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER JOBQ, JOBU, JOBV INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P DOUBLE PRECISION TOLA, TOLB ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), & TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGGSVP computes orthogonal matrices U, V and Q such that ! ! N-K-L K L ! U'*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0; ! L ( 0 0 A23 ) ! M-K-L ( 0 0 0 ) ! ! N-K-L K L ! = K ( 0 A12 A13 ) if M-K-L < 0; ! M-K ( 0 0 A23 ) ! ! N-K-L K L ! V'*B*Q = L ( 0 0 B13 ) ! P-L ( 0 0 0 ) ! ! where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular ! upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, ! otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective ! numerical rank of the (M+P)-by-N matrix (A',B')'. Z' denotes the ! transpose of Z. ! ! This decomposition is the preprocessing step for computing the ! Generalized Singular Value Decomposition (GSVD), see subroutine ! DGGSVD. ! ! Arguments ! ========= ! ! JOBU (input) CHARACTER*1 ! = 'U': Orthogonal matrix U is computed; ! = 'N': U is not computed. ! ! JOBV (input) CHARACTER*1 ! = 'V': Orthogonal matrix V is computed; ! = 'N': V is not computed. ! ! JOBQ (input) CHARACTER*1 ! = 'Q': Orthogonal matrix Q is computed; ! = 'N': Q is not computed. ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! P (input) INTEGER ! The number of rows of the matrix B. P >= 0. ! ! N (input) INTEGER ! The number of columns of the matrices A and B. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, A contains the triangular (or trapezoidal) matrix ! described in the Purpose section. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,N) ! On entry, the P-by-N matrix B. ! On exit, B contains the triangular matrix described in ! the Purpose section. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,P). ! ! TOLA (input) DOUBLE PRECISION ! TOLB (input) DOUBLE PRECISION ! TOLA and TOLB are the thresholds to determine the effective ! numerical rank of matrix B and a subblock of A. Generally, ! they are set to ! TOLA = MAX(M,N)*norm(A)*MAZHEPS, ! TOLB = MAX(P,N)*norm(B)*MAZHEPS. ! The size of TOLA and TOLB may affect the size of backward ! errors of the decomposition. ! ! K (output) INTEGER ! L (output) INTEGER ! On exit, K and L specify the dimension of the subblocks ! described in Purpose. ! K + L = effective numerical rank of (A',B')'. ! ! U (output) DOUBLE PRECISION array, dimension (LDU,M) ! If JOBU = 'U', U contains the orthogonal matrix U. ! If JOBU = 'N', U is not referenced. ! ! LDU (input) INTEGER ! The leading dimension of the array U. LDU >= max(1,M) if ! JOBU = 'U'; LDU >= 1 otherwise. ! ! V (output) DOUBLE PRECISION array, dimension (LDV,M) ! If JOBV = 'V', V contains the orthogonal matrix V. ! If JOBV = 'N', V is not referenced. ! ! LDV (input) INTEGER ! The leading dimension of the array V. LDV >= max(1,P) if ! JOBV = 'V'; LDV >= 1 otherwise. ! ! Q (output) DOUBLE PRECISION array, dimension (LDQ,N) ! If JOBQ = 'Q', Q contains the orthogonal matrix Q. ! If JOBQ = 'N', Q is not referenced. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. LDQ >= max(1,N) if ! JOBQ = 'Q'; LDQ >= 1 otherwise. ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! TAU (workspace) DOUBLE PRECISION array, dimension (N) ! ! WORK (workspace) DOUBLE PRECISION array, dimension (max(3*N,M,P)) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! ! Further Details ! =============== ! ! The subroutine uses LAPACK subroutine DGEQPF for the QR factorization ! with column pivoting to detect the effective numerical rank of the ! a matrix. It may be replaced by a better rank determination strategy. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL FORWRD, WANTQ, WANTU, WANTV INTEGER I, J ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DGEQPF, DGEQR2, DGERQ2, DLACPY, DLAPMT, DLASET, & DORG2R, DORM2R, DORMR2, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters ! WANTU = LSAME( JOBU, 'U' ) WANTV = LSAME( JOBV, 'V' ) WANTQ = LSAME( JOBQ, 'Q' ) FORWRD = .TRUE. ! INFO = 0 IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN INFO = -3 ELSE IF( M.LT.0 ) THEN INFO = -4 ELSE IF( P.LT.0 ) THEN INFO = -5 ELSE IF( N.LT.0 ) THEN INFO = -6 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -8 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN INFO = -10 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN INFO = -16 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN INFO = -18 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN INFO = -20 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGGSVP', -INFO ) RETURN END IF ! ! QR with column pivoting of B: B*P = V*( S11 S12 ) ! ( 0 0 ) ! DO 10 I = 1, N IWORK( I ) = 0 10 CONTINUE CALL DGEQPF( P, N, B, LDB, IWORK, TAU, WORK, INFO ) ! ! Update A := A*P ! CALL DLAPMT( FORWRD, M, N, A, LDA, IWORK ) ! ! Determine the effective rank of matrix B. ! L = 0 DO 20 I = 1, MIN( P, N ) IF( ABS( B( I, I ) ).GT.TOLB ) & L = L + 1 20 CONTINUE ! IF( WANTV ) THEN ! ! Copy the details of V, and form V. ! CALL DLASET( 'Full', P, P, ZERO, ZERO, V, LDV ) IF( P.GT.1 ) & CALL DLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ), & LDV ) CALL DORG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO ) END IF ! ! Clean up B ! DO 40 J = 1, L - 1 DO 30 I = J + 1, L B( I, J ) = ZERO 30 CONTINUE 40 CONTINUE IF( P.GT.L ) & CALL DLASET( 'Full', P-L, N, ZERO, ZERO, B( L+1, 1 ), LDB ) ! IF( WANTQ ) THEN ! ! Set Q = I and Update Q := Q*P ! CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ ) CALL DLAPMT( FORWRD, N, N, Q, LDQ, IWORK ) END IF ! IF( P.GE.L .AND. N.NE.L ) THEN ! ! RQ factorization of (S11 S12): ( S11 S12 ) = ( 0 S12 )*Z ! CALL DGERQ2( L, N, B, LDB, TAU, WORK, INFO ) ! ! Update A := A*Z' ! CALL DORMR2( 'Right', 'Transpose', M, N, L, B, LDB, TAU, A, & LDA, WORK, INFO ) ! IF( WANTQ ) THEN ! ! Update Q := Q*Z' ! CALL DORMR2( 'Right', 'Transpose', N, N, L, B, LDB, TAU, Q, & LDQ, WORK, INFO ) END IF ! ! Clean up B ! CALL DLASET( 'Full', L, N-L, ZERO, ZERO, B, LDB ) DO 60 J = N - L + 1, N DO 50 I = J - N + L + 1, L B( I, J ) = ZERO 50 CONTINUE 60 CONTINUE ! END IF ! ! Let N-L L ! A = ( A11 A12 ) M, ! ! then the following does the complete QR decomposition of A11: ! ! A11 = U*( 0 T12 )*P1' ! ( 0 0 ) ! DO 70 I = 1, N - L IWORK( I ) = 0 70 CONTINUE CALL DGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, INFO ) ! ! Determine the effective rank of A11 ! K = 0 DO 80 I = 1, MIN( M, N-L ) IF( ABS( A( I, I ) ).GT.TOLA ) & K = K + 1 80 CONTINUE ! ! Update A12 := U'*A12, where A12 = A( 1:M, N-L+1:N ) ! CALL DORM2R( 'Left', 'Transpose', M, L, MIN( M, N-L ), A, LDA, & TAU, A( 1, N-L+1 ), LDA, WORK, INFO ) ! IF( WANTU ) THEN ! ! Copy the details of U, and form U ! CALL DLASET( 'Full', M, M, ZERO, ZERO, U, LDU ) IF( M.GT.1 ) & CALL DLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ), & LDU ) CALL DORG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO ) END IF ! IF( WANTQ ) THEN ! ! Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1 ! CALL DLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK ) END IF ! ! Clean up A: set the strictly lower triangular part of ! A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0. ! DO 100 J = 1, K - 1 DO 90 I = J + 1, K A( I, J ) = ZERO 90 CONTINUE 100 CONTINUE IF( M.GT.K ) & CALL DLASET( 'Full', M-K, N-L, ZERO, ZERO, A( K+1, 1 ), LDA ) ! IF( N-L.GT.K ) THEN ! ! RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1 ! CALL DGERQ2( K, N-L, A, LDA, TAU, WORK, INFO ) ! IF( WANTQ ) THEN ! ! Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1' ! CALL DORMR2( 'Right', 'Transpose', N, N-L, K, A, LDA, TAU, & Q, LDQ, WORK, INFO ) END IF ! ! Clean up A ! CALL DLASET( 'Full', K, N-L-K, ZERO, ZERO, A, LDA ) DO 120 J = N - L - K + 1, N - L DO 110 I = J - N + L + K + 1, K A( I, J ) = ZERO 110 CONTINUE 120 CONTINUE ! END IF ! IF( M.GT.K ) THEN ! ! QR factorization of A( K+1:M,N-L+1:N ) ! CALL DGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO ) ! IF( WANTU ) THEN ! ! Update U(:,K+1:M) := U(:,K+1:M)*U1 ! CALL DORM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ), & A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU, & WORK, INFO ) END IF ! ! Clean up ! DO 140 J = N - L + 1, N DO 130 I = J - N + K + L + 1, M A( I, J ) = ZERO 130 CONTINUE 140 CONTINUE ! END IF ! RETURN ! ! End of DGGSVP ! END SUBROUTINE DGTCON( NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND, & WORK, IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER NORM INTEGER INFO, N DOUBLE PRECISION ANORM, RCOND ! .. ! .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION D( * ), DL( * ), DU( * ), DU2( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DGTCON estimates the reciprocal of the condition number of a real ! tridiagonal matrix A using the LU factorization as computed by ! DGTTRF. ! ! An estimate is obtained for norm(inv(A)), and the reciprocal of the ! condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies whether the 1-norm condition number or the ! infinity-norm condition number is required: ! = '1' or 'O': 1-norm; ! = 'I': Infinity-norm. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! DL (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) multipliers that define the matrix L from the ! LU factorization of A as computed by DGTTRF. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of the upper triangular matrix U from ! the LU factorization of A. ! ! DU (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) elements of the first superdiagonal of U. ! ! DU2 (input) DOUBLE PRECISION array, dimension (N-2) ! The (n-2) elements of the second superdiagonal of U. ! ! IPIV (input) INTEGER array, dimension (N) ! The pivot indices; for 1 <= i <= n, row i of the matrix was ! interchanged with row IPIV(i). IPIV(i) will always be either ! i or i+1; IPIV(i) = i indicates a row interchange was not ! required. ! ! ANORM (input) DOUBLE PRECISION ! If NORM = '1' or 'O', the 1-norm of the original matrix A. ! If NORM = 'I', the infinity-norm of the original matrix A. ! ! RCOND (output) DOUBLE PRECISION ! The reciprocal of the condition number of the matrix A, ! computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an ! estimate of the 1-norm of inv(A) computed in this routine. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (2*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL ONENRM INTEGER I, KASE, KASE1 DOUBLE PRECISION AINVNM ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DGTTRS, DLACON, XERBLA ! .. ! .. Executable Statements .. ! ! Test the input arguments. ! INFO = 0 ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' ) IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( ANORM.LT.ZERO ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGTCON', -INFO ) RETURN END IF ! ! Quick return if possible ! RCOND = ZERO IF( N.EQ.0 ) THEN RCOND = ONE RETURN ELSE IF( ANORM.EQ.ZERO ) THEN RETURN END IF ! ! Check that D(1:N) is non-zero. ! DO 10 I = 1, N IF( D( I ).EQ.ZERO ) & RETURN 10 CONTINUE ! AINVNM = ZERO IF( ONENRM ) THEN KASE1 = 1 ELSE KASE1 = 2 END IF KASE = 0 20 CONTINUE CALL DLACON( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.KASE1 ) THEN ! ! Multiply by inv(U)*inv(L). ! CALL DGTTRS( 'No transpose', N, 1, DL, D, DU, DU2, IPIV, & WORK, N, INFO ) ELSE ! ! Multiply by inv(L')*inv(U'). ! CALL DGTTRS( 'Transpose', N, 1, DL, D, DU, DU2, IPIV, WORK, & N, INFO ) END IF GO TO 20 END IF ! ! Compute the estimate of the reciprocal condition number. ! IF( AINVNM.NE.ZERO ) & RCOND = ( ONE / AINVNM ) / ANORM ! RETURN ! ! End of DGTCON ! END SUBROUTINE DGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, & IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER TRANS INTEGER INFO, LDB, LDX, N, NRHS ! .. ! .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ), & DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ), & FERR( * ), WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DGTRFS improves the computed solution to a system of linear ! equations when the coefficient matrix is tridiagonal, and provides ! error bounds and backward error estimates for the solution. ! ! Arguments ! ========= ! ! TRANS (input) CHARACTER*1 ! Specifies the form of the system of equations: ! = 'N': A * X = B (No transpose) ! = 'T': A**T * X = B (Transpose) ! = 'C': A**H * X = B (Conjugate transpose = Transpose) ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! DL (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) subdiagonal elements of A. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The diagonal elements of A. ! ! DU (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) superdiagonal elements of A. ! ! DLF (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) multipliers that define the matrix L from the ! LU factorization of A as computed by DGTTRF. ! ! DF (input) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of the upper triangular matrix U from ! the LU factorization of A. ! ! DUF (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) elements of the first superdiagonal of U. ! ! DU2 (input) DOUBLE PRECISION array, dimension (N-2) ! The (n-2) elements of the second superdiagonal of U. ! ! IPIV (input) INTEGER array, dimension (N) ! The pivot indices; for 1 <= i <= n, row i of the matrix was ! interchanged with row IPIV(i). IPIV(i) will always be either ! i or i+1; IPIV(i) = i indicates a row interchange was not ! required. ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) ! The right hand side matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! On entry, the solution matrix X, as computed by DGTTRS. ! On exit, the improved solution matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Internal Parameters ! =================== ! ! ITMAX is the maximum number of steps of iterative refinement. ! ! ===================================================================== ! ! .. Parameters .. INTEGER ITMAX PARAMETER ( ITMAX = 5 ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D+0 ) DOUBLE PRECISION THREE PARAMETER ( THREE = 3.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOTRAN CHARACTER TRANSN, TRANST INTEGER COUNT, I, J, KASE, NZ DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DGTTRS, DLACON, DLAGTM, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 NOTRAN = LSAME( TRANS, 'N' ) IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. & LSAME( TRANS, 'C' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -13 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -15 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGTRFS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN DO 10 J = 1, NRHS FERR( J ) = ZERO BERR( J ) = ZERO 10 CONTINUE RETURN END IF ! IF( NOTRAN ) THEN TRANSN = 'N' TRANST = 'T' ELSE TRANSN = 'T' TRANST = 'N' END IF ! ! NZ = maximum number of nonzero elements in each row of A, plus 1 ! NZ = 4 EPS = DLAMCH( 'Epsilon' ) SAFMIN = DLAMCH( 'Safe minimum' ) SAFE1 = NZ*SAFMIN SAFE2 = SAFE1 / EPS ! ! Do for each right hand side ! DO 110 J = 1, NRHS ! COUNT = 1 LSTRES = THREE 20 CONTINUE ! ! Loop until stopping criterion is satisfied. ! ! Compute residual R = B - op(A) * X, ! where op(A) = A, A**T, or A**H, depending on TRANS. ! CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) CALL DLAGTM( TRANS, N, 1, -ONE, DL, D, DU, X( 1, J ), LDX, ONE, & WORK( N+1 ), N ) ! ! Compute abs(op(A))*abs(x) + abs(b) for use in the backward ! error bound. ! IF( NOTRAN ) THEN IF( N.EQ.1 ) THEN WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) ELSE WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) + & ABS( DU( 1 )*X( 2, J ) ) DO 30 I = 2, N - 1 WORK( I ) = ABS( B( I, J ) ) + & ABS( DL( I-1 )*X( I-1, J ) ) + & ABS( D( I )*X( I, J ) ) + & ABS( DU( I )*X( I+1, J ) ) 30 CONTINUE WORK( N ) = ABS( B( N, J ) ) + & ABS( DL( N-1 )*X( N-1, J ) ) + & ABS( D( N )*X( N, J ) ) END IF ELSE IF( N.EQ.1 ) THEN WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) ELSE WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) + & ABS( DL( 1 )*X( 2, J ) ) DO 40 I = 2, N - 1 WORK( I ) = ABS( B( I, J ) ) + & ABS( DU( I-1 )*X( I-1, J ) ) + & ABS( D( I )*X( I, J ) ) + & ABS( DL( I )*X( I+1, J ) ) 40 CONTINUE WORK( N ) = ABS( B( N, J ) ) + & ABS( DU( N-1 )*X( N-1, J ) ) + & ABS( D( N )*X( N, J ) ) END IF END IF ! ! Compute componentwise relative backward error from formula ! ! max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) ! ! where abs(Z) is the componentwise absolute value of the matrix ! or vector Z. If the i-th component of the denominator is less ! than SAFE2, then SAFE1 is added to the i-th components of the ! numerator and denominator before dividing. ! S = ZERO DO 50 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) ELSE S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / & ( WORK( I )+SAFE1 ) ) END IF 50 CONTINUE BERR( J ) = S ! ! Test stopping criterion. Continue iterating if ! 1) The residual BERR(J) is larger than machine epsilon, and ! 2) BERR(J) decreased by at least a factor of 2 during the ! last iteration, and ! 3) At most ITMAX iterations tried. ! IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. & COUNT.LE.ITMAX ) THEN ! ! Update solution and try again. ! CALL DGTTRS( TRANS, N, 1, DLF, DF, DUF, DU2, IPIV, & WORK( N+1 ), N, INFO ) CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) LSTRES = BERR( J ) COUNT = COUNT + 1 GO TO 20 END IF ! ! Bound error from formula ! ! norm(X - XTRUE) / norm(X) .le. FERR = ! norm( abs(inv(op(A)))* ! ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) ! ! where ! norm(Z) is the magnitude of the largest component of Z ! inv(op(A)) is the inverse of op(A) ! abs(Z) is the componentwise absolute value of the matrix or ! vector Z ! NZ is the maximum number of nonzeros in any row of A, plus 1 ! EPS is machine epsilon ! ! The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) ! is incremented by SAFE1 if the i-th component of ! abs(op(A))*abs(X) + abs(B) is less than SAFE2. ! ! Use DLACON to estimate the infinity-norm of the matrix ! inv(op(A)) * diag(W), ! where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) ! DO 60 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) ELSE WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 END IF 60 CONTINUE ! KASE = 0 70 CONTINUE CALL DLACON( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), & KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN ! ! Multiply by diag(W)*inv(op(A)**T). ! CALL DGTTRS( TRANST, N, 1, DLF, DF, DUF, DU2, IPIV, & WORK( N+1 ), N, INFO ) DO 80 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 80 CONTINUE ELSE ! ! Multiply by inv(op(A))*diag(W). ! DO 90 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 90 CONTINUE CALL DGTTRS( TRANSN, N, 1, DLF, DF, DUF, DU2, IPIV, & WORK( N+1 ), N, INFO ) END IF GO TO 70 END IF ! ! Normalize error. ! LSTRES = ZERO DO 100 I = 1, N LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 100 CONTINUE IF( LSTRES.NE.ZERO ) & FERR( J ) = FERR( J ) / LSTRES ! 110 CONTINUE ! RETURN ! ! End of DGTRFS ! END SUBROUTINE DGTSV( N, NRHS, DL, D, DU, B, LDB, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDB, N, NRHS ! .. ! .. Array Arguments .. DOUBLE PRECISION B( LDB, * ), D( * ), DL( * ), DU( * ) ! .. ! ! Purpose ! ======= ! ! DGTSV solves the equation ! ! A*X = B, ! ! where A is an n by n tridiagonal matrix, by Gaussian elimination with ! partial pivoting. ! ! Note that the equation A'*X = B may be solved by interchanging the ! order of the arguments DU and DL. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! DL (input/output) DOUBLE PRECISION array, dimension (N-1) ! On entry, DL must contain the (n-1) sub-diagonal elements of ! A. ! ! On exit, DL is overwritten by the (n-2) elements of the ! second super-diagonal of the upper triangular matrix U from ! the LU factorization of A, in DL(1), ..., DL(n-2). ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, D must contain the diagonal elements of A. ! ! On exit, D is overwritten by the n diagonal elements of U. ! ! DU (input/output) DOUBLE PRECISION array, dimension (N-1) ! On entry, DU must contain the (n-1) super-diagonal elements ! of A. ! ! On exit, DU is overwritten by the (n-1) elements of the first ! super-diagonal of U. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the N by NRHS matrix of right hand side matrix B. ! On exit, if INFO = 0, the N by NRHS solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, U(i,i) is exactly zero, and the solution ! has not been computed. The factorization has not been ! completed unless i = N. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J DOUBLE PRECISION FACT, TEMP ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Executable Statements .. ! INFO = 0 IF( N.LT.0 ) THEN INFO = -1 ELSE IF( NRHS.LT.0 ) THEN INFO = -2 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGTSV ', -INFO ) RETURN END IF ! IF( N.EQ.0 ) & RETURN ! IF( NRHS.EQ.1 ) THEN DO 10 I = 1, N - 2 IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN ! ! No row interchange required ! IF( D( I ).NE.ZERO ) THEN FACT = DL( I ) / D( I ) D( I+1 ) = D( I+1 ) - FACT*DU( I ) B( I+1, 1 ) = B( I+1, 1 ) - FACT*B( I, 1 ) ELSE INFO = I RETURN END IF DL( I ) = ZERO ELSE ! ! Interchange rows I and I+1 ! FACT = D( I ) / DL( I ) D( I ) = DL( I ) TEMP = D( I+1 ) D( I+1 ) = DU( I ) - FACT*TEMP DL( I ) = DU( I+1 ) DU( I+1 ) = -FACT*DL( I ) DU( I ) = TEMP TEMP = B( I, 1 ) B( I, 1 ) = B( I+1, 1 ) B( I+1, 1 ) = TEMP - FACT*B( I+1, 1 ) END IF 10 CONTINUE IF( N.GT.1 ) THEN I = N - 1 IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN IF( D( I ).NE.ZERO ) THEN FACT = DL( I ) / D( I ) D( I+1 ) = D( I+1 ) - FACT*DU( I ) B( I+1, 1 ) = B( I+1, 1 ) - FACT*B( I, 1 ) ELSE INFO = I RETURN END IF ELSE FACT = D( I ) / DL( I ) D( I ) = DL( I ) TEMP = D( I+1 ) D( I+1 ) = DU( I ) - FACT*TEMP DU( I ) = TEMP TEMP = B( I, 1 ) B( I, 1 ) = B( I+1, 1 ) B( I+1, 1 ) = TEMP - FACT*B( I+1, 1 ) END IF END IF IF( D( N ).EQ.ZERO ) THEN INFO = N RETURN END IF ELSE DO 40 I = 1, N - 2 IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN ! ! No row interchange required ! IF( D( I ).NE.ZERO ) THEN FACT = DL( I ) / D( I ) D( I+1 ) = D( I+1 ) - FACT*DU( I ) DO 20 J = 1, NRHS B( I+1, J ) = B( I+1, J ) - FACT*B( I, J ) 20 CONTINUE ELSE INFO = I RETURN END IF DL( I ) = ZERO ELSE ! ! Interchange rows I and I+1 ! FACT = D( I ) / DL( I ) D( I ) = DL( I ) TEMP = D( I+1 ) D( I+1 ) = DU( I ) - FACT*TEMP DL( I ) = DU( I+1 ) DU( I+1 ) = -FACT*DL( I ) DU( I ) = TEMP DO 30 J = 1, NRHS TEMP = B( I, J ) B( I, J ) = B( I+1, J ) B( I+1, J ) = TEMP - FACT*B( I+1, J ) 30 CONTINUE END IF 40 CONTINUE IF( N.GT.1 ) THEN I = N - 1 IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN IF( D( I ).NE.ZERO ) THEN FACT = DL( I ) / D( I ) D( I+1 ) = D( I+1 ) - FACT*DU( I ) DO 50 J = 1, NRHS B( I+1, J ) = B( I+1, J ) - FACT*B( I, J ) 50 CONTINUE ELSE INFO = I RETURN END IF ELSE FACT = D( I ) / DL( I ) D( I ) = DL( I ) TEMP = D( I+1 ) D( I+1 ) = DU( I ) - FACT*TEMP DU( I ) = TEMP DO 60 J = 1, NRHS TEMP = B( I, J ) B( I, J ) = B( I+1, J ) B( I+1, J ) = TEMP - FACT*B( I+1, J ) 60 CONTINUE END IF END IF IF( D( N ).EQ.ZERO ) THEN INFO = N RETURN END IF END IF ! ! Back solve with the matrix U from the factorization. ! IF( NRHS.LE.2 ) THEN J = 1 70 CONTINUE B( N, J ) = B( N, J ) / D( N ) IF( N.GT.1 ) & B( N-1, J ) = ( B( N-1, J )-DU( N-1 )*B( N, J ) ) / D( N-1 ) DO 80 I = N - 2, 1, -1 B( I, J ) = ( B( I, J )-DU( I )*B( I+1, J )-DL( I )* & B( I+2, J ) ) / D( I ) 80 CONTINUE IF( J.LT.NRHS ) THEN J = J + 1 GO TO 70 END IF ELSE DO 100 J = 1, NRHS B( N, J ) = B( N, J ) / D( N ) IF( N.GT.1 ) & B( N-1, J ) = ( B( N-1, J )-DU( N-1 )*B( N, J ) ) / & D( N-1 ) DO 90 I = N - 2, 1, -1 B( I, J ) = ( B( I, J )-DU( I )*B( I+1, J )-DL( I )* & B( I+2, J ) ) / D( I ) 90 CONTINUE 100 CONTINUE END IF ! RETURN ! ! End of DGTSV ! END SUBROUTINE DGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, & DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, & WORK, IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER FACT, TRANS INTEGER INFO, LDB, LDX, N, NRHS DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ), & DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ), & FERR( * ), WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DGTSVX uses the LU factorization to compute the solution to a real ! system of linear equations A * X = B or A**T * X = B, ! where A is a tridiagonal matrix of order N and X and B are N-by-NRHS ! matrices. ! ! Error bounds on the solution and a condition estimate are also ! provided. ! ! Description ! =========== ! ! The following steps are performed: ! ! 1. If FACT = 'N', the LU decomposition is used to factor the matrix A ! as A = L * U, where L is a product of permutation and unit lower ! bidiagonal matrices and U is upper triangular with nonzeros in ! only the main diagonal and first two superdiagonals. ! ! 2. If some U(i,i)=0, so that U is exactly singular, then the routine ! returns with INFO = i. Otherwise, the factored form of A is used ! to estimate the condition number of the matrix A. If the ! reciprocal of the condition number is less than machine precision, ! INFO = N+1 is returned as a warning, but the routine still goes on ! to solve for X and compute error bounds as described below. ! ! 3. The system of equations is solved for X using the factored form ! of A. ! ! 4. Iterative refinement is applied to improve the computed solution ! matrix and calculate error bounds and backward error estimates ! for it. ! ! Arguments ! ========= ! ! FACT (input) CHARACTER*1 ! Specifies whether or not the factored form of A has been ! supplied on entry. ! = 'F': DLF, DF, DUF, DU2, and IPIV contain the factored ! form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV ! will not be modified. ! = 'N': The matrix will be copied to DLF, DF, and DUF ! and factored. ! ! TRANS (input) CHARACTER*1 ! Specifies the form of the system of equations: ! = 'N': A * X = B (No transpose) ! = 'T': A**T * X = B (Transpose) ! = 'C': A**H * X = B (Conjugate transpose = Transpose) ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! DL (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) subdiagonal elements of A. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of A. ! ! DU (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) superdiagonal elements of A. ! ! DLF (input or output) DOUBLE PRECISION array, dimension (N-1) ! If FACT = 'F', then DLF is an input argument and on entry ! contains the (n-1) multipliers that define the matrix L from ! the LU factorization of A as computed by DGTTRF. ! ! If FACT = 'N', then DLF is an output argument and on exit ! contains the (n-1) multipliers that define the matrix L from ! the LU factorization of A. ! ! DF (input or output) DOUBLE PRECISION array, dimension (N) ! If FACT = 'F', then DF is an input argument and on entry ! contains the n diagonal elements of the upper triangular ! matrix U from the LU factorization of A. ! ! If FACT = 'N', then DF is an output argument and on exit ! contains the n diagonal elements of the upper triangular ! matrix U from the LU factorization of A. ! ! DUF (input or output) DOUBLE PRECISION array, dimension (N-1) ! If FACT = 'F', then DUF is an input argument and on entry ! contains the (n-1) elements of the first superdiagonal of U. ! ! If FACT = 'N', then DUF is an output argument and on exit ! contains the (n-1) elements of the first superdiagonal of U. ! ! DU2 (input or output) DOUBLE PRECISION array, dimension (N-2) ! If FACT = 'F', then DU2 is an input argument and on entry ! contains the (n-2) elements of the second superdiagonal of ! U. ! ! If FACT = 'N', then DU2 is an output argument and on exit ! contains the (n-2) elements of the second superdiagonal of ! U. ! ! IPIV (input or output) INTEGER array, dimension (N) ! If FACT = 'F', then IPIV is an input argument and on entry ! contains the pivot indices from the LU factorization of A as ! computed by DGTTRF. ! ! If FACT = 'N', then IPIV is an output argument and on exit ! contains the pivot indices from the LU factorization of A; ! row i of the matrix was interchanged with row IPIV(i). ! IPIV(i) will always be either i or i+1; IPIV(i) = i indicates ! a row interchange was not required. ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) ! The N-by-NRHS right hand side matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! RCOND (output) DOUBLE PRECISION ! The estimate of the reciprocal condition number of the matrix ! A. If RCOND is less than the machine precision (in ! particular, if RCOND = 0), the matrix is singular to working ! precision. This condition is indicated by a return code of ! INFO > 0. ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, and i is ! <= N: U(i,i) is exactly zero. The factorization ! has not been completed unless i = N, but the ! factor U is exactly singular, so the solution ! and error bounds could not be computed. ! RCOND = 0 is returned. ! = N+1: U is nonsingular, but RCOND is less than machine ! precision, meaning that the matrix is singular ! to working precision. Nevertheless, the ! solution and error bounds are computed because ! there are a number of situations where the ! computed solution can be more accurate than the ! value of RCOND would suggest. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOFACT, NOTRAN CHARACTER NORM DOUBLE PRECISION ANORM ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANGT EXTERNAL LSAME, DLAMCH, DLANGT ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGTCON, DGTRFS, DGTTRF, DGTTRS, DLACPY, & XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! INFO = 0 NOFACT = LSAME( FACT, 'N' ) NOTRAN = LSAME( TRANS, 'N' ) IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. & LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.0 ) THEN INFO = -4 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -14 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -16 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGTSVX', -INFO ) RETURN END IF ! IF( NOFACT ) THEN ! ! Compute the LU factorization of A. ! CALL DCOPY( N, D, 1, DF, 1 ) IF( N.GT.1 ) THEN CALL DCOPY( N-1, DL, 1, DLF, 1 ) CALL DCOPY( N-1, DU, 1, DUF, 1 ) END IF CALL DGTTRF( N, DLF, DF, DUF, DU2, IPIV, INFO ) ! ! Return if INFO is non-zero. ! IF( INFO.NE.0 ) THEN IF( INFO.GT.0 ) & RCOND = ZERO RETURN END IF END IF ! ! Compute the norm of the matrix A. ! IF( NOTRAN ) THEN NORM = '1' ELSE NORM = 'I' END IF ANORM = DLANGT( NORM, N, DL, D, DU ) ! ! Compute the reciprocal of the condition number of A. ! CALL DGTCON( NORM, N, DLF, DF, DUF, DU2, IPIV, ANORM, RCOND, WORK, & IWORK, INFO ) ! ! Set INFO = N+1 if the matrix is singular to working precision. ! IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) & INFO = N + 1 ! ! Compute the solution vectors X. ! CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL DGTTRS( TRANS, N, NRHS, DLF, DF, DUF, DU2, IPIV, X, LDX, & INFO ) ! ! Use iterative refinement to improve the computed solutions and ! compute error bounds and backward error estimates for them. ! CALL DGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, & B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO ) ! RETURN ! ! End of DGTSVX ! END SUBROUTINE DGTTRF( N, DL, D, DU, DU2, IPIV, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, N ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION D( * ), DL( * ), DU( * ), DU2( * ) ! .. ! ! Purpose ! ======= ! ! DGTTRF computes an LU factorization of a real tridiagonal matrix A ! using elimination with partial pivoting and row interchanges. ! ! The factorization has the form ! A = L * U ! where L is a product of permutation and unit lower bidiagonal ! matrices and U is upper triangular with nonzeros in only the main ! diagonal and first two superdiagonals. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix A. ! ! DL (input/output) DOUBLE PRECISION array, dimension (N-1) ! On entry, DL must contain the (n-1) sub-diagonal elements of ! A. ! ! On exit, DL is overwritten by the (n-1) multipliers that ! define the matrix L from the LU factorization of A. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, D must contain the diagonal elements of A. ! ! On exit, D is overwritten by the n diagonal elements of the ! upper triangular matrix U from the LU factorization of A. ! ! DU (input/output) DOUBLE PRECISION array, dimension (N-1) ! On entry, DU must contain the (n-1) super-diagonal elements ! of A. ! ! On exit, DU is overwritten by the (n-1) elements of the first ! super-diagonal of U. ! ! DU2 (output) DOUBLE PRECISION array, dimension (N-2) ! On exit, DU2 is overwritten by the (n-2) elements of the ! second super-diagonal of U. ! ! IPIV (output) INTEGER array, dimension (N) ! The pivot indices; for 1 <= i <= n, row i of the matrix was ! interchanged with row IPIV(i). IPIV(i) will always be either ! i or i+1; IPIV(i) = i indicates a row interchange was not ! required. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -k, the k-th argument had an illegal value ! > 0: if INFO = k, U(k,k) is exactly zero. The factorization ! has been completed, but the factor U is exactly ! singular, and division by zero will occur if it is used ! to solve a system of equations. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I DOUBLE PRECISION FACT, TEMP ! .. ! .. Intrinsic Functions .. INTRINSIC ABS ! .. ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Executable Statements .. ! INFO = 0 IF( N.LT.0 ) THEN INFO = -1 CALL XERBLA( 'DGTTRF', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Initialize IPIV(i) = i and DU2(I) = 0 ! DO 10 I = 1, N IPIV( I ) = I 10 CONTINUE DO 20 I = 1, N - 2 DU2( I ) = ZERO 20 CONTINUE ! DO 30 I = 1, N - 2 IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN ! ! No row interchange required, eliminate DL(I) ! IF( D( I ).NE.ZERO ) THEN FACT = DL( I ) / D( I ) DL( I ) = FACT D( I+1 ) = D( I+1 ) - FACT*DU( I ) END IF ELSE ! ! Interchange rows I and I+1, eliminate DL(I) ! FACT = D( I ) / DL( I ) D( I ) = DL( I ) DL( I ) = FACT TEMP = DU( I ) DU( I ) = D( I+1 ) D( I+1 ) = TEMP - FACT*D( I+1 ) DU2( I ) = DU( I+1 ) DU( I+1 ) = -FACT*DU( I+1 ) IPIV( I ) = I + 1 END IF 30 CONTINUE IF( N.GT.1 ) THEN I = N - 1 IF( ABS( D( I ) ).GE.ABS( DL( I ) ) ) THEN IF( D( I ).NE.ZERO ) THEN FACT = DL( I ) / D( I ) DL( I ) = FACT D( I+1 ) = D( I+1 ) - FACT*DU( I ) END IF ELSE FACT = D( I ) / DL( I ) D( I ) = DL( I ) DL( I ) = FACT TEMP = DU( I ) DU( I ) = D( I+1 ) D( I+1 ) = TEMP - FACT*D( I+1 ) IPIV( I ) = I + 1 END IF END IF ! ! Check for a zero on the diagonal of U. ! DO 40 I = 1, N IF( D( I ).EQ.ZERO ) THEN INFO = I GO TO 50 END IF 40 CONTINUE 50 CONTINUE ! RETURN ! ! End of DGTTRF ! END SUBROUTINE DGTTRS( TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER TRANS INTEGER INFO, LDB, N, NRHS ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION B( LDB, * ), D( * ), DL( * ), DU( * ), DU2( * ) ! .. ! ! Purpose ! ======= ! ! DGTTRS solves one of the systems of equations ! A*X = B or A'*X = B, ! with a tridiagonal matrix A using the LU factorization computed ! by DGTTRF. ! ! Arguments ! ========= ! ! TRANS (input) CHARACTER ! Specifies the form of the system of equations. ! = 'N': A * X = B (No transpose) ! = 'T': A'* X = B (Transpose) ! = 'C': A'* X = B (Conjugate transpose = Transpose) ! ! N (input) INTEGER ! The order of the matrix A. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! DL (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) multipliers that define the matrix L from the ! LU factorization of A. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of the upper triangular matrix U from ! the LU factorization of A. ! ! DU (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) elements of the first super-diagonal of U. ! ! DU2 (input) DOUBLE PRECISION array, dimension (N-2) ! The (n-2) elements of the second super-diagonal of U. ! ! IPIV (input) INTEGER array, dimension (N) ! The pivot indices; for 1 <= i <= n, row i of the matrix was ! interchanged with row IPIV(i). IPIV(i) will always be either ! i or i+1; IPIV(i) = i indicates a row interchange was not ! required. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the matrix of right hand side vectors B. ! On exit, B is overwritten by the solution vectors X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL NOTRAN INTEGER ITRANS, J, JB, NB ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. External Subroutines .. EXTERNAL DGTTS2, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! INFO = 0 NOTRAN = ( TRANS.EQ.'N' .OR. TRANS.EQ.'n' ) IF( .NOT.NOTRAN .AND. .NOT.( TRANS.EQ.'T' .OR. TRANS.EQ. & 't' ) .AND. .NOT.( TRANS.EQ.'C' .OR. TRANS.EQ.'c' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDB.LT.MAX( N, 1 ) ) THEN INFO = -10 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGTTRS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) & RETURN ! ! Decode TRANS ! IF( NOTRAN ) THEN ITRANS = 0 ELSE ITRANS = 1 END IF ! ! Determine the number of right-hand sides to solve at a time. ! IF( NRHS.EQ.1 ) THEN NB = 1 ELSE NB = MAX( 1, ILAENV( 1, 'DGTTRS', TRANS, N, NRHS, -1, -1 ) ) END IF ! IF( NB.GE.NRHS ) THEN CALL DGTTS2( ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB ) ELSE DO 10 J = 1, NRHS, NB JB = MIN( NRHS-J+1, NB ) CALL DGTTS2( ITRANS, N, JB, DL, D, DU, DU2, IPIV, B( 1, J ), & LDB ) 10 CONTINUE END IF ! ! End of DGTTRS ! END SUBROUTINE DGTTS2( ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER ITRANS, LDB, N, NRHS ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION B( LDB, * ), D( * ), DL( * ), DU( * ), DU2( * ) ! .. ! ! Purpose ! ======= ! ! DGTTS2 solves one of the systems of equations ! A*X = B or A'*X = B, ! with a tridiagonal matrix A using the LU factorization computed ! by DGTTRF. ! ! Arguments ! ========= ! ! ITRANS (input) INTEGER ! Specifies the form of the system of equations. ! = 0: A * X = B (No transpose) ! = 1: A'* X = B (Transpose) ! = 2: A'* X = B (Conjugate transpose = Transpose) ! ! N (input) INTEGER ! The order of the matrix A. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! DL (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) multipliers that define the matrix L from the ! LU factorization of A. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of the upper triangular matrix U from ! the LU factorization of A. ! ! DU (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) elements of the first super-diagonal of U. ! ! DU2 (input) DOUBLE PRECISION array, dimension (N-2) ! The (n-2) elements of the second super-diagonal of U. ! ! IPIV (input) INTEGER array, dimension (N) ! The pivot indices; for 1 <= i <= n, row i of the matrix was ! interchanged with row IPIV(i). IPIV(i) will always be either ! i or i+1; IPIV(i) = i indicates a row interchange was not ! required. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the matrix of right hand side vectors B. ! On exit, B is overwritten by the solution vectors X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER I, IP, J DOUBLE PRECISION TEMP ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) & RETURN ! IF( ITRANS.EQ.0 ) THEN ! ! Solve A*X = B using the LU factorization of A, ! overwriting each right hand side vector with its solution. ! IF( NRHS.LE.1 ) THEN J = 1 10 CONTINUE ! ! Solve L*x = b. ! DO 20 I = 1, N - 1 IP = IPIV( I ) TEMP = B( I+1-IP+I, J ) - DL( I )*B( IP, J ) B( I, J ) = B( IP, J ) B( I+1, J ) = TEMP 20 CONTINUE ! ! Solve U*x = b. ! B( N, J ) = B( N, J ) / D( N ) IF( N.GT.1 ) & B( N-1, J ) = ( B( N-1, J )-DU( N-1 )*B( N, J ) ) / & D( N-1 ) DO 30 I = N - 2, 1, -1 B( I, J ) = ( B( I, J )-DU( I )*B( I+1, J )-DU2( I )* & B( I+2, J ) ) / D( I ) 30 CONTINUE IF( J.LT.NRHS ) THEN J = J + 1 GO TO 10 END IF ELSE DO 60 J = 1, NRHS ! ! Solve L*x = b. ! DO 40 I = 1, N - 1 IF( IPIV( I ).EQ.I ) THEN B( I+1, J ) = B( I+1, J ) - DL( I )*B( I, J ) ELSE TEMP = B( I, J ) B( I, J ) = B( I+1, J ) B( I+1, J ) = TEMP - DL( I )*B( I, J ) END IF 40 CONTINUE ! ! Solve U*x = b. ! B( N, J ) = B( N, J ) / D( N ) IF( N.GT.1 ) & B( N-1, J ) = ( B( N-1, J )-DU( N-1 )*B( N, J ) ) / & D( N-1 ) DO 50 I = N - 2, 1, -1 B( I, J ) = ( B( I, J )-DU( I )*B( I+1, J )-DU2( I )* & B( I+2, J ) ) / D( I ) 50 CONTINUE 60 CONTINUE END IF ELSE ! ! Solve A' * X = B. ! IF( NRHS.LE.1 ) THEN ! ! Solve U'*x = b. ! J = 1 70 CONTINUE B( 1, J ) = B( 1, J ) / D( 1 ) IF( N.GT.1 ) & B( 2, J ) = ( B( 2, J )-DU( 1 )*B( 1, J ) ) / D( 2 ) DO 80 I = 3, N B( I, J ) = ( B( I, J )-DU( I-1 )*B( I-1, J )-DU2( I-2 )* & B( I-2, J ) ) / D( I ) 80 CONTINUE ! ! Solve L'*x = b. ! DO 90 I = N - 1, 1, -1 IP = IPIV( I ) TEMP = B( I, J ) - DL( I )*B( I+1, J ) B( I, J ) = B( IP, J ) B( IP, J ) = TEMP 90 CONTINUE IF( J.LT.NRHS ) THEN J = J + 1 GO TO 70 END IF ! ELSE DO 120 J = 1, NRHS ! ! Solve U'*x = b. ! B( 1, J ) = B( 1, J ) / D( 1 ) IF( N.GT.1 ) & B( 2, J ) = ( B( 2, J )-DU( 1 )*B( 1, J ) ) / D( 2 ) DO 100 I = 3, N B( I, J ) = ( B( I, J )-DU( I-1 )*B( I-1, J )- & DU2( I-2 )*B( I-2, J ) ) / D( I ) 100 CONTINUE DO 110 I = N - 1, 1, -1 IF( IPIV( I ).EQ.I ) THEN B( I, J ) = B( I, J ) - DL( I )*B( I+1, J ) ELSE TEMP = B( I+1, J ) B( I+1, J ) = B( I, J ) - DL( I )*TEMP B( I, J ) = TEMP END IF 110 CONTINUE 120 CONTINUE END IF END IF ! ! End of DGTTS2 ! END SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, & ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, & LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER COMPQ, COMPZ, JOB INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), & B( LDB, * ), BETA( * ), Q( LDQ, * ), WORK( * ), & Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DHGEQZ implements a single-/double-shift version of the QZ method for ! finding the generalized eigenvalues ! ! w(j)=(ALPHAR(j) + i*ALPHAI(j))/BETAR(j) of the equation ! ! det( A - w(i) B ) = 0 ! ! In addition, the pair A,B may be reduced to generalized Schur form: ! B is upper triangular, and A is block upper triangular, where the ! diagonal blocks are either 1-by-1 or 2-by-2, the 2-by-2 blocks having ! complex generalized eigenvalues (see the description of the argument ! JOB.) ! ! If JOB='S', then the pair (A,B) is simultaneously reduced to Schur ! form by applying one orthogonal tranformation (usually called Q) on ! the left and another (usually called Z) on the right. The 2-by-2 ! upper-triangular diagonal blocks of B corresponding to 2-by-2 blocks ! of A will be reduced to positive diagonal matrices. (I.e., ! if A(j+1,j) is non-zero, then B(j+1,j)=B(j,j+1)=0 and B(j,j) and ! B(j+1,j+1) will be positive.) ! ! If JOB='E', then at each iteration, the same transformations ! are computed, but they are only applied to those parts of A and B ! which are needed to compute ALPHAR, ALPHAI, and BETAR. ! ! If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the orthogonal ! transformations used to reduce (A,B) are accumulated into the arrays ! Q and Z s.t.: ! ! Q(in) A(in) Z(in)* = Q(out) A(out) Z(out)* ! Q(in) B(in) Z(in)* = Q(out) B(out) Z(out)* ! ! Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix ! Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973), ! pp. 241--256. ! ! Arguments ! ========= ! ! JOB (input) CHARACTER*1 ! = 'E': compute only ALPHAR, ALPHAI, and BETA. A and B will ! not necessarily be put into generalized Schur form. ! = 'S': put A and B into generalized Schur form, as well ! as computing ALPHAR, ALPHAI, and BETA. ! ! COMPQ (input) CHARACTER*1 ! = 'N': do not modify Q. ! = 'V': multiply the array Q on the right by the transpose of ! the orthogonal tranformation that is applied to the ! left side of A and B to reduce them to Schur form. ! = 'I': like COMPQ='V', except that Q will be initialized to ! the identity first. ! ! COMPZ (input) CHARACTER*1 ! = 'N': do not modify Z. ! = 'V': multiply the array Z on the right by the orthogonal ! tranformation that is applied to the right side of ! A and B to reduce them to Schur form. ! = 'I': like COMPZ='V', except that Z will be initialized to ! the identity first. ! ! N (input) INTEGER ! The order of the matrices A, B, Q, and Z. N >= 0. ! ! ILO (input) INTEGER ! IHI (input) INTEGER ! It is assumed that A is already upper triangular in rows and ! columns 1:ILO-1 and IHI+1:N. ! 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the N-by-N upper Hessenberg matrix A. Elements ! below the subdiagonal must be zero. ! If JOB='S', then on exit A and B will have been ! simultaneously reduced to generalized Schur form. ! If JOB='E', then on exit A will have been destroyed. ! The diagonal blocks will be correct, but the off-diagonal ! portion will be meaningless. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max( 1, N ). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB, N) ! On entry, the N-by-N upper triangular matrix B. Elements ! below the diagonal must be zero. 2-by-2 blocks in B ! corresponding to 2-by-2 blocks in A will be reduced to ! positive diagonal form. (I.e., if A(j+1,j) is non-zero, ! then B(j+1,j)=B(j,j+1)=0 and B(j,j) and B(j+1,j+1) will be ! positive.) ! If JOB='S', then on exit A and B will have been ! simultaneously reduced to Schur form. ! If JOB='E', then on exit B will have been destroyed. ! Elements corresponding to diagonal blocks of A will be ! correct, but the off-diagonal portion will be meaningless. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max( 1, N ). ! ! ALPHAR (output) DOUBLE PRECISION array, dimension (N) ! ALPHAR(1:N) will be set to real parts of the diagonal ! elements of A that would result from reducing A and B to ! Schur form and then further reducing them both to triangular ! form using unitary transformations s.t. the diagonal of B ! was non-negative real. Thus, if A(j,j) is in a 1-by-1 block ! (i.e., A(j+1,j)=A(j,j+1)=0), then ALPHAR(j)=A(j,j). ! Note that the (real or complex) values ! (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the ! generalized eigenvalues of the matrix pencil A - wB. ! ! ALPHAI (output) DOUBLE PRECISION array, dimension (N) ! ALPHAI(1:N) will be set to imaginary parts of the diagonal ! elements of A that would result from reducing A and B to ! Schur form and then further reducing them both to triangular ! form using unitary transformations s.t. the diagonal of B ! was non-negative real. Thus, if A(j,j) is in a 1-by-1 block ! (i.e., A(j+1,j)=A(j,j+1)=0), then ALPHAR(j)=0. ! Note that the (real or complex) values ! (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the ! generalized eigenvalues of the matrix pencil A - wB. ! ! BETA (output) DOUBLE PRECISION array, dimension (N) ! BETA(1:N) will be set to the (real) diagonal elements of B ! that would result from reducing A and B to Schur form and ! then further reducing them both to triangular form using ! unitary transformations s.t. the diagonal of B was ! non-negative real. Thus, if A(j,j) is in a 1-by-1 block ! (i.e., A(j+1,j)=A(j,j+1)=0), then BETA(j)=B(j,j). ! Note that the (real or complex) values ! (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the ! generalized eigenvalues of the matrix pencil A - wB. ! (Note that BETA(1:N) will always be non-negative, and no ! BETAI is necessary.) ! ! Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N) ! If COMPQ='N', then Q will not be referenced. ! If COMPQ='V' or 'I', then the transpose of the orthogonal ! transformations which are applied to A and B on the left ! will be applied to the array Q on the right. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. LDQ >= 1. ! If COMPQ='V' or 'I', then LDQ >= N. ! ! Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N) ! If COMPZ='N', then Z will not be referenced. ! If COMPZ='V' or 'I', then the orthogonal transformations ! which are applied to A and B on the right will be applied ! to the array Z on the right. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1. ! If COMPZ='V' or 'I', then LDZ >= N. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO >= 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,N). ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! = 1,...,N: the QZ iteration did not converge. (A,B) is not ! in Schur form, but ALPHAR(i), ALPHAI(i), and ! BETA(i), i=INFO+1,...,N should be correct. ! = N+1,...,2*N: the shift calculation failed. (A,B) is not ! in Schur form, but ALPHAR(i), ALPHAI(i), and ! BETA(i), i=INFO-N+1,...,N should be correct. ! > 2*N: various "impossible" errors. ! ! Further Details ! =============== ! ! Iteration counters: ! ! JITER -- counts iterations. ! IITER -- counts iterations run since ILAST was last ! changed. This is therefore reset only when a 1-by-1 or ! 2-by-2 block deflates off the bottom. ! ! ===================================================================== ! ! .. Parameters .. ! $ SAFETY = 1.0E+0 ) DOUBLE PRECISION HALF, ZERO, ONE, SAFETY PARAMETER ( HALF = 0.5D+0, ZERO = 0.0D+0, ONE = 1.0D+0, & SAFETY = 1.0D+2 ) ! .. ! .. Local Scalars .. LOGICAL ILAZR2, ILAZRO, ILPIVT, ILQ, ILSCHR, ILZ, & LQUERY INTEGER ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST, & ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER, & JR, MAXIT DOUBLE PRECISION A11, A12, A1I, A1R, A21, A22, A2I, A2R, AD11, & AD11L, AD12, AD12L, AD21, AD21L, AD22, AD22L, & AD32L, AN, ANORM, ASCALE, ATOL, B11, B1A, B1I, & B1R, B22, B2A, B2I, B2R, BN, BNORM, BSCALE, & BTOL, C, C11I, C11R, C12, C21, C22I, C22R, CL, & CQ, CR, CZ, ESHIFT, S, S1, S1INV, S2, SAFMAX, & SAFMIN, SCALE, SL, SQI, SQR, SR, SZI, SZR, T, & TAU, TEMP, TEMP2, TEMPI, TEMPR, U1, U12, U12L, & U2, ULP, VS, W11, W12, W21, W22, WABS, WI, WR, & WR2 ! .. ! .. Local Arrays .. DOUBLE PRECISION V( 3 ) ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANHS, DLAPY2, DLAPY3 EXTERNAL LSAME, DLAMCH, DLANHS, DLAPY2, DLAPY3 ! .. ! .. External Subroutines .. EXTERNAL DLAG2, DLARFG, DLARTG, DLASET, DLASV2, DROT, & XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Decode JOB, COMPQ, COMPZ ! IF( LSAME( JOB, 'E' ) ) THEN ILSCHR = .FALSE. ISCHUR = 1 ELSE IF( LSAME( JOB, 'S' ) ) THEN ILSCHR = .TRUE. ISCHUR = 2 ELSE ISCHUR = 0 END IF ! IF( LSAME( COMPQ, 'N' ) ) THEN ILQ = .FALSE. ICOMPQ = 1 ELSE IF( LSAME( COMPQ, 'V' ) ) THEN ILQ = .TRUE. ICOMPQ = 2 ELSE IF( LSAME( COMPQ, 'I' ) ) THEN ILQ = .TRUE. ICOMPQ = 3 ELSE ICOMPQ = 0 END IF ! IF( LSAME( COMPZ, 'N' ) ) THEN ILZ = .FALSE. ICOMPZ = 1 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN ILZ = .TRUE. ICOMPZ = 2 ELSE IF( LSAME( COMPZ, 'I' ) ) THEN ILZ = .TRUE. ICOMPZ = 3 ELSE ICOMPZ = 0 END IF ! ! Check Argument Values ! INFO = 0 WORK( 1 ) = MAX( 1, N ) LQUERY = ( LWORK.EQ.-1 ) IF( ISCHUR.EQ.0 ) THEN INFO = -1 ELSE IF( ICOMPQ.EQ.0 ) THEN INFO = -2 ELSE IF( ICOMPZ.EQ.0 ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( ILO.LT.1 ) THEN INFO = -5 ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN INFO = -6 ELSE IF( LDA.LT.N ) THEN INFO = -8 ELSE IF( LDB.LT.N ) THEN INFO = -10 ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN INFO = -15 ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN INFO = -17 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN INFO = -19 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DHGEQZ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.LE.0 ) THEN WORK( 1 ) = DBLE( 1 ) RETURN END IF ! ! Initialize Q and Z ! IF( ICOMPQ.EQ.3 ) & CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ ) IF( ICOMPZ.EQ.3 ) & CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ ) ! ! Machine Constants ! IN = IHI + 1 - ILO SAFMIN = DLAMCH( 'S' ) SAFMAX = ONE / SAFMIN ULP = DLAMCH( 'E' )*DLAMCH( 'B' ) ANORM = DLANHS( 'F', IN, A( ILO, ILO ), LDA, WORK ) BNORM = DLANHS( 'F', IN, B( ILO, ILO ), LDB, WORK ) ATOL = MAX( SAFMIN, ULP*ANORM ) BTOL = MAX( SAFMIN, ULP*BNORM ) ASCALE = ONE / MAX( SAFMIN, ANORM ) BSCALE = ONE / MAX( SAFMIN, BNORM ) ! ! Set Eigenvalues IHI+1:N ! DO 30 J = IHI + 1, N IF( B( J, J ).LT.ZERO ) THEN IF( ILSCHR ) THEN DO 10 JR = 1, J A( JR, J ) = -A( JR, J ) B( JR, J ) = -B( JR, J ) 10 CONTINUE ELSE A( J, J ) = -A( J, J ) B( J, J ) = -B( J, J ) END IF IF( ILZ ) THEN DO 20 JR = 1, N Z( JR, J ) = -Z( JR, J ) 20 CONTINUE END IF END IF ALPHAR( J ) = A( J, J ) ALPHAI( J ) = ZERO BETA( J ) = B( J, J ) 30 CONTINUE ! ! If IHI < ILO, skip QZ steps ! IF( IHI.LT.ILO ) & GO TO 380 ! ! MAIN QZ ITERATION LOOP ! ! Initialize dynamic indices ! ! Eigenvalues ILAST+1:N have been found. ! Column operations modify rows IFRSTM:whatever. ! Row operations modify columns whatever:ILASTM. ! ! If only eigenvalues are being computed, then ! IFRSTM is the row of the last splitting row above row ILAST; ! this is always at least ILO. ! IITER counts iterations since the last eigenvalue was found, ! to tell when to use an extraordinary shift. ! MAXIT is the maximum number of QZ sweeps allowed. ! ILAST = IHI IF( ILSCHR ) THEN IFRSTM = 1 ILASTM = N ELSE IFRSTM = ILO ILASTM = IHI END IF IITER = 0 ESHIFT = ZERO MAXIT = 30*( IHI-ILO+1 ) ! DO 360 JITER = 1, MAXIT ! ! Split the matrix if possible. ! ! Two tests: ! 1: A(j,j-1)=0 or j=ILO ! 2: B(j,j)=0 ! IF( ILAST.EQ.ILO ) THEN ! ! Special case: j=ILAST ! GO TO 80 ELSE IF( ABS( A( ILAST, ILAST-1 ) ).LE.ATOL ) THEN A( ILAST, ILAST-1 ) = ZERO GO TO 80 END IF END IF ! IF( ABS( B( ILAST, ILAST ) ).LE.BTOL ) THEN B( ILAST, ILAST ) = ZERO GO TO 70 END IF ! ! General case: j unfl ) ! __ ! (sA - wB) ( CZ -SZ ) ! ( SZ CZ ) ! C11R = S1*A11 - WR*B11 C11I = -WI*B11 C12 = S1*A12 C21 = S1*A21 C22R = S1*A22 - WR*B22 C22I = -WI*B22 ! IF( ABS( C11R )+ABS( C11I )+ABS( C12 ).GT.ABS( C21 )+ & ABS( C22R )+ABS( C22I ) ) THEN T = DLAPY3( C12, C11R, C11I ) CZ = C12 / T SZR = -C11R / T SZI = -C11I / T ELSE CZ = DLAPY2( C22R, C22I ) IF( CZ.LE.SAFMIN ) THEN CZ = ZERO SZR = ONE SZI = ZERO ELSE TEMPR = C22R / CZ TEMPI = C22I / CZ T = DLAPY2( CZ, C21 ) CZ = CZ / T SZR = -C21*TEMPR / T SZI = C21*TEMPI / T END IF END IF ! ! Compute Givens rotation on left ! ! ( CQ SQ ) ! ( __ ) A or B ! ( -SQ CQ ) ! AN = ABS( A11 ) + ABS( A12 ) + ABS( A21 ) + ABS( A22 ) BN = ABS( B11 ) + ABS( B22 ) WABS = ABS( WR ) + ABS( WI ) IF( S1*AN.GT.WABS*BN ) THEN CQ = CZ*B11 SQR = SZR*B22 SQI = -SZI*B22 ELSE A1R = CZ*A11 + SZR*A12 A1I = SZI*A12 A2R = CZ*A21 + SZR*A22 A2I = SZI*A22 CQ = DLAPY2( A1R, A1I ) IF( CQ.LE.SAFMIN ) THEN CQ = ZERO SQR = ONE SQI = ZERO ELSE TEMPR = A1R / CQ TEMPI = A1I / CQ SQR = TEMPR*A2R + TEMPI*A2I SQI = TEMPI*A2R - TEMPR*A2I END IF END IF T = DLAPY3( CQ, SQR, SQI ) CQ = CQ / T SQR = SQR / T SQI = SQI / T ! ! Compute diagonal elements of QBZ ! TEMPR = SQR*SZR - SQI*SZI TEMPI = SQR*SZI + SQI*SZR B1R = CQ*CZ*B11 + TEMPR*B22 B1I = TEMPI*B22 B1A = DLAPY2( B1R, B1I ) B2R = CQ*CZ*B22 + TEMPR*B11 B2I = -TEMPI*B11 B2A = DLAPY2( B2R, B2I ) ! ! Normalize so beta > 0, and Im( alpha1 ) > 0 ! BETA( ILAST-1 ) = B1A BETA( ILAST ) = B2A ALPHAR( ILAST-1 ) = ( WR*B1A )*S1INV ALPHAI( ILAST-1 ) = ( WI*B1A )*S1INV ALPHAR( ILAST ) = ( WR*B2A )*S1INV ALPHAI( ILAST ) = -( WI*B2A )*S1INV ! ! Step 3: Go to next block -- exit if finished. ! ILAST = IFIRST - 1 IF( ILAST.LT.ILO ) & GO TO 380 ! ! Reset counters ! IITER = 0 ESHIFT = ZERO IF( .NOT.ILSCHR ) THEN ILASTM = ILAST IF( IFRSTM.GT.ILAST ) & IFRSTM = ILO END IF GO TO 350 ELSE ! ! Usual case: 3x3 or larger block, using Francis implicit ! double-shift ! ! 2 ! Eigenvalue equation is w - c w + d = 0, ! ! -1 2 -1 ! so compute 1st column of (A B ) - c A B + d ! using the formula in QZIT (from EISPACK) ! ! We assume that the block is at least 3x3 ! AD11 = ( ASCALE*A( ILAST-1, ILAST-1 ) ) / & ( BSCALE*B( ILAST-1, ILAST-1 ) ) AD21 = ( ASCALE*A( ILAST, ILAST-1 ) ) / & ( BSCALE*B( ILAST-1, ILAST-1 ) ) AD12 = ( ASCALE*A( ILAST-1, ILAST ) ) / & ( BSCALE*B( ILAST, ILAST ) ) AD22 = ( ASCALE*A( ILAST, ILAST ) ) / & ( BSCALE*B( ILAST, ILAST ) ) U12 = B( ILAST-1, ILAST ) / B( ILAST, ILAST ) AD11L = ( ASCALE*A( IFIRST, IFIRST ) ) / & ( BSCALE*B( IFIRST, IFIRST ) ) AD21L = ( ASCALE*A( IFIRST+1, IFIRST ) ) / & ( BSCALE*B( IFIRST, IFIRST ) ) AD12L = ( ASCALE*A( IFIRST, IFIRST+1 ) ) / & ( BSCALE*B( IFIRST+1, IFIRST+1 ) ) AD22L = ( ASCALE*A( IFIRST+1, IFIRST+1 ) ) / & ( BSCALE*B( IFIRST+1, IFIRST+1 ) ) AD32L = ( ASCALE*A( IFIRST+2, IFIRST+1 ) ) / & ( BSCALE*B( IFIRST+1, IFIRST+1 ) ) U12L = B( IFIRST, IFIRST+1 ) / B( IFIRST+1, IFIRST+1 ) ! V( 1 ) = ( AD11-AD11L )*( AD22-AD11L ) - AD12*AD21 + & AD21*U12*AD11L + ( AD12L-AD11L*U12L )*AD21L V( 2 ) = ( ( AD22L-AD11L )-AD21L*U12L-( AD11-AD11L )- & ( AD22-AD11L )+AD21*U12 )*AD21L V( 3 ) = AD32L*AD21L ! ISTART = IFIRST ! CALL DLARFG( 3, V( 1 ), V( 2 ), 1, TAU ) V( 1 ) = ONE ! ! Sweep ! DO 290 J = ISTART, ILAST - 2 ! ! All but last elements: use 3x3 Householder transforms. ! ! Zero (j-1)st column of A ! IF( J.GT.ISTART ) THEN V( 1 ) = A( J, J-1 ) V( 2 ) = A( J+1, J-1 ) V( 3 ) = A( J+2, J-1 ) ! CALL DLARFG( 3, A( J, J-1 ), V( 2 ), 1, TAU ) V( 1 ) = ONE A( J+1, J-1 ) = ZERO A( J+2, J-1 ) = ZERO END IF ! DO 230 JC = J, ILASTM TEMP = TAU*( A( J, JC )+V( 2 )*A( J+1, JC )+V( 3 )* & A( J+2, JC ) ) A( J, JC ) = A( J, JC ) - TEMP A( J+1, JC ) = A( J+1, JC ) - TEMP*V( 2 ) A( J+2, JC ) = A( J+2, JC ) - TEMP*V( 3 ) TEMP2 = TAU*( B( J, JC )+V( 2 )*B( J+1, JC )+V( 3 )* & B( J+2, JC ) ) B( J, JC ) = B( J, JC ) - TEMP2 B( J+1, JC ) = B( J+1, JC ) - TEMP2*V( 2 ) B( J+2, JC ) = B( J+2, JC ) - TEMP2*V( 3 ) 230 CONTINUE IF( ILQ ) THEN DO 240 JR = 1, N TEMP = TAU*( Q( JR, J )+V( 2 )*Q( JR, J+1 )+V( 3 )* & Q( JR, J+2 ) ) Q( JR, J ) = Q( JR, J ) - TEMP Q( JR, J+1 ) = Q( JR, J+1 ) - TEMP*V( 2 ) Q( JR, J+2 ) = Q( JR, J+2 ) - TEMP*V( 3 ) 240 CONTINUE END IF ! ! Zero j-th column of B (see DLAGBC for details) ! ! Swap rows to pivot ! ILPIVT = .FALSE. TEMP = MAX( ABS( B( J+1, J+1 ) ), ABS( B( J+1, J+2 ) ) ) TEMP2 = MAX( ABS( B( J+2, J+1 ) ), ABS( B( J+2, J+2 ) ) ) IF( MAX( TEMP, TEMP2 ).LT.SAFMIN ) THEN SCALE = ZERO U1 = ONE U2 = ZERO GO TO 250 ELSE IF( TEMP.GE.TEMP2 ) THEN W11 = B( J+1, J+1 ) W21 = B( J+2, J+1 ) W12 = B( J+1, J+2 ) W22 = B( J+2, J+2 ) U1 = B( J+1, J ) U2 = B( J+2, J ) ELSE W21 = B( J+1, J+1 ) W11 = B( J+2, J+1 ) W22 = B( J+1, J+2 ) W12 = B( J+2, J+2 ) U2 = B( J+1, J ) U1 = B( J+2, J ) END IF ! ! Swap columns if nec. ! IF( ABS( W12 ).GT.ABS( W11 ) ) THEN ILPIVT = .TRUE. TEMP = W12 TEMP2 = W22 W12 = W11 W22 = W21 W11 = TEMP W21 = TEMP2 END IF ! ! LU-factor ! TEMP = W21 / W11 U2 = U2 - TEMP*U1 W22 = W22 - TEMP*W12 W21 = ZERO ! ! Compute SCALE ! SCALE = ONE IF( ABS( W22 ).LT.SAFMIN ) THEN SCALE = ZERO U2 = ONE U1 = -W12 / W11 GO TO 250 END IF IF( ABS( W22 ).LT.ABS( U2 ) ) & SCALE = ABS( W22 / U2 ) IF( ABS( W11 ).LT.ABS( U1 ) ) & SCALE = MIN( SCALE, ABS( W11 / U1 ) ) ! ! Solve ! U2 = ( SCALE*U2 ) / W22 U1 = ( SCALE*U1-W12*U2 ) / W11 ! 250 CONTINUE IF( ILPIVT ) THEN TEMP = U2 U2 = U1 U1 = TEMP END IF ! ! Compute Householder Vector ! T = SQRT( SCALE**2+U1**2+U2**2 ) TAU = ONE + SCALE / T VS = -ONE / ( SCALE+T ) V( 1 ) = ONE V( 2 ) = VS*U1 V( 3 ) = VS*U2 ! ! Apply transformations from the right. ! DO 260 JR = IFRSTM, MIN( J+3, ILAST ) TEMP = TAU*( A( JR, J )+V( 2 )*A( JR, J+1 )+V( 3 )* & A( JR, J+2 ) ) A( JR, J ) = A( JR, J ) - TEMP A( JR, J+1 ) = A( JR, J+1 ) - TEMP*V( 2 ) A( JR, J+2 ) = A( JR, J+2 ) - TEMP*V( 3 ) 260 CONTINUE DO 270 JR = IFRSTM, J + 2 TEMP = TAU*( B( JR, J )+V( 2 )*B( JR, J+1 )+V( 3 )* & B( JR, J+2 ) ) B( JR, J ) = B( JR, J ) - TEMP B( JR, J+1 ) = B( JR, J+1 ) - TEMP*V( 2 ) B( JR, J+2 ) = B( JR, J+2 ) - TEMP*V( 3 ) 270 CONTINUE IF( ILZ ) THEN DO 280 JR = 1, N TEMP = TAU*( Z( JR, J )+V( 2 )*Z( JR, J+1 )+V( 3 )* & Z( JR, J+2 ) ) Z( JR, J ) = Z( JR, J ) - TEMP Z( JR, J+1 ) = Z( JR, J+1 ) - TEMP*V( 2 ) Z( JR, J+2 ) = Z( JR, J+2 ) - TEMP*V( 3 ) 280 CONTINUE END IF B( J+1, J ) = ZERO B( J+2, J ) = ZERO 290 CONTINUE ! ! Last elements: Use Givens rotations ! ! Rotations from the left ! J = ILAST - 1 TEMP = A( J, J-1 ) CALL DLARTG( TEMP, A( J+1, J-1 ), C, S, A( J, J-1 ) ) A( J+1, J-1 ) = ZERO ! DO 300 JC = J, ILASTM TEMP = C*A( J, JC ) + S*A( J+1, JC ) A( J+1, JC ) = -S*A( J, JC ) + C*A( J+1, JC ) A( J, JC ) = TEMP TEMP2 = C*B( J, JC ) + S*B( J+1, JC ) B( J+1, JC ) = -S*B( J, JC ) + C*B( J+1, JC ) B( J, JC ) = TEMP2 300 CONTINUE IF( ILQ ) THEN DO 310 JR = 1, N TEMP = C*Q( JR, J ) + S*Q( JR, J+1 ) Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 ) Q( JR, J ) = TEMP 310 CONTINUE END IF ! ! Rotations from the right. ! TEMP = B( J+1, J+1 ) CALL DLARTG( TEMP, B( J+1, J ), C, S, B( J+1, J+1 ) ) B( J+1, J ) = ZERO ! DO 320 JR = IFRSTM, ILAST TEMP = C*A( JR, J+1 ) + S*A( JR, J ) A( JR, J ) = -S*A( JR, J+1 ) + C*A( JR, J ) A( JR, J+1 ) = TEMP 320 CONTINUE DO 330 JR = IFRSTM, ILAST - 1 TEMP = C*B( JR, J+1 ) + S*B( JR, J ) B( JR, J ) = -S*B( JR, J+1 ) + C*B( JR, J ) B( JR, J+1 ) = TEMP 330 CONTINUE IF( ILZ ) THEN DO 340 JR = 1, N TEMP = C*Z( JR, J+1 ) + S*Z( JR, J ) Z( JR, J ) = -S*Z( JR, J+1 ) + C*Z( JR, J ) Z( JR, J+1 ) = TEMP 340 CONTINUE END IF ! ! End of Double-Shift code ! END IF ! GO TO 350 ! ! End of iteration loop ! 350 CONTINUE 360 CONTINUE ! ! Drop-through = non-convergence ! 370 CONTINUE INFO = ILAST GO TO 420 ! ! Successful completion of all QZ steps ! 380 CONTINUE ! ! Set Eigenvalues 1:ILO-1 ! DO 410 J = 1, ILO - 1 IF( B( J, J ).LT.ZERO ) THEN IF( ILSCHR ) THEN DO 390 JR = 1, J A( JR, J ) = -A( JR, J ) B( JR, J ) = -B( JR, J ) 390 CONTINUE ELSE A( J, J ) = -A( J, J ) B( J, J ) = -B( J, J ) END IF IF( ILZ ) THEN DO 400 JR = 1, N Z( JR, J ) = -Z( JR, J ) 400 CONTINUE END IF END IF ALPHAR( J ) = A( J, J ) ALPHAI( J ) = ZERO BETA( J ) = B( J, J ) 410 CONTINUE ! ! Normal Termination ! INFO = 0 ! ! Exit (other than argument error) -- return optimal workspace size ! 420 CONTINUE WORK( 1 ) = DBLE( N ) RETURN ! ! End of DHGEQZ ! END SUBROUTINE DHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR, WI, & VL, LDVL, VR, LDVR, MM, M, WORK, IFAILL, & IFAILR, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER EIGSRC, INITV, SIDE INTEGER INFO, LDH, LDVL, LDVR, M, MM, N ! .. ! .. Array Arguments .. LOGICAL SELECT( * ) INTEGER IFAILL( * ), IFAILR( * ) DOUBLE PRECISION H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ), & WI( * ), WORK( * ), WR( * ) ! .. ! ! Purpose ! ======= ! ! DHSEIN uses inverse iteration to find specified right and/or left ! eigenvectors of a real upper Hessenberg matrix H. ! ! The right eigenvector x and the left eigenvector y of the matrix H ! corresponding to an eigenvalue w are defined by: ! ! H * x = w * x, y**h * H = w * y**h ! ! where y**h denotes the conjugate transpose of the vector y. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'R': compute right eigenvectors only; ! = 'L': compute left eigenvectors only; ! = 'B': compute both right and left eigenvectors. ! ! EIGSRC (input) CHARACTER*1 ! Specifies the source of eigenvalues supplied in (WR,WI): ! = 'Q': the eigenvalues were found using DHSEQR; thus, if ! H has zero subdiagonal elements, and so is ! block-triangular, then the j-th eigenvalue can be ! assumed to be an eigenvalue of the block containing ! the j-th row/column. This property allows DHSEIN to ! perform inverse iteration on just one diagonal block. ! = 'N': no assumptions are made on the correspondence ! between eigenvalues and diagonal blocks. In this ! case, DHSEIN must always perform inverse iteration ! using the whole matrix H. ! ! INITV (input) CHARACTER*1 ! = 'N': no initial vectors are supplied; ! = 'U': user-supplied initial vectors are stored in the arrays ! VL and/or VR. ! ! SELECT (input/output) LOGICAL array, dimension (N) ! Specifies the eigenvectors to be computed. To select the ! real eigenvector corresponding to a real eigenvalue WR(j), ! SELECT(j) must be set to .TRUE.. To select the complex ! eigenvector corresponding to a complex eigenvalue ! (WR(j),WI(j)), with complex conjugate (WR(j+1),WI(j+1)), ! either SELECT(j) or SELECT(j+1) or both must be set to ! .TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is ! .FALSE.. ! ! N (input) INTEGER ! The order of the matrix H. N >= 0. ! ! H (input) DOUBLE PRECISION array, dimension (LDH,N) ! The upper Hessenberg matrix H. ! ! LDH (input) INTEGER ! The leading dimension of the array H. LDH >= max(1,N). ! ! WR (input/output) DOUBLE PRECISION array, dimension (N) ! WI (input) DOUBLE PRECISION array, dimension (N) ! On entry, the real and imaginary parts of the eigenvalues of ! H; a complex conjugate pair of eigenvalues must be stored in ! consecutive elements of WR and WI. ! On exit, WR may have been altered since close eigenvalues ! are perturbed slightly in searching for independent ! eigenvectors. ! ! VL (input/output) DOUBLE PRECISION array, dimension (LDVL,MM) ! On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must ! contain starting vectors for the inverse iteration for the ! left eigenvectors; the starting vector for each eigenvector ! must be in the same column(s) in which the eigenvector will ! be stored. ! On exit, if SIDE = 'L' or 'B', the left eigenvectors ! specified by SELECT will be stored consecutively in the ! columns of VL, in the same order as their eigenvalues. A ! complex eigenvector corresponding to a complex eigenvalue is ! stored in two consecutive columns, the first holding the real ! part and the second the imaginary part. ! If SIDE = 'R', VL is not referenced. ! ! LDVL (input) INTEGER ! The leading dimension of the array VL. ! LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise. ! ! VR (input/output) DOUBLE PRECISION array, dimension (LDVR,MM) ! On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must ! contain starting vectors for the inverse iteration for the ! right eigenvectors; the starting vector for each eigenvector ! must be in the same column(s) in which the eigenvector will ! be stored. ! On exit, if SIDE = 'R' or 'B', the right eigenvectors ! specified by SELECT will be stored consecutively in the ! columns of VR, in the same order as their eigenvalues. A ! complex eigenvector corresponding to a complex eigenvalue is ! stored in two consecutive columns, the first holding the real ! part and the second the imaginary part. ! If SIDE = 'L', VR is not referenced. ! ! LDVR (input) INTEGER ! The leading dimension of the array VR. ! LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise. ! ! MM (input) INTEGER ! The number of columns in the arrays VL and/or VR. MM >= M. ! ! M (output) INTEGER ! The number of columns in the arrays VL and/or VR required to ! store the eigenvectors; each selected real eigenvector ! occupies one column and each selected complex eigenvector ! occupies two columns. ! ! WORK (workspace) DOUBLE PRECISION array, dimension ((N+2)*N) ! ! IFAILL (output) INTEGER array, dimension (MM) ! If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left ! eigenvector in the i-th column of VL (corresponding to the ! eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the ! eigenvector converged satisfactorily. If the i-th and (i+1)th ! columns of VL hold a complex eigenvector, then IFAILL(i) and ! IFAILL(i+1) are set to the same value. ! If SIDE = 'R', IFAILL is not referenced. ! ! IFAILR (output) INTEGER array, dimension (MM) ! If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right ! eigenvector in the i-th column of VR (corresponding to the ! eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the ! eigenvector converged satisfactorily. If the i-th and (i+1)th ! columns of VR hold a complex eigenvector, then IFAILR(i) and ! IFAILR(i+1) are set to the same value. ! If SIDE = 'L', IFAILR is not referenced. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, i is the number of eigenvectors which ! failed to converge; see IFAILL and IFAILR for further ! details. ! ! Further Details ! =============== ! ! Each eigenvector is normalized so that the element of largest ! magnitude has magnitude 1; here the magnitude of a complex number ! (x,y) is taken to be |x|+|y|. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL BOTHV, FROMQR, LEFTV, NOINIT, PAIR, RIGHTV INTEGER I, IINFO, K, KL, KLN, KR, KSI, KSR, LDWORK DOUBLE PRECISION BIGNUM, EPS3, HNORM, SMLNUM, ULP, UNFL, WKI, & WKR ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANHS EXTERNAL LSAME, DLAMCH, DLANHS ! .. ! .. External Subroutines .. EXTERNAL DLAEIN, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. Executable Statements .. ! ! Decode and test the input parameters. ! BOTHV = LSAME( SIDE, 'B' ) RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV ! FROMQR = LSAME( EIGSRC, 'Q' ) ! NOINIT = LSAME( INITV, 'N' ) ! ! Set M to the number of columns required to store the selected ! eigenvectors, and standardize the array SELECT. ! M = 0 PAIR = .FALSE. DO 10 K = 1, N IF( PAIR ) THEN PAIR = .FALSE. SELECT( K ) = .FALSE. ELSE IF( WI( K ).EQ.ZERO ) THEN IF( SELECT( K ) ) & M = M + 1 ELSE PAIR = .TRUE. IF( SELECT( K ) .OR. SELECT( K+1 ) ) THEN SELECT( K ) = .TRUE. M = M + 2 END IF END IF END IF 10 CONTINUE ! INFO = 0 IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN INFO = -1 ELSE IF( .NOT.FROMQR .AND. .NOT.LSAME( EIGSRC, 'N' ) ) THEN INFO = -2 ELSE IF( .NOT.NOINIT .AND. .NOT.LSAME( INITV, 'U' ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDH.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN INFO = -11 ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN INFO = -13 ELSE IF( MM.LT.M ) THEN INFO = -14 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DHSEIN', -INFO ) RETURN END IF ! ! Quick return if possible. ! IF( N.EQ.0 ) & RETURN ! ! Set machine-dependent constants. ! UNFL = DLAMCH( 'Safe minimum' ) ULP = DLAMCH( 'Precision' ) SMLNUM = UNFL*( N / ULP ) BIGNUM = ( ONE-ULP ) / SMLNUM ! LDWORK = N + 1 ! KL = 1 KLN = 0 IF( FROMQR ) THEN KR = 0 ELSE KR = N END IF KSR = 1 ! DO 120 K = 1, N IF( SELECT( K ) ) THEN ! ! Compute eigenvector(s) corresponding to W(K). ! IF( FROMQR ) THEN ! ! If affiliation of eigenvalues is known, check whether ! the matrix splits. ! ! Determine KL and KR such that 1 <= KL <= K <= KR <= N ! and H(KL,KL-1) and H(KR+1,KR) are zero (or KL = 1 or ! KR = N). ! ! Then inverse iteration can be performed with the ! submatrix H(KL:N,KL:N) for a left eigenvector, and with ! the submatrix H(1:KR,1:KR) for a right eigenvector. ! DO 20 I = K, KL + 1, -1 IF( H( I, I-1 ).EQ.ZERO ) & GO TO 30 20 CONTINUE 30 CONTINUE KL = I IF( K.GT.KR ) THEN DO 40 I = K, N - 1 IF( H( I+1, I ).EQ.ZERO ) & GO TO 50 40 CONTINUE 50 CONTINUE KR = I END IF END IF ! IF( KL.NE.KLN ) THEN KLN = KL ! ! Compute infinity-norm of submatrix H(KL:KR,KL:KR) if it ! has not ben computed before. ! HNORM = DLANHS( 'I', KR-KL+1, H( KL, KL ), LDH, WORK ) IF( HNORM.GT.ZERO ) THEN EPS3 = HNORM*ULP ELSE EPS3 = SMLNUM END IF END IF ! ! Perturb eigenvalue if it is close to any previous ! selected eigenvalues affiliated to the submatrix ! H(KL:KR,KL:KR). Close roots are modified by EPS3. ! WKR = WR( K ) WKI = WI( K ) 60 CONTINUE DO 70 I = K - 1, KL, -1 IF( SELECT( I ) .AND. ABS( WR( I )-WKR )+ & ABS( WI( I )-WKI ).LT.EPS3 ) THEN WKR = WKR + EPS3 GO TO 60 END IF 70 CONTINUE WR( K ) = WKR ! PAIR = WKI.NE.ZERO IF( PAIR ) THEN KSI = KSR + 1 ELSE KSI = KSR END IF IF( LEFTV ) THEN ! ! Compute left eigenvector. ! CALL DLAEIN( .FALSE., NOINIT, N-KL+1, H( KL, KL ), LDH, & WKR, WKI, VL( KL, KSR ), VL( KL, KSI ), & WORK, LDWORK, WORK( N*N+N+1 ), EPS3, SMLNUM, & BIGNUM, IINFO ) IF( IINFO.GT.0 ) THEN IF( PAIR ) THEN INFO = INFO + 2 ELSE INFO = INFO + 1 END IF IFAILL( KSR ) = K IFAILL( KSI ) = K ELSE IFAILL( KSR ) = 0 IFAILL( KSI ) = 0 END IF DO 80 I = 1, KL - 1 VL( I, KSR ) = ZERO 80 CONTINUE IF( PAIR ) THEN DO 90 I = 1, KL - 1 VL( I, KSI ) = ZERO 90 CONTINUE END IF END IF IF( RIGHTV ) THEN ! ! Compute right eigenvector. ! CALL DLAEIN( .TRUE., NOINIT, KR, H, LDH, WKR, WKI, & VR( 1, KSR ), VR( 1, KSI ), WORK, LDWORK, & WORK( N*N+N+1 ), EPS3, SMLNUM, BIGNUM, & IINFO ) IF( IINFO.GT.0 ) THEN IF( PAIR ) THEN INFO = INFO + 2 ELSE INFO = INFO + 1 END IF IFAILR( KSR ) = K IFAILR( KSI ) = K ELSE IFAILR( KSR ) = 0 IFAILR( KSI ) = 0 END IF DO 100 I = KR + 1, N VR( I, KSR ) = ZERO 100 CONTINUE IF( PAIR ) THEN DO 110 I = KR + 1, N VR( I, KSI ) = ZERO 110 CONTINUE END IF END IF ! IF( PAIR ) THEN KSR = KSR + 2 ELSE KSR = KSR + 1 END IF END IF 120 CONTINUE ! RETURN ! ! End of DHSEIN ! END SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, & LDZ, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER COMPZ, JOB INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N ! .. ! .. Array Arguments .. DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ), & Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DHSEQR computes the eigenvalues of a real upper Hessenberg matrix H ! and, optionally, the matrices T and Z from the Schur decomposition ! H = Z T Z**T, where T is an upper quasi-triangular matrix (the Schur ! form), and Z is the orthogonal matrix of Schur vectors. ! ! Optionally Z may be postmultiplied into an input orthogonal matrix Q, ! so that this routine can give the Schur factorization of a matrix A ! which has been reduced to the Hessenberg form H by the orthogonal ! matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T. ! ! Arguments ! ========= ! ! JOB (input) CHARACTER*1 ! = 'E': compute eigenvalues only; ! = 'S': compute eigenvalues and the Schur form T. ! ! COMPZ (input) CHARACTER*1 ! = 'N': no Schur vectors are computed; ! = 'I': Z is initialized to the unit matrix and the matrix Z ! of Schur vectors of H is returned; ! = 'V': Z must contain an orthogonal matrix Q on entry, and ! the product Q*Z is returned. ! ! N (input) INTEGER ! The order of the matrix H. N >= 0. ! ! ILO (input) INTEGER ! IHI (input) INTEGER ! It is assumed that H is already upper triangular in rows ! and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally ! set by a previous call to DGEBAL, and then passed to SGEHRD ! when the matrix output by DGEBAL is reduced to Hessenberg ! form. Otherwise ILO and IHI should be set to 1 and N ! respectively. ! 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. ! ! H (input/output) DOUBLE PRECISION array, dimension (LDH,N) ! On entry, the upper Hessenberg matrix H. ! On exit, if JOB = 'S', H contains the upper quasi-triangular ! matrix T from the Schur decomposition (the Schur form); ! 2-by-2 diagonal blocks (corresponding to complex conjugate ! pairs of eigenvalues) are returned in standard form, with ! H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0. If JOB = 'E', ! the contents of H are unspecified on exit. ! ! LDH (input) INTEGER ! The leading dimension of the array H. LDH >= max(1,N). ! ! WR (output) DOUBLE PRECISION array, dimension (N) ! WI (output) DOUBLE PRECISION array, dimension (N) ! The real and imaginary parts, respectively, of the computed ! eigenvalues. If two eigenvalues are computed as a complex ! conjugate pair, they are stored in consecutive elements of ! WR and WI, say the i-th and (i+1)th, with WI(i) > 0 and ! WI(i+1) < 0. If JOB = 'S', the eigenvalues are stored in the ! same order as on the diagonal of the Schur form returned in ! H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 ! diagonal block, WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and ! WI(i+1) = -WI(i). ! ! Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N) ! If COMPZ = 'N': Z is not referenced. ! If COMPZ = 'I': on entry, Z need not be set, and on exit, Z ! contains the orthogonal matrix Z of the Schur vectors of H. ! If COMPZ = 'V': on entry Z must contain an N-by-N matrix Q, ! which is assumed to be equal to the unit matrix except for ! the submatrix Z(ILO:IHI,ILO:IHI); on exit Z contains Q*Z. ! Normally Q is the orthogonal matrix generated by DORGHR after ! the call to DGEHRD which formed the Hessenberg matrix H. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. ! LDZ >= max(1,N) if COMPZ = 'I' or 'V'; LDZ >= 1 otherwise. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,N). ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, DHSEQR failed to compute all of the ! eigenvalues in a total of 30*(IHI-ILO+1) iterations; ! elements 1:ilo-1 and i+1:n of WR and WI contain those ! eigenvalues which have been successfully computed. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 ) DOUBLE PRECISION CONST PARAMETER ( CONST = 1.5D+0 ) INTEGER NSMAX, LDS PARAMETER ( NSMAX = 15, LDS = NSMAX ) ! .. ! .. Local Scalars .. LOGICAL INITZ, LQUERY, WANTT, WANTZ INTEGER I, I1, I2, IERR, II, ITEMP, ITN, ITS, J, K, L, & MAXB, NH, NR, NS, NV DOUBLE PRECISION ABSW, OVFL, SMLNUM, TAU, TEMP, TST1, ULP, UNFL ! .. ! .. Local Arrays .. DOUBLE PRECISION S( LDS, NSMAX ), V( NSMAX+1 ), VV( NSMAX+1 ) ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX, ILAENV DOUBLE PRECISION DLAMCH, DLANHS, DLAPY2 EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DLANHS, DLAPY2 ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEMV, DLACPY, DLAHQR, DLARFG, DLARFX, & DLASET, DSCAL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Executable Statements .. ! ! Decode and test the input parameters ! WANTT = LSAME( JOB, 'S' ) INITZ = LSAME( COMPZ, 'I' ) WANTZ = INITZ .OR. LSAME( COMPZ, 'V' ) ! INFO = 0 WORK( 1 ) = MAX( 1, N ) LQUERY = ( LWORK.EQ.-1 ) IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN INFO = -1 ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN INFO = -5 ELSE IF( LDH.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDZ.LT.1 .OR. WANTZ .AND. LDZ.LT.MAX( 1, N ) ) THEN INFO = -11 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN INFO = -13 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DHSEQR', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Initialize Z, if necessary ! IF( INITZ ) & CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ ) ! ! Store the eigenvalues isolated by DGEBAL. ! DO 10 I = 1, ILO - 1 WR( I ) = H( I, I ) WI( I ) = ZERO 10 CONTINUE DO 20 I = IHI + 1, N WR( I ) = H( I, I ) WI( I ) = ZERO 20 CONTINUE ! ! Quick return if possible. ! IF( N.EQ.0 ) & RETURN IF( ILO.EQ.IHI ) THEN WR( ILO ) = H( ILO, ILO ) WI( ILO ) = ZERO RETURN END IF ! ! Set rows and columns ILO to IHI to zero below the first ! subdiagonal. ! DO 40 J = ILO, IHI - 2 DO 30 I = J + 2, N H( I, J ) = ZERO 30 CONTINUE 40 CONTINUE NH = IHI - ILO + 1 ! ! Determine the order of the multi-shift QR algorithm to be used. ! NS = ILAENV( 4, 'DHSEQR', JOB // COMPZ, N, ILO, IHI, -1 ) MAXB = ILAENV( 8, 'DHSEQR', JOB // COMPZ, N, ILO, IHI, -1 ) IF( NS.LE.2 .OR. NS.GT.NH .OR. MAXB.GE.NH ) THEN ! ! Use the standard double-shift algorithm ! CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO, & IHI, Z, LDZ, INFO ) RETURN END IF MAXB = MAX( 3, MAXB ) NS = MIN( NS, MAXB, NSMAX ) ! ! Now 2 < NS <= MAXB < NH. ! ! Set machine-dependent constants for the stopping criterion. ! If norm(H) <= sqrt(OVFL), overflow should not occur. ! UNFL = DLAMCH( 'Safe minimum' ) OVFL = ONE / UNFL CALL DLABAD( UNFL, OVFL ) ULP = DLAMCH( 'Precision' ) SMLNUM = UNFL*( NH / ULP ) ! ! I1 and I2 are the indices of the first row and last column of H ! to which transformations must be applied. If eigenvalues only are ! being computed, I1 and I2 are set inside the main loop. ! IF( WANTT ) THEN I1 = 1 I2 = N END IF ! ! ITN is the total number of multiple-shift QR iterations allowed. ! ITN = 30*NH ! ! The main loop begins here. I is the loop index and decreases from ! IHI to ILO in steps of at most MAXB. Each iteration of the loop ! works with the active submatrix in rows and columns L to I. ! Eigenvalues I+1 to IHI have already converged. Either L = ILO or ! H(L,L-1) is negligible so that the matrix splits. ! I = IHI 50 CONTINUE L = ILO IF( I.LT.ILO ) & GO TO 170 ! ! Perform multiple-shift QR iterations on rows and columns ILO to I ! until a submatrix of order at most MAXB splits off at the bottom ! because a subdiagonal element has become negligible. ! DO 150 ITS = 0, ITN ! ! Look for a single small subdiagonal element. ! DO 60 K = I, L + 1, -1 TST1 = ABS( H( K-1, K-1 ) ) + ABS( H( K, K ) ) IF( TST1.EQ.ZERO ) & TST1 = DLANHS( '1', I-L+1, H( L, L ), LDH, WORK ) IF( ABS( H( K, K-1 ) ).LE.MAX( ULP*TST1, SMLNUM ) ) & GO TO 70 60 CONTINUE 70 CONTINUE L = K IF( L.GT.ILO ) THEN ! ! H(L,L-1) is negligible. ! H( L, L-1 ) = ZERO END IF ! ! Exit from loop if a submatrix of order <= MAXB has split off. ! IF( L.GE.I-MAXB+1 ) & GO TO 160 ! ! Now the active submatrix is in rows and columns L to I. If ! eigenvalues only are being computed, only the active submatrix ! need be transformed. ! IF( .NOT.WANTT ) THEN I1 = L I2 = I END IF ! IF( ITS.EQ.20 .OR. ITS.EQ.30 ) THEN ! ! Exceptional shifts. ! DO 80 II = I - NS + 1, I WR( II ) = CONST*( ABS( H( II, II-1 ) )+ & ABS( H( II, II ) ) ) WI( II ) = ZERO 80 CONTINUE ELSE ! ! Use eigenvalues of trailing submatrix of order NS as shifts. ! CALL DLACPY( 'Full', NS, NS, H( I-NS+1, I-NS+1 ), LDH, S, & LDS ) CALL DLAHQR( .FALSE., .FALSE., NS, 1, NS, S, LDS, & WR( I-NS+1 ), WI( I-NS+1 ), 1, NS, Z, LDZ, & IERR ) IF( IERR.GT.0 ) THEN ! ! If DLAHQR failed to compute all NS eigenvalues, use the ! unconverged diagonal elements as the remaining shifts. ! DO 90 II = 1, IERR WR( I-NS+II ) = S( II, II ) WI( I-NS+II ) = ZERO 90 CONTINUE END IF END IF ! ! Form the first column of (G-w(1)) (G-w(2)) . . . (G-w(ns)) ! where G is the Hessenberg submatrix H(L:I,L:I) and w is ! the vector of shifts (stored in WR and WI). The result is ! stored in the local array V. ! V( 1 ) = ONE DO 100 II = 2, NS + 1 V( II ) = ZERO 100 CONTINUE NV = 1 DO 120 J = I - NS + 1, I IF( WI( J ).GE.ZERO ) THEN IF( WI( J ).EQ.ZERO ) THEN ! ! real shift ! CALL DCOPY( NV+1, V, 1, VV, 1 ) CALL DGEMV( 'No transpose', NV+1, NV, ONE, H( L, L ), & LDH, VV, 1, -WR( J ), V, 1 ) NV = NV + 1 ELSE IF( WI( J ).GT.ZERO ) THEN ! ! complex conjugate pair of shifts ! CALL DCOPY( NV+1, V, 1, VV, 1 ) CALL DGEMV( 'No transpose', NV+1, NV, ONE, H( L, L ), & LDH, V, 1, -TWO*WR( J ), VV, 1 ) ITEMP = IDAMAX( NV+1, VV, 1 ) TEMP = ONE / MAX( ABS( VV( ITEMP ) ), SMLNUM ) CALL DSCAL( NV+1, TEMP, VV, 1 ) ABSW = DLAPY2( WR( J ), WI( J ) ) TEMP = ( TEMP*ABSW )*ABSW CALL DGEMV( 'No transpose', NV+2, NV+1, ONE, & H( L, L ), LDH, VV, 1, TEMP, V, 1 ) NV = NV + 2 END IF ! ! Scale V(1:NV) so that max(abs(V(i))) = 1. If V is zero, ! reset it to the unit vector. ! ITEMP = IDAMAX( NV, V, 1 ) TEMP = ABS( V( ITEMP ) ) IF( TEMP.EQ.ZERO ) THEN V( 1 ) = ONE DO 110 II = 2, NV V( II ) = ZERO 110 CONTINUE ELSE TEMP = MAX( TEMP, SMLNUM ) CALL DSCAL( NV, ONE / TEMP, V, 1 ) END IF END IF 120 CONTINUE ! ! Multiple-shift QR step ! DO 140 K = L, I - 1 ! ! The first iteration of this loop determines a reflection G ! from the vector V and applies it from left and right to H, ! thus creating a nonzero bulge below the subdiagonal. ! ! Each subsequent iteration determines a reflection G to ! restore the Hessenberg form in the (K-1)th column, and thus ! chases the bulge one step toward the bottom of the active ! submatrix. NR is the order of G. ! NR = MIN( NS+1, I-K+1 ) IF( K.GT.L ) & CALL DCOPY( NR, H( K, K-1 ), 1, V, 1 ) CALL DLARFG( NR, V( 1 ), V( 2 ), 1, TAU ) IF( K.GT.L ) THEN H( K, K-1 ) = V( 1 ) DO 130 II = K + 1, I H( II, K-1 ) = ZERO 130 CONTINUE END IF V( 1 ) = ONE ! ! Apply G from the left to transform the rows of the matrix in ! columns K to I2. ! CALL DLARFX( 'Left', NR, I2-K+1, V, TAU, H( K, K ), LDH, & WORK ) ! ! Apply G from the right to transform the columns of the ! matrix in rows I1 to min(K+NR,I). ! CALL DLARFX( 'Right', MIN( K+NR, I )-I1+1, NR, V, TAU, & H( I1, K ), LDH, WORK ) ! IF( WANTZ ) THEN ! ! Accumulate transformations in the matrix Z ! CALL DLARFX( 'Right', NH, NR, V, TAU, Z( ILO, K ), LDZ, & WORK ) END IF 140 CONTINUE ! 150 CONTINUE ! ! Failure to converge in remaining number of iterations ! INFO = I RETURN ! 160 CONTINUE ! ! A submatrix of order <= MAXB in rows and columns L to I has split ! off. Use the double-shift QR algorithm to handle it. ! CALL DLAHQR( WANTT, WANTZ, N, L, I, H, LDH, WR, WI, ILO, IHI, Z, & LDZ, INFO ) IF( INFO.GT.0 ) & RETURN ! ! Decrement number of remaining iterations, and return to start of ! the main loop with a new value of I. ! ITN = ITN - ITS I = L - 1 GO TO 50 ! 170 CONTINUE WORK( 1 ) = MAX( 1, N ) RETURN ! ! End of DHSEQR ! END SUBROUTINE DLABAD( SMALL, LARGE ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. DOUBLE PRECISION LARGE, SMALL ! .. ! ! Purpose ! ======= ! ! DLABAD takes as input the values computed by DLAMCH for underflow and ! overflow, and returns the square root of each of these values if the ! log of LARGE is sufficiently large. This subroutine is intended to ! identify machines with a large exponent range, such as the Crays, and ! redefine the underflow and overflow limits to be the square roots of ! the values computed by DLAMCH. This subroutine is needed because ! DLAMCH does not compensate for poor arithmetic in the upper half of ! the exponent range, as is found on a Cray. ! ! Arguments ! ========= ! ! SMALL (input/output) DOUBLE PRECISION ! On entry, the underflow threshold as computed by DLAMCH. ! On exit, if LOG10(LARGE) is sufficiently large, the square ! root of SMALL, otherwise unchanged. ! ! LARGE (input/output) DOUBLE PRECISION ! On entry, the overflow threshold as computed by DLAMCH. ! On exit, if LOG10(LARGE) is sufficiently large, the square ! root of LARGE, otherwise unchanged. ! ! ===================================================================== ! ! .. Intrinsic Functions .. INTRINSIC LOG10, SQRT ! .. ! .. Executable Statements .. ! ! If it looks like we're on a Cray, take the square root of ! SMALL and LARGE to avoid overflow and underflow problems. ! IF( LOG10( LARGE ).GT.2000.D0 ) THEN SMALL = SQRT( SMALL ) LARGE = SQRT( LARGE ) END IF ! RETURN ! ! End of DLABAD ! END SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, & LDY ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. INTEGER LDA, LDX, LDY, M, N, NB ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), & TAUQ( * ), X( LDX, * ), Y( LDY, * ) ! .. ! ! Purpose ! ======= ! ! DLABRD reduces the first NB rows and columns of a real general ! m by n matrix A to upper or lower bidiagonal form by an orthogonal ! transformation Q' * A * P, and returns the matrices X and Y which ! are needed to apply the transformation to the unreduced part of A. ! ! If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower ! bidiagonal form. ! ! This is an auxiliary routine called by DGEBRD ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows in the matrix A. ! ! N (input) INTEGER ! The number of columns in the matrix A. ! ! NB (input) INTEGER ! The number of leading rows and columns of A to be reduced. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the m by n general matrix to be reduced. ! On exit, the first NB rows and columns of the matrix are ! overwritten; the rest of the array is unchanged. ! If m >= n, elements on and below the diagonal in the first NB ! columns, with the array TAUQ, represent the orthogonal ! matrix Q as a product of elementary reflectors; and ! elements above the diagonal in the first NB rows, with the ! array TAUP, represent the orthogonal matrix P as a product ! of elementary reflectors. ! If m < n, elements below the diagonal in the first NB ! columns, with the array TAUQ, represent the orthogonal ! matrix Q as a product of elementary reflectors, and ! elements on and above the diagonal in the first NB rows, ! with the array TAUP, represent the orthogonal matrix P as ! a product of elementary reflectors. ! See Further Details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! D (output) DOUBLE PRECISION array, dimension (NB) ! The diagonal elements of the first NB rows and columns of ! the reduced matrix. D(i) = A(i,i). ! ! E (output) DOUBLE PRECISION array, dimension (NB) ! The off-diagonal elements of the first NB rows and columns of ! the reduced matrix. ! ! TAUQ (output) DOUBLE PRECISION array dimension (NB) ! The scalar factors of the elementary reflectors which ! represent the orthogonal matrix Q. See Further Details. ! ! TAUP (output) DOUBLE PRECISION array, dimension (NB) ! The scalar factors of the elementary reflectors which ! represent the orthogonal matrix P. See Further Details. ! ! X (output) DOUBLE PRECISION array, dimension (LDX,NB) ! The m-by-nb matrix X required to update the unreduced part ! of A. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= M. ! ! Y (output) DOUBLE PRECISION array, dimension (LDY,NB) ! The n-by-nb matrix Y required to update the unreduced part ! of A. ! ! LDY (output) INTEGER ! The leading dimension of the array Y. LDY >= N. ! ! Further Details ! =============== ! ! The matrices Q and P are represented as products of elementary ! reflectors: ! ! Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) ! ! Each H(i) and G(i) has the form: ! ! H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' ! ! where tauq and taup are real scalars, and v and u are real vectors. ! ! If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in ! A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in ! A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). ! ! If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in ! A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in ! A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). ! ! The elements of the vectors v and u together form the m-by-nb matrix ! V and the nb-by-n matrix U' which are needed, with X and Y, to apply ! the transformation to the unreduced part of the matrix, using a block ! update of the form: A := A - V*Y' - X*U'. ! ! The contents of A on exit are illustrated by the following examples ! with nb = 2: ! ! m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ! ! ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 ) ! ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 ) ! ( v1 v2 a a a ) ( v1 1 a a a a ) ! ( v1 v2 a a a ) ( v1 v2 a a a a ) ! ( v1 v2 a a a ) ( v1 v2 a a a a ) ! ( v1 v2 a a a ) ! ! where a denotes an element of the original matrix which is unchanged, ! vi denotes an element of the vector defining H(i), and ui an element ! of the vector defining G(i). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. INTEGER I ! .. ! .. External Subroutines .. EXTERNAL DGEMV, DLARFG, DSCAL ! .. ! .. Intrinsic Functions .. INTRINSIC MIN ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( M.LE.0 .OR. N.LE.0 ) & RETURN ! IF( M.GE.N ) THEN ! ! Reduce to upper bidiagonal form ! DO 10 I = 1, NB ! ! Update A(i:m,i) ! CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ), & LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 ) CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ), & LDX, A( 1, I ), 1, ONE, A( I, I ), 1 ) ! ! Generate reflection Q(i) to annihilate A(i+1:m,i) ! CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1, & TAUQ( I ) ) D( I ) = A( I, I ) IF( I.LT.N ) THEN A( I, I ) = ONE ! ! Compute Y(i+1:n,i) ! CALL DGEMV( 'Transpose', M-I+1, N-I, ONE, A( I, I+1 ), & LDA, A( I, I ), 1, ZERO, Y( I+1, I ), 1 ) CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, A( I, 1 ), LDA, & A( I, I ), 1, ZERO, Y( 1, I ), 1 ) CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ), & LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 ) CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, X( I, 1 ), LDX, & A( I, I ), 1, ZERO, Y( 1, I ), 1 ) CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ), & LDA, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 ) CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 ) ! ! Update A(i,i+1:n) ! CALL DGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ), & LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA ) CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ), & LDA, X( I, 1 ), LDX, ONE, A( I, I+1 ), LDA ) ! ! Generate reflection P(i) to annihilate A(i,i+2:n) ! CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ), & LDA, TAUP( I ) ) E( I ) = A( I, I+1 ) A( I, I+1 ) = ONE ! ! Compute X(i+1:m,i) ! CALL DGEMV( 'No transpose', M-I, N-I, ONE, A( I+1, I+1 ), & LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 ) CALL DGEMV( 'Transpose', N-I, I, ONE, Y( I+1, 1 ), LDY, & A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 ) CALL DGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ), & LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 ) CALL DGEMV( 'No transpose', I-1, N-I, ONE, A( 1, I+1 ), & LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 ) CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ), & LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 ) CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 ) END IF 10 CONTINUE ELSE ! ! Reduce to lower bidiagonal form ! DO 20 I = 1, NB ! ! Update A(i,i:n) ! CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ), & LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA ) CALL DGEMV( 'Transpose', I-1, N-I+1, -ONE, A( 1, I ), LDA, & X( I, 1 ), LDX, ONE, A( I, I ), LDA ) ! ! Generate reflection P(i) to annihilate A(i,i+1:n) ! CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA, & TAUP( I ) ) D( I ) = A( I, I ) IF( I.LT.M ) THEN A( I, I ) = ONE ! ! Compute X(i+1:m,i) ! CALL DGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1, I ), & LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 ) CALL DGEMV( 'Transpose', N-I+1, I-1, ONE, Y( I, 1 ), LDY, & A( I, I ), LDA, ZERO, X( 1, I ), 1 ) CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ), & LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 ) CALL DGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1, I ), & LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 ) CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ), & LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 ) CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 ) ! ! Update A(i+1:m,i) ! CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ), & LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 ) CALL DGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ), & LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 ) ! ! Generate reflection Q(i) to annihilate A(i+2:m,i) ! CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1, & TAUQ( I ) ) E( I ) = A( I+1, I ) A( I+1, I ) = ONE ! ! Compute Y(i+1:n,i) ! CALL DGEMV( 'Transpose', M-I, N-I, ONE, A( I+1, I+1 ), & LDA, A( I+1, I ), 1, ZERO, Y( I+1, I ), 1 ) CALL DGEMV( 'Transpose', M-I, I-1, ONE, A( I+1, 1 ), LDA, & A( I+1, I ), 1, ZERO, Y( 1, I ), 1 ) CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ), & LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 ) CALL DGEMV( 'Transpose', M-I, I, ONE, X( I+1, 1 ), LDX, & A( I+1, I ), 1, ZERO, Y( 1, I ), 1 ) CALL DGEMV( 'Transpose', I, N-I, -ONE, A( 1, I+1 ), LDA, & Y( 1, I ), 1, ONE, Y( I+1, I ), 1 ) CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 ) END IF 20 CONTINUE END IF RETURN ! ! End of DLABRD ! END SUBROUTINE DLACON( N, V, X, ISGN, EST, KASE ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. INTEGER KASE, N DOUBLE PRECISION EST ! .. ! .. Array Arguments .. INTEGER ISGN( * ) DOUBLE PRECISION V( * ), X( * ) ! .. ! ! Purpose ! ======= ! ! DLACON estimates the 1-norm of a square, real matrix A. ! Reverse communication is used for evaluating matrix-vector products. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix. N >= 1. ! ! V (workspace) DOUBLE PRECISION array, dimension (N) ! On the final return, V = A*W, where EST = norm(V)/norm(W) ! (W is not returned). ! ! X (input/output) DOUBLE PRECISION array, dimension (N) ! On an intermediate return, X should be overwritten by ! A * X, if KASE=1, ! A' * X, if KASE=2, ! and DLACON must be re-called with all the other parameters ! unchanged. ! ! ISGN (workspace) INTEGER array, dimension (N) ! ! EST (output) DOUBLE PRECISION ! An estimate (a lower bound) for norm(A). ! ! KASE (input/output) INTEGER ! On the initial call to DLACON, KASE should be 0. ! On an intermediate return, KASE will be 1 or 2, indicating ! whether X should be overwritten by A * X or A' * X. ! On the final return from DLACON, KASE will again be 0. ! ! Further Details ! ======= ======= ! ! Contributed by Nick Higham, University of Manchester. ! Originally named SONEST, dated March 16, 1988. ! ! Reference: N.J. Higham, "FORTRAN codes for estimating the one-norm of ! a real or complex matrix, with applications to condition estimation", ! ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988. ! ! ===================================================================== ! ! .. Parameters .. INTEGER ITMAX PARAMETER ( ITMAX = 5 ) DOUBLE PRECISION ZERO, ONE, TWO PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, ITER, J, JLAST, JUMP DOUBLE PRECISION ALTSGN, ESTOLD, TEMP ! .. ! .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DASUM EXTERNAL IDAMAX, DASUM ! .. ! .. External Subroutines .. EXTERNAL DCOPY ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, NINT, SIGN ! .. ! .. Save statement .. SAVE ! .. ! .. Executable Statements .. ! IF( KASE.EQ.0 ) THEN DO 10 I = 1, N X( I ) = ONE / DBLE( N ) 10 CONTINUE KASE = 1 JUMP = 1 RETURN END IF ! GO TO ( 20, 40, 70, 110, 140 )JUMP ! ! ................ ENTRY (JUMP = 1) ! FIRST ITERATION. X HAS BEEN OVERWRITTEN BY A*X. ! 20 CONTINUE IF( N.EQ.1 ) THEN V( 1 ) = X( 1 ) EST = ABS( V( 1 ) ) ! ... QUIT GO TO 150 END IF EST = DASUM( N, X, 1 ) ! DO 30 I = 1, N X( I ) = SIGN( ONE, X( I ) ) ISGN( I ) = NINT( X( I ) ) 30 CONTINUE KASE = 2 JUMP = 2 RETURN ! ! ................ ENTRY (JUMP = 2) ! FIRST ITERATION. X HAS BEEN OVERWRITTEN BY TRANDPOSE(A)*X. ! 40 CONTINUE J = IDAMAX( N, X, 1 ) ITER = 2 ! ! MAIN LOOP - ITERATIONS 2,3,...,ITMAX. ! 50 CONTINUE DO 60 I = 1, N X( I ) = ZERO 60 CONTINUE X( J ) = ONE KASE = 1 JUMP = 3 RETURN ! ! ................ ENTRY (JUMP = 3) ! X HAS BEEN OVERWRITTEN BY A*X. ! 70 CONTINUE CALL DCOPY( N, X, 1, V, 1 ) ESTOLD = EST EST = DASUM( N, V, 1 ) DO 80 I = 1, N IF( NINT( SIGN( ONE, X( I ) ) ).NE.ISGN( I ) ) & GO TO 90 80 CONTINUE ! REPEATED SIGN VECTOR DETECTED, HENCE ALGORITHM HAS CONVERGED. GO TO 120 ! 90 CONTINUE ! TEST FOR CYCLING. IF( EST.LE.ESTOLD ) & GO TO 120 ! DO 100 I = 1, N X( I ) = SIGN( ONE, X( I ) ) ISGN( I ) = NINT( X( I ) ) 100 CONTINUE KASE = 2 JUMP = 4 RETURN ! ! ................ ENTRY (JUMP = 4) ! X HAS BEEN OVERWRITTEN BY TRANDPOSE(A)*X. ! 110 CONTINUE JLAST = J J = IDAMAX( N, X, 1 ) IF( ( X( JLAST ).NE.ABS( X( J ) ) ) .AND. ( ITER.LT.ITMAX ) ) THEN ITER = ITER + 1 GO TO 50 END IF ! ! ITERATION COMPLETE. FINAL STAGE. ! 120 CONTINUE ALTSGN = ONE DO 130 I = 1, N X( I ) = ALTSGN*( ONE+DBLE( I-1 ) / DBLE( N-1 ) ) ALTSGN = -ALTSGN 130 CONTINUE KASE = 1 JUMP = 5 RETURN ! ! ................ ENTRY (JUMP = 5) ! X HAS BEEN OVERWRITTEN BY A*X. ! 140 CONTINUE TEMP = TWO*( DASUM( N, X, 1 ) / DBLE( 3*N ) ) IF( TEMP.GT.EST ) THEN CALL DCOPY( N, X, 1, V, 1 ) EST = TEMP END IF ! 150 CONTINUE KASE = 0 RETURN ! ! End of DLACON ! END SUBROUTINE DLACPY( UPLO, M, N, A, LDA, B, LDB ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER LDA, LDB, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DLACPY copies all or part of a two-dimensional matrix A to another ! matrix B. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies the part of the matrix A to be copied to B. ! = 'U': Upper triangular part ! = 'L': Lower triangular part ! Otherwise: All of the matrix A ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The m by n matrix A. If UPLO = 'U', only the upper triangle ! or trapezoid is accessed; if UPLO = 'L', only the lower ! triangle or trapezoid is accessed. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! B (output) DOUBLE PRECISION array, dimension (LDB,N) ! On exit, B = A in the locations specified by UPLO. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,M). ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER I, J ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Intrinsic Functions .. INTRINSIC MIN ! .. ! .. Executable Statements .. ! IF( LSAME( UPLO, 'U' ) ) THEN DO 20 J = 1, N DO 10 I = 1, MIN( J, M ) B( I, J ) = A( I, J ) 10 CONTINUE 20 CONTINUE ELSE IF( LSAME( UPLO, 'L' ) ) THEN DO 40 J = 1, N DO 30 I = J, M B( I, J ) = A( I, J ) 30 CONTINUE 40 CONTINUE ELSE DO 60 J = 1, N DO 50 I = 1, M B( I, J ) = A( I, J ) 50 CONTINUE 60 CONTINUE END IF RETURN ! ! End of DLACPY ! END SUBROUTINE DLADIV( A, B, C, D, P, Q ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. DOUBLE PRECISION A, B, C, D, P, Q ! .. ! ! Purpose ! ======= ! ! DLADIV performs complex division in real arithmetic ! ! a + i*b ! p + i*q = --------- ! c + i*d ! ! The algorithm is due to Robert L. Smith and can be found ! in D. Knuth, The art of Computer Programming, Vol.2, p.195 ! ! Arguments ! ========= ! ! A (input) DOUBLE PRECISION ! B (input) DOUBLE PRECISION ! C (input) DOUBLE PRECISION ! D (input) DOUBLE PRECISION ! The scalars a, b, c, and d in the above expression. ! ! P (output) DOUBLE PRECISION ! Q (output) DOUBLE PRECISION ! The scalars p and q in the above expression. ! ! ===================================================================== ! ! .. Local Scalars .. DOUBLE PRECISION E, F ! .. ! .. Intrinsic Functions .. INTRINSIC ABS ! .. ! .. Executable Statements .. ! IF( ABS( D ).LT.ABS( C ) ) THEN E = D / C F = C + D*E P = ( A+B*E ) / F Q = ( B-A*E ) / F ELSE E = C / D F = D + C*E P = ( B+A*E ) / F Q = ( -A+B*E ) / F END IF ! RETURN ! ! End of DLADIV ! END SUBROUTINE DLAE2( A, B, C, RT1, RT2 ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. DOUBLE PRECISION A, B, C, RT1, RT2 ! .. ! ! Purpose ! ======= ! ! DLAE2 computes the eigenvalues of a 2-by-2 symmetric matrix ! [ A B ] ! [ B C ]. ! On return, RT1 is the eigenvalue of larger absolute value, and RT2 ! is the eigenvalue of smaller absolute value. ! ! Arguments ! ========= ! ! A (input) DOUBLE PRECISION ! The (1,1) element of the 2-by-2 matrix. ! ! B (input) DOUBLE PRECISION ! The (1,2) and (2,1) elements of the 2-by-2 matrix. ! ! C (input) DOUBLE PRECISION ! The (2,2) element of the 2-by-2 matrix. ! ! RT1 (output) DOUBLE PRECISION ! The eigenvalue of larger absolute value. ! ! RT2 (output) DOUBLE PRECISION ! The eigenvalue of smaller absolute value. ! ! Further Details ! =============== ! ! RT1 is accurate to a few ulps barring over/underflow. ! ! RT2 may be inaccurate if there is massive cancellation in the ! determinant A*C-B*B; higher precision or correctly rounded or ! correctly truncated arithmetic would be needed to compute RT2 ! accurately in all cases. ! ! Overflow is possible only if RT1 is within a factor of 5 of overflow. ! Underflow is harmless if the input data is 0 or exceeds ! underflow_threshold / macheps. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D0 ) DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D0 ) DOUBLE PRECISION HALF PARAMETER ( HALF = 0.5D0 ) ! .. ! .. Local Scalars .. DOUBLE PRECISION AB, ACMN, ACMX, ADF, DF, RT, SM, TB ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, SQRT ! .. ! .. Executable Statements .. ! ! Compute the eigenvalues ! SM = A + C DF = A - C ADF = ABS( DF ) TB = B + B AB = ABS( TB ) IF( ABS( A ).GT.ABS( C ) ) THEN ACMX = A ACMN = C ELSE ACMX = C ACMN = A END IF IF( ADF.GT.AB ) THEN RT = ADF*SQRT( ONE+( AB / ADF )**2 ) ELSE IF( ADF.LT.AB ) THEN RT = AB*SQRT( ONE+( ADF / AB )**2 ) ELSE ! ! Includes case AB=ADF=0 ! RT = AB*SQRT( TWO ) END IF IF( SM.LT.ZERO ) THEN RT1 = HALF*( SM-RT ) ! ! Order of execution important. ! To get fully accurate smaller eigenvalue, ! next line needs to be executed in higher precision. ! RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B ELSE IF( SM.GT.ZERO ) THEN RT1 = HALF*( SM+RT ) ! ! Order of execution important. ! To get fully accurate smaller eigenvalue, ! next line needs to be executed in higher precision. ! RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B ELSE ! ! Includes case RT1 = RT2 = 0 ! RT1 = HALF*RT RT2 = -HALF*RT END IF RETURN ! ! End of DLAE2 ! END SUBROUTINE DLAEBZ( IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL, & RELTOL, PIVMIN, D, E, E2, NVAL, AB, C, MOUT, & NAB, WORK, IWORK, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER IJOB, INFO, MINP, MMAX, MOUT, N, NBMIN, NITMAX DOUBLE PRECISION ABSTOL, PIVMIN, RELTOL ! .. ! .. Array Arguments .. INTEGER IWORK( * ), NAB( MMAX, * ), NVAL( * ) DOUBLE PRECISION AB( MMAX, * ), C( * ), D( * ), E( * ), E2( * ), & WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLAEBZ contains the iteration loops which compute and use the ! function N(w), which is the count of eigenvalues of a symmetric ! tridiagonal matrix T less than or equal to its argument w. It ! performs a choice of two types of loops: ! ! IJOB=1, followed by ! IJOB=2: It takes as input a list of intervals and returns a list of ! sufficiently small intervals whose union contains the same ! eigenvalues as the union of the original intervals. ! The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP. ! The output interval (AB(j,1),AB(j,2)] will contain ! eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT. ! ! IJOB=3: It performs a binary search in each input interval ! (AB(j,1),AB(j,2)] for a point w(j) such that ! N(w(j))=NVAL(j), and uses C(j) as the starting point of ! the search. If such a w(j) is found, then on output ! AB(j,1)=AB(j,2)=w. If no such w(j) is found, then on output ! (AB(j,1),AB(j,2)] will be a small interval containing the ! point where N(w) jumps through NVAL(j), unless that point ! lies outside the initial interval. ! ! Note that the intervals are in all cases half-open intervals, ! i.e., of the form (a,b] , which includes b but not a . ! ! To avoid underflow, the matrix should be scaled so that its largest ! element is no greater than overflow**(1/2) * underflow**(1/4) ! in absolute value. To assure the most accurate computation ! of small eigenvalues, the matrix should be scaled to be ! not much smaller than that, either. ! ! See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal ! Matrix", Report CS41, Computer Science Dept., Stanford ! University, July 21, 1966 ! ! Note: the arguments are, in general, *not* checked for unreasonable ! values. ! ! Arguments ! ========= ! ! IJOB (input) INTEGER ! Specifies what is to be done: ! = 1: Compute NAB for the initial intervals. ! = 2: Perform bisection iteration to find eigenvalues of T. ! = 3: Perform bisection iteration to invert N(w), i.e., ! to find a point which has a specified number of ! eigenvalues of T to its left. ! Other values will cause DLAEBZ to return with INFO=-1. ! ! NITMAX (input) INTEGER ! The maximum number of "levels" of bisection to be ! performed, i.e., an interval of width W will not be made ! smaller than 2^(-NITMAX) * W. If not all intervals ! have converged after NITMAX iterations, then INFO is set ! to the number of non-converged intervals. ! ! N (input) INTEGER ! The dimension n of the tridiagonal matrix T. It must be at ! least 1. ! ! MMAX (input) INTEGER ! The maximum number of intervals. If more than MMAX intervals ! are generated, then DLAEBZ will quit with INFO=MMAX+1. ! ! MINP (input) INTEGER ! The initial number of intervals. It may not be greater than ! MMAX. ! ! NBMIN (input) INTEGER ! The smallest number of intervals that should be processed ! using a vector loop. If zero, then only the scalar loop ! will be used. ! ! ABSTOL (input) DOUBLE PRECISION ! The minimum (absolute) width of an interval. When an ! interval is narrower than ABSTOL, or than RELTOL times the ! larger (in magnitude) endpoint, then it is considered to be ! sufficiently small, i.e., converged. This must be at least ! zero. ! ! RELTOL (input) DOUBLE PRECISION ! The minimum relative width of an interval. When an interval ! is narrower than ABSTOL, or than RELTOL times the larger (in ! magnitude) endpoint, then it is considered to be ! sufficiently small, i.e., converged. Note: this should ! always be at least radix*machine epsilon. ! ! PIVMIN (input) DOUBLE PRECISION ! The minimum absolute value of a "pivot" in the Sturm ! sequence loop. This *must* be at least max |e(j)**2| * ! safe_min and at least safe_min, where safe_min is at least ! the smallest number that can divide one without overflow. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The diagonal elements of the tridiagonal matrix T. ! ! E (input) DOUBLE PRECISION array, dimension (N) ! The offdiagonal elements of the tridiagonal matrix T in ! positions 1 through N-1. E(N) is arbitrary. ! ! E2 (input) DOUBLE PRECISION array, dimension (N) ! The squares of the offdiagonal elements of the tridiagonal ! matrix T. E2(N) is ignored. ! ! NVAL (input/output) INTEGER array, dimension (MINP) ! If IJOB=1 or 2, not referenced. ! If IJOB=3, the desired values of N(w). The elements of NVAL ! will be reordered to correspond with the intervals in AB. ! Thus, NVAL(j) on output will not, in general be the same as ! NVAL(j) on input, but it will correspond with the interval ! (AB(j,1),AB(j,2)] on output. ! ! AB (input/output) DOUBLE PRECISION array, dimension (MMAX,2) ! The endpoints of the intervals. AB(j,1) is a(j), the left ! endpoint of the j-th interval, and AB(j,2) is b(j), the ! right endpoint of the j-th interval. The input intervals ! will, in general, be modified, split, and reordered by the ! calculation. ! ! C (input/output) DOUBLE PRECISION array, dimension (MMAX) ! If IJOB=1, ignored. ! If IJOB=2, workspace. ! If IJOB=3, then on input C(j) should be initialized to the ! first search point in the binary search. ! ! MOUT (output) INTEGER ! If IJOB=1, the number of eigenvalues in the intervals. ! If IJOB=2 or 3, the number of intervals output. ! If IJOB=3, MOUT will equal MINP. ! ! NAB (input/output) INTEGER array, dimension (MMAX,2) ! If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)). ! If IJOB=2, then on input, NAB(i,j) should be set. It must ! satisfy the condition: ! N(AB(i,1)) <= NAB(i,1) <= NAB(i,2) <= N(AB(i,2)), ! which means that in interval i only eigenvalues ! NAB(i,1)+1,...,NAB(i,2) will be considered. Usually, ! NAB(i,j)=N(AB(i,j)), from a previous call to DLAEBZ with ! IJOB=1. ! On output, NAB(i,j) will contain ! max(na(k),min(nb(k),N(AB(i,j)))), where k is the index of ! the input interval that the output interval ! (AB(j,1),AB(j,2)] came from, and na(k) and nb(k) are the ! the input values of NAB(k,1) and NAB(k,2). ! If IJOB=3, then on output, NAB(i,j) contains N(AB(i,j)), ! unless N(w) > NVAL(i) for all search points w , in which ! case NAB(i,1) will not be modified, i.e., the output ! value will be the same as the input value (modulo ! reorderings -- see NVAL and AB), or unless N(w) < NVAL(i) ! for all search points w , in which case NAB(i,2) will ! not be modified. Normally, NAB should be set to some ! distinctive value(s) before DLAEBZ is called. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (MMAX) ! Workspace. ! ! IWORK (workspace) INTEGER array, dimension (MMAX) ! Workspace. ! ! INFO (output) INTEGER ! = 0: All intervals converged. ! = 1--MMAX: The last INFO intervals did not converge. ! = MMAX+1: More than MMAX intervals were generated. ! ! Further Details ! =============== ! ! This routine is intended to be called only by other LAPACK ! routines, thus the interface is less user-friendly. It is intended ! for two purposes: ! ! (a) finding eigenvalues. In this case, DLAEBZ should have one or ! more initial intervals set up in AB, and DLAEBZ should be called ! with IJOB=1. This sets up NAB, and also counts the eigenvalues. ! Intervals with no eigenvalues would usually be thrown out at ! this point. Also, if not all the eigenvalues in an interval i ! are desired, NAB(i,1) can be increased or NAB(i,2) decreased. ! For example, set NAB(i,1)=NAB(i,2)-1 to get the largest ! eigenvalue. DLAEBZ is then called with IJOB=2 and MMAX ! no smaller than the value of MOUT returned by the call with ! IJOB=1. After this (IJOB=2) call, eigenvalues NAB(i,1)+1 ! through NAB(i,2) are approximately AB(i,1) (or AB(i,2)) to the ! tolerance specified by ABSTOL and RELTOL. ! ! (b) finding an interval (a',b'] containing eigenvalues w(f),...,w(l). ! In this case, start with a Gershgorin interval (a,b). Set up ! AB to contain 2 search intervals, both initially (a,b). One ! NVAL element should contain f-1 and the other should contain l ! , while C should contain a and b, resp. NAB(i,1) should be -1 ! and NAB(i,2) should be N+1, to flag an error if the desired ! interval does not lie in (a,b). DLAEBZ is then called with ! IJOB=3. On exit, if w(f-1) < w(f), then one of the intervals -- ! j -- will have AB(j,1)=AB(j,2) and NAB(j,1)=NAB(j,2)=f-1, while ! if, to the specified tolerance, w(f-k)=...=w(f+r), k > 0 and r ! >= 0, then the interval will have N(AB(j,1))=NAB(j,1)=f-k and ! N(AB(j,2))=NAB(j,2)=f+r. The cases w(l) < w(l+1) and ! w(l-r)=...=w(l+k) are handled similarly. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, TWO, HALF PARAMETER ( ZERO = 0.0D0, TWO = 2.0D0, & HALF = 1.0D0 / TWO ) ! .. ! .. Local Scalars .. INTEGER ITMP1, ITMP2, J, JI, JIT, JP, KF, KFNEW, KL, & KLNEW DOUBLE PRECISION TMP1, TMP2 ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Executable Statements .. ! ! Check for Errors ! INFO = 0 IF( IJOB.LT.1 .OR. IJOB.GT.3 ) THEN INFO = -1 RETURN END IF ! ! Initialize NAB ! IF( IJOB.EQ.1 ) THEN ! ! Compute the number of eigenvalues in the initial intervals. ! MOUT = 0 !DIR$ NOVECTOR DO 30 JI = 1, MINP DO 20 JP = 1, 2 TMP1 = D( 1 ) - AB( JI, JP ) IF( ABS( TMP1 ).LT.PIVMIN ) & TMP1 = -PIVMIN NAB( JI, JP ) = 0 IF( TMP1.LE.ZERO ) & NAB( JI, JP ) = 1 ! DO 10 J = 2, N TMP1 = D( J ) - E2( J-1 ) / TMP1 - AB( JI, JP ) IF( ABS( TMP1 ).LT.PIVMIN ) & TMP1 = -PIVMIN IF( TMP1.LE.ZERO ) & NAB( JI, JP ) = NAB( JI, JP ) + 1 10 CONTINUE 20 CONTINUE MOUT = MOUT + NAB( JI, 2 ) - NAB( JI, 1 ) 30 CONTINUE RETURN END IF ! ! Initialize for loop ! ! KF and KL have the following meaning: ! Intervals 1,...,KF-1 have converged. ! Intervals KF,...,KL still need to be refined. ! KF = 1 KL = MINP ! ! If IJOB=2, initialize C. ! If IJOB=3, use the user-supplied starting point. ! IF( IJOB.EQ.2 ) THEN DO 40 JI = 1, MINP C( JI ) = HALF*( AB( JI, 1 )+AB( JI, 2 ) ) 40 CONTINUE END IF ! ! Iteration loop ! DO 130 JIT = 1, NITMAX ! ! Loop over intervals ! IF( KL-KF+1.GE.NBMIN .AND. NBMIN.GT.0 ) THEN ! ! Begin of Parallel Version of the loop ! DO 60 JI = KF, KL ! ! Compute N(c), the number of eigenvalues less than c ! WORK( JI ) = D( 1 ) - C( JI ) IWORK( JI ) = 0 IF( WORK( JI ).LE.PIVMIN ) THEN IWORK( JI ) = 1 WORK( JI ) = MIN( WORK( JI ), -PIVMIN ) END IF ! DO 50 J = 2, N WORK( JI ) = D( J ) - E2( J-1 ) / WORK( JI ) - C( JI ) IF( WORK( JI ).LE.PIVMIN ) THEN IWORK( JI ) = IWORK( JI ) + 1 WORK( JI ) = MIN( WORK( JI ), -PIVMIN ) END IF 50 CONTINUE 60 CONTINUE ! IF( IJOB.LE.2 ) THEN ! ! IJOB=2: Choose all intervals containing eigenvalues. ! KLNEW = KL DO 70 JI = KF, KL ! ! Insure that N(w) is monotone ! IWORK( JI ) = MIN( NAB( JI, 2 ), & MAX( NAB( JI, 1 ), IWORK( JI ) ) ) ! ! Update the Queue -- add intervals if both halves ! contain eigenvalues. ! IF( IWORK( JI ).EQ.NAB( JI, 2 ) ) THEN ! ! No eigenvalue in the upper interval: ! just use the lower interval. ! AB( JI, 2 ) = C( JI ) ! ELSE IF( IWORK( JI ).EQ.NAB( JI, 1 ) ) THEN ! ! No eigenvalue in the lower interval: ! just use the upper interval. ! AB( JI, 1 ) = C( JI ) ELSE KLNEW = KLNEW + 1 IF( KLNEW.LE.MMAX ) THEN ! ! Eigenvalue in both intervals -- add upper to ! queue. ! AB( KLNEW, 2 ) = AB( JI, 2 ) NAB( KLNEW, 2 ) = NAB( JI, 2 ) AB( KLNEW, 1 ) = C( JI ) NAB( KLNEW, 1 ) = IWORK( JI ) AB( JI, 2 ) = C( JI ) NAB( JI, 2 ) = IWORK( JI ) ELSE INFO = MMAX + 1 END IF END IF 70 CONTINUE IF( INFO.NE.0 ) & RETURN KL = KLNEW ELSE ! ! IJOB=3: Binary search. Keep only the interval containing ! w s.t. N(w) = NVAL ! DO 80 JI = KF, KL IF( IWORK( JI ).LE.NVAL( JI ) ) THEN AB( JI, 1 ) = C( JI ) NAB( JI, 1 ) = IWORK( JI ) END IF IF( IWORK( JI ).GE.NVAL( JI ) ) THEN AB( JI, 2 ) = C( JI ) NAB( JI, 2 ) = IWORK( JI ) END IF 80 CONTINUE END IF ! ELSE ! ! End of Parallel Version of the loop ! ! Begin of Serial Version of the loop ! KLNEW = KL DO 100 JI = KF, KL ! ! Compute N(w), the number of eigenvalues less than w ! TMP1 = C( JI ) TMP2 = D( 1 ) - TMP1 ITMP1 = 0 IF( TMP2.LE.PIVMIN ) THEN ITMP1 = 1 TMP2 = MIN( TMP2, -PIVMIN ) END IF ! ! A series of compiler directives to defeat vectorization ! for the next loop ! !$PL$ CMCHAR=' ' !DIR$ NEXTSCALAR !$DIR SCALAR !DIR$ NEXT SCALAR !VD$L NOVECTOR !DEC$ NOVECTOR !VD$ NOVECTOR !VDIR NOVECTOR !VOCL LOOP,SCALAR !IBM PREFER SCALAR !$PL$ CMCHAR='*' ! DO 90 J = 2, N TMP2 = D( J ) - E2( J-1 ) / TMP2 - TMP1 IF( TMP2.LE.PIVMIN ) THEN ITMP1 = ITMP1 + 1 TMP2 = MIN( TMP2, -PIVMIN ) END IF 90 CONTINUE ! IF( IJOB.LE.2 ) THEN ! ! IJOB=2: Choose all intervals containing eigenvalues. ! ! Insure that N(w) is monotone ! ITMP1 = MIN( NAB( JI, 2 ), & MAX( NAB( JI, 1 ), ITMP1 ) ) ! ! Update the Queue -- add intervals if both halves ! contain eigenvalues. ! IF( ITMP1.EQ.NAB( JI, 2 ) ) THEN ! ! No eigenvalue in the upper interval: ! just use the lower interval. ! AB( JI, 2 ) = TMP1 ! ELSE IF( ITMP1.EQ.NAB( JI, 1 ) ) THEN ! ! No eigenvalue in the lower interval: ! just use the upper interval. ! AB( JI, 1 ) = TMP1 ELSE IF( KLNEW.LT.MMAX ) THEN ! ! Eigenvalue in both intervals -- add upper to queue. ! KLNEW = KLNEW + 1 AB( KLNEW, 2 ) = AB( JI, 2 ) NAB( KLNEW, 2 ) = NAB( JI, 2 ) AB( KLNEW, 1 ) = TMP1 NAB( KLNEW, 1 ) = ITMP1 AB( JI, 2 ) = TMP1 NAB( JI, 2 ) = ITMP1 ELSE INFO = MMAX + 1 RETURN END IF ELSE ! ! IJOB=3: Binary search. Keep only the interval ! containing w s.t. N(w) = NVAL ! IF( ITMP1.LE.NVAL( JI ) ) THEN AB( JI, 1 ) = TMP1 NAB( JI, 1 ) = ITMP1 END IF IF( ITMP1.GE.NVAL( JI ) ) THEN AB( JI, 2 ) = TMP1 NAB( JI, 2 ) = ITMP1 END IF END IF 100 CONTINUE KL = KLNEW ! ! End of Serial Version of the loop ! END IF ! ! Check for convergence ! KFNEW = KF DO 110 JI = KF, KL TMP1 = ABS( AB( JI, 2 )-AB( JI, 1 ) ) TMP2 = MAX( ABS( AB( JI, 2 ) ), ABS( AB( JI, 1 ) ) ) IF( TMP1.LT.MAX( ABSTOL, PIVMIN, RELTOL*TMP2 ) .OR. & NAB( JI, 1 ).GE.NAB( JI, 2 ) ) THEN ! ! Converged -- Swap with position KFNEW, ! then increment KFNEW ! IF( JI.GT.KFNEW ) THEN TMP1 = AB( JI, 1 ) TMP2 = AB( JI, 2 ) ITMP1 = NAB( JI, 1 ) ITMP2 = NAB( JI, 2 ) AB( JI, 1 ) = AB( KFNEW, 1 ) AB( JI, 2 ) = AB( KFNEW, 2 ) NAB( JI, 1 ) = NAB( KFNEW, 1 ) NAB( JI, 2 ) = NAB( KFNEW, 2 ) AB( KFNEW, 1 ) = TMP1 AB( KFNEW, 2 ) = TMP2 NAB( KFNEW, 1 ) = ITMP1 NAB( KFNEW, 2 ) = ITMP2 IF( IJOB.EQ.3 ) THEN ITMP1 = NVAL( JI ) NVAL( JI ) = NVAL( KFNEW ) NVAL( KFNEW ) = ITMP1 END IF END IF KFNEW = KFNEW + 1 END IF 110 CONTINUE KF = KFNEW ! ! Choose Midpoints ! DO 120 JI = KF, KL C( JI ) = HALF*( AB( JI, 1 )+AB( JI, 2 ) ) 120 CONTINUE ! ! If no more intervals to refine, quit. ! IF( KF.GT.KL ) & GO TO 140 130 CONTINUE ! ! Converged ! 140 CONTINUE INFO = MAX( KL+1-KF, 0 ) MOUT = KL ! RETURN ! ! End of DLAEBZ ! END SUBROUTINE DLAED0( ICOMPQ, QSIZ, N, D, E, Q, LDQ, QSTORE, LDQS, & WORK, IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER ICOMPQ, INFO, LDQ, LDQS, N, QSIZ ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION D( * ), E( * ), Q( LDQ, * ), QSTORE( LDQS, * ), & WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLAED0 computes all eigenvalues and corresponding eigenvectors of a ! symmetric tridiagonal matrix using the divide and conquer method. ! ! Arguments ! ========= ! ! ICOMPQ (input) INTEGER ! = 0: Compute eigenvalues only. ! = 1: Compute eigenvectors of original dense symmetric matrix ! also. On entry, Q contains the orthogonal matrix used ! to reduce the original matrix to tridiagonal form. ! = 2: Compute eigenvalues and eigenvectors of tridiagonal ! matrix. ! ! QSIZ (input) INTEGER ! The dimension of the orthogonal matrix used to reduce ! the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1. ! ! N (input) INTEGER ! The dimension of the symmetric tridiagonal matrix. N >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the main diagonal of the tridiagonal matrix. ! On exit, its eigenvalues. ! ! E (input) DOUBLE PRECISION array, dimension (N-1) ! The off-diagonal elements of the tridiagonal matrix. ! On exit, E has been destroyed. ! ! Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N) ! On entry, Q must contain an N-by-N orthogonal matrix. ! If ICOMPQ = 0 Q is not referenced. ! If ICOMPQ = 1 On entry, Q is a subset of the columns of the ! orthogonal matrix used to reduce the full ! matrix to tridiagonal form corresponding to ! the subset of the full matrix which is being ! decomposed at this time. ! If ICOMPQ = 2 On entry, Q will be the identity matrix. ! On exit, Q contains the eigenvectors of the ! tridiagonal matrix. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. If eigenvectors are ! desired, then LDQ >= max(1,N). In any case, LDQ >= 1. ! ! QSTORE (workspace) DOUBLE PRECISION array, dimension (LDQS, N) ! Referenced only when ICOMPQ = 1. Used to store parts of ! the eigenvector matrix when the updating matrix multiplies ! take place. ! ! LDQS (input) INTEGER ! The leading dimension of the array QSTORE. If ICOMPQ = 1, ! then LDQS >= max(1,N). In any case, LDQS >= 1. ! ! WORK (workspace) DOUBLE PRECISION array, ! If ICOMPQ = 0 or 1, the dimension of WORK must be at least ! 1 + 3*N + 2*N*lg N + 2*N**2 ! ( lg( N ) = smallest integer k ! such that 2^k >= N ) ! If ICOMPQ = 2, the dimension of WORK must be at least ! 4*N + N**2. ! ! IWORK (workspace) INTEGER array, ! If ICOMPQ = 0 or 1, the dimension of IWORK must be at least ! 6 + 6*N + 5*N*lg N. ! ( lg( N ) = smallest integer k ! such that 2^k >= N ) ! If ICOMPQ = 2, the dimension of IWORK must be at least ! 3 + 5*N. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: The algorithm failed to compute an eigenvalue while ! working on the submatrix lying in rows and columns ! INFO/(N+1) through mod(INFO,N+1). ! ! Further Details ! =============== ! ! Based on contributions by ! Jeff Rutter, Computer Science Division, University of California ! at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO PARAMETER ( ZERO = 0.D0, ONE = 1.D0, TWO = 2.D0 ) ! .. ! .. Local Scalars .. INTEGER CURLVL, CURPRB, CURR, I, IGIVCL, IGIVNM, & IGIVPT, INDXQ, IPERM, IPRMPT, IQ, IQPTR, IWREM, & J, K, LGN, MATSIZ, MSD2, SMLSIZ, SMM1, SPM1, & SPM2, SUBMAT, SUBPBS, TLVLS DOUBLE PRECISION TEMP ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEMM, DLACPY, DLAED1, DLAED7, DSTEQR, & XERBLA ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, INT, LOG, MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.2 ) THEN INFO = -1 ELSE IF( ( ICOMPQ.EQ.1 ) .AND. ( QSIZ.LT.MAX( 0, N ) ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDQS.LT.MAX( 1, N ) ) THEN INFO = -9 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLAED0', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! SMLSIZ = ILAENV( 9, 'DLAED0', ' ', 0, 0, 0, 0 ) ! ! Determine the size and placement of the submatrices, and save in ! the leading elements of IWORK. ! IWORK( 1 ) = N SUBPBS = 1 TLVLS = 0 10 CONTINUE IF( IWORK( SUBPBS ).GT.SMLSIZ ) THEN DO 20 J = SUBPBS, 1, -1 IWORK( 2*J ) = ( IWORK( J )+1 ) / 2 IWORK( 2*J-1 ) = IWORK( J ) / 2 20 CONTINUE TLVLS = TLVLS + 1 SUBPBS = 2*SUBPBS GO TO 10 END IF DO 30 J = 2, SUBPBS IWORK( J ) = IWORK( J ) + IWORK( J-1 ) 30 CONTINUE ! ! Divide the matrix into SUBPBS submatrices of size at most SMLSIZ+1 ! using rank-1 modifications (cuts). ! SPM1 = SUBPBS - 1 DO 40 I = 1, SPM1 SUBMAT = IWORK( I ) + 1 SMM1 = SUBMAT - 1 D( SMM1 ) = D( SMM1 ) - ABS( E( SMM1 ) ) D( SUBMAT ) = D( SUBMAT ) - ABS( E( SMM1 ) ) 40 CONTINUE ! INDXQ = 4*N + 3 IF( ICOMPQ.NE.2 ) THEN ! ! Set up workspaces for eigenvalues only/accumulate new vectors ! routine ! TEMP = LOG( DBLE( N ) ) / LOG( TWO ) LGN = INT( TEMP ) IF( 2**LGN.LT.N ) & LGN = LGN + 1 IF( 2**LGN.LT.N ) & LGN = LGN + 1 IPRMPT = INDXQ + N + 1 IPERM = IPRMPT + N*LGN IQPTR = IPERM + N*LGN IGIVPT = IQPTR + N + 2 IGIVCL = IGIVPT + N*LGN ! IGIVNM = 1 IQ = IGIVNM + 2*N*LGN IWREM = IQ + N**2 + 1 ! ! Initialize pointers ! DO 50 I = 0, SUBPBS IWORK( IPRMPT+I ) = 1 IWORK( IGIVPT+I ) = 1 50 CONTINUE IWORK( IQPTR ) = 1 END IF ! ! Solve each submatrix eigenproblem at the bottom of the divide and ! conquer tree. ! CURR = 0 DO 70 I = 0, SPM1 IF( I.EQ.0 ) THEN SUBMAT = 1 MATSIZ = IWORK( 1 ) ELSE SUBMAT = IWORK( I ) + 1 MATSIZ = IWORK( I+1 ) - IWORK( I ) END IF IF( ICOMPQ.EQ.2 ) THEN CALL DSTEQR( 'I', MATSIZ, D( SUBMAT ), E( SUBMAT ), & Q( SUBMAT, SUBMAT ), LDQ, WORK, INFO ) IF( INFO.NE.0 ) & GO TO 130 ELSE CALL DSTEQR( 'I', MATSIZ, D( SUBMAT ), E( SUBMAT ), & WORK( IQ-1+IWORK( IQPTR+CURR ) ), MATSIZ, WORK, & INFO ) IF( INFO.NE.0 ) & GO TO 130 IF( ICOMPQ.EQ.1 ) THEN CALL DGEMM( 'N', 'N', QSIZ, MATSIZ, MATSIZ, ONE, & Q( 1, SUBMAT ), LDQ, WORK( IQ-1+IWORK( IQPTR+ & CURR ) ), MATSIZ, ZERO, QSTORE( 1, SUBMAT ), & LDQS ) END IF IWORK( IQPTR+CURR+1 ) = IWORK( IQPTR+CURR ) + MATSIZ**2 CURR = CURR + 1 END IF K = 1 DO 60 J = SUBMAT, IWORK( I+1 ) IWORK( INDXQ+J ) = K K = K + 1 60 CONTINUE 70 CONTINUE ! ! Successively merge eigensystems of adjacent submatrices ! into eigensystem for the corresponding larger matrix. ! ! while ( SUBPBS > 1 ) ! CURLVL = 1 80 CONTINUE IF( SUBPBS.GT.1 ) THEN SPM2 = SUBPBS - 2 DO 90 I = 0, SPM2, 2 IF( I.EQ.0 ) THEN SUBMAT = 1 MATSIZ = IWORK( 2 ) MSD2 = IWORK( 1 ) CURPRB = 0 ELSE SUBMAT = IWORK( I ) + 1 MATSIZ = IWORK( I+2 ) - IWORK( I ) MSD2 = MATSIZ / 2 CURPRB = CURPRB + 1 END IF ! ! Merge lower order eigensystems (of size MSD2 and MATSIZ - MSD2) ! into an eigensystem of size MATSIZ. ! DLAED1 is used only for the full eigensystem of a tridiagonal ! matrix. ! DLAED7 handles the cases in which eigenvalues only or eigenvalues ! and eigenvectors of a full symmetric matrix (which was reduced to ! tridiagonal form) are desired. ! IF( ICOMPQ.EQ.2 ) THEN CALL DLAED1( MATSIZ, D( SUBMAT ), Q( SUBMAT, SUBMAT ), & LDQ, IWORK( INDXQ+SUBMAT ), & E( SUBMAT+MSD2-1 ), MSD2, WORK, & IWORK( SUBPBS+1 ), INFO ) ELSE CALL DLAED7( ICOMPQ, MATSIZ, QSIZ, TLVLS, CURLVL, CURPRB, & D( SUBMAT ), QSTORE( 1, SUBMAT ), LDQS, & IWORK( INDXQ+SUBMAT ), E( SUBMAT+MSD2-1 ), & MSD2, WORK( IQ ), IWORK( IQPTR ), & IWORK( IPRMPT ), IWORK( IPERM ), & IWORK( IGIVPT ), IWORK( IGIVCL ), & WORK( IGIVNM ), WORK( IWREM ), & IWORK( SUBPBS+1 ), INFO ) END IF IF( INFO.NE.0 ) & GO TO 130 IWORK( I / 2+1 ) = IWORK( I+2 ) 90 CONTINUE SUBPBS = SUBPBS / 2 CURLVL = CURLVL + 1 GO TO 80 END IF ! ! end while ! ! Re-merge the eigenvalues/vectors which were deflated at the final ! merge step. ! IF( ICOMPQ.EQ.1 ) THEN DO 100 I = 1, N J = IWORK( INDXQ+I ) WORK( I ) = D( J ) CALL DCOPY( QSIZ, QSTORE( 1, J ), 1, Q( 1, I ), 1 ) 100 CONTINUE CALL DCOPY( N, WORK, 1, D, 1 ) ELSE IF( ICOMPQ.EQ.2 ) THEN DO 110 I = 1, N J = IWORK( INDXQ+I ) WORK( I ) = D( J ) CALL DCOPY( N, Q( 1, J ), 1, WORK( N*I+1 ), 1 ) 110 CONTINUE CALL DCOPY( N, WORK, 1, D, 1 ) CALL DLACPY( 'A', N, N, WORK( N+1 ), N, Q, LDQ ) ELSE DO 120 I = 1, N J = IWORK( INDXQ+I ) WORK( I ) = D( J ) 120 CONTINUE CALL DCOPY( N, WORK, 1, D, 1 ) END IF GO TO 140 ! 130 CONTINUE INFO = SUBMAT*( N+1 ) + SUBMAT + MATSIZ - 1 ! 140 CONTINUE RETURN ! ! End of DLAED0 ! END SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER CUTPNT, INFO, LDQ, N DOUBLE PRECISION RHO ! .. ! .. Array Arguments .. INTEGER INDXQ( * ), IWORK( * ) DOUBLE PRECISION D( * ), Q( LDQ, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLAED1 computes the updated eigensystem of a diagonal ! matrix after modification by a rank-one symmetric matrix. This ! routine is used only for the eigenproblem which requires all ! eigenvalues and eigenvectors of a tridiagonal matrix. DLAED7 handles ! the case in which eigenvalues only or eigenvalues and eigenvectors ! of a full symmetric matrix (which was reduced to tridiagonal form) ! are desired. ! ! T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out) ! ! where Z = Q'u, u is a vector of length N with ones in the ! CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. ! ! The eigenvectors of the original matrix are stored in Q, and the ! eigenvalues are in D. The algorithm consists of three stages: ! ! The first stage consists of deflating the size of the problem ! when there are multiple eigenvalues or if there is a zero in ! the Z vector. For each such occurence the dimension of the ! secular equation problem is reduced by one. This stage is ! performed by the routine DLAED2. ! ! The second stage consists of calculating the updated ! eigenvalues. This is done by finding the roots of the secular ! equation via the routine DLAED4 (as called by DLAED3). ! This routine also calculates the eigenvectors of the current ! problem. ! ! The final stage consists of computing the updated eigenvectors ! directly using the updated eigenvalues. The eigenvectors for ! the current problem are multiplied with the eigenvectors from ! the overall problem. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The dimension of the symmetric tridiagonal matrix. N >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the eigenvalues of the rank-1-perturbed matrix. ! On exit, the eigenvalues of the repaired matrix. ! ! Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) ! On entry, the eigenvectors of the rank-1-perturbed matrix. ! On exit, the eigenvectors of the repaired tridiagonal matrix. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. LDQ >= max(1,N). ! ! INDXQ (input/output) INTEGER array, dimension (N) ! On entry, the permutation which separately sorts the two ! subproblems in D into ascending order. ! On exit, the permutation which will reintegrate the ! subproblems back into sorted order, ! i.e. D( INDXQ( I = 1, N ) ) will be in ascending order. ! ! RHO (input) DOUBLE PRECISION ! The subdiagonal entry used to create the rank-1 modification. ! ! CUTPNT (input) INTEGER ! The location of the last eigenvalue in the leading sub-matrix. ! min(1,N) <= CUTPNT <= N/2. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (4*N + N**2) ! ! IWORK (workspace) INTEGER array, dimension (4*N) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = 1, an eigenvalue did not converge ! ! Further Details ! =============== ! ! Based on contributions by ! Jeff Rutter, Computer Science Division, University of California ! at Berkeley, USA ! Modified by Francoise Tisseur, University of Tennessee. ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER COLTYP, I, IDLMDA, INDX, INDXC, INDXP, IQ2, IS, & IW, IZ, K, N1, N2, ZPP1 ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLAED2, DLAED3, DLAMRG, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( N.LT.0 ) THEN INFO = -1 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( MIN( 1, N / 2 ).GT.CUTPNT .OR. ( N / 2 ).LT.CUTPNT ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLAED1', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! The following values are integer pointers which indicate ! the portion of the workspace ! used by a particular array in DLAED2 and DLAED3. ! IZ = 1 IDLMDA = IZ + N IW = IDLMDA + N IQ2 = IW + N ! INDX = 1 INDXC = INDX + N COLTYP = INDXC + N INDXP = COLTYP + N ! ! ! Form the z-vector which consists of the last row of Q_1 and the ! first row of Q_2. ! CALL DCOPY( CUTPNT, Q( CUTPNT, 1 ), LDQ, WORK( IZ ), 1 ) ZPP1 = CUTPNT + 1 CALL DCOPY( N-CUTPNT, Q( ZPP1, ZPP1 ), LDQ, WORK( IZ+CUTPNT ), 1 ) ! ! Deflate eigenvalues. ! CALL DLAED2( K, N, CUTPNT, D, Q, LDQ, INDXQ, RHO, WORK( IZ ), & WORK( IDLMDA ), WORK( IW ), WORK( IQ2 ), & IWORK( INDX ), IWORK( INDXC ), IWORK( INDXP ), & IWORK( COLTYP ), INFO ) ! IF( INFO.NE.0 ) & GO TO 20 ! ! Solve Secular Equation. ! IF( K.NE.0 ) THEN IS = ( IWORK( COLTYP )+IWORK( COLTYP+1 ) )*CUTPNT + & ( IWORK( COLTYP+1 )+IWORK( COLTYP+2 ) )*( N-CUTPNT ) + IQ2 CALL DLAED3( K, N, CUTPNT, D, Q, LDQ, RHO, WORK( IDLMDA ), & WORK( IQ2 ), IWORK( INDXC ), IWORK( COLTYP ), & WORK( IW ), WORK( IS ), INFO ) IF( INFO.NE.0 ) & GO TO 20 ! ! Prepare the INDXQ sorting permutation. ! N1 = K N2 = N - K CALL DLAMRG( N1, N2, D, 1, -1, INDXQ ) ELSE DO 10 I = 1, N INDXQ( I ) = I 10 CONTINUE END IF ! 20 CONTINUE RETURN ! ! End of DLAED1 ! END SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W, & Q2, INDX, INDXC, INDXP, COLTYP, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, K, LDQ, N, N1 DOUBLE PRECISION RHO ! .. ! .. Array Arguments .. INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ), & INDXQ( * ) DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ), & W( * ), Z( * ) ! .. ! ! Purpose ! ======= ! ! DLAED2 merges the two sets of eigenvalues together into a single ! sorted set. Then it tries to deflate the size of the problem. ! There are two ways in which deflation can occur: when two or more ! eigenvalues are close together or if there is a tiny entry in the ! Z vector. For each such occurrence the order of the related secular ! equation problem is reduced by one. ! ! Arguments ! ========= ! ! K (output) INTEGER ! The number of non-deflated eigenvalues, and the order of the ! related secular equation. 0 <= K <=N. ! ! N (input) INTEGER ! The dimension of the symmetric tridiagonal matrix. N >= 0. ! ! N1 (input) INTEGER ! The location of the last eigenvalue in the leading sub-matrix. ! min(1,N) <= N1 <= N/2. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, D contains the eigenvalues of the two submatrices to ! be combined. ! On exit, D contains the trailing (N-K) updated eigenvalues ! (those which were deflated) sorted into increasing order. ! ! Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N) ! On entry, Q contains the eigenvectors of two submatrices in ! the two square blocks with corners at (1,1), (N1,N1) ! and (N1+1, N1+1), (N,N). ! On exit, Q contains the trailing (N-K) updated eigenvectors ! (those which were deflated) in its last N-K columns. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. LDQ >= max(1,N). ! ! INDXQ (input/output) INTEGER array, dimension (N) ! The permutation which separately sorts the two sub-problems ! in D into ascending order. Note that elements in the second ! half of this permutation must first have N1 added to their ! values. Destroyed on exit. ! ! RHO (input/output) DOUBLE PRECISION ! On entry, the off-diagonal element associated with the rank-1 ! cut which originally split the two submatrices which are now ! being recombined. ! On exit, RHO has been modified to the value required by ! DLAED3. ! ! Z (input) DOUBLE PRECISION array, dimension (N) ! On entry, Z contains the updating vector (the last ! row of the first sub-eigenvector matrix and the first row of ! the second sub-eigenvector matrix). ! On exit, the contents of Z have been destroyed by the updating ! process. ! ! DLAMDA (output) DOUBLE PRECISION array, dimension (N) ! A copy of the first K eigenvalues which will be used by ! DLAED3 to form the secular equation. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! The first k values of the final deflation-altered z-vector ! which will be passed to DLAED3. ! ! Q2 (output) DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2) ! A copy of the first K eigenvectors which will be used by ! DLAED3 in a matrix multiply (DGEMM) to solve for the new ! eigenvectors. ! ! INDX (workspace) INTEGER array, dimension (N) ! The permutation used to sort the contents of DLAMDA into ! ascending order. ! ! INDXC (output) INTEGER array, dimension (N) ! The permutation used to arrange the columns of the deflated ! Q matrix into three groups: the first group contains non-zero ! elements only at and above N1, the second contains ! non-zero elements only below N1, and the third is dense. ! ! INDXP (workspace) INTEGER array, dimension (N) ! The permutation used to place deflated values of D at the end ! of the array. INDXP(1:K) points to the nondeflated D-values ! and INDXP(K+1:N) points to the deflated eigenvalues. ! ! COLTYP (workspace/output) INTEGER array, dimension (N) ! During execution, a label which will indicate which of the ! following types a column in the Q2 matrix is: ! 1 : non-zero in the upper half only; ! 2 : dense; ! 3 : non-zero in the lower half only; ! 4 : deflated. ! On exit, COLTYP(i) is the number of columns of type i, ! for i=1 to 4 only. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! Based on contributions by ! Jeff Rutter, Computer Science Division, University of California ! at Berkeley, USA ! Modified by Francoise Tisseur, University of Tennessee. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION MONE, ZERO, ONE, TWO, EIGHT PARAMETER ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0, & TWO = 2.0D0, EIGHT = 8.0D0 ) ! .. ! .. Local Arrays .. INTEGER CTOT( 4 ), PSM( 4 ) ! .. ! .. Local Scalars .. INTEGER CT, I, IMAX, IQ1, IQ2, J, JMAX, JS, K2, N1P1, & N2, NJ, PJ DOUBLE PRECISION C, EPS, S, T, TAU, TOL ! .. ! .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DLAMCH, DLAPY2 EXTERNAL IDAMAX, DLAMCH, DLAPY2 ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLACPY, DLAMRG, DROT, DSCAL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( MIN( 1, ( N / 2 ) ).GT.N1 .OR. ( N / 2 ).LT.N1 ) THEN INFO = -3 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLAED2', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! N2 = N - N1 N1P1 = N1 + 1 ! IF( RHO.LT.ZERO ) THEN CALL DSCAL( N2, MONE, Z( N1P1 ), 1 ) END IF ! ! Normalize z so that norm(z) = 1. Since z is the concatenation of ! two normalized vectors, norm2(z) = sqrt(2). ! T = ONE / SQRT( TWO ) CALL DSCAL( N, T, Z, 1 ) ! ! RHO = ABS( norm(z)**2 * RHO ) ! RHO = ABS( TWO*RHO ) ! ! Sort the eigenvalues into increasing order ! DO 10 I = N1P1, N INDXQ( I ) = INDXQ( I ) + N1 10 CONTINUE ! ! re-integrate the deflated parts from the last pass ! DO 20 I = 1, N DLAMDA( I ) = D( INDXQ( I ) ) 20 CONTINUE CALL DLAMRG( N1, N2, DLAMDA, 1, 1, INDXC ) DO 30 I = 1, N INDX( I ) = INDXQ( INDXC( I ) ) 30 CONTINUE ! ! Calculate the allowable deflation tolerance ! IMAX = IDAMAX( N, Z, 1 ) JMAX = IDAMAX( N, D, 1 ) EPS = DLAMCH( 'Epsilon' ) TOL = EIGHT*EPS*MAX( ABS( D( JMAX ) ), ABS( Z( IMAX ) ) ) ! ! If the rank-1 modifier is small enough, no more needs to be done ! except to reorganize Q so that its columns correspond with the ! elements in D. ! IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN K = 0 IQ2 = 1 DO 40 J = 1, N I = INDX( J ) CALL DCOPY( N, Q( 1, I ), 1, Q2( IQ2 ), 1 ) DLAMDA( J ) = D( I ) IQ2 = IQ2 + N 40 CONTINUE CALL DLACPY( 'A', N, N, Q2, N, Q, LDQ ) CALL DCOPY( N, DLAMDA, 1, D, 1 ) GO TO 190 END IF ! ! If there are multiple eigenvalues then the problem deflates. Here ! the number of equal eigenvalues are found. As each equal ! eigenvalue is found, an elementary reflector is computed to rotate ! the corresponding eigensubspace so that the corresponding ! components of Z are zero in this new basis. ! DO 50 I = 1, N1 COLTYP( I ) = 1 50 CONTINUE DO 60 I = N1P1, N COLTYP( I ) = 3 60 CONTINUE ! ! K = 0 K2 = N + 1 DO 70 J = 1, N NJ = INDX( J ) IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN ! ! Deflate due to small z component. ! K2 = K2 - 1 COLTYP( NJ ) = 4 INDXP( K2 ) = NJ IF( J.EQ.N ) & GO TO 100 ELSE PJ = NJ GO TO 80 END IF 70 CONTINUE 80 CONTINUE J = J + 1 NJ = INDX( J ) IF( J.GT.N ) & GO TO 100 IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN ! ! Deflate due to small z component. ! K2 = K2 - 1 COLTYP( NJ ) = 4 INDXP( K2 ) = NJ ELSE ! ! Check if eigenvalues are close enough to allow deflation. ! S = Z( PJ ) C = Z( NJ ) ! ! Find sqrt(a**2+b**2) without overflow or ! destructive underflow. ! TAU = DLAPY2( C, S ) T = D( NJ ) - D( PJ ) C = C / TAU S = -S / TAU IF( ABS( T*C*S ).LE.TOL ) THEN ! ! Deflation is possible. ! Z( NJ ) = TAU Z( PJ ) = ZERO IF( COLTYP( NJ ).NE.COLTYP( PJ ) ) & COLTYP( NJ ) = 2 COLTYP( PJ ) = 4 CALL DROT( N, Q( 1, PJ ), 1, Q( 1, NJ ), 1, C, S ) T = D( PJ )*C**2 + D( NJ )*S**2 D( NJ ) = D( PJ )*S**2 + D( NJ )*C**2 D( PJ ) = T K2 = K2 - 1 I = 1 90 CONTINUE IF( K2+I.LE.N ) THEN IF( D( PJ ).LT.D( INDXP( K2+I ) ) ) THEN INDXP( K2+I-1 ) = INDXP( K2+I ) INDXP( K2+I ) = PJ I = I + 1 GO TO 90 ELSE INDXP( K2+I-1 ) = PJ END IF ELSE INDXP( K2+I-1 ) = PJ END IF PJ = NJ ELSE K = K + 1 DLAMDA( K ) = D( PJ ) W( K ) = Z( PJ ) INDXP( K ) = PJ PJ = NJ END IF END IF GO TO 80 100 CONTINUE ! ! Record the last eigenvalue. ! K = K + 1 DLAMDA( K ) = D( PJ ) W( K ) = Z( PJ ) INDXP( K ) = PJ ! ! Count up the total number of the various types of columns, then ! form a permutation which positions the four column types into ! four uniform groups (although one or more of these groups may be ! empty). ! DO 110 J = 1, 4 CTOT( J ) = 0 110 CONTINUE DO 120 J = 1, N CT = COLTYP( J ) CTOT( CT ) = CTOT( CT ) + 1 120 CONTINUE ! ! PSM(*) = Position in SubMatrix (of types 1 through 4) ! PSM( 1 ) = 1 PSM( 2 ) = 1 + CTOT( 1 ) PSM( 3 ) = PSM( 2 ) + CTOT( 2 ) PSM( 4 ) = PSM( 3 ) + CTOT( 3 ) K = N - CTOT( 4 ) ! ! Fill out the INDXC array so that the permutation which it induces ! will place all type-1 columns first, all type-2 columns next, ! then all type-3's, and finally all type-4's. ! DO 130 J = 1, N JS = INDXP( J ) CT = COLTYP( JS ) INDX( PSM( CT ) ) = JS INDXC( PSM( CT ) ) = J PSM( CT ) = PSM( CT ) + 1 130 CONTINUE ! ! Sort the eigenvalues and corresponding eigenvectors into DLAMDA ! and Q2 respectively. The eigenvalues/vectors which were not ! deflated go into the first K slots of DLAMDA and Q2 respectively, ! while those which were deflated go into the last N - K slots. ! I = 1 IQ1 = 1 IQ2 = 1 + ( CTOT( 1 )+CTOT( 2 ) )*N1 DO 140 J = 1, CTOT( 1 ) JS = INDX( I ) CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 ) Z( I ) = D( JS ) I = I + 1 IQ1 = IQ1 + N1 140 CONTINUE ! DO 150 J = 1, CTOT( 2 ) JS = INDX( I ) CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 ) CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 ) Z( I ) = D( JS ) I = I + 1 IQ1 = IQ1 + N1 IQ2 = IQ2 + N2 150 CONTINUE ! DO 160 J = 1, CTOT( 3 ) JS = INDX( I ) CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 ) Z( I ) = D( JS ) I = I + 1 IQ2 = IQ2 + N2 160 CONTINUE ! IQ1 = IQ2 DO 170 J = 1, CTOT( 4 ) JS = INDX( I ) CALL DCOPY( N, Q( 1, JS ), 1, Q2( IQ2 ), 1 ) IQ2 = IQ2 + N Z( I ) = D( JS ) I = I + 1 170 CONTINUE ! ! The deflated eigenvalues and their corresponding vectors go back ! into the last N - K slots of D and Q respectively. ! CALL DLACPY( 'A', N, CTOT( 4 ), Q2( IQ1 ), N, Q( 1, K+1 ), LDQ ) CALL DCOPY( N-K, Z( K+1 ), 1, D( K+1 ), 1 ) ! ! Copy CTOT into COLTYP for referencing in DLAED3. ! DO 180 J = 1, 4 COLTYP( J ) = CTOT( J ) 180 CONTINUE ! 190 CONTINUE RETURN ! ! End of DLAED2 ! END SUBROUTINE DLAED3( K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX, & CTOT, W, S, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab, ! Courant Institute, NAG Ltd., and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, K, LDQ, N, N1 DOUBLE PRECISION RHO ! .. ! .. Array Arguments .. INTEGER CTOT( * ), INDX( * ) DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ), & S( * ), W( * ) ! .. ! ! Purpose ! ======= ! ! DLAED3 finds the roots of the secular equation, as defined by the ! values in D, W, and RHO, between 1 and K. It makes the ! appropriate calls to DLAED4 and then updates the eigenvectors by ! multiplying the matrix of eigenvectors of the pair of eigensystems ! being combined by the matrix of eigenvectors of the K-by-K system ! which is solved here. ! ! This code makes very mild assumptions about floating point ! arithmetic. It will work on machines with a guard digit in ! add/subtract, or on those binary machines without guard digits ! which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. ! It could conceivably fail on hexadecimal or decimal machines ! without guard digits, but we know of none. ! ! Arguments ! ========= ! ! K (input) INTEGER ! The number of terms in the rational function to be solved by ! DLAED4. K >= 0. ! ! N (input) INTEGER ! The number of rows and columns in the Q matrix. ! N >= K (deflation may result in N>K). ! ! N1 (input) INTEGER ! The location of the last eigenvalue in the leading submatrix. ! min(1,N) <= N1 <= N/2. ! ! D (output) DOUBLE PRECISION array, dimension (N) ! D(I) contains the updated eigenvalues for ! 1 <= I <= K. ! ! Q (output) DOUBLE PRECISION array, dimension (LDQ,N) ! Initially the first K columns are used as workspace. ! On output the columns 1 to K contain ! the updated eigenvectors. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. LDQ >= max(1,N). ! ! RHO (input) DOUBLE PRECISION ! The value of the parameter in the rank one update equation. ! RHO >= 0 required. ! ! DLAMDA (input/output) DOUBLE PRECISION array, dimension (K) ! The first K elements of this array contain the old roots ! of the deflated updating problem. These are the poles ! of the secular equation. May be changed on output by ! having lowest order bit set to zero on Cray X-MP, Cray Y-MP, ! Cray-2, or Cray C-90, as described above. ! ! Q2 (input) DOUBLE PRECISION array, dimension (LDQ2, N) ! The first K columns of this matrix contain the non-deflated ! eigenvectors for the split problem. ! ! INDX (input) INTEGER array, dimension (N) ! The permutation used to arrange the columns of the deflated ! Q matrix into three groups (see DLAED2). ! The rows of the eigenvectors found by DLAED4 must be likewise ! permuted before the matrix multiply can take place. ! ! CTOT (input) INTEGER array, dimension (4) ! A count of the total number of the various types of columns ! in Q, as described in INDX. The fourth column type is any ! column which has been deflated. ! ! W (input/output) DOUBLE PRECISION array, dimension (K) ! The first K elements of this array contain the components ! of the deflation-adjusted updating vector. Destroyed on ! output. ! ! S (workspace) DOUBLE PRECISION array, dimension (N1 + 1)*K ! Will contain the eigenvectors of the repaired matrix which ! will be multiplied by the previously accumulated eigenvectors ! to update the system. ! ! LDS (input) INTEGER ! The leading dimension of S. LDS >= max(1,K). ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = 1, an eigenvalue did not converge ! ! Further Details ! =============== ! ! Based on contributions by ! Jeff Rutter, Computer Science Division, University of California ! at Berkeley, USA ! Modified by Francoise Tisseur, University of Tennessee. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 ) ! .. ! .. Local Scalars .. INTEGER I, II, IQ2, J, N12, N2, N23 DOUBLE PRECISION TEMP ! .. ! .. External Functions .. DOUBLE PRECISION DLAMC3, DNRM2 EXTERNAL DLAMC3, DNRM2 ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEMM, DLACPY, DLAED4, DLASET, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, SIGN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( K.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.K ) THEN INFO = -2 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLAED3', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( K.EQ.0 ) & RETURN ! ! Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can ! be computed with high relative accuracy (barring over/underflow). ! This is a problem on machines without a guard digit in ! add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2). ! The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I), ! which on any of these machines zeros out the bottommost ! bit of DLAMDA(I) if it is 1; this makes the subsequent ! subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation ! occurs. On binary machines with a guard digit (almost all ! machines) it does not change DLAMDA(I) at all. On hexadecimal ! and decimal machines with a guard digit, it slightly ! changes the bottommost bits of DLAMDA(I). It does not account ! for hexadecimal or decimal machines without guard digits ! (we know of none). We use a subroutine call to compute ! 2*DLAMBDA(I) to prevent optimizing compilers from eliminating ! this code. ! DO 10 I = 1, K DLAMDA( I ) = DLAMC3( DLAMDA( I ), DLAMDA( I ) ) - DLAMDA( I ) 10 CONTINUE ! DO 20 J = 1, K CALL DLAED4( K, J, DLAMDA, W, Q( 1, J ), RHO, D( J ), INFO ) ! ! If the zero finder fails, the computation is terminated. ! IF( INFO.NE.0 ) & GO TO 120 20 CONTINUE ! IF( K.EQ.1 ) & GO TO 110 IF( K.EQ.2 ) THEN DO 30 J = 1, K W( 1 ) = Q( 1, J ) W( 2 ) = Q( 2, J ) II = INDX( 1 ) Q( 1, J ) = W( II ) II = INDX( 2 ) Q( 2, J ) = W( II ) 30 CONTINUE GO TO 110 END IF ! ! Compute updated W. ! CALL DCOPY( K, W, 1, S, 1 ) ! ! Initialize W(I) = Q(I,I) ! CALL DCOPY( K, Q, LDQ+1, W, 1 ) DO 60 J = 1, K DO 40 I = 1, J - 1 W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) ) 40 CONTINUE DO 50 I = J + 1, K W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) ) 50 CONTINUE 60 CONTINUE DO 70 I = 1, K W( I ) = SIGN( SQRT( -W( I ) ), S( I ) ) 70 CONTINUE ! ! Compute eigenvectors of the modified rank-1 modification. ! DO 100 J = 1, K DO 80 I = 1, K S( I ) = W( I ) / Q( I, J ) 80 CONTINUE TEMP = DNRM2( K, S, 1 ) DO 90 I = 1, K II = INDX( I ) Q( I, J ) = S( II ) / TEMP 90 CONTINUE 100 CONTINUE ! ! Compute the updated eigenvectors. ! 110 CONTINUE ! N2 = N - N1 N12 = CTOT( 1 ) + CTOT( 2 ) N23 = CTOT( 2 ) + CTOT( 3 ) ! CALL DLACPY( 'A', N23, K, Q( CTOT( 1 )+1, 1 ), LDQ, S, N23 ) IQ2 = N1*N12 + 1 IF( N23.NE.0 ) THEN CALL DGEMM( 'N', 'N', N2, K, N23, ONE, Q2( IQ2 ), N2, S, N23, & ZERO, Q( N1+1, 1 ), LDQ ) ELSE CALL DLASET( 'A', N2, K, ZERO, ZERO, Q( N1+1, 1 ), LDQ ) END IF ! CALL DLACPY( 'A', N12, K, Q, LDQ, S, N12 ) IF( N12.NE.0 ) THEN CALL DGEMM( 'N', 'N', N1, K, N12, ONE, Q2, N1, S, N12, ZERO, Q, & LDQ ) ELSE CALL DLASET( 'A', N1, K, ZERO, ZERO, Q( 1, 1 ), LDQ ) END IF ! ! 120 CONTINUE RETURN ! ! End of DLAED3 ! END SUBROUTINE DLAED4( N, I, D, Z, DELTA, RHO, DLAM, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab, ! Courant Institute, NAG Ltd., and Rice University ! December 23, 1999 ! ! .. Scalar Arguments .. INTEGER I, INFO, N DOUBLE PRECISION DLAM, RHO ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ), DELTA( * ), Z( * ) ! .. ! ! Purpose ! ======= ! ! This subroutine computes the I-th updated eigenvalue of a symmetric ! rank-one modification to a diagonal matrix whose elements are ! given in the array d, and that ! ! D(i) < D(j) for i < j ! ! and that RHO > 0. This is arranged by the calling routine, and is ! no loss in generality. The rank-one modified system is thus ! ! diag( D ) + RHO * Z * Z_transpose. ! ! where we assume the Euclidean norm of Z is 1. ! ! The method consists of approximating the rational functions in the ! secular equation by simpler interpolating rational functions. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The length of all arrays. ! ! I (input) INTEGER ! The index of the eigenvalue to be computed. 1 <= I <= N. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The original eigenvalues. It is assumed that they are in ! order, D(I) < D(J) for I < J. ! ! Z (input) DOUBLE PRECISION array, dimension (N) ! The components of the updating vector. ! ! DELTA (output) DOUBLE PRECISION array, dimension (N) ! If N .ne. 1, DELTA contains (D(j) - lambda_I) in its j-th ! component. If N = 1, then DELTA(1) = 1. The vector DELTA ! contains the information necessary to construct the ! eigenvectors. ! ! RHO (input) DOUBLE PRECISION ! The scalar in the symmetric updating formula. ! ! DLAM (output) DOUBLE PRECISION ! The computed lambda_I, the I-th updated eigenvalue. ! ! INFO (output) INTEGER ! = 0: successful exit ! > 0: if INFO = 1, the updating process failed. ! ! Internal Parameters ! =================== ! ! Logical variable ORGATI (origin-at-i?) is used for distinguishing ! whether D(i) or D(i+1) is treated as the origin. ! ! ORGATI = .true. origin at i ! ORGATI = .false. origin at i+1 ! ! Logical variable SWTCH3 (switch-for-3-poles?) is for noting ! if we are working with THREE poles! ! ! MAXIT is the maximum number of iterations allowed for each ! eigenvalue. ! ! Further Details ! =============== ! ! Based on contributions by ! Ren-Cang Li, Computer Science Division, University of California ! at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. INTEGER MAXIT PARAMETER ( MAXIT = 30 ) DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, EIGHT, TEN PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0, & THREE = 3.0D0, FOUR = 4.0D0, EIGHT = 8.0D0, & TEN = 10.0D0 ) ! .. ! .. Local Scalars .. LOGICAL ORGATI, SWTCH, SWTCH3 INTEGER II, IIM1, IIP1, IP1, ITER, J, NITER DOUBLE PRECISION A, B, C, DEL, DLTLB, DLTUB, DPHI, DPSI, DW, & EPS, ERRETM, ETA, MIDPT, PHI, PREW, PSI, & RHOINV, TAU, TEMP, TEMP1, W ! .. ! .. Local Arrays .. DOUBLE PRECISION ZZ( 3 ) ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DLAED5, DLAED6 ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Since this routine is called in an inner loop, we do no argument ! checking. ! ! Quick return for N=1 and 2. ! INFO = 0 IF( N.EQ.1 ) THEN ! ! Presumably, I=1 upon entry ! DLAM = D( 1 ) + RHO*Z( 1 )*Z( 1 ) DELTA( 1 ) = ONE RETURN END IF IF( N.EQ.2 ) THEN CALL DLAED5( I, D, Z, DELTA, RHO, DLAM ) RETURN END IF ! ! Compute machine epsilon ! EPS = DLAMCH( 'Epsilon' ) RHOINV = ONE / RHO ! ! The case I = N ! IF( I.EQ.N ) THEN ! ! Initialize some basic variables ! II = N - 1 NITER = 1 ! ! Calculate initial guess ! MIDPT = RHO / TWO ! ! If ||Z||_2 is not one, then TEMP should be set to ! RHO * ||Z||_2^2 / TWO ! DO 10 J = 1, N DELTA( J ) = ( D( J )-D( I ) ) - MIDPT 10 CONTINUE ! PSI = ZERO DO 20 J = 1, N - 2 PSI = PSI + Z( J )*Z( J ) / DELTA( J ) 20 CONTINUE ! C = RHOINV + PSI W = C + Z( II )*Z( II ) / DELTA( II ) + & Z( N )*Z( N ) / DELTA( N ) ! IF( W.LE.ZERO ) THEN TEMP = Z( N-1 )*Z( N-1 ) / ( D( N )-D( N-1 )+RHO ) + & Z( N )*Z( N ) / RHO IF( C.LE.TEMP ) THEN TAU = RHO ELSE DEL = D( N ) - D( N-1 ) A = -C*DEL + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N ) B = Z( N )*Z( N )*DEL IF( A.LT.ZERO ) THEN TAU = TWO*B / ( SQRT( A*A+FOUR*B*C )-A ) ELSE TAU = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C ) END IF END IF ! ! It can be proved that ! D(N)+RHO/2 <= LAMBDA(N) < D(N)+TAU <= D(N)+RHO ! DLTLB = MIDPT DLTUB = RHO ELSE DEL = D( N ) - D( N-1 ) A = -C*DEL + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N ) B = Z( N )*Z( N )*DEL IF( A.LT.ZERO ) THEN TAU = TWO*B / ( SQRT( A*A+FOUR*B*C )-A ) ELSE TAU = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C ) END IF ! ! It can be proved that ! D(N) < D(N)+TAU < LAMBDA(N) < D(N)+RHO/2 ! DLTLB = ZERO DLTUB = MIDPT END IF ! DO 30 J = 1, N DELTA( J ) = ( D( J )-D( I ) ) - TAU 30 CONTINUE ! ! Evaluate PSI and the derivative DPSI ! DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 40 J = 1, II TEMP = Z( J ) / DELTA( J ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 40 CONTINUE ERRETM = ABS( ERRETM ) ! ! Evaluate PHI and the derivative DPHI ! TEMP = Z( N ) / DELTA( N ) PHI = Z( N )*TEMP DPHI = TEMP*TEMP ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV + & ABS( TAU )*( DPSI+DPHI ) ! W = RHOINV + PHI + PSI ! ! Test for convergence ! IF( ABS( W ).LE.EPS*ERRETM ) THEN DLAM = D( I ) + TAU GO TO 250 END IF ! IF( W.LE.ZERO ) THEN DLTLB = MAX( DLTLB, TAU ) ELSE DLTUB = MIN( DLTUB, TAU ) END IF ! ! Calculate the new step ! NITER = NITER + 1 C = W - DELTA( N-1 )*DPSI - DELTA( N )*DPHI A = ( DELTA( N-1 )+DELTA( N ) )*W - & DELTA( N-1 )*DELTA( N )*( DPSI+DPHI ) B = DELTA( N-1 )*DELTA( N )*W IF( C.LT.ZERO ) & C = ABS( C ) IF( C.EQ.ZERO ) THEN ! ETA = B/A ! ETA = RHO - TAU ETA = DLTUB - TAU ELSE IF( A.GE.ZERO ) THEN ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) ELSE ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) END IF ! ! Note, eta should be positive if w is negative, and ! eta should be negative otherwise. However, ! if for some reason caused by roundoff, eta*w > 0, ! we simply use one Newton step instead. This way ! will guarantee eta*w < 0. ! IF( W*ETA.GT.ZERO ) & ETA = -W / ( DPSI+DPHI ) TEMP = TAU + ETA IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN IF( W.LT.ZERO ) THEN ETA = ( DLTUB-TAU ) / TWO ELSE ETA = ( DLTLB-TAU ) / TWO END IF END IF DO 50 J = 1, N DELTA( J ) = DELTA( J ) - ETA 50 CONTINUE ! TAU = TAU + ETA ! ! Evaluate PSI and the derivative DPSI ! DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 60 J = 1, II TEMP = Z( J ) / DELTA( J ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 60 CONTINUE ERRETM = ABS( ERRETM ) ! ! Evaluate PHI and the derivative DPHI ! TEMP = Z( N ) / DELTA( N ) PHI = Z( N )*TEMP DPHI = TEMP*TEMP ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV + & ABS( TAU )*( DPSI+DPHI ) ! W = RHOINV + PHI + PSI ! ! Main loop to update the values of the array DELTA ! ITER = NITER + 1 ! DO 90 NITER = ITER, MAXIT ! ! Test for convergence ! IF( ABS( W ).LE.EPS*ERRETM ) THEN DLAM = D( I ) + TAU GO TO 250 END IF ! IF( W.LE.ZERO ) THEN DLTLB = MAX( DLTLB, TAU ) ELSE DLTUB = MIN( DLTUB, TAU ) END IF ! ! Calculate the new step ! C = W - DELTA( N-1 )*DPSI - DELTA( N )*DPHI A = ( DELTA( N-1 )+DELTA( N ) )*W - & DELTA( N-1 )*DELTA( N )*( DPSI+DPHI ) B = DELTA( N-1 )*DELTA( N )*W IF( A.GE.ZERO ) THEN ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) ELSE ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) END IF ! ! Note, eta should be positive if w is negative, and ! eta should be negative otherwise. However, ! if for some reason caused by roundoff, eta*w > 0, ! we simply use one Newton step instead. This way ! will guarantee eta*w < 0. ! IF( W*ETA.GT.ZERO ) & ETA = -W / ( DPSI+DPHI ) TEMP = TAU + ETA IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN IF( W.LT.ZERO ) THEN ETA = ( DLTUB-TAU ) / TWO ELSE ETA = ( DLTLB-TAU ) / TWO END IF END IF DO 70 J = 1, N DELTA( J ) = DELTA( J ) - ETA 70 CONTINUE ! TAU = TAU + ETA ! ! Evaluate PSI and the derivative DPSI ! DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 80 J = 1, II TEMP = Z( J ) / DELTA( J ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 80 CONTINUE ERRETM = ABS( ERRETM ) ! ! Evaluate PHI and the derivative DPHI ! TEMP = Z( N ) / DELTA( N ) PHI = Z( N )*TEMP DPHI = TEMP*TEMP ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV + & ABS( TAU )*( DPSI+DPHI ) ! W = RHOINV + PHI + PSI 90 CONTINUE ! ! Return with INFO = 1, NITER = MAXIT and not converged ! INFO = 1 DLAM = D( I ) + TAU GO TO 250 ! ! End for the case I = N ! ELSE ! ! The case for I < N ! NITER = 1 IP1 = I + 1 ! ! Calculate initial guess ! DEL = D( IP1 ) - D( I ) MIDPT = DEL / TWO DO 100 J = 1, N DELTA( J ) = ( D( J )-D( I ) ) - MIDPT 100 CONTINUE ! PSI = ZERO DO 110 J = 1, I - 1 PSI = PSI + Z( J )*Z( J ) / DELTA( J ) 110 CONTINUE ! PHI = ZERO DO 120 J = N, I + 2, -1 PHI = PHI + Z( J )*Z( J ) / DELTA( J ) 120 CONTINUE C = RHOINV + PSI + PHI W = C + Z( I )*Z( I ) / DELTA( I ) + & Z( IP1 )*Z( IP1 ) / DELTA( IP1 ) ! IF( W.GT.ZERO ) THEN ! ! d(i)< the ith eigenvalue < (d(i)+d(i+1))/2 ! ! We choose d(i) as origin. ! ORGATI = .TRUE. A = C*DEL + Z( I )*Z( I ) + Z( IP1 )*Z( IP1 ) B = Z( I )*Z( I )*DEL IF( A.GT.ZERO ) THEN TAU = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) ELSE TAU = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) END IF DLTLB = ZERO DLTUB = MIDPT ELSE ! ! (d(i)+d(i+1))/2 <= the ith eigenvalue < d(i+1) ! ! We choose d(i+1) as origin. ! ORGATI = .FALSE. A = C*DEL - Z( I )*Z( I ) - Z( IP1 )*Z( IP1 ) B = Z( IP1 )*Z( IP1 )*DEL IF( A.LT.ZERO ) THEN TAU = TWO*B / ( A-SQRT( ABS( A*A+FOUR*B*C ) ) ) ELSE TAU = -( A+SQRT( ABS( A*A+FOUR*B*C ) ) ) / ( TWO*C ) END IF DLTLB = -MIDPT DLTUB = ZERO END IF ! IF( ORGATI ) THEN DO 130 J = 1, N DELTA( J ) = ( D( J )-D( I ) ) - TAU 130 CONTINUE ELSE DO 140 J = 1, N DELTA( J ) = ( D( J )-D( IP1 ) ) - TAU 140 CONTINUE END IF IF( ORGATI ) THEN II = I ELSE II = I + 1 END IF IIM1 = II - 1 IIP1 = II + 1 ! ! Evaluate PSI and the derivative DPSI ! DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 150 J = 1, IIM1 TEMP = Z( J ) / DELTA( J ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 150 CONTINUE ERRETM = ABS( ERRETM ) ! ! Evaluate PHI and the derivative DPHI ! DPHI = ZERO PHI = ZERO DO 160 J = N, IIP1, -1 TEMP = Z( J ) / DELTA( J ) PHI = PHI + Z( J )*TEMP DPHI = DPHI + TEMP*TEMP ERRETM = ERRETM + PHI 160 CONTINUE ! W = RHOINV + PHI + PSI ! ! W is the value of the secular function with ! its ii-th element removed. ! SWTCH3 = .FALSE. IF( ORGATI ) THEN IF( W.LT.ZERO ) & SWTCH3 = .TRUE. ELSE IF( W.GT.ZERO ) & SWTCH3 = .TRUE. END IF IF( II.EQ.1 .OR. II.EQ.N ) & SWTCH3 = .FALSE. ! TEMP = Z( II ) / DELTA( II ) DW = DPSI + DPHI + TEMP*TEMP TEMP = Z( II )*TEMP W = W + TEMP ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV + & THREE*ABS( TEMP ) + ABS( TAU )*DW ! ! Test for convergence ! IF( ABS( W ).LE.EPS*ERRETM ) THEN IF( ORGATI ) THEN DLAM = D( I ) + TAU ELSE DLAM = D( IP1 ) + TAU END IF GO TO 250 END IF ! IF( W.LE.ZERO ) THEN DLTLB = MAX( DLTLB, TAU ) ELSE DLTUB = MIN( DLTUB, TAU ) END IF ! ! Calculate the new step ! NITER = NITER + 1 IF( .NOT.SWTCH3 ) THEN IF( ORGATI ) THEN C = W - DELTA( IP1 )*DW - ( D( I )-D( IP1 ) )* & ( Z( I ) / DELTA( I ) )**2 ELSE C = W - DELTA( I )*DW - ( D( IP1 )-D( I ) )* & ( Z( IP1 ) / DELTA( IP1 ) )**2 END IF A = ( DELTA( I )+DELTA( IP1 ) )*W - & DELTA( I )*DELTA( IP1 )*DW B = DELTA( I )*DELTA( IP1 )*W IF( C.EQ.ZERO ) THEN IF( A.EQ.ZERO ) THEN IF( ORGATI ) THEN A = Z( I )*Z( I ) + DELTA( IP1 )*DELTA( IP1 )* & ( DPSI+DPHI ) ELSE A = Z( IP1 )*Z( IP1 ) + DELTA( I )*DELTA( I )* & ( DPSI+DPHI ) END IF END IF ETA = B / A ELSE IF( A.LE.ZERO ) THEN ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) ELSE ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) END IF ELSE ! ! Interpolation using THREE most relevant poles ! TEMP = RHOINV + PSI + PHI IF( ORGATI ) THEN TEMP1 = Z( IIM1 ) / DELTA( IIM1 ) TEMP1 = TEMP1*TEMP1 C = TEMP - DELTA( IIP1 )*( DPSI+DPHI ) - & ( D( IIM1 )-D( IIP1 ) )*TEMP1 ZZ( 1 ) = Z( IIM1 )*Z( IIM1 ) ZZ( 3 ) = DELTA( IIP1 )*DELTA( IIP1 )* & ( ( DPSI-TEMP1 )+DPHI ) ELSE TEMP1 = Z( IIP1 ) / DELTA( IIP1 ) TEMP1 = TEMP1*TEMP1 C = TEMP - DELTA( IIM1 )*( DPSI+DPHI ) - & ( D( IIP1 )-D( IIM1 ) )*TEMP1 ZZ( 1 ) = DELTA( IIM1 )*DELTA( IIM1 )* & ( DPSI+( DPHI-TEMP1 ) ) ZZ( 3 ) = Z( IIP1 )*Z( IIP1 ) END IF ZZ( 2 ) = Z( II )*Z( II ) CALL DLAED6( NITER, ORGATI, C, DELTA( IIM1 ), ZZ, W, ETA, & INFO ) IF( INFO.NE.0 ) & GO TO 250 END IF ! ! Note, eta should be positive if w is negative, and ! eta should be negative otherwise. However, ! if for some reason caused by roundoff, eta*w > 0, ! we simply use one Newton step instead. This way ! will guarantee eta*w < 0. ! IF( W*ETA.GE.ZERO ) & ETA = -W / DW TEMP = TAU + ETA IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN IF( W.LT.ZERO ) THEN ETA = ( DLTUB-TAU ) / TWO ELSE ETA = ( DLTLB-TAU ) / TWO END IF END IF ! PREW = W ! 170 CONTINUE DO 180 J = 1, N DELTA( J ) = DELTA( J ) - ETA 180 CONTINUE ! ! Evaluate PSI and the derivative DPSI ! DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 190 J = 1, IIM1 TEMP = Z( J ) / DELTA( J ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 190 CONTINUE ERRETM = ABS( ERRETM ) ! ! Evaluate PHI and the derivative DPHI ! DPHI = ZERO PHI = ZERO DO 200 J = N, IIP1, -1 TEMP = Z( J ) / DELTA( J ) PHI = PHI + Z( J )*TEMP DPHI = DPHI + TEMP*TEMP ERRETM = ERRETM + PHI 200 CONTINUE ! TEMP = Z( II ) / DELTA( II ) DW = DPSI + DPHI + TEMP*TEMP TEMP = Z( II )*TEMP W = RHOINV + PHI + PSI + TEMP ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV + & THREE*ABS( TEMP ) + ABS( TAU+ETA )*DW ! SWTCH = .FALSE. IF( ORGATI ) THEN IF( -W.GT.ABS( PREW ) / TEN ) & SWTCH = .TRUE. ELSE IF( W.GT.ABS( PREW ) / TEN ) & SWTCH = .TRUE. END IF ! TAU = TAU + ETA ! ! Main loop to update the values of the array DELTA ! ITER = NITER + 1 ! DO 240 NITER = ITER, MAXIT ! ! Test for convergence ! IF( ABS( W ).LE.EPS*ERRETM ) THEN IF( ORGATI ) THEN DLAM = D( I ) + TAU ELSE DLAM = D( IP1 ) + TAU END IF GO TO 250 END IF ! IF( W.LE.ZERO ) THEN DLTLB = MAX( DLTLB, TAU ) ELSE DLTUB = MIN( DLTUB, TAU ) END IF ! ! Calculate the new step ! IF( .NOT.SWTCH3 ) THEN IF( .NOT.SWTCH ) THEN IF( ORGATI ) THEN C = W - DELTA( IP1 )*DW - & ( D( I )-D( IP1 ) )*( Z( I ) / DELTA( I ) )**2 ELSE C = W - DELTA( I )*DW - ( D( IP1 )-D( I ) )* & ( Z( IP1 ) / DELTA( IP1 ) )**2 END IF ELSE TEMP = Z( II ) / DELTA( II ) IF( ORGATI ) THEN DPSI = DPSI + TEMP*TEMP ELSE DPHI = DPHI + TEMP*TEMP END IF C = W - DELTA( I )*DPSI - DELTA( IP1 )*DPHI END IF A = ( DELTA( I )+DELTA( IP1 ) )*W - & DELTA( I )*DELTA( IP1 )*DW B = DELTA( I )*DELTA( IP1 )*W IF( C.EQ.ZERO ) THEN IF( A.EQ.ZERO ) THEN IF( .NOT.SWTCH ) THEN IF( ORGATI ) THEN A = Z( I )*Z( I ) + DELTA( IP1 )* & DELTA( IP1 )*( DPSI+DPHI ) ELSE A = Z( IP1 )*Z( IP1 ) + & DELTA( I )*DELTA( I )*( DPSI+DPHI ) END IF ELSE A = DELTA( I )*DELTA( I )*DPSI + & DELTA( IP1 )*DELTA( IP1 )*DPHI END IF END IF ETA = B / A ELSE IF( A.LE.ZERO ) THEN ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) ELSE ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) END IF ELSE ! ! Interpolation using THREE most relevant poles ! TEMP = RHOINV + PSI + PHI IF( SWTCH ) THEN C = TEMP - DELTA( IIM1 )*DPSI - DELTA( IIP1 )*DPHI ZZ( 1 ) = DELTA( IIM1 )*DELTA( IIM1 )*DPSI ZZ( 3 ) = DELTA( IIP1 )*DELTA( IIP1 )*DPHI ELSE IF( ORGATI ) THEN TEMP1 = Z( IIM1 ) / DELTA( IIM1 ) TEMP1 = TEMP1*TEMP1 C = TEMP - DELTA( IIP1 )*( DPSI+DPHI ) - & ( D( IIM1 )-D( IIP1 ) )*TEMP1 ZZ( 1 ) = Z( IIM1 )*Z( IIM1 ) ZZ( 3 ) = DELTA( IIP1 )*DELTA( IIP1 )* & ( ( DPSI-TEMP1 )+DPHI ) ELSE TEMP1 = Z( IIP1 ) / DELTA( IIP1 ) TEMP1 = TEMP1*TEMP1 C = TEMP - DELTA( IIM1 )*( DPSI+DPHI ) - & ( D( IIP1 )-D( IIM1 ) )*TEMP1 ZZ( 1 ) = DELTA( IIM1 )*DELTA( IIM1 )* & ( DPSI+( DPHI-TEMP1 ) ) ZZ( 3 ) = Z( IIP1 )*Z( IIP1 ) END IF END IF CALL DLAED6( NITER, ORGATI, C, DELTA( IIM1 ), ZZ, W, ETA, & INFO ) IF( INFO.NE.0 ) & GO TO 250 END IF ! ! Note, eta should be positive if w is negative, and ! eta should be negative otherwise. However, ! if for some reason caused by roundoff, eta*w > 0, ! we simply use one Newton step instead. This way ! will guarantee eta*w < 0. ! IF( W*ETA.GE.ZERO ) & ETA = -W / DW TEMP = TAU + ETA IF( TEMP.GT.DLTUB .OR. TEMP.LT.DLTLB ) THEN IF( W.LT.ZERO ) THEN ETA = ( DLTUB-TAU ) / TWO ELSE ETA = ( DLTLB-TAU ) / TWO END IF END IF ! DO 210 J = 1, N DELTA( J ) = DELTA( J ) - ETA 210 CONTINUE ! TAU = TAU + ETA PREW = W ! ! Evaluate PSI and the derivative DPSI ! DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 220 J = 1, IIM1 TEMP = Z( J ) / DELTA( J ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 220 CONTINUE ERRETM = ABS( ERRETM ) ! ! Evaluate PHI and the derivative DPHI ! DPHI = ZERO PHI = ZERO DO 230 J = N, IIP1, -1 TEMP = Z( J ) / DELTA( J ) PHI = PHI + Z( J )*TEMP DPHI = DPHI + TEMP*TEMP ERRETM = ERRETM + PHI 230 CONTINUE ! TEMP = Z( II ) / DELTA( II ) DW = DPSI + DPHI + TEMP*TEMP TEMP = Z( II )*TEMP W = RHOINV + PHI + PSI + TEMP ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV + & THREE*ABS( TEMP ) + ABS( TAU )*DW IF( W*PREW.GT.ZERO .AND. ABS( W ).GT.ABS( PREW ) / TEN ) & SWTCH = .NOT.SWTCH ! 240 CONTINUE ! ! Return with INFO = 1, NITER = MAXIT and not converged ! INFO = 1 IF( ORGATI ) THEN DLAM = D( I ) + TAU ELSE DLAM = D( IP1 ) + TAU END IF ! END IF ! 250 CONTINUE ! RETURN ! ! End of DLAED4 ! END SUBROUTINE DLAED5( I, D, Z, DELTA, RHO, DLAM ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab, ! Courant Institute, NAG Ltd., and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. INTEGER I DOUBLE PRECISION DLAM, RHO ! .. ! .. Array Arguments .. DOUBLE PRECISION D( 2 ), DELTA( 2 ), Z( 2 ) ! .. ! ! Purpose ! ======= ! ! This subroutine computes the I-th eigenvalue of a symmetric rank-one ! modification of a 2-by-2 diagonal matrix ! ! diag( D ) + RHO * Z * transpose(Z) . ! ! The diagonal elements in the array D are assumed to satisfy ! ! D(i) < D(j) for i < j . ! ! We also assume RHO > 0 and that the Euclidean norm of the vector ! Z is one. ! ! Arguments ! ========= ! ! I (input) INTEGER ! The index of the eigenvalue to be computed. I = 1 or I = 2. ! ! D (input) DOUBLE PRECISION array, dimension (2) ! The original eigenvalues. We assume D(1) < D(2). ! ! Z (input) DOUBLE PRECISION array, dimension (2) ! The components of the updating vector. ! ! DELTA (output) DOUBLE PRECISION array, dimension (2) ! The vector DELTA contains the information necessary ! to construct the eigenvectors. ! ! RHO (input) DOUBLE PRECISION ! The scalar in the symmetric updating formula. ! ! DLAM (output) DOUBLE PRECISION ! The computed lambda_I, the I-th updated eigenvalue. ! ! Further Details ! =============== ! ! Based on contributions by ! Ren-Cang Li, Computer Science Division, University of California ! at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO, FOUR PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0, & FOUR = 4.0D0 ) ! .. ! .. Local Scalars .. DOUBLE PRECISION B, C, DEL, TAU, TEMP, W ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, SQRT ! .. ! .. Executable Statements .. ! DEL = D( 2 ) - D( 1 ) IF( I.EQ.1 ) THEN W = ONE + TWO*RHO*( Z( 2 )*Z( 2 )-Z( 1 )*Z( 1 ) ) / DEL IF( W.GT.ZERO ) THEN B = DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) ) C = RHO*Z( 1 )*Z( 1 )*DEL ! ! B > ZERO, always ! TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) ) DLAM = D( 1 ) + TAU DELTA( 1 ) = -Z( 1 ) / TAU DELTA( 2 ) = Z( 2 ) / ( DEL-TAU ) ELSE B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) ) C = RHO*Z( 2 )*Z( 2 )*DEL IF( B.GT.ZERO ) THEN TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) ) ELSE TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO END IF DLAM = D( 2 ) + TAU DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU ) DELTA( 2 ) = -Z( 2 ) / TAU END IF TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) ) DELTA( 1 ) = DELTA( 1 ) / TEMP DELTA( 2 ) = DELTA( 2 ) / TEMP ELSE ! ! Now I=2 ! B = -DEL + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) ) C = RHO*Z( 2 )*Z( 2 )*DEL IF( B.GT.ZERO ) THEN TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO ELSE TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) ) END IF DLAM = D( 2 ) + TAU DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU ) DELTA( 2 ) = -Z( 2 ) / TAU TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) ) DELTA( 1 ) = DELTA( 1 ) / TEMP DELTA( 2 ) = DELTA( 2 ) / TEMP END IF RETURN ! ! End OF DLAED5 ! END SUBROUTINE DLAED6( KNITER, ORGATI, RHO, D, Z, FINIT, TAU, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab, ! Courant Institute, NAG Ltd., and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. LOGICAL ORGATI INTEGER INFO, KNITER DOUBLE PRECISION FINIT, RHO, TAU ! .. ! .. Array Arguments .. DOUBLE PRECISION D( 3 ), Z( 3 ) ! .. ! ! Purpose ! ======= ! ! DLAED6 computes the positive or negative root (closest to the origin) ! of ! z(1) z(2) z(3) ! f(x) = rho + --------- + ---------- + --------- ! d(1)-x d(2)-x d(3)-x ! ! It is assumed that ! ! if ORGATI = .true. the root is between d(2) and d(3); ! otherwise it is between d(1) and d(2) ! ! This routine will be called by DLAED4 when necessary. In most cases, ! the root sought is the smallest in magnitude, though it might not be ! in some extremely rare situations. ! ! Arguments ! ========= ! ! KNITER (input) INTEGER ! Refer to DLAED4 for its significance. ! ! ORGATI (input) LOGICAL ! If ORGATI is true, the needed root is between d(2) and ! d(3); otherwise it is between d(1) and d(2). See ! DLAED4 for further details. ! ! RHO (input) DOUBLE PRECISION ! Refer to the equation f(x) above. ! ! D (input) DOUBLE PRECISION array, dimension (3) ! D satisfies d(1) < d(2) < d(3). ! ! Z (input) DOUBLE PRECISION array, dimension (3) ! Each of the elements in z must be positive. ! ! FINIT (input) DOUBLE PRECISION ! The value of f at 0. It is more accurate than the one ! evaluated inside this routine (if someone wants to do ! so). ! ! TAU (output) DOUBLE PRECISION ! The root of the equation f(x). ! ! INFO (output) INTEGER ! = 0: successful exit ! > 0: if INFO = 1, failure to converge ! ! Further Details ! =============== ! ! Based on contributions by ! Ren-Cang Li, Computer Science Division, University of California ! at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. INTEGER MAXIT PARAMETER ( MAXIT = 20 ) DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, EIGHT PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0, & THREE = 3.0D0, FOUR = 4.0D0, EIGHT = 8.0D0 ) ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. Local Arrays .. DOUBLE PRECISION DSCALE( 3 ), ZSCALE( 3 ) ! .. ! .. Local Scalars .. LOGICAL FIRST, SCALE INTEGER I, ITER, NITER DOUBLE PRECISION A, B, BASE, C, DDF, DF, EPS, ERRETM, ETA, F, & FC, SCLFAC, SCLINV, SMALL1, SMALL2, SMINV1, & SMINV2, TEMP, TEMP1, TEMP2, TEMP3, TEMP4 ! .. ! .. Save statement .. SAVE FIRST, SMALL1, SMINV1, SMALL2, SMINV2, EPS ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, INT, LOG, MAX, MIN, SQRT ! .. ! .. Data statements .. DATA FIRST / .TRUE. / ! .. ! .. Executable Statements .. ! INFO = 0 ! NITER = 1 TAU = ZERO IF( KNITER.EQ.2 ) THEN IF( ORGATI ) THEN TEMP = ( D( 3 )-D( 2 ) ) / TWO C = RHO + Z( 1 ) / ( ( D( 1 )-D( 2 ) )-TEMP ) A = C*( D( 2 )+D( 3 ) ) + Z( 2 ) + Z( 3 ) B = C*D( 2 )*D( 3 ) + Z( 2 )*D( 3 ) + Z( 3 )*D( 2 ) ELSE TEMP = ( D( 1 )-D( 2 ) ) / TWO C = RHO + Z( 3 ) / ( ( D( 3 )-D( 2 ) )-TEMP ) A = C*( D( 1 )+D( 2 ) ) + Z( 1 ) + Z( 2 ) B = C*D( 1 )*D( 2 ) + Z( 1 )*D( 2 ) + Z( 2 )*D( 1 ) END IF TEMP = MAX( ABS( A ), ABS( B ), ABS( C ) ) A = A / TEMP B = B / TEMP C = C / TEMP IF( C.EQ.ZERO ) THEN TAU = B / A ELSE IF( A.LE.ZERO ) THEN TAU = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) ELSE TAU = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) END IF TEMP = RHO + Z( 1 ) / ( D( 1 )-TAU ) + & Z( 2 ) / ( D( 2 )-TAU ) + Z( 3 ) / ( D( 3 )-TAU ) IF( ABS( FINIT ).LE.ABS( TEMP ) ) & TAU = ZERO END IF ! ! On first call to routine, get machine parameters for ! possible scaling to avoid overflow ! IF( FIRST ) THEN EPS = DLAMCH( 'Epsilon' ) BASE = DLAMCH( 'Base' ) SMALL1 = BASE**( INT( LOG( DLAMCH( 'SafMin' ) ) / LOG( BASE ) / & THREE ) ) SMINV1 = ONE / SMALL1 SMALL2 = SMALL1*SMALL1 SMINV2 = SMINV1*SMINV1 FIRST = .FALSE. END IF ! ! Determine if scaling of inputs necessary to avoid overflow ! when computing 1/TEMP**3 ! IF( ORGATI ) THEN TEMP = MIN( ABS( D( 2 )-TAU ), ABS( D( 3 )-TAU ) ) ELSE TEMP = MIN( ABS( D( 1 )-TAU ), ABS( D( 2 )-TAU ) ) END IF SCALE = .FALSE. IF( TEMP.LE.SMALL1 ) THEN SCALE = .TRUE. IF( TEMP.LE.SMALL2 ) THEN ! ! Scale up by power of radix nearest 1/SAFMIN**(2/3) ! SCLFAC = SMINV2 SCLINV = SMALL2 ELSE ! ! Scale up by power of radix nearest 1/SAFMIN**(1/3) ! SCLFAC = SMINV1 SCLINV = SMALL1 END IF ! ! Scaling up safe because D, Z, TAU scaled elsewhere to be O(1) ! DO 10 I = 1, 3 DSCALE( I ) = D( I )*SCLFAC ZSCALE( I ) = Z( I )*SCLFAC 10 CONTINUE TAU = TAU*SCLFAC ELSE ! ! Copy D and Z to DSCALE and ZSCALE ! DO 20 I = 1, 3 DSCALE( I ) = D( I ) ZSCALE( I ) = Z( I ) 20 CONTINUE END IF ! FC = ZERO DF = ZERO DDF = ZERO DO 30 I = 1, 3 TEMP = ONE / ( DSCALE( I )-TAU ) TEMP1 = ZSCALE( I )*TEMP TEMP2 = TEMP1*TEMP TEMP3 = TEMP2*TEMP FC = FC + TEMP1 / DSCALE( I ) DF = DF + TEMP2 DDF = DDF + TEMP3 30 CONTINUE F = FINIT + TAU*FC ! IF( ABS( F ).LE.ZERO ) & GO TO 60 ! ! Iteration begins ! ! It is not hard to see that ! ! 1) Iterations will go up monotonically ! if FINIT < 0; ! ! 2) Iterations will go down monotonically ! if FINIT > 0. ! ITER = NITER + 1 ! DO 50 NITER = ITER, MAXIT ! IF( ORGATI ) THEN TEMP1 = DSCALE( 2 ) - TAU TEMP2 = DSCALE( 3 ) - TAU ELSE TEMP1 = DSCALE( 1 ) - TAU TEMP2 = DSCALE( 2 ) - TAU END IF A = ( TEMP1+TEMP2 )*F - TEMP1*TEMP2*DF B = TEMP1*TEMP2*F C = F - ( TEMP1+TEMP2 )*DF + TEMP1*TEMP2*DDF TEMP = MAX( ABS( A ), ABS( B ), ABS( C ) ) A = A / TEMP B = B / TEMP C = C / TEMP IF( C.EQ.ZERO ) THEN ETA = B / A ELSE IF( A.LE.ZERO ) THEN ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) ELSE ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) END IF IF( F*ETA.GE.ZERO ) THEN ETA = -F / DF END IF ! TEMP = ETA + TAU IF( ORGATI ) THEN IF( ETA.GT.ZERO .AND. TEMP.GE.DSCALE( 3 ) ) & ETA = ( DSCALE( 3 )-TAU ) / TWO IF( ETA.LT.ZERO .AND. TEMP.LE.DSCALE( 2 ) ) & ETA = ( DSCALE( 2 )-TAU ) / TWO ELSE IF( ETA.GT.ZERO .AND. TEMP.GE.DSCALE( 2 ) ) & ETA = ( DSCALE( 2 )-TAU ) / TWO IF( ETA.LT.ZERO .AND. TEMP.LE.DSCALE( 1 ) ) & ETA = ( DSCALE( 1 )-TAU ) / TWO END IF TAU = TAU + ETA ! FC = ZERO ERRETM = ZERO DF = ZERO DDF = ZERO DO 40 I = 1, 3 TEMP = ONE / ( DSCALE( I )-TAU ) TEMP1 = ZSCALE( I )*TEMP TEMP2 = TEMP1*TEMP TEMP3 = TEMP2*TEMP TEMP4 = TEMP1 / DSCALE( I ) FC = FC + TEMP4 ERRETM = ERRETM + ABS( TEMP4 ) DF = DF + TEMP2 DDF = DDF + TEMP3 40 CONTINUE F = FINIT + TAU*FC ERRETM = EIGHT*( ABS( FINIT )+ABS( TAU )*ERRETM ) + & ABS( TAU )*DF IF( ABS( F ).LE.EPS*ERRETM ) & GO TO 60 50 CONTINUE INFO = 1 60 CONTINUE ! ! Undo scaling ! IF( SCALE ) & TAU = TAU*SCLINV RETURN ! ! End of DLAED6 ! END SUBROUTINE DLAED7( ICOMPQ, N, QSIZ, TLVLS, CURLVL, CURPBM, D, Q, & LDQ, INDXQ, RHO, CUTPNT, QSTORE, QPTR, PRMPTR, & PERM, GIVPTR, GIVCOL, GIVNUM, WORK, IWORK, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. INTEGER CURLVL, CURPBM, CUTPNT, ICOMPQ, INFO, LDQ, N, & QSIZ, TLVLS DOUBLE PRECISION RHO ! .. ! .. Array Arguments .. INTEGER GIVCOL( 2, * ), GIVPTR( * ), INDXQ( * ), & IWORK( * ), PERM( * ), PRMPTR( * ), QPTR( * ) DOUBLE PRECISION D( * ), GIVNUM( 2, * ), Q( LDQ, * ), & QSTORE( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLAED7 computes the updated eigensystem of a diagonal ! matrix after modification by a rank-one symmetric matrix. This ! routine is used only for the eigenproblem which requires all ! eigenvalues and optionally eigenvectors of a dense symmetric matrix ! that has been reduced to tridiagonal form. DLAED1 handles ! the case in which all eigenvalues and eigenvectors of a symmetric ! tridiagonal matrix are desired. ! ! T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out) ! ! where Z = Q'u, u is a vector of length N with ones in the ! CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. ! ! The eigenvectors of the original matrix are stored in Q, and the ! eigenvalues are in D. The algorithm consists of three stages: ! ! The first stage consists of deflating the size of the problem ! when there are multiple eigenvalues or if there is a zero in ! the Z vector. For each such occurence the dimension of the ! secular equation problem is reduced by one. This stage is ! performed by the routine DLAED8. ! ! The second stage consists of calculating the updated ! eigenvalues. This is done by finding the roots of the secular ! equation via the routine DLAED4 (as called by DLAED9). ! This routine also calculates the eigenvectors of the current ! problem. ! ! The final stage consists of computing the updated eigenvectors ! directly using the updated eigenvalues. The eigenvectors for ! the current problem are multiplied with the eigenvectors from ! the overall problem. ! ! Arguments ! ========= ! ! ICOMPQ (input) INTEGER ! = 0: Compute eigenvalues only. ! = 1: Compute eigenvectors of original dense symmetric matrix ! also. On entry, Q contains the orthogonal matrix used ! to reduce the original matrix to tridiagonal form. ! ! N (input) INTEGER ! The dimension of the symmetric tridiagonal matrix. N >= 0. ! ! QSIZ (input) INTEGER ! The dimension of the orthogonal matrix used to reduce ! the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1. ! ! TLVLS (input) INTEGER ! The total number of merging levels in the overall divide and ! conquer tree. ! ! CURLVL (input) INTEGER ! The current level in the overall merge routine, ! 0 <= CURLVL <= TLVLS. ! ! CURPBM (input) INTEGER ! The current problem in the current level in the overall ! merge routine (counting from upper left to lower right). ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the eigenvalues of the rank-1-perturbed matrix. ! On exit, the eigenvalues of the repaired matrix. ! ! Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N) ! On entry, the eigenvectors of the rank-1-perturbed matrix. ! On exit, the eigenvectors of the repaired tridiagonal matrix. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. LDQ >= max(1,N). ! ! INDXQ (output) INTEGER array, dimension (N) ! The permutation which will reintegrate the subproblem just ! solved back into sorted order, i.e., D( INDXQ( I = 1, N ) ) ! will be in ascending order. ! ! RHO (input) DOUBLE PRECISION ! The subdiagonal element used to create the rank-1 ! modification. ! ! CUTPNT (input) INTEGER ! Contains the location of the last eigenvalue in the leading ! sub-matrix. min(1,N) <= CUTPNT <= N. ! ! QSTORE (input/output) DOUBLE PRECISION array, dimension (N**2+1) ! Stores eigenvectors of submatrices encountered during ! divide and conquer, packed together. QPTR points to ! beginning of the submatrices. ! ! QPTR (input/output) INTEGER array, dimension (N+2) ! List of indices pointing to beginning of submatrices stored ! in QSTORE. The submatrices are numbered starting at the ! bottom left of the divide and conquer tree, from left to ! right and bottom to top. ! ! PRMPTR (input) INTEGER array, dimension (N lg N) ! Contains a list of pointers which indicate where in PERM a ! level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) ! indicates the size of the permutation and also the size of ! the full, non-deflated problem. ! ! PERM (input) INTEGER array, dimension (N lg N) ! Contains the permutations (from deflation and sorting) to be ! applied to each eigenblock. ! ! GIVPTR (input) INTEGER array, dimension (N lg N) ! Contains a list of pointers which indicate where in GIVCOL a ! level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) ! indicates the number of Givens rotations. ! ! GIVCOL (input) INTEGER array, dimension (2, N lg N) ! Each pair of numbers indicates a pair of columns to take place ! in a Givens rotation. ! ! GIVNUM (input) DOUBLE PRECISION array, dimension (2, N lg N) ! Each number indicates the S value to be used in the ! corresponding Givens rotation. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N+QSIZ*N) ! ! IWORK (workspace) INTEGER array, dimension (4*N) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = 1, an eigenvalue did not converge ! ! Further Details ! =============== ! ! Based on contributions by ! Jeff Rutter, Computer Science Division, University of California ! at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 ) ! .. ! .. Local Scalars .. INTEGER COLTYP, CURR, I, IDLMDA, INDX, INDXC, INDXP, & IQ2, IS, IW, IZ, K, LDQ2, N1, N2, PTR ! .. ! .. External Subroutines .. EXTERNAL DGEMM, DLAED8, DLAED9, DLAEDA, DLAMRG, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN INFO = -4 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( MIN( 1, N ).GT.CUTPNT .OR. N.LT.CUTPNT ) THEN INFO = -12 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLAED7', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! The following values are for bookkeeping purposes only. They are ! integer pointers which indicate the portion of the workspace ! used by a particular array in DLAED8 and DLAED9. ! IF( ICOMPQ.EQ.1 ) THEN LDQ2 = QSIZ ELSE LDQ2 = N END IF ! IZ = 1 IDLMDA = IZ + N IW = IDLMDA + N IQ2 = IW + N IS = IQ2 + N*LDQ2 ! INDX = 1 INDXC = INDX + N COLTYP = INDXC + N INDXP = COLTYP + N ! ! Form the z-vector which consists of the last row of Q_1 and the ! first row of Q_2. ! PTR = 1 + 2**TLVLS DO 10 I = 1, CURLVL - 1 PTR = PTR + 2**( TLVLS-I ) 10 CONTINUE CURR = PTR + CURPBM CALL DLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR, & GIVCOL, GIVNUM, QSTORE, QPTR, WORK( IZ ), & WORK( IZ+N ), INFO ) ! ! When solving the final problem, we no longer need the stored data, ! so we will overwrite the data from this level onto the previously ! used storage space. ! IF( CURLVL.EQ.TLVLS ) THEN QPTR( CURR ) = 1 PRMPTR( CURR ) = 1 GIVPTR( CURR ) = 1 END IF ! ! Sort and Deflate eigenvalues. ! CALL DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, CUTPNT, & WORK( IZ ), WORK( IDLMDA ), WORK( IQ2 ), LDQ2, & WORK( IW ), PERM( PRMPTR( CURR ) ), GIVPTR( CURR+1 ), & GIVCOL( 1, GIVPTR( CURR ) ), & GIVNUM( 1, GIVPTR( CURR ) ), IWORK( INDXP ), & IWORK( INDX ), INFO ) PRMPTR( CURR+1 ) = PRMPTR( CURR ) + N GIVPTR( CURR+1 ) = GIVPTR( CURR+1 ) + GIVPTR( CURR ) ! ! Solve Secular Equation. ! IF( K.NE.0 ) THEN CALL DLAED9( K, 1, K, N, D, WORK( IS ), K, RHO, WORK( IDLMDA ), & WORK( IW ), QSTORE( QPTR( CURR ) ), K, INFO ) IF( INFO.NE.0 ) & GO TO 30 IF( ICOMPQ.EQ.1 ) THEN CALL DGEMM( 'N', 'N', QSIZ, K, K, ONE, WORK( IQ2 ), LDQ2, & QSTORE( QPTR( CURR ) ), K, ZERO, Q, LDQ ) END IF QPTR( CURR+1 ) = QPTR( CURR ) + K**2 ! ! Prepare the INDXQ sorting permutation. ! N1 = K N2 = N - K CALL DLAMRG( N1, N2, D, 1, -1, INDXQ ) ELSE QPTR( CURR+1 ) = QPTR( CURR ) DO 20 I = 1, N INDXQ( I ) = I 20 CONTINUE END IF ! 30 CONTINUE RETURN ! ! End of DLAED7 ! END SUBROUTINE DLAED8( ICOMPQ, K, N, QSIZ, D, Q, LDQ, INDXQ, RHO, & CUTPNT, Z, DLAMDA, Q2, LDQ2, W, PERM, GIVPTR, & GIVCOL, GIVNUM, INDXP, INDX, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab, ! Courant Institute, NAG Ltd., and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N, & QSIZ DOUBLE PRECISION RHO ! .. ! .. Array Arguments .. INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ), & INDXQ( * ), PERM( * ) DOUBLE PRECISION D( * ), DLAMDA( * ), GIVNUM( 2, * ), & Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * ) ! .. ! ! Purpose ! ======= ! ! DLAED8 merges the two sets of eigenvalues together into a single ! sorted set. Then it tries to deflate the size of the problem. ! There are two ways in which deflation can occur: when two or more ! eigenvalues are close together or if there is a tiny element in the ! Z vector. For each such occurrence the order of the related secular ! equation problem is reduced by one. ! ! Arguments ! ========= ! ! ICOMPQ (input) INTEGER ! = 0: Compute eigenvalues only. ! = 1: Compute eigenvectors of original dense symmetric matrix ! also. On entry, Q contains the orthogonal matrix used ! to reduce the original matrix to tridiagonal form. ! ! K (output) INTEGER ! The number of non-deflated eigenvalues, and the order of the ! related secular equation. ! ! N (input) INTEGER ! The dimension of the symmetric tridiagonal matrix. N >= 0. ! ! QSIZ (input) INTEGER ! The dimension of the orthogonal matrix used to reduce ! the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the eigenvalues of the two submatrices to be ! combined. On exit, the trailing (N-K) updated eigenvalues ! (those which were deflated) sorted into increasing order. ! ! Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) ! If ICOMPQ = 0, Q is not referenced. Otherwise, ! on entry, Q contains the eigenvectors of the partially solved ! system which has been previously updated in matrix ! multiplies with other partially solved eigensystems. ! On exit, Q contains the trailing (N-K) updated eigenvectors ! (those which were deflated) in its last N-K columns. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. LDQ >= max(1,N). ! ! INDXQ (input) INTEGER array, dimension (N) ! The permutation which separately sorts the two sub-problems ! in D into ascending order. Note that elements in the second ! half of this permutation must first have CUTPNT added to ! their values in order to be accurate. ! ! RHO (input/output) DOUBLE PRECISION ! On entry, the off-diagonal element associated with the rank-1 ! cut which originally split the two submatrices which are now ! being recombined. ! On exit, RHO has been modified to the value required by ! DLAED3. ! ! CUTPNT (input) INTEGER ! The location of the last eigenvalue in the leading ! sub-matrix. min(1,N) <= CUTPNT <= N. ! ! Z (input) DOUBLE PRECISION array, dimension (N) ! On entry, Z contains the updating vector (the last row of ! the first sub-eigenvector matrix and the first row of the ! second sub-eigenvector matrix). ! On exit, the contents of Z are destroyed by the updating ! process. ! ! DLAMDA (output) DOUBLE PRECISION array, dimension (N) ! A copy of the first K eigenvalues which will be used by ! DLAED3 to form the secular equation. ! ! Q2 (output) DOUBLE PRECISION array, dimension (LDQ2,N) ! If ICOMPQ = 0, Q2 is not referenced. Otherwise, ! a copy of the first K eigenvectors which will be used by ! DLAED7 in a matrix multiply (DGEMM) to update the new ! eigenvectors. ! ! LDQ2 (input) INTEGER ! The leading dimension of the array Q2. LDQ2 >= max(1,N). ! ! W (output) DOUBLE PRECISION array, dimension (N) ! The first k values of the final deflation-altered z-vector and ! will be passed to DLAED3. ! ! PERM (output) INTEGER array, dimension (N) ! The permutations (from deflation and sorting) to be applied ! to each eigenblock. ! ! GIVPTR (output) INTEGER ! The number of Givens rotations which took place in this ! subproblem. ! ! GIVCOL (output) INTEGER array, dimension (2, N) ! Each pair of numbers indicates a pair of columns to take place ! in a Givens rotation. ! ! GIVNUM (output) DOUBLE PRECISION array, dimension (2, N) ! Each number indicates the S value to be used in the ! corresponding Givens rotation. ! ! INDXP (workspace) INTEGER array, dimension (N) ! The permutation used to place deflated values of D at the end ! of the array. INDXP(1:K) points to the nondeflated D-values ! and INDXP(K+1:N) points to the deflated eigenvalues. ! ! INDX (workspace) INTEGER array, dimension (N) ! The permutation used to sort the contents of D into ascending ! order. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! Based on contributions by ! Jeff Rutter, Computer Science Division, University of California ! at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION MONE, ZERO, ONE, TWO, EIGHT PARAMETER ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0, & TWO = 2.0D0, EIGHT = 8.0D0 ) ! .. ! .. Local Scalars .. ! INTEGER I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2 DOUBLE PRECISION C, EPS, S, T, TAU, TOL ! .. ! .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DLAMCH, DLAPY2 EXTERNAL IDAMAX, DLAMCH, DLAPY2 ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLACPY, DLAMRG, DROT, DSCAL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( ICOMPQ.LT.0 .OR. ICOMPQ.GT.1 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( ICOMPQ.EQ.1 .AND. QSIZ.LT.N ) THEN INFO = -4 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( CUTPNT.LT.MIN( 1, N ) .OR. CUTPNT.GT.N ) THEN INFO = -10 ELSE IF( LDQ2.LT.MAX( 1, N ) ) THEN INFO = -14 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLAED8', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! N1 = CUTPNT N2 = N - N1 N1P1 = N1 + 1 ! IF( RHO.LT.ZERO ) THEN CALL DSCAL( N2, MONE, Z( N1P1 ), 1 ) END IF ! ! Normalize z so that norm(z) = 1 ! T = ONE / SQRT( TWO ) DO 10 J = 1, N INDX( J ) = J 10 CONTINUE CALL DSCAL( N, T, Z, 1 ) RHO = ABS( TWO*RHO ) ! ! Sort the eigenvalues into increasing order ! DO 20 I = CUTPNT + 1, N INDXQ( I ) = INDXQ( I ) + CUTPNT 20 CONTINUE DO 30 I = 1, N DLAMDA( I ) = D( INDXQ( I ) ) W( I ) = Z( INDXQ( I ) ) 30 CONTINUE I = 1 J = CUTPNT + 1 CALL DLAMRG( N1, N2, DLAMDA, 1, 1, INDX ) DO 40 I = 1, N D( I ) = DLAMDA( INDX( I ) ) Z( I ) = W( INDX( I ) ) 40 CONTINUE ! ! Calculate the allowable deflation tolerence ! IMAX = IDAMAX( N, Z, 1 ) JMAX = IDAMAX( N, D, 1 ) EPS = DLAMCH( 'Epsilon' ) TOL = EIGHT*EPS*ABS( D( JMAX ) ) ! ! If the rank-1 modifier is small enough, no more needs to be done ! except to reorganize Q so that its columns correspond with the ! elements in D. ! IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN K = 0 IF( ICOMPQ.EQ.0 ) THEN DO 50 J = 1, N PERM( J ) = INDXQ( INDX( J ) ) 50 CONTINUE ELSE DO 60 J = 1, N PERM( J ) = INDXQ( INDX( J ) ) CALL DCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 ) 60 CONTINUE CALL DLACPY( 'A', QSIZ, N, Q2( 1, 1 ), LDQ2, Q( 1, 1 ), & LDQ ) END IF RETURN END IF ! ! If there are multiple eigenvalues then the problem deflates. Here ! the number of equal eigenvalues are found. As each equal ! eigenvalue is found, an elementary reflector is computed to rotate ! the corresponding eigensubspace so that the corresponding ! components of Z are zero in this new basis. ! K = 0 GIVPTR = 0 K2 = N + 1 DO 70 J = 1, N IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN ! ! Deflate due to small z component. ! K2 = K2 - 1 INDXP( K2 ) = J IF( J.EQ.N ) & GO TO 110 ELSE JLAM = J GO TO 80 END IF 70 CONTINUE 80 CONTINUE J = J + 1 IF( J.GT.N ) & GO TO 100 IF( RHO*ABS( Z( J ) ).LE.TOL ) THEN ! ! Deflate due to small z component. ! K2 = K2 - 1 INDXP( K2 ) = J ELSE ! ! Check if eigenvalues are close enough to allow deflation. ! S = Z( JLAM ) C = Z( J ) ! ! Find sqrt(a**2+b**2) without overflow or ! destructive underflow. ! TAU = DLAPY2( C, S ) T = D( J ) - D( JLAM ) C = C / TAU S = -S / TAU IF( ABS( T*C*S ).LE.TOL ) THEN ! ! Deflation is possible. ! Z( J ) = TAU Z( JLAM ) = ZERO ! ! Record the appropriate Givens rotation ! GIVPTR = GIVPTR + 1 GIVCOL( 1, GIVPTR ) = INDXQ( INDX( JLAM ) ) GIVCOL( 2, GIVPTR ) = INDXQ( INDX( J ) ) GIVNUM( 1, GIVPTR ) = C GIVNUM( 2, GIVPTR ) = S IF( ICOMPQ.EQ.1 ) THEN CALL DROT( QSIZ, Q( 1, INDXQ( INDX( JLAM ) ) ), 1, & Q( 1, INDXQ( INDX( J ) ) ), 1, C, S ) END IF T = D( JLAM )*C*C + D( J )*S*S D( J ) = D( JLAM )*S*S + D( J )*C*C D( JLAM ) = T K2 = K2 - 1 I = 1 90 CONTINUE IF( K2+I.LE.N ) THEN IF( D( JLAM ).LT.D( INDXP( K2+I ) ) ) THEN INDXP( K2+I-1 ) = INDXP( K2+I ) INDXP( K2+I ) = JLAM I = I + 1 GO TO 90 ELSE INDXP( K2+I-1 ) = JLAM END IF ELSE INDXP( K2+I-1 ) = JLAM END IF JLAM = J ELSE K = K + 1 W( K ) = Z( JLAM ) DLAMDA( K ) = D( JLAM ) INDXP( K ) = JLAM JLAM = J END IF END IF GO TO 80 100 CONTINUE ! ! Record the last eigenvalue. ! K = K + 1 W( K ) = Z( JLAM ) DLAMDA( K ) = D( JLAM ) INDXP( K ) = JLAM ! 110 CONTINUE ! ! Sort the eigenvalues and corresponding eigenvectors into DLAMDA ! and Q2 respectively. The eigenvalues/vectors which were not ! deflated go into the first K slots of DLAMDA and Q2 respectively, ! while those which were deflated go into the last N - K slots. ! IF( ICOMPQ.EQ.0 ) THEN DO 120 J = 1, N JP = INDXP( J ) DLAMDA( J ) = D( JP ) PERM( J ) = INDXQ( INDX( JP ) ) 120 CONTINUE ELSE DO 130 J = 1, N JP = INDXP( J ) DLAMDA( J ) = D( JP ) PERM( J ) = INDXQ( INDX( JP ) ) CALL DCOPY( QSIZ, Q( 1, PERM( J ) ), 1, Q2( 1, J ), 1 ) 130 CONTINUE END IF ! ! The deflated eigenvalues and their corresponding vectors go back ! into the last N - K slots of D and Q respectively. ! IF( K.LT.N ) THEN IF( ICOMPQ.EQ.0 ) THEN CALL DCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 ) ELSE CALL DCOPY( N-K, DLAMDA( K+1 ), 1, D( K+1 ), 1 ) CALL DLACPY( 'A', QSIZ, N-K, Q2( 1, K+1 ), LDQ2, & Q( 1, K+1 ), LDQ ) END IF END IF ! RETURN ! ! End of DLAED8 ! END SUBROUTINE DLAED9( K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W, & S, LDS, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab, ! Courant Institute, NAG Ltd., and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. INTEGER INFO, K, KSTART, KSTOP, LDQ, LDS, N DOUBLE PRECISION RHO ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), S( LDS, * ), & W( * ) ! .. ! ! Purpose ! ======= ! ! DLAED9 finds the roots of the secular equation, as defined by the ! values in D, Z, and RHO, between KSTART and KSTOP. It makes the ! appropriate calls to DLAED4 and then stores the new matrix of ! eigenvectors for use in calculating the next level of Z vectors. ! ! Arguments ! ========= ! ! K (input) INTEGER ! The number of terms in the rational function to be solved by ! DLAED4. K >= 0. ! ! KSTART (input) INTEGER ! KSTOP (input) INTEGER ! The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP ! are to be computed. 1 <= KSTART <= KSTOP <= K. ! ! N (input) INTEGER ! The number of rows and columns in the Q matrix. ! N >= K (delation may result in N > K). ! ! D (output) DOUBLE PRECISION array, dimension (N) ! D(I) contains the updated eigenvalues ! for KSTART <= I <= KSTOP. ! ! Q (workspace) DOUBLE PRECISION array, dimension (LDQ,N) ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. LDQ >= max( 1, N ). ! ! RHO (input) DOUBLE PRECISION ! The value of the parameter in the rank one update equation. ! RHO >= 0 required. ! ! DLAMDA (input) DOUBLE PRECISION array, dimension (K) ! The first K elements of this array contain the old roots ! of the deflated updating problem. These are the poles ! of the secular equation. ! ! W (input) DOUBLE PRECISION array, dimension (K) ! The first K elements of this array contain the components ! of the deflation-adjusted updating vector. ! ! S (output) DOUBLE PRECISION array, dimension (LDS, K) ! Will contain the eigenvectors of the repaired matrix which ! will be stored for subsequent Z vector calculation and ! multiplied by the previously accumulated eigenvectors ! to update the system. ! ! LDS (input) INTEGER ! The leading dimension of S. LDS >= max( 1, K ). ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = 1, an eigenvalue did not converge ! ! Further Details ! =============== ! ! Based on contributions by ! Jeff Rutter, Computer Science Division, University of California ! at Berkeley, USA ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER I, J DOUBLE PRECISION TEMP ! .. ! .. External Functions .. DOUBLE PRECISION DLAMC3, DNRM2 EXTERNAL DLAMC3, DNRM2 ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLAED4, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, SIGN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( K.LT.0 ) THEN INFO = -1 ELSE IF( KSTART.LT.1 .OR. KSTART.GT.MAX( 1, K ) ) THEN INFO = -2 ELSE IF( MAX( 1, KSTOP ).LT.KSTART .OR. KSTOP.GT.MAX( 1, K ) ) & THEN INFO = -3 ELSE IF( N.LT.K ) THEN INFO = -4 ELSE IF( LDQ.LT.MAX( 1, K ) ) THEN INFO = -7 ELSE IF( LDS.LT.MAX( 1, K ) ) THEN INFO = -12 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLAED9', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( K.EQ.0 ) & RETURN ! ! Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can ! be computed with high relative accuracy (barring over/underflow). ! This is a problem on machines without a guard digit in ! add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2). ! The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I), ! which on any of these machines zeros out the bottommost ! bit of DLAMDA(I) if it is 1; this makes the subsequent ! subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation ! occurs. On binary machines with a guard digit (almost all ! machines) it does not change DLAMDA(I) at all. On hexadecimal ! and decimal machines with a guard digit, it slightly ! changes the bottommost bits of DLAMDA(I). It does not account ! for hexadecimal or decimal machines without guard digits ! (we know of none). We use a subroutine call to compute ! 2*DLAMBDA(I) to prevent optimizing compilers from eliminating ! this code. ! DO 10 I = 1, N DLAMDA( I ) = DLAMC3( DLAMDA( I ), DLAMDA( I ) ) - DLAMDA( I ) 10 CONTINUE ! DO 20 J = KSTART, KSTOP CALL DLAED4( K, J, DLAMDA, W, Q( 1, J ), RHO, D( J ), INFO ) ! ! If the zero finder fails, the computation is terminated. ! IF( INFO.NE.0 ) & GO TO 120 20 CONTINUE ! IF( K.EQ.1 .OR. K.EQ.2 ) THEN DO 40 I = 1, K DO 30 J = 1, K S( J, I ) = Q( J, I ) 30 CONTINUE 40 CONTINUE GO TO 120 END IF ! ! Compute updated W. ! CALL DCOPY( K, W, 1, S, 1 ) ! ! Initialize W(I) = Q(I,I) ! CALL DCOPY( K, Q, LDQ+1, W, 1 ) DO 70 J = 1, K DO 50 I = 1, J - 1 W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) ) 50 CONTINUE DO 60 I = J + 1, K W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) ) 60 CONTINUE 70 CONTINUE DO 80 I = 1, K W( I ) = SIGN( SQRT( -W( I ) ), S( I, 1 ) ) 80 CONTINUE ! ! Compute eigenvectors of the modified rank-1 modification. ! DO 110 J = 1, K DO 90 I = 1, K Q( I, J ) = W( I ) / Q( I, J ) 90 CONTINUE TEMP = DNRM2( K, Q( 1, J ), 1 ) DO 100 I = 1, K S( I, J ) = Q( I, J ) / TEMP 100 CONTINUE 110 CONTINUE ! 120 CONTINUE RETURN ! ! End of DLAED9 ! END SUBROUTINE DLAEDA( N, TLVLS, CURLVL, CURPBM, PRMPTR, PERM, GIVPTR, & GIVCOL, GIVNUM, Q, QPTR, Z, ZTEMP, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. INTEGER CURLVL, CURPBM, INFO, N, TLVLS ! .. ! .. Array Arguments .. INTEGER GIVCOL( 2, * ), GIVPTR( * ), PERM( * ), & PRMPTR( * ), QPTR( * ) DOUBLE PRECISION GIVNUM( 2, * ), Q( * ), Z( * ), ZTEMP( * ) ! .. ! ! Purpose ! ======= ! ! DLAEDA computes the Z vector corresponding to the merge step in the ! CURLVLth step of the merge process with TLVLS steps for the CURPBMth ! problem. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The dimension of the symmetric tridiagonal matrix. N >= 0. ! ! TLVLS (input) INTEGER ! The total number of merging levels in the overall divide and ! conquer tree. ! ! CURLVL (input) INTEGER ! The current level in the overall merge routine, ! 0 <= curlvl <= tlvls. ! ! CURPBM (input) INTEGER ! The current problem in the current level in the overall ! merge routine (counting from upper left to lower right). ! ! PRMPTR (input) INTEGER array, dimension (N lg N) ! Contains a list of pointers which indicate where in PERM a ! level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) ! indicates the size of the permutation and incidentally the ! size of the full, non-deflated problem. ! ! PERM (input) INTEGER array, dimension (N lg N) ! Contains the permutations (from deflation and sorting) to be ! applied to each eigenblock. ! ! GIVPTR (input) INTEGER array, dimension (N lg N) ! Contains a list of pointers which indicate where in GIVCOL a ! level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) ! indicates the number of Givens rotations. ! ! GIVCOL (input) INTEGER array, dimension (2, N lg N) ! Each pair of numbers indicates a pair of columns to take place ! in a Givens rotation. ! ! GIVNUM (input) DOUBLE PRECISION array, dimension (2, N lg N) ! Each number indicates the S value to be used in the ! corresponding Givens rotation. ! ! Q (input) DOUBLE PRECISION array, dimension (N**2) ! Contains the square eigenblocks from previous levels, the ! starting positions for blocks are given by QPTR. ! ! QPTR (input) INTEGER array, dimension (N+2) ! Contains a list of pointers which indicate where in Q an ! eigenblock is stored. SQRT( QPTR(i+1) - QPTR(i) ) indicates ! the size of the block. ! ! Z (output) DOUBLE PRECISION array, dimension (N) ! On output this vector contains the updating vector (the last ! row of the first sub-eigenvector matrix and the first row of ! the second sub-eigenvector matrix). ! ! ZTEMP (workspace) DOUBLE PRECISION array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! Based on contributions by ! Jeff Rutter, Computer Science Division, University of California ! at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, HALF, ONE PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. INTEGER BSIZ1, BSIZ2, CURR, I, K, MID, PSIZ1, PSIZ2, & PTR, ZPTR1 ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEMV, DROT, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC DBLE, INT, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( N.LT.0 ) THEN INFO = -1 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLAEDA', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Determine location of first number in second half. ! MID = N / 2 + 1 ! ! Gather last/first rows of appropriate eigenblocks into center of Z ! PTR = 1 ! ! Determine location of lowest level subproblem in the full storage ! scheme ! CURR = PTR + CURPBM*2**CURLVL + 2**( CURLVL-1 ) - 1 ! ! Determine size of these matrices. We add HALF to the value of ! the SQRT in case the machine underestimates one of these square ! roots. ! BSIZ1 = INT( HALF+SQRT( DBLE( QPTR( CURR+1 )-QPTR( CURR ) ) ) ) BSIZ2 = INT( HALF+SQRT( DBLE( QPTR( CURR+2 )-QPTR( CURR+1 ) ) ) ) DO 10 K = 1, MID - BSIZ1 - 1 Z( K ) = ZERO 10 CONTINUE CALL DCOPY( BSIZ1, Q( QPTR( CURR )+BSIZ1-1 ), BSIZ1, & Z( MID-BSIZ1 ), 1 ) CALL DCOPY( BSIZ2, Q( QPTR( CURR+1 ) ), BSIZ2, Z( MID ), 1 ) DO 20 K = MID + BSIZ2, N Z( K ) = ZERO 20 CONTINUE ! ! Loop thru remaining levels 1 -> CURLVL applying the Givens ! rotations and permutation and then multiplying the center matrices ! against the current Z. ! PTR = 2**TLVLS + 1 DO 70 K = 1, CURLVL - 1 CURR = PTR + CURPBM*2**( CURLVL-K ) + 2**( CURLVL-K-1 ) - 1 PSIZ1 = PRMPTR( CURR+1 ) - PRMPTR( CURR ) PSIZ2 = PRMPTR( CURR+2 ) - PRMPTR( CURR+1 ) ZPTR1 = MID - PSIZ1 ! ! Apply Givens at CURR and CURR+1 ! DO 30 I = GIVPTR( CURR ), GIVPTR( CURR+1 ) - 1 CALL DROT( 1, Z( ZPTR1+GIVCOL( 1, I )-1 ), 1, & Z( ZPTR1+GIVCOL( 2, I )-1 ), 1, GIVNUM( 1, I ), & GIVNUM( 2, I ) ) 30 CONTINUE DO 40 I = GIVPTR( CURR+1 ), GIVPTR( CURR+2 ) - 1 CALL DROT( 1, Z( MID-1+GIVCOL( 1, I ) ), 1, & Z( MID-1+GIVCOL( 2, I ) ), 1, GIVNUM( 1, I ), & GIVNUM( 2, I ) ) 40 CONTINUE PSIZ1 = PRMPTR( CURR+1 ) - PRMPTR( CURR ) PSIZ2 = PRMPTR( CURR+2 ) - PRMPTR( CURR+1 ) DO 50 I = 0, PSIZ1 - 1 ZTEMP( I+1 ) = Z( ZPTR1+PERM( PRMPTR( CURR )+I )-1 ) 50 CONTINUE DO 60 I = 0, PSIZ2 - 1 ZTEMP( PSIZ1+I+1 ) = Z( MID+PERM( PRMPTR( CURR+1 )+I )-1 ) 60 CONTINUE ! ! Multiply Blocks at CURR and CURR+1 ! ! Determine size of these matrices. We add HALF to the value of ! the SQRT in case the machine underestimates one of these ! square roots. ! BSIZ1 = INT( HALF+SQRT( DBLE( QPTR( CURR+1 )-QPTR( CURR ) ) ) ) BSIZ2 = INT( HALF+SQRT( DBLE( QPTR( CURR+2 )-QPTR( CURR+ & 1 ) ) ) ) IF( BSIZ1.GT.0 ) THEN CALL DGEMV( 'T', BSIZ1, BSIZ1, ONE, Q( QPTR( CURR ) ), & BSIZ1, ZTEMP( 1 ), 1, ZERO, Z( ZPTR1 ), 1 ) END IF CALL DCOPY( PSIZ1-BSIZ1, ZTEMP( BSIZ1+1 ), 1, Z( ZPTR1+BSIZ1 ), & 1 ) IF( BSIZ2.GT.0 ) THEN CALL DGEMV( 'T', BSIZ2, BSIZ2, ONE, Q( QPTR( CURR+1 ) ), & BSIZ2, ZTEMP( PSIZ1+1 ), 1, ZERO, Z( MID ), 1 ) END IF CALL DCOPY( PSIZ2-BSIZ2, ZTEMP( PSIZ1+BSIZ2+1 ), 1, & Z( MID+BSIZ2 ), 1 ) ! PTR = PTR + 2**( TLVLS-K ) 70 CONTINUE ! RETURN ! ! End of DLAEDA ! END SUBROUTINE DLAEIN( RIGHTV, NOINIT, N, H, LDH, WR, WI, VR, VI, B, & LDB, WORK, EPS3, SMLNUM, BIGNUM, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. LOGICAL NOINIT, RIGHTV INTEGER INFO, LDB, LDH, N DOUBLE PRECISION BIGNUM, EPS3, SMLNUM, WI, WR ! .. ! .. Array Arguments .. DOUBLE PRECISION B( LDB, * ), H( LDH, * ), VI( * ), VR( * ), & WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLAEIN uses inverse iteration to find a right or left eigenvector ! corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg ! matrix H. ! ! Arguments ! ========= ! ! RIGHTV (input) LOGICAL ! = .TRUE. : compute right eigenvector; ! = .FALSE.: compute left eigenvector. ! ! NOINIT (input) LOGICAL ! = .TRUE. : no initial vector supplied in (VR,VI). ! = .FALSE.: initial vector supplied in (VR,VI). ! ! N (input) INTEGER ! The order of the matrix H. N >= 0. ! ! H (input) DOUBLE PRECISION array, dimension (LDH,N) ! The upper Hessenberg matrix H. ! ! LDH (input) INTEGER ! The leading dimension of the array H. LDH >= max(1,N). ! ! WR (input) DOUBLE PRECISION ! WI (input) DOUBLE PRECISION ! The real and imaginary parts of the eigenvalue of H whose ! corresponding right or left eigenvector is to be computed. ! ! VR (input/output) DOUBLE PRECISION array, dimension (N) ! VI (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain ! a real starting vector for inverse iteration using the real ! eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI ! must contain the real and imaginary parts of a complex ! starting vector for inverse iteration using the complex ! eigenvalue (WR,WI); otherwise VR and VI need not be set. ! On exit, if WI = 0.0 (real eigenvalue), VR contains the ! computed real eigenvector; if WI.ne.0.0 (complex eigenvalue), ! VR and VI contain the real and imaginary parts of the ! computed complex eigenvector. The eigenvector is normalized ! so that the component of largest magnitude has magnitude 1; ! here the magnitude of a complex number (x,y) is taken to be ! |x| + |y|. ! VI is not referenced if WI = 0.0. ! ! B (workspace) DOUBLE PRECISION array, dimension (LDB,N) ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= N+1. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (N) ! ! EPS3 (input) DOUBLE PRECISION ! A small machine-dependent value which is used to perturb ! close eigenvalues, and to replace zero pivots. ! ! SMLNUM (input) DOUBLE PRECISION ! A machine-dependent value close to the underflow threshold. ! ! BIGNUM (input) DOUBLE PRECISION ! A machine-dependent value close to the overflow threshold. ! ! INFO (output) INTEGER ! = 0: successful exit ! = 1: inverse iteration did not converge; VR is set to the ! last iterate, and so is VI if WI.ne.0.0. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TENTH PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TENTH = 1.0D-1 ) ! .. ! .. Local Scalars .. CHARACTER NORMIN, TRANS INTEGER I, I1, I2, I3, IERR, ITS, J DOUBLE PRECISION ABSBII, ABSBJJ, EI, EJ, GROWTO, NORM, NRMSML, & REC, ROOTN, SCALE, TEMP, VCRIT, VMAX, VNORM, W, & W1, X, XI, XR, Y ! .. ! .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DASUM, DLAPY2, DNRM2 EXTERNAL IDAMAX, DASUM, DLAPY2, DNRM2 ! .. ! .. External Subroutines .. EXTERNAL DLADIV, DLATRS, DSCAL ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX, SQRT ! .. ! .. Executable Statements .. ! INFO = 0 ! ! GROWTO is the threshold used in the acceptance test for an ! eigenvector. ! ROOTN = SQRT( DBLE( N ) ) GROWTO = TENTH / ROOTN NRMSML = MAX( ONE, EPS3*ROOTN )*SMLNUM ! ! Form B = H - (WR,WI)*I (except that the subdiagonal elements and ! the imaginary parts of the diagonal elements are not stored). ! DO 20 J = 1, N DO 10 I = 1, J - 1 B( I, J ) = H( I, J ) 10 CONTINUE B( J, J ) = H( J, J ) - WR 20 CONTINUE ! IF( WI.EQ.ZERO ) THEN ! ! Real eigenvalue. ! IF( NOINIT ) THEN ! ! Set initial vector. ! DO 30 I = 1, N VR( I ) = EPS3 30 CONTINUE ELSE ! ! Scale supplied initial vector. ! VNORM = DNRM2( N, VR, 1 ) CALL DSCAL( N, ( EPS3*ROOTN ) / MAX( VNORM, NRMSML ), VR, & 1 ) END IF ! IF( RIGHTV ) THEN ! ! LU decomposition with partial pivoting of B, replacing zero ! pivots by EPS3. ! DO 60 I = 1, N - 1 EI = H( I+1, I ) IF( ABS( B( I, I ) ).LT.ABS( EI ) ) THEN ! ! Interchange rows and eliminate. ! X = B( I, I ) / EI B( I, I ) = EI DO 40 J = I + 1, N TEMP = B( I+1, J ) B( I+1, J ) = B( I, J ) - X*TEMP B( I, J ) = TEMP 40 CONTINUE ELSE ! ! Eliminate without interchange. ! IF( B( I, I ).EQ.ZERO ) & B( I, I ) = EPS3 X = EI / B( I, I ) IF( X.NE.ZERO ) THEN DO 50 J = I + 1, N B( I+1, J ) = B( I+1, J ) - X*B( I, J ) 50 CONTINUE END IF END IF 60 CONTINUE IF( B( N, N ).EQ.ZERO ) & B( N, N ) = EPS3 ! TRANS = 'N' ! ELSE ! ! UL decomposition with partial pivoting of B, replacing zero ! pivots by EPS3. ! DO 90 J = N, 2, -1 EJ = H( J, J-1 ) IF( ABS( B( J, J ) ).LT.ABS( EJ ) ) THEN ! ! Interchange columns and eliminate. ! X = B( J, J ) / EJ B( J, J ) = EJ DO 70 I = 1, J - 1 TEMP = B( I, J-1 ) B( I, J-1 ) = B( I, J ) - X*TEMP B( I, J ) = TEMP 70 CONTINUE ELSE ! ! Eliminate without interchange. ! IF( B( J, J ).EQ.ZERO ) & B( J, J ) = EPS3 X = EJ / B( J, J ) IF( X.NE.ZERO ) THEN DO 80 I = 1, J - 1 B( I, J-1 ) = B( I, J-1 ) - X*B( I, J ) 80 CONTINUE END IF END IF 90 CONTINUE IF( B( 1, 1 ).EQ.ZERO ) & B( 1, 1 ) = EPS3 ! TRANS = 'T' ! END IF ! NORMIN = 'N' DO 110 ITS = 1, N ! ! Solve U*x = scale*v for a right eigenvector ! or U'*x = scale*v for a left eigenvector, ! overwriting x on v. ! CALL DLATRS( 'Upper', TRANS, 'Nonunit', NORMIN, N, B, LDB, & VR, SCALE, WORK, IERR ) NORMIN = 'Y' ! ! Test for sufficient growth in the norm of v. ! VNORM = DASUM( N, VR, 1 ) IF( VNORM.GE.GROWTO*SCALE ) & GO TO 120 ! ! Choose new orthogonal starting vector and try again. ! TEMP = EPS3 / ( ROOTN+ONE ) VR( 1 ) = EPS3 DO 100 I = 2, N VR( I ) = TEMP 100 CONTINUE VR( N-ITS+1 ) = VR( N-ITS+1 ) - EPS3*ROOTN 110 CONTINUE ! ! Failure to find eigenvector in N iterations. ! INFO = 1 ! 120 CONTINUE ! ! Normalize eigenvector. ! I = IDAMAX( N, VR, 1 ) CALL DSCAL( N, ONE / ABS( VR( I ) ), VR, 1 ) ELSE ! ! Complex eigenvalue. ! IF( NOINIT ) THEN ! ! Set initial vector. ! DO 130 I = 1, N VR( I ) = EPS3 VI( I ) = ZERO 130 CONTINUE ELSE ! ! Scale supplied initial vector. ! NORM = DLAPY2( DNRM2( N, VR, 1 ), DNRM2( N, VI, 1 ) ) REC = ( EPS3*ROOTN ) / MAX( NORM, NRMSML ) CALL DSCAL( N, REC, VR, 1 ) CALL DSCAL( N, REC, VI, 1 ) END IF ! IF( RIGHTV ) THEN ! ! LU decomposition with partial pivoting of B, replacing zero ! pivots by EPS3. ! ! The imaginary part of the (i,j)-th element of U is stored in ! B(j+1,i). ! B( 2, 1 ) = -WI DO 140 I = 2, N B( I+1, 1 ) = ZERO 140 CONTINUE ! DO 170 I = 1, N - 1 ABSBII = DLAPY2( B( I, I ), B( I+1, I ) ) EI = H( I+1, I ) IF( ABSBII.LT.ABS( EI ) ) THEN ! ! Interchange rows and eliminate. ! XR = B( I, I ) / EI XI = B( I+1, I ) / EI B( I, I ) = EI B( I+1, I ) = ZERO DO 150 J = I + 1, N TEMP = B( I+1, J ) B( I+1, J ) = B( I, J ) - XR*TEMP B( J+1, I+1 ) = B( J+1, I ) - XI*TEMP B( I, J ) = TEMP B( J+1, I ) = ZERO 150 CONTINUE B( I+2, I ) = -WI B( I+1, I+1 ) = B( I+1, I+1 ) - XI*WI B( I+2, I+1 ) = B( I+2, I+1 ) + XR*WI ELSE ! ! Eliminate without interchanging rows. ! IF( ABSBII.EQ.ZERO ) THEN B( I, I ) = EPS3 B( I+1, I ) = ZERO ABSBII = EPS3 END IF EI = ( EI / ABSBII ) / ABSBII XR = B( I, I )*EI XI = -B( I+1, I )*EI DO 160 J = I + 1, N B( I+1, J ) = B( I+1, J ) - XR*B( I, J ) + & XI*B( J+1, I ) B( J+1, I+1 ) = -XR*B( J+1, I ) - XI*B( I, J ) 160 CONTINUE B( I+2, I+1 ) = B( I+2, I+1 ) - WI END IF ! ! Compute 1-norm of offdiagonal elements of i-th row. ! WORK( I ) = DASUM( N-I, B( I, I+1 ), LDB ) + & DASUM( N-I, B( I+2, I ), 1 ) 170 CONTINUE IF( B( N, N ).EQ.ZERO .AND. B( N+1, N ).EQ.ZERO ) & B( N, N ) = EPS3 WORK( N ) = ZERO ! I1 = N I2 = 1 I3 = -1 ELSE ! ! UL decomposition with partial pivoting of conjg(B), ! replacing zero pivots by EPS3. ! ! The imaginary part of the (i,j)-th element of U is stored in ! B(j+1,i). ! B( N+1, N ) = WI DO 180 J = 1, N - 1 B( N+1, J ) = ZERO 180 CONTINUE ! DO 210 J = N, 2, -1 EJ = H( J, J-1 ) ABSBJJ = DLAPY2( B( J, J ), B( J+1, J ) ) IF( ABSBJJ.LT.ABS( EJ ) ) THEN ! ! Interchange columns and eliminate ! XR = B( J, J ) / EJ XI = B( J+1, J ) / EJ B( J, J ) = EJ B( J+1, J ) = ZERO DO 190 I = 1, J - 1 TEMP = B( I, J-1 ) B( I, J-1 ) = B( I, J ) - XR*TEMP B( J, I ) = B( J+1, I ) - XI*TEMP B( I, J ) = TEMP B( J+1, I ) = ZERO 190 CONTINUE B( J+1, J-1 ) = WI B( J-1, J-1 ) = B( J-1, J-1 ) + XI*WI B( J, J-1 ) = B( J, J-1 ) - XR*WI ELSE ! ! Eliminate without interchange. ! IF( ABSBJJ.EQ.ZERO ) THEN B( J, J ) = EPS3 B( J+1, J ) = ZERO ABSBJJ = EPS3 END IF EJ = ( EJ / ABSBJJ ) / ABSBJJ XR = B( J, J )*EJ XI = -B( J+1, J )*EJ DO 200 I = 1, J - 1 B( I, J-1 ) = B( I, J-1 ) - XR*B( I, J ) + & XI*B( J+1, I ) B( J, I ) = -XR*B( J+1, I ) - XI*B( I, J ) 200 CONTINUE B( J, J-1 ) = B( J, J-1 ) + WI END IF ! ! Compute 1-norm of offdiagonal elements of j-th column. ! WORK( J ) = DASUM( J-1, B( 1, J ), 1 ) + & DASUM( J-1, B( J+1, 1 ), LDB ) 210 CONTINUE IF( B( 1, 1 ).EQ.ZERO .AND. B( 2, 1 ).EQ.ZERO ) & B( 1, 1 ) = EPS3 WORK( 1 ) = ZERO ! I1 = 1 I2 = N I3 = 1 END IF ! DO 270 ITS = 1, N SCALE = ONE VMAX = ONE VCRIT = BIGNUM ! ! Solve U*(xr,xi) = scale*(vr,vi) for a right eigenvector, ! or U'*(xr,xi) = scale*(vr,vi) for a left eigenvector, ! overwriting (xr,xi) on (vr,vi). ! DO 250 I = I1, I2, I3 ! IF( WORK( I ).GT.VCRIT ) THEN REC = ONE / VMAX CALL DSCAL( N, REC, VR, 1 ) CALL DSCAL( N, REC, VI, 1 ) SCALE = SCALE*REC VMAX = ONE VCRIT = BIGNUM END IF ! XR = VR( I ) XI = VI( I ) IF( RIGHTV ) THEN DO 220 J = I + 1, N XR = XR - B( I, J )*VR( J ) + B( J+1, I )*VI( J ) XI = XI - B( I, J )*VI( J ) - B( J+1, I )*VR( J ) 220 CONTINUE ELSE DO 230 J = 1, I - 1 XR = XR - B( J, I )*VR( J ) + B( I+1, J )*VI( J ) XI = XI - B( J, I )*VI( J ) - B( I+1, J )*VR( J ) 230 CONTINUE END IF ! W = ABS( B( I, I ) ) + ABS( B( I+1, I ) ) IF( W.GT.SMLNUM ) THEN IF( W.LT.ONE ) THEN W1 = ABS( XR ) + ABS( XI ) IF( W1.GT.W*BIGNUM ) THEN REC = ONE / W1 CALL DSCAL( N, REC, VR, 1 ) CALL DSCAL( N, REC, VI, 1 ) XR = VR( I ) XI = VI( I ) SCALE = SCALE*REC VMAX = VMAX*REC END IF END IF ! ! Divide by diagonal element of B. ! CALL DLADIV( XR, XI, B( I, I ), B( I+1, I ), VR( I ), & VI( I ) ) VMAX = MAX( ABS( VR( I ) )+ABS( VI( I ) ), VMAX ) VCRIT = BIGNUM / VMAX ELSE DO 240 J = 1, N VR( J ) = ZERO VI( J ) = ZERO 240 CONTINUE VR( I ) = ONE VI( I ) = ONE SCALE = ZERO VMAX = ONE VCRIT = BIGNUM END IF 250 CONTINUE ! ! Test for sufficient growth in the norm of (VR,VI). ! VNORM = DASUM( N, VR, 1 ) + DASUM( N, VI, 1 ) IF( VNORM.GE.GROWTO*SCALE ) & GO TO 280 ! ! Choose a new orthogonal starting vector and try again. ! Y = EPS3 / ( ROOTN+ONE ) VR( 1 ) = EPS3 VI( 1 ) = ZERO ! DO 260 I = 2, N VR( I ) = Y VI( I ) = ZERO 260 CONTINUE VR( N-ITS+1 ) = VR( N-ITS+1 ) - EPS3*ROOTN 270 CONTINUE ! ! Failure to find eigenvector in N iterations ! INFO = 1 ! 280 CONTINUE ! ! Normalize eigenvector. ! VNORM = ZERO DO 290 I = 1, N VNORM = MAX( VNORM, ABS( VR( I ) )+ABS( VI( I ) ) ) 290 CONTINUE CALL DSCAL( N, ONE / VNORM, VR, 1 ) CALL DSCAL( N, ONE / VNORM, VI, 1 ) ! END IF ! RETURN ! ! End of DLAEIN ! END SUBROUTINE DLAEV2( A, B, C, RT1, RT2, CS1, SN1 ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. DOUBLE PRECISION A, B, C, CS1, RT1, RT2, SN1 ! .. ! ! Purpose ! ======= ! ! DLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix ! [ A B ] ! [ B C ]. ! On return, RT1 is the eigenvalue of larger absolute value, RT2 is the ! eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right ! eigenvector for RT1, giving the decomposition ! ! [ CS1 SN1 ] [ A B ] [ CS1 -SN1 ] = [ RT1 0 ] ! [-SN1 CS1 ] [ B C ] [ SN1 CS1 ] [ 0 RT2 ]. ! ! Arguments ! ========= ! ! A (input) DOUBLE PRECISION ! The (1,1) element of the 2-by-2 matrix. ! ! B (input) DOUBLE PRECISION ! The (1,2) element and the conjugate of the (2,1) element of ! the 2-by-2 matrix. ! ! C (input) DOUBLE PRECISION ! The (2,2) element of the 2-by-2 matrix. ! ! RT1 (output) DOUBLE PRECISION ! The eigenvalue of larger absolute value. ! ! RT2 (output) DOUBLE PRECISION ! The eigenvalue of smaller absolute value. ! ! CS1 (output) DOUBLE PRECISION ! SN1 (output) DOUBLE PRECISION ! The vector (CS1, SN1) is a unit right eigenvector for RT1. ! ! Further Details ! =============== ! ! RT1 is accurate to a few ulps barring over/underflow. ! ! RT2 may be inaccurate if there is massive cancellation in the ! determinant A*C-B*B; higher precision or correctly rounded or ! correctly truncated arithmetic would be needed to compute RT2 ! accurately in all cases. ! ! CS1 and SN1 are accurate to a few ulps barring over/underflow. ! ! Overflow is possible only if RT1 is within a factor of 5 of overflow. ! Underflow is harmless if the input data is 0 or exceeds ! underflow_threshold / macheps. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D0 ) DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D0 ) DOUBLE PRECISION HALF PARAMETER ( HALF = 0.5D0 ) ! .. ! .. Local Scalars .. INTEGER SGN1, SGN2 DOUBLE PRECISION AB, ACMN, ACMX, ACS, ADF, CS, CT, DF, RT, SM, & TB, TN ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, SQRT ! .. ! .. Executable Statements .. ! ! Compute the eigenvalues ! SM = A + C DF = A - C ADF = ABS( DF ) TB = B + B AB = ABS( TB ) IF( ABS( A ).GT.ABS( C ) ) THEN ACMX = A ACMN = C ELSE ACMX = C ACMN = A END IF IF( ADF.GT.AB ) THEN RT = ADF*SQRT( ONE+( AB / ADF )**2 ) ELSE IF( ADF.LT.AB ) THEN RT = AB*SQRT( ONE+( ADF / AB )**2 ) ELSE ! ! Includes case AB=ADF=0 ! RT = AB*SQRT( TWO ) END IF IF( SM.LT.ZERO ) THEN RT1 = HALF*( SM-RT ) SGN1 = -1 ! ! Order of execution important. ! To get fully accurate smaller eigenvalue, ! next line needs to be executed in higher precision. ! RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B ELSE IF( SM.GT.ZERO ) THEN RT1 = HALF*( SM+RT ) SGN1 = 1 ! ! Order of execution important. ! To get fully accurate smaller eigenvalue, ! next line needs to be executed in higher precision. ! RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B ELSE ! ! Includes case RT1 = RT2 = 0 ! RT1 = HALF*RT RT2 = -HALF*RT SGN1 = 1 END IF ! ! Compute the eigenvector ! IF( DF.GE.ZERO ) THEN CS = DF + RT SGN2 = 1 ELSE CS = DF - RT SGN2 = -1 END IF ACS = ABS( CS ) IF( ACS.GT.AB ) THEN CT = -TB / CS SN1 = ONE / SQRT( ONE+CT*CT ) CS1 = CT*SN1 ELSE IF( AB.EQ.ZERO ) THEN CS1 = ONE SN1 = ZERO ELSE TN = -CS / TB CS1 = ONE / SQRT( ONE+TN*TN ) SN1 = TN*CS1 END IF END IF IF( SGN1.EQ.SGN2 ) THEN TN = CS1 CS1 = -SN1 SN1 = TN END IF RETURN ! ! End of DLAEV2 ! END SUBROUTINE DLAEXC( WANTQ, N, T, LDT, Q, LDQ, J1, N1, N2, WORK, & INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. LOGICAL WANTQ INTEGER INFO, J1, LDQ, LDT, N, N1, N2 ! .. ! .. Array Arguments .. DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in ! an upper quasi-triangular matrix T by an orthogonal similarity ! transformation. ! ! T must be in Schur canonical form, that is, block upper triangular ! with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block ! has its diagonal elemnts equal and its off-diagonal elements of ! opposite sign. ! ! Arguments ! ========= ! ! WANTQ (input) LOGICAL ! = .TRUE. : accumulate the transformation in the matrix Q; ! = .FALSE.: do not accumulate the transformation. ! ! N (input) INTEGER ! The order of the matrix T. N >= 0. ! ! T (input/output) DOUBLE PRECISION array, dimension (LDT,N) ! On entry, the upper quasi-triangular matrix T, in Schur ! canonical form. ! On exit, the updated matrix T, again in Schur canonical form. ! ! LDT (input) INTEGER ! The leading dimension of the array T. LDT >= max(1,N). ! ! Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) ! On entry, if WANTQ is .TRUE., the orthogonal matrix Q. ! On exit, if WANTQ is .TRUE., the updated matrix Q. ! If WANTQ is .FALSE., Q is not referenced. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. ! LDQ >= 1; and if WANTQ is .TRUE., LDQ >= N. ! ! J1 (input) INTEGER ! The index of the first row of the first block T11. ! ! N1 (input) INTEGER ! The order of the first block T11. N1 = 0, 1 or 2. ! ! N2 (input) INTEGER ! The order of the second block T22. N2 = 0, 1 or 2. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! = 1: the transformed matrix T would be too far from Schur ! form; the blocks are not swapped and T and Q are ! unchanged. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) DOUBLE PRECISION TEN PARAMETER ( TEN = 1.0D+1 ) INTEGER LDD, LDX PARAMETER ( LDD = 4, LDX = 2 ) ! .. ! .. Local Scalars .. INTEGER IERR, J2, J3, J4, K, ND DOUBLE PRECISION CS, DNORM, EPS, SCALE, SMLNUM, SN, T11, T22, & T33, TAU, TAU1, TAU2, TEMP, THRESH, WI1, WI2, & WR1, WR2, XNORM ! .. ! .. Local Arrays .. DOUBLE PRECISION D( LDD, 4 ), U( 3 ), U1( 3 ), U2( 3 ), & X( LDX, 2 ) ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL DLAMCH, DLANGE ! .. ! .. External Subroutines .. EXTERNAL DLACPY, DLANV2, DLARFG, DLARFX, DLARTG, DLASY2, & DROT ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. Executable Statements .. ! INFO = 0 ! ! Quick return if possible ! IF( N.EQ.0 .OR. N1.EQ.0 .OR. N2.EQ.0 ) & RETURN IF( J1+N1.GT.N ) & RETURN ! J2 = J1 + 1 J3 = J1 + 2 J4 = J1 + 3 ! IF( N1.EQ.1 .AND. N2.EQ.1 ) THEN ! ! Swap two 1-by-1 blocks. ! T11 = T( J1, J1 ) T22 = T( J2, J2 ) ! ! Determine the transformation to perform the interchange. ! CALL DLARTG( T( J1, J2 ), T22-T11, CS, SN, TEMP ) ! ! Apply transformation to the matrix T. ! IF( J3.LE.N ) & CALL DROT( N-J1-1, T( J1, J3 ), LDT, T( J2, J3 ), LDT, CS, & SN ) CALL DROT( J1-1, T( 1, J1 ), 1, T( 1, J2 ), 1, CS, SN ) ! T( J1, J1 ) = T22 T( J2, J2 ) = T11 ! IF( WANTQ ) THEN ! ! Accumulate transformation in the matrix Q. ! CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J2 ), 1, CS, SN ) END IF ! ELSE ! ! Swapping involves at least one 2-by-2 block. ! ! Copy the diagonal block of order N1+N2 to the local array D ! and compute its norm. ! ND = N1 + N2 CALL DLACPY( 'Full', ND, ND, T( J1, J1 ), LDT, D, LDD ) DNORM = DLANGE( 'Max', ND, ND, D, LDD, WORK ) ! ! Compute machine-dependent threshold for test for accepting ! swap. ! EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) / EPS THRESH = MAX( TEN*EPS*DNORM, SMLNUM ) ! ! Solve T11*X - X*T22 = scale*T12 for X. ! CALL DLASY2( .FALSE., .FALSE., -1, N1, N2, D, LDD, & D( N1+1, N1+1 ), LDD, D( 1, N1+1 ), LDD, SCALE, X, & LDX, XNORM, IERR ) ! ! Swap the adjacent diagonal blocks. ! K = N1 + N1 + N2 - 3 GO TO ( 10, 20, 30 )K ! 10 CONTINUE ! ! N1 = 1, N2 = 2: generate elementary reflector H so that: ! ! ( scale, X11, X12 ) H = ( 0, 0, * ) ! U( 1 ) = SCALE U( 2 ) = X( 1, 1 ) U( 3 ) = X( 1, 2 ) CALL DLARFG( 3, U( 3 ), U, 1, TAU ) U( 3 ) = ONE T11 = T( J1, J1 ) ! ! Perform swap provisionally on diagonal block in D. ! CALL DLARFX( 'L', 3, 3, U, TAU, D, LDD, WORK ) CALL DLARFX( 'R', 3, 3, U, TAU, D, LDD, WORK ) ! ! Test whether to reject swap. ! IF( MAX( ABS( D( 3, 1 ) ), ABS( D( 3, 2 ) ), ABS( D( 3, & 3 )-T11 ) ).GT.THRESH )GO TO 50 ! ! Accept swap: apply transformation to the entire matrix T. ! CALL DLARFX( 'L', 3, N-J1+1, U, TAU, T( J1, J1 ), LDT, WORK ) CALL DLARFX( 'R', J2, 3, U, TAU, T( 1, J1 ), LDT, WORK ) ! T( J3, J1 ) = ZERO T( J3, J2 ) = ZERO T( J3, J3 ) = T11 ! IF( WANTQ ) THEN ! ! Accumulate transformation in the matrix Q. ! CALL DLARFX( 'R', N, 3, U, TAU, Q( 1, J1 ), LDQ, WORK ) END IF GO TO 40 ! 20 CONTINUE ! ! N1 = 2, N2 = 1: generate elementary reflector H so that: ! ! H ( -X11 ) = ( * ) ! ( -X21 ) = ( 0 ) ! ( scale ) = ( 0 ) ! U( 1 ) = -X( 1, 1 ) U( 2 ) = -X( 2, 1 ) U( 3 ) = SCALE CALL DLARFG( 3, U( 1 ), U( 2 ), 1, TAU ) U( 1 ) = ONE T33 = T( J3, J3 ) ! ! Perform swap provisionally on diagonal block in D. ! CALL DLARFX( 'L', 3, 3, U, TAU, D, LDD, WORK ) CALL DLARFX( 'R', 3, 3, U, TAU, D, LDD, WORK ) ! ! Test whether to reject swap. ! IF( MAX( ABS( D( 2, 1 ) ), ABS( D( 3, 1 ) ), ABS( D( 1, & 1 )-T33 ) ).GT.THRESH )GO TO 50 ! ! Accept swap: apply transformation to the entire matrix T. ! CALL DLARFX( 'R', J3, 3, U, TAU, T( 1, J1 ), LDT, WORK ) CALL DLARFX( 'L', 3, N-J1, U, TAU, T( J1, J2 ), LDT, WORK ) ! T( J1, J1 ) = T33 T( J2, J1 ) = ZERO T( J3, J1 ) = ZERO ! IF( WANTQ ) THEN ! ! Accumulate transformation in the matrix Q. ! CALL DLARFX( 'R', N, 3, U, TAU, Q( 1, J1 ), LDQ, WORK ) END IF GO TO 40 ! 30 CONTINUE ! ! N1 = 2, N2 = 2: generate elementary reflectors H(1) and H(2) so ! that: ! ! H(2) H(1) ( -X11 -X12 ) = ( * * ) ! ( -X21 -X22 ) ( 0 * ) ! ( scale 0 ) ( 0 0 ) ! ( 0 scale ) ( 0 0 ) ! U1( 1 ) = -X( 1, 1 ) U1( 2 ) = -X( 2, 1 ) U1( 3 ) = SCALE CALL DLARFG( 3, U1( 1 ), U1( 2 ), 1, TAU1 ) U1( 1 ) = ONE ! TEMP = -TAU1*( X( 1, 2 )+U1( 2 )*X( 2, 2 ) ) U2( 1 ) = -TEMP*U1( 2 ) - X( 2, 2 ) U2( 2 ) = -TEMP*U1( 3 ) U2( 3 ) = SCALE CALL DLARFG( 3, U2( 1 ), U2( 2 ), 1, TAU2 ) U2( 1 ) = ONE ! ! Perform swap provisionally on diagonal block in D. ! CALL DLARFX( 'L', 3, 4, U1, TAU1, D, LDD, WORK ) CALL DLARFX( 'R', 4, 3, U1, TAU1, D, LDD, WORK ) CALL DLARFX( 'L', 3, 4, U2, TAU2, D( 2, 1 ), LDD, WORK ) CALL DLARFX( 'R', 4, 3, U2, TAU2, D( 1, 2 ), LDD, WORK ) ! ! Test whether to reject swap. ! IF( MAX( ABS( D( 3, 1 ) ), ABS( D( 3, 2 ) ), ABS( D( 4, 1 ) ), & ABS( D( 4, 2 ) ) ).GT.THRESH )GO TO 50 ! ! Accept swap: apply transformation to the entire matrix T. ! CALL DLARFX( 'L', 3, N-J1+1, U1, TAU1, T( J1, J1 ), LDT, WORK ) CALL DLARFX( 'R', J4, 3, U1, TAU1, T( 1, J1 ), LDT, WORK ) CALL DLARFX( 'L', 3, N-J1+1, U2, TAU2, T( J2, J1 ), LDT, WORK ) CALL DLARFX( 'R', J4, 3, U2, TAU2, T( 1, J2 ), LDT, WORK ) ! T( J3, J1 ) = ZERO T( J3, J2 ) = ZERO T( J4, J1 ) = ZERO T( J4, J2 ) = ZERO ! IF( WANTQ ) THEN ! ! Accumulate transformation in the matrix Q. ! CALL DLARFX( 'R', N, 3, U1, TAU1, Q( 1, J1 ), LDQ, WORK ) CALL DLARFX( 'R', N, 3, U2, TAU2, Q( 1, J2 ), LDQ, WORK ) END IF ! 40 CONTINUE ! IF( N2.EQ.2 ) THEN ! ! Standardize new 2-by-2 block T11 ! CALL DLANV2( T( J1, J1 ), T( J1, J2 ), T( J2, J1 ), & T( J2, J2 ), WR1, WI1, WR2, WI2, CS, SN ) CALL DROT( N-J1-1, T( J1, J1+2 ), LDT, T( J2, J1+2 ), LDT, & CS, SN ) CALL DROT( J1-1, T( 1, J1 ), 1, T( 1, J2 ), 1, CS, SN ) IF( WANTQ ) & CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J2 ), 1, CS, SN ) END IF ! IF( N1.EQ.2 ) THEN ! ! Standardize new 2-by-2 block T22 ! J3 = J1 + N2 J4 = J3 + 1 CALL DLANV2( T( J3, J3 ), T( J3, J4 ), T( J4, J3 ), & T( J4, J4 ), WR1, WI1, WR2, WI2, CS, SN ) IF( J3+2.LE.N ) & CALL DROT( N-J3-1, T( J3, J3+2 ), LDT, T( J4, J3+2 ), & LDT, CS, SN ) CALL DROT( J3-1, T( 1, J3 ), 1, T( 1, J4 ), 1, CS, SN ) IF( WANTQ ) & CALL DROT( N, Q( 1, J3 ), 1, Q( 1, J4 ), 1, CS, SN ) END IF ! END IF RETURN ! ! Exit with INFO = 1 if swap was rejected. ! 50 CONTINUE INFO = 1 RETURN ! ! End of DLAEXC ! END SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, & WR2, WI ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. INTEGER LDA, LDB DOUBLE PRECISION SAFMIN, SCALE1, SCALE2, WI, WR1, WR2 ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue ! problem A - w B, with scaling as necessary to avoid over-/underflow. ! ! The scaling factor "s" results in a modified eigenvalue equation ! ! s A - w B ! ! where s is a non-negative scaling factor chosen so that w, w B, ! and s A do not overflow and, if possible, do not underflow, either. ! ! Arguments ! ========= ! ! A (input) DOUBLE PRECISION array, dimension (LDA, 2) ! On entry, the 2 x 2 matrix A. It is assumed that its 1-norm ! is less than 1/SAFMIN. Entries less than ! sqrt(SAFMIN)*norm(A) are subject to being treated as zero. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= 2. ! ! B (input) DOUBLE PRECISION array, dimension (LDB, 2) ! On entry, the 2 x 2 upper triangular matrix B. It is ! assumed that the one-norm of B is less than 1/SAFMIN. The ! diagonals should be at least sqrt(SAFMIN) times the largest ! element of B (in absolute value); if a diagonal is smaller ! than that, then +/- sqrt(SAFMIN) will be used instead of ! that diagonal. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= 2. ! ! SAFMIN (input) DOUBLE PRECISION ! The smallest positive number s.t. 1/SAFMIN does not ! overflow. (This should always be DLAMCH('S') -- it is an ! argument in order to avoid having to call DLAMCH frequently.) ! ! SCALE1 (output) DOUBLE PRECISION ! A scaling factor used to avoid over-/underflow in the ! eigenvalue equation which defines the first eigenvalue. If ! the eigenvalues are complex, then the eigenvalues are ! ( WR1 +/- WI i ) / SCALE1 (which may lie outside the ! exponent range of the machine), SCALE1=SCALE2, and SCALE1 ! will always be positive. If the eigenvalues are real, then ! the first (real) eigenvalue is WR1 / SCALE1 , but this may ! overflow or underflow, and in fact, SCALE1 may be zero or ! less than the underflow threshhold if the exact eigenvalue ! is sufficiently large. ! ! SCALE2 (output) DOUBLE PRECISION ! A scaling factor used to avoid over-/underflow in the ! eigenvalue equation which defines the second eigenvalue. If ! the eigenvalues are complex, then SCALE2=SCALE1. If the ! eigenvalues are real, then the second (real) eigenvalue is ! WR2 / SCALE2 , but this may overflow or underflow, and in ! fact, SCALE2 may be zero or less than the underflow ! threshhold if the exact eigenvalue is sufficiently large. ! ! WR1 (output) DOUBLE PRECISION ! If the eigenvalue is real, then WR1 is SCALE1 times the ! eigenvalue closest to the (2,2) element of A B**(-1). If the ! eigenvalue is complex, then WR1=WR2 is SCALE1 times the real ! part of the eigenvalues. ! ! WR2 (output) DOUBLE PRECISION ! If the eigenvalue is real, then WR2 is SCALE2 times the ! other eigenvalue. If the eigenvalue is complex, then ! WR1=WR2 is SCALE1 times the real part of the eigenvalues. ! ! WI (output) DOUBLE PRECISION ! If the eigenvalue is real, then WI is zero. If the ! eigenvalue is complex, then WI is SCALE1 times the imaginary ! part of the eigenvalues. WI will always be non-negative. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 ) DOUBLE PRECISION HALF PARAMETER ( HALF = ONE / TWO ) DOUBLE PRECISION FUZZY1 PARAMETER ( FUZZY1 = ONE+1.0D-5 ) ! .. ! .. Local Scalars .. DOUBLE PRECISION A11, A12, A21, A22, ABI22, ANORM, AS11, AS12, & AS22, ASCALE, B11, B12, B22, BINV11, BINV22, & BMIN, BNORM, BSCALE, BSIZE, C1, C2, C3, C4, C5, & DIFF, DISCR, PP, QQ, R, RTMAX, RTMIN, S1, S2, & SAFMAX, SHIFT, SS, SUM, WABS, WBIG, WDET, & WSCALE, WSIZE, WSMALL ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SIGN, SQRT ! .. ! .. Executable Statements .. ! RTMIN = SQRT( SAFMIN ) RTMAX = ONE / RTMIN SAFMAX = ONE / SAFMIN ! ! Scale A ! ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ), & ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN ) ASCALE = ONE / ANORM A11 = ASCALE*A( 1, 1 ) A21 = ASCALE*A( 2, 1 ) A12 = ASCALE*A( 1, 2 ) A22 = ASCALE*A( 2, 2 ) ! ! Perturb B if necessary to insure non-singularity ! B11 = B( 1, 1 ) B12 = B( 1, 2 ) B22 = B( 2, 2 ) BMIN = RTMIN*MAX( ABS( B11 ), ABS( B12 ), ABS( B22 ), RTMIN ) IF( ABS( B11 ).LT.BMIN ) & B11 = SIGN( BMIN, B11 ) IF( ABS( B22 ).LT.BMIN ) & B22 = SIGN( BMIN, B22 ) ! ! Scale B ! BNORM = MAX( ABS( B11 ), ABS( B12 )+ABS( B22 ), SAFMIN ) BSIZE = MAX( ABS( B11 ), ABS( B22 ) ) BSCALE = ONE / BSIZE B11 = B11*BSCALE B12 = B12*BSCALE B22 = B22*BSCALE ! ! Compute larger eigenvalue by method described by C. van Loan ! ! ( AS is A shifted by -SHIFT*B ) ! BINV11 = ONE / B11 BINV22 = ONE / B22 S1 = A11*BINV11 S2 = A22*BINV22 IF( ABS( S1 ).LE.ABS( S2 ) ) THEN AS12 = A12 - S1*B12 AS22 = A22 - S1*B22 SS = A21*( BINV11*BINV22 ) ABI22 = AS22*BINV22 - SS*B12 PP = HALF*ABI22 SHIFT = S1 ELSE AS12 = A12 - S2*B12 AS11 = A11 - S2*B11 SS = A21*( BINV11*BINV22 ) ABI22 = -SS*B12 PP = HALF*( AS11*BINV11+ABI22 ) SHIFT = S2 END IF QQ = SS*AS12 IF( ABS( PP*RTMIN ).GE.ONE ) THEN DISCR = ( RTMIN*PP )**2 + QQ*SAFMIN R = SQRT( ABS( DISCR ) )*RTMAX ELSE IF( PP**2+ABS( QQ ).LE.SAFMIN ) THEN DISCR = ( RTMAX*PP )**2 + QQ*SAFMAX R = SQRT( ABS( DISCR ) )*RTMIN ELSE DISCR = PP**2 + QQ R = SQRT( ABS( DISCR ) ) END IF END IF ! ! Note: the test of R in the following IF is to cover the case when ! DISCR is small and negative and is flushed to zero during ! the calculation of R. On machines which have a consistent ! flush-to-zero threshhold and handle numbers above that ! threshhold correctly, it would not be necessary. ! IF( DISCR.GE.ZERO .OR. R.EQ.ZERO ) THEN SUM = PP + SIGN( R, PP ) DIFF = PP - SIGN( R, PP ) WBIG = SHIFT + SUM ! ! Compute smaller eigenvalue ! WSMALL = SHIFT + DIFF IF( HALF*ABS( WBIG ).GT.MAX( ABS( WSMALL ), SAFMIN ) ) THEN WDET = ( A11*A22-A12*A21 )*( BINV11*BINV22 ) WSMALL = WDET / WBIG END IF ! ! Choose (real) eigenvalue closest to 2,2 element of A*B**(-1) ! for WR1. ! IF( PP.GT.ABI22 ) THEN WR1 = MIN( WBIG, WSMALL ) WR2 = MAX( WBIG, WSMALL ) ELSE WR1 = MAX( WBIG, WSMALL ) WR2 = MIN( WBIG, WSMALL ) END IF WI = ZERO ELSE ! ! Complex eigenvalues ! WR1 = SHIFT + PP WR2 = WR1 WI = R END IF ! ! Further scaling to avoid underflow and overflow in computing ! SCALE1 and overflow in computing w*B. ! ! This scale factor (WSCALE) is bounded from above using C1 and C2, ! and from below using C3 and C4. ! C1 implements the condition s A must never overflow. ! C2 implements the condition w B must never overflow. ! C3, with C2, ! implement the condition that s A - w B must never overflow. ! C4 implements the condition s should not underflow. ! C5 implements the condition max(s,|w|) should be at least 2. ! C1 = BSIZE*( SAFMIN*MAX( ONE, ASCALE ) ) C2 = SAFMIN*MAX( ONE, BNORM ) C3 = BSIZE*SAFMIN IF( ASCALE.LE.ONE .AND. BSIZE.LE.ONE ) THEN C4 = MIN( ONE, ( ASCALE / SAFMIN )*BSIZE ) ELSE C4 = ONE END IF IF( ASCALE.LE.ONE .OR. BSIZE.LE.ONE ) THEN C5 = MIN( ONE, ASCALE*BSIZE ) ELSE C5 = ONE END IF ! ! Scale first eigenvalue ! WABS = ABS( WR1 ) + ABS( WI ) WSIZE = MAX( SAFMIN, C1, FUZZY1*( WABS*C2+C3 ), & MIN( C4, HALF*MAX( WABS, C5 ) ) ) IF( WSIZE.NE.ONE ) THEN WSCALE = ONE / WSIZE IF( WSIZE.GT.ONE ) THEN SCALE1 = ( MAX( ASCALE, BSIZE )*WSCALE )* & MIN( ASCALE, BSIZE ) ELSE SCALE1 = ( MIN( ASCALE, BSIZE )*WSCALE )* & MAX( ASCALE, BSIZE ) END IF WR1 = WR1*WSCALE IF( WI.NE.ZERO ) THEN WI = WI*WSCALE WR2 = WR1 SCALE2 = SCALE1 END IF ELSE SCALE1 = ASCALE*BSIZE SCALE2 = SCALE1 END IF ! ! Scale second eigenvalue (if real) ! IF( WI.EQ.ZERO ) THEN WSIZE = MAX( SAFMIN, C1, FUZZY1*( ABS( WR2 )*C2+C3 ), & MIN( C4, HALF*MAX( ABS( WR2 ), C5 ) ) ) IF( WSIZE.NE.ONE ) THEN WSCALE = ONE / WSIZE IF( WSIZE.GT.ONE ) THEN SCALE2 = ( MAX( ASCALE, BSIZE )*WSCALE )* & MIN( ASCALE, BSIZE ) ELSE SCALE2 = ( MIN( ASCALE, BSIZE )*WSCALE )* & MAX( ASCALE, BSIZE ) END IF WR2 = WR2*WSCALE ELSE SCALE2 = ASCALE*BSIZE END IF END IF ! ! End of DLAG2 ! RETURN END SUBROUTINE DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, & SNV, CSQ, SNQ ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. LOGICAL UPPER DOUBLE PRECISION A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, SNQ, & SNU, SNV ! .. ! ! Purpose ! ======= ! ! DLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such ! that if ( UPPER ) then ! ! U'*A*Q = U'*( A1 A2 )*Q = ( x 0 ) ! ( 0 A3 ) ( x x ) ! and ! V'*B*Q = V'*( B1 B2 )*Q = ( x 0 ) ! ( 0 B3 ) ( x x ) ! ! or if ( .NOT.UPPER ) then ! ! U'*A*Q = U'*( A1 0 )*Q = ( x x ) ! ( A2 A3 ) ( 0 x ) ! and ! V'*B*Q = V'*( B1 0 )*Q = ( x x ) ! ( B2 B3 ) ( 0 x ) ! ! The rows of the transformed A and B are parallel, where ! ! U = ( CSU SNU ), V = ( CSV SNV ), Q = ( CSQ SNQ ) ! ( -SNU CSU ) ( -SNV CSV ) ( -SNQ CSQ ) ! ! Z' denotes the transpose of Z. ! ! ! Arguments ! ========= ! ! UPPER (input) LOGICAL ! = .TRUE.: the input matrices A and B are upper triangular. ! = .FALSE.: the input matrices A and B are lower triangular. ! ! A1 (input) DOUBLE PRECISION ! A2 (input) DOUBLE PRECISION ! A3 (input) DOUBLE PRECISION ! On entry, A1, A2 and A3 are elements of the input 2-by-2 ! upper (lower) triangular matrix A. ! ! B1 (input) DOUBLE PRECISION ! B2 (input) DOUBLE PRECISION ! B3 (input) DOUBLE PRECISION ! On entry, B1, B2 and B3 are elements of the input 2-by-2 ! upper (lower) triangular matrix B. ! ! CSU (output) DOUBLE PRECISION ! SNU (output) DOUBLE PRECISION ! The desired orthogonal matrix U. ! ! CSV (output) DOUBLE PRECISION ! SNV (output) DOUBLE PRECISION ! The desired orthogonal matrix V. ! ! CSQ (output) DOUBLE PRECISION ! SNQ (output) DOUBLE PRECISION ! The desired orthogonal matrix Q. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. DOUBLE PRECISION A, AUA11, AUA12, AUA21, AUA22, AVB11, AVB12, & AVB21, AVB22, B, C, CSL, CSR, D, R, S1, S2, & SNL, SNR, UA11, UA11R, UA12, UA21, UA22, UA22R, & VB11, VB11R, VB12, VB21, VB22, VB22R ! .. ! .. External Subroutines .. EXTERNAL DLARTG, DLASV2 ! .. ! .. Intrinsic Functions .. INTRINSIC ABS ! .. ! .. Executable Statements .. ! IF( UPPER ) THEN ! ! Input matrices A and B are upper triangular matrices ! ! Form matrix C = A*adj(B) = ( a b ) ! ( 0 d ) ! A = A1*B3 D = A3*B1 B = A2*B1 - A1*B2 ! ! The SVD of real 2-by-2 triangular C ! ! ( CSL -SNL )*( A B )*( CSR SNR ) = ( R 0 ) ! ( SNL CSL ) ( 0 D ) ( -SNR CSR ) ( 0 T ) ! CALL DLASV2( A, B, D, S1, S2, SNR, CSR, SNL, CSL ) ! IF( ABS( CSL ).GE.ABS( SNL ) .OR. ABS( CSR ).GE.ABS( SNR ) ) & THEN ! ! Compute the (1,1) and (1,2) elements of U'*A and V'*B, ! and (1,2) element of |U|'*|A| and |V|'*|B|. ! UA11R = CSL*A1 UA12 = CSL*A2 + SNL*A3 ! VB11R = CSR*B1 VB12 = CSR*B2 + SNR*B3 ! AUA12 = ABS( CSL )*ABS( A2 ) + ABS( SNL )*ABS( A3 ) AVB12 = ABS( CSR )*ABS( B2 ) + ABS( SNR )*ABS( B3 ) ! ! zero (1,2) elements of U'*A and V'*B ! IF( ( ABS( UA11R )+ABS( UA12 ) ).NE.ZERO ) THEN IF( AUA12 / ( ABS( UA11R )+ABS( UA12 ) ).LE.AVB12 / & ( ABS( VB11R )+ABS( VB12 ) ) ) THEN CALL DLARTG( -UA11R, UA12, CSQ, SNQ, R ) ELSE CALL DLARTG( -VB11R, VB12, CSQ, SNQ, R ) END IF ELSE CALL DLARTG( -VB11R, VB12, CSQ, SNQ, R ) END IF ! CSU = CSL SNU = -SNL CSV = CSR SNV = -SNR ! ELSE ! ! Compute the (2,1) and (2,2) elements of U'*A and V'*B, ! and (2,2) element of |U|'*|A| and |V|'*|B|. ! UA21 = -SNL*A1 UA22 = -SNL*A2 + CSL*A3 ! VB21 = -SNR*B1 VB22 = -SNR*B2 + CSR*B3 ! AUA22 = ABS( SNL )*ABS( A2 ) + ABS( CSL )*ABS( A3 ) AVB22 = ABS( SNR )*ABS( B2 ) + ABS( CSR )*ABS( B3 ) ! ! zero (2,2) elements of U'*A and V'*B, and then swap. ! IF( ( ABS( UA21 )+ABS( UA22 ) ).NE.ZERO ) THEN IF( AUA22 / ( ABS( UA21 )+ABS( UA22 ) ).LE.AVB22 / & ( ABS( VB21 )+ABS( VB22 ) ) ) THEN CALL DLARTG( -UA21, UA22, CSQ, SNQ, R ) ELSE CALL DLARTG( -VB21, VB22, CSQ, SNQ, R ) END IF ELSE CALL DLARTG( -VB21, VB22, CSQ, SNQ, R ) END IF ! CSU = SNL SNU = CSL CSV = SNR SNV = CSR ! END IF ! ELSE ! ! Input matrices A and B are lower triangular matrices ! ! Form matrix C = A*adj(B) = ( a 0 ) ! ( c d ) ! A = A1*B3 D = A3*B1 C = A2*B3 - A3*B2 ! ! The SVD of real 2-by-2 triangular C ! ! ( CSL -SNL )*( A 0 )*( CSR SNR ) = ( R 0 ) ! ( SNL CSL ) ( C D ) ( -SNR CSR ) ( 0 T ) ! CALL DLASV2( A, C, D, S1, S2, SNR, CSR, SNL, CSL ) ! IF( ABS( CSR ).GE.ABS( SNR ) .OR. ABS( CSL ).GE.ABS( SNL ) ) & THEN ! ! Compute the (2,1) and (2,2) elements of U'*A and V'*B, ! and (2,1) element of |U|'*|A| and |V|'*|B|. ! UA21 = -SNR*A1 + CSR*A2 UA22R = CSR*A3 ! VB21 = -SNL*B1 + CSL*B2 VB22R = CSL*B3 ! AUA21 = ABS( SNR )*ABS( A1 ) + ABS( CSR )*ABS( A2 ) AVB21 = ABS( SNL )*ABS( B1 ) + ABS( CSL )*ABS( B2 ) ! ! zero (2,1) elements of U'*A and V'*B. ! IF( ( ABS( UA21 )+ABS( UA22R ) ).NE.ZERO ) THEN IF( AUA21 / ( ABS( UA21 )+ABS( UA22R ) ).LE.AVB21 / & ( ABS( VB21 )+ABS( VB22R ) ) ) THEN CALL DLARTG( UA22R, UA21, CSQ, SNQ, R ) ELSE CALL DLARTG( VB22R, VB21, CSQ, SNQ, R ) END IF ELSE CALL DLARTG( VB22R, VB21, CSQ, SNQ, R ) END IF ! CSU = CSR SNU = -SNR CSV = CSL SNV = -SNL ! ELSE ! ! Compute the (1,1) and (1,2) elements of U'*A and V'*B, ! and (1,1) element of |U|'*|A| and |V|'*|B|. ! UA11 = CSR*A1 + SNR*A2 UA12 = SNR*A3 ! VB11 = CSL*B1 + SNL*B2 VB12 = SNL*B3 ! AUA11 = ABS( CSR )*ABS( A1 ) + ABS( SNR )*ABS( A2 ) AVB11 = ABS( CSL )*ABS( B1 ) + ABS( SNL )*ABS( B2 ) ! ! zero (1,1) elements of U'*A and V'*B, and then swap. ! IF( ( ABS( UA11 )+ABS( UA12 ) ).NE.ZERO ) THEN IF( AUA11 / ( ABS( UA11 )+ABS( UA12 ) ).LE.AVB11 / & ( ABS( VB11 )+ABS( VB12 ) ) ) THEN CALL DLARTG( UA12, UA11, CSQ, SNQ, R ) ELSE CALL DLARTG( VB12, VB11, CSQ, SNQ, R ) END IF ELSE CALL DLARTG( VB12, VB11, CSQ, SNQ, R ) END IF ! CSU = SNR SNU = CSR CSV = SNL SNV = CSL ! END IF ! END IF ! RETURN ! ! End of DLAGS2 ! END SUBROUTINE DLAGTF( N, A, LAMBDA, B, C, TOL, D, IN, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, N DOUBLE PRECISION LAMBDA, TOL ! .. ! .. Array Arguments .. INTEGER IN( * ) DOUBLE PRECISION A( * ), B( * ), C( * ), D( * ) ! .. ! ! Purpose ! ======= ! ! DLAGTF factorizes the matrix (T - lambda*I), where T is an n by n ! tridiagonal matrix and lambda is a scalar, as ! ! T - lambda*I = PLU, ! ! where P is a permutation matrix, L is a unit lower tridiagonal matrix ! with at most one non-zero sub-diagonal elements per column and U is ! an upper triangular matrix with at most two non-zero super-diagonal ! elements per column. ! ! The factorization is obtained by Gaussian elimination with partial ! pivoting and implicit row scaling. ! ! The parameter LAMBDA is included in the routine so that DLAGTF may ! be used, in conjunction with DLAGTS, to obtain eigenvectors of T by ! inverse iteration. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix T. ! ! A (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, A must contain the diagonal elements of T. ! ! On exit, A is overwritten by the n diagonal elements of the ! upper triangular matrix U of the factorization of T. ! ! LAMBDA (input) DOUBLE PRECISION ! On entry, the scalar lambda. ! ! B (input/output) DOUBLE PRECISION array, dimension (N-1) ! On entry, B must contain the (n-1) super-diagonal elements of ! T. ! ! On exit, B is overwritten by the (n-1) super-diagonal ! elements of the matrix U of the factorization of T. ! ! C (input/output) DOUBLE PRECISION array, dimension (N-1) ! On entry, C must contain the (n-1) sub-diagonal elements of ! T. ! ! On exit, C is overwritten by the (n-1) sub-diagonal elements ! of the matrix L of the factorization of T. ! ! TOL (input) DOUBLE PRECISION ! On entry, a relative tolerance used to indicate whether or ! not the matrix (T - lambda*I) is nearly singular. TOL should ! normally be chose as approximately the largest relative error ! in the elements of T. For example, if the elements of T are ! correct to about 4 significant figures, then TOL should be ! set to about 5*10**(-4). If TOL is supplied as less than eps, ! where eps is the relative machine precision, then the value ! eps is used in place of TOL. ! ! D (output) DOUBLE PRECISION array, dimension (N-2) ! On exit, D is overwritten by the (n-2) second super-diagonal ! elements of the matrix U of the factorization of T. ! ! IN (output) INTEGER array, dimension (N) ! On exit, IN contains details of the permutation matrix P. If ! an interchange occurred at the kth step of the elimination, ! then IN(k) = 1, otherwise IN(k) = 0. The element IN(n) ! returns the smallest positive integer j such that ! ! abs( u(j,j) ).le. norm( (T - lambda*I)(j) )*TOL, ! ! where norm( A(j) ) denotes the sum of the absolute values of ! the jth row of the matrix A. If no such j exists then IN(n) ! is returned as zero. If IN(n) is returned as positive, then a ! diagonal element of U is small, indicating that ! (T - lambda*I) is singular or nearly singular, ! ! INFO (output) INTEGER ! = 0 : successful exit ! .lt. 0: if INFO = -k, the kth argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER K DOUBLE PRECISION EPS, MULT, PIV1, PIV2, SCALE1, SCALE2, TEMP, TL ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Executable Statements .. ! INFO = 0 IF( N.LT.0 ) THEN INFO = -1 CALL XERBLA( 'DLAGTF', -INFO ) RETURN END IF ! IF( N.EQ.0 ) & RETURN ! A( 1 ) = A( 1 ) - LAMBDA IN( N ) = 0 IF( N.EQ.1 ) THEN IF( A( 1 ).EQ.ZERO ) & IN( 1 ) = 1 RETURN END IF ! EPS = DLAMCH( 'Epsilon' ) ! TL = MAX( TOL, EPS ) SCALE1 = ABS( A( 1 ) ) + ABS( B( 1 ) ) DO 10 K = 1, N - 1 A( K+1 ) = A( K+1 ) - LAMBDA SCALE2 = ABS( C( K ) ) + ABS( A( K+1 ) ) IF( K.LT.( N-1 ) ) & SCALE2 = SCALE2 + ABS( B( K+1 ) ) IF( A( K ).EQ.ZERO ) THEN PIV1 = ZERO ELSE PIV1 = ABS( A( K ) ) / SCALE1 END IF IF( C( K ).EQ.ZERO ) THEN IN( K ) = 0 PIV2 = ZERO SCALE1 = SCALE2 IF( K.LT.( N-1 ) ) & D( K ) = ZERO ELSE PIV2 = ABS( C( K ) ) / SCALE2 IF( PIV2.LE.PIV1 ) THEN IN( K ) = 0 SCALE1 = SCALE2 C( K ) = C( K ) / A( K ) A( K+1 ) = A( K+1 ) - C( K )*B( K ) IF( K.LT.( N-1 ) ) & D( K ) = ZERO ELSE IN( K ) = 1 MULT = A( K ) / C( K ) A( K ) = C( K ) TEMP = A( K+1 ) A( K+1 ) = B( K ) - MULT*TEMP IF( K.LT.( N-1 ) ) THEN D( K ) = B( K+1 ) B( K+1 ) = -MULT*D( K ) END IF B( K ) = TEMP C( K ) = MULT END IF END IF IF( ( MAX( PIV1, PIV2 ).LE.TL ) .AND. ( IN( N ).EQ.0 ) ) & IN( N ) = K 10 CONTINUE IF( ( ABS( A( N ) ).LE.SCALE1*TL ) .AND. ( IN( N ).EQ.0 ) ) & IN( N ) = N ! RETURN ! ! End of DLAGTF ! END SUBROUTINE DLAGTM( TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX, BETA, & B, LDB ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER TRANS INTEGER LDB, LDX, N, NRHS DOUBLE PRECISION ALPHA, BETA ! .. ! .. Array Arguments .. DOUBLE PRECISION B( LDB, * ), D( * ), DL( * ), DU( * ), & X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DLAGTM performs a matrix-vector product of the form ! ! B := alpha * A * X + beta * B ! ! where A is a tridiagonal matrix of order N, B and X are N by NRHS ! matrices, and alpha and beta are real scalars, each of which may be ! 0., 1., or -1. ! ! Arguments ! ========= ! ! TRANS (input) CHARACTER ! Specifies the operation applied to A. ! = 'N': No transpose, B := alpha * A * X + beta * B ! = 'T': Transpose, B := alpha * A'* X + beta * B ! = 'C': Conjugate transpose = Transpose ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices X and B. ! ! ALPHA (input) DOUBLE PRECISION ! The scalar alpha. ALPHA must be 0., 1., or -1.; otherwise, ! it is assumed to be 0. ! ! DL (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) sub-diagonal elements of T. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The diagonal elements of T. ! ! DU (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) super-diagonal elements of T. ! ! X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) ! The N by NRHS matrix X. ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(N,1). ! ! BETA (input) DOUBLE PRECISION ! The scalar beta. BETA must be 0., 1., or -1.; otherwise, ! it is assumed to be 1. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the N by NRHS matrix B. ! On exit, B is overwritten by the matrix expression ! B := alpha * A * X + beta * B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(N,1). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Executable Statements .. ! IF( N.EQ.0 ) & RETURN ! ! Multiply B by BETA if BETA.NE.1. ! IF( BETA.EQ.ZERO ) THEN DO 20 J = 1, NRHS DO 10 I = 1, N B( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ELSE IF( BETA.EQ.-ONE ) THEN DO 40 J = 1, NRHS DO 30 I = 1, N B( I, J ) = -B( I, J ) 30 CONTINUE 40 CONTINUE END IF ! IF( ALPHA.EQ.ONE ) THEN IF( LSAME( TRANS, 'N' ) ) THEN ! ! Compute B := B + A*X ! DO 60 J = 1, NRHS IF( N.EQ.1 ) THEN B( 1, J ) = B( 1, J ) + D( 1 )*X( 1, J ) ELSE B( 1, J ) = B( 1, J ) + D( 1 )*X( 1, J ) + & DU( 1 )*X( 2, J ) B( N, J ) = B( N, J ) + DL( N-1 )*X( N-1, J ) + & D( N )*X( N, J ) DO 50 I = 2, N - 1 B( I, J ) = B( I, J ) + DL( I-1 )*X( I-1, J ) + & D( I )*X( I, J ) + DU( I )*X( I+1, J ) 50 CONTINUE END IF 60 CONTINUE ELSE ! ! Compute B := B + A'*X ! DO 80 J = 1, NRHS IF( N.EQ.1 ) THEN B( 1, J ) = B( 1, J ) + D( 1 )*X( 1, J ) ELSE B( 1, J ) = B( 1, J ) + D( 1 )*X( 1, J ) + & DL( 1 )*X( 2, J ) B( N, J ) = B( N, J ) + DU( N-1 )*X( N-1, J ) + & D( N )*X( N, J ) DO 70 I = 2, N - 1 B( I, J ) = B( I, J ) + DU( I-1 )*X( I-1, J ) + & D( I )*X( I, J ) + DL( I )*X( I+1, J ) 70 CONTINUE END IF 80 CONTINUE END IF ELSE IF( ALPHA.EQ.-ONE ) THEN IF( LSAME( TRANS, 'N' ) ) THEN ! ! Compute B := B - A*X ! DO 100 J = 1, NRHS IF( N.EQ.1 ) THEN B( 1, J ) = B( 1, J ) - D( 1 )*X( 1, J ) ELSE B( 1, J ) = B( 1, J ) - D( 1 )*X( 1, J ) - & DU( 1 )*X( 2, J ) B( N, J ) = B( N, J ) - DL( N-1 )*X( N-1, J ) - & D( N )*X( N, J ) DO 90 I = 2, N - 1 B( I, J ) = B( I, J ) - DL( I-1 )*X( I-1, J ) - & D( I )*X( I, J ) - DU( I )*X( I+1, J ) 90 CONTINUE END IF 100 CONTINUE ELSE ! ! Compute B := B - A'*X ! DO 120 J = 1, NRHS IF( N.EQ.1 ) THEN B( 1, J ) = B( 1, J ) - D( 1 )*X( 1, J ) ELSE B( 1, J ) = B( 1, J ) - D( 1 )*X( 1, J ) - & DL( 1 )*X( 2, J ) B( N, J ) = B( N, J ) - DU( N-1 )*X( N-1, J ) - & D( N )*X( N, J ) DO 110 I = 2, N - 1 B( I, J ) = B( I, J ) - DU( I-1 )*X( I-1, J ) - & D( I )*X( I, J ) - DL( I )*X( I+1, J ) 110 CONTINUE END IF 120 CONTINUE END IF END IF RETURN ! ! End of DLAGTM ! END SUBROUTINE DLAGTS( JOB, N, A, B, C, D, IN, Y, TOL, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. INTEGER INFO, JOB, N DOUBLE PRECISION TOL ! .. ! .. Array Arguments .. INTEGER IN( * ) DOUBLE PRECISION A( * ), B( * ), C( * ), D( * ), Y( * ) ! .. ! ! Purpose ! ======= ! ! DLAGTS may be used to solve one of the systems of equations ! ! (T - lambda*I)*x = y or (T - lambda*I)'*x = y, ! ! where T is an n by n tridiagonal matrix, for x, following the ! factorization of (T - lambda*I) as ! ! (T - lambda*I) = P*L*U , ! ! by routine DLAGTF. The choice of equation to be solved is ! controlled by the argument JOB, and in each case there is an option ! to perturb zero or very small diagonal elements of U, this option ! being intended for use in applications such as inverse iteration. ! ! Arguments ! ========= ! ! JOB (input) INTEGER ! Specifies the job to be performed by DLAGTS as follows: ! = 1: The equations (T - lambda*I)x = y are to be solved, ! but diagonal elements of U are not to be perturbed. ! = -1: The equations (T - lambda*I)x = y are to be solved ! and, if overflow would otherwise occur, the diagonal ! elements of U are to be perturbed. See argument TOL ! below. ! = 2: The equations (T - lambda*I)'x = y are to be solved, ! but diagonal elements of U are not to be perturbed. ! = -2: The equations (T - lambda*I)'x = y are to be solved ! and, if overflow would otherwise occur, the diagonal ! elements of U are to be perturbed. See argument TOL ! below. ! ! N (input) INTEGER ! The order of the matrix T. ! ! A (input) DOUBLE PRECISION array, dimension (N) ! On entry, A must contain the diagonal elements of U as ! returned from DLAGTF. ! ! B (input) DOUBLE PRECISION array, dimension (N-1) ! On entry, B must contain the first super-diagonal elements of ! U as returned from DLAGTF. ! ! C (input) DOUBLE PRECISION array, dimension (N-1) ! On entry, C must contain the sub-diagonal elements of L as ! returned from DLAGTF. ! ! D (input) DOUBLE PRECISION array, dimension (N-2) ! On entry, D must contain the second super-diagonal elements ! of U as returned from DLAGTF. ! ! IN (input) INTEGER array, dimension (N) ! On entry, IN must contain details of the matrix P as returned ! from DLAGTF. ! ! Y (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the right hand side vector y. ! On exit, Y is overwritten by the solution vector x. ! ! TOL (input/output) DOUBLE PRECISION ! On entry, with JOB .lt. 0, TOL should be the minimum ! perturbation to be made to very small diagonal elements of U. ! TOL should normally be chosen as about eps*norm(U), where eps ! is the relative machine precision, but if TOL is supplied as ! non-positive, then it is reset to eps*max( abs( u(i,j) ) ). ! If JOB .gt. 0 then TOL is not referenced. ! ! On exit, TOL is changed as described above, only if TOL is ! non-positive on entry. Otherwise TOL is unchanged. ! ! INFO (output) INTEGER ! = 0 : successful exit ! .lt. 0: if INFO = -i, the i-th argument had an illegal value ! .gt. 0: overflow would occur when computing the INFO(th) ! element of the solution vector x. This can only occur ! when JOB is supplied as positive and either means ! that a diagonal element of U is very small, or that ! the elements of the right-hand side vector y are very ! large. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER K DOUBLE PRECISION ABSAK, AK, BIGNUM, EPS, PERT, SFMIN, TEMP ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SIGN ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Executable Statements .. ! INFO = 0 IF( ( ABS( JOB ).GT.2 ) .OR. ( JOB.EQ.0 ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLAGTS', -INFO ) RETURN END IF ! IF( N.EQ.0 ) & RETURN ! EPS = DLAMCH( 'Epsilon' ) SFMIN = DLAMCH( 'Safe minimum' ) BIGNUM = ONE / SFMIN ! IF( JOB.LT.0 ) THEN IF( TOL.LE.ZERO ) THEN TOL = ABS( A( 1 ) ) IF( N.GT.1 ) & TOL = MAX( TOL, ABS( A( 2 ) ), ABS( B( 1 ) ) ) DO 10 K = 3, N TOL = MAX( TOL, ABS( A( K ) ), ABS( B( K-1 ) ), & ABS( D( K-2 ) ) ) 10 CONTINUE TOL = TOL*EPS IF( TOL.EQ.ZERO ) & TOL = EPS END IF END IF ! IF( ABS( JOB ).EQ.1 ) THEN DO 20 K = 2, N IF( IN( K-1 ).EQ.0 ) THEN Y( K ) = Y( K ) - C( K-1 )*Y( K-1 ) ELSE TEMP = Y( K-1 ) Y( K-1 ) = Y( K ) Y( K ) = TEMP - C( K-1 )*Y( K ) END IF 20 CONTINUE IF( JOB.EQ.1 ) THEN DO 30 K = N, 1, -1 IF( K.LE.N-2 ) THEN TEMP = Y( K ) - B( K )*Y( K+1 ) - D( K )*Y( K+2 ) ELSE IF( K.EQ.N-1 ) THEN TEMP = Y( K ) - B( K )*Y( K+1 ) ELSE TEMP = Y( K ) END IF AK = A( K ) ABSAK = ABS( AK ) IF( ABSAK.LT.ONE ) THEN IF( ABSAK.LT.SFMIN ) THEN IF( ABSAK.EQ.ZERO .OR. ABS( TEMP )*SFMIN.GT.ABSAK ) & THEN INFO = K RETURN ELSE TEMP = TEMP*BIGNUM AK = AK*BIGNUM END IF ELSE IF( ABS( TEMP ).GT.ABSAK*BIGNUM ) THEN INFO = K RETURN END IF END IF Y( K ) = TEMP / AK 30 CONTINUE ELSE DO 50 K = N, 1, -1 IF( K.LE.N-2 ) THEN TEMP = Y( K ) - B( K )*Y( K+1 ) - D( K )*Y( K+2 ) ELSE IF( K.EQ.N-1 ) THEN TEMP = Y( K ) - B( K )*Y( K+1 ) ELSE TEMP = Y( K ) END IF AK = A( K ) PERT = SIGN( TOL, AK ) 40 CONTINUE ABSAK = ABS( AK ) IF( ABSAK.LT.ONE ) THEN IF( ABSAK.LT.SFMIN ) THEN IF( ABSAK.EQ.ZERO .OR. ABS( TEMP )*SFMIN.GT.ABSAK ) & THEN AK = AK + PERT PERT = 2*PERT GO TO 40 ELSE TEMP = TEMP*BIGNUM AK = AK*BIGNUM END IF ELSE IF( ABS( TEMP ).GT.ABSAK*BIGNUM ) THEN AK = AK + PERT PERT = 2*PERT GO TO 40 END IF END IF Y( K ) = TEMP / AK 50 CONTINUE END IF ELSE ! ! Come to here if JOB = 2 or -2 ! IF( JOB.EQ.2 ) THEN DO 60 K = 1, N IF( K.GE.3 ) THEN TEMP = Y( K ) - B( K-1 )*Y( K-1 ) - D( K-2 )*Y( K-2 ) ELSE IF( K.EQ.2 ) THEN TEMP = Y( K ) - B( K-1 )*Y( K-1 ) ELSE TEMP = Y( K ) END IF AK = A( K ) ABSAK = ABS( AK ) IF( ABSAK.LT.ONE ) THEN IF( ABSAK.LT.SFMIN ) THEN IF( ABSAK.EQ.ZERO .OR. ABS( TEMP )*SFMIN.GT.ABSAK ) & THEN INFO = K RETURN ELSE TEMP = TEMP*BIGNUM AK = AK*BIGNUM END IF ELSE IF( ABS( TEMP ).GT.ABSAK*BIGNUM ) THEN INFO = K RETURN END IF END IF Y( K ) = TEMP / AK 60 CONTINUE ELSE DO 80 K = 1, N IF( K.GE.3 ) THEN TEMP = Y( K ) - B( K-1 )*Y( K-1 ) - D( K-2 )*Y( K-2 ) ELSE IF( K.EQ.2 ) THEN TEMP = Y( K ) - B( K-1 )*Y( K-1 ) ELSE TEMP = Y( K ) END IF AK = A( K ) PERT = SIGN( TOL, AK ) 70 CONTINUE ABSAK = ABS( AK ) IF( ABSAK.LT.ONE ) THEN IF( ABSAK.LT.SFMIN ) THEN IF( ABSAK.EQ.ZERO .OR. ABS( TEMP )*SFMIN.GT.ABSAK ) & THEN AK = AK + PERT PERT = 2*PERT GO TO 70 ELSE TEMP = TEMP*BIGNUM AK = AK*BIGNUM END IF ELSE IF( ABS( TEMP ).GT.ABSAK*BIGNUM ) THEN AK = AK + PERT PERT = 2*PERT GO TO 70 END IF END IF Y( K ) = TEMP / AK 80 CONTINUE END IF ! DO 90 K = N, 2, -1 IF( IN( K-1 ).EQ.0 ) THEN Y( K-1 ) = Y( K-1 ) - C( K-1 )*Y( K ) ELSE TEMP = Y( K-1 ) Y( K-1 ) = Y( K ) Y( K ) = TEMP - C( K-1 )*Y( K ) END IF 90 CONTINUE END IF ! ! End of DLAGTS ! END SUBROUTINE DLAGV2( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL, & CSR, SNR ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER LDA, LDB DOUBLE PRECISION CSL, CSR, SNL, SNR ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), ALPHAI( 2 ), ALPHAR( 2 ), & B( LDB, * ), BETA( 2 ) ! .. ! ! Purpose ! ======= ! ! DLAGV2 computes the Generalized Schur factorization of a real 2-by-2 ! matrix pencil (A,B) where B is upper triangular. This routine ! computes orthogonal (rotation) matrices given by CSL, SNL and CSR, ! SNR such that ! ! 1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0 ! types), then ! ! [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ] ! [ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ] ! ! [ b11 b12 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ] ! [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ], ! ! 2) if the pencil (A,B) has a pair of complex conjugate eigenvalues, ! then ! ! [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ] ! [ a21 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ] ! ! [ b11 0 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ] ! [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ] ! ! where b11 >= b22 > 0. ! ! ! Arguments ! ========= ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, 2) ! On entry, the 2 x 2 matrix A. ! On exit, A is overwritten by the ``A-part'' of the ! generalized Schur form. ! ! LDA (input) INTEGER ! THe leading dimension of the array A. LDA >= 2. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB, 2) ! On entry, the upper triangular 2 x 2 matrix B. ! On exit, B is overwritten by the ``B-part'' of the ! generalized Schur form. ! ! LDB (input) INTEGER ! THe leading dimension of the array B. LDB >= 2. ! ! ALPHAR (output) DOUBLE PRECISION array, dimension (2) ! ALPHAI (output) DOUBLE PRECISION array, dimension (2) ! BETA (output) DOUBLE PRECISION array, dimension (2) ! (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the ! pencil (A,B), k=1,2, i = sqrt(-1). Note that BETA(k) may ! be zero. ! ! CSL (output) DOUBLE PRECISION ! The cosine of the left rotation matrix. ! ! SNL (output) DOUBLE PRECISION ! The sine of the left rotation matrix. ! ! CSR (output) DOUBLE PRECISION ! The cosine of the right rotation matrix. ! ! SNR (output) DOUBLE PRECISION ! The sine of the right rotation matrix. ! ! Further Details ! =============== ! ! Based on contributions by ! Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. DOUBLE PRECISION ANORM, ASCALE, BNORM, BSCALE, H1, H2, H3, QQ, & R, RR, SAFMIN, SCALE1, SCALE2, T, ULP, WI, WR1, & WR2 ! .. ! .. External Subroutines .. EXTERNAL DLAG2, DLARTG, DLASV2, DROT ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH, DLAPY2 EXTERNAL DLAMCH, DLAPY2 ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. Executable Statements .. ! SAFMIN = DLAMCH( 'S' ) ULP = DLAMCH( 'P' ) ! ! Scale A ! ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ), & ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN ) ASCALE = ONE / ANORM A( 1, 1 ) = ASCALE*A( 1, 1 ) A( 1, 2 ) = ASCALE*A( 1, 2 ) A( 2, 1 ) = ASCALE*A( 2, 1 ) A( 2, 2 ) = ASCALE*A( 2, 2 ) ! ! Scale B ! BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 1, 2 ) )+ABS( B( 2, 2 ) ), & SAFMIN ) BSCALE = ONE / BNORM B( 1, 1 ) = BSCALE*B( 1, 1 ) B( 1, 2 ) = BSCALE*B( 1, 2 ) B( 2, 2 ) = BSCALE*B( 2, 2 ) ! ! Check if A can be deflated ! IF( ABS( A( 2, 1 ) ).LE.ULP ) THEN CSL = ONE SNL = ZERO CSR = ONE SNR = ZERO A( 2, 1 ) = ZERO B( 2, 1 ) = ZERO ! ! Check if B is singular ! ELSE IF( ABS( B( 1, 1 ) ).LE.ULP ) THEN CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R ) CSR = ONE SNR = ZERO CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL ) CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL ) A( 2, 1 ) = ZERO B( 1, 1 ) = ZERO B( 2, 1 ) = ZERO ! ELSE IF( ABS( B( 2, 2 ) ).LE.ULP ) THEN CALL DLARTG( A( 2, 2 ), A( 2, 1 ), CSR, SNR, T ) SNR = -SNR CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR ) CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR ) CSL = ONE SNL = ZERO A( 2, 1 ) = ZERO B( 2, 1 ) = ZERO B( 2, 2 ) = ZERO ! ELSE ! ! B is nonsingular, first compute the eigenvalues of (A,B) ! CALL DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2, & WI ) ! IF( WI.EQ.ZERO ) THEN ! ! two real eigenvalues, compute s*A-w*B ! H1 = SCALE1*A( 1, 1 ) - WR1*B( 1, 1 ) H2 = SCALE1*A( 1, 2 ) - WR1*B( 1, 2 ) H3 = SCALE1*A( 2, 2 ) - WR1*B( 2, 2 ) ! RR = DLAPY2( H1, H2 ) QQ = DLAPY2( SCALE1*A( 2, 1 ), H3 ) ! IF( RR.GT.QQ ) THEN ! ! find right rotation matrix to zero 1,1 element of ! (sA - wB) ! CALL DLARTG( H2, H1, CSR, SNR, T ) ! ELSE ! ! find right rotation matrix to zero 2,1 element of ! (sA - wB) ! CALL DLARTG( H3, SCALE1*A( 2, 1 ), CSR, SNR, T ) ! END IF ! SNR = -SNR CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR ) CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR ) ! ! compute inf norms of A and B ! H1 = MAX( ABS( A( 1, 1 ) )+ABS( A( 1, 2 ) ), & ABS( A( 2, 1 ) )+ABS( A( 2, 2 ) ) ) H2 = MAX( ABS( B( 1, 1 ) )+ABS( B( 1, 2 ) ), & ABS( B( 2, 1 ) )+ABS( B( 2, 2 ) ) ) ! IF( ( SCALE1*H1 ).GE.ABS( WR1 )*H2 ) THEN ! ! find left rotation matrix Q to zero out B(2,1) ! CALL DLARTG( B( 1, 1 ), B( 2, 1 ), CSL, SNL, R ) ! ELSE ! ! find left rotation matrix Q to zero out A(2,1) ! CALL DLARTG( A( 1, 1 ), A( 2, 1 ), CSL, SNL, R ) ! END IF ! CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL ) CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL ) ! A( 2, 1 ) = ZERO B( 2, 1 ) = ZERO ! ELSE ! ! a pair of complex conjugate eigenvalues ! first compute the SVD of the matrix B ! CALL DLASV2( B( 1, 1 ), B( 1, 2 ), B( 2, 2 ), R, T, SNR, & CSR, SNL, CSL ) ! ! Form (A,B) := Q(A,B)Z' where Q is left rotation matrix and ! Z is right rotation matrix computed from DLASV2 ! CALL DROT( 2, A( 1, 1 ), LDA, A( 2, 1 ), LDA, CSL, SNL ) CALL DROT( 2, B( 1, 1 ), LDB, B( 2, 1 ), LDB, CSL, SNL ) CALL DROT( 2, A( 1, 1 ), 1, A( 1, 2 ), 1, CSR, SNR ) CALL DROT( 2, B( 1, 1 ), 1, B( 1, 2 ), 1, CSR, SNR ) ! B( 2, 1 ) = ZERO B( 1, 2 ) = ZERO ! END IF ! END IF ! ! Unscaling ! A( 1, 1 ) = ANORM*A( 1, 1 ) A( 2, 1 ) = ANORM*A( 2, 1 ) A( 1, 2 ) = ANORM*A( 1, 2 ) A( 2, 2 ) = ANORM*A( 2, 2 ) B( 1, 1 ) = BNORM*B( 1, 1 ) B( 2, 1 ) = BNORM*B( 2, 1 ) B( 1, 2 ) = BNORM*B( 1, 2 ) B( 2, 2 ) = BNORM*B( 2, 2 ) ! IF( WI.EQ.ZERO ) THEN ALPHAR( 1 ) = A( 1, 1 ) ALPHAR( 2 ) = A( 2, 2 ) ALPHAI( 1 ) = ZERO ALPHAI( 2 ) = ZERO BETA( 1 ) = B( 1, 1 ) BETA( 2 ) = B( 2, 2 ) ELSE ALPHAR( 1 ) = ANORM*WR1 / SCALE1 / BNORM ALPHAI( 1 ) = ANORM*WI / SCALE1 / BNORM ALPHAR( 2 ) = ALPHAR( 1 ) ALPHAI( 2 ) = -ALPHAI( 1 ) BETA( 1 ) = ONE BETA( 2 ) = ONE END IF ! 10 CONTINUE ! RETURN ! ! End of DLAGV2 ! END SUBROUTINE DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, & ILOZ, IHIZ, Z, LDZ, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. LOGICAL WANTT, WANTZ INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N ! .. ! .. Array Arguments .. DOUBLE PRECISION H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DLAHQR is an auxiliary routine called by DHSEQR to update the ! eigenvalues and Schur decomposition already computed by DHSEQR, by ! dealing with the Hessenberg submatrix in rows and columns ILO to IHI. ! ! Arguments ! ========= ! ! WANTT (input) LOGICAL ! = .TRUE. : the full Schur form T is required; ! = .FALSE.: only eigenvalues are required. ! ! WANTZ (input) LOGICAL ! = .TRUE. : the matrix of Schur vectors Z is required; ! = .FALSE.: Schur vectors are not required. ! ! N (input) INTEGER ! The order of the matrix H. N >= 0. ! ! ILO (input) INTEGER ! IHI (input) INTEGER ! It is assumed that H is already upper quasi-triangular in ! rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ! ILO = 1). DLAHQR works primarily with the Hessenberg ! submatrix in rows and columns ILO to IHI, but applies ! transformations to all of H if WANTT is .TRUE.. ! 1 <= ILO <= max(1,IHI); IHI <= N. ! ! H (input/output) DOUBLE PRECISION array, dimension (LDH,N) ! On entry, the upper Hessenberg matrix H. ! On exit, if WANTT is .TRUE., H is upper quasi-triangular in ! rows and columns ILO:IHI, with any 2-by-2 diagonal blocks in ! standard form. If WANTT is .FALSE., the contents of H are ! unspecified on exit. ! ! LDH (input) INTEGER ! The leading dimension of the array H. LDH >= max(1,N). ! ! WR (output) DOUBLE PRECISION array, dimension (N) ! WI (output) DOUBLE PRECISION array, dimension (N) ! The real and imaginary parts, respectively, of the computed ! eigenvalues ILO to IHI are stored in the corresponding ! elements of WR and WI. If two eigenvalues are computed as a ! complex conjugate pair, they are stored in consecutive ! elements of WR and WI, say the i-th and (i+1)th, with ! WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the ! eigenvalues are stored in the same order as on the diagonal ! of the Schur form returned in H, with WR(i) = H(i,i), and, if ! H(i:i+1,i:i+1) is a 2-by-2 diagonal block, ! WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i). ! ! ILOZ (input) INTEGER ! IHIZ (input) INTEGER ! Specify the rows of Z to which transformations must be ! applied if WANTZ is .TRUE.. ! 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. ! ! Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N) ! If WANTZ is .TRUE., on entry Z must contain the current ! matrix Z of transformations accumulated by DHSEQR, and on ! exit Z has been updated; transformations are applied only to ! the submatrix Z(ILOZ:IHIZ,ILO:IHI). ! If WANTZ is .FALSE., Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! > 0: DLAHQR failed to compute all the eigenvalues ILO to IHI ! in a total of 30*(IHI-ILO+1) iterations; if INFO = i, ! elements i+1:ihi of WR and WI contain those eigenvalues ! which have been successfully computed. ! ! Further Details ! =============== ! ! 2-96 Based on modifications by ! David Day, Sandia National Laboratory, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, HALF PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, HALF = 0.5D0 ) DOUBLE PRECISION DAT1, DAT2 PARAMETER ( DAT1 = 0.75D+0, DAT2 = -0.4375D+0 ) ! .. ! .. Local Scalars .. INTEGER I, I1, I2, ITN, ITS, J, K, L, M, NH, NR, NZ DOUBLE PRECISION AVE, CS, DISC, H00, H10, H11, H12, H21, H22, & H33, H33S, H43H34, H44, H44S, OVFL, S, SMLNUM, & SN, SUM, T1, T2, T3, TST1, ULP, UNFL, V1, V2, & V3 ! .. ! .. Local Arrays .. DOUBLE PRECISION V( 3 ), WORK( 1 ) ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH, DLANHS EXTERNAL DLAMCH, DLANHS ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLANV2, DLARFG, DROT ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SIGN, SQRT ! .. ! .. Executable Statements .. ! INFO = 0 ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN IF( ILO.EQ.IHI ) THEN WR( ILO ) = H( ILO, ILO ) WI( ILO ) = ZERO RETURN END IF ! NH = IHI - ILO + 1 NZ = IHIZ - ILOZ + 1 ! ! Set machine-dependent constants for the stopping criterion. ! If norm(H) <= sqrt(OVFL), overflow should not occur. ! UNFL = DLAMCH( 'Safe minimum' ) OVFL = ONE / UNFL CALL DLABAD( UNFL, OVFL ) ULP = DLAMCH( 'Precision' ) SMLNUM = UNFL*( NH / ULP ) ! ! I1 and I2 are the indices of the first row and last column of H ! to which transformations must be applied. If eigenvalues only are ! being computed, I1 and I2 are set inside the main loop. ! IF( WANTT ) THEN I1 = 1 I2 = N END IF ! ! ITN is the total number of QR iterations allowed. ! ITN = 30*NH ! ! The main loop begins here. I is the loop index and decreases from ! IHI to ILO in steps of 1 or 2. Each iteration of the loop works ! with the active submatrix in rows and columns L to I. ! Eigenvalues I+1 to IHI have already converged. Either L = ILO or ! H(L,L-1) is negligible so that the matrix splits. ! I = IHI 10 CONTINUE L = ILO IF( I.LT.ILO ) & GO TO 150 ! ! Perform QR iterations on rows and columns ILO to I until a ! submatrix of order 1 or 2 splits off at the bottom because a ! subdiagonal element has become negligible. ! DO 130 ITS = 0, ITN ! ! Look for a single small subdiagonal element. ! DO 20 K = I, L + 1, -1 TST1 = ABS( H( K-1, K-1 ) ) + ABS( H( K, K ) ) IF( TST1.EQ.ZERO ) & TST1 = DLANHS( '1', I-L+1, H( L, L ), LDH, WORK ) IF( ABS( H( K, K-1 ) ).LE.MAX( ULP*TST1, SMLNUM ) ) & GO TO 30 20 CONTINUE 30 CONTINUE L = K IF( L.GT.ILO ) THEN ! ! H(L,L-1) is negligible ! H( L, L-1 ) = ZERO END IF ! ! Exit from loop if a submatrix of order 1 or 2 has split off. ! IF( L.GE.I-1 ) & GO TO 140 ! ! Now the active submatrix is in rows and columns L to I. If ! eigenvalues only are being computed, only the active submatrix ! need be transformed. ! IF( .NOT.WANTT ) THEN I1 = L I2 = I END IF ! IF( ITS.EQ.10 .OR. ITS.EQ.20 ) THEN ! ! Exceptional shift. ! S = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) ) H44 = DAT1*S + H( I, I ) H33 = H44 H43H34 = DAT2*S*S ELSE ! ! Prepare to use Francis' double shift ! (i.e. 2nd degree generalized Rayleigh quotient) ! H44 = H( I, I ) H33 = H( I-1, I-1 ) H43H34 = H( I, I-1 )*H( I-1, I ) S = H( I-1, I-2 )*H( I-1, I-2 ) DISC = ( H33-H44 )*HALF DISC = DISC*DISC + H43H34 IF( DISC.GT.ZERO ) THEN ! ! Real roots: use Wilkinson's shift twice ! DISC = SQRT( DISC ) AVE = HALF*( H33+H44 ) IF( ABS( H33 )-ABS( H44 ).GT.ZERO ) THEN H33 = H33*H44 - H43H34 H44 = H33 / ( SIGN( DISC, AVE )+AVE ) ELSE H44 = SIGN( DISC, AVE ) + AVE END IF H33 = H44 H43H34 = ZERO END IF END IF ! ! Look for two consecutive small subdiagonal elements. ! DO 40 M = I - 2, L, -1 ! Determine the effect of starting the double-shift QR ! iteration at row M, and see if this would make H(M,M-1) ! negligible. ! H11 = H( M, M ) H22 = H( M+1, M+1 ) H21 = H( M+1, M ) H12 = H( M, M+1 ) H44S = H44 - H11 H33S = H33 - H11 V1 = ( H33S*H44S-H43H34 ) / H21 + H12 V2 = H22 - H11 - H33S - H44S V3 = H( M+2, M+1 ) S = ABS( V1 ) + ABS( V2 ) + ABS( V3 ) V1 = V1 / S V2 = V2 / S V3 = V3 / S V( 1 ) = V1 V( 2 ) = V2 V( 3 ) = V3 IF( M.EQ.L ) & GO TO 50 H00 = H( M-1, M-1 ) H10 = H( M, M-1 ) TST1 = ABS( V1 )*( ABS( H00 )+ABS( H11 )+ABS( H22 ) ) IF( ABS( H10 )*( ABS( V2 )+ABS( V3 ) ).LE.ULP*TST1 ) & GO TO 50 40 CONTINUE 50 CONTINUE ! ! Double-shift QR step ! DO 120 K = M, I - 1 ! ! The first iteration of this loop determines a reflection G ! from the vector V and applies it from left and right to H, ! thus creating a nonzero bulge below the subdiagonal. ! ! Each subsequent iteration determines a reflection G to ! restore the Hessenberg form in the (K-1)th column, and thus ! chases the bulge one step toward the bottom of the active ! submatrix. NR is the order of G. ! NR = MIN( 3, I-K+1 ) IF( K.GT.M ) & CALL DCOPY( NR, H( K, K-1 ), 1, V, 1 ) CALL DLARFG( NR, V( 1 ), V( 2 ), 1, T1 ) IF( K.GT.M ) THEN H( K, K-1 ) = V( 1 ) H( K+1, K-1 ) = ZERO IF( K.LT.I-1 ) & H( K+2, K-1 ) = ZERO ELSE IF( M.GT.L ) THEN H( K, K-1 ) = -H( K, K-1 ) END IF V2 = V( 2 ) T2 = T1*V2 IF( NR.EQ.3 ) THEN V3 = V( 3 ) T3 = T1*V3 ! ! Apply G from the left to transform the rows of the matrix ! in columns K to I2. ! DO 60 J = K, I2 SUM = H( K, J ) + V2*H( K+1, J ) + V3*H( K+2, J ) H( K, J ) = H( K, J ) - SUM*T1 H( K+1, J ) = H( K+1, J ) - SUM*T2 H( K+2, J ) = H( K+2, J ) - SUM*T3 60 CONTINUE ! ! Apply G from the right to transform the columns of the ! matrix in rows I1 to min(K+3,I). ! DO 70 J = I1, MIN( K+3, I ) SUM = H( J, K ) + V2*H( J, K+1 ) + V3*H( J, K+2 ) H( J, K ) = H( J, K ) - SUM*T1 H( J, K+1 ) = H( J, K+1 ) - SUM*T2 H( J, K+2 ) = H( J, K+2 ) - SUM*T3 70 CONTINUE ! IF( WANTZ ) THEN ! ! Accumulate transformations in the matrix Z ! DO 80 J = ILOZ, IHIZ SUM = Z( J, K ) + V2*Z( J, K+1 ) + V3*Z( J, K+2 ) Z( J, K ) = Z( J, K ) - SUM*T1 Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2 Z( J, K+2 ) = Z( J, K+2 ) - SUM*T3 80 CONTINUE END IF ELSE IF( NR.EQ.2 ) THEN ! ! Apply G from the left to transform the rows of the matrix ! in columns K to I2. ! DO 90 J = K, I2 SUM = H( K, J ) + V2*H( K+1, J ) H( K, J ) = H( K, J ) - SUM*T1 H( K+1, J ) = H( K+1, J ) - SUM*T2 90 CONTINUE ! ! Apply G from the right to transform the columns of the ! matrix in rows I1 to min(K+3,I). ! DO 100 J = I1, I SUM = H( J, K ) + V2*H( J, K+1 ) H( J, K ) = H( J, K ) - SUM*T1 H( J, K+1 ) = H( J, K+1 ) - SUM*T2 100 CONTINUE ! IF( WANTZ ) THEN ! ! Accumulate transformations in the matrix Z ! DO 110 J = ILOZ, IHIZ SUM = Z( J, K ) + V2*Z( J, K+1 ) Z( J, K ) = Z( J, K ) - SUM*T1 Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2 110 CONTINUE END IF END IF 120 CONTINUE ! 130 CONTINUE ! ! Failure to converge in remaining number of iterations ! INFO = I RETURN ! 140 CONTINUE ! IF( L.EQ.I ) THEN ! ! H(I,I-1) is negligible: one eigenvalue has converged. ! WR( I ) = H( I, I ) WI( I ) = ZERO ELSE IF( L.EQ.I-1 ) THEN ! ! H(I-1,I-2) is negligible: a pair of eigenvalues have converged. ! ! Transform the 2-by-2 submatrix to standard Schur form, ! and compute and store the eigenvalues. ! CALL DLANV2( H( I-1, I-1 ), H( I-1, I ), H( I, I-1 ), & H( I, I ), WR( I-1 ), WI( I-1 ), WR( I ), WI( I ), & CS, SN ) ! IF( WANTT ) THEN ! ! Apply the transformation to the rest of H. ! IF( I2.GT.I ) & CALL DROT( I2-I, H( I-1, I+1 ), LDH, H( I, I+1 ), LDH, & CS, SN ) CALL DROT( I-I1-1, H( I1, I-1 ), 1, H( I1, I ), 1, CS, SN ) END IF IF( WANTZ ) THEN ! ! Apply the transformation to Z. ! CALL DROT( NZ, Z( ILOZ, I-1 ), 1, Z( ILOZ, I ), 1, CS, SN ) END IF END IF ! ! Decrement number of remaining iterations, and return to start of ! the main loop with new value of I. ! ITN = ITN - ITS I = L - 1 GO TO 10 ! 150 CONTINUE RETURN ! ! End of DLAHQR ! END SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER K, LDA, LDT, LDY, N, NB ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ), & Y( LDY, NB ) ! .. ! ! Purpose ! ======= ! ! DLAHRD reduces the first NB columns of a real general n-by-(n-k+1) ! matrix A so that elements below the k-th subdiagonal are zero. The ! reduction is performed by an orthogonal similarity transformation ! Q' * A * Q. The routine returns the matrices V and T which determine ! Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. ! ! This is an auxiliary routine called by DGEHRD. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix A. ! ! K (input) INTEGER ! The offset for the reduction. Elements below the k-th ! subdiagonal in the first NB columns are reduced to zero. ! ! NB (input) INTEGER ! The number of columns to be reduced. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N-K+1) ! On entry, the n-by-(n-k+1) general matrix A. ! On exit, the elements on and above the k-th subdiagonal in ! the first NB columns are overwritten with the corresponding ! elements of the reduced matrix; the elements below the k-th ! subdiagonal, with the array TAU, represent the matrix Q as a ! product of elementary reflectors. The other columns of A are ! unchanged. See Further Details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! TAU (output) DOUBLE PRECISION array, dimension (NB) ! The scalar factors of the elementary reflectors. See Further ! Details. ! ! T (output) DOUBLE PRECISION array, dimension (LDT,NB) ! The upper triangular matrix T. ! ! LDT (input) INTEGER ! The leading dimension of the array T. LDT >= NB. ! ! Y (output) DOUBLE PRECISION array, dimension (LDY,NB) ! The n-by-nb matrix Y. ! ! LDY (input) INTEGER ! The leading dimension of the array Y. LDY >= N. ! ! Further Details ! =============== ! ! The matrix Q is represented as a product of nb elementary reflectors ! ! Q = H(1) H(2) . . . H(nb). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in ! A(i+k+1:n,i), and tau in TAU(i). ! ! The elements of the vectors v together form the (n-k+1)-by-nb matrix ! V which is needed, with T and Y, to apply the transformation to the ! unreduced part of the matrix, using an update of the form: ! A := (I - V*T*V') * (A - Y*V'). ! ! The contents of A on exit are illustrated by the following example ! with n = 7, k = 3 and nb = 2: ! ! ( a h a a a ) ! ( a h a a a ) ! ( a h a a a ) ! ( h h a a a ) ! ( v1 h a a a ) ! ( v1 v2 a a a ) ! ( v1 v2 a a a ) ! ! where a denotes an element of the original matrix A, h denotes a ! modified element of the upper Hessenberg matrix H, and vi denotes an ! element of the vector defining H(i). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I DOUBLE PRECISION EI ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DGEMV, DLARFG, DSCAL, DTRMV ! .. ! .. Intrinsic Functions .. INTRINSIC MIN ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( N.LE.1 ) & RETURN ! DO 10 I = 1, NB IF( I.GT.1 ) THEN ! ! Update A(1:n,i) ! ! Compute i-th column of A - Y * V' ! CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, & A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 ) ! ! Apply I - V * T' * V' to this column (call it b) from the ! left, using the last column of T as workspace ! ! Let V = ( V1 ) and b = ( b1 ) (first I-1 rows) ! ( V2 ) ( b2 ) ! ! where V1 is unit lower triangular ! ! w := V1' * b1 ! CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) CALL DTRMV( 'Lower', 'Transpose', 'Unit', I-1, A( K+1, 1 ), & LDA, T( 1, NB ), 1 ) ! ! w := w + V2'*b2 ! CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ), & LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 ) ! ! w := T'*w ! CALL DTRMV( 'Upper', 'Transpose', 'Non-unit', I-1, T, LDT, & T( 1, NB ), 1 ) ! ! b2 := b2 - V2*w ! CALL DGEMV( 'No transpose', N-K-I+1, I-1, -ONE, A( K+I, 1 ), & LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 ) ! ! b1 := b1 - V1*w ! CALL DTRMV( 'Lower', 'No transpose', 'Unit', I-1, & A( K+1, 1 ), LDA, T( 1, NB ), 1 ) CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 ) ! A( K+I-1, I-1 ) = EI END IF ! ! Generate the elementary reflector H(i) to annihilate ! A(k+i+1:n,i) ! CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1, & TAU( I ) ) EI = A( K+I, I ) A( K+I, I ) = ONE ! ! Compute Y(1:n,i) ! CALL DGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA, & A( K+I, I ), 1, ZERO, Y( 1, I ), 1 ) CALL DGEMV( 'Transpose', N-K-I+1, I-1, ONE, A( K+I, 1 ), LDA, & A( K+I, I ), 1, ZERO, T( 1, I ), 1 ) CALL DGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1, & ONE, Y( 1, I ), 1 ) CALL DSCAL( N, TAU( I ), Y( 1, I ), 1 ) ! ! Compute T(1:i,i) ! CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 ) CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, LDT, & T( 1, I ), 1 ) T( I, I ) = TAU( I ) ! 10 CONTINUE A( K+NB, NB ) = EI ! RETURN ! ! End of DLAHRD ! END SUBROUTINE DLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. INTEGER J, JOB DOUBLE PRECISION C, GAMMA, S, SEST, SESTPR ! .. ! .. Array Arguments .. DOUBLE PRECISION W( J ), X( J ) ! .. ! ! Purpose ! ======= ! ! DLAIC1 applies one step of incremental condition estimation in ! its simplest version: ! ! Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j ! lower triangular matrix L, such that ! twonorm(L*x) = sest ! Then DLAIC1 computes sestpr, s, c such that ! the vector ! [ s*x ] ! xhat = [ c ] ! is an approximate singular vector of ! [ L 0 ] ! Lhat = [ w' gamma ] ! in the sense that ! twonorm(Lhat*xhat) = sestpr. ! ! Depending on JOB, an estimate for the largest or smallest singular ! value is computed. ! ! Note that [s c]' and sestpr**2 is an eigenpair of the system ! ! diag(sest*sest, 0) + [alpha gamma] * [ alpha ] ! [ gamma ] ! ! where alpha = x'*w. ! ! Arguments ! ========= ! ! JOB (input) INTEGER ! = 1: an estimate for the largest singular value is computed. ! = 2: an estimate for the smallest singular value is computed. ! ! J (input) INTEGER ! Length of X and W ! ! X (input) DOUBLE PRECISION array, dimension (J) ! The j-vector x. ! ! SEST (input) DOUBLE PRECISION ! Estimated singular value of j by j matrix L ! ! W (input) DOUBLE PRECISION array, dimension (J) ! The j-vector w. ! ! GAMMA (input) DOUBLE PRECISION ! The diagonal element gamma. ! ! SEDTPR (output) DOUBLE PRECISION ! Estimated singular value of (j+1) by (j+1) matrix Lhat. ! ! S (output) DOUBLE PRECISION ! Sine needed in forming xhat. ! ! C (output) DOUBLE PRECISION ! Cosine needed in forming xhat. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 ) DOUBLE PRECISION HALF, FOUR PARAMETER ( HALF = 0.5D0, FOUR = 4.0D0 ) ! .. ! .. Local Scalars .. DOUBLE PRECISION ABSALP, ABSEST, ABSGAM, ALPHA, B, COSINE, EPS, & NORMA, S1, S2, SINE, T, TEST, TMP, ZETA1, ZETA2 ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SIGN, SQRT ! .. ! .. External Functions .. DOUBLE PRECISION DDOT, DLAMCH EXTERNAL DDOT, DLAMCH ! .. ! .. Executable Statements .. ! EPS = DLAMCH( 'Epsilon' ) ALPHA = DDOT( J, X, 1, W, 1 ) ! ABSALP = ABS( ALPHA ) ABSGAM = ABS( GAMMA ) ABSEST = ABS( SEST ) ! IF( JOB.EQ.1 ) THEN ! ! Estimating largest singular value ! ! special cases ! IF( SEST.EQ.ZERO ) THEN S1 = MAX( ABSGAM, ABSALP ) IF( S1.EQ.ZERO ) THEN S = ZERO C = ONE SESTPR = ZERO ELSE S = ALPHA / S1 C = GAMMA / S1 TMP = SQRT( S*S+C*C ) S = S / TMP C = C / TMP SESTPR = S1*TMP END IF RETURN ELSE IF( ABSGAM.LE.EPS*ABSEST ) THEN S = ONE C = ZERO TMP = MAX( ABSEST, ABSALP ) S1 = ABSEST / TMP S2 = ABSALP / TMP SESTPR = TMP*SQRT( S1*S1+S2*S2 ) RETURN ELSE IF( ABSALP.LE.EPS*ABSEST ) THEN S1 = ABSGAM S2 = ABSEST IF( S1.LE.S2 ) THEN S = ONE C = ZERO SESTPR = S2 ELSE S = ZERO C = ONE SESTPR = S1 END IF RETURN ELSE IF( ABSEST.LE.EPS*ABSALP .OR. ABSEST.LE.EPS*ABSGAM ) THEN S1 = ABSGAM S2 = ABSALP IF( S1.LE.S2 ) THEN TMP = S1 / S2 S = SQRT( ONE+TMP*TMP ) SESTPR = S2*S C = ( GAMMA / S2 ) / S S = SIGN( ONE, ALPHA ) / S ELSE TMP = S2 / S1 C = SQRT( ONE+TMP*TMP ) SESTPR = S1*C S = ( ALPHA / S1 ) / C C = SIGN( ONE, GAMMA ) / C END IF RETURN ELSE ! ! normal case ! ZETA1 = ALPHA / ABSEST ZETA2 = GAMMA / ABSEST ! B = ( ONE-ZETA1*ZETA1-ZETA2*ZETA2 )*HALF C = ZETA1*ZETA1 IF( B.GT.ZERO ) THEN T = C / ( B+SQRT( B*B+C ) ) ELSE T = SQRT( B*B+C ) - B END IF ! SINE = -ZETA1 / T COSINE = -ZETA2 / ( ONE+T ) TMP = SQRT( SINE*SINE+COSINE*COSINE ) S = SINE / TMP C = COSINE / TMP SESTPR = SQRT( T+ONE )*ABSEST RETURN END IF ! ELSE IF( JOB.EQ.2 ) THEN ! ! Estimating smallest singular value ! ! special cases ! IF( SEST.EQ.ZERO ) THEN SESTPR = ZERO IF( MAX( ABSGAM, ABSALP ).EQ.ZERO ) THEN SINE = ONE COSINE = ZERO ELSE SINE = -GAMMA COSINE = ALPHA END IF S1 = MAX( ABS( SINE ), ABS( COSINE ) ) S = SINE / S1 C = COSINE / S1 TMP = SQRT( S*S+C*C ) S = S / TMP C = C / TMP RETURN ELSE IF( ABSGAM.LE.EPS*ABSEST ) THEN S = ZERO C = ONE SESTPR = ABSGAM RETURN ELSE IF( ABSALP.LE.EPS*ABSEST ) THEN S1 = ABSGAM S2 = ABSEST IF( S1.LE.S2 ) THEN S = ZERO C = ONE SESTPR = S1 ELSE S = ONE C = ZERO SESTPR = S2 END IF RETURN ELSE IF( ABSEST.LE.EPS*ABSALP .OR. ABSEST.LE.EPS*ABSGAM ) THEN S1 = ABSGAM S2 = ABSALP IF( S1.LE.S2 ) THEN TMP = S1 / S2 C = SQRT( ONE+TMP*TMP ) SESTPR = ABSEST*( TMP / C ) S = -( GAMMA / S2 ) / C C = SIGN( ONE, ALPHA ) / C ELSE TMP = S2 / S1 S = SQRT( ONE+TMP*TMP ) SESTPR = ABSEST / S C = ( ALPHA / S1 ) / S S = -SIGN( ONE, GAMMA ) / S END IF RETURN ELSE ! ! normal case ! ZETA1 = ALPHA / ABSEST ZETA2 = GAMMA / ABSEST ! NORMA = MAX( ONE+ZETA1*ZETA1+ABS( ZETA1*ZETA2 ), & ABS( ZETA1*ZETA2 )+ZETA2*ZETA2 ) ! ! See if root is closer to zero or to ONE ! TEST = ONE + TWO*( ZETA1-ZETA2 )*( ZETA1+ZETA2 ) IF( TEST.GE.ZERO ) THEN ! ! root is close to zero, compute directly ! B = ( ZETA1*ZETA1+ZETA2*ZETA2+ONE )*HALF C = ZETA2*ZETA2 T = C / ( B+SQRT( ABS( B*B-C ) ) ) SINE = ZETA1 / ( ONE-T ) COSINE = -ZETA2 / T SESTPR = SQRT( T+FOUR*EPS*EPS*NORMA )*ABSEST ELSE ! ! root is closer to ONE, shift by that amount ! B = ( ZETA2*ZETA2+ZETA1*ZETA1-ONE )*HALF C = ZETA1*ZETA1 IF( B.GE.ZERO ) THEN T = -C / ( B+SQRT( B*B+C ) ) ELSE T = B - SQRT( B*B+C ) END IF SINE = -ZETA1 / T COSINE = -ZETA2 / ( ONE+T ) SESTPR = SQRT( ONE+T+FOUR*EPS*EPS*NORMA )*ABSEST END IF TMP = SQRT( SINE*SINE+COSINE*COSINE ) S = SINE / TMP C = COSINE / TMP RETURN ! END IF END IF RETURN ! ! End of DLAIC1 ! END SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B, & LDB, WR, WI, X, LDX, SCALE, XNORM, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. LOGICAL LTRANS INTEGER INFO, LDA, LDB, LDX, NA, NW DOUBLE PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DLALN2 solves a system of the form (ca A - w D ) X = s B ! or (ca A' - w D) X = s B with possible scaling ("s") and ! perturbation of A. (A' means A-transpose.) ! ! A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA ! real diagonal matrix, w is a real or complex value, and X and B are ! NA x 1 matrices -- real if w is real, complex if w is complex. NA ! may be 1 or 2. ! ! If w is complex, X and B are represented as NA x 2 matrices, ! the first column of each being the real part and the second ! being the imaginary part. ! ! "s" is a scaling factor (.LE. 1), computed by DLALN2, which is ! so chosen that X can be computed without overflow. X is further ! scaled if necessary to assure that norm(ca A - w D)*norm(X) is less ! than overflow. ! ! If both singular values of (ca A - w D) are less than SMIN, ! SMIN*identity will be used instead of (ca A - w D). If only one ! singular value is less than SMIN, one element of (ca A - w D) will be ! perturbed enough to make the smallest singular value roughly SMIN. ! If both singular values are at least SMIN, (ca A - w D) will not be ! perturbed. In any case, the perturbation will be at most some small ! multiple of max( SMIN, ulp*norm(ca A - w D) ). The singular values ! are computed by infinity-norm approximations, and thus will only be ! correct to a factor of 2 or so. ! ! Note: all input quantities are assumed to be smaller than overflow ! by a reasonable factor. (See BIGNUM.) ! ! Arguments ! ========== ! ! LTRANS (input) LOGICAL ! =.TRUE.: A-transpose will be used. ! =.FALSE.: A will be used (not transposed.) ! ! NA (input) INTEGER ! The size of the matrix A. It may (only) be 1 or 2. ! ! NW (input) INTEGER ! 1 if "w" is real, 2 if "w" is complex. It may only be 1 ! or 2. ! ! SMIN (input) DOUBLE PRECISION ! The desired lower bound on the singular values of A. This ! should be a safe distance away from underflow or overflow, ! say, between (underflow/machine precision) and (machine ! precision * overflow ). (See BIGNUM and ULP.) ! ! CA (input) DOUBLE PRECISION ! The coefficient c, which A is multiplied by. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,NA) ! The NA x NA matrix A. ! ! LDA (input) INTEGER ! The leading dimension of A. It must be at least NA. ! ! D1 (input) DOUBLE PRECISION ! The 1,1 element in the diagonal matrix D. ! ! D2 (input) DOUBLE PRECISION ! The 2,2 element in the diagonal matrix D. Not used if NW=1. ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NW) ! The NA x NW matrix B (right-hand side). If NW=2 ("w" is ! complex), column 1 contains the real part of B and column 2 ! contains the imaginary part. ! ! LDB (input) INTEGER ! The leading dimension of B. It must be at least NA. ! ! WR (input) DOUBLE PRECISION ! The real part of the scalar "w". ! ! WI (input) DOUBLE PRECISION ! The imaginary part of the scalar "w". Not used if NW=1. ! ! X (output) DOUBLE PRECISION array, dimension (LDX,NW) ! The NA x NW matrix X (unknowns), as computed by DLALN2. ! If NW=2 ("w" is complex), on exit, column 1 will contain ! the real part of X and column 2 will contain the imaginary ! part. ! ! LDX (input) INTEGER ! The leading dimension of X. It must be at least NA. ! ! SCALE (output) DOUBLE PRECISION ! The scale factor that B must be multiplied by to insure ! that overflow does not occur when computing X. Thus, ! (ca A - w D) X will be SCALE*B, not B (ignoring ! perturbations of A.) It will be at most 1. ! ! XNORM (output) DOUBLE PRECISION ! The infinity-norm of X, when X is regarded as an NA x NW ! real matrix. ! ! INFO (output) INTEGER ! An error flag. It will be set to zero if no error occurs, ! a negative number if an argument is in error, or a positive ! number if ca A - w D had to be perturbed. ! The possible values are: ! = 0: No error occurred, and (ca A - w D) did not have to be ! perturbed. ! = 1: (ca A - w D) had to be perturbed to make its smallest ! (or only) singular value greater than SMIN. ! NOTE: In the interests of speed, this routine does not ! check the inputs for errors. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D0 ) ! .. ! .. Local Scalars .. INTEGER ICMAX, J DOUBLE PRECISION BBND, BI1, BI2, BIGNUM, BNORM, BR1, BR2, CI21, & CI22, CMAX, CNORM, CR21, CR22, CSI, CSR, LI21, & LR21, SMINI, SMLNUM, TEMP, U22ABS, UI11, UI11R, & UI12, UI12S, UI22, UR11, UR11R, UR12, UR12S, & UR22, XI1, XI2, XR1, XR2 ! .. ! .. Local Arrays .. LOGICAL RSWAP( 4 ), ZSWAP( 4 ) INTEGER IPIVOT( 4, 4 ) DOUBLE PRECISION CI( 2, 2 ), CIV( 4 ), CR( 2, 2 ), CRV( 4 ) ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DLADIV ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. Equivalences .. EQUIVALENCE ( CI( 1, 1 ), CIV( 1 ) ), & ( CR( 1, 1 ), CRV( 1 ) ) ! .. ! .. Data statements .. DATA ZSWAP / .FALSE., .FALSE., .TRUE., .TRUE. / DATA RSWAP / .FALSE., .TRUE., .FALSE., .TRUE. / DATA IPIVOT / 1, 2, 3, 4, 2, 1, 4, 3, 3, 4, 1, 2, 4, & 3, 2, 1 / ! .. ! .. Executable Statements .. ! ! Compute BIGNUM ! SMLNUM = TWO*DLAMCH( 'Safe minimum' ) BIGNUM = ONE / SMLNUM SMINI = MAX( SMIN, SMLNUM ) ! ! Don't check for input errors ! INFO = 0 ! ! Standard Initializations ! SCALE = ONE ! IF( NA.EQ.1 ) THEN ! ! 1 x 1 (i.e., scalar) system C X = B ! IF( NW.EQ.1 ) THEN ! ! Real 1x1 system. ! ! C = ca A - w D ! CSR = CA*A( 1, 1 ) - WR*D1 CNORM = ABS( CSR ) ! ! If | C | < SMINI, use C = SMINI ! IF( CNORM.LT.SMINI ) THEN CSR = SMINI CNORM = SMINI INFO = 1 END IF ! ! Check scaling for X = B / C ! BNORM = ABS( B( 1, 1 ) ) IF( CNORM.LT.ONE .AND. BNORM.GT.ONE ) THEN IF( BNORM.GT.BIGNUM*CNORM ) & SCALE = ONE / BNORM END IF ! ! Compute X ! X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / CSR XNORM = ABS( X( 1, 1 ) ) ELSE ! ! Complex 1x1 system (w is complex) ! ! C = ca A - w D ! CSR = CA*A( 1, 1 ) - WR*D1 CSI = -WI*D1 CNORM = ABS( CSR ) + ABS( CSI ) ! ! If | C | < SMINI, use C = SMINI ! IF( CNORM.LT.SMINI ) THEN CSR = SMINI CSI = ZERO CNORM = SMINI INFO = 1 END IF ! ! Check scaling for X = B / C ! BNORM = ABS( B( 1, 1 ) ) + ABS( B( 1, 2 ) ) IF( CNORM.LT.ONE .AND. BNORM.GT.ONE ) THEN IF( BNORM.GT.BIGNUM*CNORM ) & SCALE = ONE / BNORM END IF ! ! Compute X ! CALL DLADIV( SCALE*B( 1, 1 ), SCALE*B( 1, 2 ), CSR, CSI, & X( 1, 1 ), X( 1, 2 ) ) XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) ) END IF ! ELSE ! ! 2x2 System ! ! Compute the real part of C = ca A - w D (or ca A' - w D ) ! CR( 1, 1 ) = CA*A( 1, 1 ) - WR*D1 CR( 2, 2 ) = CA*A( 2, 2 ) - WR*D2 IF( LTRANS ) THEN CR( 1, 2 ) = CA*A( 2, 1 ) CR( 2, 1 ) = CA*A( 1, 2 ) ELSE CR( 2, 1 ) = CA*A( 2, 1 ) CR( 1, 2 ) = CA*A( 1, 2 ) END IF ! IF( NW.EQ.1 ) THEN ! ! Real 2x2 system (w is real) ! ! Find the largest element in C ! CMAX = ZERO ICMAX = 0 ! DO 10 J = 1, 4 IF( ABS( CRV( J ) ).GT.CMAX ) THEN CMAX = ABS( CRV( J ) ) ICMAX = J END IF 10 CONTINUE ! ! If norm(C) < SMINI, use SMINI*identity. ! IF( CMAX.LT.SMINI ) THEN BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 2, 1 ) ) ) IF( SMINI.LT.ONE .AND. BNORM.GT.ONE ) THEN IF( BNORM.GT.BIGNUM*SMINI ) & SCALE = ONE / BNORM END IF TEMP = SCALE / SMINI X( 1, 1 ) = TEMP*B( 1, 1 ) X( 2, 1 ) = TEMP*B( 2, 1 ) XNORM = TEMP*BNORM INFO = 1 RETURN END IF ! ! Gaussian elimination with complete pivoting. ! UR11 = CRV( ICMAX ) CR21 = CRV( IPIVOT( 2, ICMAX ) ) UR12 = CRV( IPIVOT( 3, ICMAX ) ) CR22 = CRV( IPIVOT( 4, ICMAX ) ) UR11R = ONE / UR11 LR21 = UR11R*CR21 UR22 = CR22 - UR12*LR21 ! ! If smaller pivot < SMINI, use SMINI ! IF( ABS( UR22 ).LT.SMINI ) THEN UR22 = SMINI INFO = 1 END IF IF( RSWAP( ICMAX ) ) THEN BR1 = B( 2, 1 ) BR2 = B( 1, 1 ) ELSE BR1 = B( 1, 1 ) BR2 = B( 2, 1 ) END IF BR2 = BR2 - LR21*BR1 BBND = MAX( ABS( BR1*( UR22*UR11R ) ), ABS( BR2 ) ) IF( BBND.GT.ONE .AND. ABS( UR22 ).LT.ONE ) THEN IF( BBND.GE.BIGNUM*ABS( UR22 ) ) & SCALE = ONE / BBND END IF ! XR2 = ( BR2*SCALE ) / UR22 XR1 = ( SCALE*BR1 )*UR11R - XR2*( UR11R*UR12 ) IF( ZSWAP( ICMAX ) ) THEN X( 1, 1 ) = XR2 X( 2, 1 ) = XR1 ELSE X( 1, 1 ) = XR1 X( 2, 1 ) = XR2 END IF XNORM = MAX( ABS( XR1 ), ABS( XR2 ) ) ! ! Further scaling if norm(A) norm(X) > overflow ! IF( XNORM.GT.ONE .AND. CMAX.GT.ONE ) THEN IF( XNORM.GT.BIGNUM / CMAX ) THEN TEMP = CMAX / BIGNUM X( 1, 1 ) = TEMP*X( 1, 1 ) X( 2, 1 ) = TEMP*X( 2, 1 ) XNORM = TEMP*XNORM SCALE = TEMP*SCALE END IF END IF ELSE ! ! Complex 2x2 system (w is complex) ! ! Find the largest element in C ! CI( 1, 1 ) = -WI*D1 CI( 2, 1 ) = ZERO CI( 1, 2 ) = ZERO CI( 2, 2 ) = -WI*D2 CMAX = ZERO ICMAX = 0 ! DO 20 J = 1, 4 IF( ABS( CRV( J ) )+ABS( CIV( J ) ).GT.CMAX ) THEN CMAX = ABS( CRV( J ) ) + ABS( CIV( J ) ) ICMAX = J END IF 20 CONTINUE ! ! If norm(C) < SMINI, use SMINI*identity. ! IF( CMAX.LT.SMINI ) THEN BNORM = MAX( ABS( B( 1, 1 ) )+ABS( B( 1, 2 ) ), & ABS( B( 2, 1 ) )+ABS( B( 2, 2 ) ) ) IF( SMINI.LT.ONE .AND. BNORM.GT.ONE ) THEN IF( BNORM.GT.BIGNUM*SMINI ) & SCALE = ONE / BNORM END IF TEMP = SCALE / SMINI X( 1, 1 ) = TEMP*B( 1, 1 ) X( 2, 1 ) = TEMP*B( 2, 1 ) X( 1, 2 ) = TEMP*B( 1, 2 ) X( 2, 2 ) = TEMP*B( 2, 2 ) XNORM = TEMP*BNORM INFO = 1 RETURN END IF ! ! Gaussian elimination with complete pivoting. ! UR11 = CRV( ICMAX ) UI11 = CIV( ICMAX ) CR21 = CRV( IPIVOT( 2, ICMAX ) ) CI21 = CIV( IPIVOT( 2, ICMAX ) ) UR12 = CRV( IPIVOT( 3, ICMAX ) ) UI12 = CIV( IPIVOT( 3, ICMAX ) ) CR22 = CRV( IPIVOT( 4, ICMAX ) ) CI22 = CIV( IPIVOT( 4, ICMAX ) ) IF( ICMAX.EQ.1 .OR. ICMAX.EQ.4 ) THEN ! ! Code when off-diagonals of pivoted C are real ! IF( ABS( UR11 ).GT.ABS( UI11 ) ) THEN TEMP = UI11 / UR11 UR11R = ONE / ( UR11*( ONE+TEMP**2 ) ) UI11R = -TEMP*UR11R ELSE TEMP = UR11 / UI11 UI11R = -ONE / ( UI11*( ONE+TEMP**2 ) ) UR11R = -TEMP*UI11R END IF LR21 = CR21*UR11R LI21 = CR21*UI11R UR12S = UR12*UR11R UI12S = UR12*UI11R UR22 = CR22 - UR12*LR21 UI22 = CI22 - UR12*LI21 ELSE ! ! Code when diagonals of pivoted C are real ! UR11R = ONE / UR11 UI11R = ZERO LR21 = CR21*UR11R LI21 = CI21*UR11R UR12S = UR12*UR11R UI12S = UI12*UR11R UR22 = CR22 - UR12*LR21 + UI12*LI21 UI22 = -UR12*LI21 - UI12*LR21 END IF U22ABS = ABS( UR22 ) + ABS( UI22 ) ! ! If smaller pivot < SMINI, use SMINI ! IF( U22ABS.LT.SMINI ) THEN UR22 = SMINI UI22 = ZERO INFO = 1 END IF IF( RSWAP( ICMAX ) ) THEN BR2 = B( 1, 1 ) BR1 = B( 2, 1 ) BI2 = B( 1, 2 ) BI1 = B( 2, 2 ) ELSE BR1 = B( 1, 1 ) BR2 = B( 2, 1 ) BI1 = B( 1, 2 ) BI2 = B( 2, 2 ) END IF BR2 = BR2 - LR21*BR1 + LI21*BI1 BI2 = BI2 - LI21*BR1 - LR21*BI1 BBND = MAX( ( ABS( BR1 )+ABS( BI1 ) )* & ( U22ABS*( ABS( UR11R )+ABS( UI11R ) ) ), & ABS( BR2 )+ABS( BI2 ) ) IF( BBND.GT.ONE .AND. U22ABS.LT.ONE ) THEN IF( BBND.GE.BIGNUM*U22ABS ) THEN SCALE = ONE / BBND BR1 = SCALE*BR1 BI1 = SCALE*BI1 BR2 = SCALE*BR2 BI2 = SCALE*BI2 END IF END IF ! CALL DLADIV( BR2, BI2, UR22, UI22, XR2, XI2 ) XR1 = UR11R*BR1 - UI11R*BI1 - UR12S*XR2 + UI12S*XI2 XI1 = UI11R*BR1 + UR11R*BI1 - UI12S*XR2 - UR12S*XI2 IF( ZSWAP( ICMAX ) ) THEN X( 1, 1 ) = XR2 X( 2, 1 ) = XR1 X( 1, 2 ) = XI2 X( 2, 2 ) = XI1 ELSE X( 1, 1 ) = XR1 X( 2, 1 ) = XR2 X( 1, 2 ) = XI1 X( 2, 2 ) = XI2 END IF XNORM = MAX( ABS( XR1 )+ABS( XI1 ), ABS( XR2 )+ABS( XI2 ) ) ! ! Further scaling if norm(A) norm(X) > overflow ! IF( XNORM.GT.ONE .AND. CMAX.GT.ONE ) THEN IF( XNORM.GT.BIGNUM / CMAX ) THEN TEMP = CMAX / BIGNUM X( 1, 1 ) = TEMP*X( 1, 1 ) X( 2, 1 ) = TEMP*X( 2, 1 ) X( 1, 2 ) = TEMP*X( 1, 2 ) X( 2, 2 ) = TEMP*X( 2, 2 ) XNORM = TEMP*XNORM SCALE = TEMP*SCALE END IF END IF END IF END IF ! RETURN ! ! End of DLALN2 ! END SUBROUTINE DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX, & PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, & POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! December 1, 1999 ! ! .. Scalar Arguments .. INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL, & LDGNUM, NL, NR, NRHS, SQRE DOUBLE PRECISION C, S ! .. ! .. Array Arguments .. INTEGER GIVCOL( LDGCOL, * ), PERM( * ) DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ), DIFL( * ), & DIFR( LDGNUM, * ), GIVNUM( LDGNUM, * ), & POLES( LDGNUM, * ), WORK( * ), Z( * ) ! .. ! ! Purpose ! ======= ! ! DLALS0 applies back the multiplying factors of either the left or the ! right singular vector matrix of a diagonal matrix appended by a row ! to the right hand side matrix B in solving the least squares problem ! using the divide-and-conquer SVD approach. ! ! For the left singular vector matrix, three types of orthogonal ! matrices are involved: ! ! (1L) Givens rotations: the number of such rotations is GIVPTR; the ! pairs of columns/rows they were applied to are stored in GIVCOL; ! and the C- and S-values of these rotations are stored in GIVNUM. ! ! (2L) Permutation. The (NL+1)-st row of B is to be moved to the first ! row, and for J=2:N, PERM(J)-th row of B is to be moved to the ! J-th row. ! ! (3L) The left singular vector matrix of the remaining matrix. ! ! For the right singular vector matrix, four types of orthogonal ! matrices are involved: ! ! (1R) The right singular vector matrix of the remaining matrix. ! ! (2R) If SQRE = 1, one extra Givens rotation to generate the right ! null space. ! ! (3R) The inverse transformation of (2L). ! ! (4R) The inverse transformation of (1L). ! ! Arguments ! ========= ! ! ICOMPQ (input) INTEGER ! Specifies whether singular vectors are to be computed in ! factored form: ! = 0: Left singular vector matrix. ! = 1: Right singular vector matrix. ! ! NL (input) INTEGER ! The row dimension of the upper block. NL >= 1. ! ! NR (input) INTEGER ! The row dimension of the lower block. NR >= 1. ! ! SQRE (input) INTEGER ! = 0: the lower block is an NR-by-NR square matrix. ! = 1: the lower block is an NR-by-(NR+1) rectangular matrix. ! ! The bidiagonal matrix has row dimension N = NL + NR + 1, ! and column dimension M = N + SQRE. ! ! NRHS (input) INTEGER ! The number of columns of B and BX. NRHS must be at least 1. ! ! B (input/output) DOUBLE PRECISION array, dimension ( LDB, NRHS ) ! On input, B contains the right hand sides of the least ! squares problem in rows 1 through M. On output, B contains ! the solution X in rows 1 through N. ! ! LDB (input) INTEGER ! The leading dimension of B. LDB must be at least ! max(1,MAX( M, N ) ). ! ! BX (workspace) DOUBLE PRECISION array, dimension ( LDBX, NRHS ) ! ! LDBX (input) INTEGER ! The leading dimension of BX. ! ! PERM (input) INTEGER array, dimension ( N ) ! The permutations (from deflation and sorting) applied ! to the two blocks. ! ! GIVPTR (input) INTEGER ! The number of Givens rotations which took place in this ! subproblem. ! ! GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 ) ! Each pair of numbers indicates a pair of rows/columns ! involved in a Givens rotation. ! ! LDGCOL (input) INTEGER ! The leading dimension of GIVCOL, must be at least N. ! ! GIVNUM (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) ! Each number indicates the C or S value used in the ! corresponding Givens rotation. ! ! LDGNUM (input) INTEGER ! The leading dimension of arrays DIFR, POLES and ! GIVNUM, must be at least K. ! ! POLES (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) ! On entry, POLES(1:K, 1) contains the new singular ! values obtained from solving the secular equation, and ! POLES(1:K, 2) is an array containing the poles in the secular ! equation. ! ! DIFL (input) DOUBLE PRECISION array, dimension ( K ). ! On entry, DIFL(I) is the distance between I-th updated ! (undeflated) singular value and the I-th (undeflated) old ! singular value. ! ! DIFR (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ). ! On entry, DIFR(I, 1) contains the distances between I-th ! updated (undeflated) singular value and the I+1-th ! (undeflated) old singular value. And DIFR(I, 2) is the ! normalizing factor for the I-th right singular vector. ! ! Z (input) DOUBLE PRECISION array, dimension ( K ) ! Contain the components of the deflation-adjusted updating row ! vector. ! ! K (input) INTEGER ! Contains the dimension of the non-deflated matrix, ! This is the order of the related secular equation. 1 <= K <=N. ! ! C (input) DOUBLE PRECISION ! C contains garbage if SQRE =0 and the C-value of a Givens ! rotation related to the right null space if SQRE = 1. ! ! S (input) DOUBLE PRECISION ! S contains garbage if SQRE =0 and the S-value of a Givens ! rotation related to the right null space if SQRE = 1. ! ! WORK (workspace) DOUBLE PRECISION array, dimension ( K ) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Ren-Cang Li, Computer Science Division, University of ! California at Berkeley, USA ! Osni Marques, LBNL/NERSC, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO, NEGONE PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0, NEGONE = -1.0D0 ) ! .. ! .. Local Scalars .. INTEGER I, J, M, N, NLP1 DOUBLE PRECISION DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, TEMP ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEMV, DLACPY, DLASCL, DROT, DSCAL, & XERBLA ! .. ! .. External Functions .. DOUBLE PRECISION DLAMC3, DNRM2 EXTERNAL DLAMC3, DNRM2 ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN INFO = -1 ELSE IF( NL.LT.1 ) THEN INFO = -2 ELSE IF( NR.LT.1 ) THEN INFO = -3 ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN INFO = -4 END IF ! N = NL + NR + 1 ! IF( NRHS.LT.1 ) THEN INFO = -5 ELSE IF( LDB.LT.N ) THEN INFO = -7 ELSE IF( LDBX.LT.N ) THEN INFO = -9 ELSE IF( GIVPTR.LT.0 ) THEN INFO = -11 ELSE IF( LDGCOL.LT.N ) THEN INFO = -13 ELSE IF( LDGNUM.LT.N ) THEN INFO = -15 ELSE IF( K.LT.1 ) THEN INFO = -20 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLALS0', -INFO ) RETURN END IF ! M = N + SQRE NLP1 = NL + 1 ! IF( ICOMPQ.EQ.0 ) THEN ! ! Apply back orthogonal transformations from the left. ! ! Step (1L): apply back the Givens rotations performed. ! DO 10 I = 1, GIVPTR CALL DROT( NRHS, B( GIVCOL( I, 2 ), 1 ), LDB, & B( GIVCOL( I, 1 ), 1 ), LDB, GIVNUM( I, 2 ), & GIVNUM( I, 1 ) ) 10 CONTINUE ! ! Step (2L): permute rows of B. ! CALL DCOPY( NRHS, B( NLP1, 1 ), LDB, BX( 1, 1 ), LDBX ) DO 20 I = 2, N CALL DCOPY( NRHS, B( PERM( I ), 1 ), LDB, BX( I, 1 ), LDBX ) 20 CONTINUE ! ! Step (3L): apply the inverse of the left singular vector ! matrix to BX. ! IF( K.EQ.1 ) THEN CALL DCOPY( NRHS, BX, LDBX, B, LDB ) IF( Z( 1 ).LT.ZERO ) THEN CALL DSCAL( NRHS, NEGONE, B, LDB ) END IF ELSE DO 50 J = 1, K DIFLJ = DIFL( J ) DJ = POLES( J, 1 ) DSIGJ = -POLES( J, 2 ) IF( J.LT.K ) THEN DIFRJ = -DIFR( J, 1 ) DSIGJP = -POLES( J+1, 2 ) END IF IF( ( Z( J ).EQ.ZERO ) .OR. ( POLES( J, 2 ).EQ.ZERO ) ) & THEN WORK( J ) = ZERO ELSE WORK( J ) = -POLES( J, 2 )*Z( J ) / DIFLJ / & ( POLES( J, 2 )+DJ ) END IF DO 30 I = 1, J - 1 IF( ( Z( I ).EQ.ZERO ) .OR. & ( POLES( I, 2 ).EQ.ZERO ) ) THEN WORK( I ) = ZERO ELSE WORK( I ) = POLES( I, 2 )*Z( I ) / & ( DLAMC3( POLES( I, 2 ), DSIGJ )- & DIFLJ ) / ( POLES( I, 2 )+DJ ) END IF 30 CONTINUE DO 40 I = J + 1, K IF( ( Z( I ).EQ.ZERO ) .OR. & ( POLES( I, 2 ).EQ.ZERO ) ) THEN WORK( I ) = ZERO ELSE WORK( I ) = POLES( I, 2 )*Z( I ) / & ( DLAMC3( POLES( I, 2 ), DSIGJP )+ & DIFRJ ) / ( POLES( I, 2 )+DJ ) END IF 40 CONTINUE WORK( 1 ) = NEGONE TEMP = DNRM2( K, WORK, 1 ) CALL DGEMV( 'T', K, NRHS, ONE, BX, LDBX, WORK, 1, ZERO, & B( J, 1 ), LDB ) CALL DLASCL( 'G', 0, 0, TEMP, ONE, 1, NRHS, B( J, 1 ), & LDB, INFO ) 50 CONTINUE END IF ! ! Move the deflated rows of BX to B also. ! IF( K.LT.MAX( M, N ) ) & CALL DLACPY( 'A', N-K, NRHS, BX( K+1, 1 ), LDBX, & B( K+1, 1 ), LDB ) ELSE ! ! Apply back the right orthogonal transformations. ! ! Step (1R): apply back the new right singular vector matrix ! to B. ! IF( K.EQ.1 ) THEN CALL DCOPY( NRHS, B, LDB, BX, LDBX ) ELSE DO 80 J = 1, K DSIGJ = POLES( J, 2 ) IF( Z( J ).EQ.ZERO ) THEN WORK( J ) = ZERO ELSE WORK( J ) = -Z( J ) / DIFL( J ) / & ( DSIGJ+POLES( J, 1 ) ) / DIFR( J, 2 ) END IF DO 60 I = 1, J - 1 IF( Z( J ).EQ.ZERO ) THEN WORK( I ) = ZERO ELSE WORK( I ) = Z( J ) / ( DLAMC3( DSIGJ, -POLES( I+1, & 2 ) )-DIFR( I, 1 ) ) / & ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 ) END IF 60 CONTINUE DO 70 I = J + 1, K IF( Z( J ).EQ.ZERO ) THEN WORK( I ) = ZERO ELSE WORK( I ) = Z( J ) / ( DLAMC3( DSIGJ, -POLES( I, & 2 ) )-DIFL( I ) ) / & ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 ) END IF 70 CONTINUE CALL DGEMV( 'T', K, NRHS, ONE, B, LDB, WORK, 1, ZERO, & BX( J, 1 ), LDBX ) 80 CONTINUE END IF ! ! Step (2R): if SQRE = 1, apply back the rotation that is ! related to the right null space of the subproblem. ! IF( SQRE.EQ.1 ) THEN CALL DCOPY( NRHS, B( M, 1 ), LDB, BX( M, 1 ), LDBX ) CALL DROT( NRHS, BX( 1, 1 ), LDBX, BX( M, 1 ), LDBX, C, S ) END IF IF( K.LT.MAX( M, N ) ) & CALL DLACPY( 'A', N-K, NRHS, B( K+1, 1 ), LDB, BX( K+1, 1 ), & LDBX ) ! ! Step (3R): permute rows of B. ! CALL DCOPY( NRHS, BX( 1, 1 ), LDBX, B( NLP1, 1 ), LDB ) IF( SQRE.EQ.1 ) THEN CALL DCOPY( NRHS, BX( M, 1 ), LDBX, B( M, 1 ), LDB ) END IF DO 90 I = 2, N CALL DCOPY( NRHS, BX( I, 1 ), LDBX, B( PERM( I ), 1 ), LDB ) 90 CONTINUE ! ! Step (4R): apply back the Givens rotations performed. ! DO 100 I = GIVPTR, 1, -1 CALL DROT( NRHS, B( GIVCOL( I, 2 ), 1 ), LDB, & B( GIVCOL( I, 1 ), 1 ), LDB, GIVNUM( I, 2 ), & -GIVNUM( I, 1 ) ) 100 CONTINUE END IF ! RETURN ! ! End of DLALS0 ! END SUBROUTINE DLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U, & LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, & GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK, & IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS, & SMLSIZ ! .. ! .. Array Arguments .. INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), & K( * ), PERM( LDGCOL, * ) DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ), C( * ), & DIFL( LDU, * ), DIFR( LDU, * ), & GIVNUM( LDU, * ), POLES( LDU, * ), S( * ), & U( LDU, * ), VT( LDU, * ), WORK( * ), & Z( LDU, * ) ! .. ! ! Purpose ! ======= ! ! DLALSA is an itermediate step in solving the least squares problem ! by computing the SVD of the coefficient matrix in compact form (The ! singular vectors are computed as products of simple orthorgonal ! matrices.). ! ! If ICOMPQ = 0, DLALSA applies the inverse of the left singular vector ! matrix of an upper bidiagonal matrix to the right hand side; and if ! ICOMPQ = 1, DLALSA applies the right singular vector matrix to the ! right hand side. The singular vector matrices were generated in ! compact form by DLALSA. ! ! Arguments ! ========= ! ! ! ICOMPQ (input) INTEGER ! Specifies whether the left or the right singular vector ! matrix is involved. ! = 0: Left singular vector matrix ! = 1: Right singular vector matrix ! ! SMLSIZ (input) INTEGER ! The maximum size of the subproblems at the bottom of the ! computation tree. ! ! N (input) INTEGER ! The row and column dimensions of the upper bidiagonal matrix. ! ! NRHS (input) INTEGER ! The number of columns of B and BX. NRHS must be at least 1. ! ! B (input) DOUBLE PRECISION array, dimension ( LDB, NRHS ) ! On input, B contains the right hand sides of the least ! squares problem in rows 1 through M. On output, B contains ! the solution X in rows 1 through N. ! ! LDB (input) INTEGER ! The leading dimension of B in the calling subprogram. ! LDB must be at least max(1,MAX( M, N ) ). ! ! BX (output) DOUBLE PRECISION array, dimension ( LDBX, NRHS ) ! On exit, the result of applying the left or right singular ! vector matrix to B. ! ! LDBX (input) INTEGER ! The leading dimension of BX. ! ! U (input) DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ). ! On entry, U contains the left singular vector matrices of all ! subproblems at the bottom level. ! ! LDU (input) INTEGER, LDU = > N. ! The leading dimension of arrays U, VT, DIFL, DIFR, ! POLES, GIVNUM, and Z. ! ! VT (input) DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ). ! On entry, VT' contains the right singular vector matrices of ! all subproblems at the bottom level. ! ! K (input) INTEGER array, dimension ( N ). ! ! DIFL (input) DOUBLE PRECISION array, dimension ( LDU, NLVL ). ! where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1. ! ! DIFR (input) DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ). ! On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record ! distances between singular values on the I-th level and ! singular values on the (I -1)-th level, and DIFR(*, 2 * I) ! record the normalizing factors of the right singular vectors ! matrices of subproblems on I-th level. ! ! Z (input) DOUBLE PRECISION array, dimension ( LDU, NLVL ). ! On entry, Z(1, I) contains the components of the deflation- ! adjusted updating row vector for subproblems on the I-th ! level. ! ! POLES (input) DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ). ! On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old ! singular values involved in the secular equations on the I-th ! level. ! ! GIVPTR (input) INTEGER array, dimension ( N ). ! On entry, GIVPTR( I ) records the number of Givens ! rotations performed on the I-th problem on the computation ! tree. ! ! GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 * NLVL ). ! On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the ! locations of Givens rotations performed on the I-th level on ! the computation tree. ! ! LDGCOL (input) INTEGER, LDGCOL = > N. ! The leading dimension of arrays GIVCOL and PERM. ! ! PERM (input) INTEGER array, dimension ( LDGCOL, NLVL ). ! On entry, PERM(*, I) records permutations done on the I-th ! level of the computation tree. ! ! GIVNUM (input) DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ). ! On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S- ! values of Givens rotations performed on the I-th level on the ! computation tree. ! ! C (input) DOUBLE PRECISION array, dimension ( N ). ! On entry, if the I-th subproblem is not square, ! C( I ) contains the C-value of a Givens rotation related to ! the right null space of the I-th subproblem. ! ! S (input) DOUBLE PRECISION array, dimension ( N ). ! On entry, if the I-th subproblem is not square, ! S( I ) contains the S-value of a Givens rotation related to ! the right null space of the I-th subproblem. ! ! WORK (workspace) DOUBLE PRECISION array. ! The dimension must be at least N. ! ! IWORK (workspace) INTEGER array. ! The dimension must be at least 3 * N ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Ren-Cang Li, Computer Science Division, University of ! California at Berkeley, USA ! Osni Marques, LBNL/NERSC, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. INTEGER I, I1, IC, IM1, INODE, J, LF, LL, LVL, LVL2, & ND, NDB1, NDIML, NDIMR, NL, NLF, NLP1, NLVL, & NR, NRF, NRP1, SQRE ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEMM, DLALS0, DLASDT, XERBLA ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN INFO = -1 ELSE IF( SMLSIZ.LT.3 ) THEN INFO = -2 ELSE IF( N.LT.SMLSIZ ) THEN INFO = -3 ELSE IF( NRHS.LT.1 ) THEN INFO = -4 ELSE IF( LDB.LT.N ) THEN INFO = -6 ELSE IF( LDBX.LT.N ) THEN INFO = -8 ELSE IF( LDU.LT.N ) THEN INFO = -10 ELSE IF( LDGCOL.LT.N ) THEN INFO = -19 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLALSA', -INFO ) RETURN END IF ! ! Book-keeping and setting up the computation tree. ! INODE = 1 NDIML = INODE + N NDIMR = NDIML + N ! CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ), & IWORK( NDIMR ), SMLSIZ ) ! ! The following code applies back the left singular vector factors. ! For applying back the right singular vector factors, go to 50. ! IF( ICOMPQ.EQ.1 ) THEN GO TO 50 END IF ! ! The nodes on the bottom level of the tree were solved ! by DLASDQ. The corresponding left and right singular vector ! matrices are in explicit form. First apply back the left ! singular vector matrices. ! NDB1 = ( ND+1 ) / 2 DO 10 I = NDB1, ND ! ! IC : center row of each node ! NL : number of rows of left subproblem ! NR : number of rows of right subproblem ! NLF: starting row of the left subproblem ! NRF: starting row of the right subproblem ! I1 = I - 1 IC = IWORK( INODE+I1 ) NL = IWORK( NDIML+I1 ) NR = IWORK( NDIMR+I1 ) NLF = IC - NL NRF = IC + 1 CALL DGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU, & B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX ) CALL DGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU, & B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX ) 10 CONTINUE ! ! Next copy the rows of B that correspond to unchanged rows ! in the bidiagonal matrix to BX. ! DO 20 I = 1, ND IC = IWORK( INODE+I-1 ) CALL DCOPY( NRHS, B( IC, 1 ), LDB, BX( IC, 1 ), LDBX ) 20 CONTINUE ! ! Finally go through the left singular vector matrices of all ! the other subproblems bottom-up on the tree. ! J = 2**NLVL SQRE = 0 ! DO 40 LVL = NLVL, 1, -1 LVL2 = 2*LVL - 1 ! ! find the first node LF and last node LL on ! the current level LVL ! IF( LVL.EQ.1 ) THEN LF = 1 LL = 1 ELSE LF = 2**( LVL-1 ) LL = 2*LF - 1 END IF DO 30 I = LF, LL IM1 = I - 1 IC = IWORK( INODE+IM1 ) NL = IWORK( NDIML+IM1 ) NR = IWORK( NDIMR+IM1 ) NLF = IC - NL NRF = IC + 1 J = J - 1 CALL DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, BX( NLF, 1 ), LDBX, & B( NLF, 1 ), LDB, PERM( NLF, LVL ), & GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL, & GIVNUM( NLF, LVL2 ), LDU, POLES( NLF, LVL2 ), & DIFL( NLF, LVL ), DIFR( NLF, LVL2 ), & Z( NLF, LVL ), K( J ), C( J ), S( J ), WORK, & INFO ) 30 CONTINUE 40 CONTINUE GO TO 90 ! ! ICOMPQ = 1: applying back the right singular vector factors. ! 50 CONTINUE ! ! First now go through the right singular vector matrices of all ! the tree nodes top-down. ! J = 0 DO 70 LVL = 1, NLVL LVL2 = 2*LVL - 1 ! ! Find the first node LF and last node LL on ! the current level LVL. ! IF( LVL.EQ.1 ) THEN LF = 1 LL = 1 ELSE LF = 2**( LVL-1 ) LL = 2*LF - 1 END IF DO 60 I = LL, LF, -1 IM1 = I - 1 IC = IWORK( INODE+IM1 ) NL = IWORK( NDIML+IM1 ) NR = IWORK( NDIMR+IM1 ) NLF = IC - NL NRF = IC + 1 IF( I.EQ.LL ) THEN SQRE = 0 ELSE SQRE = 1 END IF J = J + 1 CALL DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B( NLF, 1 ), LDB, & BX( NLF, 1 ), LDBX, PERM( NLF, LVL ), & GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL, & GIVNUM( NLF, LVL2 ), LDU, POLES( NLF, LVL2 ), & DIFL( NLF, LVL ), DIFR( NLF, LVL2 ), & Z( NLF, LVL ), K( J ), C( J ), S( J ), WORK, & INFO ) 60 CONTINUE 70 CONTINUE ! ! The nodes on the bottom level of the tree were solved ! by DLASDQ. The corresponding right singular vector ! matrices are in explicit form. Apply them back. ! NDB1 = ( ND+1 ) / 2 DO 80 I = NDB1, ND I1 = I - 1 IC = IWORK( INODE+I1 ) NL = IWORK( NDIML+I1 ) NR = IWORK( NDIMR+I1 ) NLP1 = NL + 1 IF( I.EQ.ND ) THEN NRP1 = NR ELSE NRP1 = NR + 1 END IF NLF = IC - NL NRF = IC + 1 CALL DGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ), LDU, & B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX ) CALL DGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ), LDU, & B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX ) 80 CONTINUE ! 90 CONTINUE ! RETURN ! ! End of DLALSA ! END SUBROUTINE DLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, & RANK, WORK, IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDB, N, NRHS, RANK, SMLSIZ DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION B( LDB, * ), D( * ), E( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLALSD uses the singular value decomposition of A to solve the least ! squares problem of finding X to minimize the Euclidean norm of each ! column of A*X-B, where A is N-by-N upper bidiagonal, and X and B ! are N-by-NRHS. The solution X overwrites B. ! ! The singular values of A smaller than RCOND times the largest ! singular value are treated as zero in solving the least squares ! problem; in this case a minimum norm solution is returned. ! The actual singular values are returned in D in ascending order. ! ! This code makes very mild assumptions about floating point ! arithmetic. It will work on machines with a guard digit in ! add/subtract, or on those binary machines without guard digits ! which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2. ! It could conceivably fail on hexadecimal or decimal machines ! without guard digits, but we know of none. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': D and E define an upper bidiagonal matrix. ! = 'L': D and E define a lower bidiagonal matrix. ! ! SMLSIZ (input) INTEGER ! The maximum size of the subproblems at the bottom of the ! computation tree. ! ! N (input) INTEGER ! The dimension of the bidiagonal matrix. N >= 0. ! ! NRHS (input) INTEGER ! The number of columns of B. NRHS must be at least 1. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry D contains the main diagonal of the bidiagonal ! matrix. On exit, if INFO = 0, D contains its singular values. ! ! E (input) DOUBLE PRECISION array, dimension (N-1) ! Contains the super-diagonal entries of the bidiagonal matrix. ! On exit, E has been destroyed. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On input, B contains the right hand sides of the least ! squares problem. On output, B contains the solution X. ! ! LDB (input) INTEGER ! The leading dimension of B in the calling subprogram. ! LDB must be at least max(1,N). ! ! RCOND (input) DOUBLE PRECISION ! The singular values of A less than or equal to RCOND times ! the largest singular value are treated as zero in solving ! the least squares problem. If RCOND is negative, ! machine precision is used instead. ! For example, if diag(S)*X=B were the least squares problem, ! where diag(S) is a diagonal matrix of singular values, the ! solution would be X(i) = B(i) / S(i) if S(i) is greater than ! RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to ! RCOND*max(S). ! ! RANK (output) INTEGER ! The number of singular values of A greater than RCOND times ! the largest singular value. ! ! WORK (workspace) DOUBLE PRECISION array, dimension at least ! (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2), ! where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1). ! ! IWORK (workspace) INTEGER array, dimension at least ! (3*N*NLVL + 11*N) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: The algorithm failed to compute an singular value while ! working on the submatrix lying in rows and columns ! INFO/(N+1) through MOD(INFO,N+1). ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Ren-Cang Li, Computer Science Division, University of ! California at Berkeley, USA ! Osni Marques, LBNL/NERSC, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 ) ! .. ! .. Local Scalars .. INTEGER BX, BXST, C, DIFL, DIFR, GIVCOL, GIVNUM, & GIVPTR, I, ICMPQ1, ICMPQ2, IWK, J, K, NLVL, & NM1, NSIZE, NSUB, NWORK, PERM, POLES, S, SIZEI, & SMLSZP, SQRE, ST, ST1, U, VT, Z DOUBLE PRECISION CS, EPS, ORGNRM, R, SN, TOL ! .. ! .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DLAMCH, DLANST EXTERNAL IDAMAX, DLAMCH, DLANST ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEMM, DLACPY, DLALSA, DLARTG, DLASCL, & DLASDA, DLASDQ, DLASET, DLASRT, DROT, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, INT, LOG, SIGN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( N.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.1 ) THEN INFO = -4 ELSE IF( ( LDB.LT.1 ) .OR. ( LDB.LT.N ) ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLALSD', -INFO ) RETURN END IF ! EPS = DLAMCH( 'Epsilon' ) ! ! Set up the tolerance. ! IF( ( RCOND.LE.ZERO ) .OR. ( RCOND.GE.ONE ) ) THEN RCOND = EPS END IF ! RANK = 0 ! ! Quick return if possible. ! IF( N.EQ.0 ) THEN RETURN ELSE IF( N.EQ.1 ) THEN IF( D( 1 ).EQ.ZERO ) THEN CALL DLASET( 'A', 1, NRHS, ZERO, ZERO, B, LDB ) ELSE RANK = 1 CALL DLASCL( 'G', 0, 0, D( 1 ), ONE, 1, NRHS, B, LDB, INFO ) D( 1 ) = ABS( D( 1 ) ) END IF RETURN END IF ! ! Rotate the matrix if it is lower bidiagonal. ! IF( UPLO.EQ.'L' ) THEN DO 10 I = 1, N - 1 CALL DLARTG( D( I ), E( I ), CS, SN, R ) D( I ) = R E( I ) = SN*D( I+1 ) D( I+1 ) = CS*D( I+1 ) IF( NRHS.EQ.1 ) THEN CALL DROT( 1, B( I, 1 ), 1, B( I+1, 1 ), 1, CS, SN ) ELSE WORK( I*2-1 ) = CS WORK( I*2 ) = SN END IF 10 CONTINUE IF( NRHS.GT.1 ) THEN DO 30 I = 1, NRHS DO 20 J = 1, N - 1 CS = WORK( J*2-1 ) SN = WORK( J*2 ) CALL DROT( 1, B( J, I ), 1, B( J+1, I ), 1, CS, SN ) 20 CONTINUE 30 CONTINUE END IF END IF ! ! Scale. ! NM1 = N - 1 ORGNRM = DLANST( 'M', N, D, E ) IF( ORGNRM.EQ.ZERO ) THEN CALL DLASET( 'A', N, NRHS, ZERO, ZERO, B, LDB ) RETURN END IF ! CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO ) CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, NM1, 1, E, NM1, INFO ) ! ! If N is smaller than the minimum divide size SMLSIZ, then solve ! the problem with another solver. ! IF( N.LE.SMLSIZ ) THEN NWORK = 1 + N*N CALL DLASET( 'A', N, N, ZERO, ONE, WORK, N ) CALL DLASDQ( 'U', 0, N, N, 0, NRHS, D, E, WORK, N, WORK, N, B, & LDB, WORK( NWORK ), INFO ) IF( INFO.NE.0 ) THEN RETURN END IF TOL = RCOND*ABS( D( IDAMAX( N, D, 1 ) ) ) DO 40 I = 1, N IF( D( I ).LE.TOL ) THEN CALL DLASET( 'A', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB ) ELSE CALL DLASCL( 'G', 0, 0, D( I ), ONE, 1, NRHS, B( I, 1 ), & LDB, INFO ) RANK = RANK + 1 END IF 40 CONTINUE CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, WORK, N, B, LDB, ZERO, & WORK( NWORK ), N ) CALL DLACPY( 'A', N, NRHS, WORK( NWORK ), N, B, LDB ) ! ! Unscale. ! CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO ) CALL DLASRT( 'D', N, D, INFO ) CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, NRHS, B, LDB, INFO ) ! RETURN END IF ! ! Book-keeping and setting up some constants. ! NLVL = INT( LOG( DBLE( N ) / DBLE( SMLSIZ+1 ) ) / LOG( TWO ) ) + 1 ! SMLSZP = SMLSIZ + 1 ! U = 1 VT = 1 + SMLSIZ*N DIFL = VT + SMLSZP*N DIFR = DIFL + NLVL*N Z = DIFR + NLVL*N*2 C = Z + NLVL*N S = C + N POLES = S + N GIVNUM = POLES + 2*NLVL*N BX = GIVNUM + 2*NLVL*N NWORK = BX + N*NRHS ! SIZEI = 1 + N K = SIZEI + N GIVPTR = K + N PERM = GIVPTR + N GIVCOL = PERM + NLVL*N IWK = GIVCOL + NLVL*N*2 ! ST = 1 SQRE = 0 ICMPQ1 = 1 ICMPQ2 = 0 NSUB = 0 ! DO 50 I = 1, N IF( ABS( D( I ) ).LT.EPS ) THEN D( I ) = SIGN( EPS, D( I ) ) END IF 50 CONTINUE ! DO 60 I = 1, NM1 IF( ( ABS( E( I ) ).LT.EPS ) .OR. ( I.EQ.NM1 ) ) THEN NSUB = NSUB + 1 IWORK( NSUB ) = ST ! ! Subproblem found. First determine its size and then ! apply divide and conquer on it. ! IF( I.LT.NM1 ) THEN ! ! A subproblem with E(I) small for I < NM1. ! NSIZE = I - ST + 1 IWORK( SIZEI+NSUB-1 ) = NSIZE ELSE IF( ABS( E( I ) ).GE.EPS ) THEN ! ! A subproblem with E(NM1) not too small but I = NM1. ! NSIZE = N - ST + 1 IWORK( SIZEI+NSUB-1 ) = NSIZE ELSE ! ! A subproblem with E(NM1) small. This implies an ! 1-by-1 subproblem at D(N), which is not solved ! explicitly. ! NSIZE = I - ST + 1 IWORK( SIZEI+NSUB-1 ) = NSIZE NSUB = NSUB + 1 IWORK( NSUB ) = N IWORK( SIZEI+NSUB-1 ) = 1 CALL DCOPY( NRHS, B( N, 1 ), LDB, WORK( BX+NM1 ), N ) END IF ST1 = ST - 1 IF( NSIZE.EQ.1 ) THEN ! ! This is a 1-by-1 subproblem and is not solved ! explicitly. ! CALL DCOPY( NRHS, B( ST, 1 ), LDB, WORK( BX+ST1 ), N ) ELSE IF( NSIZE.LE.SMLSIZ ) THEN ! ! This is a small subproblem and is solved by DLASDQ. ! CALL DLASET( 'A', NSIZE, NSIZE, ZERO, ONE, & WORK( VT+ST1 ), N ) CALL DLASDQ( 'U', 0, NSIZE, NSIZE, 0, NRHS, D( ST ), & E( ST ), WORK( VT+ST1 ), N, WORK( NWORK ), & N, B( ST, 1 ), LDB, WORK( NWORK ), INFO ) IF( INFO.NE.0 ) THEN RETURN END IF CALL DLACPY( 'A', NSIZE, NRHS, B( ST, 1 ), LDB, & WORK( BX+ST1 ), N ) ELSE ! ! A large problem. Solve it using divide and conquer. ! CALL DLASDA( ICMPQ1, SMLSIZ, NSIZE, SQRE, D( ST ), & E( ST ), WORK( U+ST1 ), N, WORK( VT+ST1 ), & IWORK( K+ST1 ), WORK( DIFL+ST1 ), & WORK( DIFR+ST1 ), WORK( Z+ST1 ), & WORK( POLES+ST1 ), IWORK( GIVPTR+ST1 ), & IWORK( GIVCOL+ST1 ), N, IWORK( PERM+ST1 ), & WORK( GIVNUM+ST1 ), WORK( C+ST1 ), & WORK( S+ST1 ), WORK( NWORK ), IWORK( IWK ), & INFO ) IF( INFO.NE.0 ) THEN RETURN END IF BXST = BX + ST1 CALL DLALSA( ICMPQ2, SMLSIZ, NSIZE, NRHS, B( ST, 1 ), & LDB, WORK( BXST ), N, WORK( U+ST1 ), N, & WORK( VT+ST1 ), IWORK( K+ST1 ), & WORK( DIFL+ST1 ), WORK( DIFR+ST1 ), & WORK( Z+ST1 ), WORK( POLES+ST1 ), & IWORK( GIVPTR+ST1 ), IWORK( GIVCOL+ST1 ), N, & IWORK( PERM+ST1 ), WORK( GIVNUM+ST1 ), & WORK( C+ST1 ), WORK( S+ST1 ), WORK( NWORK ), & IWORK( IWK ), INFO ) IF( INFO.NE.0 ) THEN RETURN END IF END IF ST = I + 1 END IF 60 CONTINUE ! ! Apply the singular values and treat the tiny ones as zero. ! TOL = RCOND*ABS( D( IDAMAX( N, D, 1 ) ) ) ! DO 70 I = 1, N ! ! Some of the elements in D can be negative because 1-by-1 ! subproblems were not solved explicitly. ! IF( ABS( D( I ) ).LE.TOL ) THEN CALL DLASET( 'A', 1, NRHS, ZERO, ZERO, WORK( BX+I-1 ), N ) ELSE RANK = RANK + 1 CALL DLASCL( 'G', 0, 0, D( I ), ONE, 1, NRHS, & WORK( BX+I-1 ), N, INFO ) END IF D( I ) = ABS( D( I ) ) 70 CONTINUE ! ! Now apply back the right singular vectors. ! ICMPQ2 = 1 DO 80 I = 1, NSUB ST = IWORK( I ) ST1 = ST - 1 NSIZE = IWORK( SIZEI+I-1 ) BXST = BX + ST1 IF( NSIZE.EQ.1 ) THEN CALL DCOPY( NRHS, WORK( BXST ), N, B( ST, 1 ), LDB ) ELSE IF( NSIZE.LE.SMLSIZ ) THEN CALL DGEMM( 'T', 'N', NSIZE, NRHS, NSIZE, ONE, & WORK( VT+ST1 ), N, WORK( BXST ), N, ZERO, & B( ST, 1 ), LDB ) ELSE CALL DLALSA( ICMPQ2, SMLSIZ, NSIZE, NRHS, WORK( BXST ), N, & B( ST, 1 ), LDB, WORK( U+ST1 ), N, & WORK( VT+ST1 ), IWORK( K+ST1 ), & WORK( DIFL+ST1 ), WORK( DIFR+ST1 ), & WORK( Z+ST1 ), WORK( POLES+ST1 ), & IWORK( GIVPTR+ST1 ), IWORK( GIVCOL+ST1 ), N, & IWORK( PERM+ST1 ), WORK( GIVNUM+ST1 ), & WORK( C+ST1 ), WORK( S+ST1 ), WORK( NWORK ), & IWORK( IWK ), INFO ) IF( INFO.NE.0 ) THEN RETURN END IF END IF 80 CONTINUE ! ! Unscale and sort the singular values. ! CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO ) CALL DLASRT( 'D', N, D, INFO ) CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, NRHS, B, LDB, INFO ) ! RETURN ! ! End of DLALSD ! END DOUBLE PRECISION FUNCTION DLAMCH( CMACH ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER CMACH ! .. ! ! Purpose ! ======= ! ! DLAMCH determines double precision machine parameters. ! ! Arguments ! ========= ! ! CMACH (input) CHARACTER*1 ! Specifies the value to be returned by DLAMCH: ! = 'E' or 'e', DLAMCH := eps ! = 'S' or 's , DLAMCH := sfmin ! = 'B' or 'b', DLAMCH := base ! = 'P' or 'p', DLAMCH := eps*base ! = 'N' or 'n', DLAMCH := t ! = 'R' or 'r', DLAMCH := rnd ! = 'M' or 'm', DLAMCH := emin ! = 'U' or 'u', DLAMCH := rmin ! = 'L' or 'l', DLAMCH := emax ! = 'O' or 'o', DLAMCH := rmax ! ! where ! ! eps = relative machine precision ! sfmin = safe minimum, such that 1/sfmin does not overflow ! base = base of the machine ! prec = eps*base ! t = number of (base) digits in the mantissa ! rnd = 1.0 when rounding occurs in addition, 0.0 otherwise ! emin = minimum exponent before (gradual) underflow ! rmin = underflow threshold - base**(emin-1) ! emax = largest exponent before overflow ! rmax = overflow threshold - (base**emax)*(1-eps) ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL FIRST, LRND INTEGER BETA, IMAX, IMIN, IT DOUBLE PRECISION BASE, EMAX, EMIN, EPS, PREC, RMACH, RMAX, RMIN, & RND, SFMIN, SMALL, T ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DLAMC2 ! .. ! .. Save statement .. SAVE FIRST, EPS, SFMIN, BASE, T, RND, EMIN, RMIN, & EMAX, RMAX, PREC ! .. ! .. Data statements .. DATA FIRST / .TRUE. / ! .. ! .. Executable Statements .. ! IF( FIRST ) THEN FIRST = .FALSE. CALL DLAMC2( BETA, IT, LRND, EPS, IMIN, RMIN, IMAX, RMAX ) BASE = BETA T = IT IF( LRND ) THEN RND = ONE EPS = ( BASE**( 1-IT ) ) / 2 ELSE RND = ZERO EPS = BASE**( 1-IT ) END IF PREC = EPS*BASE EMIN = IMIN EMAX = IMAX SFMIN = RMIN SMALL = ONE / RMAX IF( SMALL.GE.SFMIN ) THEN ! ! Use SMALL plus a bit, to avoid the possibility of rounding ! causing overflow when computing 1/sfmin. ! SFMIN = SMALL*( ONE+EPS ) END IF END IF ! IF( LSAME( CMACH, 'E' ) ) THEN RMACH = EPS ELSE IF( LSAME( CMACH, 'S' ) ) THEN RMACH = SFMIN ELSE IF( LSAME( CMACH, 'B' ) ) THEN RMACH = BASE ELSE IF( LSAME( CMACH, 'P' ) ) THEN RMACH = PREC ELSE IF( LSAME( CMACH, 'N' ) ) THEN RMACH = T ELSE IF( LSAME( CMACH, 'R' ) ) THEN RMACH = RND ELSE IF( LSAME( CMACH, 'M' ) ) THEN RMACH = EMIN ELSE IF( LSAME( CMACH, 'U' ) ) THEN RMACH = RMIN ELSE IF( LSAME( CMACH, 'L' ) ) THEN RMACH = EMAX ELSE IF( LSAME( CMACH, 'O' ) ) THEN RMACH = RMAX END IF ! DLAMCH = RMACH RETURN ! ! End of DLAMCH ! END ! !*********************************************************************** ! SUBROUTINE DLAMC1( BETA, T, RND, IEEE1 ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. LOGICAL IEEE1, RND INTEGER BETA, T ! .. ! ! Purpose ! ======= ! ! DLAMC1 determines the machine parameters given by BETA, T, RND, and ! IEEE1. ! ! Arguments ! ========= ! ! BETA (output) INTEGER ! The base of the machine. ! ! T (output) INTEGER ! The number of ( BETA ) digits in the mantissa. ! ! RND (output) LOGICAL ! Specifies whether proper rounding ( RND = .TRUE. ) or ! chopping ( RND = .FALSE. ) occurs in addition. This may not ! be a reliable guide to the way in which the machine performs ! its arithmetic. ! ! IEEE1 (output) LOGICAL ! Specifies whether rounding appears to be done in the IEEE ! 'round to nearest' style. ! ! Further Details ! =============== ! ! The routine is based on the routine ENVRON by Malcolm and ! incorporates suggestions by Gentleman and Marovich. See ! ! Malcolm M. A. (1972) Algorithms to reveal properties of ! floating-point arithmetic. Comms. of the ACM, 15, 949-951. ! ! Gentleman W. M. and Marovich S. B. (1974) More on algorithms ! that reveal properties of floating point arithmetic units. ! Comms. of the ACM, 17, 276-277. ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL FIRST, LIEEE1, LRND INTEGER LBETA, LT DOUBLE PRECISION A, B, C, F, ONE, QTR, SAVEC, T1, T2 ! .. ! .. External Functions .. DOUBLE PRECISION DLAMC3 EXTERNAL DLAMC3 ! .. ! .. Save statement .. SAVE FIRST, LIEEE1, LBETA, LRND, LT ! .. ! .. Data statements .. DATA FIRST / .TRUE. / ! .. ! .. Executable Statements .. ! IF( FIRST ) THEN FIRST = .FALSE. ONE = 1 ! ! LBETA, LIEEE1, LT and LRND are the local values of BETA, ! IEEE1, T and RND. ! ! Throughout this routine we use the function DLAMC3 to ensure ! that relevant values are stored and not held in registers, or ! are not affected by optimizers. ! ! Compute a = 2.0**m with the smallest positive integer m such ! that ! ! fl( a + 1.0 ) = a. ! A = 1 C = 1 ! !+ WHILE( C.EQ.ONE )LOOP 10 CONTINUE IF( C.EQ.ONE ) THEN A = 2*A C = DLAMC3( A, ONE ) C = DLAMC3( C, -A ) GO TO 10 END IF !+ END WHILE ! ! Now compute b = 2.0**m with the smallest positive integer m ! such that ! ! fl( a + b ) .gt. a. ! B = 1 C = DLAMC3( A, B ) ! !+ WHILE( C.EQ.A )LOOP 20 CONTINUE IF( C.EQ.A ) THEN B = 2*B C = DLAMC3( A, B ) GO TO 20 END IF !+ END WHILE ! ! Now compute the base. a and c are neighbouring floating point ! numbers in the interval ( beta**t, beta**( t + 1 ) ) and so ! their difference is beta. Adding 0.25 to c is to ensure that it ! is truncated to beta and not ( beta - 1 ). ! QTR = ONE / 4 SAVEC = C C = DLAMC3( C, -A ) LBETA = C + QTR ! ! Now determine whether rounding or chopping occurs, by adding a ! bit less than beta/2 and a bit more than beta/2 to a. ! B = LBETA F = DLAMC3( B / 2, -B / 100 ) C = DLAMC3( F, A ) IF( C.EQ.A ) THEN LRND = .TRUE. ELSE LRND = .FALSE. END IF F = DLAMC3( B / 2, B / 100 ) C = DLAMC3( F, A ) IF( ( LRND ) .AND. ( C.EQ.A ) ) & LRND = .FALSE. ! ! Try and decide whether rounding is done in the IEEE 'round to ! nearest' style. B/2 is half a unit in the last place of the two ! numbers A and SAVEC. Furthermore, A is even, i.e. has last bit ! zero, and SAVEC is odd. Thus adding B/2 to A should not change ! A, but adding B/2 to SAVEC should change SAVEC. ! T1 = DLAMC3( B / 2, A ) T2 = DLAMC3( B / 2, SAVEC ) LIEEE1 = ( T1.EQ.A ) .AND. ( T2.GT.SAVEC ) .AND. LRND ! ! Now find the mantissa, t. It should be the integer part of ! log to the base beta of a, however it is safer to determine t ! by powering. So we find t as the smallest positive integer for ! which ! ! fl( beta**t + 1.0 ) = 1.0. ! LT = 0 A = 1 C = 1 ! !+ WHILE( C.EQ.ONE )LOOP 30 CONTINUE IF( C.EQ.ONE ) THEN LT = LT + 1 A = A*LBETA C = DLAMC3( A, ONE ) C = DLAMC3( C, -A ) GO TO 30 END IF !+ END WHILE ! END IF ! BETA = LBETA T = LT RND = LRND IEEE1 = LIEEE1 RETURN ! ! End of DLAMC1 ! END ! !*********************************************************************** ! SUBROUTINE DLAMC2( BETA, T, RND, EPS, EMIN, RMIN, EMAX, RMAX ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. LOGICAL RND INTEGER BETA, EMAX, EMIN, T DOUBLE PRECISION EPS, RMAX, RMIN ! .. ! ! Purpose ! ======= ! ! DLAMC2 determines the machine parameters specified in its argument ! list. ! ! Arguments ! ========= ! ! BETA (output) INTEGER ! The base of the machine. ! ! T (output) INTEGER ! The number of ( BETA ) digits in the mantissa. ! ! RND (output) LOGICAL ! Specifies whether proper rounding ( RND = .TRUE. ) or ! chopping ( RND = .FALSE. ) occurs in addition. This may not ! be a reliable guide to the way in which the machine performs ! its arithmetic. ! ! EPS (output) DOUBLE PRECISION ! The smallest positive number such that ! ! fl( 1.0 - EPS ) .LT. 1.0, ! ! where fl denotes the computed value. ! ! EMIN (output) INTEGER ! The minimum exponent before (gradual) underflow occurs. ! ! RMIN (output) DOUBLE PRECISION ! The smallest normalized number for the machine, given by ! BASE**( EMIN - 1 ), where BASE is the floating point value ! of BETA. ! ! EMAX (output) INTEGER ! The maximum exponent before overflow occurs. ! ! RMAX (output) DOUBLE PRECISION ! The largest positive number for the machine, given by ! BASE**EMAX * ( 1 - EPS ), where BASE is the floating point ! value of BETA. ! ! Further Details ! =============== ! ! The computation of EPS is based on a routine PARANOIA by ! W. Kahan of the University of California at Berkeley. ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL FIRST, IEEE, IWARN, LIEEE1, LRND INTEGER GNMIN, GPMIN, I, LBETA, LEMAX, LEMIN, LT, & NGNMIN, NGPMIN DOUBLE PRECISION A, B, C, HALF, LEPS, LRMAX, LRMIN, ONE, RBASE, & SIXTH, SMALL, THIRD, TWO, ZERO ! .. ! .. External Functions .. DOUBLE PRECISION DLAMC3 EXTERNAL DLAMC3 ! .. ! .. External Subroutines .. EXTERNAL DLAMC1, DLAMC4, DLAMC5 ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Save statement .. SAVE FIRST, IWARN, LBETA, LEMAX, LEMIN, LEPS, LRMAX, & LRMIN, LT ! .. ! .. Data statements .. DATA FIRST / .TRUE. / , IWARN / .FALSE. / ! .. ! .. Executable Statements .. ! IF( FIRST ) THEN FIRST = .FALSE. ZERO = 0 ONE = 1 TWO = 2 ! ! LBETA, LT, LRND, LEPS, LEMIN and LRMIN are the local values of ! BETA, T, RND, EPS, EMIN and RMIN. ! ! Throughout this routine we use the function DLAMC3 to ensure ! that relevant values are stored and not held in registers, or ! are not affected by optimizers. ! ! DLAMC1 returns the parameters LBETA, LT, LRND and LIEEE1. ! CALL DLAMC1( LBETA, LT, LRND, LIEEE1 ) ! ! Start to find EPS. ! B = LBETA A = B**( -LT ) LEPS = A ! ! Try some tricks to see whether or not this is the correct EPS. ! B = TWO / 3 HALF = ONE / 2 SIXTH = DLAMC3( B, -HALF ) THIRD = DLAMC3( SIXTH, SIXTH ) B = DLAMC3( THIRD, -HALF ) B = DLAMC3( B, SIXTH ) B = ABS( B ) IF( B.LT.LEPS ) & B = LEPS ! LEPS = 1 ! !+ WHILE( ( LEPS.GT.B ).AND.( B.GT.ZERO ) )LOOP 10 CONTINUE IF( ( LEPS.GT.B ) .AND. ( B.GT.ZERO ) ) THEN LEPS = B C = DLAMC3( HALF*LEPS, ( TWO**5 )*( LEPS**2 ) ) C = DLAMC3( HALF, -C ) B = DLAMC3( HALF, C ) C = DLAMC3( HALF, -B ) B = DLAMC3( HALF, C ) GO TO 10 END IF !+ END WHILE ! IF( A.LT.LEPS ) & LEPS = A ! ! Computation of EPS complete. ! ! Now find EMIN. Let A = + or - 1, and + or - (1 + BASE**(-3)). ! Keep dividing A by BETA until (gradual) underflow occurs. This ! is detected when we cannot recover the previous A. ! RBASE = ONE / LBETA SMALL = ONE DO 20 I = 1, 3 SMALL = DLAMC3( SMALL*RBASE, ZERO ) 20 CONTINUE A = DLAMC3( ONE, SMALL ) CALL DLAMC4( NGPMIN, ONE, LBETA ) CALL DLAMC4( NGNMIN, -ONE, LBETA ) CALL DLAMC4( GPMIN, A, LBETA ) CALL DLAMC4( GNMIN, -A, LBETA ) IEEE = .FALSE. ! IF( ( NGPMIN.EQ.NGNMIN ) .AND. ( GPMIN.EQ.GNMIN ) ) THEN IF( NGPMIN.EQ.GPMIN ) THEN LEMIN = NGPMIN ! ( Non twos-complement machines, no gradual underflow; ! e.g., VAX ) ELSE IF( ( GPMIN-NGPMIN ).EQ.3 ) THEN LEMIN = NGPMIN - 1 + LT IEEE = .TRUE. ! ( Non twos-complement machines, with gradual underflow; ! e.g., IEEE standard followers ) ELSE LEMIN = MIN( NGPMIN, GPMIN ) ! ( A guess; no known machine ) IWARN = .TRUE. END IF ! ELSE IF( ( NGPMIN.EQ.GPMIN ) .AND. ( NGNMIN.EQ.GNMIN ) ) THEN IF( ABS( NGPMIN-NGNMIN ).EQ.1 ) THEN LEMIN = MAX( NGPMIN, NGNMIN ) ! ( Twos-complement machines, no gradual underflow; ! e.g., CYBER 205 ) ELSE LEMIN = MIN( NGPMIN, NGNMIN ) ! ( A guess; no known machine ) IWARN = .TRUE. END IF ! ELSE IF( ( ABS( NGPMIN-NGNMIN ).EQ.1 ) .AND. & ( GPMIN.EQ.GNMIN ) ) THEN IF( ( GPMIN-MIN( NGPMIN, NGNMIN ) ).EQ.3 ) THEN LEMIN = MAX( NGPMIN, NGNMIN ) - 1 + LT ! ( Twos-complement machines with gradual underflow; ! no known machine ) ELSE LEMIN = MIN( NGPMIN, NGNMIN ) ! ( A guess; no known machine ) IWARN = .TRUE. END IF ! ELSE LEMIN = MIN( NGPMIN, NGNMIN, GPMIN, GNMIN ) ! ( A guess; no known machine ) IWARN = .TRUE. END IF !** ! Comment out this if block if EMIN is ok IF( IWARN ) THEN FIRST = .TRUE. WRITE( 6, FMT = 9999 )LEMIN END IF !** ! ! Assume IEEE arithmetic if we found denormalised numbers above, ! or if arithmetic seems to round in the IEEE style, determined ! in routine DLAMC1. A true IEEE machine should have both things ! true; however, faulty machines may have one or the other. ! IEEE = IEEE .OR. LIEEE1 ! ! Compute RMIN by successive division by BETA. We could compute ! RMIN as BASE**( EMIN - 1 ), but some machines underflow during ! this computation. ! LRMIN = 1 DO 30 I = 1, 1 - LEMIN LRMIN = DLAMC3( LRMIN*RBASE, ZERO ) 30 CONTINUE ! ! Finally, call DLAMC5 to compute EMAX and RMAX. ! CALL DLAMC5( LBETA, LT, LEMIN, IEEE, LEMAX, LRMAX ) END IF ! BETA = LBETA T = LT RND = LRND EPS = LEPS EMIN = LEMIN RMIN = LRMIN EMAX = LEMAX RMAX = LRMAX ! RETURN ! 9999 FORMAT( / / ' WARNING. The value EMIN may be incorrect:-', & ' EMIN = ', I8, / & ' If, after inspection, the value EMIN looks', & ' acceptable please comment out ', & / ' the IF block as marked within the code of routine', & ' DLAMC2,', / ' otherwise supply EMIN explicitly.', / ) ! ! End of DLAMC2 ! END ! !*********************************************************************** ! DOUBLE PRECISION FUNCTION DLAMC3( A, B ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. DOUBLE PRECISION A, B ! .. ! ! Purpose ! ======= ! ! DLAMC3 is intended to force A and B to be stored prior to doing ! the addition of A and B , for use in situations where optimizers ! might hold one of these in a register. ! ! Arguments ! ========= ! ! A, B (input) DOUBLE PRECISION ! The values A and B. ! ! ===================================================================== ! ! .. Executable Statements .. ! DLAMC3 = A + B ! RETURN ! ! End of DLAMC3 ! END ! !*********************************************************************** ! SUBROUTINE DLAMC4( EMIN, START, BASE ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. INTEGER BASE, EMIN DOUBLE PRECISION START ! .. ! ! Purpose ! ======= ! ! DLAMC4 is a service routine for DLAMC2. ! ! Arguments ! ========= ! ! EMIN (output) EMIN ! The minimum exponent before (gradual) underflow, computed by ! setting A = START and dividing by BASE until the previous A ! can not be recovered. ! ! START (input) DOUBLE PRECISION ! The starting point for determining EMIN. ! ! BASE (input) INTEGER ! The base of the machine. ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER I DOUBLE PRECISION A, B1, B2, C1, C2, D1, D2, ONE, RBASE, ZERO ! .. ! .. External Functions .. DOUBLE PRECISION DLAMC3 EXTERNAL DLAMC3 ! .. ! .. Executable Statements .. ! A = START ONE = 1 RBASE = ONE / BASE ZERO = 0 EMIN = 1 B1 = DLAMC3( A*RBASE, ZERO ) C1 = A C2 = A D1 = A D2 = A !+ WHILE( ( C1.EQ.A ).AND.( C2.EQ.A ).AND. ! $ ( D1.EQ.A ).AND.( D2.EQ.A ) )LOOP 10 CONTINUE IF( ( C1.EQ.A ) .AND. ( C2.EQ.A ) .AND. ( D1.EQ.A ) .AND. & ( D2.EQ.A ) ) THEN EMIN = EMIN - 1 A = B1 B1 = DLAMC3( A / BASE, ZERO ) C1 = DLAMC3( B1*BASE, ZERO ) D1 = ZERO DO 20 I = 1, BASE D1 = D1 + B1 20 CONTINUE B2 = DLAMC3( A*RBASE, ZERO ) C2 = DLAMC3( B2 / RBASE, ZERO ) D2 = ZERO DO 30 I = 1, BASE D2 = D2 + B2 30 CONTINUE GO TO 10 END IF !+ END WHILE ! RETURN ! ! End of DLAMC4 ! END ! !*********************************************************************** ! SUBROUTINE DLAMC5( BETA, P, EMIN, IEEE, EMAX, RMAX ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. LOGICAL IEEE INTEGER BETA, EMAX, EMIN, P DOUBLE PRECISION RMAX ! .. ! ! Purpose ! ======= ! ! DLAMC5 attempts to compute RMAX, the largest machine floating-point ! number, without overflow. It assumes that EMAX + abs(EMIN) sum ! approximately to a power of 2. It will fail on machines where this ! assumption does not hold, for example, the Cyber 205 (EMIN = -28625, ! EMAX = 28718). It will also fail if the value supplied for EMIN is ! too large (i.e. too close to zero), probably with overflow. ! ! Arguments ! ========= ! ! BETA (input) INTEGER ! The base of floating-point arithmetic. ! ! P (input) INTEGER ! The number of base BETA digits in the mantissa of a ! floating-point value. ! ! EMIN (input) INTEGER ! The minimum exponent before (gradual) underflow. ! ! IEEE (input) LOGICAL ! A logical flag specifying whether or not the arithmetic ! system is thought to comply with the IEEE standard. ! ! EMAX (output) INTEGER ! The largest exponent before overflow ! ! RMAX (output) DOUBLE PRECISION ! The largest machine floating-point number. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. INTEGER EXBITS, EXPSUM, I, LEXP, NBITS, TRY, UEXP DOUBLE PRECISION OLDY, RECBAS, Y, Z ! .. ! .. External Functions .. DOUBLE PRECISION DLAMC3 EXTERNAL DLAMC3 ! .. ! .. Intrinsic Functions .. INTRINSIC MOD ! .. ! .. Executable Statements .. ! ! First compute LEXP and UEXP, two powers of 2 that bound ! abs(EMIN). We then assume that EMAX + abs(EMIN) will sum ! approximately to the bound that is closest to abs(EMIN). ! (EMAX is the exponent of the required number RMAX). ! LEXP = 1 EXBITS = 1 10 CONTINUE TRY = LEXP*2 IF( TRY.LE.( -EMIN ) ) THEN LEXP = TRY EXBITS = EXBITS + 1 GO TO 10 END IF IF( LEXP.EQ.-EMIN ) THEN UEXP = LEXP ELSE UEXP = TRY EXBITS = EXBITS + 1 END IF ! ! Now -LEXP is less than or equal to EMIN, and -UEXP is greater ! than or equal to EMIN. EXBITS is the number of bits needed to ! store the exponent. ! IF( ( UEXP+EMIN ).GT.( -LEXP-EMIN ) ) THEN EXPSUM = 2*LEXP ELSE EXPSUM = 2*UEXP END IF ! ! EXPSUM is the exponent range, approximately equal to ! EMAX - EMIN + 1 . ! EMAX = EXPSUM + EMIN - 1 NBITS = 1 + EXBITS + P ! ! NBITS is the total number of bits needed to store a ! floating-point number. ! IF( ( MOD( NBITS, 2 ).EQ.1 ) .AND. ( BETA.EQ.2 ) ) THEN ! ! Either there are an odd number of bits used to store a ! floating-point number, which is unlikely, or some bits are ! not used in the representation of numbers, which is possible, ! (e.g. Cray machines) or the mantissa has an implicit bit, ! (e.g. IEEE machines, Dec Vax machines), which is perhaps the ! most likely. We have to assume the last alternative. ! If this is true, then we need to reduce EMAX by one because ! there must be some way of representing zero in an implicit-bit ! system. On machines like Cray, we are reducing EMAX by one ! unnecessarily. ! EMAX = EMAX - 1 END IF ! IF( IEEE ) THEN ! ! Assume we are on an IEEE machine which reserves one exponent ! for infinity and NaN. ! EMAX = EMAX - 1 END IF ! ! Now create RMAX, the largest machine number, which should ! be equal to (1.0 - BETA**(-P)) * BETA**EMAX . ! ! First compute 1.0 - BETA**(-P), being careful that the ! result is less than 1.0 . ! RECBAS = ONE / BETA Z = BETA - ONE Y = ZERO DO 20 I = 1, P Z = Z*RECBAS IF( Y.LT.ONE ) & OLDY = Y Y = DLAMC3( Y, Z ) 20 CONTINUE IF( Y.GE.ONE ) & Y = OLDY ! ! Now multiply by BETA**EMAX to get RMAX. ! DO 30 I = 1, EMAX Y = DLAMC3( Y*BETA, ZERO ) 30 CONTINUE ! RMAX = Y RETURN ! ! End of DLAMC5 ! END SUBROUTINE DLAMRG( N1, N2, A, DTRD1, DTRD2, INDEX ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. INTEGER DTRD1, DTRD2, N1, N2 ! .. ! .. Array Arguments .. INTEGER INDEX( * ) DOUBLE PRECISION A( * ) ! .. ! ! Purpose ! ======= ! ! DLAMRG will create a permutation list which will merge the elements ! of A (which is composed of two independently sorted sets) into a ! single set which is sorted in ascending order. ! ! Arguments ! ========= ! ! N1 (input) INTEGER ! N2 (input) INTEGER ! These arguements contain the respective lengths of the two ! sorted lists to be merged. ! ! A (input) DOUBLE PRECISION array, dimension (N1+N2) ! The first N1 elements of A contain a list of numbers which ! are sorted in either ascending or descending order. Likewise ! for the final N2 elements. ! ! DTRD1 (input) INTEGER ! DTRD2 (input) INTEGER ! These are the strides to be taken through the array A. ! Allowable strides are 1 and -1. They indicate whether a ! subset of A is sorted in ascending (DTRDx = 1) or descending ! (DTRDx = -1) order. ! ! INDEX (output) INTEGER array, dimension (N1+N2) ! On exit this array will contain a permutation such that ! if B( I ) = A( INDEX( I ) ) for I=1,N1+N2, then B will be ! sorted in ascending order. ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER I, IND1, IND2, N1SV, N2SV ! .. ! .. Executable Statements .. ! N1SV = N1 N2SV = N2 IF( DTRD1.GT.0 ) THEN IND1 = 1 ELSE IND1 = N1 END IF IF( DTRD2.GT.0 ) THEN IND2 = 1 + N1 ELSE IND2 = N1 + N2 END IF I = 1 ! while ( (N1SV > 0) & (N2SV > 0) ) 10 CONTINUE IF( N1SV.GT.0 .AND. N2SV.GT.0 ) THEN IF( A( IND1 ).LE.A( IND2 ) ) THEN INDEX( I ) = IND1 I = I + 1 IND1 = IND1 + DTRD1 N1SV = N1SV - 1 ELSE INDEX( I ) = IND2 I = I + 1 IND2 = IND2 + DTRD2 N2SV = N2SV - 1 END IF GO TO 10 END IF ! end while IF( N1SV.EQ.0 ) THEN DO 20 N1SV = 1, N2SV INDEX( I ) = IND2 I = I + 1 IND2 = IND2 + DTRD2 20 CONTINUE ELSE ! N2SV .EQ. 0 DO 30 N2SV = 1, N1SV INDEX( I ) = IND1 I = I + 1 IND1 = IND1 + DTRD1 30 CONTINUE END IF ! RETURN ! ! End of DLAMRG ! END DOUBLE PRECISION FUNCTION DLANGB( NORM, N, KL, KU, AB, LDAB, & WORK ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER NORM INTEGER KL, KU, LDAB, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLANGB returns the value of the one norm, or the Frobenius norm, or ! the infinity norm, or the element of largest absolute value of an ! n by n band matrix A, with kl sub-diagonals and ku super-diagonals. ! ! Description ! =========== ! ! DLANGB returns the value ! ! DLANGB = ( max(abs(A(i,j))), NORM = 'M' or 'm' ! ( ! ( norm1(A), NORM = '1', 'O' or 'o' ! ( ! ( normI(A), NORM = 'I' or 'i' ! ( ! ( normF(A), NORM = 'F', 'f', 'E' or 'e' ! ! where norm1 denotes the one norm of a matrix (maximum column sum), ! normI denotes the infinity norm of a matrix (maximum row sum) and ! normF denotes the Frobenius norm of a matrix (square root of sum of ! squares). Note that max(abs(A(i,j))) is not a matrix norm. ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies the value to be returned in DLANGB as described ! above. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. When N = 0, DLANGB is ! set to zero. ! ! KL (input) INTEGER ! The number of sub-diagonals of the matrix A. KL >= 0. ! ! KU (input) INTEGER ! The number of super-diagonals of the matrix A. KU >= 0. ! ! AB (input) DOUBLE PRECISION array, dimension (LDAB,N) ! The band matrix A, stored in rows 1 to KL+KU+1. The j-th ! column of A is stored in the j-th column of the array AB as ! follows: ! AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KL+KU+1. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (LWORK), ! where LWORK >= N when NORM = 'I'; otherwise, WORK is not ! referenced. ! ! ===================================================================== ! ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J, K, L DOUBLE PRECISION SCALE, SUM, VALUE ! .. ! .. External Subroutines .. EXTERNAL DLASSQ ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! IF( N.EQ.0 ) THEN VALUE = ZERO ELSE IF( LSAME( NORM, 'M' ) ) THEN ! ! Find max(abs(A(i,j))). ! VALUE = ZERO DO 20 J = 1, N DO 10 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 ) VALUE = MAX( VALUE, ABS( AB( I, J ) ) ) 10 CONTINUE 20 CONTINUE ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN ! ! Find norm1(A). ! VALUE = ZERO DO 40 J = 1, N SUM = ZERO DO 30 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 ) SUM = SUM + ABS( AB( I, J ) ) 30 CONTINUE VALUE = MAX( VALUE, SUM ) 40 CONTINUE ELSE IF( LSAME( NORM, 'I' ) ) THEN ! ! Find normI(A). ! DO 50 I = 1, N WORK( I ) = ZERO 50 CONTINUE DO 70 J = 1, N K = KU + 1 - J DO 60 I = MAX( 1, J-KU ), MIN( N, J+KL ) WORK( I ) = WORK( I ) + ABS( AB( K+I, J ) ) 60 CONTINUE 70 CONTINUE VALUE = ZERO DO 80 I = 1, N VALUE = MAX( VALUE, WORK( I ) ) 80 CONTINUE ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN ! ! Find normF(A). ! SCALE = ZERO SUM = ONE DO 90 J = 1, N L = MAX( 1, J-KU ) K = KU + 1 - J + L CALL DLASSQ( MIN( N, J+KL )-L+1, AB( K, J ), 1, SCALE, SUM ) 90 CONTINUE VALUE = SCALE*SQRT( SUM ) END IF ! DLANGB = VALUE RETURN ! ! End of DLANGB ! END DOUBLE PRECISION FUNCTION DLANGE( NORM, M, N, A, LDA, WORK ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER NORM INTEGER LDA, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLANGE returns the value of the one norm, or the Frobenius norm, or ! the infinity norm, or the element of largest absolute value of a ! real matrix A. ! ! Description ! =========== ! ! DLANGE returns the value ! ! DLANGE = ( max(abs(A(i,j))), NORM = 'M' or 'm' ! ( ! ( norm1(A), NORM = '1', 'O' or 'o' ! ( ! ( normI(A), NORM = 'I' or 'i' ! ( ! ( normF(A), NORM = 'F', 'f', 'E' or 'e' ! ! where norm1 denotes the one norm of a matrix (maximum column sum), ! normI denotes the infinity norm of a matrix (maximum row sum) and ! normF denotes the Frobenius norm of a matrix (square root of sum of ! squares). Note that max(abs(A(i,j))) is not a matrix norm. ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies the value to be returned in DLANGE as described ! above. ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. When M = 0, ! DLANGE is set to zero. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. When N = 0, ! DLANGE is set to zero. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The m by n matrix A. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(M,1). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (LWORK), ! where LWORK >= M when NORM = 'I'; otherwise, WORK is not ! referenced. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J DOUBLE PRECISION SCALE, SUM, VALUE ! .. ! .. External Subroutines .. EXTERNAL DLASSQ ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! IF( MIN( M, N ).EQ.0 ) THEN VALUE = ZERO ELSE IF( LSAME( NORM, 'M' ) ) THEN ! ! Find max(abs(A(i,j))). ! VALUE = ZERO DO 20 J = 1, N DO 10 I = 1, M VALUE = MAX( VALUE, ABS( A( I, J ) ) ) 10 CONTINUE 20 CONTINUE ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN ! ! Find norm1(A). ! VALUE = ZERO DO 40 J = 1, N SUM = ZERO DO 30 I = 1, M SUM = SUM + ABS( A( I, J ) ) 30 CONTINUE VALUE = MAX( VALUE, SUM ) 40 CONTINUE ELSE IF( LSAME( NORM, 'I' ) ) THEN ! ! Find normI(A). ! DO 50 I = 1, M WORK( I ) = ZERO 50 CONTINUE DO 70 J = 1, N DO 60 I = 1, M WORK( I ) = WORK( I ) + ABS( A( I, J ) ) 60 CONTINUE 70 CONTINUE VALUE = ZERO DO 80 I = 1, M VALUE = MAX( VALUE, WORK( I ) ) 80 CONTINUE ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN ! ! Find normF(A). ! SCALE = ZERO SUM = ONE DO 90 J = 1, N CALL DLASSQ( M, A( 1, J ), 1, SCALE, SUM ) 90 CONTINUE VALUE = SCALE*SQRT( SUM ) END IF ! DLANGE = VALUE RETURN ! ! End of DLANGE ! END DOUBLE PRECISION FUNCTION DLANGT( NORM, N, DL, D, DU ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER NORM INTEGER N ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ), DL( * ), DU( * ) ! .. ! ! Purpose ! ======= ! ! DLANGT returns the value of the one norm, or the Frobenius norm, or ! the infinity norm, or the element of largest absolute value of a ! real tridiagonal matrix A. ! ! Description ! =========== ! ! DLANGT returns the value ! ! DLANGT = ( max(abs(A(i,j))), NORM = 'M' or 'm' ! ( ! ( norm1(A), NORM = '1', 'O' or 'o' ! ( ! ( normI(A), NORM = 'I' or 'i' ! ( ! ( normF(A), NORM = 'F', 'f', 'E' or 'e' ! ! where norm1 denotes the one norm of a matrix (maximum column sum), ! normI denotes the infinity norm of a matrix (maximum row sum) and ! normF denotes the Frobenius norm of a matrix (square root of sum of ! squares). Note that max(abs(A(i,j))) is not a matrix norm. ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies the value to be returned in DLANGT as described ! above. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. When N = 0, DLANGT is ! set to zero. ! ! DL (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) sub-diagonal elements of A. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The diagonal elements of A. ! ! DU (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) super-diagonal elements of A. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I DOUBLE PRECISION ANORM, SCALE, SUM ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DLASSQ ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! IF( N.LE.0 ) THEN ANORM = ZERO ELSE IF( LSAME( NORM, 'M' ) ) THEN ! ! Find max(abs(A(i,j))). ! ANORM = ABS( D( N ) ) DO 10 I = 1, N - 1 ANORM = MAX( ANORM, ABS( DL( I ) ) ) ANORM = MAX( ANORM, ABS( D( I ) ) ) ANORM = MAX( ANORM, ABS( DU( I ) ) ) 10 CONTINUE ELSE IF( LSAME( NORM, 'O' ) .OR. NORM.EQ.'1' ) THEN ! ! Find norm1(A). ! IF( N.EQ.1 ) THEN ANORM = ABS( D( 1 ) ) ELSE ANORM = MAX( ABS( D( 1 ) )+ABS( DL( 1 ) ), & ABS( D( N ) )+ABS( DU( N-1 ) ) ) DO 20 I = 2, N - 1 ANORM = MAX( ANORM, ABS( D( I ) )+ABS( DL( I ) )+ & ABS( DU( I-1 ) ) ) 20 CONTINUE END IF ELSE IF( LSAME( NORM, 'I' ) ) THEN ! ! Find normI(A). ! IF( N.EQ.1 ) THEN ANORM = ABS( D( 1 ) ) ELSE ANORM = MAX( ABS( D( 1 ) )+ABS( DU( 1 ) ), & ABS( D( N ) )+ABS( DL( N-1 ) ) ) DO 30 I = 2, N - 1 ANORM = MAX( ANORM, ABS( D( I ) )+ABS( DU( I ) )+ & ABS( DL( I-1 ) ) ) 30 CONTINUE END IF ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN ! ! Find normF(A). ! SCALE = ZERO SUM = ONE CALL DLASSQ( N, D, 1, SCALE, SUM ) IF( N.GT.1 ) THEN CALL DLASSQ( N-1, DL, 1, SCALE, SUM ) CALL DLASSQ( N-1, DU, 1, SCALE, SUM ) END IF ANORM = SCALE*SQRT( SUM ) END IF ! DLANGT = ANORM RETURN ! ! End of DLANGT ! END DOUBLE PRECISION FUNCTION DLANHS( NORM, N, A, LDA, WORK ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER NORM INTEGER LDA, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLANHS returns the value of the one norm, or the Frobenius norm, or ! the infinity norm, or the element of largest absolute value of a ! Hessenberg matrix A. ! ! Description ! =========== ! ! DLANHS returns the value ! ! DLANHS = ( max(abs(A(i,j))), NORM = 'M' or 'm' ! ( ! ( norm1(A), NORM = '1', 'O' or 'o' ! ( ! ( normI(A), NORM = 'I' or 'i' ! ( ! ( normF(A), NORM = 'F', 'f', 'E' or 'e' ! ! where norm1 denotes the one norm of a matrix (maximum column sum), ! normI denotes the infinity norm of a matrix (maximum row sum) and ! normF denotes the Frobenius norm of a matrix (square root of sum of ! squares). Note that max(abs(A(i,j))) is not a matrix norm. ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies the value to be returned in DLANHS as described ! above. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. When N = 0, DLANHS is ! set to zero. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The n by n upper Hessenberg matrix A; the part of A below the ! first sub-diagonal is not referenced. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(N,1). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (LWORK), ! where LWORK >= N when NORM = 'I'; otherwise, WORK is not ! referenced. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J DOUBLE PRECISION SCALE, SUM, VALUE ! .. ! .. External Subroutines .. EXTERNAL DLASSQ ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! IF( N.EQ.0 ) THEN VALUE = ZERO ELSE IF( LSAME( NORM, 'M' ) ) THEN ! ! Find max(abs(A(i,j))). ! VALUE = ZERO DO 20 J = 1, N DO 10 I = 1, MIN( N, J+1 ) VALUE = MAX( VALUE, ABS( A( I, J ) ) ) 10 CONTINUE 20 CONTINUE ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN ! ! Find norm1(A). ! VALUE = ZERO DO 40 J = 1, N SUM = ZERO DO 30 I = 1, MIN( N, J+1 ) SUM = SUM + ABS( A( I, J ) ) 30 CONTINUE VALUE = MAX( VALUE, SUM ) 40 CONTINUE ELSE IF( LSAME( NORM, 'I' ) ) THEN ! ! Find normI(A). ! DO 50 I = 1, N WORK( I ) = ZERO 50 CONTINUE DO 70 J = 1, N DO 60 I = 1, MIN( N, J+1 ) WORK( I ) = WORK( I ) + ABS( A( I, J ) ) 60 CONTINUE 70 CONTINUE VALUE = ZERO DO 80 I = 1, N VALUE = MAX( VALUE, WORK( I ) ) 80 CONTINUE ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN ! ! Find normF(A). ! SCALE = ZERO SUM = ONE DO 90 J = 1, N CALL DLASSQ( MIN( N, J+1 ), A( 1, J ), 1, SCALE, SUM ) 90 CONTINUE VALUE = SCALE*SQRT( SUM ) END IF ! DLANHS = VALUE RETURN ! ! End of DLANHS ! END DOUBLE PRECISION FUNCTION DLANSB( NORM, UPLO, N, K, AB, LDAB, & WORK ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER NORM, UPLO INTEGER K, LDAB, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLANSB returns the value of the one norm, or the Frobenius norm, or ! the infinity norm, or the element of largest absolute value of an ! n by n symmetric band matrix A, with k super-diagonals. ! ! Description ! =========== ! ! DLANSB returns the value ! ! DLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm' ! ( ! ( norm1(A), NORM = '1', 'O' or 'o' ! ( ! ( normI(A), NORM = 'I' or 'i' ! ( ! ( normF(A), NORM = 'F', 'f', 'E' or 'e' ! ! where norm1 denotes the one norm of a matrix (maximum column sum), ! normI denotes the infinity norm of a matrix (maximum row sum) and ! normF denotes the Frobenius norm of a matrix (square root of sum of ! squares). Note that max(abs(A(i,j))) is not a matrix norm. ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies the value to be returned in DLANSB as described ! above. ! ! UPLO (input) CHARACTER*1 ! Specifies whether the upper or lower triangular part of the ! band matrix A is supplied. ! = 'U': Upper triangular part is supplied ! = 'L': Lower triangular part is supplied ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. When N = 0, DLANSB is ! set to zero. ! ! K (input) INTEGER ! The number of super-diagonals or sub-diagonals of the ! band matrix A. K >= 0. ! ! AB (input) DOUBLE PRECISION array, dimension (LDAB,N) ! The upper or lower triangle of the symmetric band matrix A, ! stored in the first K+1 rows of AB. The j-th column of A is ! stored in the j-th column of the array AB as follows: ! if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k). ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= K+1. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (LWORK), ! where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, ! WORK is not referenced. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J, L DOUBLE PRECISION ABSA, SCALE, SUM, VALUE ! .. ! .. External Subroutines .. EXTERNAL DLASSQ ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! IF( N.EQ.0 ) THEN VALUE = ZERO ELSE IF( LSAME( NORM, 'M' ) ) THEN ! ! Find max(abs(A(i,j))). ! VALUE = ZERO IF( LSAME( UPLO, 'U' ) ) THEN DO 20 J = 1, N DO 10 I = MAX( K+2-J, 1 ), K + 1 VALUE = MAX( VALUE, ABS( AB( I, J ) ) ) 10 CONTINUE 20 CONTINUE ELSE DO 40 J = 1, N DO 30 I = 1, MIN( N+1-J, K+1 ) VALUE = MAX( VALUE, ABS( AB( I, J ) ) ) 30 CONTINUE 40 CONTINUE END IF ELSE IF( ( LSAME( NORM, 'I' ) ) .OR. ( LSAME( NORM, 'O' ) ) .OR. & ( NORM.EQ.'1' ) ) THEN ! ! Find normI(A) ( = norm1(A), since A is symmetric). ! VALUE = ZERO IF( LSAME( UPLO, 'U' ) ) THEN DO 60 J = 1, N SUM = ZERO L = K + 1 - J DO 50 I = MAX( 1, J-K ), J - 1 ABSA = ABS( AB( L+I, J ) ) SUM = SUM + ABSA WORK( I ) = WORK( I ) + ABSA 50 CONTINUE WORK( J ) = SUM + ABS( AB( K+1, J ) ) 60 CONTINUE DO 70 I = 1, N VALUE = MAX( VALUE, WORK( I ) ) 70 CONTINUE ELSE DO 80 I = 1, N WORK( I ) = ZERO 80 CONTINUE DO 100 J = 1, N SUM = WORK( J ) + ABS( AB( 1, J ) ) L = 1 - J DO 90 I = J + 1, MIN( N, J+K ) ABSA = ABS( AB( L+I, J ) ) SUM = SUM + ABSA WORK( I ) = WORK( I ) + ABSA 90 CONTINUE VALUE = MAX( VALUE, SUM ) 100 CONTINUE END IF ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN ! ! Find normF(A). ! SCALE = ZERO SUM = ONE IF( K.GT.0 ) THEN IF( LSAME( UPLO, 'U' ) ) THEN DO 110 J = 2, N CALL DLASSQ( MIN( J-1, K ), AB( MAX( K+2-J, 1 ), J ), & 1, SCALE, SUM ) 110 CONTINUE L = K + 1 ELSE DO 120 J = 1, N - 1 CALL DLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE, & SUM ) 120 CONTINUE L = 1 END IF SUM = 2*SUM ELSE L = 1 END IF CALL DLASSQ( N, AB( L, 1 ), LDAB, SCALE, SUM ) VALUE = SCALE*SQRT( SUM ) END IF ! DLANSB = VALUE RETURN ! ! End of DLANSB ! END DOUBLE PRECISION FUNCTION DLANSP( NORM, UPLO, N, AP, WORK ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER NORM, UPLO INTEGER N ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLANSP returns the value of the one norm, or the Frobenius norm, or ! the infinity norm, or the element of largest absolute value of a ! real symmetric matrix A, supplied in packed form. ! ! Description ! =========== ! ! DLANSP returns the value ! ! DLANSP = ( max(abs(A(i,j))), NORM = 'M' or 'm' ! ( ! ( norm1(A), NORM = '1', 'O' or 'o' ! ( ! ( normI(A), NORM = 'I' or 'i' ! ( ! ( normF(A), NORM = 'F', 'f', 'E' or 'e' ! ! where norm1 denotes the one norm of a matrix (maximum column sum), ! normI denotes the infinity norm of a matrix (maximum row sum) and ! normF denotes the Frobenius norm of a matrix (square root of sum of ! squares). Note that max(abs(A(i,j))) is not a matrix norm. ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies the value to be returned in DLANSP as described ! above. ! ! UPLO (input) CHARACTER*1 ! Specifies whether the upper or lower triangular part of the ! symmetric matrix A is supplied. ! = 'U': Upper triangular part of A is supplied ! = 'L': Lower triangular part of A is supplied ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. When N = 0, DLANSP is ! set to zero. ! ! AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The upper or lower triangle of the symmetric matrix A, packed ! columnwise in a linear array. The j-th column of A is stored ! in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (LWORK), ! where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, ! WORK is not referenced. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J, K DOUBLE PRECISION ABSA, SCALE, SUM, VALUE ! .. ! .. External Subroutines .. EXTERNAL DLASSQ ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! IF( N.EQ.0 ) THEN VALUE = ZERO ELSE IF( LSAME( NORM, 'M' ) ) THEN ! ! Find max(abs(A(i,j))). ! VALUE = ZERO IF( LSAME( UPLO, 'U' ) ) THEN K = 1 DO 20 J = 1, N DO 10 I = K, K + J - 1 VALUE = MAX( VALUE, ABS( AP( I ) ) ) 10 CONTINUE K = K + J 20 CONTINUE ELSE K = 1 DO 40 J = 1, N DO 30 I = K, K + N - J VALUE = MAX( VALUE, ABS( AP( I ) ) ) 30 CONTINUE K = K + N - J + 1 40 CONTINUE END IF ELSE IF( ( LSAME( NORM, 'I' ) ) .OR. ( LSAME( NORM, 'O' ) ) .OR. & ( NORM.EQ.'1' ) ) THEN ! ! Find normI(A) ( = norm1(A), since A is symmetric). ! VALUE = ZERO K = 1 IF( LSAME( UPLO, 'U' ) ) THEN DO 60 J = 1, N SUM = ZERO DO 50 I = 1, J - 1 ABSA = ABS( AP( K ) ) SUM = SUM + ABSA WORK( I ) = WORK( I ) + ABSA K = K + 1 50 CONTINUE WORK( J ) = SUM + ABS( AP( K ) ) K = K + 1 60 CONTINUE DO 70 I = 1, N VALUE = MAX( VALUE, WORK( I ) ) 70 CONTINUE ELSE DO 80 I = 1, N WORK( I ) = ZERO 80 CONTINUE DO 100 J = 1, N SUM = WORK( J ) + ABS( AP( K ) ) K = K + 1 DO 90 I = J + 1, N ABSA = ABS( AP( K ) ) SUM = SUM + ABSA WORK( I ) = WORK( I ) + ABSA K = K + 1 90 CONTINUE VALUE = MAX( VALUE, SUM ) 100 CONTINUE END IF ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN ! ! Find normF(A). ! SCALE = ZERO SUM = ONE K = 2 IF( LSAME( UPLO, 'U' ) ) THEN DO 110 J = 2, N CALL DLASSQ( J-1, AP( K ), 1, SCALE, SUM ) K = K + J 110 CONTINUE ELSE DO 120 J = 1, N - 1 CALL DLASSQ( N-J, AP( K ), 1, SCALE, SUM ) K = K + N - J + 1 120 CONTINUE END IF SUM = 2*SUM K = 1 DO 130 I = 1, N IF( AP( K ).NE.ZERO ) THEN ABSA = ABS( AP( K ) ) IF( SCALE.LT.ABSA ) THEN SUM = ONE + SUM*( SCALE / ABSA )**2 SCALE = ABSA ELSE SUM = SUM + ( ABSA / SCALE )**2 END IF END IF IF( LSAME( UPLO, 'U' ) ) THEN K = K + I + 1 ELSE K = K + N - I + 1 END IF 130 CONTINUE VALUE = SCALE*SQRT( SUM ) END IF ! DLANSP = VALUE RETURN ! ! End of DLANSP ! END DOUBLE PRECISION FUNCTION DLANST( NORM, N, D, E ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER NORM INTEGER N ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ), E( * ) ! .. ! ! Purpose ! ======= ! ! DLANST returns the value of the one norm, or the Frobenius norm, or ! the infinity norm, or the element of largest absolute value of a ! real symmetric tridiagonal matrix A. ! ! Description ! =========== ! ! DLANST returns the value ! ! DLANST = ( max(abs(A(i,j))), NORM = 'M' or 'm' ! ( ! ( norm1(A), NORM = '1', 'O' or 'o' ! ( ! ( normI(A), NORM = 'I' or 'i' ! ( ! ( normF(A), NORM = 'F', 'f', 'E' or 'e' ! ! where norm1 denotes the one norm of a matrix (maximum column sum), ! normI denotes the infinity norm of a matrix (maximum row sum) and ! normF denotes the Frobenius norm of a matrix (square root of sum of ! squares). Note that max(abs(A(i,j))) is not a matrix norm. ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies the value to be returned in DLANST as described ! above. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. When N = 0, DLANST is ! set to zero. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The diagonal elements of A. ! ! E (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) sub-diagonal or super-diagonal elements of A. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I DOUBLE PRECISION ANORM, SCALE, SUM ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DLASSQ ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! IF( N.LE.0 ) THEN ANORM = ZERO ELSE IF( LSAME( NORM, 'M' ) ) THEN ! ! Find max(abs(A(i,j))). ! ANORM = ABS( D( N ) ) DO 10 I = 1, N - 1 ANORM = MAX( ANORM, ABS( D( I ) ) ) ANORM = MAX( ANORM, ABS( E( I ) ) ) 10 CONTINUE ELSE IF( LSAME( NORM, 'O' ) .OR. NORM.EQ.'1' .OR. & LSAME( NORM, 'I' ) ) THEN ! ! Find norm1(A). ! IF( N.EQ.1 ) THEN ANORM = ABS( D( 1 ) ) ELSE ANORM = MAX( ABS( D( 1 ) )+ABS( E( 1 ) ), & ABS( E( N-1 ) )+ABS( D( N ) ) ) DO 20 I = 2, N - 1 ANORM = MAX( ANORM, ABS( D( I ) )+ABS( E( I ) )+ & ABS( E( I-1 ) ) ) 20 CONTINUE END IF ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN ! ! Find normF(A). ! SCALE = ZERO SUM = ONE IF( N.GT.1 ) THEN CALL DLASSQ( N-1, E, 1, SCALE, SUM ) SUM = 2*SUM END IF CALL DLASSQ( N, D, 1, SCALE, SUM ) ANORM = SCALE*SQRT( SUM ) END IF ! DLANST = ANORM RETURN ! ! End of DLANST ! END DOUBLE PRECISION FUNCTION DLANSY( NORM, UPLO, N, A, LDA, WORK ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER NORM, UPLO INTEGER LDA, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLANSY returns the value of the one norm, or the Frobenius norm, or ! the infinity norm, or the element of largest absolute value of a ! real symmetric matrix A. ! ! Description ! =========== ! ! DLANSY returns the value ! ! DLANSY = ( max(abs(A(i,j))), NORM = 'M' or 'm' ! ( ! ( norm1(A), NORM = '1', 'O' or 'o' ! ( ! ( normI(A), NORM = 'I' or 'i' ! ( ! ( normF(A), NORM = 'F', 'f', 'E' or 'e' ! ! where norm1 denotes the one norm of a matrix (maximum column sum), ! normI denotes the infinity norm of a matrix (maximum row sum) and ! normF denotes the Frobenius norm of a matrix (square root of sum of ! squares). Note that max(abs(A(i,j))) is not a matrix norm. ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies the value to be returned in DLANSY as described ! above. ! ! UPLO (input) CHARACTER*1 ! Specifies whether the upper or lower triangular part of the ! symmetric matrix A is to be referenced. ! = 'U': Upper triangular part of A is referenced ! = 'L': Lower triangular part of A is referenced ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. When N = 0, DLANSY is ! set to zero. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The symmetric matrix A. If UPLO = 'U', the leading n by n ! upper triangular part of A contains the upper triangular part ! of the matrix A, and the strictly lower triangular part of A ! is not referenced. If UPLO = 'L', the leading n by n lower ! triangular part of A contains the lower triangular part of ! the matrix A, and the strictly upper triangular part of A is ! not referenced. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(N,1). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (LWORK), ! where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, ! WORK is not referenced. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J DOUBLE PRECISION ABSA, SCALE, SUM, VALUE ! .. ! .. External Subroutines .. EXTERNAL DLASSQ ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! IF( N.EQ.0 ) THEN VALUE = ZERO ELSE IF( LSAME( NORM, 'M' ) ) THEN ! ! Find max(abs(A(i,j))). ! VALUE = ZERO IF( LSAME( UPLO, 'U' ) ) THEN DO 20 J = 1, N DO 10 I = 1, J VALUE = MAX( VALUE, ABS( A( I, J ) ) ) 10 CONTINUE 20 CONTINUE ELSE DO 40 J = 1, N DO 30 I = J, N VALUE = MAX( VALUE, ABS( A( I, J ) ) ) 30 CONTINUE 40 CONTINUE END IF ELSE IF( ( LSAME( NORM, 'I' ) ) .OR. ( LSAME( NORM, 'O' ) ) .OR. & ( NORM.EQ.'1' ) ) THEN ! ! Find normI(A) ( = norm1(A), since A is symmetric). ! VALUE = ZERO IF( LSAME( UPLO, 'U' ) ) THEN DO 60 J = 1, N SUM = ZERO DO 50 I = 1, J - 1 ABSA = ABS( A( I, J ) ) SUM = SUM + ABSA WORK( I ) = WORK( I ) + ABSA 50 CONTINUE WORK( J ) = SUM + ABS( A( J, J ) ) 60 CONTINUE DO 70 I = 1, N VALUE = MAX( VALUE, WORK( I ) ) 70 CONTINUE ELSE DO 80 I = 1, N WORK( I ) = ZERO 80 CONTINUE DO 100 J = 1, N SUM = WORK( J ) + ABS( A( J, J ) ) DO 90 I = J + 1, N ABSA = ABS( A( I, J ) ) SUM = SUM + ABSA WORK( I ) = WORK( I ) + ABSA 90 CONTINUE VALUE = MAX( VALUE, SUM ) 100 CONTINUE END IF ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN ! ! Find normF(A). ! SCALE = ZERO SUM = ONE IF( LSAME( UPLO, 'U' ) ) THEN DO 110 J = 2, N CALL DLASSQ( J-1, A( 1, J ), 1, SCALE, SUM ) 110 CONTINUE ELSE DO 120 J = 1, N - 1 CALL DLASSQ( N-J, A( J+1, J ), 1, SCALE, SUM ) 120 CONTINUE END IF SUM = 2*SUM CALL DLASSQ( N, A, LDA+1, SCALE, SUM ) VALUE = SCALE*SQRT( SUM ) END IF ! DLANSY = VALUE RETURN ! ! End of DLANSY ! END DOUBLE PRECISION FUNCTION DLANTB( NORM, UPLO, DIAG, N, K, AB, & LDAB, WORK ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER DIAG, NORM, UPLO INTEGER K, LDAB, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLANTB returns the value of the one norm, or the Frobenius norm, or ! the infinity norm, or the element of largest absolute value of an ! n by n triangular band matrix A, with ( k + 1 ) diagonals. ! ! Description ! =========== ! ! DLANTB returns the value ! ! DLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm' ! ( ! ( norm1(A), NORM = '1', 'O' or 'o' ! ( ! ( normI(A), NORM = 'I' or 'i' ! ( ! ( normF(A), NORM = 'F', 'f', 'E' or 'e' ! ! where norm1 denotes the one norm of a matrix (maximum column sum), ! normI denotes the infinity norm of a matrix (maximum row sum) and ! normF denotes the Frobenius norm of a matrix (square root of sum of ! squares). Note that max(abs(A(i,j))) is not a matrix norm. ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies the value to be returned in DLANTB as described ! above. ! ! UPLO (input) CHARACTER*1 ! Specifies whether the matrix A is upper or lower triangular. ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! DIAG (input) CHARACTER*1 ! Specifies whether or not the matrix A is unit triangular. ! = 'N': Non-unit triangular ! = 'U': Unit triangular ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. When N = 0, DLANTB is ! set to zero. ! ! K (input) INTEGER ! The number of super-diagonals of the matrix A if UPLO = 'U', ! or the number of sub-diagonals of the matrix A if UPLO = 'L'. ! K >= 0. ! ! AB (input) DOUBLE PRECISION array, dimension (LDAB,N) ! The upper or lower triangular band matrix A, stored in the ! first k+1 rows of AB. The j-th column of A is stored ! in the j-th column of the array AB as follows: ! if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k). ! Note that when DIAG = 'U', the elements of the array AB ! corresponding to the diagonal elements of the matrix A are ! not referenced, but are assumed to be one. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= K+1. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (LWORK), ! where LWORK >= N when NORM = 'I'; otherwise, WORK is not ! referenced. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UDIAG INTEGER I, J, L DOUBLE PRECISION SCALE, SUM, VALUE ! .. ! .. External Subroutines .. EXTERNAL DLASSQ ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! IF( N.EQ.0 ) THEN VALUE = ZERO ELSE IF( LSAME( NORM, 'M' ) ) THEN ! ! Find max(abs(A(i,j))). ! IF( LSAME( DIAG, 'U' ) ) THEN VALUE = ONE IF( LSAME( UPLO, 'U' ) ) THEN DO 20 J = 1, N DO 10 I = MAX( K+2-J, 1 ), K VALUE = MAX( VALUE, ABS( AB( I, J ) ) ) 10 CONTINUE 20 CONTINUE ELSE DO 40 J = 1, N DO 30 I = 2, MIN( N+1-J, K+1 ) VALUE = MAX( VALUE, ABS( AB( I, J ) ) ) 30 CONTINUE 40 CONTINUE END IF ELSE VALUE = ZERO IF( LSAME( UPLO, 'U' ) ) THEN DO 60 J = 1, N DO 50 I = MAX( K+2-J, 1 ), K + 1 VALUE = MAX( VALUE, ABS( AB( I, J ) ) ) 50 CONTINUE 60 CONTINUE ELSE DO 80 J = 1, N DO 70 I = 1, MIN( N+1-J, K+1 ) VALUE = MAX( VALUE, ABS( AB( I, J ) ) ) 70 CONTINUE 80 CONTINUE END IF END IF ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN ! ! Find norm1(A). ! VALUE = ZERO UDIAG = LSAME( DIAG, 'U' ) IF( LSAME( UPLO, 'U' ) ) THEN DO 110 J = 1, N IF( UDIAG ) THEN SUM = ONE DO 90 I = MAX( K+2-J, 1 ), K SUM = SUM + ABS( AB( I, J ) ) 90 CONTINUE ELSE SUM = ZERO DO 100 I = MAX( K+2-J, 1 ), K + 1 SUM = SUM + ABS( AB( I, J ) ) 100 CONTINUE END IF VALUE = MAX( VALUE, SUM ) 110 CONTINUE ELSE DO 140 J = 1, N IF( UDIAG ) THEN SUM = ONE DO 120 I = 2, MIN( N+1-J, K+1 ) SUM = SUM + ABS( AB( I, J ) ) 120 CONTINUE ELSE SUM = ZERO DO 130 I = 1, MIN( N+1-J, K+1 ) SUM = SUM + ABS( AB( I, J ) ) 130 CONTINUE END IF VALUE = MAX( VALUE, SUM ) 140 CONTINUE END IF ELSE IF( LSAME( NORM, 'I' ) ) THEN ! ! Find normI(A). ! VALUE = ZERO IF( LSAME( UPLO, 'U' ) ) THEN IF( LSAME( DIAG, 'U' ) ) THEN DO 150 I = 1, N WORK( I ) = ONE 150 CONTINUE DO 170 J = 1, N L = K + 1 - J DO 160 I = MAX( 1, J-K ), J - 1 WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) ) 160 CONTINUE 170 CONTINUE ELSE DO 180 I = 1, N WORK( I ) = ZERO 180 CONTINUE DO 200 J = 1, N L = K + 1 - J DO 190 I = MAX( 1, J-K ), J WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) ) 190 CONTINUE 200 CONTINUE END IF ELSE IF( LSAME( DIAG, 'U' ) ) THEN DO 210 I = 1, N WORK( I ) = ONE 210 CONTINUE DO 230 J = 1, N L = 1 - J DO 220 I = J + 1, MIN( N, J+K ) WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) ) 220 CONTINUE 230 CONTINUE ELSE DO 240 I = 1, N WORK( I ) = ZERO 240 CONTINUE DO 260 J = 1, N L = 1 - J DO 250 I = J, MIN( N, J+K ) WORK( I ) = WORK( I ) + ABS( AB( L+I, J ) ) 250 CONTINUE 260 CONTINUE END IF END IF DO 270 I = 1, N VALUE = MAX( VALUE, WORK( I ) ) 270 CONTINUE ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN ! ! Find normF(A). ! IF( LSAME( UPLO, 'U' ) ) THEN IF( LSAME( DIAG, 'U' ) ) THEN SCALE = ONE SUM = N IF( K.GT.0 ) THEN DO 280 J = 2, N CALL DLASSQ( MIN( J-1, K ), & AB( MAX( K+2-J, 1 ), J ), 1, SCALE, & SUM ) 280 CONTINUE END IF ELSE SCALE = ZERO SUM = ONE DO 290 J = 1, N CALL DLASSQ( MIN( J, K+1 ), AB( MAX( K+2-J, 1 ), J ), & 1, SCALE, SUM ) 290 CONTINUE END IF ELSE IF( LSAME( DIAG, 'U' ) ) THEN SCALE = ONE SUM = N IF( K.GT.0 ) THEN DO 300 J = 1, N - 1 CALL DLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE, & SUM ) 300 CONTINUE END IF ELSE SCALE = ZERO SUM = ONE DO 310 J = 1, N CALL DLASSQ( MIN( N-J+1, K+1 ), AB( 1, J ), 1, SCALE, & SUM ) 310 CONTINUE END IF END IF VALUE = SCALE*SQRT( SUM ) END IF ! DLANTB = VALUE RETURN ! ! End of DLANTB ! END DOUBLE PRECISION FUNCTION DLANTP( NORM, UPLO, DIAG, N, AP, WORK ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER DIAG, NORM, UPLO INTEGER N ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLANTP returns the value of the one norm, or the Frobenius norm, or ! the infinity norm, or the element of largest absolute value of a ! triangular matrix A, supplied in packed form. ! ! Description ! =========== ! ! DLANTP returns the value ! ! DLANTP = ( max(abs(A(i,j))), NORM = 'M' or 'm' ! ( ! ( norm1(A), NORM = '1', 'O' or 'o' ! ( ! ( normI(A), NORM = 'I' or 'i' ! ( ! ( normF(A), NORM = 'F', 'f', 'E' or 'e' ! ! where norm1 denotes the one norm of a matrix (maximum column sum), ! normI denotes the infinity norm of a matrix (maximum row sum) and ! normF denotes the Frobenius norm of a matrix (square root of sum of ! squares). Note that max(abs(A(i,j))) is not a matrix norm. ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies the value to be returned in DLANTP as described ! above. ! ! UPLO (input) CHARACTER*1 ! Specifies whether the matrix A is upper or lower triangular. ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! DIAG (input) CHARACTER*1 ! Specifies whether or not the matrix A is unit triangular. ! = 'N': Non-unit triangular ! = 'U': Unit triangular ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. When N = 0, DLANTP is ! set to zero. ! ! AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The upper or lower triangular matrix A, packed columnwise in ! a linear array. The j-th column of A is stored in the array ! AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. ! Note that when DIAG = 'U', the elements of the array AP ! corresponding to the diagonal elements of the matrix A are ! not referenced, but are assumed to be one. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (LWORK), ! where LWORK >= N when NORM = 'I'; otherwise, WORK is not ! referenced. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UDIAG INTEGER I, J, K DOUBLE PRECISION SCALE, SUM, VALUE ! .. ! .. External Subroutines .. EXTERNAL DLASSQ ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! IF( N.EQ.0 ) THEN VALUE = ZERO ELSE IF( LSAME( NORM, 'M' ) ) THEN ! ! Find max(abs(A(i,j))). ! K = 1 IF( LSAME( DIAG, 'U' ) ) THEN VALUE = ONE IF( LSAME( UPLO, 'U' ) ) THEN DO 20 J = 1, N DO 10 I = K, K + J - 2 VALUE = MAX( VALUE, ABS( AP( I ) ) ) 10 CONTINUE K = K + J 20 CONTINUE ELSE DO 40 J = 1, N DO 30 I = K + 1, K + N - J VALUE = MAX( VALUE, ABS( AP( I ) ) ) 30 CONTINUE K = K + N - J + 1 40 CONTINUE END IF ELSE VALUE = ZERO IF( LSAME( UPLO, 'U' ) ) THEN DO 60 J = 1, N DO 50 I = K, K + J - 1 VALUE = MAX( VALUE, ABS( AP( I ) ) ) 50 CONTINUE K = K + J 60 CONTINUE ELSE DO 80 J = 1, N DO 70 I = K, K + N - J VALUE = MAX( VALUE, ABS( AP( I ) ) ) 70 CONTINUE K = K + N - J + 1 80 CONTINUE END IF END IF ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN ! ! Find norm1(A). ! VALUE = ZERO K = 1 UDIAG = LSAME( DIAG, 'U' ) IF( LSAME( UPLO, 'U' ) ) THEN DO 110 J = 1, N IF( UDIAG ) THEN SUM = ONE DO 90 I = K, K + J - 2 SUM = SUM + ABS( AP( I ) ) 90 CONTINUE ELSE SUM = ZERO DO 100 I = K, K + J - 1 SUM = SUM + ABS( AP( I ) ) 100 CONTINUE END IF K = K + J VALUE = MAX( VALUE, SUM ) 110 CONTINUE ELSE DO 140 J = 1, N IF( UDIAG ) THEN SUM = ONE DO 120 I = K + 1, K + N - J SUM = SUM + ABS( AP( I ) ) 120 CONTINUE ELSE SUM = ZERO DO 130 I = K, K + N - J SUM = SUM + ABS( AP( I ) ) 130 CONTINUE END IF K = K + N - J + 1 VALUE = MAX( VALUE, SUM ) 140 CONTINUE END IF ELSE IF( LSAME( NORM, 'I' ) ) THEN ! ! Find normI(A). ! K = 1 IF( LSAME( UPLO, 'U' ) ) THEN IF( LSAME( DIAG, 'U' ) ) THEN DO 150 I = 1, N WORK( I ) = ONE 150 CONTINUE DO 170 J = 1, N DO 160 I = 1, J - 1 WORK( I ) = WORK( I ) + ABS( AP( K ) ) K = K + 1 160 CONTINUE K = K + 1 170 CONTINUE ELSE DO 180 I = 1, N WORK( I ) = ZERO 180 CONTINUE DO 200 J = 1, N DO 190 I = 1, J WORK( I ) = WORK( I ) + ABS( AP( K ) ) K = K + 1 190 CONTINUE 200 CONTINUE END IF ELSE IF( LSAME( DIAG, 'U' ) ) THEN DO 210 I = 1, N WORK( I ) = ONE 210 CONTINUE DO 230 J = 1, N K = K + 1 DO 220 I = J + 1, N WORK( I ) = WORK( I ) + ABS( AP( K ) ) K = K + 1 220 CONTINUE 230 CONTINUE ELSE DO 240 I = 1, N WORK( I ) = ZERO 240 CONTINUE DO 260 J = 1, N DO 250 I = J, N WORK( I ) = WORK( I ) + ABS( AP( K ) ) K = K + 1 250 CONTINUE 260 CONTINUE END IF END IF VALUE = ZERO DO 270 I = 1, N VALUE = MAX( VALUE, WORK( I ) ) 270 CONTINUE ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN ! ! Find normF(A). ! IF( LSAME( UPLO, 'U' ) ) THEN IF( LSAME( DIAG, 'U' ) ) THEN SCALE = ONE SUM = N K = 2 DO 280 J = 2, N CALL DLASSQ( J-1, AP( K ), 1, SCALE, SUM ) K = K + J 280 CONTINUE ELSE SCALE = ZERO SUM = ONE K = 1 DO 290 J = 1, N CALL DLASSQ( J, AP( K ), 1, SCALE, SUM ) K = K + J 290 CONTINUE END IF ELSE IF( LSAME( DIAG, 'U' ) ) THEN SCALE = ONE SUM = N K = 2 DO 300 J = 1, N - 1 CALL DLASSQ( N-J, AP( K ), 1, SCALE, SUM ) K = K + N - J + 1 300 CONTINUE ELSE SCALE = ZERO SUM = ONE K = 1 DO 310 J = 1, N CALL DLASSQ( N-J+1, AP( K ), 1, SCALE, SUM ) K = K + N - J + 1 310 CONTINUE END IF END IF VALUE = SCALE*SQRT( SUM ) END IF ! DLANTP = VALUE RETURN ! ! End of DLANTP ! END DOUBLE PRECISION FUNCTION DLANTR( NORM, UPLO, DIAG, M, N, A, LDA, & WORK ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER DIAG, NORM, UPLO INTEGER LDA, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLANTR returns the value of the one norm, or the Frobenius norm, or ! the infinity norm, or the element of largest absolute value of a ! trapezoidal or triangular matrix A. ! ! Description ! =========== ! ! DLANTR returns the value ! ! DLANTR = ( max(abs(A(i,j))), NORM = 'M' or 'm' ! ( ! ( norm1(A), NORM = '1', 'O' or 'o' ! ( ! ( normI(A), NORM = 'I' or 'i' ! ( ! ( normF(A), NORM = 'F', 'f', 'E' or 'e' ! ! where norm1 denotes the one norm of a matrix (maximum column sum), ! normI denotes the infinity norm of a matrix (maximum row sum) and ! normF denotes the Frobenius norm of a matrix (square root of sum of ! squares). Note that max(abs(A(i,j))) is not a matrix norm. ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies the value to be returned in DLANTR as described ! above. ! ! UPLO (input) CHARACTER*1 ! Specifies whether the matrix A is upper or lower trapezoidal. ! = 'U': Upper trapezoidal ! = 'L': Lower trapezoidal ! Note that A is triangular instead of trapezoidal if M = N. ! ! DIAG (input) CHARACTER*1 ! Specifies whether or not the matrix A has unit diagonal. ! = 'N': Non-unit diagonal ! = 'U': Unit diagonal ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0, and if ! UPLO = 'U', M <= N. When M = 0, DLANTR is set to zero. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0, and if ! UPLO = 'L', N <= M. When N = 0, DLANTR is set to zero. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The trapezoidal matrix A (A is triangular if M = N). ! If UPLO = 'U', the leading m by n upper trapezoidal part of ! the array A contains the upper trapezoidal matrix, and the ! strictly lower triangular part of A is not referenced. ! If UPLO = 'L', the leading m by n lower trapezoidal part of ! the array A contains the lower trapezoidal matrix, and the ! strictly upper triangular part of A is not referenced. Note ! that when DIAG = 'U', the diagonal elements of A are not ! referenced and are assumed to be one. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(M,1). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (LWORK), ! where LWORK >= M when NORM = 'I'; otherwise, WORK is not ! referenced. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UDIAG INTEGER I, J DOUBLE PRECISION SCALE, SUM, VALUE ! .. ! .. External Subroutines .. EXTERNAL DLASSQ ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! IF( MIN( M, N ).EQ.0 ) THEN VALUE = ZERO ELSE IF( LSAME( NORM, 'M' ) ) THEN ! ! Find max(abs(A(i,j))). ! IF( LSAME( DIAG, 'U' ) ) THEN VALUE = ONE IF( LSAME( UPLO, 'U' ) ) THEN DO 20 J = 1, N DO 10 I = 1, MIN( M, J-1 ) VALUE = MAX( VALUE, ABS( A( I, J ) ) ) 10 CONTINUE 20 CONTINUE ELSE DO 40 J = 1, N DO 30 I = J + 1, M VALUE = MAX( VALUE, ABS( A( I, J ) ) ) 30 CONTINUE 40 CONTINUE END IF ELSE VALUE = ZERO IF( LSAME( UPLO, 'U' ) ) THEN DO 60 J = 1, N DO 50 I = 1, MIN( M, J ) VALUE = MAX( VALUE, ABS( A( I, J ) ) ) 50 CONTINUE 60 CONTINUE ELSE DO 80 J = 1, N DO 70 I = J, M VALUE = MAX( VALUE, ABS( A( I, J ) ) ) 70 CONTINUE 80 CONTINUE END IF END IF ELSE IF( ( LSAME( NORM, 'O' ) ) .OR. ( NORM.EQ.'1' ) ) THEN ! ! Find norm1(A). ! VALUE = ZERO UDIAG = LSAME( DIAG, 'U' ) IF( LSAME( UPLO, 'U' ) ) THEN DO 110 J = 1, N IF( ( UDIAG ) .AND. ( J.LE.M ) ) THEN SUM = ONE DO 90 I = 1, J - 1 SUM = SUM + ABS( A( I, J ) ) 90 CONTINUE ELSE SUM = ZERO DO 100 I = 1, MIN( M, J ) SUM = SUM + ABS( A( I, J ) ) 100 CONTINUE END IF VALUE = MAX( VALUE, SUM ) 110 CONTINUE ELSE DO 140 J = 1, N IF( UDIAG ) THEN SUM = ONE DO 120 I = J + 1, M SUM = SUM + ABS( A( I, J ) ) 120 CONTINUE ELSE SUM = ZERO DO 130 I = J, M SUM = SUM + ABS( A( I, J ) ) 130 CONTINUE END IF VALUE = MAX( VALUE, SUM ) 140 CONTINUE END IF ELSE IF( LSAME( NORM, 'I' ) ) THEN ! ! Find normI(A). ! IF( LSAME( UPLO, 'U' ) ) THEN IF( LSAME( DIAG, 'U' ) ) THEN DO 150 I = 1, M WORK( I ) = ONE 150 CONTINUE DO 170 J = 1, N DO 160 I = 1, MIN( M, J-1 ) WORK( I ) = WORK( I ) + ABS( A( I, J ) ) 160 CONTINUE 170 CONTINUE ELSE DO 180 I = 1, M WORK( I ) = ZERO 180 CONTINUE DO 200 J = 1, N DO 190 I = 1, MIN( M, J ) WORK( I ) = WORK( I ) + ABS( A( I, J ) ) 190 CONTINUE 200 CONTINUE END IF ELSE IF( LSAME( DIAG, 'U' ) ) THEN DO 210 I = 1, N WORK( I ) = ONE 210 CONTINUE DO 220 I = N + 1, M WORK( I ) = ZERO 220 CONTINUE DO 240 J = 1, N DO 230 I = J + 1, M WORK( I ) = WORK( I ) + ABS( A( I, J ) ) 230 CONTINUE 240 CONTINUE ELSE DO 250 I = 1, M WORK( I ) = ZERO 250 CONTINUE DO 270 J = 1, N DO 260 I = J, M WORK( I ) = WORK( I ) + ABS( A( I, J ) ) 260 CONTINUE 270 CONTINUE END IF END IF VALUE = ZERO DO 280 I = 1, M VALUE = MAX( VALUE, WORK( I ) ) 280 CONTINUE ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN ! ! Find normF(A). ! IF( LSAME( UPLO, 'U' ) ) THEN IF( LSAME( DIAG, 'U' ) ) THEN SCALE = ONE SUM = MIN( M, N ) DO 290 J = 2, N CALL DLASSQ( MIN( M, J-1 ), A( 1, J ), 1, SCALE, SUM ) 290 CONTINUE ELSE SCALE = ZERO SUM = ONE DO 300 J = 1, N CALL DLASSQ( MIN( M, J ), A( 1, J ), 1, SCALE, SUM ) 300 CONTINUE END IF ELSE IF( LSAME( DIAG, 'U' ) ) THEN SCALE = ONE SUM = MIN( M, N ) DO 310 J = 1, N CALL DLASSQ( M-J, A( MIN( M, J+1 ), J ), 1, SCALE, & SUM ) 310 CONTINUE ELSE SCALE = ZERO SUM = ONE DO 320 J = 1, N CALL DLASSQ( M-J+1, A( J, J ), 1, SCALE, SUM ) 320 CONTINUE END IF END IF VALUE = SCALE*SQRT( SUM ) END IF ! DLANTR = VALUE RETURN ! ! End of DLANTR ! END SUBROUTINE DLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. DOUBLE PRECISION A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN ! .. ! ! Purpose ! ======= ! ! DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric ! matrix in standard form: ! ! [ A B ] = [ CS -SN ] [ AA BB ] [ CS SN ] ! [ C D ] [ SN CS ] [ CC DD ] [-SN CS ] ! ! where either ! 1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or ! 2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex ! conjugate eigenvalues. ! ! Arguments ! ========= ! ! A (input/output) DOUBLE PRECISION ! B (input/output) DOUBLE PRECISION ! C (input/output) DOUBLE PRECISION ! D (input/output) DOUBLE PRECISION ! On entry, the elements of the input matrix. ! On exit, they are overwritten by the elements of the ! standardised Schur form. ! ! RT1R (output) DOUBLE PRECISION ! RT1I (output) DOUBLE PRECISION ! RT2R (output) DOUBLE PRECISION ! RT2I (output) DOUBLE PRECISION ! The real and imaginary parts of the eigenvalues. If the ! eigenvalues are a complex conjugate pair, RT1I > 0. ! ! CS (output) DOUBLE PRECISION ! SN (output) DOUBLE PRECISION ! Parameters of the rotation matrix. ! ! Further Details ! =============== ! ! Modified by V. Sima, Research Institute for Informatics, Bucharest, ! Romania, to reduce the risk of cancellation errors, ! when computing real eigenvalues, and to ensure, if possible, that ! abs(RT1R) >= abs(RT2R). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, HALF, ONE PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 ) DOUBLE PRECISION MULTPL PARAMETER ( MULTPL = 4.0D+0 ) ! .. ! .. Local Scalars .. DOUBLE PRECISION AA, BB, BCMAX, BCMIS, CC, CS1, DD, EPS, P, SAB, & SAC, SCALE, SIGMA, SN1, TAU, TEMP, Z ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH, DLAPY2 EXTERNAL DLAMCH, DLAPY2 ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SIGN, SQRT ! .. ! .. Executable Statements .. ! EPS = DLAMCH( 'P' ) IF( C.EQ.ZERO ) THEN CS = ONE SN = ZERO GO TO 10 ! ELSE IF( B.EQ.ZERO ) THEN ! ! Swap rows and columns ! CS = ZERO SN = ONE TEMP = D D = A A = TEMP B = -C C = ZERO GO TO 10 ELSE IF( ( A-D ).EQ.ZERO .AND. SIGN( ONE, B ).NE.SIGN( ONE, C ) ) & THEN CS = ONE SN = ZERO GO TO 10 ELSE ! TEMP = A - D P = HALF*TEMP BCMAX = MAX( ABS( B ), ABS( C ) ) BCMIS = MIN( ABS( B ), ABS( C ) )*SIGN( ONE, B )*SIGN( ONE, C ) SCALE = MAX( ABS( P ), BCMAX ) Z = ( P / SCALE )*P + ( BCMAX / SCALE )*BCMIS ! ! If Z is of the order of the machine accuracy, postpone the ! decision on the nature of eigenvalues ! IF( Z.GE.MULTPL*EPS ) THEN ! ! Real eigenvalues. Compute A and D. ! Z = P + SIGN( SQRT( SCALE )*SQRT( Z ), P ) A = D + Z D = D - ( BCMAX / Z )*BCMIS ! ! Compute B and the rotation matrix ! TAU = DLAPY2( C, Z ) CS = Z / TAU SN = C / TAU B = B - C C = ZERO ELSE ! ! Complex eigenvalues, or real (almost) equal eigenvalues. ! Make diagonal elements equal. ! SIGMA = B + C TAU = DLAPY2( SIGMA, TEMP ) CS = SQRT( HALF*( ONE+ABS( SIGMA ) / TAU ) ) SN = -( P / ( TAU*CS ) )*SIGN( ONE, SIGMA ) ! ! Compute [ AA BB ] = [ A B ] [ CS -SN ] ! [ CC DD ] [ C D ] [ SN CS ] ! AA = A*CS + B*SN BB = -A*SN + B*CS CC = C*CS + D*SN DD = -C*SN + D*CS ! ! Compute [ A B ] = [ CS SN ] [ AA BB ] ! [ C D ] [-SN CS ] [ CC DD ] ! A = AA*CS + CC*SN B = BB*CS + DD*SN C = -AA*SN + CC*CS D = -BB*SN + DD*CS ! TEMP = HALF*( A+D ) A = TEMP D = TEMP ! IF( C.NE.ZERO ) THEN IF( B.NE.ZERO ) THEN IF( SIGN( ONE, B ).EQ.SIGN( ONE, C ) ) THEN ! ! Real eigenvalues: reduce to upper triangular form ! SAB = SQRT( ABS( B ) ) SAC = SQRT( ABS( C ) ) P = SIGN( SAB*SAC, C ) TAU = ONE / SQRT( ABS( B+C ) ) A = TEMP + P D = TEMP - P B = B - C C = ZERO CS1 = SAB*TAU SN1 = SAC*TAU TEMP = CS*CS1 - SN*SN1 SN = CS*SN1 + SN*CS1 CS = TEMP END IF ELSE B = -C C = ZERO TEMP = CS CS = -SN SN = TEMP END IF END IF END IF ! END IF ! 10 CONTINUE ! ! Store eigenvalues in (RT1R,RT1I) and (RT2R,RT2I). ! RT1R = A RT2R = D IF( C.EQ.ZERO ) THEN RT1I = ZERO RT2I = ZERO ELSE RT1I = SQRT( ABS( B ) )*SQRT( ABS( C ) ) RT2I = -RT1I END IF RETURN ! ! End of DLANV2 ! END SUBROUTINE DLAPLL( N, X, INCX, Y, INCY, SSMIN ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. INTEGER INCX, INCY, N DOUBLE PRECISION SSMIN ! .. ! .. Array Arguments .. DOUBLE PRECISION X( * ), Y( * ) ! .. ! ! Purpose ! ======= ! ! Given two column vectors X and Y, let ! ! A = ( X Y ). ! ! The subroutine first computes the QR factorization of A = Q*R, ! and then computes the SVD of the 2-by-2 upper triangular matrix R. ! The smaller singular value of R is returned in SSMIN, which is used ! as the measurement of the linear dependency of the vectors X and Y. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The length of the vectors X and Y. ! ! X (input/output) DOUBLE PRECISION array, ! dimension (1+(N-1)*INCX) ! On entry, X contains the N-vector X. ! On exit, X is overwritten. ! ! INCX (input) INTEGER ! The increment between successive elements of X. INCX > 0. ! ! Y (input/output) DOUBLE PRECISION array, ! dimension (1+(N-1)*INCY) ! On entry, Y contains the N-vector Y. ! On exit, Y is overwritten. ! ! INCY (input) INTEGER ! The increment between successive elements of Y. INCY > 0. ! ! SSMIN (output) DOUBLE PRECISION ! The smallest singular value of the N-by-2 matrix A = ( X Y ). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. DOUBLE PRECISION A11, A12, A22, C, SSMAX, TAU ! .. ! .. External Functions .. DOUBLE PRECISION DDOT EXTERNAL DDOT ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DLARFG, DLAS2 ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( N.LE.1 ) THEN SSMIN = ZERO RETURN END IF ! ! Compute the QR factorization of the N-by-2 matrix ( X Y ) ! CALL DLARFG( N, X( 1 ), X( 1+INCX ), INCX, TAU ) A11 = X( 1 ) X( 1 ) = ONE ! C = -TAU*DDOT( N, X, INCX, Y, INCY ) CALL DAXPY( N, C, X, INCX, Y, INCY ) ! CALL DLARFG( N-1, Y( 1+INCY ), Y( 1+2*INCY ), INCY, TAU ) ! A12 = Y( 1 ) A22 = Y( 1+INCY ) ! ! Compute the SVD of 2-by-2 Upper triangular matrix. ! CALL DLAS2( A11, A12, A22, SSMIN, SSMAX ) ! RETURN ! ! End of DLAPLL ! END SUBROUTINE DLAPMT( FORWRD, M, N, X, LDX, K ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. LOGICAL FORWRD INTEGER LDX, M, N ! .. ! .. Array Arguments .. INTEGER K( * ) DOUBLE PRECISION X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DLAPMT rearranges the columns of the M by N matrix X as specified ! by the permutation K(1),K(2),...,K(N) of the integers 1,...,N. ! If FORWRD = .TRUE., forward permutation: ! ! X(*,K(J)) is moved X(*,J) for J = 1,2,...,N. ! ! If FORWRD = .FALSE., backward permutation: ! ! X(*,J) is moved to X(*,K(J)) for J = 1,2,...,N. ! ! Arguments ! ========= ! ! FORWRD (input) LOGICAL ! = .TRUE., forward permutation ! = .FALSE., backward permutation ! ! M (input) INTEGER ! The number of rows of the matrix X. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix X. N >= 0. ! ! X (input/output) DOUBLE PRECISION array, dimension (LDX,N) ! On entry, the M by N matrix X. ! On exit, X contains the permuted matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X, LDX >= MAX(1,M). ! ! K (input) INTEGER array, dimension (N) ! On entry, K contains the permutation vector. ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER I, II, IN, J DOUBLE PRECISION TEMP ! .. ! .. Executable Statements .. ! IF( N.LE.1 ) & RETURN ! DO 10 I = 1, N K( I ) = -K( I ) 10 CONTINUE ! IF( FORWRD ) THEN ! ! Forward permutation ! DO 50 I = 1, N ! IF( K( I ).GT.0 ) & GO TO 40 ! J = I K( J ) = -K( J ) IN = K( J ) ! 20 CONTINUE IF( K( IN ).GT.0 ) & GO TO 40 ! DO 30 II = 1, M TEMP = X( II, J ) X( II, J ) = X( II, IN ) X( II, IN ) = TEMP 30 CONTINUE ! K( IN ) = -K( IN ) J = IN IN = K( IN ) GO TO 20 ! 40 CONTINUE ! 50 CONTINUE ! ELSE ! ! Backward permutation ! DO 90 I = 1, N ! IF( K( I ).GT.0 ) & GO TO 80 ! K( I ) = -K( I ) J = K( I ) 60 CONTINUE IF( J.EQ.I ) & GO TO 80 ! DO 70 II = 1, M TEMP = X( II, I ) X( II, I ) = X( II, J ) X( II, J ) = TEMP 70 CONTINUE ! K( J ) = -K( J ) J = K( J ) GO TO 60 ! 80 CONTINUE ! 90 CONTINUE ! END IF ! RETURN ! ! End of DLAPMT ! END DOUBLE PRECISION FUNCTION DLAPY2( X, Y ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. DOUBLE PRECISION X, Y ! .. ! ! Purpose ! ======= ! ! DLAPY2 returns sqrt(x**2+y**2), taking care not to cause unnecessary ! overflow. ! ! Arguments ! ========= ! ! X (input) DOUBLE PRECISION ! Y (input) DOUBLE PRECISION ! X and Y specify the values x and y. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D0 ) ! .. ! .. Local Scalars .. DOUBLE PRECISION W, XABS, YABS, Z ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! XABS = ABS( X ) YABS = ABS( Y ) W = MAX( XABS, YABS ) Z = MIN( XABS, YABS ) IF( Z.EQ.ZERO ) THEN DLAPY2 = W ELSE DLAPY2 = W*SQRT( ONE+( Z / W )**2 ) END IF RETURN ! ! End of DLAPY2 ! END DOUBLE PRECISION FUNCTION DLAPY3( X, Y, Z ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. DOUBLE PRECISION X, Y, Z ! .. ! ! Purpose ! ======= ! ! DLAPY3 returns sqrt(x**2+y**2+z**2), taking care not to cause ! unnecessary overflow. ! ! Arguments ! ========= ! ! X (input) DOUBLE PRECISION ! Y (input) DOUBLE PRECISION ! Z (input) DOUBLE PRECISION ! X, Y and Z specify the values x, y and z. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D0 ) ! .. ! .. Local Scalars .. DOUBLE PRECISION W, XABS, YABS, ZABS ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! XABS = ABS( X ) YABS = ABS( Y ) ZABS = ABS( Z ) W = MAX( XABS, YABS, ZABS ) IF( W.EQ.ZERO ) THEN DLAPY3 = ZERO ELSE DLAPY3 = W*SQRT( ( XABS / W )**2+( YABS / W )**2+ & ( ZABS / W )**2 ) END IF RETURN ! ! End of DLAPY3 ! END SUBROUTINE DLAQGB( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, & AMAX, EQUED ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER EQUED INTEGER KL, KU, LDAB, M, N DOUBLE PRECISION AMAX, COLCND, ROWCND ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), C( * ), R( * ) ! .. ! ! Purpose ! ======= ! ! DLAQGB equilibrates a general M by N band matrix A with KL ! subdiagonals and KU superdiagonals using the row and scaling factors ! in the vectors R and C. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! KL (input) INTEGER ! The number of subdiagonals within the band of A. KL >= 0. ! ! KU (input) INTEGER ! The number of superdiagonals within the band of A. KU >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) ! On entry, the matrix A in band storage, in rows 1 to KL+KU+1. ! The j-th column of A is stored in the j-th column of the ! array AB as follows: ! AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl) ! ! On exit, the equilibrated matrix, in the same storage format ! as A. See EQUED for the form of the equilibrated matrix. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDA >= KL+KU+1. ! ! R (output) DOUBLE PRECISION array, dimension (M) ! The row scale factors for A. ! ! C (output) DOUBLE PRECISION array, dimension (N) ! The column scale factors for A. ! ! ROWCND (output) DOUBLE PRECISION ! Ratio of the smallest R(i) to the largest R(i). ! ! COLCND (output) DOUBLE PRECISION ! Ratio of the smallest C(i) to the largest C(i). ! ! AMAX (input) DOUBLE PRECISION ! Absolute value of largest matrix entry. ! ! EQUED (output) CHARACTER*1 ! Specifies the form of equilibration that was done. ! = 'N': No equilibration ! = 'R': Row equilibration, i.e., A has been premultiplied by ! diag(R). ! = 'C': Column equilibration, i.e., A has been postmultiplied ! by diag(C). ! = 'B': Both row and column equilibration, i.e., A has been ! replaced by diag(R) * A * diag(C). ! ! Internal Parameters ! =================== ! ! THRESH is a threshold value used to decide if row or column scaling ! should be done based on the ratio of the row or column scaling ! factors. If ROWCND < THRESH, row scaling is done, and if ! COLCND < THRESH, column scaling is done. ! ! LARGE and SMALL are threshold values used to decide if row scaling ! should be done based on the absolute size of the largest matrix ! element. If AMAX > LARGE or AMAX < SMALL, row scaling is done. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, THRESH PARAMETER ( ONE = 1.0D+0, THRESH = 0.1D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J DOUBLE PRECISION CJ, LARGE, SMALL ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( M.LE.0 .OR. N.LE.0 ) THEN EQUED = 'N' RETURN END IF ! ! Initialize LARGE and SMALL. ! SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' ) LARGE = ONE / SMALL ! IF( ROWCND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE ) & THEN ! ! No row scaling ! IF( COLCND.GE.THRESH ) THEN ! ! No column scaling ! EQUED = 'N' ELSE ! ! Column scaling ! DO 20 J = 1, N CJ = C( J ) DO 10 I = MAX( 1, J-KU ), MIN( M, J+KL ) AB( KU+1+I-J, J ) = CJ*AB( KU+1+I-J, J ) 10 CONTINUE 20 CONTINUE EQUED = 'C' END IF ELSE IF( COLCND.GE.THRESH ) THEN ! ! Row scaling, no column scaling ! DO 40 J = 1, N DO 30 I = MAX( 1, J-KU ), MIN( M, J+KL ) AB( KU+1+I-J, J ) = R( I )*AB( KU+1+I-J, J ) 30 CONTINUE 40 CONTINUE EQUED = 'R' ELSE ! ! Row and column scaling ! DO 60 J = 1, N CJ = C( J ) DO 50 I = MAX( 1, J-KU ), MIN( M, J+KL ) AB( KU+1+I-J, J ) = CJ*R( I )*AB( KU+1+I-J, J ) 50 CONTINUE 60 CONTINUE EQUED = 'B' END IF ! RETURN ! ! End of DLAQGB ! END SUBROUTINE DLAQGE( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, & EQUED ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER EQUED INTEGER LDA, M, N DOUBLE PRECISION AMAX, COLCND, ROWCND ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( * ), R( * ) ! .. ! ! Purpose ! ======= ! ! DLAQGE equilibrates a general M by N matrix A using the row and ! scaling factors in the vectors R and C. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M by N matrix A. ! On exit, the equilibrated matrix. See EQUED for the form of ! the equilibrated matrix. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(M,1). ! ! R (input) DOUBLE PRECISION array, dimension (M) ! The row scale factors for A. ! ! C (input) DOUBLE PRECISION array, dimension (N) ! The column scale factors for A. ! ! ROWCND (input) DOUBLE PRECISION ! Ratio of the smallest R(i) to the largest R(i). ! ! COLCND (input) DOUBLE PRECISION ! Ratio of the smallest C(i) to the largest C(i). ! ! AMAX (input) DOUBLE PRECISION ! Absolute value of largest matrix entry. ! ! EQUED (output) CHARACTER*1 ! Specifies the form of equilibration that was done. ! = 'N': No equilibration ! = 'R': Row equilibration, i.e., A has been premultiplied by ! diag(R). ! = 'C': Column equilibration, i.e., A has been postmultiplied ! by diag(C). ! = 'B': Both row and column equilibration, i.e., A has been ! replaced by diag(R) * A * diag(C). ! ! Internal Parameters ! =================== ! ! THRESH is a threshold value used to decide if row or column scaling ! should be done based on the ratio of the row or column scaling ! factors. If ROWCND < THRESH, row scaling is done, and if ! COLCND < THRESH, column scaling is done. ! ! LARGE and SMALL are threshold values used to decide if row scaling ! should be done based on the absolute size of the largest matrix ! element. If AMAX > LARGE or AMAX < SMALL, row scaling is done. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, THRESH PARAMETER ( ONE = 1.0D+0, THRESH = 0.1D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J DOUBLE PRECISION CJ, LARGE, SMALL ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( M.LE.0 .OR. N.LE.0 ) THEN EQUED = 'N' RETURN END IF ! ! Initialize LARGE and SMALL. ! SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' ) LARGE = ONE / SMALL ! IF( ROWCND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE ) & THEN ! ! No row scaling ! IF( COLCND.GE.THRESH ) THEN ! ! No column scaling ! EQUED = 'N' ELSE ! ! Column scaling ! DO 20 J = 1, N CJ = C( J ) DO 10 I = 1, M A( I, J ) = CJ*A( I, J ) 10 CONTINUE 20 CONTINUE EQUED = 'C' END IF ELSE IF( COLCND.GE.THRESH ) THEN ! ! Row scaling, no column scaling ! DO 40 J = 1, N DO 30 I = 1, M A( I, J ) = R( I )*A( I, J ) 30 CONTINUE 40 CONTINUE EQUED = 'R' ELSE ! ! Row and column scaling ! DO 60 J = 1, N CJ = C( J ) DO 50 I = 1, M A( I, J ) = CJ*R( I )*A( I, J ) 50 CONTINUE 60 CONTINUE EQUED = 'B' END IF ! RETURN ! ! End of DLAQGE ! END SUBROUTINE DLAQP2( M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2, & WORK ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER LDA, M, N, OFFSET ! .. ! .. Array Arguments .. INTEGER JPVT( * ) DOUBLE PRECISION A( LDA, * ), TAU( * ), VN1( * ), VN2( * ), & WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLAQP2 computes a QR factorization with column pivoting of ! the block A(OFFSET+1:M,1:N). ! The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! OFFSET (input) INTEGER ! The number of rows of the matrix A that must be pivoted ! but no factorized. OFFSET >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, the upper triangle of block A(OFFSET+1:M,1:N) is ! the triangular factor obtained; the elements in block ! A(OFFSET+1:M,1:N) below the diagonal, together with the ! array TAU, represent the orthogonal matrix Q as a product of ! elementary reflectors. Block A(1:OFFSET,1:N) has been ! accordingly pivoted, but no factorized. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! JPVT (input/output) INTEGER array, dimension (N) ! On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted ! to the front of A*P (a leading column); if JPVT(i) = 0, ! the i-th column of A is a free column. ! On exit, if JPVT(i) = k, then the i-th column of A*P ! was the k-th column of A. ! ! TAU (output) DOUBLE PRECISION array, dimension (min(M,N)) ! The scalar factors of the elementary reflectors. ! ! VN1 (input/output) DOUBLE PRECISION array, dimension (N) ! The vector with the partial column norms. ! ! VN2 (input/output) DOUBLE PRECISION array, dimension (N) ! The vector with the exact column norms. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (N) ! ! Further Details ! =============== ! ! Based on contributions by ! G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain ! X. Sun, Computer Science Dept., Duke University, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, ITEMP, J, MN, OFFPI, PVT DOUBLE PRECISION AII, TEMP, TEMP2 ! .. ! .. External Subroutines .. EXTERNAL DLARF, DLARFG, DSWAP ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT ! .. ! .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DNRM2 EXTERNAL IDAMAX, DNRM2 ! .. ! .. Executable Statements .. ! MN = MIN( M-OFFSET, N ) ! ! Compute factorization. ! DO 20 I = 1, MN ! OFFPI = OFFSET + I ! ! Determine ith pivot column and swap if necessary. ! PVT = ( I-1 ) + IDAMAX( N-I+1, VN1( I ), 1 ) ! IF( PVT.NE.I ) THEN CALL DSWAP( M, A( 1, PVT ), 1, A( 1, I ), 1 ) ITEMP = JPVT( PVT ) JPVT( PVT ) = JPVT( I ) JPVT( I ) = ITEMP VN1( PVT ) = VN1( I ) VN2( PVT ) = VN2( I ) END IF ! ! Generate elementary reflector H(i). ! IF( OFFPI.LT.M ) THEN CALL DLARFG( M-OFFPI+1, A( OFFPI, I ), A( OFFPI+1, I ), 1, & TAU( I ) ) ELSE CALL DLARFG( 1, A( M, I ), A( M, I ), 1, TAU( I ) ) END IF ! IF( I.LT.N ) THEN ! ! Apply H(i)' to A(offset+i:m,i+1:n) from the left. ! AII = A( OFFPI, I ) A( OFFPI, I ) = ONE CALL DLARF( 'Left', M-OFFPI+1, N-I, A( OFFPI, I ), 1, & TAU( I ), A( OFFPI, I+1 ), LDA, WORK( 1 ) ) A( OFFPI, I ) = AII END IF ! ! Update partial column norms. ! DO 10 J = I + 1, N IF( VN1( J ).NE.ZERO ) THEN TEMP = ONE - ( ABS( A( OFFPI, J ) ) / VN1( J ) )**2 TEMP = MAX( TEMP, ZERO ) TEMP2 = ONE + 0.05D0*TEMP*( VN1( J ) / VN2( J ) )**2 IF( TEMP2.EQ.ONE ) THEN IF( OFFPI.LT.M ) THEN VN1( J ) = DNRM2( M-OFFPI, A( OFFPI+1, J ), 1 ) VN2( J ) = VN1( J ) ELSE VN1( J ) = ZERO VN2( J ) = ZERO END IF ELSE VN1( J ) = VN1( J )*SQRT( TEMP ) END IF END IF 10 CONTINUE ! 20 CONTINUE ! RETURN ! ! End of DLAQP2 ! END SUBROUTINE DLAQPS( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, & VN2, AUXV, F, LDF ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER KB, LDA, LDF, M, N, NB, OFFSET ! .. ! .. Array Arguments .. INTEGER JPVT( * ) DOUBLE PRECISION A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ), & VN1( * ), VN2( * ) ! .. ! ! Purpose ! ======= ! ! DLAQPS computes a step of QR factorization with column pivoting ! of a real M-by-N matrix A by using Blas-3. It tries to factorize ! NB columns from A starting from the row OFFSET+1, and updates all ! of the matrix with Blas-3 xGEMM. ! ! In some cases, due to catastrophic cancellations, it cannot ! factorize NB columns. Hence, the actual number of factorized ! columns is returned in KB. ! ! Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0 ! ! OFFSET (input) INTEGER ! The number of rows of A that have been factorized in ! previous steps. ! ! NB (input) INTEGER ! The number of columns to factorize. ! ! KB (output) INTEGER ! The number of columns actually factorized. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, block A(OFFSET+1:M,1:KB) is the triangular ! factor obtained and block A(1:OFFSET,1:N) has been ! accordingly pivoted, but no factorized. ! The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has ! been updated. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! JPVT (input/output) INTEGER array, dimension (N) ! JPVT(I) = K <==> Column K of the full matrix A has been ! permuted into position I in AP. ! ! TAU (output) DOUBLE PRECISION array, dimension (KB) ! The scalar factors of the elementary reflectors. ! ! VN1 (input/output) DOUBLE PRECISION array, dimension (N) ! The vector with the partial column norms. ! ! VN2 (input/output) DOUBLE PRECISION array, dimension (N) ! The vector with the exact column norms. ! ! AUXV (input/output) DOUBLE PRECISION array, dimension (NB) ! Auxiliar vector. ! ! F (input/output) DOUBLE PRECISION array, dimension (LDF,NB) ! Matrix F' = L*Y'*A. ! ! LDF (input) INTEGER ! The leading dimension of the array F. LDF >= max(1,N). ! ! Further Details ! =============== ! ! Based on contributions by ! G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain ! X. Sun, Computer Science Dept., Duke University, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER ITEMP, J, K, LASTRK, LSTICC, PVT, RK DOUBLE PRECISION AKK, TEMP, TEMP2 ! .. ! .. External Subroutines .. EXTERNAL DGEMM, DGEMV, DLARFG, DSWAP ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX, MIN, NINT, SQRT ! .. ! .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DNRM2 EXTERNAL IDAMAX, DNRM2 ! .. ! .. Executable Statements .. ! LASTRK = MIN( M, N+OFFSET ) LSTICC = 0 K = 0 ! ! Beginning of while loop. ! 10 CONTINUE IF( ( K.LT.NB ) .AND. ( LSTICC.EQ.0 ) ) THEN K = K + 1 RK = OFFSET + K ! ! Determine ith pivot column and swap if necessary ! PVT = ( K-1 ) + IDAMAX( N-K+1, VN1( K ), 1 ) IF( PVT.NE.K ) THEN CALL DSWAP( M, A( 1, PVT ), 1, A( 1, K ), 1 ) CALL DSWAP( K-1, F( PVT, 1 ), LDF, F( K, 1 ), LDF ) ITEMP = JPVT( PVT ) JPVT( PVT ) = JPVT( K ) JPVT( K ) = ITEMP VN1( PVT ) = VN1( K ) VN2( PVT ) = VN2( K ) END IF ! ! Apply previous Householder reflectors to column K: ! A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)'. ! IF( K.GT.1 ) THEN CALL DGEMV( 'No transpose', M-RK+1, K-1, -ONE, A( RK, 1 ), & LDA, F( K, 1 ), LDF, ONE, A( RK, K ), 1 ) END IF ! ! Generate elementary reflector H(k). ! IF( RK.LT.M ) THEN CALL DLARFG( M-RK+1, A( RK, K ), A( RK+1, K ), 1, TAU( K ) ) ELSE CALL DLARFG( 1, A( RK, K ), A( RK, K ), 1, TAU( K ) ) END IF ! AKK = A( RK, K ) A( RK, K ) = ONE ! ! Compute Kth column of F: ! ! Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)'*A(RK:M,K). ! IF( K.LT.N ) THEN CALL DGEMV( 'Transpose', M-RK+1, N-K, TAU( K ), & A( RK, K+1 ), LDA, A( RK, K ), 1, ZERO, & F( K+1, K ), 1 ) END IF ! ! Padding F(1:K,K) with zeros. ! DO 20 J = 1, K F( J, K ) = ZERO 20 CONTINUE ! ! Incremental updating of F: ! F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)' ! *A(RK:M,K). ! IF( K.GT.1 ) THEN CALL DGEMV( 'Transpose', M-RK+1, K-1, -TAU( K ), A( RK, 1 ), & LDA, A( RK, K ), 1, ZERO, AUXV( 1 ), 1 ) ! CALL DGEMV( 'No transpose', N, K-1, ONE, F( 1, 1 ), LDF, & AUXV( 1 ), 1, ONE, F( 1, K ), 1 ) END IF ! ! Update the current row of A: ! A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)'. ! IF( K.LT.N ) THEN CALL DGEMV( 'No transpose', N-K, K, -ONE, F( K+1, 1 ), LDF, & A( RK, 1 ), LDA, ONE, A( RK, K+1 ), LDA ) END IF ! ! Update partial column norms. ! IF( RK.LT.LASTRK ) THEN DO 30 J = K + 1, N IF( VN1( J ).NE.ZERO ) THEN TEMP = ABS( A( RK, J ) ) / VN1( J ) TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) ) TEMP2 = ONE + 0.05D0*TEMP*( VN1( J ) / VN2( J ) )**2 IF( TEMP2.EQ.ONE ) THEN VN2( J ) = DBLE( LSTICC ) LSTICC = J ELSE VN1( J ) = VN1( J )*SQRT( TEMP ) END IF END IF 30 CONTINUE END IF ! A( RK, K ) = AKK ! ! End of while loop. ! GO TO 10 END IF KB = K RK = OFFSET + KB ! ! Apply the block reflector to the rest of the matrix: ! A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) - ! A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)'. ! IF( KB.LT.MIN( N, M-OFFSET ) ) THEN CALL DGEMM( 'No transpose', 'Transpose', M-RK, N-KB, KB, -ONE, & A( RK+1, 1 ), LDA, F( KB+1, 1 ), LDF, ONE, & A( RK+1, KB+1 ), LDA ) END IF ! ! Recomputation of difficult columns. ! 40 CONTINUE IF( LSTICC.GT.0 ) THEN ITEMP = NINT( VN2( LSTICC ) ) VN1( LSTICC ) = DNRM2( M-RK, A( RK+1, LSTICC ), 1 ) VN2( LSTICC ) = VN1( LSTICC ) LSTICC = ITEMP GO TO 40 END IF ! RETURN ! ! End of DLAQPS ! END SUBROUTINE DLAQSB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER EQUED, UPLO INTEGER KD, LDAB, N DOUBLE PRECISION AMAX, SCOND ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), S( * ) ! .. ! ! Purpose ! ======= ! ! DLAQSB equilibrates a symmetric band matrix A using the scaling ! factors in the vector S. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the upper or lower triangular part of the ! symmetric matrix A is stored. ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of super-diagonals of the matrix A if UPLO = 'U', ! or the number of sub-diagonals if UPLO = 'L'. KD >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) ! On entry, the upper or lower triangle of the symmetric band ! matrix A, stored in the first KD+1 rows of the array. The ! j-th column of A is stored in the j-th column of the array AB ! as follows: ! if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). ! ! On exit, if INFO = 0, the triangular factor U or L from the ! Cholesky factorization A = U'*U or A = L*L' of the band ! matrix A, in the same storage format as A. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KD+1. ! ! S (output) DOUBLE PRECISION array, dimension (N) ! The scale factors for A. ! ! SCOND (input) DOUBLE PRECISION ! Ratio of the smallest S(i) to the largest S(i). ! ! AMAX (input) DOUBLE PRECISION ! Absolute value of largest matrix entry. ! ! EQUED (output) CHARACTER*1 ! Specifies whether or not equilibration was done. ! = 'N': No equilibration. ! = 'Y': Equilibration was done, i.e., A has been replaced by ! diag(S) * A * diag(S). ! ! Internal Parameters ! =================== ! ! THRESH is a threshold value used to decide if scaling should be done ! based on the ratio of the scaling factors. If SCOND < THRESH, ! scaling is done. ! ! LARGE and SMALL are threshold values used to decide if scaling should ! be done based on the absolute size of the largest matrix element. ! If AMAX > LARGE or AMAX < SMALL, scaling is done. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, THRESH PARAMETER ( ONE = 1.0D+0, THRESH = 0.1D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J DOUBLE PRECISION CJ, LARGE, SMALL ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( N.LE.0 ) THEN EQUED = 'N' RETURN END IF ! ! Initialize LARGE and SMALL. ! SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' ) LARGE = ONE / SMALL ! IF( SCOND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE ) THEN ! ! No equilibration ! EQUED = 'N' ELSE ! ! Replace A by diag(S) * A * diag(S). ! IF( LSAME( UPLO, 'U' ) ) THEN ! ! Upper triangle of A is stored in band format. ! DO 20 J = 1, N CJ = S( J ) DO 10 I = MAX( 1, J-KD ), J AB( KD+1+I-J, J ) = CJ*S( I )*AB( KD+1+I-J, J ) 10 CONTINUE 20 CONTINUE ELSE ! ! Lower triangle of A is stored. ! DO 40 J = 1, N CJ = S( J ) DO 30 I = J, MIN( N, J+KD ) AB( 1+I-J, J ) = CJ*S( I )*AB( 1+I-J, J ) 30 CONTINUE 40 CONTINUE END IF EQUED = 'Y' END IF ! RETURN ! ! End of DLAQSB ! END SUBROUTINE DLAQSP( UPLO, N, AP, S, SCOND, AMAX, EQUED ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER EQUED, UPLO INTEGER N DOUBLE PRECISION AMAX, SCOND ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ), S( * ) ! .. ! ! Purpose ! ======= ! ! DLAQSP equilibrates a symmetric matrix A using the scaling factors ! in the vector S. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the upper or lower triangular part of the ! symmetric matrix A is stored. ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! A, packed columnwise in a linear array. The j-th column of A ! is stored in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. ! ! On exit, the equilibrated matrix: diag(S) * A * diag(S), in ! the same storage format as A. ! ! S (input) DOUBLE PRECISION array, dimension (N) ! The scale factors for A. ! ! SCOND (input) DOUBLE PRECISION ! Ratio of the smallest S(i) to the largest S(i). ! ! AMAX (input) DOUBLE PRECISION ! Absolute value of largest matrix entry. ! ! EQUED (output) CHARACTER*1 ! Specifies whether or not equilibration was done. ! = 'N': No equilibration. ! = 'Y': Equilibration was done, i.e., A has been replaced by ! diag(S) * A * diag(S). ! ! Internal Parameters ! =================== ! ! THRESH is a threshold value used to decide if scaling should be done ! based on the ratio of the scaling factors. If SCOND < THRESH, ! scaling is done. ! ! LARGE and SMALL are threshold values used to decide if scaling should ! be done based on the absolute size of the largest matrix element. ! If AMAX > LARGE or AMAX < SMALL, scaling is done. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, THRESH PARAMETER ( ONE = 1.0D+0, THRESH = 0.1D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J, JC DOUBLE PRECISION CJ, LARGE, SMALL ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( N.LE.0 ) THEN EQUED = 'N' RETURN END IF ! ! Initialize LARGE and SMALL. ! SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' ) LARGE = ONE / SMALL ! IF( SCOND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE ) THEN ! ! No equilibration ! EQUED = 'N' ELSE ! ! Replace A by diag(S) * A * diag(S). ! IF( LSAME( UPLO, 'U' ) ) THEN ! ! Upper triangle of A is stored. ! JC = 1 DO 20 J = 1, N CJ = S( J ) DO 10 I = 1, J AP( JC+I-1 ) = CJ*S( I )*AP( JC+I-1 ) 10 CONTINUE JC = JC + J 20 CONTINUE ELSE ! ! Lower triangle of A is stored. ! JC = 1 DO 40 J = 1, N CJ = S( J ) DO 30 I = J, N AP( JC+I-J ) = CJ*S( I )*AP( JC+I-J ) 30 CONTINUE JC = JC + N - J + 1 40 CONTINUE END IF EQUED = 'Y' END IF ! RETURN ! ! End of DLAQSP ! END SUBROUTINE DLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER EQUED, UPLO INTEGER LDA, N DOUBLE PRECISION AMAX, SCOND ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), S( * ) ! .. ! ! Purpose ! ======= ! ! DLAQSY equilibrates a symmetric matrix A using the scaling factors ! in the vector S. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the upper or lower triangular part of the ! symmetric matrix A is stored. ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the symmetric matrix A. If UPLO = 'U', the leading ! n by n upper triangular part of A contains the upper ! triangular part of the matrix A, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading n by n lower triangular part of A contains the lower ! triangular part of the matrix A, and the strictly upper ! triangular part of A is not referenced. ! ! On exit, if EQUED = 'Y', the equilibrated matrix: ! diag(S) * A * diag(S). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(N,1). ! ! S (input) DOUBLE PRECISION array, dimension (N) ! The scale factors for A. ! ! SCOND (input) DOUBLE PRECISION ! Ratio of the smallest S(i) to the largest S(i). ! ! AMAX (input) DOUBLE PRECISION ! Absolute value of largest matrix entry. ! ! EQUED (output) CHARACTER*1 ! Specifies whether or not equilibration was done. ! = 'N': No equilibration. ! = 'Y': Equilibration was done, i.e., A has been replaced by ! diag(S) * A * diag(S). ! ! Internal Parameters ! =================== ! ! THRESH is a threshold value used to decide if scaling should be done ! based on the ratio of the scaling factors. If SCOND < THRESH, ! scaling is done. ! ! LARGE and SMALL are threshold values used to decide if scaling should ! be done based on the absolute size of the largest matrix element. ! If AMAX > LARGE or AMAX < SMALL, scaling is done. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, THRESH PARAMETER ( ONE = 1.0D+0, THRESH = 0.1D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J DOUBLE PRECISION CJ, LARGE, SMALL ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( N.LE.0 ) THEN EQUED = 'N' RETURN END IF ! ! Initialize LARGE and SMALL. ! SMALL = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' ) LARGE = ONE / SMALL ! IF( SCOND.GE.THRESH .AND. AMAX.GE.SMALL .AND. AMAX.LE.LARGE ) THEN ! ! No equilibration ! EQUED = 'N' ELSE ! ! Replace A by diag(S) * A * diag(S). ! IF( LSAME( UPLO, 'U' ) ) THEN ! ! Upper triangle of A is stored. ! DO 20 J = 1, N CJ = S( J ) DO 10 I = 1, J A( I, J ) = CJ*S( I )*A( I, J ) 10 CONTINUE 20 CONTINUE ELSE ! ! Lower triangle of A is stored. ! DO 40 J = 1, N CJ = S( J ) DO 30 I = J, N A( I, J ) = CJ*S( I )*A( I, J ) 30 CONTINUE 40 CONTINUE END IF EQUED = 'Y' END IF ! RETURN ! ! End of DLAQSY ! END SUBROUTINE DLAQTR( LTRAN, LREAL, N, T, LDT, B, W, SCALE, X, WORK, & INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. LOGICAL LREAL, LTRAN INTEGER INFO, LDT, N DOUBLE PRECISION SCALE, W ! .. ! .. Array Arguments .. DOUBLE PRECISION B( * ), T( LDT, * ), WORK( * ), X( * ) ! .. ! ! Purpose ! ======= ! ! DLAQTR solves the real quasi-triangular system ! ! op(T)*p = scale*c, if LREAL = .TRUE. ! ! or the complex quasi-triangular systems ! ! op(T + iB)*(p+iq) = scale*(c+id), if LREAL = .FALSE. ! ! in real arithmetic, where T is upper quasi-triangular. ! If LREAL = .FALSE., then the first diagonal block of T must be ! 1 by 1, B is the specially structured matrix ! ! B = [ b(1) b(2) ... b(n) ] ! [ w ] ! [ w ] ! [ . ] ! [ w ] ! ! op(A) = A or A', A' denotes the conjugate transpose of ! matrix A. ! ! On input, X = [ c ]. On output, X = [ p ]. ! [ d ] [ q ] ! ! This subroutine is designed for the condition number estimation ! in routine DTRSNA. ! ! Arguments ! ========= ! ! LTRAN (input) LOGICAL ! On entry, LTRAN specifies the option of conjugate transpose: ! = .FALSE., op(T+i*B) = T+i*B, ! = .TRUE., op(T+i*B) = (T+i*B)'. ! ! LREAL (input) LOGICAL ! On entry, LREAL specifies the input matrix structure: ! = .FALSE., the input is complex ! = .TRUE., the input is real ! ! N (input) INTEGER ! On entry, N specifies the order of T+i*B. N >= 0. ! ! T (input) DOUBLE PRECISION array, dimension (LDT,N) ! On entry, T contains a matrix in Schur canonical form. ! If LREAL = .FALSE., then the first diagonal block of T mu ! be 1 by 1. ! ! LDT (input) INTEGER ! The leading dimension of the matrix T. LDT >= max(1,N). ! ! B (input) DOUBLE PRECISION array, dimension (N) ! On entry, B contains the elements to form the matrix ! B as described above. ! If LREAL = .TRUE., B is not referenced. ! ! W (input) DOUBLE PRECISION ! On entry, W is the diagonal element of the matrix B. ! If LREAL = .TRUE., W is not referenced. ! ! SCALE (output) DOUBLE PRECISION ! On exit, SCALE is the scale factor. ! ! X (input/output) DOUBLE PRECISION array, dimension (2*N) ! On entry, X contains the right hand side of the system. ! On exit, X is overwritten by the solution. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (N) ! ! INFO (output) INTEGER ! On exit, INFO is set to ! 0: successful exit. ! 1: the some diagonal 1 by 1 block has been perturbed by ! a small number SMIN to keep nonsingularity. ! 2: the some diagonal 2 by 2 block has been perturbed by ! a small number in DLALN2 to keep nonsingularity. ! NOTE: In the interests of speed, this routine does not ! check the inputs for errors. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOTRAN INTEGER I, IERR, J, J1, J2, JNEXT, K, N1, N2 DOUBLE PRECISION BIGNUM, EPS, REC, SCALOC, SI, SMIN, SMINW, & SMLNUM, SR, TJJ, TMP, XJ, XMAX, XNORM, Z ! .. ! .. Local Arrays .. DOUBLE PRECISION D( 2, 2 ), V( 2, 2 ) ! .. ! .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DASUM, DDOT, DLAMCH, DLANGE EXTERNAL IDAMAX, DASUM, DDOT, DLAMCH, DLANGE ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DLADIV, DLALN2, DSCAL ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. Executable Statements .. ! ! Do not test the input parameters for errors ! NOTRAN = .NOT.LTRAN INFO = 0 ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Set constants to control overflow ! EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) / EPS BIGNUM = ONE / SMLNUM ! XNORM = DLANGE( 'M', N, N, T, LDT, D ) IF( .NOT.LREAL ) & XNORM = MAX( XNORM, ABS( W ), DLANGE( 'M', N, 1, B, N, D ) ) SMIN = MAX( SMLNUM, EPS*XNORM ) ! ! Compute 1-norm of each column of strictly upper triangular ! part of T to control overflow in triangular solver. ! WORK( 1 ) = ZERO DO 10 J = 2, N WORK( J ) = DASUM( J-1, T( 1, J ), 1 ) 10 CONTINUE ! IF( .NOT.LREAL ) THEN DO 20 I = 2, N WORK( I ) = WORK( I ) + ABS( B( I ) ) 20 CONTINUE END IF ! N2 = 2*N N1 = N IF( .NOT.LREAL ) & N1 = N2 K = IDAMAX( N1, X, 1 ) XMAX = ABS( X( K ) ) SCALE = ONE ! IF( XMAX.GT.BIGNUM ) THEN SCALE = BIGNUM / XMAX CALL DSCAL( N1, SCALE, X, 1 ) XMAX = BIGNUM END IF ! IF( LREAL ) THEN ! IF( NOTRAN ) THEN ! ! Solve T*p = scale*c ! JNEXT = N DO 30 J = N, 1, -1 IF( J.GT.JNEXT ) & GO TO 30 J1 = J J2 = J JNEXT = J - 1 IF( J.GT.1 ) THEN IF( T( J, J-1 ).NE.ZERO ) THEN J1 = J - 1 JNEXT = J - 2 END IF END IF ! IF( J1.EQ.J2 ) THEN ! ! Meet 1 by 1 diagonal block ! ! Scale to avoid overflow when computing ! x(j) = b(j)/T(j,j) ! XJ = ABS( X( J1 ) ) TJJ = ABS( T( J1, J1 ) ) TMP = T( J1, J1 ) IF( TJJ.LT.SMIN ) THEN TMP = SMIN TJJ = SMIN INFO = 1 END IF ! IF( XJ.EQ.ZERO ) & GO TO 30 ! IF( TJJ.LT.ONE ) THEN IF( XJ.GT.BIGNUM*TJJ ) THEN REC = ONE / XJ CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF X( J1 ) = X( J1 ) / TMP XJ = ABS( X( J1 ) ) ! ! Scale x if necessary to avoid overflow when adding a ! multiple of column j1 of T. ! IF( XJ.GT.ONE ) THEN REC = ONE / XJ IF( WORK( J1 ).GT.( BIGNUM-XMAX )*REC ) THEN CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC END IF END IF IF( J1.GT.1 ) THEN CALL DAXPY( J1-1, -X( J1 ), T( 1, J1 ), 1, X, 1 ) K = IDAMAX( J1-1, X, 1 ) XMAX = ABS( X( K ) ) END IF ! ELSE ! ! Meet 2 by 2 diagonal block ! ! Call 2 by 2 linear system solve, to take ! care of possible overflow by scaling factor. ! D( 1, 1 ) = X( J1 ) D( 2, 1 ) = X( J2 ) CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, T( J1, J1 ), & LDT, ONE, ONE, D, 2, ZERO, ZERO, V, 2, & SCALOC, XNORM, IERR ) IF( IERR.NE.0 ) & INFO = 2 ! IF( SCALOC.NE.ONE ) THEN CALL DSCAL( N, SCALOC, X, 1 ) SCALE = SCALE*SCALOC END IF X( J1 ) = V( 1, 1 ) X( J2 ) = V( 2, 1 ) ! ! Scale V(1,1) (= X(J1)) and/or V(2,1) (=X(J2)) ! to avoid overflow in updating right-hand side. ! XJ = MAX( ABS( V( 1, 1 ) ), ABS( V( 2, 1 ) ) ) IF( XJ.GT.ONE ) THEN REC = ONE / XJ IF( MAX( WORK( J1 ), WORK( J2 ) ).GT. & ( BIGNUM-XMAX )*REC ) THEN CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC END IF END IF ! ! Update right-hand side ! IF( J1.GT.1 ) THEN CALL DAXPY( J1-1, -X( J1 ), T( 1, J1 ), 1, X, 1 ) CALL DAXPY( J1-1, -X( J2 ), T( 1, J2 ), 1, X, 1 ) K = IDAMAX( J1-1, X, 1 ) XMAX = ABS( X( K ) ) END IF ! END IF ! 30 CONTINUE ! ELSE ! ! Solve T'*p = scale*c ! JNEXT = 1 DO 40 J = 1, N IF( J.LT.JNEXT ) & GO TO 40 J1 = J J2 = J JNEXT = J + 1 IF( J.LT.N ) THEN IF( T( J+1, J ).NE.ZERO ) THEN J2 = J + 1 JNEXT = J + 2 END IF END IF ! IF( J1.EQ.J2 ) THEN ! ! 1 by 1 diagonal block ! ! Scale if necessary to avoid overflow in forming the ! right-hand side element by inner product. ! XJ = ABS( X( J1 ) ) IF( XMAX.GT.ONE ) THEN REC = ONE / XMAX IF( WORK( J1 ).GT.( BIGNUM-XJ )*REC ) THEN CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF ! X( J1 ) = X( J1 ) - DDOT( J1-1, T( 1, J1 ), 1, X, 1 ) ! XJ = ABS( X( J1 ) ) TJJ = ABS( T( J1, J1 ) ) TMP = T( J1, J1 ) IF( TJJ.LT.SMIN ) THEN TMP = SMIN TJJ = SMIN INFO = 1 END IF ! IF( TJJ.LT.ONE ) THEN IF( XJ.GT.BIGNUM*TJJ ) THEN REC = ONE / XJ CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF X( J1 ) = X( J1 ) / TMP XMAX = MAX( XMAX, ABS( X( J1 ) ) ) ! ELSE ! ! 2 by 2 diagonal block ! ! Scale if necessary to avoid overflow in forming the ! right-hand side elements by inner product. ! XJ = MAX( ABS( X( J1 ) ), ABS( X( J2 ) ) ) IF( XMAX.GT.ONE ) THEN REC = ONE / XMAX IF( MAX( WORK( J2 ), WORK( J1 ) ).GT.( BIGNUM-XJ )* & REC ) THEN CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF ! D( 1, 1 ) = X( J1 ) - DDOT( J1-1, T( 1, J1 ), 1, X, & 1 ) D( 2, 1 ) = X( J2 ) - DDOT( J1-1, T( 1, J2 ), 1, X, & 1 ) ! CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, T( J1, J1 ), & LDT, ONE, ONE, D, 2, ZERO, ZERO, V, 2, & SCALOC, XNORM, IERR ) IF( IERR.NE.0 ) & INFO = 2 ! IF( SCALOC.NE.ONE ) THEN CALL DSCAL( N, SCALOC, X, 1 ) SCALE = SCALE*SCALOC END IF X( J1 ) = V( 1, 1 ) X( J2 ) = V( 2, 1 ) XMAX = MAX( ABS( X( J1 ) ), ABS( X( J2 ) ), XMAX ) ! END IF 40 CONTINUE END IF ! ELSE ! SMINW = MAX( EPS*ABS( W ), SMIN ) IF( NOTRAN ) THEN ! ! Solve (T + iB)*(p+iq) = c+id ! JNEXT = N DO 70 J = N, 1, -1 IF( J.GT.JNEXT ) & GO TO 70 J1 = J J2 = J JNEXT = J - 1 IF( J.GT.1 ) THEN IF( T( J, J-1 ).NE.ZERO ) THEN J1 = J - 1 JNEXT = J - 2 END IF END IF ! IF( J1.EQ.J2 ) THEN ! ! 1 by 1 diagonal block ! ! Scale if necessary to avoid overflow in division ! Z = W IF( J1.EQ.1 ) & Z = B( 1 ) XJ = ABS( X( J1 ) ) + ABS( X( N+J1 ) ) TJJ = ABS( T( J1, J1 ) ) + ABS( Z ) TMP = T( J1, J1 ) IF( TJJ.LT.SMINW ) THEN TMP = SMINW TJJ = SMINW INFO = 1 END IF ! IF( XJ.EQ.ZERO ) & GO TO 70 ! IF( TJJ.LT.ONE ) THEN IF( XJ.GT.BIGNUM*TJJ ) THEN REC = ONE / XJ CALL DSCAL( N2, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF CALL DLADIV( X( J1 ), X( N+J1 ), TMP, Z, SR, SI ) X( J1 ) = SR X( N+J1 ) = SI XJ = ABS( X( J1 ) ) + ABS( X( N+J1 ) ) ! ! Scale x if necessary to avoid overflow when adding a ! multiple of column j1 of T. ! IF( XJ.GT.ONE ) THEN REC = ONE / XJ IF( WORK( J1 ).GT.( BIGNUM-XMAX )*REC ) THEN CALL DSCAL( N2, REC, X, 1 ) SCALE = SCALE*REC END IF END IF ! IF( J1.GT.1 ) THEN CALL DAXPY( J1-1, -X( J1 ), T( 1, J1 ), 1, X, 1 ) CALL DAXPY( J1-1, -X( N+J1 ), T( 1, J1 ), 1, & X( N+1 ), 1 ) ! X( 1 ) = X( 1 ) + B( J1 )*X( N+J1 ) X( N+1 ) = X( N+1 ) - B( J1 )*X( J1 ) ! XMAX = ZERO DO 50 K = 1, J1 - 1 XMAX = MAX( XMAX, ABS( X( K ) )+ & ABS( X( K+N ) ) ) 50 CONTINUE END IF ! ELSE ! ! Meet 2 by 2 diagonal block ! D( 1, 1 ) = X( J1 ) D( 2, 1 ) = X( J2 ) D( 1, 2 ) = X( N+J1 ) D( 2, 2 ) = X( N+J2 ) CALL DLALN2( .FALSE., 2, 2, SMINW, ONE, T( J1, J1 ), & LDT, ONE, ONE, D, 2, ZERO, -W, V, 2, & SCALOC, XNORM, IERR ) IF( IERR.NE.0 ) & INFO = 2 ! IF( SCALOC.NE.ONE ) THEN CALL DSCAL( 2*N, SCALOC, X, 1 ) SCALE = SCALOC*SCALE END IF X( J1 ) = V( 1, 1 ) X( J2 ) = V( 2, 1 ) X( N+J1 ) = V( 1, 2 ) X( N+J2 ) = V( 2, 2 ) ! ! Scale X(J1), .... to avoid overflow in ! updating right hand side. ! XJ = MAX( ABS( V( 1, 1 ) )+ABS( V( 1, 2 ) ), & ABS( V( 2, 1 ) )+ABS( V( 2, 2 ) ) ) IF( XJ.GT.ONE ) THEN REC = ONE / XJ IF( MAX( WORK( J1 ), WORK( J2 ) ).GT. & ( BIGNUM-XMAX )*REC ) THEN CALL DSCAL( N2, REC, X, 1 ) SCALE = SCALE*REC END IF END IF ! ! Update the right-hand side. ! IF( J1.GT.1 ) THEN CALL DAXPY( J1-1, -X( J1 ), T( 1, J1 ), 1, X, 1 ) CALL DAXPY( J1-1, -X( J2 ), T( 1, J2 ), 1, X, 1 ) ! CALL DAXPY( J1-1, -X( N+J1 ), T( 1, J1 ), 1, & X( N+1 ), 1 ) CALL DAXPY( J1-1, -X( N+J2 ), T( 1, J2 ), 1, & X( N+1 ), 1 ) ! X( 1 ) = X( 1 ) + B( J1 )*X( N+J1 ) + & B( J2 )*X( N+J2 ) X( N+1 ) = X( N+1 ) - B( J1 )*X( J1 ) - & B( J2 )*X( J2 ) ! XMAX = ZERO DO 60 K = 1, J1 - 1 XMAX = MAX( ABS( X( K ) )+ABS( X( K+N ) ), & XMAX ) 60 CONTINUE END IF ! END IF 70 CONTINUE ! ELSE ! ! Solve (T + iB)'*(p+iq) = c+id ! JNEXT = 1 DO 80 J = 1, N IF( J.LT.JNEXT ) & GO TO 80 J1 = J J2 = J JNEXT = J + 1 IF( J.LT.N ) THEN IF( T( J+1, J ).NE.ZERO ) THEN J2 = J + 1 JNEXT = J + 2 END IF END IF ! IF( J1.EQ.J2 ) THEN ! ! 1 by 1 diagonal block ! ! Scale if necessary to avoid overflow in forming the ! right-hand side element by inner product. ! XJ = ABS( X( J1 ) ) + ABS( X( J1+N ) ) IF( XMAX.GT.ONE ) THEN REC = ONE / XMAX IF( WORK( J1 ).GT.( BIGNUM-XJ )*REC ) THEN CALL DSCAL( N2, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF ! X( J1 ) = X( J1 ) - DDOT( J1-1, T( 1, J1 ), 1, X, 1 ) X( N+J1 ) = X( N+J1 ) - DDOT( J1-1, T( 1, J1 ), 1, & X( N+1 ), 1 ) IF( J1.GT.1 ) THEN X( J1 ) = X( J1 ) - B( J1 )*X( N+1 ) X( N+J1 ) = X( N+J1 ) + B( J1 )*X( 1 ) END IF XJ = ABS( X( J1 ) ) + ABS( X( J1+N ) ) ! Z = W IF( J1.EQ.1 ) & Z = B( 1 ) ! ! Scale if necessary to avoid overflow in ! complex division ! TJJ = ABS( T( J1, J1 ) ) + ABS( Z ) TMP = T( J1, J1 ) IF( TJJ.LT.SMINW ) THEN TMP = SMINW TJJ = SMINW INFO = 1 END IF ! IF( TJJ.LT.ONE ) THEN IF( XJ.GT.BIGNUM*TJJ ) THEN REC = ONE / XJ CALL DSCAL( N2, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF CALL DLADIV( X( J1 ), X( N+J1 ), TMP, -Z, SR, SI ) X( J1 ) = SR X( J1+N ) = SI XMAX = MAX( ABS( X( J1 ) )+ABS( X( J1+N ) ), XMAX ) ! ELSE ! ! 2 by 2 diagonal block ! ! Scale if necessary to avoid overflow in forming the ! right-hand side element by inner product. ! XJ = MAX( ABS( X( J1 ) )+ABS( X( N+J1 ) ), & ABS( X( J2 ) )+ABS( X( N+J2 ) ) ) IF( XMAX.GT.ONE ) THEN REC = ONE / XMAX IF( MAX( WORK( J1 ), WORK( J2 ) ).GT. & ( BIGNUM-XJ ) / XMAX ) THEN CALL DSCAL( N2, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF ! D( 1, 1 ) = X( J1 ) - DDOT( J1-1, T( 1, J1 ), 1, X, & 1 ) D( 2, 1 ) = X( J2 ) - DDOT( J1-1, T( 1, J2 ), 1, X, & 1 ) D( 1, 2 ) = X( N+J1 ) - DDOT( J1-1, T( 1, J1 ), 1, & X( N+1 ), 1 ) D( 2, 2 ) = X( N+J2 ) - DDOT( J1-1, T( 1, J2 ), 1, & X( N+1 ), 1 ) D( 1, 1 ) = D( 1, 1 ) - B( J1 )*X( N+1 ) D( 2, 1 ) = D( 2, 1 ) - B( J2 )*X( N+1 ) D( 1, 2 ) = D( 1, 2 ) + B( J1 )*X( 1 ) D( 2, 2 ) = D( 2, 2 ) + B( J2 )*X( 1 ) ! CALL DLALN2( .TRUE., 2, 2, SMINW, ONE, T( J1, J1 ), & LDT, ONE, ONE, D, 2, ZERO, W, V, 2, & SCALOC, XNORM, IERR ) IF( IERR.NE.0 ) & INFO = 2 ! IF( SCALOC.NE.ONE ) THEN CALL DSCAL( N2, SCALOC, X, 1 ) SCALE = SCALOC*SCALE END IF X( J1 ) = V( 1, 1 ) X( J2 ) = V( 2, 1 ) X( N+J1 ) = V( 1, 2 ) X( N+J2 ) = V( 2, 2 ) XMAX = MAX( ABS( X( J1 ) )+ABS( X( N+J1 ) ), & ABS( X( J2 ) )+ABS( X( N+J2 ) ), XMAX ) ! END IF ! 80 CONTINUE ! END IF ! END IF ! RETURN ! ! End of DLAQTR ! END SUBROUTINE DLAR1V( N, B1, BN, SIGMA, D, L, LD, LLD, GERSCH, Z, & ZTZ, MINGMA, R, ISUPPZ, WORK ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER B1, BN, N, R DOUBLE PRECISION MINGMA, SIGMA, ZTZ ! .. ! .. Array Arguments .. INTEGER ISUPPZ( * ) DOUBLE PRECISION D( * ), GERSCH( * ), L( * ), LD( * ), LLD( * ), & WORK( * ), Z( * ) ! .. ! ! Purpose ! ======= ! ! DLAR1V computes the (scaled) r-th column of the inverse of ! the sumbmatrix in rows B1 through BN of the tridiagonal matrix ! L D L^T - sigma I. The following steps accomplish this computation : ! (a) Stationary qd transform, L D L^T - sigma I = L(+) D(+) L(+)^T, ! (b) Progressive qd transform, L D L^T - sigma I = U(-) D(-) U(-)^T, ! (c) Computation of the diagonal elements of the inverse of ! L D L^T - sigma I by combining the above transforms, and choosing ! r as the index where the diagonal of the inverse is (one of the) ! largest in magnitude. ! (d) Computation of the (scaled) r-th column of the inverse using the ! twisted factorization obtained by combining the top part of the ! the stationary and the bottom part of the progressive transform. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix L D L^T. ! ! B1 (input) INTEGER ! First index of the submatrix of L D L^T. ! ! BN (input) INTEGER ! Last index of the submatrix of L D L^T. ! ! SIGMA (input) DOUBLE PRECISION ! The shift. Initially, when R = 0, SIGMA should be a good ! approximation to an eigenvalue of L D L^T. ! ! L (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) subdiagonal elements of the unit bidiagonal matrix ! L, in elements 1 to N-1. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of the diagonal matrix D. ! ! LD (input) DOUBLE PRECISION array, dimension (N-1) ! The n-1 elements L(i)*D(i). ! ! LLD (input) DOUBLE PRECISION array, dimension (N-1) ! The n-1 elements L(i)*L(i)*D(i). ! ! GERSCH (input) DOUBLE PRECISION array, dimension (2*N) ! The n Gerschgorin intervals. These are used to restrict ! the initial search for R, when R is input as 0. ! ! Z (output) DOUBLE PRECISION array, dimension (N) ! The (scaled) r-th column of the inverse. Z(R) is returned ! to be 1. ! ! ZTZ (output) DOUBLE PRECISION ! The square of the norm of Z. ! ! MINGMA (output) DOUBLE PRECISION ! The reciprocal of the largest (in magnitude) diagonal ! element of the inverse of L D L^T - sigma I. ! ! R (input/output) INTEGER ! Initially, R should be input to be 0 and is then output as ! the index where the diagonal element of the inverse is ! largest in magnitude. In later iterations, this same value ! of R should be input. ! ! ISUPPZ (output) INTEGER array, dimension (2) ! The support of the vector in Z, i.e., the vector Z is ! nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (4*N) ! ! Further Details ! =============== ! ! Based on contributions by ! Inderjit Dhillon, IBM Almaden, USA ! Osni Marques, LBNL/NERSC, USA ! ! ===================================================================== ! ! .. Parameters .. INTEGER BLKSIZ PARAMETER ( BLKSIZ = 32 ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL SAWNAN INTEGER FROM, I, INDP, INDS, INDUMN, J, R1, R2, TO DOUBLE PRECISION DMINUS, DPLUS, EPS, S, TMP ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Executable Statements .. ! EPS = DLAMCH( 'Precision' ) IF( R.EQ.0 ) THEN ! ! Eliminate the top and bottom indices from the possible values ! of R where the desired eigenvector is largest in magnitude. ! R1 = B1 DO 10 I = B1, BN IF( SIGMA.GE.GERSCH( 2*I-1 ) .OR. SIGMA.LE.GERSCH( 2*I ) ) & THEN R1 = I GO TO 20 END IF 10 CONTINUE 20 CONTINUE R2 = BN DO 30 I = BN, B1, -1 IF( SIGMA.GE.GERSCH( 2*I-1 ) .OR. SIGMA.LE.GERSCH( 2*I ) ) & THEN R2 = I GO TO 40 END IF 30 CONTINUE 40 CONTINUE ELSE R1 = R R2 = R END IF ! INDUMN = N INDS = 2*N + 1 INDP = 3*N + 1 SAWNAN = .FALSE. ! ! Compute the stationary transform (using the differential form) ! untill the index R2 ! IF( B1.EQ.1 ) THEN WORK( INDS ) = ZERO ELSE WORK( INDS ) = LLD( B1-1 ) END IF S = WORK( INDS ) - SIGMA DO 50 I = B1, R2 - 1 DPLUS = D( I ) + S WORK( I ) = LD( I ) / DPLUS WORK( INDS+I ) = S*WORK( I )*L( I ) S = WORK( INDS+I ) - SIGMA 50 CONTINUE ! IF( .NOT.( S.GT.ZERO .OR. S.LT.ONE ) ) THEN ! ! Run a slower version of the above loop if a NaN is detected ! SAWNAN = .TRUE. J = B1 + 1 60 CONTINUE IF( WORK( INDS+J ).GT.ZERO .OR. WORK( INDS+J ).LT.ONE ) THEN J = J + 1 GO TO 60 END IF WORK( INDS+J ) = LLD( J ) S = WORK( INDS+J ) - SIGMA DO 70 I = J + 1, R2 - 1 DPLUS = D( I ) + S WORK( I ) = LD( I ) / DPLUS IF( WORK( I ).EQ.ZERO ) THEN WORK( INDS+I ) = LLD( I ) ELSE WORK( INDS+I ) = S*WORK( I )*L( I ) END IF S = WORK( INDS+I ) - SIGMA 70 CONTINUE END IF WORK( INDP+BN-1 ) = D( BN ) - SIGMA DO 80 I = BN - 1, R1, -1 DMINUS = LLD( I ) + WORK( INDP+I ) TMP = D( I ) / DMINUS WORK( INDUMN+I ) = L( I )*TMP WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - SIGMA 80 CONTINUE TMP = WORK( INDP+R1-1 ) IF( .NOT.( TMP.GT.ZERO .OR. TMP.LT.ONE ) ) THEN ! ! Run a slower version of the above loop if a NaN is detected ! SAWNAN = .TRUE. J = BN - 3 90 CONTINUE IF( WORK( INDP+J ).GT.ZERO .OR. WORK( INDP+J ).LT.ONE ) THEN J = J - 1 GO TO 90 END IF WORK( INDP+J ) = D( J+1 ) - SIGMA DO 100 I = J, R1, -1 DMINUS = LLD( I ) + WORK( INDP+I ) TMP = D( I ) / DMINUS WORK( INDUMN+I ) = L( I )*TMP IF( TMP.EQ.ZERO ) THEN WORK( INDP+I-1 ) = D( I ) - SIGMA ELSE WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - SIGMA END IF 100 CONTINUE END IF ! ! Find the index (from R1 to R2) of the largest (in magnitude) ! diagonal element of the inverse ! MINGMA = WORK( INDS+R1-1 ) + WORK( INDP+R1-1 ) IF( MINGMA.EQ.ZERO ) & MINGMA = EPS*WORK( INDS+R1-1 ) R = R1 DO 110 I = R1, R2 - 1 TMP = WORK( INDS+I ) + WORK( INDP+I ) IF( TMP.EQ.ZERO ) & TMP = EPS*WORK( INDS+I ) IF( ABS( TMP ).LT.ABS( MINGMA ) ) THEN MINGMA = TMP R = I + 1 END IF 110 CONTINUE ! ! Compute the (scaled) r-th column of the inverse ! ISUPPZ( 1 ) = B1 ISUPPZ( 2 ) = BN Z( R ) = ONE ZTZ = ONE IF( .NOT.SAWNAN ) THEN FROM = R - 1 TO = MAX( R-BLKSIZ, B1 ) 120 CONTINUE IF( FROM.GE.B1 ) THEN DO 130 I = FROM, TO, -1 Z( I ) = -( WORK( I )*Z( I+1 ) ) ZTZ = ZTZ + Z( I )*Z( I ) 130 CONTINUE IF( ABS( Z( TO ) ).LE.EPS .AND. ABS( Z( TO+1 ) ).LE.EPS ) & THEN ISUPPZ( 1 ) = TO + 2 ELSE FROM = TO - 1 TO = MAX( TO-BLKSIZ, B1 ) GO TO 120 END IF END IF FROM = R + 1 TO = MIN( R+BLKSIZ, BN ) 140 CONTINUE IF( FROM.LE.BN ) THEN DO 150 I = FROM, TO Z( I ) = -( WORK( INDUMN+I-1 )*Z( I-1 ) ) ZTZ = ZTZ + Z( I )*Z( I ) 150 CONTINUE IF( ABS( Z( TO ) ).LE.EPS .AND. ABS( Z( TO-1 ) ).LE.EPS ) & THEN ISUPPZ( 2 ) = TO - 2 ELSE FROM = TO + 1 TO = MIN( TO+BLKSIZ, BN ) GO TO 140 END IF END IF ELSE DO 160 I = R - 1, B1, -1 IF( Z( I+1 ).EQ.ZERO ) THEN Z( I ) = -( LD( I+1 ) / LD( I ) )*Z( I+2 ) ELSE IF( ABS( Z( I+1 ) ).LE.EPS .AND. ABS( Z( I+2 ) ).LE. & EPS ) THEN ISUPPZ( 1 ) = I + 3 GO TO 170 ELSE Z( I ) = -( WORK( I )*Z( I+1 ) ) END IF ZTZ = ZTZ + Z( I )*Z( I ) 160 CONTINUE 170 CONTINUE DO 180 I = R, BN - 1 IF( Z( I ).EQ.ZERO ) THEN Z( I+1 ) = -( LD( I-1 ) / LD( I ) )*Z( I-1 ) ELSE IF( ABS( Z( I ) ).LE.EPS .AND. ABS( Z( I-1 ) ).LE.EPS ) & THEN ISUPPZ( 2 ) = I - 2 GO TO 190 ELSE Z( I+1 ) = -( WORK( INDUMN+I )*Z( I ) ) END IF ZTZ = ZTZ + Z( I+1 )*Z( I+1 ) 180 CONTINUE 190 CONTINUE END IF DO 200 I = B1, ISUPPZ( 1 ) - 3 Z( I ) = ZERO 200 CONTINUE DO 210 I = ISUPPZ( 2 ) + 3, BN Z( I ) = ZERO 210 CONTINUE ! RETURN ! ! End of DLAR1V ! END SUBROUTINE DLAR2V( N, X, Y, Z, INCX, C, S, INCC ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. INTEGER INCC, INCX, N ! .. ! .. Array Arguments .. DOUBLE PRECISION C( * ), S( * ), X( * ), Y( * ), Z( * ) ! .. ! ! Purpose ! ======= ! ! DLAR2V applies a vector of real plane rotations from both sides to ! a sequence of 2-by-2 real symmetric matrices, defined by the elements ! of the vectors x, y and z. For i = 1,2,...,n ! ! ( x(i) z(i) ) := ( c(i) s(i) ) ( x(i) z(i) ) ( c(i) -s(i) ) ! ( z(i) y(i) ) ( -s(i) c(i) ) ( z(i) y(i) ) ( s(i) c(i) ) ! ! Arguments ! ========= ! ! N (input) INTEGER ! The number of plane rotations to be applied. ! ! X (input/output) DOUBLE PRECISION array, ! dimension (1+(N-1)*INCX) ! The vector x. ! ! Y (input/output) DOUBLE PRECISION array, ! dimension (1+(N-1)*INCX) ! The vector y. ! ! Z (input/output) DOUBLE PRECISION array, ! dimension (1+(N-1)*INCX) ! The vector z. ! ! INCX (input) INTEGER ! The increment between elements of X, Y and Z. INCX > 0. ! ! C (input) DOUBLE PRECISION array, dimension (1+(N-1)*INCC) ! The cosines of the plane rotations. ! ! S (input) DOUBLE PRECISION array, dimension (1+(N-1)*INCC) ! The sines of the plane rotations. ! ! INCC (input) INTEGER ! The increment between elements of C and S. INCC > 0. ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER I, IC, IX DOUBLE PRECISION CI, SI, T1, T2, T3, T4, T5, T6, XI, YI, ZI ! .. ! .. Executable Statements .. ! IX = 1 IC = 1 DO 10 I = 1, N XI = X( IX ) YI = Y( IX ) ZI = Z( IX ) CI = C( IC ) SI = S( IC ) T1 = SI*ZI T2 = CI*ZI T3 = T2 - SI*XI T4 = T2 + SI*YI T5 = CI*XI + T1 T6 = CI*YI - T1 X( IX ) = CI*T5 + SI*T4 Y( IX ) = CI*T6 - SI*T3 Z( IX ) = CI*T4 - SI*T5 IX = IX + INCX IC = IC + INCC 10 CONTINUE ! ! End of DLAR2V ! RETURN END SUBROUTINE DLARF( SIDE, M, N, V, INCV, TAU, C, LDC, WORK ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER SIDE INTEGER INCV, LDC, M, N DOUBLE PRECISION TAU ! .. ! .. Array Arguments .. DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLARF applies a real elementary reflector H to a real m by n matrix ! C, from either the left or the right. H is represented in the form ! ! H = I - tau * v * v' ! ! where tau is a real scalar and v is a real vector. ! ! If tau = 0, then H is taken to be the unit matrix. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': form H * C ! = 'R': form C * H ! ! M (input) INTEGER ! The number of rows of the matrix C. ! ! N (input) INTEGER ! The number of columns of the matrix C. ! ! V (input) DOUBLE PRECISION array, dimension ! (1 + (M-1)*abs(INCV)) if SIDE = 'L' ! or (1 + (N-1)*abs(INCV)) if SIDE = 'R' ! The vector v in the representation of H. V is not used if ! TAU = 0. ! ! INCV (input) INTEGER ! The increment between elements of v. INCV <> 0. ! ! TAU (input) DOUBLE PRECISION ! The value tau in the representation of H. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the m by n matrix C. ! On exit, C is overwritten by the matrix H * C if SIDE = 'L', ! or C * H if SIDE = 'R'. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace) DOUBLE PRECISION array, dimension ! (N) if SIDE = 'L' ! or (M) if SIDE = 'R' ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. External Subroutines .. EXTERNAL DGEMV, DGER ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Executable Statements .. ! IF( LSAME( SIDE, 'L' ) ) THEN ! ! Form H * C ! IF( TAU.NE.ZERO ) THEN ! ! w := C' * v ! CALL DGEMV( 'Transpose', M, N, ONE, C, LDC, V, INCV, ZERO, & WORK, 1 ) ! ! C := C - v * w' ! CALL DGER( M, N, -TAU, V, INCV, WORK, 1, C, LDC ) END IF ELSE ! ! Form C * H ! IF( TAU.NE.ZERO ) THEN ! ! w := C * v ! CALL DGEMV( 'No transpose', M, N, ONE, C, LDC, V, INCV, & ZERO, WORK, 1 ) ! ! C := C - w * v' ! CALL DGER( M, N, -TAU, WORK, 1, V, INCV, C, LDC ) END IF END IF RETURN ! ! End of DLARF ! END SUBROUTINE DLARFB( SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, & T, LDT, C, LDC, WORK, LDWORK ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER DIRECT, SIDE, STOREV, TRANS INTEGER K, LDC, LDT, LDV, LDWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION C( LDC, * ), T( LDT, * ), V( LDV, * ), & WORK( LDWORK, * ) ! .. ! ! Purpose ! ======= ! ! DLARFB applies a real block reflector H or its transpose H' to a ! real m by n matrix C, from either the left or the right. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': apply H or H' from the Left ! = 'R': apply H or H' from the Right ! ! TRANS (input) CHARACTER*1 ! = 'N': apply H (No transpose) ! = 'T': apply H' (Transpose) ! ! DIRECT (input) CHARACTER*1 ! Indicates how H is formed from a product of elementary ! reflectors ! = 'F': H = H(1) H(2) . . . H(k) (Forward) ! = 'B': H = H(k) . . . H(2) H(1) (Backward) ! ! STOREV (input) CHARACTER*1 ! Indicates how the vectors which define the elementary ! reflectors are stored: ! = 'C': Columnwise ! = 'R': Rowwise ! ! M (input) INTEGER ! The number of rows of the matrix C. ! ! N (input) INTEGER ! The number of columns of the matrix C. ! ! K (input) INTEGER ! The order of the matrix T (= the number of elementary ! reflectors whose product defines the block reflector). ! ! V (input) DOUBLE PRECISION array, dimension ! (LDV,K) if STOREV = 'C' ! (LDV,M) if STOREV = 'R' and SIDE = 'L' ! (LDV,N) if STOREV = 'R' and SIDE = 'R' ! The matrix V. See further details. ! ! LDV (input) INTEGER ! The leading dimension of the array V. ! If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M); ! if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N); ! if STOREV = 'R', LDV >= K. ! ! T (input) DOUBLE PRECISION array, dimension (LDT,K) ! The triangular k by k matrix T in the representation of the ! block reflector. ! ! LDT (input) INTEGER ! The leading dimension of the array T. LDT >= K. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the m by n matrix C. ! On exit, C is overwritten by H*C or H'*C or C*H or C*H'. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDA >= max(1,M). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (LDWORK,K) ! ! LDWORK (input) INTEGER ! The leading dimension of the array WORK. ! If SIDE = 'L', LDWORK >= max(1,N); ! if SIDE = 'R', LDWORK >= max(1,M). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. CHARACTER TRANST INTEGER I, J ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEMM, DTRMM ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( M.LE.0 .OR. N.LE.0 ) & RETURN ! IF( LSAME( TRANS, 'N' ) ) THEN TRANST = 'T' ELSE TRANST = 'N' END IF ! IF( LSAME( STOREV, 'C' ) ) THEN ! IF( LSAME( DIRECT, 'F' ) ) THEN ! ! Let V = ( V1 ) (first K rows) ! ( V2 ) ! where V1 is unit lower triangular. ! IF( LSAME( SIDE, 'L' ) ) THEN ! ! Form H * C or H' * C where C = ( C1 ) ! ( C2 ) ! ! W := C' * V = (C1'*V1 + C2'*V2) (stored in WORK) ! ! W := C1' ! DO 10 J = 1, K CALL DCOPY( N, C( J, 1 ), LDC, WORK( 1, J ), 1 ) 10 CONTINUE ! ! W := W * V1 ! CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit', N, & K, ONE, V, LDV, WORK, LDWORK ) IF( M.GT.K ) THEN ! ! W := W + C2'*V2 ! CALL DGEMM( 'Transpose', 'No transpose', N, K, M-K, & ONE, C( K+1, 1 ), LDC, V( K+1, 1 ), LDV, & ONE, WORK, LDWORK ) END IF ! ! W := W * T' or W * T ! CALL DTRMM( 'Right', 'Upper', TRANST, 'Non-unit', N, K, & ONE, T, LDT, WORK, LDWORK ) ! ! C := C - V * W' ! IF( M.GT.K ) THEN ! ! C2 := C2 - V2 * W' ! CALL DGEMM( 'No transpose', 'Transpose', M-K, N, K, & -ONE, V( K+1, 1 ), LDV, WORK, LDWORK, ONE, & C( K+1, 1 ), LDC ) END IF ! ! W := W * V1' ! CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', N, K, & ONE, V, LDV, WORK, LDWORK ) ! ! C1 := C1 - W' ! DO 30 J = 1, K DO 20 I = 1, N C( J, I ) = C( J, I ) - WORK( I, J ) 20 CONTINUE 30 CONTINUE ! ELSE IF( LSAME( SIDE, 'R' ) ) THEN ! ! Form C * H or C * H' where C = ( C1 C2 ) ! ! W := C * V = (C1*V1 + C2*V2) (stored in WORK) ! ! W := C1 ! DO 40 J = 1, K CALL DCOPY( M, C( 1, J ), 1, WORK( 1, J ), 1 ) 40 CONTINUE ! ! W := W * V1 ! CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit', M, & K, ONE, V, LDV, WORK, LDWORK ) IF( N.GT.K ) THEN ! ! W := W + C2 * V2 ! CALL DGEMM( 'No transpose', 'No transpose', M, K, N-K, & ONE, C( 1, K+1 ), LDC, V( K+1, 1 ), LDV, & ONE, WORK, LDWORK ) END IF ! ! W := W * T or W * T' ! CALL DTRMM( 'Right', 'Upper', TRANS, 'Non-unit', M, K, & ONE, T, LDT, WORK, LDWORK ) ! ! C := C - W * V' ! IF( N.GT.K ) THEN ! ! C2 := C2 - W * V2' ! CALL DGEMM( 'No transpose', 'Transpose', M, N-K, K, & -ONE, WORK, LDWORK, V( K+1, 1 ), LDV, ONE, & C( 1, K+1 ), LDC ) END IF ! ! W := W * V1' ! CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', M, K, & ONE, V, LDV, WORK, LDWORK ) ! ! C1 := C1 - W ! DO 60 J = 1, K DO 50 I = 1, M C( I, J ) = C( I, J ) - WORK( I, J ) 50 CONTINUE 60 CONTINUE END IF ! ELSE ! ! Let V = ( V1 ) ! ( V2 ) (last K rows) ! where V2 is unit upper triangular. ! IF( LSAME( SIDE, 'L' ) ) THEN ! ! Form H * C or H' * C where C = ( C1 ) ! ( C2 ) ! ! W := C' * V = (C1'*V1 + C2'*V2) (stored in WORK) ! ! W := C2' ! DO 70 J = 1, K CALL DCOPY( N, C( M-K+J, 1 ), LDC, WORK( 1, J ), 1 ) 70 CONTINUE ! ! W := W * V2 ! CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit', N, & K, ONE, V( M-K+1, 1 ), LDV, WORK, LDWORK ) IF( M.GT.K ) THEN ! ! W := W + C1'*V1 ! CALL DGEMM( 'Transpose', 'No transpose', N, K, M-K, & ONE, C, LDC, V, LDV, ONE, WORK, LDWORK ) END IF ! ! W := W * T' or W * T ! CALL DTRMM( 'Right', 'Lower', TRANST, 'Non-unit', N, K, & ONE, T, LDT, WORK, LDWORK ) ! ! C := C - V * W' ! IF( M.GT.K ) THEN ! ! C1 := C1 - V1 * W' ! CALL DGEMM( 'No transpose', 'Transpose', M-K, N, K, & -ONE, V, LDV, WORK, LDWORK, ONE, C, LDC ) END IF ! ! W := W * V2' ! CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', N, K, & ONE, V( M-K+1, 1 ), LDV, WORK, LDWORK ) ! ! C2 := C2 - W' ! DO 90 J = 1, K DO 80 I = 1, N C( M-K+J, I ) = C( M-K+J, I ) - WORK( I, J ) 80 CONTINUE 90 CONTINUE ! ELSE IF( LSAME( SIDE, 'R' ) ) THEN ! ! Form C * H or C * H' where C = ( C1 C2 ) ! ! W := C * V = (C1*V1 + C2*V2) (stored in WORK) ! ! W := C2 ! DO 100 J = 1, K CALL DCOPY( M, C( 1, N-K+J ), 1, WORK( 1, J ), 1 ) 100 CONTINUE ! ! W := W * V2 ! CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit', M, & K, ONE, V( N-K+1, 1 ), LDV, WORK, LDWORK ) IF( N.GT.K ) THEN ! ! W := W + C1 * V1 ! CALL DGEMM( 'No transpose', 'No transpose', M, K, N-K, & ONE, C, LDC, V, LDV, ONE, WORK, LDWORK ) END IF ! ! W := W * T or W * T' ! CALL DTRMM( 'Right', 'Lower', TRANS, 'Non-unit', M, K, & ONE, T, LDT, WORK, LDWORK ) ! ! C := C - W * V' ! IF( N.GT.K ) THEN ! ! C1 := C1 - W * V1' ! CALL DGEMM( 'No transpose', 'Transpose', M, N-K, K, & -ONE, WORK, LDWORK, V, LDV, ONE, C, LDC ) END IF ! ! W := W * V2' ! CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', M, K, & ONE, V( N-K+1, 1 ), LDV, WORK, LDWORK ) ! ! C2 := C2 - W ! DO 120 J = 1, K DO 110 I = 1, M C( I, N-K+J ) = C( I, N-K+J ) - WORK( I, J ) 110 CONTINUE 120 CONTINUE END IF END IF ! ELSE IF( LSAME( STOREV, 'R' ) ) THEN ! IF( LSAME( DIRECT, 'F' ) ) THEN ! ! Let V = ( V1 V2 ) (V1: first K columns) ! where V1 is unit upper triangular. ! IF( LSAME( SIDE, 'L' ) ) THEN ! ! Form H * C or H' * C where C = ( C1 ) ! ( C2 ) ! ! W := C' * V' = (C1'*V1' + C2'*V2') (stored in WORK) ! ! W := C1' ! DO 130 J = 1, K CALL DCOPY( N, C( J, 1 ), LDC, WORK( 1, J ), 1 ) 130 CONTINUE ! ! W := W * V1' ! CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', N, K, & ONE, V, LDV, WORK, LDWORK ) IF( M.GT.K ) THEN ! ! W := W + C2'*V2' ! CALL DGEMM( 'Transpose', 'Transpose', N, K, M-K, ONE, & C( K+1, 1 ), LDC, V( 1, K+1 ), LDV, ONE, & WORK, LDWORK ) END IF ! ! W := W * T' or W * T ! CALL DTRMM( 'Right', 'Upper', TRANST, 'Non-unit', N, K, & ONE, T, LDT, WORK, LDWORK ) ! ! C := C - V' * W' ! IF( M.GT.K ) THEN ! ! C2 := C2 - V2' * W' ! CALL DGEMM( 'Transpose', 'Transpose', M-K, N, K, -ONE, & V( 1, K+1 ), LDV, WORK, LDWORK, ONE, & C( K+1, 1 ), LDC ) END IF ! ! W := W * V1 ! CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit', N, & K, ONE, V, LDV, WORK, LDWORK ) ! ! C1 := C1 - W' ! DO 150 J = 1, K DO 140 I = 1, N C( J, I ) = C( J, I ) - WORK( I, J ) 140 CONTINUE 150 CONTINUE ! ELSE IF( LSAME( SIDE, 'R' ) ) THEN ! ! Form C * H or C * H' where C = ( C1 C2 ) ! ! W := C * V' = (C1*V1' + C2*V2') (stored in WORK) ! ! W := C1 ! DO 160 J = 1, K CALL DCOPY( M, C( 1, J ), 1, WORK( 1, J ), 1 ) 160 CONTINUE ! ! W := W * V1' ! CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Unit', M, K, & ONE, V, LDV, WORK, LDWORK ) IF( N.GT.K ) THEN ! ! W := W + C2 * V2' ! CALL DGEMM( 'No transpose', 'Transpose', M, K, N-K, & ONE, C( 1, K+1 ), LDC, V( 1, K+1 ), LDV, & ONE, WORK, LDWORK ) END IF ! ! W := W * T or W * T' ! CALL DTRMM( 'Right', 'Upper', TRANS, 'Non-unit', M, K, & ONE, T, LDT, WORK, LDWORK ) ! ! C := C - W * V ! IF( N.GT.K ) THEN ! ! C2 := C2 - W * V2 ! CALL DGEMM( 'No transpose', 'No transpose', M, N-K, K, & -ONE, WORK, LDWORK, V( 1, K+1 ), LDV, ONE, & C( 1, K+1 ), LDC ) END IF ! ! W := W * V1 ! CALL DTRMM( 'Right', 'Upper', 'No transpose', 'Unit', M, & K, ONE, V, LDV, WORK, LDWORK ) ! ! C1 := C1 - W ! DO 180 J = 1, K DO 170 I = 1, M C( I, J ) = C( I, J ) - WORK( I, J ) 170 CONTINUE 180 CONTINUE ! END IF ! ELSE ! ! Let V = ( V1 V2 ) (V2: last K columns) ! where V2 is unit lower triangular. ! IF( LSAME( SIDE, 'L' ) ) THEN ! ! Form H * C or H' * C where C = ( C1 ) ! ( C2 ) ! ! W := C' * V' = (C1'*V1' + C2'*V2') (stored in WORK) ! ! W := C2' ! DO 190 J = 1, K CALL DCOPY( N, C( M-K+J, 1 ), LDC, WORK( 1, J ), 1 ) 190 CONTINUE ! ! W := W * V2' ! CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', N, K, & ONE, V( 1, M-K+1 ), LDV, WORK, LDWORK ) IF( M.GT.K ) THEN ! ! W := W + C1'*V1' ! CALL DGEMM( 'Transpose', 'Transpose', N, K, M-K, ONE, & C, LDC, V, LDV, ONE, WORK, LDWORK ) END IF ! ! W := W * T' or W * T ! CALL DTRMM( 'Right', 'Lower', TRANST, 'Non-unit', N, K, & ONE, T, LDT, WORK, LDWORK ) ! ! C := C - V' * W' ! IF( M.GT.K ) THEN ! ! C1 := C1 - V1' * W' ! CALL DGEMM( 'Transpose', 'Transpose', M-K, N, K, -ONE, & V, LDV, WORK, LDWORK, ONE, C, LDC ) END IF ! ! W := W * V2 ! CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit', N, & K, ONE, V( 1, M-K+1 ), LDV, WORK, LDWORK ) ! ! C2 := C2 - W' ! DO 210 J = 1, K DO 200 I = 1, N C( M-K+J, I ) = C( M-K+J, I ) - WORK( I, J ) 200 CONTINUE 210 CONTINUE ! ELSE IF( LSAME( SIDE, 'R' ) ) THEN ! ! Form C * H or C * H' where C = ( C1 C2 ) ! ! W := C * V' = (C1*V1' + C2*V2') (stored in WORK) ! ! W := C2 ! DO 220 J = 1, K CALL DCOPY( M, C( 1, N-K+J ), 1, WORK( 1, J ), 1 ) 220 CONTINUE ! ! W := W * V2' ! CALL DTRMM( 'Right', 'Lower', 'Transpose', 'Unit', M, K, & ONE, V( 1, N-K+1 ), LDV, WORK, LDWORK ) IF( N.GT.K ) THEN ! ! W := W + C1 * V1' ! CALL DGEMM( 'No transpose', 'Transpose', M, K, N-K, & ONE, C, LDC, V, LDV, ONE, WORK, LDWORK ) END IF ! ! W := W * T or W * T' ! CALL DTRMM( 'Right', 'Lower', TRANS, 'Non-unit', M, K, & ONE, T, LDT, WORK, LDWORK ) ! ! C := C - W * V ! IF( N.GT.K ) THEN ! ! C1 := C1 - W * V1 ! CALL DGEMM( 'No transpose', 'No transpose', M, N-K, K, & -ONE, WORK, LDWORK, V, LDV, ONE, C, LDC ) END IF ! ! W := W * V2 ! CALL DTRMM( 'Right', 'Lower', 'No transpose', 'Unit', M, & K, ONE, V( 1, N-K+1 ), LDV, WORK, LDWORK ) ! ! C1 := C1 - W ! DO 240 J = 1, K DO 230 I = 1, M C( I, N-K+J ) = C( I, N-K+J ) - WORK( I, J ) 230 CONTINUE 240 CONTINUE ! END IF ! END IF END IF ! RETURN ! ! End of DLARFB ! END SUBROUTINE DLARFG( N, ALPHA, X, INCX, TAU ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. INTEGER INCX, N DOUBLE PRECISION ALPHA, TAU ! .. ! .. Array Arguments .. DOUBLE PRECISION X( * ) ! .. ! ! Purpose ! ======= ! ! DLARFG generates a real elementary reflector H of order n, such ! that ! ! H * ( alpha ) = ( beta ), H' * H = I. ! ( x ) ( 0 ) ! ! where alpha and beta are scalars, and x is an (n-1)-element real ! vector. H is represented in the form ! ! H = I - tau * ( 1 ) * ( 1 v' ) , ! ( v ) ! ! where tau is a real scalar and v is a real (n-1)-element ! vector. ! ! If the elements of x are all zero, then tau = 0 and H is taken to be ! the unit matrix. ! ! Otherwise 1 <= tau <= 2. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the elementary reflector. ! ! ALPHA (input/output) DOUBLE PRECISION ! On entry, the value alpha. ! On exit, it is overwritten with the value beta. ! ! X (input/output) DOUBLE PRECISION array, dimension ! (1+(N-2)*abs(INCX)) ! On entry, the vector x. ! On exit, it is overwritten with the vector v. ! ! INCX (input) INTEGER ! The increment between elements of X. INCX > 0. ! ! TAU (output) DOUBLE PRECISION ! The value tau. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER J, KNT DOUBLE PRECISION BETA, RSAFMN, SAFMIN, XNORM ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH, DLAPY2, DNRM2 EXTERNAL DLAMCH, DLAPY2, DNRM2 ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, SIGN ! .. ! .. External Subroutines .. EXTERNAL DSCAL ! .. ! .. Executable Statements .. ! IF( N.LE.1 ) THEN TAU = ZERO RETURN END IF ! XNORM = DNRM2( N-1, X, INCX ) ! IF( XNORM.EQ.ZERO ) THEN ! ! H = I ! TAU = ZERO ELSE ! ! general case ! BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA ) SAFMIN = DLAMCH( 'S' ) / DLAMCH( 'E' ) IF( ABS( BETA ).LT.SAFMIN ) THEN ! ! XNORM, BETA may be inaccurate; scale X and recompute them ! RSAFMN = ONE / SAFMIN KNT = 0 10 CONTINUE KNT = KNT + 1 CALL DSCAL( N-1, RSAFMN, X, INCX ) BETA = BETA*RSAFMN ALPHA = ALPHA*RSAFMN IF( ABS( BETA ).LT.SAFMIN ) & GO TO 10 ! ! New BETA is at most 1, at least SAFMIN ! XNORM = DNRM2( N-1, X, INCX ) BETA = -SIGN( DLAPY2( ALPHA, XNORM ), ALPHA ) TAU = ( BETA-ALPHA ) / BETA CALL DSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX ) ! ! If ALPHA is subnormal, it may lose relative accuracy ! ALPHA = BETA DO 20 J = 1, KNT ALPHA = ALPHA*SAFMIN 20 CONTINUE ELSE TAU = ( BETA-ALPHA ) / BETA CALL DSCAL( N-1, ONE / ( ALPHA-BETA ), X, INCX ) ALPHA = BETA END IF END IF ! RETURN ! ! End of DLARFG ! END SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER DIRECT, STOREV INTEGER K, LDT, LDV, N ! .. ! .. Array Arguments .. DOUBLE PRECISION T( LDT, * ), TAU( * ), V( LDV, * ) ! .. ! ! Purpose ! ======= ! ! DLARFT forms the triangular factor T of a real block reflector H ! of order n, which is defined as a product of k elementary reflectors. ! ! If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; ! ! If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. ! ! If STOREV = 'C', the vector which defines the elementary reflector ! H(i) is stored in the i-th column of the array V, and ! ! H = I - V * T * V' ! ! If STOREV = 'R', the vector which defines the elementary reflector ! H(i) is stored in the i-th row of the array V, and ! ! H = I - V' * T * V ! ! Arguments ! ========= ! ! DIRECT (input) CHARACTER*1 ! Specifies the order in which the elementary reflectors are ! multiplied to form the block reflector: ! = 'F': H = H(1) H(2) . . . H(k) (Forward) ! = 'B': H = H(k) . . . H(2) H(1) (Backward) ! ! STOREV (input) CHARACTER*1 ! Specifies how the vectors which define the elementary ! reflectors are stored (see also Further Details): ! = 'C': columnwise ! = 'R': rowwise ! ! N (input) INTEGER ! The order of the block reflector H. N >= 0. ! ! K (input) INTEGER ! The order of the triangular factor T (= the number of ! elementary reflectors). K >= 1. ! ! V (input/output) DOUBLE PRECISION array, dimension ! (LDV,K) if STOREV = 'C' ! (LDV,N) if STOREV = 'R' ! The matrix V. See further details. ! ! LDV (input) INTEGER ! The leading dimension of the array V. ! If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K. ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i). ! ! T (output) DOUBLE PRECISION array, dimension (LDT,K) ! The k by k triangular factor T of the block reflector. ! If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is ! lower triangular. The rest of the array is not used. ! ! LDT (input) INTEGER ! The leading dimension of the array T. LDT >= K. ! ! Further Details ! =============== ! ! The shape of the matrix V and the storage of the vectors which define ! the H(i) is best illustrated by the following example with n = 5 and ! k = 3. The elements equal to 1 are not stored; the corresponding ! array elements are modified but restored on exit. The rest of the ! array is not used. ! ! DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': ! ! V = ( 1 ) V = ( 1 v1 v1 v1 v1 ) ! ( v1 1 ) ( 1 v2 v2 v2 ) ! ( v1 v2 1 ) ( 1 v3 v3 ) ! ( v1 v2 v3 ) ! ( v1 v2 v3 ) ! ! DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': ! ! V = ( v1 v2 v3 ) V = ( v1 v1 1 ) ! ( v1 v2 v3 ) ( v2 v2 v2 1 ) ! ( 1 v2 v3 ) ( v3 v3 v3 v3 1 ) ! ( 1 v3 ) ! ( 1 ) ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J DOUBLE PRECISION VII ! .. ! .. External Subroutines .. EXTERNAL DGEMV, DTRMV ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( LSAME( DIRECT, 'F' ) ) THEN DO 20 I = 1, K IF( TAU( I ).EQ.ZERO ) THEN ! ! H(i) = I ! DO 10 J = 1, I T( J, I ) = ZERO 10 CONTINUE ELSE ! ! general case ! VII = V( I, I ) V( I, I ) = ONE IF( LSAME( STOREV, 'C' ) ) THEN ! ! T(1:i-1,i) := - tau(i) * V(i:n,1:i-1)' * V(i:n,i) ! CALL DGEMV( 'Transpose', N-I+1, I-1, -TAU( I ), & V( I, 1 ), LDV, V( I, I ), 1, ZERO, & T( 1, I ), 1 ) ELSE ! ! T(1:i-1,i) := - tau(i) * V(1:i-1,i:n) * V(i,i:n)' ! CALL DGEMV( 'No transpose', I-1, N-I+1, -TAU( I ), & V( 1, I ), LDV, V( I, I ), LDV, ZERO, & T( 1, I ), 1 ) END IF V( I, I ) = VII ! ! T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i) ! CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T, & LDT, T( 1, I ), 1 ) T( I, I ) = TAU( I ) END IF 20 CONTINUE ELSE DO 40 I = K, 1, -1 IF( TAU( I ).EQ.ZERO ) THEN ! ! H(i) = I ! DO 30 J = I, K T( J, I ) = ZERO 30 CONTINUE ELSE ! ! general case ! IF( I.LT.K ) THEN IF( LSAME( STOREV, 'C' ) ) THEN VII = V( N-K+I, I ) V( N-K+I, I ) = ONE ! ! T(i+1:k,i) := ! - tau(i) * V(1:n-k+i,i+1:k)' * V(1:n-k+i,i) ! CALL DGEMV( 'Transpose', N-K+I, K-I, -TAU( I ), & V( 1, I+1 ), LDV, V( 1, I ), 1, ZERO, & T( I+1, I ), 1 ) V( N-K+I, I ) = VII ELSE VII = V( I, N-K+I ) V( I, N-K+I ) = ONE ! ! T(i+1:k,i) := ! - tau(i) * V(i+1:k,1:n-k+i) * V(i,1:n-k+i)' ! CALL DGEMV( 'No transpose', K-I, N-K+I, -TAU( I ), & V( I+1, 1 ), LDV, V( I, 1 ), LDV, ZERO, & T( I+1, I ), 1 ) V( I, N-K+I ) = VII END IF ! ! T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i) ! CALL DTRMV( 'Lower', 'No transpose', 'Non-unit', K-I, & T( I+1, I+1 ), LDT, T( I+1, I ), 1 ) END IF T( I, I ) = TAU( I ) END IF 40 CONTINUE END IF RETURN ! ! End of DLARFT ! END SUBROUTINE DLARFX( SIDE, M, N, V, TAU, C, LDC, WORK ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER SIDE INTEGER LDC, M, N DOUBLE PRECISION TAU ! .. ! .. Array Arguments .. DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLARFX applies a real elementary reflector H to a real m by n ! matrix C, from either the left or the right. H is represented in the ! form ! ! H = I - tau * v * v' ! ! where tau is a real scalar and v is a real vector. ! ! If tau = 0, then H is taken to be the unit matrix ! ! This version uses inline code if H has order < 11. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': form H * C ! = 'R': form C * H ! ! M (input) INTEGER ! The number of rows of the matrix C. ! ! N (input) INTEGER ! The number of columns of the matrix C. ! ! V (input) DOUBLE PRECISION array, dimension (M) if SIDE = 'L' ! or (N) if SIDE = 'R' ! The vector v in the representation of H. ! ! TAU (input) DOUBLE PRECISION ! The value tau in the representation of H. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the m by n matrix C. ! On exit, C is overwritten by the matrix H * C if SIDE = 'L', ! or C * H if SIDE = 'R'. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDA >= (1,M). ! ! WORK (workspace) DOUBLE PRECISION array, dimension ! (N) if SIDE = 'L' ! or (M) if SIDE = 'R' ! WORK is not referenced if H has order < 11. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER J DOUBLE PRECISION SUM, T1, T10, T2, T3, T4, T5, T6, T7, T8, T9, & V1, V10, V2, V3, V4, V5, V6, V7, V8, V9 ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DGEMV, DGER ! .. ! .. Executable Statements .. ! IF( TAU.EQ.ZERO ) & RETURN IF( LSAME( SIDE, 'L' ) ) THEN ! ! Form H * C, where H has order m. ! GO TO ( 10, 30, 50, 70, 90, 110, 130, 150, & 170, 190 )M ! ! Code for general M ! ! w := C'*v ! CALL DGEMV( 'Transpose', M, N, ONE, C, LDC, V, 1, ZERO, WORK, & 1 ) ! ! C := C - tau * v * w' ! CALL DGER( M, N, -TAU, V, 1, WORK, 1, C, LDC ) GO TO 410 10 CONTINUE ! ! Special code for 1 x 1 Householder ! T1 = ONE - TAU*V( 1 )*V( 1 ) DO 20 J = 1, N C( 1, J ) = T1*C( 1, J ) 20 CONTINUE GO TO 410 30 CONTINUE ! ! Special code for 2 x 2 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 DO 40 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 40 CONTINUE GO TO 410 50 CONTINUE ! ! Special code for 3 x 3 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 DO 60 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 C( 3, J ) = C( 3, J ) - SUM*T3 60 CONTINUE GO TO 410 70 CONTINUE ! ! Special code for 4 x 4 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 DO 80 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) + & V4*C( 4, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 C( 3, J ) = C( 3, J ) - SUM*T3 C( 4, J ) = C( 4, J ) - SUM*T4 80 CONTINUE GO TO 410 90 CONTINUE ! ! Special code for 5 x 5 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 DO 100 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) + & V4*C( 4, J ) + V5*C( 5, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 C( 3, J ) = C( 3, J ) - SUM*T3 C( 4, J ) = C( 4, J ) - SUM*T4 C( 5, J ) = C( 5, J ) - SUM*T5 100 CONTINUE GO TO 410 110 CONTINUE ! ! Special code for 6 x 6 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 DO 120 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) + & V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 C( 3, J ) = C( 3, J ) - SUM*T3 C( 4, J ) = C( 4, J ) - SUM*T4 C( 5, J ) = C( 5, J ) - SUM*T5 C( 6, J ) = C( 6, J ) - SUM*T6 120 CONTINUE GO TO 410 130 CONTINUE ! ! Special code for 7 x 7 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 V7 = V( 7 ) T7 = TAU*V7 DO 140 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) + & V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) + & V7*C( 7, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 C( 3, J ) = C( 3, J ) - SUM*T3 C( 4, J ) = C( 4, J ) - SUM*T4 C( 5, J ) = C( 5, J ) - SUM*T5 C( 6, J ) = C( 6, J ) - SUM*T6 C( 7, J ) = C( 7, J ) - SUM*T7 140 CONTINUE GO TO 410 150 CONTINUE ! ! Special code for 8 x 8 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 V7 = V( 7 ) T7 = TAU*V7 V8 = V( 8 ) T8 = TAU*V8 DO 160 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) + & V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) + & V7*C( 7, J ) + V8*C( 8, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 C( 3, J ) = C( 3, J ) - SUM*T3 C( 4, J ) = C( 4, J ) - SUM*T4 C( 5, J ) = C( 5, J ) - SUM*T5 C( 6, J ) = C( 6, J ) - SUM*T6 C( 7, J ) = C( 7, J ) - SUM*T7 C( 8, J ) = C( 8, J ) - SUM*T8 160 CONTINUE GO TO 410 170 CONTINUE ! ! Special code for 9 x 9 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 V7 = V( 7 ) T7 = TAU*V7 V8 = V( 8 ) T8 = TAU*V8 V9 = V( 9 ) T9 = TAU*V9 DO 180 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) + & V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) + & V7*C( 7, J ) + V8*C( 8, J ) + V9*C( 9, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 C( 3, J ) = C( 3, J ) - SUM*T3 C( 4, J ) = C( 4, J ) - SUM*T4 C( 5, J ) = C( 5, J ) - SUM*T5 C( 6, J ) = C( 6, J ) - SUM*T6 C( 7, J ) = C( 7, J ) - SUM*T7 C( 8, J ) = C( 8, J ) - SUM*T8 C( 9, J ) = C( 9, J ) - SUM*T9 180 CONTINUE GO TO 410 190 CONTINUE ! ! Special code for 10 x 10 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 V7 = V( 7 ) T7 = TAU*V7 V8 = V( 8 ) T8 = TAU*V8 V9 = V( 9 ) T9 = TAU*V9 V10 = V( 10 ) T10 = TAU*V10 DO 200 J = 1, N SUM = V1*C( 1, J ) + V2*C( 2, J ) + V3*C( 3, J ) + & V4*C( 4, J ) + V5*C( 5, J ) + V6*C( 6, J ) + & V7*C( 7, J ) + V8*C( 8, J ) + V9*C( 9, J ) + & V10*C( 10, J ) C( 1, J ) = C( 1, J ) - SUM*T1 C( 2, J ) = C( 2, J ) - SUM*T2 C( 3, J ) = C( 3, J ) - SUM*T3 C( 4, J ) = C( 4, J ) - SUM*T4 C( 5, J ) = C( 5, J ) - SUM*T5 C( 6, J ) = C( 6, J ) - SUM*T6 C( 7, J ) = C( 7, J ) - SUM*T7 C( 8, J ) = C( 8, J ) - SUM*T8 C( 9, J ) = C( 9, J ) - SUM*T9 C( 10, J ) = C( 10, J ) - SUM*T10 200 CONTINUE GO TO 410 ELSE ! ! Form C * H, where H has order n. ! GO TO ( 210, 230, 250, 270, 290, 310, 330, 350, & 370, 390 )N ! ! Code for general N ! ! w := C * v ! CALL DGEMV( 'No transpose', M, N, ONE, C, LDC, V, 1, ZERO, & WORK, 1 ) ! ! C := C - tau * w * v' ! CALL DGER( M, N, -TAU, WORK, 1, V, 1, C, LDC ) GO TO 410 210 CONTINUE ! ! Special code for 1 x 1 Householder ! T1 = ONE - TAU*V( 1 )*V( 1 ) DO 220 J = 1, M C( J, 1 ) = T1*C( J, 1 ) 220 CONTINUE GO TO 410 230 CONTINUE ! ! Special code for 2 x 2 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 DO 240 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 240 CONTINUE GO TO 410 250 CONTINUE ! ! Special code for 3 x 3 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 DO 260 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 C( J, 3 ) = C( J, 3 ) - SUM*T3 260 CONTINUE GO TO 410 270 CONTINUE ! ! Special code for 4 x 4 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 DO 280 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) + & V4*C( J, 4 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 C( J, 3 ) = C( J, 3 ) - SUM*T3 C( J, 4 ) = C( J, 4 ) - SUM*T4 280 CONTINUE GO TO 410 290 CONTINUE ! ! Special code for 5 x 5 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 DO 300 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) + & V4*C( J, 4 ) + V5*C( J, 5 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 C( J, 3 ) = C( J, 3 ) - SUM*T3 C( J, 4 ) = C( J, 4 ) - SUM*T4 C( J, 5 ) = C( J, 5 ) - SUM*T5 300 CONTINUE GO TO 410 310 CONTINUE ! ! Special code for 6 x 6 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 DO 320 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) + & V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 C( J, 3 ) = C( J, 3 ) - SUM*T3 C( J, 4 ) = C( J, 4 ) - SUM*T4 C( J, 5 ) = C( J, 5 ) - SUM*T5 C( J, 6 ) = C( J, 6 ) - SUM*T6 320 CONTINUE GO TO 410 330 CONTINUE ! ! Special code for 7 x 7 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 V7 = V( 7 ) T7 = TAU*V7 DO 340 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) + & V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) + & V7*C( J, 7 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 C( J, 3 ) = C( J, 3 ) - SUM*T3 C( J, 4 ) = C( J, 4 ) - SUM*T4 C( J, 5 ) = C( J, 5 ) - SUM*T5 C( J, 6 ) = C( J, 6 ) - SUM*T6 C( J, 7 ) = C( J, 7 ) - SUM*T7 340 CONTINUE GO TO 410 350 CONTINUE ! ! Special code for 8 x 8 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 V7 = V( 7 ) T7 = TAU*V7 V8 = V( 8 ) T8 = TAU*V8 DO 360 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) + & V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) + & V7*C( J, 7 ) + V8*C( J, 8 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 C( J, 3 ) = C( J, 3 ) - SUM*T3 C( J, 4 ) = C( J, 4 ) - SUM*T4 C( J, 5 ) = C( J, 5 ) - SUM*T5 C( J, 6 ) = C( J, 6 ) - SUM*T6 C( J, 7 ) = C( J, 7 ) - SUM*T7 C( J, 8 ) = C( J, 8 ) - SUM*T8 360 CONTINUE GO TO 410 370 CONTINUE ! ! Special code for 9 x 9 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 V7 = V( 7 ) T7 = TAU*V7 V8 = V( 8 ) T8 = TAU*V8 V9 = V( 9 ) T9 = TAU*V9 DO 380 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) + & V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) + & V7*C( J, 7 ) + V8*C( J, 8 ) + V9*C( J, 9 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 C( J, 3 ) = C( J, 3 ) - SUM*T3 C( J, 4 ) = C( J, 4 ) - SUM*T4 C( J, 5 ) = C( J, 5 ) - SUM*T5 C( J, 6 ) = C( J, 6 ) - SUM*T6 C( J, 7 ) = C( J, 7 ) - SUM*T7 C( J, 8 ) = C( J, 8 ) - SUM*T8 C( J, 9 ) = C( J, 9 ) - SUM*T9 380 CONTINUE GO TO 410 390 CONTINUE ! ! Special code for 10 x 10 Householder ! V1 = V( 1 ) T1 = TAU*V1 V2 = V( 2 ) T2 = TAU*V2 V3 = V( 3 ) T3 = TAU*V3 V4 = V( 4 ) T4 = TAU*V4 V5 = V( 5 ) T5 = TAU*V5 V6 = V( 6 ) T6 = TAU*V6 V7 = V( 7 ) T7 = TAU*V7 V8 = V( 8 ) T8 = TAU*V8 V9 = V( 9 ) T9 = TAU*V9 V10 = V( 10 ) T10 = TAU*V10 DO 400 J = 1, M SUM = V1*C( J, 1 ) + V2*C( J, 2 ) + V3*C( J, 3 ) + & V4*C( J, 4 ) + V5*C( J, 5 ) + V6*C( J, 6 ) + & V7*C( J, 7 ) + V8*C( J, 8 ) + V9*C( J, 9 ) + & V10*C( J, 10 ) C( J, 1 ) = C( J, 1 ) - SUM*T1 C( J, 2 ) = C( J, 2 ) - SUM*T2 C( J, 3 ) = C( J, 3 ) - SUM*T3 C( J, 4 ) = C( J, 4 ) - SUM*T4 C( J, 5 ) = C( J, 5 ) - SUM*T5 C( J, 6 ) = C( J, 6 ) - SUM*T6 C( J, 7 ) = C( J, 7 ) - SUM*T7 C( J, 8 ) = C( J, 8 ) - SUM*T8 C( J, 9 ) = C( J, 9 ) - SUM*T9 C( J, 10 ) = C( J, 10 ) - SUM*T10 400 CONTINUE GO TO 410 END IF 410 CONTINUE RETURN ! ! End of DLARFX ! END SUBROUTINE DLARGV( N, X, INCX, Y, INCY, C, INCC ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INCC, INCX, INCY, N ! .. ! .. Array Arguments .. DOUBLE PRECISION C( * ), X( * ), Y( * ) ! .. ! ! Purpose ! ======= ! ! DLARGV generates a vector of real plane rotations, determined by ! elements of the real vectors x and y. For i = 1,2,...,n ! ! ( c(i) s(i) ) ( x(i) ) = ( a(i) ) ! ( -s(i) c(i) ) ( y(i) ) = ( 0 ) ! ! Arguments ! ========= ! ! N (input) INTEGER ! The number of plane rotations to be generated. ! ! X (input/output) DOUBLE PRECISION array, ! dimension (1+(N-1)*INCX) ! On entry, the vector x. ! On exit, x(i) is overwritten by a(i), for i = 1,...,n. ! ! INCX (input) INTEGER ! The increment between elements of X. INCX > 0. ! ! Y (input/output) DOUBLE PRECISION array, ! dimension (1+(N-1)*INCY) ! On entry, the vector y. ! On exit, the sines of the plane rotations. ! ! INCY (input) INTEGER ! The increment between elements of Y. INCY > 0. ! ! C (output) DOUBLE PRECISION array, dimension (1+(N-1)*INCC) ! The cosines of the plane rotations. ! ! INCC (input) INTEGER ! The increment between elements of C. INCC > 0. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, IC, IX, IY DOUBLE PRECISION F, G, T, TT ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, SQRT ! .. ! .. Executable Statements .. ! IX = 1 IY = 1 IC = 1 DO 10 I = 1, N F = X( IX ) G = Y( IY ) IF( G.EQ.ZERO ) THEN C( IC ) = ONE ELSE IF( F.EQ.ZERO ) THEN C( IC ) = ZERO Y( IY ) = ONE X( IX ) = G ELSE IF( ABS( F ).GT.ABS( G ) ) THEN T = G / F TT = SQRT( ONE+T*T ) C( IC ) = ONE / TT Y( IY ) = T*C( IC ) X( IX ) = F*TT ELSE T = F / G TT = SQRT( ONE+T*T ) Y( IY ) = ONE / TT C( IC ) = T*Y( IY ) X( IX ) = G*TT END IF IC = IC + INCC IY = IY + INCY IX = IX + INCX 10 CONTINUE RETURN ! ! End of DLARGV ! END SUBROUTINE DLARNV( IDIST, ISEED, N, X ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. INTEGER IDIST, N ! .. ! .. Array Arguments .. INTEGER ISEED( 4 ) DOUBLE PRECISION X( * ) ! .. ! ! Purpose ! ======= ! ! DLARNV returns a vector of n random real numbers from a uniform or ! normal distribution. ! ! Arguments ! ========= ! ! IDIST (input) INTEGER ! Specifies the distribution of the random numbers: ! = 1: uniform (0,1) ! = 2: uniform (-1,1) ! = 3: normal (0,1) ! ! ISEED (input/output) INTEGER array, dimension (4) ! On entry, the seed of the random number generator; the array ! elements must be between 0 and 4095, and ISEED(4) must be ! odd. ! On exit, the seed is updated. ! ! N (input) INTEGER ! The number of random numbers to be generated. ! ! X (output) DOUBLE PRECISION array, dimension (N) ! The generated random numbers. ! ! Further Details ! =============== ! ! This routine calls the auxiliary routine DLARUV to generate random ! real numbers from a uniform (0,1) distribution, in batches of up to ! 128 using vectorisable code. The Box-Muller method is used to ! transform numbers from a uniform to a normal distribution. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, TWO PARAMETER ( ONE = 1.0D+0, TWO = 2.0D+0 ) INTEGER LV PARAMETER ( LV = 128 ) DOUBLE PRECISION TWOPI PARAMETER ( TWOPI = 6.2831853071795864769252867663D+0 ) ! .. ! .. Local Scalars .. INTEGER I, IL, IL2, IV ! .. ! .. Local Arrays .. DOUBLE PRECISION U( LV ) ! .. ! .. Intrinsic Functions .. INTRINSIC COS, LOG, MIN, SQRT ! .. ! .. External Subroutines .. EXTERNAL DLARUV ! .. ! .. Executable Statements .. ! DO 40 IV = 1, N, LV / 2 IL = MIN( LV / 2, N-IV+1 ) IF( IDIST.EQ.3 ) THEN IL2 = 2*IL ELSE IL2 = IL END IF ! ! Call DLARUV to generate IL2 numbers from a uniform (0,1) ! distribution (IL2 <= LV) ! CALL DLARUV( ISEED, IL2, U ) ! IF( IDIST.EQ.1 ) THEN ! ! Copy generated numbers ! DO 10 I = 1, IL X( IV+I-1 ) = U( I ) 10 CONTINUE ELSE IF( IDIST.EQ.2 ) THEN ! ! Convert generated numbers to uniform (-1,1) distribution ! DO 20 I = 1, IL X( IV+I-1 ) = TWO*U( I ) - ONE 20 CONTINUE ELSE IF( IDIST.EQ.3 ) THEN ! ! Convert generated numbers to normal (0,1) distribution ! DO 30 I = 1, IL X( IV+I-1 ) = SQRT( -TWO*LOG( U( 2*I-1 ) ) )* & COS( TWOPI*U( 2*I ) ) 30 CONTINUE END IF 40 CONTINUE RETURN ! ! End of DLARNV ! END SUBROUTINE DLARRB( N, D, L, LD, LLD, IFIRST, ILAST, SIGMA, RELTOL, & W, WGAP, WERR, WORK, IWORK, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER IFIRST, ILAST, INFO, N DOUBLE PRECISION RELTOL, SIGMA ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION D( * ), L( * ), LD( * ), LLD( * ), W( * ), & WERR( * ), WGAP( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! Given the relatively robust representation(RRR) L D L^T, DLARRB ! does ``limited'' bisection to locate the eigenvalues of L D L^T, ! W( IFIRST ) thru' W( ILAST ), to more accuracy. Intervals ! [left, right] are maintained by storing their mid-points and ! semi-widths in the arrays W and WERR respectively. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of the diagonal matrix D. ! ! L (input) DOUBLE PRECISION array, dimension (N-1) ! The n-1 subdiagonal elements of the unit bidiagonal matrix L. ! ! LD (input) DOUBLE PRECISION array, dimension (N-1) ! The n-1 elements L(i)*D(i). ! ! LLD (input) DOUBLE PRECISION array, dimension (N-1) ! The n-1 elements L(i)*L(i)*D(i). ! ! IFIRST (input) INTEGER ! The index of the first eigenvalue in the cluster. ! ! ILAST (input) INTEGER ! The index of the last eigenvalue in the cluster. ! ! SIGMA (input) DOUBLE PRECISION ! The shift used to form L D L^T (see DLARRF). ! ! RELTOL (input) DOUBLE PRECISION ! The relative tolerance. ! ! W (input/output) DOUBLE PRECISION array, dimension (N) ! On input, W( IFIRST ) thru' W( ILAST ) are estimates of the ! corresponding eigenvalues of L D L^T. ! On output, these estimates are ``refined''. ! ! WGAP (input/output) DOUBLE PRECISION array, dimension (N) ! The gaps between the eigenvalues of L D L^T. Very small ! gaps are changed on output. ! ! WERR (input/output) DOUBLE PRECISION array, dimension (N) ! On input, WERR( IFIRST ) thru' WERR( ILAST ) are the errors ! in the estimates W( IFIRST ) thru' W( ILAST ). ! On output, these are the ``refined'' errors. ! !****Reminder to Inder --- WORK is never used in this subroutine ***** ! WORK (input) DOUBLE PRECISION array, dimension (???) ! Workspace. ! ! IWORK (input) INTEGER array, dimension (2*N) ! Workspace. ! !****Reminder to Inder --- INFO is never set in this subroutine ****** ! INFO (output) INTEGER ! Error flag. ! ! Further Details ! =============== ! ! Based on contributions by ! Inderjit Dhillon, IBM Almaden, USA ! Osni Marques, LBNL/NERSC, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, TWO, HALF PARAMETER ( ZERO = 0.0D0, TWO = 2.0D0, HALF = 0.5D0 ) ! .. ! .. Local Scalars .. INTEGER CNT, I, I1, I2, INITI1, INITI2, J, K, NCNVRG, & NEIG, NINT, NRIGHT, OLNINT DOUBLE PRECISION DELTA, EPS, GAP, LEFT, MID, PERT, RIGHT, S, & THRESH, TMP, WIDTH ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Executable Statements .. ! EPS = DLAMCH( 'Precision' ) I1 = IFIRST I2 = IFIRST NEIG = ILAST - IFIRST + 1 NCNVRG = 0 THRESH = RELTOL DO 10 I = IFIRST, ILAST IWORK( I ) = 0 PERT = EPS*( ABS( SIGMA )+ABS( W( I ) ) ) WERR( I ) = WERR( I ) + PERT IF( WGAP( I ).LT.PERT ) & WGAP( I ) = PERT 10 CONTINUE DO 20 I = I1, ILAST IF( I.EQ.1 ) THEN GAP = WGAP( I ) ELSE IF( I.EQ.N ) THEN GAP = WGAP( I-1 ) ELSE GAP = MIN( WGAP( I-1 ), WGAP( I ) ) END IF IF( WERR( I ).LT.THRESH*GAP ) THEN NCNVRG = NCNVRG + 1 IWORK( I ) = 1 IF( I1.EQ.I ) & I1 = I1 + 1 ELSE I2 = I END IF 20 CONTINUE ! ! Initialize the unconverged intervals. ! I = I1 NINT = 0 RIGHT = ZERO 30 CONTINUE IF( I.LE.I2 ) THEN IF( IWORK( I ).EQ.0 ) THEN DELTA = EPS LEFT = W( I ) - WERR( I ) ! ! Do while( CNT(LEFT).GT.I-1 ) ! 40 CONTINUE IF( I.GT.I1 .AND. LEFT.LE.RIGHT ) THEN LEFT = RIGHT CNT = I - 1 ELSE S = -LEFT CNT = 0 DO 50 J = 1, N - 1 TMP = D( J ) + S S = S*( LD( J ) / TMP )*L( J ) - LEFT IF( TMP.LT.ZERO ) & CNT = CNT + 1 50 CONTINUE TMP = D( N ) + S IF( TMP.LT.ZERO ) & CNT = CNT + 1 IF( CNT.GT.I-1 ) THEN DELTA = TWO*DELTA LEFT = LEFT - ( ABS( SIGMA )+ABS( LEFT ) )*DELTA GO TO 40 END IF END IF DELTA = EPS RIGHT = W( I ) + WERR( I ) ! ! Do while( CNT(RIGHT).LT.I ) ! 60 CONTINUE S = -RIGHT CNT = 0 DO 70 J = 1, N - 1 TMP = D( J ) + S S = S*( LD( J ) / TMP )*L( J ) - RIGHT IF( TMP.LT.ZERO ) & CNT = CNT + 1 70 CONTINUE TMP = D( N ) + S IF( TMP.LT.ZERO ) & CNT = CNT + 1 IF( CNT.LT.I ) THEN DELTA = TWO*DELTA RIGHT = RIGHT + ( ABS( SIGMA )+ABS( RIGHT ) )*DELTA GO TO 60 END IF WERR( I ) = LEFT W( I ) = RIGHT IWORK( N+I ) = CNT NINT = NINT + 1 I = CNT + 1 ELSE I = I + 1 END IF GO TO 30 END IF ! ! While( NCNVRG.LT.NEIG ) ! INITI1 = I1 INITI2 = I2 80 CONTINUE IF( NCNVRG.LT.NEIG ) THEN OLNINT = NINT I = I1 DO 100 K = 1, OLNINT NRIGHT = IWORK( N+I ) IF( IWORK( I ).EQ.0 ) THEN MID = HALF*( WERR( I )+W( I ) ) S = -MID CNT = 0 DO 90 J = 1, N - 1 TMP = D( J ) + S S = S*( LD( J ) / TMP )*L( J ) - MID IF( TMP.LT.ZERO ) & CNT = CNT + 1 90 CONTINUE TMP = D( N ) + S IF( TMP.LT.ZERO ) & CNT = CNT + 1 CNT = MAX( I-1, MIN( NRIGHT, CNT ) ) IF( I.EQ.NRIGHT ) THEN IF( I.EQ.IFIRST ) THEN GAP = WERR( I+1 ) - W( I ) ELSE IF( I.EQ.ILAST ) THEN GAP = WERR( I ) - W( I-1 ) ELSE GAP = MIN( WERR( I+1 )-W( I ), WERR( I )-W( I-1 ) ) END IF WIDTH = W( I ) - MID IF( WIDTH.LT.THRESH*GAP ) THEN NCNVRG = NCNVRG + 1 IWORK( I ) = 1 IF( I1.EQ.I ) THEN I1 = I1 + 1 NINT = NINT - 1 END IF END IF END IF IF( IWORK( I ).EQ.0 ) & I2 = K IF( CNT.EQ.I-1 ) THEN WERR( I ) = MID ELSE IF( CNT.EQ.NRIGHT ) THEN W( I ) = MID ELSE IWORK( N+I ) = CNT NINT = NINT + 1 WERR( CNT+1 ) = MID W( CNT+1 ) = W( I ) W( I ) = MID I = CNT + 1 IWORK( N+I ) = NRIGHT END IF END IF I = NRIGHT + 1 100 CONTINUE NINT = NINT - OLNINT + I2 GO TO 80 END IF DO 110 I = INITI1, INITI2 W( I ) = HALF*( WERR( I )+W( I ) ) WERR( I ) = W( I ) - WERR( I ) 110 CONTINUE ! RETURN ! ! End of DLARRB ! END SUBROUTINE DLARRE( N, D, E, TOL, NSPLIT, ISPLIT, M, W, WOFF, & GERSCH, WORK, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, M, N, NSPLIT DOUBLE PRECISION TOL ! .. ! .. Array Arguments .. INTEGER ISPLIT( * ) DOUBLE PRECISION D( * ), E( * ), GERSCH( * ), W( * ), WOFF( * ), & WORK( * ) ! .. ! ! Purpose ! ======= ! ! Given the tridiagonal matrix T, DLARRE sets "small" off-diagonal ! elements to zero, and for each unreduced block T_i, it finds ! (i) the numbers sigma_i ! (ii) the base T_i - sigma_i I = L_i D_i L_i^T representations and ! (iii) eigenvalues of each L_i D_i L_i^T. ! The representations and eigenvalues found are then used by ! DSTEGR to compute the eigenvectors of a symmetric tridiagonal ! matrix. Currently, the base representations are limited to being ! positive or negative definite, and the eigenvalues of the definite ! matrices are found by the dqds algorithm (subroutine DLASQ2). As ! an added benefit, DLARRE also outputs the n Gerschgorin ! intervals for each L_i D_i L_i^T. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the n diagonal elements of the tridiagonal ! matrix T. ! On exit, the n diagonal elements of the diagonal ! matrices D_i. ! ! E (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the (n-1) subdiagonal elements of the tridiagonal ! matrix T; E(N) need not be set. ! On exit, the subdiagonal elements of the unit bidiagonal ! matrices L_i. ! ! TOL (input) DOUBLE PRECISION ! The threshold for splitting. If on input |E(i)| < TOL, then ! the matrix T is split into smaller blocks. ! ! NSPLIT (input) INTEGER ! The number of blocks T splits into. 1 <= NSPLIT <= N. ! ! ISPLIT (output) INTEGER array, dimension (2*N) ! The splitting points, at which T breaks up into submatrices. ! The first submatrix consists of rows/columns 1 to ISPLIT(1), ! the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), ! etc., and the NSPLIT-th consists of rows/columns ! ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. ! ! M (output) INTEGER ! The total number of eigenvalues (of all the L_i D_i L_i^T) ! found. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! The first M elements contain the eigenvalues. The ! eigenvalues of each of the blocks, L_i D_i L_i^T, are ! sorted in ascending order. ! ! WOFF (output) DOUBLE PRECISION array, dimension (N) ! The NSPLIT base points sigma_i. ! ! GERSCH (output) DOUBLE PRECISION array, dimension (2*N) ! The n Gerschgorin intervals. ! ! WORK (input) DOUBLE PRECISION array, dimension (4*N???) ! Workspace. ! ! INFO (output) INTEGER ! Output error code from DLASQ2 ! ! Further Details ! =============== ! ! Based on contributions by ! Inderjit Dhillon, IBM Almaden, USA ! Osni Marques, LBNL/NERSC, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO, FOUR, FOURTH PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0, & FOUR = 4.0D0, FOURTH = ONE / FOUR ) ! .. ! .. Local Scalars .. INTEGER CNT, I, IBEGIN, IEND, IN, J, JBLK, MAXCNT DOUBLE PRECISION DELTA, EPS, GL, GU, NRM, OFFD, S, SGNDEF, & SIGMA, TAU, TMP1, WIDTH ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLASQ2 ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Executable Statements .. ! INFO = 0 EPS = DLAMCH( 'Precision' ) ! ! Compute Splitting Points ! NSPLIT = 1 DO 10 I = 1, N - 1 IF( ABS( E( I ) ).LE.TOL ) THEN ISPLIT( NSPLIT ) = I NSPLIT = NSPLIT + 1 END IF 10 CONTINUE ISPLIT( NSPLIT ) = N ! IBEGIN = 1 DO 170 JBLK = 1, NSPLIT IEND = ISPLIT( JBLK ) IF( IBEGIN.EQ.IEND ) THEN W( IBEGIN ) = D( IBEGIN ) WOFF( JBLK ) = ZERO IBEGIN = IEND + 1 GO TO 170 END IF IN = IEND - IBEGIN + 1 ! ! Form the n Gerschgorin intervals ! GL = D( IBEGIN ) - ABS( E( IBEGIN ) ) GU = D( IBEGIN ) + ABS( E( IBEGIN ) ) GERSCH( 2*IBEGIN-1 ) = GL GERSCH( 2*IBEGIN ) = GU GERSCH( 2*IEND-1 ) = D( IEND ) - ABS( E( IEND-1 ) ) GERSCH( 2*IEND ) = D( IEND ) + ABS( E( IEND-1 ) ) GL = MIN( GERSCH( 2*IEND-1 ), GL ) GU = MAX( GERSCH( 2*IEND ), GU ) DO 20 I = IBEGIN + 1, IEND - 1 OFFD = ABS( E( I-1 ) ) + ABS( E( I ) ) GERSCH( 2*I-1 ) = D( I ) - OFFD GL = MIN( GERSCH( 2*I-1 ), GL ) GERSCH( 2*I ) = D( I ) + OFFD GU = MAX( GERSCH( 2*I ), GU ) 20 CONTINUE NRM = MAX( ABS( GL ), ABS( GU ) ) ! ! Find the number SIGMA where the base representation ! T - sigma I = L D L^T is to be formed. ! WIDTH = GU - GL DO 30 I = IBEGIN, IEND - 1 WORK( I ) = E( I )*E( I ) 30 CONTINUE DO 50 J = 1, 2 IF( J.EQ.1 ) THEN TAU = GL + FOURTH*WIDTH ELSE TAU = GU - FOURTH*WIDTH END IF TMP1 = D( IBEGIN ) - TAU IF( TMP1.LT.ZERO ) THEN CNT = 1 ELSE CNT = 0 END IF DO 40 I = IBEGIN + 1, IEND TMP1 = D( I ) - TAU - WORK( I-1 ) / TMP1 IF( TMP1.LT.ZERO ) & CNT = CNT + 1 40 CONTINUE IF( CNT.EQ.0 ) THEN GL = TAU ELSE IF( CNT.EQ.IN ) THEN GU = TAU END IF IF( J.EQ.1 ) THEN MAXCNT = CNT SIGMA = GL SGNDEF = ONE ELSE IF( IN-CNT.GT.MAXCNT ) THEN SIGMA = GU SGNDEF = -ONE END IF END IF 50 CONTINUE ! ! Find the base L D L^T representation ! WORK( 3*IN ) = ONE DELTA = EPS TAU = SGNDEF*NRM 60 CONTINUE SIGMA = SIGMA - DELTA*TAU WORK( 1 ) = D( IBEGIN ) - SIGMA J = IBEGIN DO 70 I = 1, IN - 1 WORK( 2*IN+I ) = ONE / WORK( 2*I-1 ) TMP1 = E( J )*WORK( 2*IN+I ) WORK( 2*I+1 ) = ( D( J+1 )-SIGMA ) - TMP1*E( J ) WORK( 2*I ) = TMP1 J = J + 1 70 CONTINUE DO 80 I = IN, 1, -1 TMP1 = SGNDEF*WORK( 2*I-1 ) IF( TMP1.LT.ZERO .OR. WORK( 2*IN+I ).EQ.ZERO .OR. .NOT. & ( TMP1.GT.ZERO .OR. TMP1.LT.ONE ) ) THEN DELTA = TWO*DELTA GO TO 60 END IF J = J - 1 80 CONTINUE ! J = IBEGIN D( IBEGIN ) = WORK( 1 ) WORK( 1 ) = ABS( WORK( 1 ) ) DO 90 I = 1, IN - 1 TMP1 = E( J ) E( J ) = WORK( 2*I ) WORK( 2*I ) = ABS( TMP1*WORK( 2*I ) ) J = J + 1 D( J ) = WORK( 2*I+1 ) WORK( 2*I+1 ) = ABS( WORK( 2*I+1 ) ) 90 CONTINUE ! CALL DLASQ2( IN, WORK, INFO ) ! TAU = SGNDEF*WORK( IN ) WORK( 3*IN ) = ONE DELTA = TWO*EPS 100 CONTINUE TAU = TAU*( ONE-DELTA ) ! S = -TAU J = IBEGIN DO 110 I = 1, IN - 1 WORK( I ) = D( J ) + S WORK( 2*IN+I ) = ONE / WORK( I ) ! WORK( N+I ) = ( E( I ) * D( I ) ) / WORK( I ) WORK( IN+I ) = ( E( J )*D( J ) )*WORK( 2*IN+I ) S = S*WORK( IN+I )*E( J ) - TAU J = J + 1 110 CONTINUE WORK( IN ) = D( IEND ) + S ! ! Checking to see if all the diagonal elements of the new ! L D L^T representation have the same sign ! DO 120 I = IN, 1, -1 TMP1 = SGNDEF*WORK( I ) IF( TMP1.LT.ZERO .OR. WORK( 2*IN+I ).EQ.ZERO .OR. .NOT. & ( TMP1.GT.ZERO .OR. TMP1.LT.ONE ) ) THEN DELTA = TWO*DELTA GO TO 100 END IF 120 CONTINUE ! SIGMA = SIGMA + TAU CALL DCOPY( IN, WORK, 1, D( IBEGIN ), 1 ) CALL DCOPY( IN-1, WORK( IN+1 ), 1, E( IBEGIN ), 1 ) WOFF( JBLK ) = SIGMA ! ! Update the n Gerschgorin intervals ! DO 130 I = IBEGIN, IEND GERSCH( 2*I-1 ) = GERSCH( 2*I-1 ) - SIGMA GERSCH( 2*I ) = GERSCH( 2*I ) - SIGMA 130 CONTINUE ! ! Compute the eigenvalues of L D L^T. ! J = IBEGIN DO 140 I = 1, IN - 1 WORK( 2*I-1 ) = ABS( D( J ) ) WORK( 2*I ) = E( J )*E( J )*WORK( 2*I-1 ) J = J + 1 140 CONTINUE WORK( 2*IN-1 ) = ABS( D( IEND ) ) ! CALL DLASQ2( IN, WORK, INFO ) ! J = IBEGIN IF( SGNDEF.GT.ZERO ) THEN DO 150 I = 1, IN W( J ) = WORK( IN-I+1 ) J = J + 1 150 CONTINUE ELSE DO 160 I = 1, IN W( J ) = -WORK( I ) J = J + 1 160 CONTINUE END IF IBEGIN = IEND + 1 170 CONTINUE M = N ! RETURN ! ! End of DLARRE ! END SUBROUTINE DLARRF( N, D, L, LD, LLD, IFIRST, ILAST, W, DPLUS, & LPLUS, WORK, IWORK, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER IFIRST, ILAST, INFO, N ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION D( * ), DPLUS( * ), L( * ), LD( * ), LLD( * ), & LPLUS( * ), W( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! Given the initial representation L D L^T and its cluster of close ! eigenvalues (in a relative measure), W( IFIRST ), W( IFIRST+1 ), ... ! W( ILAST ), DLARRF finds a new relatively robust representation ! L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the ! eigenvalues of L(+) D(+) L(+)^T is relatively isolated. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of the diagonal matrix D. ! ! L (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) subdiagonal elements of the unit bidiagonal ! matrix L. ! ! LD (input) DOUBLE PRECISION array, dimension (N-1) ! The n-1 elements L(i)*D(i). ! ! LLD (input) DOUBLE PRECISION array, dimension (N-1) ! The n-1 elements L(i)*L(i)*D(i). ! ! IFIRST (input) INTEGER ! The index of the first eigenvalue in the cluster. ! ! ILAST (input) INTEGER ! The index of the last eigenvalue in the cluster. ! ! W (input/output) DOUBLE PRECISION array, dimension (N) ! On input, the eigenvalues of L D L^T in ascending order. ! W( IFIRST ) through W( ILAST ) form the cluster of relatively ! close eigenalues. ! On output, W( IFIRST ) thru' W( ILAST ) are estimates of the ! corresponding eigenvalues of L(+) D(+) L(+)^T. ! ! SIGMA (input) DOUBLE PRECISION ! The shift used to form L(+) D(+) L(+)^T. ! ! DPLUS (output) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of the diagonal matrix D(+). ! ! LPLUS (output) DOUBLE PRECISION array, dimension (N) ! The first (n-1) elements of LPLUS contain the subdiagonal ! elements of the unit bidiagonal matrix L(+). LPLUS( N ) is ! set to SIGMA. ! ! WORK (input) DOUBLE PRECISION array, dimension (???) ! Workspace. ! ! Further Details ! =============== ! ! Based on contributions by ! Inderjit Dhillon, IBM Almaden, USA ! Osni Marques, LBNL/NERSC, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, TWO PARAMETER ( ZERO = 0.0D0, TWO = 2.0D0 ) ! .. ! .. Local Scalars .. INTEGER I DOUBLE PRECISION DELTA, EPS, S, SIGMA ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. Intrinsic Functions .. INTRINSIC ABS ! .. ! .. Executable Statements .. ! INFO = 0 EPS = DLAMCH( 'Precision' ) IF( IFIRST.EQ.1 ) THEN SIGMA = W( IFIRST ) ELSE IF( ILAST.EQ.N ) THEN SIGMA = W( ILAST ) ELSE INFO = 1 RETURN END IF ! ! Compute the new relatively robust representation (RRR) ! DELTA = TWO*EPS 10 CONTINUE IF( IFIRST.EQ.1 ) THEN SIGMA = SIGMA - ABS( SIGMA )*DELTA ELSE SIGMA = SIGMA + ABS( SIGMA )*DELTA END IF S = -SIGMA DO 20 I = 1, N - 1 DPLUS( I ) = D( I ) + S LPLUS( I ) = LD( I ) / DPLUS( I ) S = S*LPLUS( I )*L( I ) - SIGMA 20 CONTINUE DPLUS( N ) = D( N ) + S IF( IFIRST.EQ.1 ) THEN DO 30 I = 1, N IF( DPLUS( I ).LT.ZERO ) THEN DELTA = TWO*DELTA GO TO 10 END IF 30 CONTINUE ELSE DO 40 I = 1, N IF( DPLUS( I ).GT.ZERO ) THEN DELTA = TWO*DELTA GO TO 10 END IF 40 CONTINUE END IF DO 50 I = IFIRST, ILAST W( I ) = W( I ) - SIGMA 50 CONTINUE LPLUS( N ) = SIGMA ! RETURN ! ! End of DLARRF ! END SUBROUTINE DLARRV( N, D, L, ISPLIT, M, W, IBLOCK, GERSCH, TOL, Z, & LDZ, ISUPPZ, WORK, IWORK, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDZ, M, N DOUBLE PRECISION TOL ! .. ! .. Array Arguments .. INTEGER IBLOCK( * ), ISPLIT( * ), ISUPPZ( * ), & IWORK( * ) DOUBLE PRECISION D( * ), GERSCH( * ), L( * ), W( * ), WORK( * ), & Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DLARRV computes the eigenvectors of the tridiagonal matrix ! T = L D L^T given L, D and the eigenvalues of L D L^T. ! The input eigenvalues should have high relative accuracy with ! respect to the entries of L and D. The desired accuracy of the ! output can be specified by the input parameter TOL. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix. N >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the n diagonal elements of the diagonal matrix D. ! On exit, D may be overwritten. ! ! L (input/output) DOUBLE PRECISION array, dimension (N-1) ! On entry, the (n-1) subdiagonal elements of the unit ! bidiagonal matrix L in elements 1 to N-1 of L. L(N) need ! not be set. On exit, L is overwritten. ! ! ISPLIT (input) INTEGER array, dimension (N) ! The splitting points, at which T breaks up into submatrices. ! The first submatrix consists of rows/columns 1 to ! ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 ! through ISPLIT( 2 ), etc. ! ! TOL (input) DOUBLE PRECISION ! The absolute error tolerance for the ! eigenvalues/eigenvectors. ! Errors in the input eigenvalues must be bounded by TOL. ! The eigenvectors output have residual norms ! bounded by TOL, and the dot products between different ! eigenvectors are bounded by TOL. TOL must be at least ! N*EPS*|T|, where EPS is the machine precision and |T| is ! the 1-norm of the tridiagonal matrix. ! ! M (input) INTEGER ! The total number of eigenvalues found. 0 <= M <= N. ! If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. ! ! W (input) DOUBLE PRECISION array, dimension (N) ! The first M elements of W contain the eigenvalues for ! which eigenvectors are to be computed. The eigenvalues ! should be grouped by split-off block and ordered from ! smallest to largest within the block ( The output array ! W from DLARRE is expected here ). ! Errors in W must be bounded by TOL (see above). ! ! IBLOCK (input) INTEGER array, dimension (N) ! The submatrix indices associated with the corresponding ! eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to ! the first submatrix from the top, =2 if W(i) belongs to ! the second submatrix, etc. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) ) ! If JOBZ = 'V', then if INFO = 0, the first M columns of Z ! contain the orthonormal eigenvectors of the matrix T ! corresponding to the selected eigenvalues, with the i-th ! column of Z holding the eigenvector associated with W(i). ! If JOBZ = 'N', then Z is not referenced. ! Note: the user must ensure that at least max(1,M) columns are ! supplied in the array Z; if RANGE = 'V', the exact value of M ! is not known in advance and an upper bound must be used. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) ) ! The support of the eigenvectors in Z, i.e., the indices ! indicating the nonzero elements in Z. The i-th eigenvector ! is nonzero only in elements ISUPPZ( 2*i-1 ) through ! ISUPPZ( 2*i ). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (13*N) ! ! IWORK (workspace) INTEGER array, dimension (6*N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = 1, internal error in DLARRB ! if INFO = 2, internal error in DSTEIN ! ! Further Details ! =============== ! ! Based on contributions by ! Inderjit Dhillon, IBM Almaden, USA ! Osni Marques, LBNL/NERSC, USA ! ! ===================================================================== ! ! .. Parameters .. INTEGER MGSSIZ PARAMETER ( MGSSIZ = 20 ) DOUBLE PRECISION ZERO, ONE, FOUR PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, FOUR = 4.0D0 ) ! .. ! .. Local Scalars .. LOGICAL MGSCLS INTEGER I, IBEGIN, IEND, IINDC1, IINDC2, IINDR, IINDWK, & IINFO, IM, IN, INDERR, INDGAP, INDLD, INDLLD, & INDWRK, ITER, ITMP1, ITMP2, J, JBLK, K, KTOT, & LSBDPT, MAXITR, NCLUS, NDEPTH, NDONE, NEWCLS, & NEWFRS, NEWFTT, NEWLST, NEWSIZ, NSPLIT, OLDCLS, & OLDFST, OLDIEN, OLDLST, OLDNCL, P, Q DOUBLE PRECISION EPS, GAP, LAMBDA, MGSTOL, MINGMA, MINRGP, & NRMINV, RELGAP, RELTOL, RESID, RQCORR, SIGMA, & TMP1, ZTZ ! .. ! .. External Functions .. DOUBLE PRECISION DDOT, DLAMCH, DNRM2 EXTERNAL DDOT, DLAMCH, DNRM2 ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DLAR1V, DLARRB, DLARRF, DLASET, & DSCAL, DSTEIN ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX, MIN, SQRT ! .. ! .. Local Arrays .. INTEGER TEMP( 1 ) ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INDERR = N + 1 INDLD = 2*N INDLLD = 3*N INDGAP = 4*N INDWRK = 5*N + 1 ! IINDR = N IINDC1 = 2*N IINDC2 = 3*N IINDWK = 4*N + 1 ! EPS = DLAMCH( 'Precision' ) ! DO 10 I = 1, 2*N IWORK( I ) = 0 10 CONTINUE DO 20 I = 1, M WORK( INDERR+I-1 ) = EPS*ABS( W( I ) ) 20 CONTINUE CALL DLASET( 'Full', N, N, ZERO, ZERO, Z, LDZ ) MGSTOL = 5.0D0*EPS ! NSPLIT = IBLOCK( M ) IBEGIN = 1 DO 170 JBLK = 1, NSPLIT IEND = ISPLIT( JBLK ) ! ! Find the eigenvectors of the submatrix indexed IBEGIN ! through IEND. ! IF( IBEGIN.EQ.IEND ) THEN Z( IBEGIN, IBEGIN ) = ONE ISUPPZ( 2*IBEGIN-1 ) = IBEGIN ISUPPZ( 2*IBEGIN ) = IBEGIN IBEGIN = IEND + 1 GO TO 170 END IF OLDIEN = IBEGIN - 1 IN = IEND - OLDIEN RELTOL = MIN( 1.0D-2, ONE / DBLE( IN ) ) IM = IN CALL DCOPY( IM, W( IBEGIN ), 1, WORK, 1 ) DO 30 I = 1, IN - 1 WORK( INDGAP+I ) = WORK( I+1 ) - WORK( I ) 30 CONTINUE WORK( INDGAP+IN ) = MAX( ABS( WORK( IN ) ), EPS ) NDONE = 0 ! NDEPTH = 0 LSBDPT = 1 NCLUS = 1 IWORK( IINDC1+1 ) = 1 IWORK( IINDC1+2 ) = IN ! ! While( NDONE.LT.IM ) do ! 40 CONTINUE IF( NDONE.LT.IM ) THEN OLDNCL = NCLUS NCLUS = 0 LSBDPT = 1 - LSBDPT DO 150 I = 1, OLDNCL IF( LSBDPT.EQ.0 ) THEN OLDCLS = IINDC1 NEWCLS = IINDC2 ELSE OLDCLS = IINDC2 NEWCLS = IINDC1 END IF ! ! If NDEPTH > 1, retrieve the relatively robust ! representation (RRR) and perform limited bisection ! (if necessary) to get approximate eigenvalues. ! J = OLDCLS + 2*I OLDFST = IWORK( J-1 ) OLDLST = IWORK( J ) IF( NDEPTH.GT.0 ) THEN J = OLDIEN + OLDFST CALL DCOPY( IN, Z( IBEGIN, J ), 1, D( IBEGIN ), 1 ) CALL DCOPY( IN, Z( IBEGIN, J+1 ), 1, L( IBEGIN ), 1 ) SIGMA = L( IEND ) END IF K = IBEGIN DO 50 J = 1, IN - 1 WORK( INDLD+J ) = D( K )*L( K ) WORK( INDLLD+J ) = WORK( INDLD+J )*L( K ) K = K + 1 50 CONTINUE IF( NDEPTH.GT.0 ) THEN CALL DLARRB( IN, D( IBEGIN ), L( IBEGIN ), & WORK( INDLD+1 ), WORK( INDLLD+1 ), & OLDFST, OLDLST, SIGMA, RELTOL, WORK, & WORK( INDGAP+1 ), WORK( INDERR ), & WORK( INDWRK ), IWORK( IINDWK ), IINFO ) IF( IINFO.NE.0 ) THEN INFO = 1 RETURN END IF END IF ! ! Classify eigenvalues of the current representation (RRR) ! as (i) isolated, (ii) loosely clustered or (iii) tightly ! clustered ! NEWFRS = OLDFST DO 140 J = OLDFST, OLDLST IF( J.EQ.OLDLST .OR. WORK( INDGAP+J ).GE.RELTOL* & ABS( WORK( J ) ) ) THEN NEWLST = J ELSE ! ! continue (to the next loop) ! RELGAP = WORK( INDGAP+J ) / ABS( WORK( J ) ) IF( J.EQ.NEWFRS ) THEN MINRGP = RELGAP ELSE MINRGP = MIN( MINRGP, RELGAP ) END IF GO TO 140 END IF NEWSIZ = NEWLST - NEWFRS + 1 MAXITR = 10 NEWFTT = OLDIEN + NEWFRS IF( NEWSIZ.GT.1 ) THEN MGSCLS = NEWSIZ.LE.MGSSIZ .AND. MINRGP.GE.MGSTOL IF( .NOT.MGSCLS ) THEN CALL DLARRF( IN, D( IBEGIN ), L( IBEGIN ), & WORK( INDLD+1 ), WORK( INDLLD+1 ), & NEWFRS, NEWLST, WORK, & Z( IBEGIN, NEWFTT ), & Z( IBEGIN, NEWFTT+1 ), & WORK( INDWRK ), IWORK( IINDWK ), & INFO ) IF( INFO.EQ.0 ) THEN NCLUS = NCLUS + 1 K = NEWCLS + 2*NCLUS IWORK( K-1 ) = NEWFRS IWORK( K ) = NEWLST ELSE INFO = 0 IF( MINRGP.GE.MGSTOL ) THEN MGSCLS = .TRUE. ELSE ! ! Call DSTEIN to process this tight cluster. ! This happens only if MINRGP <= MGSTOL ! and DLARRF returns INFO = 1. The latter ! means that a new RRR to "break" the ! cluster could not be found. ! WORK( INDWRK ) = D( IBEGIN ) DO 60 K = 1, IN - 1 WORK( INDWRK+K ) = D( IBEGIN+K ) + & WORK( INDLLD+K ) 60 CONTINUE DO 70 K = 1, NEWSIZ IWORK( IINDWK+K-1 ) = 1 70 CONTINUE DO 80 K = NEWFRS, NEWLST ISUPPZ( 2*( IBEGIN+K )-3 ) = 1 ISUPPZ( 2*( IBEGIN+K )-2 ) = IN 80 CONTINUE TEMP( 1 ) = IN CALL DSTEIN( IN, WORK( INDWRK ), & WORK( INDLD+1 ), NEWSIZ, & WORK( NEWFRS ), & IWORK( IINDWK ), TEMP( 1 ), & Z( IBEGIN, NEWFTT ), LDZ, & WORK( INDWRK+IN ), & IWORK( IINDWK+IN ), & IWORK( IINDWK+2*IN ), IINFO ) IF( IINFO.NE.0 ) THEN INFO = 2 RETURN END IF NDONE = NDONE + NEWSIZ END IF END IF END IF ELSE MGSCLS = .FALSE. END IF IF( NEWSIZ.EQ.1 .OR. MGSCLS ) THEN KTOT = NEWFTT DO 100 K = NEWFRS, NEWLST ITER = 0 90 CONTINUE LAMBDA = WORK( K ) CALL DLAR1V( IN, 1, IN, LAMBDA, D( IBEGIN ), & L( IBEGIN ), WORK( INDLD+1 ), & WORK( INDLLD+1 ), & GERSCH( 2*OLDIEN+1 ), & Z( IBEGIN, KTOT ), ZTZ, MINGMA, & IWORK( IINDR+KTOT ), & ISUPPZ( 2*KTOT-1 ), & WORK( INDWRK ) ) TMP1 = ONE / ZTZ NRMINV = SQRT( TMP1 ) RESID = ABS( MINGMA )*NRMINV RQCORR = MINGMA*TMP1 IF( K.EQ.IN ) THEN GAP = WORK( INDGAP+K-1 ) ELSE IF( K.EQ.1 ) THEN GAP = WORK( INDGAP+K ) ELSE GAP = MIN( WORK( INDGAP+K-1 ), & WORK( INDGAP+K ) ) END IF ITER = ITER + 1 IF( RESID.GT.TOL*GAP .AND. ABS( RQCORR ).GT. & FOUR*EPS*ABS( LAMBDA ) ) THEN WORK( K ) = LAMBDA + RQCORR IF( ITER.LT.MAXITR ) THEN GO TO 90 END IF END IF IWORK( KTOT ) = 1 IF( NEWSIZ.EQ.1 ) & NDONE = NDONE + 1 CALL DSCAL( IN, NRMINV, Z( IBEGIN, KTOT ), 1 ) KTOT = KTOT + 1 100 CONTINUE IF( NEWSIZ.GT.1 ) THEN ITMP1 = ISUPPZ( 2*NEWFTT-1 ) ITMP2 = ISUPPZ( 2*NEWFTT ) KTOT = OLDIEN + NEWLST DO 120 P = NEWFTT + 1, KTOT DO 110 Q = NEWFTT, P - 1 TMP1 = -DDOT( IN, Z( IBEGIN, P ), 1, & Z( IBEGIN, Q ), 1 ) CALL DAXPY( IN, TMP1, Z( IBEGIN, Q ), 1, & Z( IBEGIN, P ), 1 ) 110 CONTINUE TMP1 = ONE / DNRM2( IN, Z( IBEGIN, P ), 1 ) CALL DSCAL( IN, TMP1, Z( IBEGIN, P ), 1 ) ITMP1 = MIN( ITMP1, ISUPPZ( 2*P-1 ) ) ITMP2 = MAX( ITMP2, ISUPPZ( 2*P ) ) 120 CONTINUE DO 130 P = NEWFTT, KTOT ISUPPZ( 2*P-1 ) = ITMP1 ISUPPZ( 2*P ) = ITMP2 130 CONTINUE NDONE = NDONE + NEWSIZ END IF END IF NEWFRS = J + 1 140 CONTINUE 150 CONTINUE NDEPTH = NDEPTH + 1 GO TO 40 END IF J = 2*IBEGIN DO 160 I = IBEGIN, IEND ISUPPZ( J-1 ) = ISUPPZ( J-1 ) + OLDIEN ISUPPZ( J ) = ISUPPZ( J ) + OLDIEN J = J + 2 160 CONTINUE IBEGIN = IEND + 1 170 CONTINUE ! RETURN ! ! End of DLARRV ! END SUBROUTINE DLARTG( F, G, CS, SN, R ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. DOUBLE PRECISION CS, F, G, R, SN ! .. ! ! Purpose ! ======= ! ! DLARTG generate a plane rotation so that ! ! [ CS SN ] . [ F ] = [ R ] where CS**2 + SN**2 = 1. ! [ -SN CS ] [ G ] [ 0 ] ! ! This is a slower, more accurate version of the BLAS1 routine DROTG, ! with the following other differences: ! F and G are unchanged on return. ! If G=0, then CS=1 and SN=0. ! If F=0 and (G .ne. 0), then CS=0 and SN=1 without doing any ! floating point operations (saves work in DBDSQR when ! there are zeros on the diagonal). ! ! If F exceeds G in magnitude, CS will be positive. ! ! Arguments ! ========= ! ! F (input) DOUBLE PRECISION ! The first component of vector to be rotated. ! ! G (input) DOUBLE PRECISION ! The second component of vector to be rotated. ! ! CS (output) DOUBLE PRECISION ! The cosine of the rotation. ! ! SN (output) DOUBLE PRECISION ! The sine of the rotation. ! ! R (output) DOUBLE PRECISION ! The nonzero component of the rotated vector. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D0 ) ! .. ! .. Local Scalars .. LOGICAL FIRST INTEGER COUNT, I DOUBLE PRECISION EPS, F1, G1, SAFMIN, SAFMN2, SAFMX2, SCALE ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, INT, LOG, MAX, SQRT ! .. ! .. Save statement .. SAVE FIRST, SAFMX2, SAFMIN, SAFMN2 ! .. ! .. Data statements .. DATA FIRST / .TRUE. / ! .. ! .. Executable Statements .. ! IF( FIRST ) THEN FIRST = .FALSE. SAFMIN = DLAMCH( 'S' ) EPS = DLAMCH( 'E' ) SAFMN2 = DLAMCH( 'B' )**INT( LOG( SAFMIN / EPS ) / & LOG( DLAMCH( 'B' ) ) / TWO ) SAFMX2 = ONE / SAFMN2 END IF IF( G.EQ.ZERO ) THEN CS = ONE SN = ZERO R = F ELSE IF( F.EQ.ZERO ) THEN CS = ZERO SN = ONE R = G ELSE F1 = F G1 = G SCALE = MAX( ABS( F1 ), ABS( G1 ) ) IF( SCALE.GE.SAFMX2 ) THEN COUNT = 0 10 CONTINUE COUNT = COUNT + 1 F1 = F1*SAFMN2 G1 = G1*SAFMN2 SCALE = MAX( ABS( F1 ), ABS( G1 ) ) IF( SCALE.GE.SAFMX2 ) & GO TO 10 R = SQRT( F1**2+G1**2 ) CS = F1 / R SN = G1 / R DO 20 I = 1, COUNT R = R*SAFMX2 20 CONTINUE ELSE IF( SCALE.LE.SAFMN2 ) THEN COUNT = 0 30 CONTINUE COUNT = COUNT + 1 F1 = F1*SAFMX2 G1 = G1*SAFMX2 SCALE = MAX( ABS( F1 ), ABS( G1 ) ) IF( SCALE.LE.SAFMN2 ) & GO TO 30 R = SQRT( F1**2+G1**2 ) CS = F1 / R SN = G1 / R DO 40 I = 1, COUNT R = R*SAFMN2 40 CONTINUE ELSE R = SQRT( F1**2+G1**2 ) CS = F1 / R SN = G1 / R END IF IF( ABS( F ).GT.ABS( G ) .AND. CS.LT.ZERO ) THEN CS = -CS SN = -SN R = -R END IF END IF RETURN ! ! End of DLARTG ! END SUBROUTINE DLARTV( N, X, INCX, Y, INCY, C, S, INCC ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. INTEGER INCC, INCX, INCY, N ! .. ! .. Array Arguments .. DOUBLE PRECISION C( * ), S( * ), X( * ), Y( * ) ! .. ! ! Purpose ! ======= ! ! DLARTV applies a vector of real plane rotations to elements of the ! real vectors x and y. For i = 1,2,...,n ! ! ( x(i) ) := ( c(i) s(i) ) ( x(i) ) ! ( y(i) ) ( -s(i) c(i) ) ( y(i) ) ! ! Arguments ! ========= ! ! N (input) INTEGER ! The number of plane rotations to be applied. ! ! X (input/output) DOUBLE PRECISION array, ! dimension (1+(N-1)*INCX) ! The vector x. ! ! INCX (input) INTEGER ! The increment between elements of X. INCX > 0. ! ! Y (input/output) DOUBLE PRECISION array, ! dimension (1+(N-1)*INCY) ! The vector y. ! ! INCY (input) INTEGER ! The increment between elements of Y. INCY > 0. ! ! C (input) DOUBLE PRECISION array, dimension (1+(N-1)*INCC) ! The cosines of the plane rotations. ! ! S (input) DOUBLE PRECISION array, dimension (1+(N-1)*INCC) ! The sines of the plane rotations. ! ! INCC (input) INTEGER ! The increment between elements of C and S. INCC > 0. ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER I, IC, IX, IY DOUBLE PRECISION XI, YI ! .. ! .. Executable Statements .. ! IX = 1 IY = 1 IC = 1 DO 10 I = 1, N XI = X( IX ) YI = Y( IY ) X( IX ) = C( IC )*XI + S( IC )*YI Y( IY ) = C( IC )*YI - S( IC )*XI IX = IX + INCX IY = IY + INCY IC = IC + INCC 10 CONTINUE RETURN ! ! End of DLARTV ! END SUBROUTINE DLARUV( ISEED, N, X ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. INTEGER N ! .. ! .. Array Arguments .. INTEGER ISEED( 4 ) DOUBLE PRECISION X( N ) ! .. ! ! Purpose ! ======= ! ! DLARUV returns a vector of n random real numbers from a uniform (0,1) ! distribution (n <= 128). ! ! This is an auxiliary routine called by DLARNV and ZLARNV. ! ! Arguments ! ========= ! ! ISEED (input/output) INTEGER array, dimension (4) ! On entry, the seed of the random number generator; the array ! elements must be between 0 and 4095, and ISEED(4) must be ! odd. ! On exit, the seed is updated. ! ! N (input) INTEGER ! The number of random numbers to be generated. N <= 128. ! ! X (output) DOUBLE PRECISION array, dimension (N) ! The generated random numbers. ! ! Further Details ! =============== ! ! This routine uses a multiplicative congruential method with modulus ! 2**48 and multiplier 33952834046453 (see G.S.Fishman, ! 'Multiplicative congruential random number generators with modulus ! 2**b: an exhaustive analysis for b = 32 and a partial analysis for ! b = 48', Math. Comp. 189, pp 331-344, 1990). ! ! 48-bit integers are stored in 4 integer array elements with 12 bits ! per element. Hence the routine is portable across machines with ! integers of 32 bits or more. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D0 ) INTEGER LV, IPW2 DOUBLE PRECISION R PARAMETER ( LV = 128, IPW2 = 4096, R = ONE / IPW2 ) ! .. ! .. Local Scalars .. INTEGER I, I1, I2, I3, I4, IT1, IT2, IT3, IT4, J ! .. ! .. Local Arrays .. INTEGER MM( LV, 4 ) ! .. ! .. Intrinsic Functions .. INTRINSIC DBLE, MIN, MOD ! .. ! .. Data statements .. DATA ( MM( 1, J ), J = 1, 4 ) / 494, 322, 2508, & 2549 / DATA ( MM( 2, J ), J = 1, 4 ) / 2637, 789, 3754, & 1145 / DATA ( MM( 3, J ), J = 1, 4 ) / 255, 1440, 1766, & 2253 / DATA ( MM( 4, J ), J = 1, 4 ) / 2008, 752, 3572, & 305 / DATA ( MM( 5, J ), J = 1, 4 ) / 1253, 2859, 2893, & 3301 / DATA ( MM( 6, J ), J = 1, 4 ) / 3344, 123, 307, & 1065 / DATA ( MM( 7, J ), J = 1, 4 ) / 4084, 1848, 1297, & 3133 / DATA ( MM( 8, J ), J = 1, 4 ) / 1739, 643, 3966, & 2913 / DATA ( MM( 9, J ), J = 1, 4 ) / 3143, 2405, 758, & 3285 / DATA ( MM( 10, J ), J = 1, 4 ) / 3468, 2638, 2598, & 1241 / DATA ( MM( 11, J ), J = 1, 4 ) / 688, 2344, 3406, & 1197 / DATA ( MM( 12, J ), J = 1, 4 ) / 1657, 46, 2922, & 3729 / DATA ( MM( 13, J ), J = 1, 4 ) / 1238, 3814, 1038, & 2501 / DATA ( MM( 14, J ), J = 1, 4 ) / 3166, 913, 2934, & 1673 / DATA ( MM( 15, J ), J = 1, 4 ) / 1292, 3649, 2091, & 541 / DATA ( MM( 16, J ), J = 1, 4 ) / 3422, 339, 2451, & 2753 / DATA ( MM( 17, J ), J = 1, 4 ) / 1270, 3808, 1580, & 949 / DATA ( MM( 18, J ), J = 1, 4 ) / 2016, 822, 1958, & 2361 / DATA ( MM( 19, J ), J = 1, 4 ) / 154, 2832, 2055, & 1165 / DATA ( MM( 20, J ), J = 1, 4 ) / 2862, 3078, 1507, & 4081 / DATA ( MM( 21, J ), J = 1, 4 ) / 697, 3633, 1078, & 2725 / DATA ( MM( 22, J ), J = 1, 4 ) / 1706, 2970, 3273, & 3305 / DATA ( MM( 23, J ), J = 1, 4 ) / 491, 637, 17, & 3069 / DATA ( MM( 24, J ), J = 1, 4 ) / 931, 2249, 854, & 3617 / DATA ( MM( 25, J ), J = 1, 4 ) / 1444, 2081, 2916, & 3733 / DATA ( MM( 26, J ), J = 1, 4 ) / 444, 4019, 3971, & 409 / DATA ( MM( 27, J ), J = 1, 4 ) / 3577, 1478, 2889, & 2157 / DATA ( MM( 28, J ), J = 1, 4 ) / 3944, 242, 3831, & 1361 / DATA ( MM( 29, J ), J = 1, 4 ) / 2184, 481, 2621, & 3973 / DATA ( MM( 30, J ), J = 1, 4 ) / 1661, 2075, 1541, & 1865 / DATA ( MM( 31, J ), J = 1, 4 ) / 3482, 4058, 893, & 2525 / DATA ( MM( 32, J ), J = 1, 4 ) / 657, 622, 736, & 1409 / DATA ( MM( 33, J ), J = 1, 4 ) / 3023, 3376, 3992, & 3445 / DATA ( MM( 34, J ), J = 1, 4 ) / 3618, 812, 787, & 3577 / DATA ( MM( 35, J ), J = 1, 4 ) / 1267, 234, 2125, & 77 / DATA ( MM( 36, J ), J = 1, 4 ) / 1828, 641, 2364, & 3761 / DATA ( MM( 37, J ), J = 1, 4 ) / 164, 4005, 2460, & 2149 / DATA ( MM( 38, J ), J = 1, 4 ) / 3798, 1122, 257, & 1449 / DATA ( MM( 39, J ), J = 1, 4 ) / 3087, 3135, 1574, & 3005 / DATA ( MM( 40, J ), J = 1, 4 ) / 2400, 2640, 3912, & 225 / DATA ( MM( 41, J ), J = 1, 4 ) / 2870, 2302, 1216, & 85 / DATA ( MM( 42, J ), J = 1, 4 ) / 3876, 40, 3248, & 3673 / DATA ( MM( 43, J ), J = 1, 4 ) / 1905, 1832, 3401, & 3117 / DATA ( MM( 44, J ), J = 1, 4 ) / 1593, 2247, 2124, & 3089 / DATA ( MM( 45, J ), J = 1, 4 ) / 1797, 2034, 2762, & 1349 / DATA ( MM( 46, J ), J = 1, 4 ) / 1234, 2637, 149, & 2057 / DATA ( MM( 47, J ), J = 1, 4 ) / 3460, 1287, 2245, & 413 / DATA ( MM( 48, J ), J = 1, 4 ) / 328, 1691, 166, & 65 / DATA ( MM( 49, J ), J = 1, 4 ) / 2861, 496, 466, & 1845 / DATA ( MM( 50, J ), J = 1, 4 ) / 1950, 1597, 4018, & 697 / DATA ( MM( 51, J ), J = 1, 4 ) / 617, 2394, 1399, & 3085 / DATA ( MM( 52, J ), J = 1, 4 ) / 2070, 2584, 190, & 3441 / DATA ( MM( 53, J ), J = 1, 4 ) / 3331, 1843, 2879, & 1573 / DATA ( MM( 54, J ), J = 1, 4 ) / 769, 336, 153, & 3689 / DATA ( MM( 55, J ), J = 1, 4 ) / 1558, 1472, 2320, & 2941 / DATA ( MM( 56, J ), J = 1, 4 ) / 2412, 2407, 18, & 929 / DATA ( MM( 57, J ), J = 1, 4 ) / 2800, 433, 712, & 533 / DATA ( MM( 58, J ), J = 1, 4 ) / 189, 2096, 2159, & 2841 / DATA ( MM( 59, J ), J = 1, 4 ) / 287, 1761, 2318, & 4077 / DATA ( MM( 60, J ), J = 1, 4 ) / 2045, 2810, 2091, & 721 / DATA ( MM( 61, J ), J = 1, 4 ) / 1227, 566, 3443, & 2821 / DATA ( MM( 62, J ), J = 1, 4 ) / 2838, 442, 1510, & 2249 / DATA ( MM( 63, J ), J = 1, 4 ) / 209, 41, 449, & 2397 / DATA ( MM( 64, J ), J = 1, 4 ) / 2770, 1238, 1956, & 2817 / DATA ( MM( 65, J ), J = 1, 4 ) / 3654, 1086, 2201, & 245 / DATA ( MM( 66, J ), J = 1, 4 ) / 3993, 603, 3137, & 1913 / DATA ( MM( 67, J ), J = 1, 4 ) / 192, 840, 3399, & 1997 / DATA ( MM( 68, J ), J = 1, 4 ) / 2253, 3168, 1321, & 3121 / DATA ( MM( 69, J ), J = 1, 4 ) / 3491, 1499, 2271, & 997 / DATA ( MM( 70, J ), J = 1, 4 ) / 2889, 1084, 3667, & 1833 / DATA ( MM( 71, J ), J = 1, 4 ) / 2857, 3438, 2703, & 2877 / DATA ( MM( 72, J ), J = 1, 4 ) / 2094, 2408, 629, & 1633 / DATA ( MM( 73, J ), J = 1, 4 ) / 1818, 1589, 2365, & 981 / DATA ( MM( 74, J ), J = 1, 4 ) / 688, 2391, 2431, & 2009 / DATA ( MM( 75, J ), J = 1, 4 ) / 1407, 288, 1113, & 941 / DATA ( MM( 76, J ), J = 1, 4 ) / 634, 26, 3922, & 2449 / DATA ( MM( 77, J ), J = 1, 4 ) / 3231, 512, 2554, & 197 / DATA ( MM( 78, J ), J = 1, 4 ) / 815, 1456, 184, & 2441 / DATA ( MM( 79, J ), J = 1, 4 ) / 3524, 171, 2099, & 285 / DATA ( MM( 80, J ), J = 1, 4 ) / 1914, 1677, 3228, & 1473 / DATA ( MM( 81, J ), J = 1, 4 ) / 516, 2657, 4012, & 2741 / DATA ( MM( 82, J ), J = 1, 4 ) / 164, 2270, 1921, & 3129 / DATA ( MM( 83, J ), J = 1, 4 ) / 303, 2587, 3452, & 909 / DATA ( MM( 84, J ), J = 1, 4 ) / 2144, 2961, 3901, & 2801 / DATA ( MM( 85, J ), J = 1, 4 ) / 3480, 1970, 572, & 421 / DATA ( MM( 86, J ), J = 1, 4 ) / 119, 1817, 3309, & 4073 / DATA ( MM( 87, J ), J = 1, 4 ) / 3357, 676, 3171, & 2813 / DATA ( MM( 88, J ), J = 1, 4 ) / 837, 1410, 817, & 2337 / DATA ( MM( 89, J ), J = 1, 4 ) / 2826, 3723, 3039, & 1429 / DATA ( MM( 90, J ), J = 1, 4 ) / 2332, 2803, 1696, & 1177 / DATA ( MM( 91, J ), J = 1, 4 ) / 2089, 3185, 1256, & 1901 / DATA ( MM( 92, J ), J = 1, 4 ) / 3780, 184, 3715, & 81 / DATA ( MM( 93, J ), J = 1, 4 ) / 1700, 663, 2077, & 1669 / DATA ( MM( 94, J ), J = 1, 4 ) / 3712, 499, 3019, & 2633 / DATA ( MM( 95, J ), J = 1, 4 ) / 150, 3784, 1497, & 2269 / DATA ( MM( 96, J ), J = 1, 4 ) / 2000, 1631, 1101, & 129 / DATA ( MM( 97, J ), J = 1, 4 ) / 3375, 1925, 717, & 1141 / DATA ( MM( 98, J ), J = 1, 4 ) / 1621, 3912, 51, & 249 / DATA ( MM( 99, J ), J = 1, 4 ) / 3090, 1398, 981, & 3917 / DATA ( MM( 100, J ), J = 1, 4 ) / 3765, 1349, 1978, & 2481 / DATA ( MM( 101, J ), J = 1, 4 ) / 1149, 1441, 1813, & 3941 / DATA ( MM( 102, J ), J = 1, 4 ) / 3146, 2224, 3881, & 2217 / DATA ( MM( 103, J ), J = 1, 4 ) / 33, 2411, 76, & 2749 / DATA ( MM( 104, J ), J = 1, 4 ) / 3082, 1907, 3846, & 3041 / DATA ( MM( 105, J ), J = 1, 4 ) / 2741, 3192, 3694, & 1877 / DATA ( MM( 106, J ), J = 1, 4 ) / 359, 2786, 1682, & 345 / DATA ( MM( 107, J ), J = 1, 4 ) / 3316, 382, 124, & 2861 / DATA ( MM( 108, J ), J = 1, 4 ) / 1749, 37, 1660, & 1809 / DATA ( MM( 109, J ), J = 1, 4 ) / 185, 759, 3997, & 3141 / DATA ( MM( 110, J ), J = 1, 4 ) / 2784, 2948, 479, & 2825 / DATA ( MM( 111, J ), J = 1, 4 ) / 2202, 1862, 1141, & 157 / DATA ( MM( 112, J ), J = 1, 4 ) / 2199, 3802, 886, & 2881 / DATA ( MM( 113, J ), J = 1, 4 ) / 1364, 2423, 3514, & 3637 / DATA ( MM( 114, J ), J = 1, 4 ) / 1244, 2051, 1301, & 1465 / DATA ( MM( 115, J ), J = 1, 4 ) / 2020, 2295, 3604, & 2829 / DATA ( MM( 116, J ), J = 1, 4 ) / 3160, 1332, 1888, & 2161 / DATA ( MM( 117, J ), J = 1, 4 ) / 2785, 1832, 1836, & 3365 / DATA ( MM( 118, J ), J = 1, 4 ) / 2772, 2405, 1990, & 361 / DATA ( MM( 119, J ), J = 1, 4 ) / 1217, 3638, 2058, & 2685 / DATA ( MM( 120, J ), J = 1, 4 ) / 1822, 3661, 692, & 3745 / DATA ( MM( 121, J ), J = 1, 4 ) / 1245, 327, 1194, & 2325 / DATA ( MM( 122, J ), J = 1, 4 ) / 2252, 3660, 20, & 3609 / DATA ( MM( 123, J ), J = 1, 4 ) / 3904, 716, 3285, & 3821 / DATA ( MM( 124, J ), J = 1, 4 ) / 2774, 1842, 2046, & 3537 / DATA ( MM( 125, J ), J = 1, 4 ) / 997, 3987, 2107, & 517 / DATA ( MM( 126, J ), J = 1, 4 ) / 2573, 1368, 3508, & 3017 / DATA ( MM( 127, J ), J = 1, 4 ) / 1148, 1848, 3525, & 2141 / DATA ( MM( 128, J ), J = 1, 4 ) / 545, 2366, 3801, & 1537 / ! .. ! .. Executable Statements .. ! I1 = ISEED( 1 ) I2 = ISEED( 2 ) I3 = ISEED( 3 ) I4 = ISEED( 4 ) ! DO 10 I = 1, MIN( N, LV ) ! ! Multiply the seed by i-th power of the multiplier modulo 2**48 ! IT4 = I4*MM( I, 4 ) IT3 = IT4 / IPW2 IT4 = IT4 - IPW2*IT3 IT3 = IT3 + I3*MM( I, 4 ) + I4*MM( I, 3 ) IT2 = IT3 / IPW2 IT3 = IT3 - IPW2*IT2 IT2 = IT2 + I2*MM( I, 4 ) + I3*MM( I, 3 ) + I4*MM( I, 2 ) IT1 = IT2 / IPW2 IT2 = IT2 - IPW2*IT1 IT1 = IT1 + I1*MM( I, 4 ) + I2*MM( I, 3 ) + I3*MM( I, 2 ) + & I4*MM( I, 1 ) IT1 = MOD( IT1, IPW2 ) ! ! Convert 48-bit integer to a real number in the interval (0,1) ! X( I ) = R*( DBLE( IT1 )+R*( DBLE( IT2 )+R*( DBLE( IT3 )+R* & DBLE( IT4 ) ) ) ) 10 CONTINUE ! ! Return final value of seed ! ISEED( 1 ) = IT1 ISEED( 2 ) = IT2 ISEED( 3 ) = IT3 ISEED( 4 ) = IT4 RETURN ! ! End of DLARUV ! END SUBROUTINE DLARZ( SIDE, M, N, L, V, INCV, TAU, C, LDC, WORK ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER SIDE INTEGER INCV, L, LDC, M, N DOUBLE PRECISION TAU ! .. ! .. Array Arguments .. DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLARZ applies a real elementary reflector H to a real M-by-N ! matrix C, from either the left or the right. H is represented in the ! form ! ! H = I - tau * v * v' ! ! where tau is a real scalar and v is a real vector. ! ! If tau = 0, then H is taken to be the unit matrix. ! ! ! H is a product of k elementary reflectors as returned by DTZRZF. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': form H * C ! = 'R': form C * H ! ! M (input) INTEGER ! The number of rows of the matrix C. ! ! N (input) INTEGER ! The number of columns of the matrix C. ! ! L (input) INTEGER ! The number of entries of the vector V containing ! the meaningful part of the Householder vectors. ! If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. ! ! V (input) DOUBLE PRECISION array, dimension (1+(L-1)*abs(INCV)) ! The vector v in the representation of H as returned by ! DTZRZF. V is not used if TAU = 0. ! ! INCV (input) INTEGER ! The increment between elements of v. INCV <> 0. ! ! TAU (input) DOUBLE PRECISION ! The value tau in the representation of H. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the M-by-N matrix C. ! On exit, C is overwritten by the matrix H * C if SIDE = 'L', ! or C * H if SIDE = 'R'. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace) DOUBLE PRECISION array, dimension ! (N) if SIDE = 'L' ! or (M) if SIDE = 'R' ! ! Further Details ! =============== ! ! Based on contributions by ! A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DGEMV, DGER ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Executable Statements .. ! IF( LSAME( SIDE, 'L' ) ) THEN ! ! Form H * C ! IF( TAU.NE.ZERO ) THEN ! ! w( 1:n ) = C( 1, 1:n ) ! CALL DCOPY( N, C, LDC, WORK, 1 ) ! ! w( 1:n ) = w( 1:n ) + C( m-l+1:m, 1:n )' * v( 1:l ) ! CALL DGEMV( 'Transpose', L, N, ONE, C( M-L+1, 1 ), LDC, V, & INCV, ONE, WORK, 1 ) ! ! C( 1, 1:n ) = C( 1, 1:n ) - tau * w( 1:n ) ! CALL DAXPY( N, -TAU, WORK, 1, C, LDC ) ! ! C( m-l+1:m, 1:n ) = C( m-l+1:m, 1:n ) - ... ! tau * v( 1:l ) * w( 1:n )' ! CALL DGER( L, N, -TAU, V, INCV, WORK, 1, C( M-L+1, 1 ), & LDC ) END IF ! ELSE ! ! Form C * H ! IF( TAU.NE.ZERO ) THEN ! ! w( 1:m ) = C( 1:m, 1 ) ! CALL DCOPY( M, C, 1, WORK, 1 ) ! ! w( 1:m ) = w( 1:m ) + C( 1:m, n-l+1:n, 1:n ) * v( 1:l ) ! CALL DGEMV( 'No transpose', M, L, ONE, C( 1, N-L+1 ), LDC, & V, INCV, ONE, WORK, 1 ) ! ! C( 1:m, 1 ) = C( 1:m, 1 ) - tau * w( 1:m ) ! CALL DAXPY( M, -TAU, WORK, 1, C, 1 ) ! ! C( 1:m, n-l+1:n ) = C( 1:m, n-l+1:n ) - ... ! tau * w( 1:m ) * v( 1:l )' ! CALL DGER( M, L, -TAU, WORK, 1, V, INCV, C( 1, N-L+1 ), & LDC ) ! END IF ! END IF ! RETURN ! ! End of DLARZ ! END SUBROUTINE DLARZB( SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, & LDV, T, LDT, C, LDC, WORK, LDWORK ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! December 1, 1999 ! ! .. Scalar Arguments .. CHARACTER DIRECT, SIDE, STOREV, TRANS INTEGER K, L, LDC, LDT, LDV, LDWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION C( LDC, * ), T( LDT, * ), V( LDV, * ), & WORK( LDWORK, * ) ! .. ! ! Purpose ! ======= ! ! DLARZB applies a real block reflector H or its transpose H**T to ! a real distributed M-by-N C from the left or the right. ! ! Currently, only STOREV = 'R' and DIRECT = 'B' are supported. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': apply H or H' from the Left ! = 'R': apply H or H' from the Right ! ! TRANS (input) CHARACTER*1 ! = 'N': apply H (No transpose) ! = 'C': apply H' (Transpose) ! ! DIRECT (input) CHARACTER*1 ! Indicates how H is formed from a product of elementary ! reflectors ! = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet) ! = 'B': H = H(k) . . . H(2) H(1) (Backward) ! ! STOREV (input) CHARACTER*1 ! Indicates how the vectors which define the elementary ! reflectors are stored: ! = 'C': Columnwise (not supported yet) ! = 'R': Rowwise ! ! M (input) INTEGER ! The number of rows of the matrix C. ! ! N (input) INTEGER ! The number of columns of the matrix C. ! ! K (input) INTEGER ! The order of the matrix T (= the number of elementary ! reflectors whose product defines the block reflector). ! ! L (input) INTEGER ! The number of columns of the matrix V containing the ! meaningful part of the Householder reflectors. ! If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. ! ! V (input) DOUBLE PRECISION array, dimension (LDV,NV). ! If STOREV = 'C', NV = K; if STOREV = 'R', NV = L. ! ! LDV (input) INTEGER ! The leading dimension of the array V. ! If STOREV = 'C', LDV >= L; if STOREV = 'R', LDV >= K. ! ! T (input) DOUBLE PRECISION array, dimension (LDT,K) ! The triangular K-by-K matrix T in the representation of the ! block reflector. ! ! LDT (input) INTEGER ! The leading dimension of the array T. LDT >= K. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the M-by-N matrix C. ! On exit, C is overwritten by H*C or H'*C or C*H or C*H'. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (LDWORK,K) ! ! LDWORK (input) INTEGER ! The leading dimension of the array WORK. ! If SIDE = 'L', LDWORK >= max(1,N); ! if SIDE = 'R', LDWORK >= max(1,M). ! ! Further Details ! =============== ! ! Based on contributions by ! A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. CHARACTER TRANST INTEGER I, INFO, J ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEMM, DTRMM, XERBLA ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( M.LE.0 .OR. N.LE.0 ) & RETURN ! ! Check for currently supported options ! INFO = 0 IF( .NOT.LSAME( DIRECT, 'B' ) ) THEN INFO = -3 ELSE IF( .NOT.LSAME( STOREV, 'R' ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLARZB', -INFO ) RETURN END IF ! IF( LSAME( TRANS, 'N' ) ) THEN TRANST = 'T' ELSE TRANST = 'N' END IF ! IF( LSAME( SIDE, 'L' ) ) THEN ! ! Form H * C or H' * C ! ! W( 1:n, 1:k ) = C( 1:k, 1:n )' ! DO 10 J = 1, K CALL DCOPY( N, C( J, 1 ), LDC, WORK( 1, J ), 1 ) 10 CONTINUE ! ! W( 1:n, 1:k ) = W( 1:n, 1:k ) + ... ! C( m-l+1:m, 1:n )' * V( 1:k, 1:l )' ! IF( L.GT.0 ) & CALL DGEMM( 'Transpose', 'Transpose', N, K, L, ONE, & C( M-L+1, 1 ), LDC, V, LDV, ONE, WORK, LDWORK ) ! ! W( 1:n, 1:k ) = W( 1:n, 1:k ) * T' or W( 1:m, 1:k ) * T ! CALL DTRMM( 'Right', 'Lower', TRANST, 'Non-unit', N, K, ONE, T, & LDT, WORK, LDWORK ) ! ! C( 1:k, 1:n ) = C( 1:k, 1:n ) - W( 1:n, 1:k )' ! DO 30 J = 1, N DO 20 I = 1, K C( I, J ) = C( I, J ) - WORK( J, I ) 20 CONTINUE 30 CONTINUE ! ! C( m-l+1:m, 1:n ) = C( m-l+1:m, 1:n ) - ... ! V( 1:k, 1:l )' * W( 1:n, 1:k )' ! IF( L.GT.0 ) & CALL DGEMM( 'Transpose', 'Transpose', L, N, K, -ONE, V, LDV, & WORK, LDWORK, ONE, C( M-L+1, 1 ), LDC ) ! ELSE IF( LSAME( SIDE, 'R' ) ) THEN ! ! Form C * H or C * H' ! ! W( 1:m, 1:k ) = C( 1:m, 1:k ) ! DO 40 J = 1, K CALL DCOPY( M, C( 1, J ), 1, WORK( 1, J ), 1 ) 40 CONTINUE ! ! W( 1:m, 1:k ) = W( 1:m, 1:k ) + ... ! C( 1:m, n-l+1:n ) * V( 1:k, 1:l )' ! IF( L.GT.0 ) & CALL DGEMM( 'No transpose', 'Transpose', M, K, L, ONE, & C( 1, N-L+1 ), LDC, V, LDV, ONE, WORK, LDWORK ) ! ! W( 1:m, 1:k ) = W( 1:m, 1:k ) * T or W( 1:m, 1:k ) * T' ! CALL DTRMM( 'Right', 'Lower', TRANS, 'Non-unit', M, K, ONE, T, & LDT, WORK, LDWORK ) ! ! C( 1:m, 1:k ) = C( 1:m, 1:k ) - W( 1:m, 1:k ) ! DO 60 J = 1, K DO 50 I = 1, M C( I, J ) = C( I, J ) - WORK( I, J ) 50 CONTINUE 60 CONTINUE ! ! C( 1:m, n-l+1:n ) = C( 1:m, n-l+1:n ) - ... ! W( 1:m, 1:k ) * V( 1:k, 1:l ) ! IF( L.GT.0 ) & CALL DGEMM( 'No transpose', 'No transpose', M, L, K, -ONE, & WORK, LDWORK, V, LDV, ONE, C( 1, N-L+1 ), LDC ) ! END IF ! RETURN ! ! End of DLARZB ! END SUBROUTINE DLARZT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER DIRECT, STOREV INTEGER K, LDT, LDV, N ! .. ! .. Array Arguments .. DOUBLE PRECISION T( LDT, * ), TAU( * ), V( LDV, * ) ! .. ! ! Purpose ! ======= ! ! DLARZT forms the triangular factor T of a real block reflector ! H of order > n, which is defined as a product of k elementary ! reflectors. ! ! If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; ! ! If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. ! ! If STOREV = 'C', the vector which defines the elementary reflector ! H(i) is stored in the i-th column of the array V, and ! ! H = I - V * T * V' ! ! If STOREV = 'R', the vector which defines the elementary reflector ! H(i) is stored in the i-th row of the array V, and ! ! H = I - V' * T * V ! ! Currently, only STOREV = 'R' and DIRECT = 'B' are supported. ! ! Arguments ! ========= ! ! DIRECT (input) CHARACTER*1 ! Specifies the order in which the elementary reflectors are ! multiplied to form the block reflector: ! = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet) ! = 'B': H = H(k) . . . H(2) H(1) (Backward) ! ! STOREV (input) CHARACTER*1 ! Specifies how the vectors which define the elementary ! reflectors are stored (see also Further Details): ! = 'C': columnwise (not supported yet) ! = 'R': rowwise ! ! N (input) INTEGER ! The order of the block reflector H. N >= 0. ! ! K (input) INTEGER ! The order of the triangular factor T (= the number of ! elementary reflectors). K >= 1. ! ! V (input/output) DOUBLE PRECISION array, dimension ! (LDV,K) if STOREV = 'C' ! (LDV,N) if STOREV = 'R' ! The matrix V. See further details. ! ! LDV (input) INTEGER ! The leading dimension of the array V. ! If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K. ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i). ! ! T (output) DOUBLE PRECISION array, dimension (LDT,K) ! The k by k triangular factor T of the block reflector. ! If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is ! lower triangular. The rest of the array is not used. ! ! LDT (input) INTEGER ! The leading dimension of the array T. LDT >= K. ! ! Further Details ! =============== ! ! Based on contributions by ! A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA ! ! The shape of the matrix V and the storage of the vectors which define ! the H(i) is best illustrated by the following example with n = 5 and ! k = 3. The elements equal to 1 are not stored; the corresponding ! array elements are modified but restored on exit. The rest of the ! array is not used. ! ! DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': ! ! ______V_____ ! ( v1 v2 v3 ) / \ ! ( v1 v2 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 ) ! V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 ) ! ( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 ) ! ( v1 v2 v3 ) ! . . . ! . . . ! 1 . . ! 1 . ! 1 ! ! DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': ! ! ______V_____ ! 1 / \ ! . 1 ( 1 . . . . v1 v1 v1 v1 v1 ) ! . . 1 ( . 1 . . . v2 v2 v2 v2 v2 ) ! . . . ( . . 1 . . v3 v3 v3 v3 v3 ) ! . . . ! ( v1 v2 v3 ) ! ( v1 v2 v3 ) ! V = ( v1 v2 v3 ) ! ( v1 v2 v3 ) ! ( v1 v2 v3 ) ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, INFO, J ! .. ! .. External Subroutines .. EXTERNAL DGEMV, DTRMV, XERBLA ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Executable Statements .. ! ! Check for currently supported options ! INFO = 0 IF( .NOT.LSAME( DIRECT, 'B' ) ) THEN INFO = -1 ELSE IF( .NOT.LSAME( STOREV, 'R' ) ) THEN INFO = -2 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLARZT', -INFO ) RETURN END IF ! DO 20 I = K, 1, -1 IF( TAU( I ).EQ.ZERO ) THEN ! ! H(i) = I ! DO 10 J = I, K T( J, I ) = ZERO 10 CONTINUE ELSE ! ! general case ! IF( I.LT.K ) THEN ! ! T(i+1:k,i) = - tau(i) * V(i+1:k,1:n) * V(i,1:n)' ! CALL DGEMV( 'No transpose', K-I, N, -TAU( I ), & V( I+1, 1 ), LDV, V( I, 1 ), LDV, ZERO, & T( I+1, I ), 1 ) ! ! T(i+1:k,i) = T(i+1:k,i+1:k) * T(i+1:k,i) ! CALL DTRMV( 'Lower', 'No transpose', 'Non-unit', K-I, & T( I+1, I+1 ), LDT, T( I+1, I ), 1 ) END IF T( I, I ) = TAU( I ) END IF 20 CONTINUE RETURN ! ! End of DLARZT ! END SUBROUTINE DLAS2( F, G, H, SSMIN, SSMAX ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. DOUBLE PRECISION F, G, H, SSMAX, SSMIN ! .. ! ! Purpose ! ======= ! ! DLAS2 computes the singular values of the 2-by-2 matrix ! [ F G ] ! [ 0 H ]. ! On return, SSMIN is the smaller singular value and SSMAX is the ! larger singular value. ! ! Arguments ! ========= ! ! F (input) DOUBLE PRECISION ! The (1,1) element of the 2-by-2 matrix. ! ! G (input) DOUBLE PRECISION ! The (1,2) element of the 2-by-2 matrix. ! ! H (input) DOUBLE PRECISION ! The (2,2) element of the 2-by-2 matrix. ! ! SSMIN (output) DOUBLE PRECISION ! The smaller singular value. ! ! SSMAX (output) DOUBLE PRECISION ! The larger singular value. ! ! Further Details ! =============== ! ! Barring over/underflow, all output quantities are correct to within ! a few units in the last place (ulps), even in the absence of a guard ! digit in addition/subtraction. ! ! In IEEE arithmetic, the code works correctly if one matrix element is ! infinite. ! ! Overflow will not occur unless the largest singular value itself ! overflows, or is within a few ulps of overflow. (On machines with ! partial overflow, like the Cray, overflow may occur if the largest ! singular value is within a factor of 2 of overflow.) ! ! Underflow is harmless if underflow is gradual. Otherwise, results ! may correspond to a matrix modified by perturbations of size near ! the underflow threshold. ! ! ==================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D0 ) ! .. ! .. Local Scalars .. DOUBLE PRECISION AS, AT, AU, C, FA, FHMN, FHMX, GA, HA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! FA = ABS( F ) GA = ABS( G ) HA = ABS( H ) FHMN = MIN( FA, HA ) FHMX = MAX( FA, HA ) IF( FHMN.EQ.ZERO ) THEN SSMIN = ZERO IF( FHMX.EQ.ZERO ) THEN SSMAX = GA ELSE SSMAX = MAX( FHMX, GA )*SQRT( ONE+ & ( MIN( FHMX, GA ) / MAX( FHMX, GA ) )**2 ) END IF ELSE IF( GA.LT.FHMX ) THEN AS = ONE + FHMN / FHMX AT = ( FHMX-FHMN ) / FHMX AU = ( GA / FHMX )**2 C = TWO / ( SQRT( AS*AS+AU )+SQRT( AT*AT+AU ) ) SSMIN = FHMN*C SSMAX = FHMX / C ELSE AU = FHMX / GA IF( AU.EQ.ZERO ) THEN ! ! Avoid possible harmful underflow if exponent range ! asymmetric (true SSMIN may not underflow even if ! AU underflows) ! SSMIN = ( FHMN*FHMX ) / GA SSMAX = GA ELSE AS = ONE + FHMN / FHMX AT = ( FHMX-FHMN ) / FHMX C = ONE / ( SQRT( ONE+( AS*AU )**2 )+ & SQRT( ONE+( AT*AU )**2 ) ) SSMIN = ( FHMN*C )*AU SSMIN = SSMIN + SSMIN SSMAX = GA / ( C+C ) END IF END IF END IF RETURN ! ! End of DLAS2 ! END SUBROUTINE DLASCL( TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER TYPE INTEGER INFO, KL, KU, LDA, M, N DOUBLE PRECISION CFROM, CTO ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ) ! .. ! ! Purpose ! ======= ! ! DLASCL multiplies the M by N real matrix A by the real scalar ! CTO/CFROM. This is done without over/underflow as long as the final ! result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that ! A may be full, upper triangular, lower triangular, upper Hessenberg, ! or banded. ! ! Arguments ! ========= ! ! TYPE (input) CHARACTER*1 ! TYPE indices the storage type of the input matrix. ! = 'G': A is a full matrix. ! = 'L': A is a lower triangular matrix. ! = 'U': A is an upper triangular matrix. ! = 'H': A is an upper Hessenberg matrix. ! = 'B': A is a symmetric band matrix with lower bandwidth KL ! and upper bandwidth KU and with the only the lower ! half stored. ! = 'Q': A is a symmetric band matrix with lower bandwidth KL ! and upper bandwidth KU and with the only the upper ! half stored. ! = 'Z': A is a band matrix with lower bandwidth KL and upper ! bandwidth KU. ! ! KL (input) INTEGER ! The lower bandwidth of A. Referenced only if TYPE = 'B', ! 'Q' or 'Z'. ! ! KU (input) INTEGER ! The upper bandwidth of A. Referenced only if TYPE = 'B', ! 'Q' or 'Z'. ! ! CFROM (input) DOUBLE PRECISION ! CTO (input) DOUBLE PRECISION ! The matrix A is multiplied by CTO/CFROM. A(I,J) is computed ! without over/underflow if the final result CTO*A(I,J)/CFROM ! can be represented without over/underflow. CFROM must be ! nonzero. ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,M) ! The matrix to be multiplied by CTO/CFROM. See TYPE for the ! storage type. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! INFO (output) INTEGER ! 0 - successful exit ! <0 - if INFO = -i, the i-th argument had an illegal value. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL DONE INTEGER I, ITYPE, J, K1, K2, K3, K4 DOUBLE PRECISION BIGNUM, CFROM1, CFROMC, CTO1, CTOC, MUL, SMLNUM ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 ! IF( LSAME( TYPE, 'G' ) ) THEN ITYPE = 0 ELSE IF( LSAME( TYPE, 'L' ) ) THEN ITYPE = 1 ELSE IF( LSAME( TYPE, 'U' ) ) THEN ITYPE = 2 ELSE IF( LSAME( TYPE, 'H' ) ) THEN ITYPE = 3 ELSE IF( LSAME( TYPE, 'B' ) ) THEN ITYPE = 4 ELSE IF( LSAME( TYPE, 'Q' ) ) THEN ITYPE = 5 ELSE IF( LSAME( TYPE, 'Z' ) ) THEN ITYPE = 6 ELSE ITYPE = -1 END IF ! IF( ITYPE.EQ.-1 ) THEN INFO = -1 ELSE IF( CFROM.EQ.ZERO ) THEN INFO = -4 ELSE IF( M.LT.0 ) THEN INFO = -6 ELSE IF( N.LT.0 .OR. ( ITYPE.EQ.4 .AND. N.NE.M ) .OR. & ( ITYPE.EQ.5 .AND. N.NE.M ) ) THEN INFO = -7 ELSE IF( ITYPE.LE.3 .AND. LDA.LT.MAX( 1, M ) ) THEN INFO = -9 ELSE IF( ITYPE.GE.4 ) THEN IF( KL.LT.0 .OR. KL.GT.MAX( M-1, 0 ) ) THEN INFO = -2 ELSE IF( KU.LT.0 .OR. KU.GT.MAX( N-1, 0 ) .OR. & ( ( ITYPE.EQ.4 .OR. ITYPE.EQ.5 ) .AND. KL.NE.KU ) ) & THEN INFO = -3 ELSE IF( ( ITYPE.EQ.4 .AND. LDA.LT.KL+1 ) .OR. & ( ITYPE.EQ.5 .AND. LDA.LT.KU+1 ) .OR. & ( ITYPE.EQ.6 .AND. LDA.LT.2*KL+KU+1 ) ) THEN INFO = -9 END IF END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLASCL', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. M.EQ.0 ) & RETURN ! ! Get machine parameters ! SMLNUM = DLAMCH( 'S' ) BIGNUM = ONE / SMLNUM ! CFROMC = CFROM CTOC = CTO ! 10 CONTINUE CFROM1 = CFROMC*SMLNUM CTO1 = CTOC / BIGNUM IF( ABS( CFROM1 ).GT.ABS( CTOC ) .AND. CTOC.NE.ZERO ) THEN MUL = SMLNUM DONE = .FALSE. CFROMC = CFROM1 ELSE IF( ABS( CTO1 ).GT.ABS( CFROMC ) ) THEN MUL = BIGNUM DONE = .FALSE. CTOC = CTO1 ELSE MUL = CTOC / CFROMC DONE = .TRUE. END IF ! IF( ITYPE.EQ.0 ) THEN ! ! Full matrix ! DO 30 J = 1, N DO 20 I = 1, M A( I, J ) = A( I, J )*MUL 20 CONTINUE 30 CONTINUE ! ELSE IF( ITYPE.EQ.1 ) THEN ! ! Lower triangular matrix ! DO 50 J = 1, N DO 40 I = J, M A( I, J ) = A( I, J )*MUL 40 CONTINUE 50 CONTINUE ! ELSE IF( ITYPE.EQ.2 ) THEN ! ! Upper triangular matrix ! DO 70 J = 1, N DO 60 I = 1, MIN( J, M ) A( I, J ) = A( I, J )*MUL 60 CONTINUE 70 CONTINUE ! ELSE IF( ITYPE.EQ.3 ) THEN ! ! Upper Hessenberg matrix ! DO 90 J = 1, N DO 80 I = 1, MIN( J+1, M ) A( I, J ) = A( I, J )*MUL 80 CONTINUE 90 CONTINUE ! ELSE IF( ITYPE.EQ.4 ) THEN ! ! Lower half of a symmetric band matrix ! K3 = KL + 1 K4 = N + 1 DO 110 J = 1, N DO 100 I = 1, MIN( K3, K4-J ) A( I, J ) = A( I, J )*MUL 100 CONTINUE 110 CONTINUE ! ELSE IF( ITYPE.EQ.5 ) THEN ! ! Upper half of a symmetric band matrix ! K1 = KU + 2 K3 = KU + 1 DO 130 J = 1, N DO 120 I = MAX( K1-J, 1 ), K3 A( I, J ) = A( I, J )*MUL 120 CONTINUE 130 CONTINUE ! ELSE IF( ITYPE.EQ.6 ) THEN ! ! Band matrix ! K1 = KL + KU + 2 K2 = KL + 1 K3 = 2*KL + KU + 1 K4 = KL + KU + 1 + M DO 150 J = 1, N DO 140 I = MAX( K1-J, K2 ), MIN( K3, K4-J ) A( I, J ) = A( I, J )*MUL 140 CONTINUE 150 CONTINUE ! END IF ! IF( .NOT.DONE ) & GO TO 10 ! RETURN ! ! End of DLASCL ! END SUBROUTINE DLASD0( N, SQRE, D, E, U, LDU, VT, LDVT, SMLSIZ, IWORK, & WORK, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDU, LDVT, N, SMLSIZ, SQRE ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION D( * ), E( * ), U( LDU, * ), VT( LDVT, * ), & WORK( * ) ! .. ! ! Purpose ! ======= ! ! Using a divide and conquer approach, DLASD0 computes the singular ! value decomposition (SVD) of a real upper bidiagonal N-by-M ! matrix B with diagonal D and offdiagonal E, where M = N + SQRE. ! The algorithm computes orthogonal matrices U and VT such that ! B = U * S * VT. The singular values S are overwritten on D. ! ! A related subroutine, DLASDA, computes only the singular values, ! and optionally, the singular vectors in compact form. ! ! Arguments ! ========= ! ! N (input) INTEGER ! On entry, the row dimension of the upper bidiagonal matrix. ! This is also the dimension of the main diagonal array D. ! ! SQRE (input) INTEGER ! Specifies the column dimension of the bidiagonal matrix. ! = 0: The bidiagonal matrix has column dimension M = N; ! = 1: The bidiagonal matrix has column dimension M = N+1; ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry D contains the main diagonal of the bidiagonal ! matrix. ! On exit D, if INFO = 0, contains its singular values. ! ! E (input) DOUBLE PRECISION array, dimension (M-1) ! Contains the subdiagonal entries of the bidiagonal matrix. ! On exit, E has been destroyed. ! ! U (output) DOUBLE PRECISION array, dimension at least (LDQ, N) ! On exit, U contains the left singular vectors. ! ! LDU (input) INTEGER ! On entry, leading dimension of U. ! ! VT (output) DOUBLE PRECISION array, dimension at least (LDVT, M) ! On exit, VT' contains the right singular vectors. ! ! LDVT (input) INTEGER ! On entry, leading dimension of VT. ! ! SMLSIZ (input) INTEGER ! On entry, maximum size of the subproblems at the ! bottom of the computation tree. ! ! IWORK INTEGER work array. ! Dimension must be at least (8 * N) ! ! WORK DOUBLE PRECISION work array. ! Dimension must be at least (3 * M**2 + 2 * M) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = 1, an singular value did not converge ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Huan Ren, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER I, I1, IC, IDXQ, IDXQC, IM1, INODE, ITEMP, IWK, & J, LF, LL, LVL, M, NCC, ND, NDB1, NDIML, NDIMR, & NL, NLF, NLP1, NLVL, NR, NRF, NRP1, SQREI DOUBLE PRECISION ALPHA, BETA ! .. ! .. External Subroutines .. EXTERNAL DLASD1, DLASDQ, DLASDT, XERBLA ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( N.LT.0 ) THEN INFO = -1 ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN INFO = -2 END IF ! M = N + SQRE ! IF( LDU.LT.N ) THEN INFO = -6 ELSE IF( LDVT.LT.M ) THEN INFO = -8 ELSE IF( SMLSIZ.LT.3 ) THEN INFO = -9 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLASD0', -INFO ) RETURN END IF ! ! If the input matrix is too small, call DLASDQ to find the SVD. ! IF( N.LE.SMLSIZ ) THEN CALL DLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDVT, U, LDU, U, & LDU, WORK, INFO ) RETURN END IF ! ! Set up the computation tree. ! INODE = 1 NDIML = INODE + N NDIMR = NDIML + N IDXQ = NDIMR + N IWK = IDXQ + N CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ), & IWORK( NDIMR ), SMLSIZ ) ! ! For the nodes on bottom level of the tree, solve ! their subproblems by DLASDQ. ! NDB1 = ( ND+1 ) / 2 NCC = 0 DO 30 I = NDB1, ND ! ! IC : center row of each node ! NL : number of rows of left subproblem ! NR : number of rows of right subproblem ! NLF: starting row of the left subproblem ! NRF: starting row of the right subproblem ! I1 = I - 1 IC = IWORK( INODE+I1 ) NL = IWORK( NDIML+I1 ) NLP1 = NL + 1 NR = IWORK( NDIMR+I1 ) NRP1 = NR + 1 NLF = IC - NL NRF = IC + 1 SQREI = 1 CALL DLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ), E( NLF ), & VT( NLF, NLF ), LDVT, U( NLF, NLF ), LDU, & U( NLF, NLF ), LDU, WORK, INFO ) IF( INFO.NE.0 ) THEN RETURN END IF ITEMP = IDXQ + NLF - 2 DO 10 J = 1, NL IWORK( ITEMP+J ) = J 10 CONTINUE IF( I.EQ.ND ) THEN SQREI = SQRE ELSE SQREI = 1 END IF NRP1 = NR + SQREI CALL DLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ), E( NRF ), & VT( NRF, NRF ), LDVT, U( NRF, NRF ), LDU, & U( NRF, NRF ), LDU, WORK, INFO ) IF( INFO.NE.0 ) THEN RETURN END IF ITEMP = IDXQ + IC DO 20 J = 1, NR IWORK( ITEMP+J-1 ) = J 20 CONTINUE 30 CONTINUE ! ! Now conquer each subproblem bottom-up. ! DO 50 LVL = NLVL, 1, -1 ! ! Find the first node LF and last node LL on the ! current level LVL. ! IF( LVL.EQ.1 ) THEN LF = 1 LL = 1 ELSE LF = 2**( LVL-1 ) LL = 2*LF - 1 END IF DO 40 I = LF, LL IM1 = I - 1 IC = IWORK( INODE+IM1 ) NL = IWORK( NDIML+IM1 ) NR = IWORK( NDIMR+IM1 ) NLF = IC - NL IF( ( SQRE.EQ.0 ) .AND. ( I.EQ.LL ) ) THEN SQREI = SQRE ELSE SQREI = 1 END IF IDXQC = IDXQ + NLF - 1 ALPHA = D( IC ) BETA = E( IC ) CALL DLASD1( NL, NR, SQREI, D( NLF ), ALPHA, BETA, & U( NLF, NLF ), LDU, VT( NLF, NLF ), LDVT, & IWORK( IDXQC ), IWORK( IWK ), WORK, INFO ) IF( INFO.NE.0 ) THEN RETURN END IF 40 CONTINUE 50 CONTINUE ! RETURN ! ! End of DLASD0 ! END SUBROUTINE DLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT, & IDXQ, IWORK, WORK, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDU, LDVT, NL, NR, SQRE DOUBLE PRECISION ALPHA, BETA ! .. ! .. Array Arguments .. INTEGER IDXQ( * ), IWORK( * ) DOUBLE PRECISION D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B, ! where N = NL + NR + 1 and M = N + SQRE. DLASD1 is called from DLASD0. ! ! A related subroutine DLASD7 handles the case in which the singular ! values (and the singular vectors in factored form) are desired. ! ! DLASD1 computes the SVD as follows: ! ! ( D1(in) 0 0 0 ) ! B = U(in) * ( Z1' a Z2' b ) * VT(in) ! ( 0 0 D2(in) 0 ) ! ! = U(out) * ( D(out) 0) * VT(out) ! ! where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M ! with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros ! elsewhere; and the entry b is empty if SQRE = 0. ! ! The left singular vectors of the original matrix are stored in U, and ! the transpose of the right singular vectors are stored in VT, and the ! singular values are in D. The algorithm consists of three stages: ! ! The first stage consists of deflating the size of the problem ! when there are multiple singular values or when there are zeros in ! the Z vector. For each such occurence the dimension of the ! secular equation problem is reduced by one. This stage is ! performed by the routine DLASD2. ! ! The second stage consists of calculating the updated ! singular values. This is done by finding the square roots of the ! roots of the secular equation via the routine DLASD4 (as called ! by DLASD3). This routine also calculates the singular vectors of ! the current problem. ! ! The final stage consists of computing the updated singular vectors ! directly using the updated singular values. The singular vectors ! for the current problem are multiplied with the singular vectors ! from the overall problem. ! ! Arguments ! ========= ! ! NL (input) INTEGER ! The row dimension of the upper block. NL >= 1. ! ! NR (input) INTEGER ! The row dimension of the lower block. NR >= 1. ! ! SQRE (input) INTEGER ! = 0: the lower block is an NR-by-NR square matrix. ! = 1: the lower block is an NR-by-(NR+1) rectangular matrix. ! ! The bidiagonal matrix has row dimension N = NL + NR + 1, ! and column dimension M = N + SQRE. ! ! D (input/output) DOUBLE PRECISION array, ! dimension (N = NL+NR+1). ! On entry D(1:NL,1:NL) contains the singular values of the ! upper block; and D(NL+2:N) contains the singular values of ! the lower block. On exit D(1:N) contains the singular values ! of the modified matrix. ! ! ALPHA (input) DOUBLE PRECISION ! Contains the diagonal element associated with the added row. ! ! BETA (input) DOUBLE PRECISION ! Contains the off-diagonal element associated with the added ! row. ! ! U (input/output) DOUBLE PRECISION array, dimension(LDU,N) ! On entry U(1:NL, 1:NL) contains the left singular vectors of ! the upper block; U(NL+2:N, NL+2:N) contains the left singular ! vectors of the lower block. On exit U contains the left ! singular vectors of the bidiagonal matrix. ! ! LDU (input) INTEGER ! The leading dimension of the array U. LDU >= max( 1, N ). ! ! VT (input/output) DOUBLE PRECISION array, dimension(LDVT,M) ! where M = N + SQRE. ! On entry VT(1:NL+1, 1:NL+1)' contains the right singular ! vectors of the upper block; VT(NL+2:M, NL+2:M)' contains ! the right singular vectors of the lower block. On exit ! VT' contains the right singular vectors of the ! bidiagonal matrix. ! ! LDVT (input) INTEGER ! The leading dimension of the array VT. LDVT >= max( 1, M ). ! ! IDXQ (output) INTEGER array, dimension(N) ! This contains the permutation which will reintegrate the ! subproblem just solved back into sorted order, i.e. ! D( IDXQ( I = 1, N ) ) will be in ascending order. ! ! IWORK (workspace) INTEGER array, dimension( 4 * N ) ! ! WORK (workspace) DOUBLE PRECISION array, dimension( 3*M**2 + 2*M ) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = 1, an singular value did not converge ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Huan Ren, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. ! DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER COLTYP, I, IDX, IDXC, IDXP, IQ, ISIGMA, IU2, & IVT2, IZ, K, LDQ, LDU2, LDVT2, M, N, N1, N2 DOUBLE PRECISION ORGNRM ! .. ! .. External Subroutines .. EXTERNAL DLAMRG, DLASCL, DLASD2, DLASD3, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( NL.LT.1 ) THEN INFO = -1 ELSE IF( NR.LT.1 ) THEN INFO = -2 ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN INFO = -3 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLASD1', -INFO ) RETURN END IF ! N = NL + NR + 1 M = N + SQRE ! ! The following values are for bookkeeping purposes only. They are ! integer pointers which indicate the portion of the workspace ! used by a particular array in DLASD2 and DLASD3. ! LDU2 = N LDVT2 = M ! IZ = 1 ISIGMA = IZ + M IU2 = ISIGMA + N IVT2 = IU2 + LDU2*N IQ = IVT2 + LDVT2*M ! IDX = 1 IDXC = IDX + N COLTYP = IDXC + N IDXP = COLTYP + N ! ! Scale. ! ORGNRM = MAX( ABS( ALPHA ), ABS( BETA ) ) D( NL+1 ) = ZERO DO 10 I = 1, N IF( ABS( D( I ) ).GT.ORGNRM ) THEN ORGNRM = ABS( D( I ) ) END IF 10 CONTINUE CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO ) ALPHA = ALPHA / ORGNRM BETA = BETA / ORGNRM ! ! Deflate singular values. ! CALL DLASD2( NL, NR, SQRE, K, D, WORK( IZ ), ALPHA, BETA, U, LDU, & VT, LDVT, WORK( ISIGMA ), WORK( IU2 ), LDU2, & WORK( IVT2 ), LDVT2, IWORK( IDXP ), IWORK( IDX ), & IWORK( IDXC ), IDXQ, IWORK( COLTYP ), INFO ) ! ! Solve Secular Equation and update singular vectors. ! LDQ = K CALL DLASD3( NL, NR, SQRE, K, D, WORK( IQ ), LDQ, WORK( ISIGMA ), & U, LDU, WORK( IU2 ), LDU2, VT, LDVT, WORK( IVT2 ), & LDVT2, IWORK( IDXC ), IWORK( COLTYP ), WORK( IZ ), & INFO ) IF( INFO.NE.0 ) THEN RETURN END IF ! ! Unscale. ! CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO ) ! ! Prepare the IDXQ sorting permutation. ! N1 = K N2 = N - K CALL DLAMRG( N1, N2, D, 1, -1, IDXQ ) ! RETURN ! ! End of DLASD1 ! END SUBROUTINE DLASD2( NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT, & LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX, & IDXC, IDXQ, COLTYP, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab, ! Courant Institute, NAG Ltd., and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE DOUBLE PRECISION ALPHA, BETA ! .. ! .. Array Arguments .. INTEGER COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ), & IDXQ( * ) DOUBLE PRECISION D( * ), DSIGMA( * ), U( LDU, * ), & U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ), & Z( * ) ! .. ! ! Purpose ! ======= ! ! DLASD2 merges the two sets of singular values together into a single ! sorted set. Then it tries to deflate the size of the problem. ! There are two ways in which deflation can occur: when two or more ! singular values are close together or if there is a tiny entry in the ! Z vector. For each such occurrence the order of the related secular ! equation problem is reduced by one. ! ! DLASD2 is called from DLASD1. ! ! Arguments ! ========= ! ! NL (input) INTEGER ! The row dimension of the upper block. NL >= 1. ! ! NR (input) INTEGER ! The row dimension of the lower block. NR >= 1. ! ! SQRE (input) INTEGER ! = 0: the lower block is an NR-by-NR square matrix. ! = 1: the lower block is an NR-by-(NR+1) rectangular matrix. ! ! The bidiagonal matrix has N = NL + NR + 1 rows and ! M = N + SQRE >= N columns. ! ! K (output) INTEGER ! Contains the dimension of the non-deflated matrix, ! This is the order of the related secular equation. 1 <= K <=N. ! ! D (input/output) DOUBLE PRECISION array, dimension(N) ! On entry D contains the singular values of the two submatrices ! to be combined. On exit D contains the trailing (N-K) updated ! singular values (those which were deflated) sorted into ! increasing order. ! ! ALPHA (input) DOUBLE PRECISION ! Contains the diagonal element associated with the added row. ! ! BETA (input) DOUBLE PRECISION ! Contains the off-diagonal element associated with the added ! row. ! ! U (input/output) DOUBLE PRECISION array, dimension(LDU,N) ! On entry U contains the left singular vectors of two ! submatrices in the two square blocks with corners at (1,1), ! (NL, NL), and (NL+2, NL+2), (N,N). ! On exit U contains the trailing (N-K) updated left singular ! vectors (those which were deflated) in its last N-K columns. ! ! LDU (input) INTEGER ! The leading dimension of the array U. LDU >= N. ! ! Z (output) DOUBLE PRECISION array, dimension(N) ! On exit Z contains the updating row vector in the secular ! equation. ! ! DSIGMA (output) DOUBLE PRECISION array, dimension (N) ! Contains a copy of the diagonal elements (K-1 singular values ! and one zero) in the secular equation. ! ! U2 (output) DOUBLE PRECISION array, dimension(LDU2,N) ! Contains a copy of the first K-1 left singular vectors which ! will be used by DLASD3 in a matrix multiply (DGEMM) to solve ! for the new left singular vectors. U2 is arranged into four ! blocks. The first block contains a column with 1 at NL+1 and ! zero everywhere else; the second block contains non-zero ! entries only at and above NL; the third contains non-zero ! entries only below NL+1; and the fourth is dense. ! ! LDU2 (input) INTEGER ! The leading dimension of the array U2. LDU2 >= N. ! ! VT (input/output) DOUBLE PRECISION array, dimension(LDVT,M) ! On entry VT' contains the right singular vectors of two ! submatrices in the two square blocks with corners at (1,1), ! (NL+1, NL+1), and (NL+2, NL+2), (M,M). ! On exit VT' contains the trailing (N-K) updated right singular ! vectors (those which were deflated) in its last N-K columns. ! In case SQRE =1, the last row of VT spans the right null ! space. ! ! LDVT (input) INTEGER ! The leading dimension of the array VT. LDVT >= M. ! ! VT2 (output) DOUBLE PRECISION array, dimension(LDVT2,N) ! VT2' contains a copy of the first K right singular vectors ! which will be used by DLASD3 in a matrix multiply (DGEMM) to ! solve for the new right singular vectors. VT2 is arranged into ! three blocks. The first block contains a row that corresponds ! to the special 0 diagonal element in SIGMA; the second block ! contains non-zeros only at and before NL +1; the third block ! contains non-zeros only at and after NL +2. ! ! LDVT2 (input) INTEGER ! The leading dimension of the array VT2. LDVT2 >= M. ! ! IDXP (workspace) INTEGER array, dimension(N) ! This will contain the permutation used to place deflated ! values of D at the end of the array. On output IDXP(2:K) ! points to the nondeflated D-values and IDXP(K+1:N) ! points to the deflated singular values. ! ! IDX (workspace) INTEGER array, dimension(N) ! This will contain the permutation used to sort the contents of ! D into ascending order. ! ! IDXC (output) INTEGER array, dimension(N) ! This will contain the permutation used to arrange the columns ! of the deflated U matrix into three groups: the first group ! contains non-zero entries only at and above NL, the second ! contains non-zero entries only below NL+2, and the third is ! dense. ! ! COLTYP (workspace/output) INTEGER array, dimension(N) ! As workspace, this will contain a label which will indicate ! which of the following types a column in the U2 matrix or a ! row in the VT2 matrix is: ! 1 : non-zero in the upper half only ! 2 : non-zero in the lower half only ! 3 : dense ! 4 : deflated ! ! On exit, it is an array of dimension 4, with COLTYP(I) being ! the dimension of the I-th type columns. ! ! IDXQ (input) INTEGER array, dimension(N) ! This contains the permutation which separately sorts the two ! sub-problems in D into ascending order. Note that entries in ! the first hlaf of this permutation must first be moved one ! position backward; and entries in the second half ! must first have NL+1 added to their values. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Huan Ren, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO, EIGHT PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0, & EIGHT = 8.0D+0 ) ! .. ! .. Local Arrays .. INTEGER CTOT( 4 ), PSM( 4 ) ! .. ! .. Local Scalars .. INTEGER CT, I, IDXI, IDXJ, IDXJP, J, JP, JPREV, K2, M, & N, NLP1, NLP2 DOUBLE PRECISION C, EPS, HLFTOL, S, TAU, TOL, Z1 ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH, DLAPY2 EXTERNAL DLAMCH, DLAPY2 ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLACPY, DLAMRG, DLASET, DROT, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( NL.LT.1 ) THEN INFO = -1 ELSE IF( NR.LT.1 ) THEN INFO = -2 ELSE IF( ( SQRE.NE.1 ) .AND. ( SQRE.NE.0 ) ) THEN INFO = -3 END IF ! N = NL + NR + 1 M = N + SQRE ! IF( LDU.LT.N ) THEN INFO = -10 ELSE IF( LDVT.LT.M ) THEN INFO = -12 ELSE IF( LDU2.LT.N ) THEN INFO = -15 ELSE IF( LDVT2.LT.M ) THEN INFO = -17 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLASD2', -INFO ) RETURN END IF ! NLP1 = NL + 1 NLP2 = NL + 2 ! ! Generate the first part of the vector Z; and move the singular ! values in the first part of D one position backward. ! Z1 = ALPHA*VT( NLP1, NLP1 ) Z( 1 ) = Z1 DO 10 I = NL, 1, -1 Z( I+1 ) = ALPHA*VT( I, NLP1 ) D( I+1 ) = D( I ) IDXQ( I+1 ) = IDXQ( I ) + 1 10 CONTINUE ! ! Generate the second part of the vector Z. ! DO 20 I = NLP2, M Z( I ) = BETA*VT( I, NLP2 ) 20 CONTINUE ! ! Initialize some reference arrays. ! DO 30 I = 2, NLP1 COLTYP( I ) = 1 30 CONTINUE DO 40 I = NLP2, N COLTYP( I ) = 2 40 CONTINUE ! ! Sort the singular values into increasing order ! DO 50 I = NLP2, N IDXQ( I ) = IDXQ( I ) + NLP1 50 CONTINUE ! ! DSIGMA, IDXC, IDXC, and the first column of U2 ! are used as storage space. ! DO 60 I = 2, N DSIGMA( I ) = D( IDXQ( I ) ) U2( I, 1 ) = Z( IDXQ( I ) ) IDXC( I ) = COLTYP( IDXQ( I ) ) 60 CONTINUE ! CALL DLAMRG( NL, NR, DSIGMA( 2 ), 1, 1, IDX( 2 ) ) ! DO 70 I = 2, N IDXI = 1 + IDX( I ) D( I ) = DSIGMA( IDXI ) Z( I ) = U2( IDXI, 1 ) COLTYP( I ) = IDXC( IDXI ) 70 CONTINUE ! ! Calculate the allowable deflation tolerance ! EPS = DLAMCH( 'Epsilon' ) TOL = MAX( ABS( ALPHA ), ABS( BETA ) ) TOL = EIGHT*EPS*MAX( ABS( D( N ) ), TOL ) ! ! There are 2 kinds of deflation -- first a value in the z-vector ! is small, second two (or more) singular values are very close ! together (their difference is small). ! ! If the value in the z-vector is small, we simply permute the ! array so that the corresponding singular value is moved to the ! end. ! ! If two values in the D-vector are close, we perform a two-sided ! rotation designed to make one of the corresponding z-vector ! entries zero, and then permute the array so that the deflated ! singular value is moved to the end. ! ! If there are multiple singular values then the problem deflates. ! Here the number of equal singular values are found. As each equal ! singular value is found, an elementary reflector is computed to ! rotate the corresponding singular subspace so that the ! corresponding components of Z are zero in this new basis. ! K = 1 K2 = N + 1 DO 80 J = 2, N IF( ABS( Z( J ) ).LE.TOL ) THEN ! ! Deflate due to small z component. ! K2 = K2 - 1 IDXP( K2 ) = J COLTYP( J ) = 4 IF( J.EQ.N ) & GO TO 120 ELSE JPREV = J GO TO 90 END IF 80 CONTINUE 90 CONTINUE J = JPREV 100 CONTINUE J = J + 1 IF( J.GT.N ) & GO TO 110 IF( ABS( Z( J ) ).LE.TOL ) THEN ! ! Deflate due to small z component. ! K2 = K2 - 1 IDXP( K2 ) = J COLTYP( J ) = 4 ELSE ! ! Check if singular values are close enough to allow deflation. ! IF( ABS( D( J )-D( JPREV ) ).LE.TOL ) THEN ! ! Deflation is possible. ! S = Z( JPREV ) C = Z( J ) ! ! Find sqrt(a**2+b**2) without overflow or ! destructive underflow. ! TAU = DLAPY2( C, S ) C = C / TAU S = -S / TAU Z( J ) = TAU Z( JPREV ) = ZERO ! ! Apply back the Givens rotation to the left and right ! singular vector matrices. ! IDXJP = IDXQ( IDX( JPREV )+1 ) IDXJ = IDXQ( IDX( J )+1 ) IF( IDXJP.LE.NLP1 ) THEN IDXJP = IDXJP - 1 END IF IF( IDXJ.LE.NLP1 ) THEN IDXJ = IDXJ - 1 END IF CALL DROT( N, U( 1, IDXJP ), 1, U( 1, IDXJ ), 1, C, S ) CALL DROT( M, VT( IDXJP, 1 ), LDVT, VT( IDXJ, 1 ), LDVT, C, & S ) IF( COLTYP( J ).NE.COLTYP( JPREV ) ) THEN COLTYP( J ) = 3 END IF COLTYP( JPREV ) = 4 K2 = K2 - 1 IDXP( K2 ) = JPREV JPREV = J ELSE K = K + 1 U2( K, 1 ) = Z( JPREV ) DSIGMA( K ) = D( JPREV ) IDXP( K ) = JPREV JPREV = J END IF END IF GO TO 100 110 CONTINUE ! ! Record the last singular value. ! K = K + 1 U2( K, 1 ) = Z( JPREV ) DSIGMA( K ) = D( JPREV ) IDXP( K ) = JPREV ! 120 CONTINUE ! ! Count up the total number of the various types of columns, then ! form a permutation which positions the four column types into ! four groups of uniform structure (although one or more of these ! groups may be empty). ! DO 130 J = 1, 4 CTOT( J ) = 0 130 CONTINUE DO 140 J = 2, N CT = COLTYP( J ) CTOT( CT ) = CTOT( CT ) + 1 140 CONTINUE ! ! PSM(*) = Position in SubMatrix (of types 1 through 4) ! PSM( 1 ) = 2 PSM( 2 ) = 2 + CTOT( 1 ) PSM( 3 ) = PSM( 2 ) + CTOT( 2 ) PSM( 4 ) = PSM( 3 ) + CTOT( 3 ) ! ! Fill out the IDXC array so that the permutation which it induces ! will place all type-1 columns first, all type-2 columns next, ! then all type-3's, and finally all type-4's, starting from the ! second column. This applies similarly to the rows of VT. ! DO 150 J = 2, N JP = IDXP( J ) CT = COLTYP( JP ) IDXC( PSM( CT ) ) = J PSM( CT ) = PSM( CT ) + 1 150 CONTINUE ! ! Sort the singular values and corresponding singular vectors into ! DSIGMA, U2, and VT2 respectively. The singular values/vectors ! which were not deflated go into the first K slots of DSIGMA, U2, ! and VT2 respectively, while those which were deflated go into the ! last N - K slots, except that the first column/row will be treated ! separately. ! DO 160 J = 2, N JP = IDXP( J ) DSIGMA( J ) = D( JP ) IDXJ = IDXQ( IDX( IDXP( IDXC( J ) ) )+1 ) IF( IDXJ.LE.NLP1 ) THEN IDXJ = IDXJ - 1 END IF CALL DCOPY( N, U( 1, IDXJ ), 1, U2( 1, J ), 1 ) CALL DCOPY( M, VT( IDXJ, 1 ), LDVT, VT2( J, 1 ), LDVT2 ) 160 CONTINUE ! ! Determine DSIGMA(1), DSIGMA(2) and Z(1) ! DSIGMA( 1 ) = ZERO HLFTOL = TOL / TWO IF( ABS( DSIGMA( 2 ) ).LE.HLFTOL ) & DSIGMA( 2 ) = HLFTOL IF( M.GT.N ) THEN Z( 1 ) = DLAPY2( Z1, Z( M ) ) IF( Z( 1 ).LE.TOL ) THEN C = ONE S = ZERO Z( 1 ) = TOL ELSE C = Z1 / Z( 1 ) S = Z( M ) / Z( 1 ) END IF ELSE IF( ABS( Z1 ).LE.TOL ) THEN Z( 1 ) = TOL ELSE Z( 1 ) = Z1 END IF END IF ! ! Move the rest of the updating row to Z. ! CALL DCOPY( K-1, U2( 2, 1 ), 1, Z( 2 ), 1 ) ! ! Determine the first column of U2, the first row of VT2 and the ! last row of VT. ! CALL DLASET( 'A', N, 1, ZERO, ZERO, U2, LDU2 ) U2( NLP1, 1 ) = ONE IF( M.GT.N ) THEN DO 170 I = 1, NLP1 VT( M, I ) = -S*VT( NLP1, I ) VT2( 1, I ) = C*VT( NLP1, I ) 170 CONTINUE DO 180 I = NLP2, M VT2( 1, I ) = S*VT( M, I ) VT( M, I ) = C*VT( M, I ) 180 CONTINUE ELSE CALL DCOPY( M, VT( NLP1, 1 ), LDVT, VT2( 1, 1 ), LDVT2 ) END IF IF( M.GT.N ) THEN CALL DCOPY( M, VT( M, 1 ), LDVT, VT2( M, 1 ), LDVT2 ) END IF ! ! The deflated singular values and their corresponding vectors go ! into the back of D, U, and V respectively. ! IF( N.GT.K ) THEN CALL DCOPY( N-K, DSIGMA( K+1 ), 1, D( K+1 ), 1 ) CALL DLACPY( 'A', N, N-K, U2( 1, K+1 ), LDU2, U( 1, K+1 ), & LDU ) CALL DLACPY( 'A', N-K, M, VT2( K+1, 1 ), LDVT2, VT( K+1, 1 ), & LDVT ) END IF ! ! Copy CTOT into COLTYP for referencing in DLASD3. ! DO 190 J = 1, 4 COLTYP( J ) = CTOT( J ) 190 CONTINUE ! RETURN ! ! End of DLASD2 ! END SUBROUTINE DLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2, & LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z, & INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab, ! Courant Institute, NAG Ltd., and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR, & SQRE ! .. ! .. Array Arguments .. INTEGER CTOT( * ), IDXC( * ) DOUBLE PRECISION D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ), & U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ), & Z( * ) ! .. ! ! Purpose ! ======= ! ! DLASD3 finds all the square roots of the roots of the secular ! equation, as defined by the values in D and Z. It makes the ! appropriate calls to DLASD4 and then updates the singular ! vectors by matrix multiplication. ! ! This code makes very mild assumptions about floating point ! arithmetic. It will work on machines with a guard digit in ! add/subtract, or on those binary machines without guard digits ! which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2. ! It could conceivably fail on hexadecimal or decimal machines ! without guard digits, but we know of none. ! ! DLASD3 is called from DLASD1. ! ! Arguments ! ========= ! ! NL (input) INTEGER ! The row dimension of the upper block. NL >= 1. ! ! NR (input) INTEGER ! The row dimension of the lower block. NR >= 1. ! ! SQRE (input) INTEGER ! = 0: the lower block is an NR-by-NR square matrix. ! = 1: the lower block is an NR-by-(NR+1) rectangular matrix. ! ! The bidiagonal matrix has N = NL + NR + 1 rows and ! M = N + SQRE >= N columns. ! ! K (input) INTEGER ! The size of the secular equation, 1 =< K = < N. ! ! D (output) DOUBLE PRECISION array, dimension(K) ! On exit the square roots of the roots of the secular equation, ! in ascending order. ! ! Q (workspace) DOUBLE PRECISION array, ! dimension at least (LDQ,K). ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. LDQ >= K. ! ! DSIGMA (input) DOUBLE PRECISION array, dimension(K) ! The first K elements of this array contain the old roots ! of the deflated updating problem. These are the poles ! of the secular equation. ! ! U (input) DOUBLE PRECISION array, dimension (LDU, N) ! The last N - K columns of this matrix contain the deflated ! left singular vectors. ! ! LDU (input) INTEGER ! The leading dimension of the array U. LDU >= N. ! ! U2 (input) DOUBLE PRECISION array, dimension (LDU2, N) ! The first K columns of this matrix contain the non-deflated ! left singular vectors for the split problem. ! ! LDU2 (input) INTEGER ! The leading dimension of the array U2. LDU2 >= N. ! ! VT (input) DOUBLE PRECISION array, dimension (LDVT, M) ! The last M - K columns of VT' contain the deflated ! right singular vectors. ! ! LDVT (input) INTEGER ! The leading dimension of the array VT. LDVT >= N. ! ! VT2 (input) DOUBLE PRECISION array, dimension (LDVT2, N) ! The first K columns of VT2' contain the non-deflated ! right singular vectors for the split problem. ! ! LDVT2 (input) INTEGER ! The leading dimension of the array VT2. LDVT2 >= N. ! ! IDXC (input) INTEGER array, dimension ( N ) ! The permutation used to arrange the columns of U (and rows of ! VT) into three groups: the first group contains non-zero ! entries only at and above (or before) NL +1; the second ! contains non-zero entries only at and below (or after) NL+2; ! and the third is dense. The first column of U and the row of ! VT are treated separately, however. ! ! The rows of the singular vectors found by DLASD4 ! must be likewise permuted before the matrix multiplies can ! take place. ! ! CTOT (input) INTEGER array, dimension ( 4 ) ! A count of the total number of the various types of columns ! in U (or rows in VT), as described in IDXC. The fourth column ! type is any column which has been deflated. ! ! Z (input) DOUBLE PRECISION array, dimension (K) ! The first K elements of this array contain the components ! of the deflation-adjusted updating row vector. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = 1, an singular value did not converge ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Huan Ren, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO, NEGONE PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0, & NEGONE = -1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER CTEMP, I, J, JC, KTEMP, M, N, NLP1, NLP2, NRP1 DOUBLE PRECISION RHO, TEMP ! .. ! .. External Functions .. DOUBLE PRECISION DLAMC3, DNRM2 EXTERNAL DLAMC3, DNRM2 ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEMM, DLACPY, DLASCL, DLASD4, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, SIGN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( NL.LT.1 ) THEN INFO = -1 ELSE IF( NR.LT.1 ) THEN INFO = -2 ELSE IF( ( SQRE.NE.1 ) .AND. ( SQRE.NE.0 ) ) THEN INFO = -3 END IF ! N = NL + NR + 1 M = N + SQRE NLP1 = NL + 1 NLP2 = NL + 2 ! IF( ( K.LT.1 ) .OR. ( K.GT.N ) ) THEN INFO = -4 ELSE IF( LDQ.LT.K ) THEN INFO = -7 ELSE IF( LDU.LT.N ) THEN INFO = -10 ELSE IF( LDU2.LT.N ) THEN INFO = -12 ELSE IF( LDVT.LT.M ) THEN INFO = -14 ELSE IF( LDVT2.LT.M ) THEN INFO = -16 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLASD3', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( K.EQ.1 ) THEN D( 1 ) = ABS( Z( 1 ) ) CALL DCOPY( M, VT2( 1, 1 ), LDVT2, VT( 1, 1 ), LDVT ) IF( Z( 1 ).GT.ZERO ) THEN CALL DCOPY( N, U2( 1, 1 ), 1, U( 1, 1 ), 1 ) ELSE DO 10 I = 1, N U( I, 1 ) = -U2( I, 1 ) 10 CONTINUE END IF RETURN END IF ! ! Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(j) can ! be computed with high relative accuracy (barring over/underflow). ! This is a problem on machines without a guard digit in ! add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2). ! The following code replaces DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I), ! which on any of these machines zeros out the bottommost ! bit of DSIGMA(I) if it is 1; this makes the subsequent ! subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation ! occurs. On binary machines with a guard digit (almost all ! machines) it does not change DSIGMA(I) at all. On hexadecimal ! and decimal machines with a guard digit, it slightly ! changes the bottommost bits of DSIGMA(I). It does not account ! for hexadecimal or decimal machines without guard digits ! (we know of none). We use a subroutine call to compute ! 2*DLAMBDA(I) to prevent optimizing compilers from eliminating ! this code. ! DO 20 I = 1, K DSIGMA( I ) = DLAMC3( DSIGMA( I ), DSIGMA( I ) ) - DSIGMA( I ) 20 CONTINUE ! ! Keep a copy of Z. ! CALL DCOPY( K, Z, 1, Q, 1 ) ! ! Normalize Z. ! RHO = DNRM2( K, Z, 1 ) CALL DLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO ) RHO = RHO*RHO ! ! Find the new singular values. ! DO 30 J = 1, K CALL DLASD4( K, J, DSIGMA, Z, U( 1, J ), RHO, D( J ), & VT( 1, J ), INFO ) ! ! If the zero finder fails, the computation is terminated. ! IF( INFO.NE.0 ) THEN RETURN END IF 30 CONTINUE ! ! Compute updated Z. ! DO 60 I = 1, K Z( I ) = U( I, K )*VT( I, K ) DO 40 J = 1, I - 1 Z( I ) = Z( I )*( U( I, J )*VT( I, J ) / & ( DSIGMA( I )-DSIGMA( J ) ) / & ( DSIGMA( I )+DSIGMA( J ) ) ) 40 CONTINUE DO 50 J = I, K - 1 Z( I ) = Z( I )*( U( I, J )*VT( I, J ) / & ( DSIGMA( I )-DSIGMA( J+1 ) ) / & ( DSIGMA( I )+DSIGMA( J+1 ) ) ) 50 CONTINUE Z( I ) = SIGN( SQRT( ABS( Z( I ) ) ), Q( I, 1 ) ) 60 CONTINUE ! ! Compute left singular vectors of the modified diagonal matrix, ! and store related information for the right singular vectors. ! DO 90 I = 1, K VT( 1, I ) = Z( 1 ) / U( 1, I ) / VT( 1, I ) U( 1, I ) = NEGONE DO 70 J = 2, K VT( J, I ) = Z( J ) / U( J, I ) / VT( J, I ) U( J, I ) = DSIGMA( J )*VT( J, I ) 70 CONTINUE TEMP = DNRM2( K, U( 1, I ), 1 ) Q( 1, I ) = U( 1, I ) / TEMP DO 80 J = 2, K JC = IDXC( J ) Q( J, I ) = U( JC, I ) / TEMP 80 CONTINUE 90 CONTINUE ! ! Update the left singular vector matrix. ! IF( K.EQ.2 ) THEN CALL DGEMM( 'N', 'N', N, K, K, ONE, U2, LDU2, Q, LDQ, ZERO, U, & LDU ) GO TO 100 END IF IF( CTOT( 1 ).GT.0 ) THEN CALL DGEMM( 'N', 'N', NL, K, CTOT( 1 ), ONE, U2( 1, 2 ), LDU2, & Q( 2, 1 ), LDQ, ZERO, U( 1, 1 ), LDU ) IF( CTOT( 3 ).GT.0 ) THEN KTEMP = 2 + CTOT( 1 ) + CTOT( 2 ) CALL DGEMM( 'N', 'N', NL, K, CTOT( 3 ), ONE, U2( 1, KTEMP ), & LDU2, Q( KTEMP, 1 ), LDQ, ONE, U( 1, 1 ), LDU ) END IF ELSE IF( CTOT( 3 ).GT.0 ) THEN KTEMP = 2 + CTOT( 1 ) + CTOT( 2 ) CALL DGEMM( 'N', 'N', NL, K, CTOT( 3 ), ONE, U2( 1, KTEMP ), & LDU2, Q( KTEMP, 1 ), LDQ, ZERO, U( 1, 1 ), LDU ) ELSE CALL DLACPY( 'F', NL, K, U2, LDU2, U, LDU ) END IF CALL DCOPY( K, Q( 1, 1 ), LDQ, U( NLP1, 1 ), LDU ) KTEMP = 2 + CTOT( 1 ) CTEMP = CTOT( 2 ) + CTOT( 3 ) CALL DGEMM( 'N', 'N', NR, K, CTEMP, ONE, U2( NLP2, KTEMP ), LDU2, & Q( KTEMP, 1 ), LDQ, ZERO, U( NLP2, 1 ), LDU ) ! ! Generate the right singular vectors. ! 100 CONTINUE DO 120 I = 1, K TEMP = DNRM2( K, VT( 1, I ), 1 ) Q( I, 1 ) = VT( 1, I ) / TEMP DO 110 J = 2, K JC = IDXC( J ) Q( I, J ) = VT( JC, I ) / TEMP 110 CONTINUE 120 CONTINUE ! ! Update the right singular vector matrix. ! IF( K.EQ.2 ) THEN CALL DGEMM( 'N', 'N', K, M, K, ONE, Q, LDQ, VT2, LDVT2, ZERO, & VT, LDVT ) RETURN END IF KTEMP = 1 + CTOT( 1 ) CALL DGEMM( 'N', 'N', K, NLP1, KTEMP, ONE, Q( 1, 1 ), LDQ, & VT2( 1, 1 ), LDVT2, ZERO, VT( 1, 1 ), LDVT ) KTEMP = 2 + CTOT( 1 ) + CTOT( 2 ) IF( KTEMP.LE.LDVT2 ) & CALL DGEMM( 'N', 'N', K, NLP1, CTOT( 3 ), ONE, Q( 1, KTEMP ), & LDQ, VT2( KTEMP, 1 ), LDVT2, ONE, VT( 1, 1 ), & LDVT ) ! KTEMP = CTOT( 1 ) + 1 NRP1 = NR + SQRE IF( KTEMP.GT.1 ) THEN DO 130 I = 1, K Q( I, KTEMP ) = Q( I, 1 ) 130 CONTINUE DO 140 I = NLP2, M VT2( KTEMP, I ) = VT2( 1, I ) 140 CONTINUE END IF CTEMP = 1 + CTOT( 2 ) + CTOT( 3 ) CALL DGEMM( 'N', 'N', K, NRP1, CTEMP, ONE, Q( 1, KTEMP ), LDQ, & VT2( KTEMP, NLP2 ), LDVT2, ZERO, VT( 1, NLP2 ), LDVT ) ! RETURN ! ! End of DLASD3 ! END SUBROUTINE DLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab, ! Courant Institute, NAG Ltd., and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. INTEGER I, INFO, N DOUBLE PRECISION RHO, SIGMA ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ), DELTA( * ), WORK( * ), Z( * ) ! .. ! ! Purpose ! ======= ! ! This subroutine computes the square root of the I-th updated ! eigenvalue of a positive symmetric rank-one modification to ! a positive diagonal matrix whose entries are given as the squares ! of the corresponding entries in the array d, and that ! ! 0 <= D(i) < D(j) for i < j ! ! and that RHO > 0. This is arranged by the calling routine, and is ! no loss in generality. The rank-one modified system is thus ! ! diag( D ) * diag( D ) + RHO * Z * Z_transpose. ! ! where we assume the Euclidean norm of Z is 1. ! ! The method consists of approximating the rational functions in the ! secular equation by simpler interpolating rational functions. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The length of all arrays. ! ! I (input) INTEGER ! The index of the eigenvalue to be computed. 1 <= I <= N. ! ! D (input) DOUBLE PRECISION array, dimension ( N ) ! The original eigenvalues. It is assumed that they are in ! order, 0 <= D(I) < D(J) for I < J. ! ! Z (input) DOUBLE PRECISION array, dimension ( N ) ! The components of the updating vector. ! ! DELTA (output) DOUBLE PRECISION array, dimension ( N ) ! If N .ne. 1, DELTA contains (D(j) - sigma_I) in its j-th ! component. If N = 1, then DELTA(1) = 1. The vector DELTA ! contains the information necessary to construct the ! (singular) eigenvectors. ! ! RHO (input) DOUBLE PRECISION ! The scalar in the symmetric updating formula. ! ! SIGMA (output) DOUBLE PRECISION ! The computed lambda_I, the I-th updated eigenvalue. ! ! WORK (workspace) DOUBLE PRECISION array, dimension ( N ) ! If N .ne. 1, WORK contains (D(j) + sigma_I) in its j-th ! component. If N = 1, then WORK( 1 ) = 1. ! ! INFO (output) INTEGER ! = 0: successful exit ! > 0: if INFO = 1, the updating process failed. ! ! Internal Parameters ! =================== ! ! Logical variable ORGATI (origin-at-i?) is used for distinguishing ! whether D(i) or D(i+1) is treated as the origin. ! ! ORGATI = .true. origin at i ! ORGATI = .false. origin at i+1 ! ! Logical variable SWTCH3 (switch-for-3-poles?) is for noting ! if we are working with THREE poles! ! ! MAXIT is the maximum number of iterations allowed for each ! eigenvalue. ! ! Further Details ! =============== ! ! Based on contributions by ! Ren-Cang Li, Computer Science Division, University of California ! at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. INTEGER MAXIT PARAMETER ( MAXIT = 20 ) DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, EIGHT, TEN PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0, & THREE = 3.0D+0, FOUR = 4.0D+0, EIGHT = 8.0D+0, & TEN = 10.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL ORGATI, SWTCH, SWTCH3 INTEGER II, IIM1, IIP1, IP1, ITER, J, NITER DOUBLE PRECISION A, B, C, DELSQ, DELSQ2, DPHI, DPSI, DTIIM, & DTIIP, DTIPSQ, DTISQ, DTNSQ, DTNSQ1, DW, EPS, & ERRETM, ETA, PHI, PREW, PSI, RHOINV, SG2LB, & SG2UB, TAU, TEMP, TEMP1, TEMP2, W ! .. ! .. Local Arrays .. DOUBLE PRECISION DD( 3 ), ZZ( 3 ) ! .. ! .. External Subroutines .. EXTERNAL DLAED6, DLASD5 ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Since this routine is called in an inner loop, we do no argument ! checking. ! ! Quick return for N=1 and 2. ! INFO = 0 IF( N.EQ.1 ) THEN ! ! Presumably, I=1 upon entry ! SIGMA = SQRT( D( 1 )*D( 1 )+RHO*Z( 1 )*Z( 1 ) ) DELTA( 1 ) = ONE WORK( 1 ) = ONE RETURN END IF IF( N.EQ.2 ) THEN CALL DLASD5( I, D, Z, DELTA, RHO, SIGMA, WORK ) RETURN END IF ! ! Compute machine epsilon ! EPS = DLAMCH( 'Epsilon' ) RHOINV = ONE / RHO ! ! The case I = N ! IF( I.EQ.N ) THEN ! ! Initialize some basic variables ! II = N - 1 NITER = 1 ! ! Calculate initial guess ! TEMP = RHO / TWO ! ! If ||Z||_2 is not one, then TEMP should be set to ! RHO * ||Z||_2^2 / TWO ! TEMP1 = TEMP / ( D( N )+SQRT( D( N )*D( N )+TEMP ) ) DO 10 J = 1, N WORK( J ) = D( J ) + D( N ) + TEMP1 DELTA( J ) = ( D( J )-D( N ) ) - TEMP1 10 CONTINUE ! PSI = ZERO DO 20 J = 1, N - 2 PSI = PSI + Z( J )*Z( J ) / ( DELTA( J )*WORK( J ) ) 20 CONTINUE ! C = RHOINV + PSI W = C + Z( II )*Z( II ) / ( DELTA( II )*WORK( II ) ) + & Z( N )*Z( N ) / ( DELTA( N )*WORK( N ) ) ! IF( W.LE.ZERO ) THEN TEMP1 = SQRT( D( N )*D( N )+RHO ) TEMP = Z( N-1 )*Z( N-1 ) / ( ( D( N-1 )+TEMP1 )* & ( D( N )-D( N-1 )+RHO / ( D( N )+TEMP1 ) ) ) + & Z( N )*Z( N ) / RHO ! ! The following TAU is to approximate ! SIGMA_n^2 - D( N )*D( N ) ! IF( C.LE.TEMP ) THEN TAU = RHO ELSE DELSQ = ( D( N )-D( N-1 ) )*( D( N )+D( N-1 ) ) A = -C*DELSQ + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N ) B = Z( N )*Z( N )*DELSQ IF( A.LT.ZERO ) THEN TAU = TWO*B / ( SQRT( A*A+FOUR*B*C )-A ) ELSE TAU = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C ) END IF END IF ! ! It can be proved that ! D(N)^2+RHO/2 <= SIGMA_n^2 < D(N)^2+TAU <= D(N)^2+RHO ! ELSE DELSQ = ( D( N )-D( N-1 ) )*( D( N )+D( N-1 ) ) A = -C*DELSQ + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N ) B = Z( N )*Z( N )*DELSQ ! ! The following TAU is to approximate ! SIGMA_n^2 - D( N )*D( N ) ! IF( A.LT.ZERO ) THEN TAU = TWO*B / ( SQRT( A*A+FOUR*B*C )-A ) ELSE TAU = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C ) END IF ! ! It can be proved that ! D(N)^2 < D(N)^2+TAU < SIGMA(N)^2 < D(N)^2+RHO/2 ! END IF ! ! The following ETA is to approximate SIGMA_n - D( N ) ! ETA = TAU / ( D( N )+SQRT( D( N )*D( N )+TAU ) ) ! SIGMA = D( N ) + ETA DO 30 J = 1, N DELTA( J ) = ( D( J )-D( I ) ) - ETA WORK( J ) = D( J ) + D( I ) + ETA 30 CONTINUE ! ! Evaluate PSI and the derivative DPSI ! DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 40 J = 1, II TEMP = Z( J ) / ( DELTA( J )*WORK( J ) ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 40 CONTINUE ERRETM = ABS( ERRETM ) ! ! Evaluate PHI and the derivative DPHI ! TEMP = Z( N ) / ( DELTA( N )*WORK( N ) ) PHI = Z( N )*TEMP DPHI = TEMP*TEMP ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV + & ABS( TAU )*( DPSI+DPHI ) ! W = RHOINV + PHI + PSI ! ! Test for convergence ! IF( ABS( W ).LE.EPS*ERRETM ) THEN GO TO 240 END IF ! ! Calculate the new step ! NITER = NITER + 1 DTNSQ1 = WORK( N-1 )*DELTA( N-1 ) DTNSQ = WORK( N )*DELTA( N ) C = W - DTNSQ1*DPSI - DTNSQ*DPHI A = ( DTNSQ+DTNSQ1 )*W - DTNSQ*DTNSQ1*( DPSI+DPHI ) B = DTNSQ*DTNSQ1*W IF( C.LT.ZERO ) & C = ABS( C ) IF( C.EQ.ZERO ) THEN ETA = RHO - SIGMA*SIGMA ELSE IF( A.GE.ZERO ) THEN ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) ELSE ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) END IF ! ! Note, eta should be positive if w is negative, and ! eta should be negative otherwise. However, ! if for some reason caused by roundoff, eta*w > 0, ! we simply use one Newton step instead. This way ! will guarantee eta*w < 0. ! IF( W*ETA.GT.ZERO ) & ETA = -W / ( DPSI+DPHI ) TEMP = ETA - DTNSQ IF( TEMP.GT.RHO ) & ETA = RHO + DTNSQ ! TAU = TAU + ETA ETA = ETA / ( SIGMA+SQRT( ETA+SIGMA*SIGMA ) ) DO 50 J = 1, N DELTA( J ) = DELTA( J ) - ETA WORK( J ) = WORK( J ) + ETA 50 CONTINUE ! SIGMA = SIGMA + ETA ! ! Evaluate PSI and the derivative DPSI ! DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 60 J = 1, II TEMP = Z( J ) / ( WORK( J )*DELTA( J ) ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 60 CONTINUE ERRETM = ABS( ERRETM ) ! ! Evaluate PHI and the derivative DPHI ! TEMP = Z( N ) / ( WORK( N )*DELTA( N ) ) PHI = Z( N )*TEMP DPHI = TEMP*TEMP ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV + & ABS( TAU )*( DPSI+DPHI ) ! W = RHOINV + PHI + PSI ! ! Main loop to update the values of the array DELTA ! ITER = NITER + 1 ! DO 90 NITER = ITER, MAXIT ! ! Test for convergence ! IF( ABS( W ).LE.EPS*ERRETM ) THEN GO TO 240 END IF ! ! Calculate the new step ! DTNSQ1 = WORK( N-1 )*DELTA( N-1 ) DTNSQ = WORK( N )*DELTA( N ) C = W - DTNSQ1*DPSI - DTNSQ*DPHI A = ( DTNSQ+DTNSQ1 )*W - DTNSQ1*DTNSQ*( DPSI+DPHI ) B = DTNSQ1*DTNSQ*W IF( A.GE.ZERO ) THEN ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) ELSE ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) END IF ! ! Note, eta should be positive if w is negative, and ! eta should be negative otherwise. However, ! if for some reason caused by roundoff, eta*w > 0, ! we simply use one Newton step instead. This way ! will guarantee eta*w < 0. ! IF( W*ETA.GT.ZERO ) & ETA = -W / ( DPSI+DPHI ) TEMP = ETA - DTNSQ IF( TEMP.LE.ZERO ) & ETA = ETA / TWO ! TAU = TAU + ETA ETA = ETA / ( SIGMA+SQRT( ETA+SIGMA*SIGMA ) ) DO 70 J = 1, N DELTA( J ) = DELTA( J ) - ETA WORK( J ) = WORK( J ) + ETA 70 CONTINUE ! SIGMA = SIGMA + ETA ! ! Evaluate PSI and the derivative DPSI ! DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 80 J = 1, II TEMP = Z( J ) / ( WORK( J )*DELTA( J ) ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 80 CONTINUE ERRETM = ABS( ERRETM ) ! ! Evaluate PHI and the derivative DPHI ! TEMP = Z( N ) / ( WORK( N )*DELTA( N ) ) PHI = Z( N )*TEMP DPHI = TEMP*TEMP ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV + & ABS( TAU )*( DPSI+DPHI ) ! W = RHOINV + PHI + PSI 90 CONTINUE ! ! Return with INFO = 1, NITER = MAXIT and not converged ! INFO = 1 GO TO 240 ! ! End for the case I = N ! ELSE ! ! The case for I < N ! NITER = 1 IP1 = I + 1 ! ! Calculate initial guess ! DELSQ = ( D( IP1 )-D( I ) )*( D( IP1 )+D( I ) ) DELSQ2 = DELSQ / TWO TEMP = DELSQ2 / ( D( I )+SQRT( D( I )*D( I )+DELSQ2 ) ) DO 100 J = 1, N WORK( J ) = D( J ) + D( I ) + TEMP DELTA( J ) = ( D( J )-D( I ) ) - TEMP 100 CONTINUE ! PSI = ZERO DO 110 J = 1, I - 1 PSI = PSI + Z( J )*Z( J ) / ( WORK( J )*DELTA( J ) ) 110 CONTINUE ! PHI = ZERO DO 120 J = N, I + 2, -1 PHI = PHI + Z( J )*Z( J ) / ( WORK( J )*DELTA( J ) ) 120 CONTINUE C = RHOINV + PSI + PHI W = C + Z( I )*Z( I ) / ( WORK( I )*DELTA( I ) ) + & Z( IP1 )*Z( IP1 ) / ( WORK( IP1 )*DELTA( IP1 ) ) ! IF( W.GT.ZERO ) THEN ! ! d(i)^2 < the ith sigma^2 < (d(i)^2+d(i+1)^2)/2 ! ! We choose d(i) as origin. ! ORGATI = .TRUE. SG2LB = ZERO SG2UB = DELSQ2 A = C*DELSQ + Z( I )*Z( I ) + Z( IP1 )*Z( IP1 ) B = Z( I )*Z( I )*DELSQ IF( A.GT.ZERO ) THEN TAU = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) ELSE TAU = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) END IF ! ! TAU now is an estimation of SIGMA^2 - D( I )^2. The ! following, however, is the corresponding estimation of ! SIGMA - D( I ). ! ETA = TAU / ( D( I )+SQRT( D( I )*D( I )+TAU ) ) ELSE ! ! (d(i)^2+d(i+1)^2)/2 <= the ith sigma^2 < d(i+1)^2/2 ! ! We choose d(i+1) as origin. ! ORGATI = .FALSE. SG2LB = -DELSQ2 SG2UB = ZERO A = C*DELSQ - Z( I )*Z( I ) - Z( IP1 )*Z( IP1 ) B = Z( IP1 )*Z( IP1 )*DELSQ IF( A.LT.ZERO ) THEN TAU = TWO*B / ( A-SQRT( ABS( A*A+FOUR*B*C ) ) ) ELSE TAU = -( A+SQRT( ABS( A*A+FOUR*B*C ) ) ) / ( TWO*C ) END IF ! ! TAU now is an estimation of SIGMA^2 - D( IP1 )^2. The ! following, however, is the corresponding estimation of ! SIGMA - D( IP1 ). ! ETA = TAU / ( D( IP1 )+SQRT( ABS( D( IP1 )*D( IP1 )+ & TAU ) ) ) END IF ! IF( ORGATI ) THEN II = I SIGMA = D( I ) + ETA DO 130 J = 1, N WORK( J ) = D( J ) + D( I ) + ETA DELTA( J ) = ( D( J )-D( I ) ) - ETA 130 CONTINUE ELSE II = I + 1 SIGMA = D( IP1 ) + ETA DO 140 J = 1, N WORK( J ) = D( J ) + D( IP1 ) + ETA DELTA( J ) = ( D( J )-D( IP1 ) ) - ETA 140 CONTINUE END IF IIM1 = II - 1 IIP1 = II + 1 ! ! Evaluate PSI and the derivative DPSI ! DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 150 J = 1, IIM1 TEMP = Z( J ) / ( WORK( J )*DELTA( J ) ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 150 CONTINUE ERRETM = ABS( ERRETM ) ! ! Evaluate PHI and the derivative DPHI ! DPHI = ZERO PHI = ZERO DO 160 J = N, IIP1, -1 TEMP = Z( J ) / ( WORK( J )*DELTA( J ) ) PHI = PHI + Z( J )*TEMP DPHI = DPHI + TEMP*TEMP ERRETM = ERRETM + PHI 160 CONTINUE ! W = RHOINV + PHI + PSI ! ! W is the value of the secular function with ! its ii-th element removed. ! SWTCH3 = .FALSE. IF( ORGATI ) THEN IF( W.LT.ZERO ) & SWTCH3 = .TRUE. ELSE IF( W.GT.ZERO ) & SWTCH3 = .TRUE. END IF IF( II.EQ.1 .OR. II.EQ.N ) & SWTCH3 = .FALSE. ! TEMP = Z( II ) / ( WORK( II )*DELTA( II ) ) DW = DPSI + DPHI + TEMP*TEMP TEMP = Z( II )*TEMP W = W + TEMP ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV + & THREE*ABS( TEMP ) + ABS( TAU )*DW ! ! Test for convergence ! IF( ABS( W ).LE.EPS*ERRETM ) THEN GO TO 240 END IF ! IF( W.LE.ZERO ) THEN SG2LB = MAX( SG2LB, TAU ) ELSE SG2UB = MIN( SG2UB, TAU ) END IF ! ! Calculate the new step ! NITER = NITER + 1 IF( .NOT.SWTCH3 ) THEN DTIPSQ = WORK( IP1 )*DELTA( IP1 ) DTISQ = WORK( I )*DELTA( I ) IF( ORGATI ) THEN C = W - DTIPSQ*DW + DELSQ*( Z( I ) / DTISQ )**2 ELSE C = W - DTISQ*DW - DELSQ*( Z( IP1 ) / DTIPSQ )**2 END IF A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW B = DTIPSQ*DTISQ*W IF( C.EQ.ZERO ) THEN IF( A.EQ.ZERO ) THEN IF( ORGATI ) THEN A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*( DPSI+DPHI ) ELSE A = Z( IP1 )*Z( IP1 ) + DTISQ*DTISQ*( DPSI+DPHI ) END IF END IF ETA = B / A ELSE IF( A.LE.ZERO ) THEN ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) ELSE ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) END IF ELSE ! ! Interpolation using THREE most relevant poles ! DTIIM = WORK( IIM1 )*DELTA( IIM1 ) DTIIP = WORK( IIP1 )*DELTA( IIP1 ) TEMP = RHOINV + PSI + PHI IF( ORGATI ) THEN TEMP1 = Z( IIM1 ) / DTIIM TEMP1 = TEMP1*TEMP1 C = ( TEMP - DTIIP*( DPSI+DPHI ) ) - & ( D( IIM1 )-D( IIP1 ) )*( D( IIM1 )+D( IIP1 ) )*TEMP1 ZZ( 1 ) = Z( IIM1 )*Z( IIM1 ) IF( DPSI.LT.TEMP1 ) THEN ZZ( 3 ) = DTIIP*DTIIP*DPHI ELSE ZZ( 3 ) = DTIIP*DTIIP*( ( DPSI-TEMP1 )+DPHI ) END IF ELSE TEMP1 = Z( IIP1 ) / DTIIP TEMP1 = TEMP1*TEMP1 C = ( TEMP - DTIIM*( DPSI+DPHI ) ) - & ( D( IIP1 )-D( IIM1 ) )*( D( IIM1 )+D( IIP1 ) )*TEMP1 IF( DPHI.LT.TEMP1 ) THEN ZZ( 1 ) = DTIIM*DTIIM*DPSI ELSE ZZ( 1 ) = DTIIM*DTIIM*( DPSI+( DPHI-TEMP1 ) ) END IF ZZ( 3 ) = Z( IIP1 )*Z( IIP1 ) END IF ZZ( 2 ) = Z( II )*Z( II ) DD( 1 ) = DTIIM DD( 2 ) = DELTA( II )*WORK( II ) DD( 3 ) = DTIIP CALL DLAED6( NITER, ORGATI, C, DD, ZZ, W, ETA, INFO ) IF( INFO.NE.0 ) & GO TO 240 END IF ! ! Note, eta should be positive if w is negative, and ! eta should be negative otherwise. However, ! if for some reason caused by roundoff, eta*w > 0, ! we simply use one Newton step instead. This way ! will guarantee eta*w < 0. ! IF( W*ETA.GE.ZERO ) & ETA = -W / DW IF( ORGATI ) THEN TEMP1 = WORK( I )*DELTA( I ) TEMP = ETA - TEMP1 ELSE TEMP1 = WORK( IP1 )*DELTA( IP1 ) TEMP = ETA - TEMP1 END IF IF( TEMP.GT.SG2UB .OR. TEMP.LT.SG2LB ) THEN IF( W.LT.ZERO ) THEN ETA = ( SG2UB-TAU ) / TWO ELSE ETA = ( SG2LB-TAU ) / TWO END IF END IF ! TAU = TAU + ETA ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) ) ! PREW = W ! SIGMA = SIGMA + ETA DO 170 J = 1, N WORK( J ) = WORK( J ) + ETA DELTA( J ) = DELTA( J ) - ETA 170 CONTINUE ! ! Evaluate PSI and the derivative DPSI ! DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 180 J = 1, IIM1 TEMP = Z( J ) / ( WORK( J )*DELTA( J ) ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 180 CONTINUE ERRETM = ABS( ERRETM ) ! ! Evaluate PHI and the derivative DPHI ! DPHI = ZERO PHI = ZERO DO 190 J = N, IIP1, -1 TEMP = Z( J ) / ( WORK( J )*DELTA( J ) ) PHI = PHI + Z( J )*TEMP DPHI = DPHI + TEMP*TEMP ERRETM = ERRETM + PHI 190 CONTINUE ! TEMP = Z( II ) / ( WORK( II )*DELTA( II ) ) DW = DPSI + DPHI + TEMP*TEMP TEMP = Z( II )*TEMP W = RHOINV + PHI + PSI + TEMP ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV + & THREE*ABS( TEMP ) + ABS( TAU )*DW ! IF( W.LE.ZERO ) THEN SG2LB = MAX( SG2LB, TAU ) ELSE SG2UB = MIN( SG2UB, TAU ) END IF ! SWTCH = .FALSE. IF( ORGATI ) THEN IF( -W.GT.ABS( PREW ) / TEN ) & SWTCH = .TRUE. ELSE IF( W.GT.ABS( PREW ) / TEN ) & SWTCH = .TRUE. END IF ! ! Main loop to update the values of the array DELTA and WORK ! ITER = NITER + 1 ! DO 230 NITER = ITER, MAXIT ! ! Test for convergence ! IF( ABS( W ).LE.EPS*ERRETM ) THEN GO TO 240 END IF ! ! Calculate the new step ! IF( .NOT.SWTCH3 ) THEN DTIPSQ = WORK( IP1 )*DELTA( IP1 ) DTISQ = WORK( I )*DELTA( I ) IF( .NOT.SWTCH ) THEN IF( ORGATI ) THEN C = W - DTIPSQ*DW + DELSQ*( Z( I ) / DTISQ )**2 ELSE C = W - DTISQ*DW - DELSQ*( Z( IP1 ) / DTIPSQ )**2 END IF ELSE TEMP = Z( II ) / ( WORK( II )*DELTA( II ) ) IF( ORGATI ) THEN DPSI = DPSI + TEMP*TEMP ELSE DPHI = DPHI + TEMP*TEMP END IF C = W - DTISQ*DPSI - DTIPSQ*DPHI END IF A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW B = DTIPSQ*DTISQ*W IF( C.EQ.ZERO ) THEN IF( A.EQ.ZERO ) THEN IF( .NOT.SWTCH ) THEN IF( ORGATI ) THEN A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ* & ( DPSI+DPHI ) ELSE A = Z( IP1 )*Z( IP1 ) + & DTISQ*DTISQ*( DPSI+DPHI ) END IF ELSE A = DTISQ*DTISQ*DPSI + DTIPSQ*DTIPSQ*DPHI END IF END IF ETA = B / A ELSE IF( A.LE.ZERO ) THEN ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C ) ELSE ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) END IF ELSE ! ! Interpolation using THREE most relevant poles ! DTIIM = WORK( IIM1 )*DELTA( IIM1 ) DTIIP = WORK( IIP1 )*DELTA( IIP1 ) TEMP = RHOINV + PSI + PHI IF( SWTCH ) THEN C = TEMP - DTIIM*DPSI - DTIIP*DPHI ZZ( 1 ) = DTIIM*DTIIM*DPSI ZZ( 3 ) = DTIIP*DTIIP*DPHI ELSE IF( ORGATI ) THEN TEMP1 = Z( IIM1 ) / DTIIM TEMP1 = TEMP1*TEMP1 TEMP2 = ( D( IIM1 )-D( IIP1 ) )* & ( D( IIM1 )+D( IIP1 ) )*TEMP1 C = TEMP - DTIIP*( DPSI+DPHI ) - TEMP2 ZZ( 1 ) = Z( IIM1 )*Z( IIM1 ) IF( DPSI.LT.TEMP1 ) THEN ZZ( 3 ) = DTIIP*DTIIP*DPHI ELSE ZZ( 3 ) = DTIIP*DTIIP*( ( DPSI-TEMP1 )+DPHI ) END IF ELSE TEMP1 = Z( IIP1 ) / DTIIP TEMP1 = TEMP1*TEMP1 TEMP2 = ( D( IIP1 )-D( IIM1 ) )* & ( D( IIM1 )+D( IIP1 ) )*TEMP1 C = TEMP - DTIIM*( DPSI+DPHI ) - TEMP2 IF( DPHI.LT.TEMP1 ) THEN ZZ( 1 ) = DTIIM*DTIIM*DPSI ELSE ZZ( 1 ) = DTIIM*DTIIM*( DPSI+( DPHI-TEMP1 ) ) END IF ZZ( 3 ) = Z( IIP1 )*Z( IIP1 ) END IF END IF DD( 1 ) = DTIIM DD( 2 ) = DELTA( II )*WORK( II ) DD( 3 ) = DTIIP CALL DLAED6( NITER, ORGATI, C, DD, ZZ, W, ETA, INFO ) IF( INFO.NE.0 ) & GO TO 240 END IF ! ! Note, eta should be positive if w is negative, and ! eta should be negative otherwise. However, ! if for some reason caused by roundoff, eta*w > 0, ! we simply use one Newton step instead. This way ! will guarantee eta*w < 0. ! IF( W*ETA.GE.ZERO ) & ETA = -W / DW IF( ORGATI ) THEN TEMP1 = WORK( I )*DELTA( I ) TEMP = ETA - TEMP1 ELSE TEMP1 = WORK( IP1 )*DELTA( IP1 ) TEMP = ETA - TEMP1 END IF IF( TEMP.GT.SG2UB .OR. TEMP.LT.SG2LB ) THEN IF( W.LT.ZERO ) THEN ETA = ( SG2UB-TAU ) / TWO ELSE ETA = ( SG2LB-TAU ) / TWO END IF END IF ! TAU = TAU + ETA ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) ) ! SIGMA = SIGMA + ETA DO 200 J = 1, N WORK( J ) = WORK( J ) + ETA DELTA( J ) = DELTA( J ) - ETA 200 CONTINUE ! PREW = W ! ! Evaluate PSI and the derivative DPSI ! DPSI = ZERO PSI = ZERO ERRETM = ZERO DO 210 J = 1, IIM1 TEMP = Z( J ) / ( WORK( J )*DELTA( J ) ) PSI = PSI + Z( J )*TEMP DPSI = DPSI + TEMP*TEMP ERRETM = ERRETM + PSI 210 CONTINUE ERRETM = ABS( ERRETM ) ! ! Evaluate PHI and the derivative DPHI ! DPHI = ZERO PHI = ZERO DO 220 J = N, IIP1, -1 TEMP = Z( J ) / ( WORK( J )*DELTA( J ) ) PHI = PHI + Z( J )*TEMP DPHI = DPHI + TEMP*TEMP ERRETM = ERRETM + PHI 220 CONTINUE ! TEMP = Z( II ) / ( WORK( II )*DELTA( II ) ) DW = DPSI + DPHI + TEMP*TEMP TEMP = Z( II )*TEMP W = RHOINV + PHI + PSI + TEMP ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV + & THREE*ABS( TEMP ) + ABS( TAU )*DW IF( W*PREW.GT.ZERO .AND. ABS( W ).GT.ABS( PREW ) / TEN ) & SWTCH = .NOT.SWTCH ! IF( W.LE.ZERO ) THEN SG2LB = MAX( SG2LB, TAU ) ELSE SG2UB = MIN( SG2UB, TAU ) END IF ! 230 CONTINUE ! ! Return with INFO = 1, NITER = MAXIT and not converged ! INFO = 1 ! END IF ! 240 CONTINUE RETURN ! ! End of DLASD4 ! END SUBROUTINE DLASD5( I, D, Z, DELTA, RHO, DSIGMA, WORK ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab, ! Courant Institute, NAG Ltd., and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER I DOUBLE PRECISION DSIGMA, RHO ! .. ! .. Array Arguments .. DOUBLE PRECISION D( 2 ), DELTA( 2 ), WORK( 2 ), Z( 2 ) ! .. ! ! Purpose ! ======= ! ! This subroutine computes the square root of the I-th eigenvalue ! of a positive symmetric rank-one modification of a 2-by-2 diagonal ! matrix ! ! diag( D ) * diag( D ) + RHO * Z * transpose(Z) . ! ! The diagonal entries in the array D are assumed to satisfy ! ! 0 <= D(i) < D(j) for i < j . ! ! We also assume RHO > 0 and that the Euclidean norm of the vector ! Z is one. ! ! Arguments ! ========= ! ! I (input) INTEGER ! The index of the eigenvalue to be computed. I = 1 or I = 2. ! ! D (input) DOUBLE PRECISION array, dimension ( 2 ) ! The original eigenvalues. We assume 0 <= D(1) < D(2). ! ! Z (input) DOUBLE PRECISION array, dimension ( 2 ) ! The components of the updating vector. ! ! DELTA (output) DOUBLE PRECISION array, dimension ( 2 ) ! Contains (D(j) - lambda_I) in its j-th component. ! The vector DELTA contains the information necessary ! to construct the eigenvectors. ! ! RHO (input) DOUBLE PRECISION ! The scalar in the symmetric updating formula. ! ! DSIGMA (output) DOUBLE PRECISION ! The computed lambda_I, the I-th updated eigenvalue. ! ! WORK (workspace) DOUBLE PRECISION array, dimension ( 2 ) ! WORK contains (D(j) + sigma_I) in its j-th component. ! ! Further Details ! =============== ! ! Based on contributions by ! Ren-Cang Li, Computer Science Division, University of California ! at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0, & THREE = 3.0D+0, FOUR = 4.0D+0 ) ! .. ! .. Local Scalars .. DOUBLE PRECISION B, C, DEL, DELSQ, TAU, W ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, SQRT ! .. ! .. Executable Statements .. ! DEL = D( 2 ) - D( 1 ) DELSQ = DEL*( D( 2 )+D( 1 ) ) IF( I.EQ.1 ) THEN W = ONE + FOUR*RHO*( Z( 2 )*Z( 2 ) / ( D( 1 )+THREE*D( 2 ) )- & Z( 1 )*Z( 1 ) / ( THREE*D( 1 )+D( 2 ) ) ) / DEL IF( W.GT.ZERO ) THEN B = DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) ) C = RHO*Z( 1 )*Z( 1 )*DELSQ ! ! B > ZERO, always ! ! The following TAU is DSIGMA * DSIGMA - D( 1 ) * D( 1 ) ! TAU = TWO*C / ( B+SQRT( ABS( B*B-FOUR*C ) ) ) ! ! The following TAU is DSIGMA - D( 1 ) ! TAU = TAU / ( D( 1 )+SQRT( D( 1 )*D( 1 )+TAU ) ) DSIGMA = D( 1 ) + TAU DELTA( 1 ) = -TAU DELTA( 2 ) = DEL - TAU WORK( 1 ) = TWO*D( 1 ) + TAU WORK( 2 ) = ( D( 1 )+TAU ) + D( 2 ) ! DELTA( 1 ) = -Z( 1 ) / TAU ! DELTA( 2 ) = Z( 2 ) / ( DEL-TAU ) ELSE B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) ) C = RHO*Z( 2 )*Z( 2 )*DELSQ ! ! The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 ) ! IF( B.GT.ZERO ) THEN TAU = -TWO*C / ( B+SQRT( B*B+FOUR*C ) ) ELSE TAU = ( B-SQRT( B*B+FOUR*C ) ) / TWO END IF ! ! The following TAU is DSIGMA - D( 2 ) ! TAU = TAU / ( D( 2 )+SQRT( ABS( D( 2 )*D( 2 )+TAU ) ) ) DSIGMA = D( 2 ) + TAU DELTA( 1 ) = -( DEL+TAU ) DELTA( 2 ) = -TAU WORK( 1 ) = D( 1 ) + TAU + D( 2 ) WORK( 2 ) = TWO*D( 2 ) + TAU ! DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU ) ! DELTA( 2 ) = -Z( 2 ) / TAU END IF ! TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) ) ! DELTA( 1 ) = DELTA( 1 ) / TEMP ! DELTA( 2 ) = DELTA( 2 ) / TEMP ELSE ! ! Now I=2 ! B = -DELSQ + RHO*( Z( 1 )*Z( 1 )+Z( 2 )*Z( 2 ) ) C = RHO*Z( 2 )*Z( 2 )*DELSQ ! ! The following TAU is DSIGMA * DSIGMA - D( 2 ) * D( 2 ) ! IF( B.GT.ZERO ) THEN TAU = ( B+SQRT( B*B+FOUR*C ) ) / TWO ELSE TAU = TWO*C / ( -B+SQRT( B*B+FOUR*C ) ) END IF ! ! The following TAU is DSIGMA - D( 2 ) ! TAU = TAU / ( D( 2 )+SQRT( D( 2 )*D( 2 )+TAU ) ) DSIGMA = D( 2 ) + TAU DELTA( 1 ) = -( DEL+TAU ) DELTA( 2 ) = -TAU WORK( 1 ) = D( 1 ) + TAU + D( 2 ) WORK( 2 ) = TWO*D( 2 ) + TAU ! DELTA( 1 ) = -Z( 1 ) / ( DEL+TAU ) ! DELTA( 2 ) = -Z( 2 ) / TAU ! TEMP = SQRT( DELTA( 1 )*DELTA( 1 )+DELTA( 2 )*DELTA( 2 ) ) ! DELTA( 1 ) = DELTA( 1 ) / TEMP ! DELTA( 2 ) = DELTA( 2 ) / TEMP END IF RETURN ! ! End of DLASD5 ! END SUBROUTINE DLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA, & IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, & LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK, & IWORK, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL, & NR, SQRE DOUBLE PRECISION ALPHA, BETA, C, S ! .. ! .. Array Arguments .. INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ), & PERM( * ) DOUBLE PRECISION D( * ), DIFL( * ), DIFR( * ), & GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ), & VF( * ), VL( * ), WORK( * ), Z( * ) ! .. ! ! Purpose ! ======= ! ! DLASD6 computes the SVD of an updated upper bidiagonal matrix B ! obtained by merging two smaller ones by appending a row. This ! routine is used only for the problem which requires all singular ! values and optionally singular vector matrices in factored form. ! B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE. ! A related subroutine, DLASD1, handles the case in which all singular ! values and singular vectors of the bidiagonal matrix are desired. ! ! DLASD6 computes the SVD as follows: ! ! ( D1(in) 0 0 0 ) ! B = U(in) * ( Z1' a Z2' b ) * VT(in) ! ( 0 0 D2(in) 0 ) ! ! = U(out) * ( D(out) 0) * VT(out) ! ! where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M ! with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros ! elsewhere; and the entry b is empty if SQRE = 0. ! ! The singular values of B can be computed using D1, D2, the first ! components of all the right singular vectors of the lower block, and ! the last components of all the right singular vectors of the upper ! block. These components are stored and updated in VF and VL, ! respectively, in DLASD6. Hence U and VT are not explicitly ! referenced. ! ! The singular values are stored in D. The algorithm consists of two ! stages: ! ! The first stage consists of deflating the size of the problem ! when there are multiple singular values or if there is a zero ! in the Z vector. For each such occurence the dimension of the ! secular equation problem is reduced by one. This stage is ! performed by the routine DLASD7. ! ! The second stage consists of calculating the updated ! singular values. This is done by finding the roots of the ! secular equation via the routine DLASD4 (as called by DLASD8). ! This routine also updates VF and VL and computes the distances ! between the updated singular values and the old singular ! values. ! ! DLASD6 is called from DLASDA. ! ! Arguments ! ========= ! ! ICOMPQ (input) INTEGER ! Specifies whether singular vectors are to be computed in ! factored form: ! = 0: Compute singular values only. ! = 1: Compute singular vectors in factored form as well. ! ! NL (input) INTEGER ! The row dimension of the upper block. NL >= 1. ! ! NR (input) INTEGER ! The row dimension of the lower block. NR >= 1. ! ! SQRE (input) INTEGER ! = 0: the lower block is an NR-by-NR square matrix. ! = 1: the lower block is an NR-by-(NR+1) rectangular matrix. ! ! The bidiagonal matrix has row dimension N = NL + NR + 1, ! and column dimension M = N + SQRE. ! ! D (input/output) DOUBLE PRECISION array, dimension ( NL+NR+1 ). ! On entry D(1:NL,1:NL) contains the singular values of the ! upper block, and D(NL+2:N) contains the singular values ! of the lower block. On exit D(1:N) contains the singular ! values of the modified matrix. ! ! VF (input/output) DOUBLE PRECISION array, dimension ( M ) ! On entry, VF(1:NL+1) contains the first components of all ! right singular vectors of the upper block; and VF(NL+2:M) ! contains the first components of all right singular vectors ! of the lower block. On exit, VF contains the first components ! of all right singular vectors of the bidiagonal matrix. ! ! VL (input/output) DOUBLE PRECISION array, dimension ( M ) ! On entry, VL(1:NL+1) contains the last components of all ! right singular vectors of the upper block; and VL(NL+2:M) ! contains the last components of all right singular vectors of ! the lower block. On exit, VL contains the last components of ! all right singular vectors of the bidiagonal matrix. ! ! ALPHA (input) DOUBLE PRECISION ! Contains the diagonal element associated with the added row. ! ! BETA (input) DOUBLE PRECISION ! Contains the off-diagonal element associated with the added ! row. ! ! IDXQ (output) INTEGER array, dimension ( N ) ! This contains the permutation which will reintegrate the ! subproblem just solved back into sorted order, i.e. ! D( IDXQ( I = 1, N ) ) will be in ascending order. ! ! PERM (output) INTEGER array, dimension ( N ) ! The permutations (from deflation and sorting) to be applied ! to each block. Not referenced if ICOMPQ = 0. ! ! GIVPTR (output) INTEGER ! The number of Givens rotations which took place in this ! subproblem. Not referenced if ICOMPQ = 0. ! ! GIVCOL (output) INTEGER array, dimension ( LDGCOL, 2 ) ! Each pair of numbers indicates a pair of columns to take place ! in a Givens rotation. Not referenced if ICOMPQ = 0. ! ! LDGCOL (input) INTEGER ! leading dimension of GIVCOL, must be at least N. ! ! GIVNUM (output) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) ! Each number indicates the C or S value to be used in the ! corresponding Givens rotation. Not referenced if ICOMPQ = 0. ! ! LDGNUM (input) INTEGER ! The leading dimension of GIVNUM and POLES, must be at least N. ! ! POLES (output) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) ! On exit, POLES(1,*) is an array containing the new singular ! values obtained from solving the secular equation, and ! POLES(2,*) is an array containing the poles in the secular ! equation. Not referenced if ICOMPQ = 0. ! ! DIFL (output) DOUBLE PRECISION array, dimension ( N ) ! On exit, DIFL(I) is the distance between I-th updated ! (undeflated) singular value and the I-th (undeflated) old ! singular value. ! ! DIFR (output) DOUBLE PRECISION array, ! dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and ! dimension ( N ) if ICOMPQ = 0. ! On exit, DIFR(I, 1) is the distance between I-th updated ! (undeflated) singular value and the I+1-th (undeflated) old ! singular value. ! ! If ICOMPQ = 1, DIFR(1:K,2) is an array containing the ! normalizing factors for the right singular vector matrix. ! ! See DLASD8 for details on DIFL and DIFR. ! ! Z (output) DOUBLE PRECISION array, dimension ( M ) ! The first elements of this array contain the components ! of the deflation-adjusted updating row vector. ! ! K (output) INTEGER ! Contains the dimension of the non-deflated matrix, ! This is the order of the related secular equation. 1 <= K <=N. ! ! C (output) DOUBLE PRECISION ! C contains garbage if SQRE =0 and the C-value of a Givens ! rotation related to the right null space if SQRE = 1. ! ! S (output) DOUBLE PRECISION ! S contains garbage if SQRE =0 and the S-value of a Givens ! rotation related to the right null space if SQRE = 1. ! ! WORK (workspace) DOUBLE PRECISION array, dimension ( 4 * M ) ! ! IWORK (workspace) INTEGER array, dimension ( 3 * N ) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = 1, an singular value did not converge ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Huan Ren, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, IDX, IDXC, IDXP, ISIGMA, IVFW, IVLW, IW, M, & N, N1, N2 DOUBLE PRECISION ORGNRM ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLAMRG, DLASCL, DLASD7, DLASD8, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 N = NL + NR + 1 M = N + SQRE ! IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN INFO = -1 ELSE IF( NL.LT.1 ) THEN INFO = -2 ELSE IF( NR.LT.1 ) THEN INFO = -3 ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN INFO = -4 ELSE IF( LDGCOL.LT.N ) THEN INFO = -14 ELSE IF( LDGNUM.LT.N ) THEN INFO = -16 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLASD6', -INFO ) RETURN END IF ! ! The following values are for bookkeeping purposes only. They are ! integer pointers which indicate the portion of the workspace ! used by a particular array in DLASD7 and DLASD8. ! ISIGMA = 1 IW = ISIGMA + N IVFW = IW + M IVLW = IVFW + M ! IDX = 1 IDXC = IDX + N IDXP = IDXC + N ! ! Scale. ! ORGNRM = MAX( ABS( ALPHA ), ABS( BETA ) ) D( NL+1 ) = ZERO DO 10 I = 1, N IF( ABS( D( I ) ).GT.ORGNRM ) THEN ORGNRM = ABS( D( I ) ) END IF 10 CONTINUE CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO ) ALPHA = ALPHA / ORGNRM BETA = BETA / ORGNRM ! ! Sort and Deflate singular values. ! CALL DLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, WORK( IW ), VF, & WORK( IVFW ), VL, WORK( IVLW ), ALPHA, BETA, & WORK( ISIGMA ), IWORK( IDX ), IWORK( IDXP ), IDXQ, & PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S, & INFO ) ! ! Solve Secular Equation, compute DIFL, DIFR, and update VF, VL. ! CALL DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDGNUM, & WORK( ISIGMA ), WORK( IW ), INFO ) ! ! Save the poles if ICOMPQ = 1. ! IF( ICOMPQ.EQ.1 ) THEN CALL DCOPY( K, D, 1, POLES( 1, 1 ), 1 ) CALL DCOPY( K, WORK( ISIGMA ), 1, POLES( 1, 2 ), 1 ) END IF ! ! Unscale. ! CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO ) ! ! Prepare the IDXQ sorting permutation. ! N1 = K N2 = N - K CALL DLAMRG( N1, N2, D, 1, -1, IDXQ ) ! RETURN ! ! End of DLASD6 ! END SUBROUTINE DLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, ZW, VF, VFW, VL, & VLW, ALPHA, BETA, DSIGMA, IDX, IDXP, IDXQ, & PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, & C, S, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab, ! Courant Institute, NAG Ltd., and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL, & NR, SQRE DOUBLE PRECISION ALPHA, BETA, C, S ! .. ! .. Array Arguments .. INTEGER GIVCOL( LDGCOL, * ), IDX( * ), IDXP( * ), & IDXQ( * ), PERM( * ) DOUBLE PRECISION D( * ), DSIGMA( * ), GIVNUM( LDGNUM, * ), & VF( * ), VFW( * ), VL( * ), VLW( * ), Z( * ), & ZW( * ) ! .. ! ! Purpose ! ======= ! ! DLASD7 merges the two sets of singular values together into a single ! sorted set. Then it tries to deflate the size of the problem. There ! are two ways in which deflation can occur: when two or more singular ! values are close together or if there is a tiny entry in the Z ! vector. For each such occurrence the order of the related ! secular equation problem is reduced by one. ! ! DLASD7 is called from DLASD6. ! ! Arguments ! ========= ! ! ICOMPQ (input) INTEGER ! Specifies whether singular vectors are to be computed ! in compact form, as follows: ! = 0: Compute singular values only. ! = 1: Compute singular vectors of upper ! bidiagonal matrix in compact form. ! ! NL (input) INTEGER ! The row dimension of the upper block. NL >= 1. ! ! NR (input) INTEGER ! The row dimension of the lower block. NR >= 1. ! ! SQRE (input) INTEGER ! = 0: the lower block is an NR-by-NR square matrix. ! = 1: the lower block is an NR-by-(NR+1) rectangular matrix. ! ! The bidiagonal matrix has ! N = NL + NR + 1 rows and ! M = N + SQRE >= N columns. ! ! K (output) INTEGER ! Contains the dimension of the non-deflated matrix, this is ! the order of the related secular equation. 1 <= K <=N. ! ! D (input/output) DOUBLE PRECISION array, dimension ( N ) ! On entry D contains the singular values of the two submatrices ! to be combined. On exit D contains the trailing (N-K) updated ! singular values (those which were deflated) sorted into ! increasing order. ! ! Z (output) DOUBLE PRECISION array, dimension ( M ) ! On exit Z contains the updating row vector in the secular ! equation. ! ! ZW (workspace) DOUBLE PRECISION array, dimension ( M ) ! Workspace for Z. ! ! VF (input/output) DOUBLE PRECISION array, dimension ( M ) ! On entry, VF(1:NL+1) contains the first components of all ! right singular vectors of the upper block; and VF(NL+2:M) ! contains the first components of all right singular vectors ! of the lower block. On exit, VF contains the first components ! of all right singular vectors of the bidiagonal matrix. ! ! VFW (workspace) DOUBLE PRECISION array, dimension ( M ) ! Workspace for VF. ! ! VL (input/output) DOUBLE PRECISION array, dimension ( M ) ! On entry, VL(1:NL+1) contains the last components of all ! right singular vectors of the upper block; and VL(NL+2:M) ! contains the last components of all right singular vectors ! of the lower block. On exit, VL contains the last components ! of all right singular vectors of the bidiagonal matrix. ! ! VLW (workspace) DOUBLE PRECISION array, dimension ( M ) ! Workspace for VL. ! ! ALPHA (input) DOUBLE PRECISION ! Contains the diagonal element associated with the added row. ! ! BETA (input) DOUBLE PRECISION ! Contains the off-diagonal element associated with the added ! row. ! ! DSIGMA (output) DOUBLE PRECISION array, dimension ( N ) ! Contains a copy of the diagonal elements (K-1 singular values ! and one zero) in the secular equation. ! ! IDX (workspace) INTEGER array, dimension ( N ) ! This will contain the permutation used to sort the contents of ! D into ascending order. ! ! IDXP (workspace) INTEGER array, dimension ( N ) ! This will contain the permutation used to place deflated ! values of D at the end of the array. On output IDXP(2:K) ! points to the nondeflated D-values and IDXP(K+1:N) ! points to the deflated singular values. ! ! IDXQ (input) INTEGER array, dimension ( N ) ! This contains the permutation which separately sorts the two ! sub-problems in D into ascending order. Note that entries in ! the first half of this permutation must first be moved one ! position backward; and entries in the second half ! must first have NL+1 added to their values. ! ! PERM (output) INTEGER array, dimension ( N ) ! The permutations (from deflation and sorting) to be applied ! to each singular block. Not referenced if ICOMPQ = 0. ! ! GIVPTR (output) INTEGER ! The number of Givens rotations which took place in this ! subproblem. Not referenced if ICOMPQ = 0. ! ! GIVCOL (output) INTEGER array, dimension ( LDGCOL, 2 ) ! Each pair of numbers indicates a pair of columns to take place ! in a Givens rotation. Not referenced if ICOMPQ = 0. ! ! LDGCOL (input) INTEGER ! The leading dimension of GIVCOL, must be at least N. ! ! GIVNUM (output) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) ! Each number indicates the C or S value to be used in the ! corresponding Givens rotation. Not referenced if ICOMPQ = 0. ! ! LDGNUM (input) INTEGER ! The leading dimension of GIVNUM, must be at least N. ! ! C (output) DOUBLE PRECISION ! C contains garbage if SQRE =0 and the C-value of a Givens ! rotation related to the right null space if SQRE = 1. ! ! S (output) DOUBLE PRECISION ! S contains garbage if SQRE =0 and the S-value of a Givens ! rotation related to the right null space if SQRE = 1. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Huan Ren, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO, EIGHT PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0, & EIGHT = 8.0D+0 ) ! .. ! .. Local Scalars .. ! INTEGER I, IDXI, IDXJ, IDXJP, J, JP, JPREV, K2, M, N, & NLP1, NLP2 DOUBLE PRECISION EPS, HLFTOL, TAU, TOL, Z1 ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLAMRG, DROT, XERBLA ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH, DLAPY2 EXTERNAL DLAMCH, DLAPY2 ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 N = NL + NR + 1 M = N + SQRE ! IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN INFO = -1 ELSE IF( NL.LT.1 ) THEN INFO = -2 ELSE IF( NR.LT.1 ) THEN INFO = -3 ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN INFO = -4 ELSE IF( LDGCOL.LT.N ) THEN INFO = -22 ELSE IF( LDGNUM.LT.N ) THEN INFO = -24 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLASD7', -INFO ) RETURN END IF ! NLP1 = NL + 1 NLP2 = NL + 2 IF( ICOMPQ.EQ.1 ) THEN GIVPTR = 0 END IF ! ! Generate the first part of the vector Z and move the singular ! values in the first part of D one position backward. ! Z1 = ALPHA*VL( NLP1 ) VL( NLP1 ) = ZERO TAU = VF( NLP1 ) DO 10 I = NL, 1, -1 Z( I+1 ) = ALPHA*VL( I ) VL( I ) = ZERO VF( I+1 ) = VF( I ) D( I+1 ) = D( I ) IDXQ( I+1 ) = IDXQ( I ) + 1 10 CONTINUE VF( 1 ) = TAU ! ! Generate the second part of the vector Z. ! DO 20 I = NLP2, M Z( I ) = BETA*VF( I ) VF( I ) = ZERO 20 CONTINUE ! ! Sort the singular values into increasing order ! DO 30 I = NLP2, N IDXQ( I ) = IDXQ( I ) + NLP1 30 CONTINUE ! ! DSIGMA, IDXC, IDXC, and ZW are used as storage space. ! DO 40 I = 2, N DSIGMA( I ) = D( IDXQ( I ) ) ZW( I ) = Z( IDXQ( I ) ) VFW( I ) = VF( IDXQ( I ) ) VLW( I ) = VL( IDXQ( I ) ) 40 CONTINUE ! CALL DLAMRG( NL, NR, DSIGMA( 2 ), 1, 1, IDX( 2 ) ) ! DO 50 I = 2, N IDXI = 1 + IDX( I ) D( I ) = DSIGMA( IDXI ) Z( I ) = ZW( IDXI ) VF( I ) = VFW( IDXI ) VL( I ) = VLW( IDXI ) 50 CONTINUE ! ! Calculate the allowable deflation tolerence ! EPS = DLAMCH( 'Epsilon' ) TOL = MAX( ABS( ALPHA ), ABS( BETA ) ) TOL = EIGHT*EIGHT*EPS*MAX( ABS( D( N ) ), TOL ) ! ! There are 2 kinds of deflation -- first a value in the z-vector ! is small, second two (or more) singular values are very close ! together (their difference is small). ! ! If the value in the z-vector is small, we simply permute the ! array so that the corresponding singular value is moved to the ! end. ! ! If two values in the D-vector are close, we perform a two-sided ! rotation designed to make one of the corresponding z-vector ! entries zero, and then permute the array so that the deflated ! singular value is moved to the end. ! ! If there are multiple singular values then the problem deflates. ! Here the number of equal singular values are found. As each equal ! singular value is found, an elementary reflector is computed to ! rotate the corresponding singular subspace so that the ! corresponding components of Z are zero in this new basis. ! K = 1 K2 = N + 1 DO 60 J = 2, N IF( ABS( Z( J ) ).LE.TOL ) THEN ! ! Deflate due to small z component. ! K2 = K2 - 1 IDXP( K2 ) = J IF( J.EQ.N ) & GO TO 100 ELSE JPREV = J GO TO 70 END IF 60 CONTINUE 70 CONTINUE J = JPREV 80 CONTINUE J = J + 1 IF( J.GT.N ) & GO TO 90 IF( ABS( Z( J ) ).LE.TOL ) THEN ! ! Deflate due to small z component. ! K2 = K2 - 1 IDXP( K2 ) = J ELSE ! ! Check if singular values are close enough to allow deflation. ! IF( ABS( D( J )-D( JPREV ) ).LE.TOL ) THEN ! ! Deflation is possible. ! S = Z( JPREV ) C = Z( J ) ! ! Find sqrt(a**2+b**2) without overflow or ! destructive underflow. ! TAU = DLAPY2( C, S ) Z( J ) = TAU Z( JPREV ) = ZERO C = C / TAU S = -S / TAU ! ! Record the appropriate Givens rotation ! IF( ICOMPQ.EQ.1 ) THEN GIVPTR = GIVPTR + 1 IDXJP = IDXQ( IDX( JPREV )+1 ) IDXJ = IDXQ( IDX( J )+1 ) IF( IDXJP.LE.NLP1 ) THEN IDXJP = IDXJP - 1 END IF IF( IDXJ.LE.NLP1 ) THEN IDXJ = IDXJ - 1 END IF GIVCOL( GIVPTR, 2 ) = IDXJP GIVCOL( GIVPTR, 1 ) = IDXJ GIVNUM( GIVPTR, 2 ) = C GIVNUM( GIVPTR, 1 ) = S END IF CALL DROT( 1, VF( JPREV ), 1, VF( J ), 1, C, S ) CALL DROT( 1, VL( JPREV ), 1, VL( J ), 1, C, S ) K2 = K2 - 1 IDXP( K2 ) = JPREV JPREV = J ELSE K = K + 1 ZW( K ) = Z( JPREV ) DSIGMA( K ) = D( JPREV ) IDXP( K ) = JPREV JPREV = J END IF END IF GO TO 80 90 CONTINUE ! ! Record the last singular value. ! K = K + 1 ZW( K ) = Z( JPREV ) DSIGMA( K ) = D( JPREV ) IDXP( K ) = JPREV ! 100 CONTINUE ! ! Sort the singular values into DSIGMA. The singular values which ! were not deflated go into the first K slots of DSIGMA, except ! that DSIGMA(1) is treated separately. ! DO 110 J = 2, N JP = IDXP( J ) DSIGMA( J ) = D( JP ) VFW( J ) = VF( JP ) VLW( J ) = VL( JP ) 110 CONTINUE IF( ICOMPQ.EQ.1 ) THEN DO 120 J = 2, N JP = IDXP( J ) PERM( J ) = IDXQ( IDX( JP )+1 ) IF( PERM( J ).LE.NLP1 ) THEN PERM( J ) = PERM( J ) - 1 END IF 120 CONTINUE END IF ! ! The deflated singular values go back into the last N - K slots of ! D. ! CALL DCOPY( N-K, DSIGMA( K+1 ), 1, D( K+1 ), 1 ) ! ! Determine DSIGMA(1), DSIGMA(2), Z(1), VF(1), VL(1), VF(M), and ! VL(M). ! DSIGMA( 1 ) = ZERO HLFTOL = TOL / TWO IF( ABS( DSIGMA( 2 ) ).LE.HLFTOL ) & DSIGMA( 2 ) = HLFTOL IF( M.GT.N ) THEN Z( 1 ) = DLAPY2( Z1, Z( M ) ) IF( Z( 1 ).LE.TOL ) THEN C = ONE S = ZERO Z( 1 ) = TOL ELSE C = Z1 / Z( 1 ) S = -Z( M ) / Z( 1 ) END IF CALL DROT( 1, VF( M ), 1, VF( 1 ), 1, C, S ) CALL DROT( 1, VL( M ), 1, VL( 1 ), 1, C, S ) ELSE IF( ABS( Z1 ).LE.TOL ) THEN Z( 1 ) = TOL ELSE Z( 1 ) = Z1 END IF END IF ! ! Restore Z, VF, and VL. ! CALL DCOPY( K-1, ZW( 2 ), 1, Z( 2 ), 1 ) CALL DCOPY( N-1, VFW( 2 ), 1, VF( 2 ), 1 ) CALL DCOPY( N-1, VLW( 2 ), 1, VL( 2 ), 1 ) ! RETURN ! ! End of DLASD7 ! END SUBROUTINE DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR, & DSIGMA, WORK, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab, ! Courant Institute, NAG Ltd., and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER ICOMPQ, INFO, K, LDDIFR ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ), DIFL( * ), DIFR( LDDIFR, * ), & DSIGMA( * ), VF( * ), VL( * ), WORK( * ), & Z( * ) ! .. ! ! Purpose ! ======= ! ! DLASD8 finds the square roots of the roots of the secular equation, ! as defined by the values in DSIGMA and Z. It makes the appropriate ! calls to DLASD4, and stores, for each element in D, the distance ! to its two nearest poles (elements in DSIGMA). It also updates ! the arrays VF and VL, the first and last components of all the ! right singular vectors of the original bidiagonal matrix. ! ! DLASD8 is called from DLASD6. ! ! Arguments ! ========= ! ! ICOMPQ (input) INTEGER ! Specifies whether singular vectors are to be computed in ! factored form in the calling routine: ! = 0: Compute singular values only. ! = 1: Compute singular vectors in factored form as well. ! ! K (input) INTEGER ! The number of terms in the rational function to be solved ! by DLASD4. K >= 1. ! ! D (output) DOUBLE PRECISION array, dimension ( K ) ! On output, D contains the updated singular values. ! ! Z (input) DOUBLE PRECISION array, dimension ( K ) ! The first K elements of this array contain the components ! of the deflation-adjusted updating row vector. ! ! VF (input/output) DOUBLE PRECISION array, dimension ( K ) ! On entry, VF contains information passed through DBEDE8. ! On exit, VF contains the first K components of the first ! components of all right singular vectors of the bidiagonal ! matrix. ! ! VL (input/output) DOUBLE PRECISION array, dimension ( K ) ! On entry, VL contains information passed through DBEDE8. ! On exit, VL contains the first K components of the last ! components of all right singular vectors of the bidiagonal ! matrix. ! ! DIFL (output) DOUBLE PRECISION array, dimension ( K ) ! On exit, DIFL(I) = D(I) - DSIGMA(I). ! ! DIFR (output) DOUBLE PRECISION array, ! dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and ! dimension ( K ) if ICOMPQ = 0. ! On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not ! defined and will not be referenced. ! ! If ICOMPQ = 1, DIFR(1:K,2) is an array containing the ! normalizing factors for the right singular vector matrix. ! ! LDDIFR (input) INTEGER ! The leading dimension of DIFR, must be at least K. ! ! DSIGMA (input) DOUBLE PRECISION array, dimension ( K ) ! The first K elements of this array contain the old roots ! of the deflated updating problem. These are the poles ! of the secular equation. ! ! WORK (workspace) DOUBLE PRECISION array, dimension at least 3 * K ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = 1, an singular value did not converge ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Huan Ren, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, IWK1, IWK2, IWK2I, IWK3, IWK3I, J DOUBLE PRECISION DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, RHO, TEMP ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLASCL, DLASD4, DLASET, XERBLA ! .. ! .. External Functions .. DOUBLE PRECISION DDOT, DLAMC3, DNRM2 EXTERNAL DDOT, DLAMC3, DNRM2 ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, SIGN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN INFO = -1 ELSE IF( K.LT.1 ) THEN INFO = -2 ELSE IF( LDDIFR.LT.K ) THEN INFO = -9 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLASD8', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( K.EQ.1 ) THEN D( 1 ) = ABS( Z( 1 ) ) DIFL( 1 ) = D( 1 ) IF( ICOMPQ.EQ.1 ) THEN DIFL( 2 ) = ONE DIFR( 1, 2 ) = ONE END IF RETURN END IF ! ! Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(j) can ! be computed with high relative accuracy (barring over/underflow). ! This is a problem on machines without a guard digit in ! add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2). ! The following code replaces DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I), ! which on any of these machines zeros out the bottommost ! bit of DSIGMA(I) if it is 1; this makes the subsequent ! subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation ! occurs. On binary machines with a guard digit (almost all ! machines) it does not change DSIGMA(I) at all. On hexadecimal ! and decimal machines with a guard digit, it slightly ! changes the bottommost bits of DSIGMA(I). It does not account ! for hexadecimal or decimal machines without guard digits ! (we know of none). We use a subroutine call to compute ! 2*DLAMBDA(I) to prevent optimizing compilers from eliminating ! this code. ! DO 10 I = 1, K DSIGMA( I ) = DLAMC3( DSIGMA( I ), DSIGMA( I ) ) - DSIGMA( I ) 10 CONTINUE ! ! Book keeping. ! IWK1 = 1 IWK2 = IWK1 + K IWK3 = IWK2 + K IWK2I = IWK2 - 1 IWK3I = IWK3 - 1 ! ! Normalize Z. ! RHO = DNRM2( K, Z, 1 ) CALL DLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO ) RHO = RHO*RHO ! ! Initialize WORK(IWK3). ! CALL DLASET( 'A', K, 1, ONE, ONE, WORK( IWK3 ), K ) ! ! Compute the updated singular values, the arrays DIFL, DIFR, ! and the updated Z. ! DO 40 J = 1, K CALL DLASD4( K, J, DSIGMA, Z, WORK( IWK1 ), RHO, D( J ), & WORK( IWK2 ), INFO ) ! ! If the root finder fails, the computation is terminated. ! IF( INFO.NE.0 ) THEN RETURN END IF WORK( IWK3I+J ) = WORK( IWK3I+J )*WORK( J )*WORK( IWK2I+J ) DIFL( J ) = -WORK( J ) DIFR( J, 1 ) = -WORK( J+1 ) DO 20 I = 1, J - 1 WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )* & WORK( IWK2I+I ) / ( DSIGMA( I )- & DSIGMA( J ) ) / ( DSIGMA( I )+ & DSIGMA( J ) ) 20 CONTINUE DO 30 I = J + 1, K WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )* & WORK( IWK2I+I ) / ( DSIGMA( I )- & DSIGMA( J ) ) / ( DSIGMA( I )+ & DSIGMA( J ) ) 30 CONTINUE 40 CONTINUE ! ! Compute updated Z. ! DO 50 I = 1, K Z( I ) = SIGN( SQRT( ABS( WORK( IWK3I+I ) ) ), Z( I ) ) 50 CONTINUE ! ! Update VF and VL. ! DO 80 J = 1, K DIFLJ = DIFL( J ) DJ = D( J ) DSIGJ = -DSIGMA( J ) IF( J.LT.K ) THEN DIFRJ = -DIFR( J, 1 ) DSIGJP = -DSIGMA( J+1 ) END IF WORK( J ) = -Z( J ) / DIFLJ / ( DSIGMA( J )+DJ ) DO 60 I = 1, J - 1 WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ), DSIGJ )-DIFLJ ) & / ( DSIGMA( I )+DJ ) 60 CONTINUE DO 70 I = J + 1, K WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ), DSIGJP )+DIFRJ ) & / ( DSIGMA( I )+DJ ) 70 CONTINUE TEMP = DNRM2( K, WORK, 1 ) WORK( IWK2I+J ) = DDOT( K, WORK, 1, VF, 1 ) / TEMP WORK( IWK3I+J ) = DDOT( K, WORK, 1, VL, 1 ) / TEMP IF( ICOMPQ.EQ.1 ) THEN DIFR( J, 2 ) = TEMP END IF 80 CONTINUE ! CALL DCOPY( K, WORK( IWK2 ), 1, VF, 1 ) CALL DCOPY( K, WORK( IWK3 ), 1, VL, 1 ) ! RETURN ! ! End of DLASD8 ! END SUBROUTINE DLASD9( ICOMPQ, LDU, K, D, Z, VF, VL, DIFL, DIFR, & DSIGMA, WORK, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab, ! Courant Institute, NAG Ltd., and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER ICOMPQ, INFO, K, LDU ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ), DIFL( * ), DIFR( LDU, * ), DSIGMA( * ), & VF( * ), VL( * ), WORK( * ), Z( * ) ! .. ! ! Purpose ! ======= ! ! DLASD9 finds the square roots of the roots of the secular equation, ! as defined by the values in DSIGMA and Z. It makes the ! appropriate calls to DLASD4, and stores, for each element in D, ! the distance to its two nearest poles (elements in DSIGMA). It also ! updates the arrays VF and VL, the first and last components of all ! the right singular vectors of the original bidiagonal matrix. ! ! DLASD9 is called from DLASD7. ! ! Arguments ! ========= ! ! ICOMPQ (input) INTEGER ! Specifies whether singular vectors are to be computed in ! factored form in the calling routine: ! ! ICOMPQ = 0 Compute singular values only. ! ! ICOMPQ = 1 Compute singular vector matrices in ! factored form also. ! K (input) INTEGER ! The number of terms in the rational function to be solved by ! DLASD4. K >= 1. ! ! D (output) DOUBLE PRECISION array, dimension(K) ! D(I) contains the updated singular values. ! ! DSIGMA (input) DOUBLE PRECISION array, dimension(K) ! The first K elements of this array contain the old roots ! of the deflated updating problem. These are the poles ! of the secular equation. ! ! Z (input) DOUBLE PRECISION array, dimension (K) ! The first K elements of this array contain the components ! of the deflation-adjusted updating row vector. ! ! VF (input/output) DOUBLE PRECISION array, dimension(K) ! On entry, VF contains information passed through SBEDE8.f ! On exit, VF contains the first K components of the first ! components of all right singular vectors of the bidiagonal ! matrix. ! ! VL (input/output) DOUBLE PRECISION array, dimension(K) ! On entry, VL contains information passed through SBEDE8.f ! On exit, VL contains the first K components of the last ! components of all right singular vectors of the bidiagonal ! matrix. ! ! DIFL (output) DOUBLE PRECISION array, dimension (K). ! On exit, DIFL(I) = D(I) - DSIGMA(I). ! ! DIFR (output) DOUBLE PRECISION array, ! dimension (LDU, 2) if ICOMPQ =1 and ! dimension (K) if ICOMPQ = 0. ! On exit, DIFR(I, 1) = D(I) - DSIGMA(I+1), DIFR(K, 1) is not ! defined and will not be referenced. ! ! If ICOMPQ = 1, DIFR(1:K, 2) is an array containing the ! normalizing factors for the right singular vector matrix. ! ! WORK (workspace) DOUBLE PRECISION array, ! dimension at least (3 * K) ! Workspace. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = 1, an singular value did not converge ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Huan Ren, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D0 ) ! .. ! .. Local Scalars .. INTEGER I, IWK1, IWK2, IWK2I, IWK3, IWK3I, J DOUBLE PRECISION DIFLJ, DIFRJ, DJ, DJP1, DSIGJ, DSIGJP, RHO, & TEMP ! .. ! .. External Functions .. DOUBLE PRECISION DDOT, DLAMC3, DNRM2 EXTERNAL DDOT, DLAMC3, DNRM2 ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLASCL, DLASD4, DLASET, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, SIGN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN INFO = -1 ELSE IF( K.LT.1 ) THEN INFO = -3 ELSE IF( LDU.LT.K ) THEN INFO = -2 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLASD9', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( K.EQ.1 ) THEN D( 1 ) = ABS( Z( 1 ) ) DIFL( 1 ) = D( 1 ) IF( ICOMPQ.EQ.1 ) THEN DIFL( 2 ) = ONE DIFR( 1, 2 ) = ONE END IF RETURN END IF ! ! Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(j) can ! be computed with high relative accuracy (barring over/underflow). ! This is a problem on machines without a guard digit in ! add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2). ! The following code replaces DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I), ! which on any of these machines zeros out the bottommost ! bit of DSIGMA(I) if it is 1; this makes the subsequent ! subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation ! occurs. On binary machines with a guard digit (almost all ! machines) it does not change DSIGMA(I) at all. On hexadecimal ! and decimal machines with a guard digit, it slightly ! changes the bottommost bits of DSIGMA(I). It does not account ! for hexadecimal or decimal machines without guard digits ! (we know of none). We use a subroutine call to compute ! 2*DLAMBDA(I) to prevent optimizing compilers from eliminating ! this code. ! DO 10 I = 1, K DSIGMA( I ) = DLAMC3( DSIGMA( I ), DSIGMA( I ) ) - DSIGMA( I ) 10 CONTINUE ! ! Book keeping. ! IWK1 = 1 IWK2 = IWK1 + K IWK3 = IWK2 + K IWK2I = IWK2 - 1 IWK3I = IWK3 - 1 ! ! Normalize Z. ! RHO = DNRM2( K, Z, 1 ) CALL DLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO ) RHO = RHO*RHO ! ! Initialize WORK(IWK3). ! CALL DLASET( 'A', K, 1, ONE, ONE, WORK( IWK3 ), K ) ! ! Compute the updated singular values, the arrays DIFL, DIFR, ! and the updated Z. ! DO 40 J = 1, K CALL DLASD4( K, J, DSIGMA, Z, WORK( IWK1 ), RHO, D( J ), & WORK( IWK2 ), INFO ) ! ! If the root finder fails, the computation is terminated. ! IF( INFO.NE.0 ) THEN RETURN END IF WORK( IWK3I+J ) = WORK( IWK3I+J )*WORK( J )*WORK( IWK2I+J ) DIFL( J ) = -WORK( J ) DIFR( J, 1 ) = -WORK( J+1 ) DO 20 I = 1, J - 1 WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )* & WORK( IWK2I+I ) / ( DSIGMA( I )- & DSIGMA( J ) ) / ( DSIGMA( I )+ & DSIGMA( J ) ) 20 CONTINUE DO 30 I = J + 1, K WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )* & WORK( IWK2I+I ) / ( DSIGMA( I )- & DSIGMA( J ) ) / ( DSIGMA( I )+ & DSIGMA( J ) ) 30 CONTINUE 40 CONTINUE ! ! Compute updated Z. ! DO 50 I = 1, K Z( I ) = SIGN( SQRT( ABS( WORK( IWK3I+I ) ) ), Z( I ) ) 50 CONTINUE ! ! Update VF and VL. ! DO 80 J = 1, K DIFLJ = DIFL( J ) DJ = D( J ) DSIGJ = -DSIGMA( J ) IF( J.LT.K ) THEN DIFRJ = -DIFR( J, 1 ) DJP1 = D( J+1 ) DSIGJP = -DSIGMA( J+1 ) END IF WORK( J ) = -Z( J ) / DIFLJ / ( DSIGMA( J )+DJ ) DO 60 I = 1, J - 1 WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ), DSIGJ )-DIFLJ ) & / ( DSIGMA( I )+DJ ) 60 CONTINUE DO 70 I = J + 1, K WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ), DSIGJP )+DIFRJ ) & / ( DSIGMA( I )+DJ ) 70 CONTINUE TEMP = DNRM2( K, WORK, 1 ) WORK( IWK2I+J ) = DDOT( K, WORK, 1, VF, 1 ) / TEMP WORK( IWK3I+J ) = DDOT( K, WORK, 1, VL, 1 ) / TEMP IF( ICOMPQ.EQ.1 ) THEN DIFR( J, 2 ) = TEMP END IF 80 CONTINUE ! CALL DCOPY( K, WORK( IWK2 ), 1, VF, 1 ) CALL DCOPY( K, WORK( IWK3 ), 1, VL, 1 ) ! RETURN ! ! End of DLASD9 ! END SUBROUTINE DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K, & DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, & PERM, GIVNUM, C, S, WORK, IWORK, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. INTEGER ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE ! .. ! .. Array Arguments .. INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), & K( * ), PERM( LDGCOL, * ) DOUBLE PRECISION C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ), & E( * ), GIVNUM( LDU, * ), POLES( LDU, * ), & S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ), & Z( LDU, * ) ! .. ! ! Purpose ! ======= ! ! Using a divide and conquer approach, DLASDA computes the singular ! value decomposition (SVD) of a real upper bidiagonal N-by-M matrix ! B with diagonal D and offdiagonal E, where M = N + SQRE. The ! algorithm computes the singular values in the SVD B = U * S * VT. ! The orthogonal matrices U and VT are optionally computed in ! compact form. ! ! A related subroutine, DLASD0, computes the singular values and ! the singular vectors in explicit form. ! ! Arguments ! ========= ! ! ICOMPQ (input) INTEGER ! Specifies whether singular vectors are to be computed ! in compact form, as follows ! = 0: Compute singular values only. ! = 1: Compute singular vectors of upper bidiagonal ! matrix in compact form. ! ! SMLSIZ (input) INTEGER ! The maximum size of the subproblems at the bottom of the ! computation tree. ! ! N (input) INTEGER ! The row dimension of the upper bidiagonal matrix. This is ! also the dimension of the main diagonal array D. ! ! SQRE (input) INTEGER ! Specifies the column dimension of the bidiagonal matrix. ! = 0: The bidiagonal matrix has column dimension M = N; ! = 1: The bidiagonal matrix has column dimension M = N + 1. ! ! D (input/output) DOUBLE PRECISION array, dimension ( N ) ! On entry D contains the main diagonal of the bidiagonal ! matrix. On exit D, if INFO = 0, contains its singular values. ! ! E (input) DOUBLE PRECISION array, dimension ( M-1 ) ! Contains the subdiagonal entries of the bidiagonal matrix. ! On exit, E has been destroyed. ! ! U (output) DOUBLE PRECISION array, ! dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced ! if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left ! singular vector matrices of all subproblems at the bottom ! level. ! ! LDU (input) INTEGER, LDU = > N. ! The leading dimension of arrays U, VT, DIFL, DIFR, POLES, ! GIVNUM, and Z. ! ! VT (output) DOUBLE PRECISION array, ! dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced ! if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT' contains the right ! singular vector matrices of all subproblems at the bottom ! level. ! ! K (output) INTEGER array, ! dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0. ! If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th ! secular equation on the computation tree. ! ! DIFL (output) DOUBLE PRECISION array, dimension ( LDU, NLVL ), ! where NLVL = floor(log_2 (N/SMLSIZ))). ! ! DIFR (output) DOUBLE PRECISION array, ! dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and ! dimension ( N ) if ICOMPQ = 0. ! If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1) ! record distances between singular values on the I-th ! level and singular values on the (I -1)-th level, and ! DIFR(1:N, 2 * I ) contains the normalizing factors for ! the right singular vector matrix. See DLASD8 for details. ! ! Z (output) DOUBLE PRECISION array, ! dimension ( LDU, NLVL ) if ICOMPQ = 1 and ! dimension ( N ) if ICOMPQ = 0. ! The first K elements of Z(1, I) contain the components of ! the deflation-adjusted updating row vector for subproblems ! on the I-th level. ! ! POLES (output) DOUBLE PRECISION array, ! dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced ! if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and ! POLES(1, 2*I) contain the new and old singular values ! involved in the secular equations on the I-th level. ! ! GIVPTR (output) INTEGER array, ! dimension ( N ) if ICOMPQ = 1, and not referenced if ! ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records ! the number of Givens rotations performed on the I-th ! problem on the computation tree. ! ! GIVCOL (output) INTEGER array, ! dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not ! referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, ! GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations ! of Givens rotations performed on the I-th level on the ! computation tree. ! ! LDGCOL (input) INTEGER, LDGCOL = > N. ! The leading dimension of arrays GIVCOL and PERM. ! ! PERM (output) INTEGER array, ! dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced ! if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records ! permutations done on the I-th level of the computation tree. ! ! GIVNUM (output) DOUBLE PRECISION array, ! dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not ! referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, ! GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S- ! values of Givens rotations performed on the I-th level on ! the computation tree. ! ! C (output) DOUBLE PRECISION array, ! dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. ! If ICOMPQ = 1 and the I-th subproblem is not square, on exit, ! C( I ) contains the C-value of a Givens rotation related to ! the right null space of the I-th subproblem. ! ! S (output) DOUBLE PRECISION array, dimension ( N ) if ! ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 ! and the I-th subproblem is not square, on exit, S( I ) ! contains the S-value of a Givens rotation related to ! the right null space of the I-th subproblem. ! ! WORK (workspace) DOUBLE PRECISION array, dimension ! (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)). ! ! IWORK (workspace) INTEGER array. ! Dimension must be at least (7 * N). ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = 1, an singular value did not converge ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Huan Ren, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, I1, IC, IDXQ, IDXQI, IM1, INODE, ITEMP, IWK, & J, LF, LL, LVL, LVL2, M, NCC, ND, NDB1, NDIML, & NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, NRU, & NWORK1, NWORK2, SMLSZP, SQREI, VF, VFI, VL, VLI DOUBLE PRECISION ALPHA, BETA ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLASD6, DLASDQ, DLASDT, DLASET, XERBLA ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN INFO = -1 ELSE IF( SMLSIZ.LT.3 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN INFO = -4 ELSE IF( LDU.LT.( N+SQRE ) ) THEN INFO = -8 ELSE IF( LDGCOL.LT.N ) THEN INFO = -17 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLASDA', -INFO ) RETURN END IF ! M = N + SQRE ! ! If the input matrix is too small, call DLASDQ to find the SVD. ! IF( N.LE.SMLSIZ ) THEN IF( ICOMPQ.EQ.0 ) THEN CALL DLASDQ( 'U', SQRE, N, 0, 0, 0, D, E, VT, LDU, U, LDU, & U, LDU, WORK, INFO ) ELSE CALL DLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDU, U, LDU, & U, LDU, WORK, INFO ) END IF RETURN END IF ! ! Book-keeping and set up the computation tree. ! INODE = 1 NDIML = INODE + N NDIMR = NDIML + N IDXQ = NDIMR + N IWK = IDXQ + N ! NCC = 0 NRU = 0 ! SMLSZP = SMLSIZ + 1 VF = 1 VL = VF + M NWORK1 = VL + M NWORK2 = NWORK1 + SMLSZP*SMLSZP ! CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ), & IWORK( NDIMR ), SMLSIZ ) ! ! for the nodes on bottom level of the tree, solve ! their subproblems by DLASDQ. ! NDB1 = ( ND+1 ) / 2 DO 30 I = NDB1, ND ! ! IC : center row of each node ! NL : number of rows of left subproblem ! NR : number of rows of right subproblem ! NLF: starting row of the left subproblem ! NRF: starting row of the right subproblem ! I1 = I - 1 IC = IWORK( INODE+I1 ) NL = IWORK( NDIML+I1 ) NLP1 = NL + 1 NR = IWORK( NDIMR+I1 ) NLF = IC - NL NRF = IC + 1 IDXQI = IDXQ + NLF - 2 VFI = VF + NLF - 1 VLI = VL + NLF - 1 SQREI = 1 IF( ICOMPQ.EQ.0 ) THEN CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, WORK( NWORK1 ), & SMLSZP ) CALL DLASDQ( 'U', SQREI, NL, NLP1, NRU, NCC, D( NLF ), & E( NLF ), WORK( NWORK1 ), SMLSZP, & WORK( NWORK2 ), NL, WORK( NWORK2 ), NL, & WORK( NWORK2 ), INFO ) ITEMP = NWORK1 + NL*SMLSZP CALL DCOPY( NLP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 ) CALL DCOPY( NLP1, WORK( ITEMP ), 1, WORK( VLI ), 1 ) ELSE CALL DLASET( 'A', NL, NL, ZERO, ONE, U( NLF, 1 ), LDU ) CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, VT( NLF, 1 ), LDU ) CALL DLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ), & E( NLF ), VT( NLF, 1 ), LDU, U( NLF, 1 ), LDU, & U( NLF, 1 ), LDU, WORK( NWORK1 ), INFO ) CALL DCOPY( NLP1, VT( NLF, 1 ), 1, WORK( VFI ), 1 ) CALL DCOPY( NLP1, VT( NLF, NLP1 ), 1, WORK( VLI ), 1 ) END IF IF( INFO.NE.0 ) THEN RETURN END IF DO 10 J = 1, NL IWORK( IDXQI+J ) = J 10 CONTINUE IF( ( I.EQ.ND ) .AND. ( SQRE.EQ.0 ) ) THEN SQREI = 0 ELSE SQREI = 1 END IF IDXQI = IDXQI + NLP1 VFI = VFI + NLP1 VLI = VLI + NLP1 NRP1 = NR + SQREI IF( ICOMPQ.EQ.0 ) THEN CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, WORK( NWORK1 ), & SMLSZP ) CALL DLASDQ( 'U', SQREI, NR, NRP1, NRU, NCC, D( NRF ), & E( NRF ), WORK( NWORK1 ), SMLSZP, & WORK( NWORK2 ), NR, WORK( NWORK2 ), NR, & WORK( NWORK2 ), INFO ) ITEMP = NWORK1 + ( NRP1-1 )*SMLSZP CALL DCOPY( NRP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 ) CALL DCOPY( NRP1, WORK( ITEMP ), 1, WORK( VLI ), 1 ) ELSE CALL DLASET( 'A', NR, NR, ZERO, ONE, U( NRF, 1 ), LDU ) CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, VT( NRF, 1 ), LDU ) CALL DLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ), & E( NRF ), VT( NRF, 1 ), LDU, U( NRF, 1 ), LDU, & U( NRF, 1 ), LDU, WORK( NWORK1 ), INFO ) CALL DCOPY( NRP1, VT( NRF, 1 ), 1, WORK( VFI ), 1 ) CALL DCOPY( NRP1, VT( NRF, NRP1 ), 1, WORK( VLI ), 1 ) END IF IF( INFO.NE.0 ) THEN RETURN END IF DO 20 J = 1, NR IWORK( IDXQI+J ) = J 20 CONTINUE 30 CONTINUE ! ! Now conquer each subproblem bottom-up. ! J = 2**NLVL DO 50 LVL = NLVL, 1, -1 LVL2 = LVL*2 - 1 ! ! Find the first node LF and last node LL on ! the current level LVL. ! IF( LVL.EQ.1 ) THEN LF = 1 LL = 1 ELSE LF = 2**( LVL-1 ) LL = 2*LF - 1 END IF DO 40 I = LF, LL IM1 = I - 1 IC = IWORK( INODE+IM1 ) NL = IWORK( NDIML+IM1 ) NR = IWORK( NDIMR+IM1 ) NLF = IC - NL NRF = IC + 1 IF( I.EQ.LL ) THEN SQREI = SQRE ELSE SQREI = 1 END IF VFI = VF + NLF - 1 VLI = VL + NLF - 1 IDXQI = IDXQ + NLF - 1 ALPHA = D( IC ) BETA = E( IC ) IF( ICOMPQ.EQ.0 ) THEN CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ), & WORK( VFI ), WORK( VLI ), ALPHA, BETA, & IWORK( IDXQI ), PERM, GIVPTR( 1 ), GIVCOL, & LDGCOL, GIVNUM, LDU, POLES, DIFL, DIFR, Z, & K( 1 ), C( 1 ), S( 1 ), WORK( NWORK1 ), & IWORK( IWK ), INFO ) ELSE J = J - 1 CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ), & WORK( VFI ), WORK( VLI ), ALPHA, BETA, & IWORK( IDXQI ), PERM( NLF, LVL ), & GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL, & GIVNUM( NLF, LVL2 ), LDU, & POLES( NLF, LVL2 ), DIFL( NLF, LVL ), & DIFR( NLF, LVL2 ), Z( NLF, LVL ), K( J ), & C( J ), S( J ), WORK( NWORK1 ), & IWORK( IWK ), INFO ) END IF IF( INFO.NE.0 ) THEN RETURN END IF 40 CONTINUE 50 CONTINUE ! RETURN ! ! End of DLASDA ! END SUBROUTINE DLASDQ( UPLO, SQRE, N, NCVT, NRU, NCC, D, E, VT, LDVT, & U, LDU, C, LDC, WORK, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU, SQRE ! .. ! .. Array Arguments .. DOUBLE PRECISION C( LDC, * ), D( * ), E( * ), U( LDU, * ), & VT( LDVT, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLASDQ computes the singular value decomposition (SVD) of a real ! (upper or lower) bidiagonal matrix with diagonal D and offdiagonal ! E, accumulating the transformations if desired. Letting B denote ! the input bidiagonal matrix, the algorithm computes orthogonal ! matrices Q and P such that B = Q * S * P' (P' denotes the transpose ! of P). The singular values S are overwritten on D. ! ! The input matrix U is changed to U * Q if desired. ! The input matrix VT is changed to P' * VT if desired. ! The input matrix C is changed to Q' * C if desired. ! ! See "Computing Small Singular Values of Bidiagonal Matrices With ! Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, ! LAPACK Working Note #3, for a detailed description of the algorithm. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! On entry, UPLO specifies whether the input bidiagonal matrix ! is upper or lower bidiagonal, and wether it is square are ! not. ! UPLO = 'U' or 'u' B is upper bidiagonal. ! UPLO = 'L' or 'l' B is lower bidiagonal. ! ! SQRE (input) INTEGER ! = 0: then the input matrix is N-by-N. ! = 1: then the input matrix is N-by-(N+1) if UPLU = 'U' and ! (N+1)-by-N if UPLU = 'L'. ! ! The bidiagonal matrix has ! N = NL + NR + 1 rows and ! M = N + SQRE >= N columns. ! ! N (input) INTEGER ! On entry, N specifies the number of rows and columns ! in the matrix. N must be at least 0. ! ! NCVT (input) INTEGER ! On entry, NCVT specifies the number of columns of ! the matrix VT. NCVT must be at least 0. ! ! NRU (input) INTEGER ! On entry, NRU specifies the number of rows of ! the matrix U. NRU must be at least 0. ! ! NCC (input) INTEGER ! On entry, NCC specifies the number of columns of ! the matrix C. NCC must be at least 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, D contains the diagonal entries of the ! bidiagonal matrix whose SVD is desired. On normal exit, ! D contains the singular values in ascending order. ! ! E (input/output) DOUBLE PRECISION array. ! dimension is (N-1) if SQRE = 0 and N if SQRE = 1. ! On entry, the entries of E contain the offdiagonal entries ! of the bidiagonal matrix whose SVD is desired. On normal ! exit, E will contain 0. If the algorithm does not converge, ! D and E will contain the diagonal and superdiagonal entries ! of a bidiagonal matrix orthogonally equivalent to the one ! given as input. ! ! VT (input/output) DOUBLE PRECISION array, dimension (LDVT, NCVT) ! On entry, contains a matrix which on exit has been ! premultiplied by P', dimension N-by-NCVT if SQRE = 0 ! and (N+1)-by-NCVT if SQRE = 1 (not referenced if NCVT=0). ! ! LDVT (input) INTEGER ! On entry, LDVT specifies the leading dimension of VT as ! declared in the calling (sub) program. LDVT must be at ! least 1. If NCVT is nonzero LDVT must also be at least N. ! ! U (input/output) DOUBLE PRECISION array, dimension (LDU, N) ! On entry, contains a matrix which on exit has been ! postmultiplied by Q, dimension NRU-by-N if SQRE = 0 ! and NRU-by-(N+1) if SQRE = 1 (not referenced if NRU=0). ! ! LDU (input) INTEGER ! On entry, LDU specifies the leading dimension of U as ! declared in the calling (sub) program. LDU must be at ! least max( 1, NRU ) . ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC, NCC) ! On entry, contains an N-by-NCC matrix which on exit ! has been premultiplied by Q' dimension N-by-NCC if SQRE = 0 ! and (N+1)-by-NCC if SQRE = 1 (not referenced if NCC=0). ! ! LDC (input) INTEGER ! On entry, LDC specifies the leading dimension of C as ! declared in the calling (sub) program. LDC must be at ! least 1. If NCC is nonzero, LDC must also be at least N. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (4*N) ! Workspace. Only referenced if one of NCVT, NRU, or NCC is ! nonzero, and if N is at least 2. ! ! INFO (output) INTEGER ! On exit, a value of 0 indicates a successful exit. ! If INFO < 0, argument number -INFO is illegal. ! If INFO > 0, the algorithm did not converge, and INFO ! specifies how many superdiagonals did not converge. ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Huan Ren, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL ROTATE INTEGER I, ISUB, IUPLO, J, NP1, SQRE1 DOUBLE PRECISION CS, R, SMIN, SN ! .. ! .. External Subroutines .. EXTERNAL DBDSQR, DLARTG, DLASR, DSWAP, XERBLA ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IUPLO = 0 IF( LSAME( UPLO, 'U' ) ) & IUPLO = 1 IF( LSAME( UPLO, 'L' ) ) & IUPLO = 2 IF( IUPLO.EQ.0 ) THEN INFO = -1 ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( NCVT.LT.0 ) THEN INFO = -4 ELSE IF( NRU.LT.0 ) THEN INFO = -5 ELSE IF( NCC.LT.0 ) THEN INFO = -6 ELSE IF( ( NCVT.EQ.0 .AND. LDVT.LT.1 ) .OR. & ( NCVT.GT.0 .AND. LDVT.LT.MAX( 1, N ) ) ) THEN INFO = -10 ELSE IF( LDU.LT.MAX( 1, NRU ) ) THEN INFO = -12 ELSE IF( ( NCC.EQ.0 .AND. LDC.LT.1 ) .OR. & ( NCC.GT.0 .AND. LDC.LT.MAX( 1, N ) ) ) THEN INFO = -14 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLASDQ', -INFO ) RETURN END IF IF( N.EQ.0 ) & RETURN ! ! ROTATE is true if any singular vectors desired, false otherwise ! ROTATE = ( NCVT.GT.0 ) .OR. ( NRU.GT.0 ) .OR. ( NCC.GT.0 ) NP1 = N + 1 SQRE1 = SQRE ! ! If matrix non-square upper bidiagonal, rotate to be lower ! bidiagonal. The rotations are on the right. ! IF( ( IUPLO.EQ.1 ) .AND. ( SQRE1.EQ.1 ) ) THEN DO 10 I = 1, N - 1 CALL DLARTG( D( I ), E( I ), CS, SN, R ) D( I ) = R E( I ) = SN*D( I+1 ) D( I+1 ) = CS*D( I+1 ) IF( ROTATE ) THEN WORK( I ) = CS WORK( N+I ) = SN END IF 10 CONTINUE CALL DLARTG( D( N ), E( N ), CS, SN, R ) D( N ) = R E( N ) = ZERO IF( ROTATE ) THEN WORK( N ) = CS WORK( N+N ) = SN END IF IUPLO = 2 SQRE1 = 0 ! ! Update singular vectors if desired. ! IF( NCVT.GT.0 ) & CALL DLASR( 'L', 'V', 'F', NP1, NCVT, WORK( 1 ), & WORK( NP1 ), VT, LDVT ) END IF ! ! If matrix lower bidiagonal, rotate to be upper bidiagonal ! by applying Givens rotations on the left. ! IF( IUPLO.EQ.2 ) THEN DO 20 I = 1, N - 1 CALL DLARTG( D( I ), E( I ), CS, SN, R ) D( I ) = R E( I ) = SN*D( I+1 ) D( I+1 ) = CS*D( I+1 ) IF( ROTATE ) THEN WORK( I ) = CS WORK( N+I ) = SN END IF 20 CONTINUE ! ! If matrix (N+1)-by-N lower bidiagonal, one additional ! rotation is needed. ! IF( SQRE1.EQ.1 ) THEN CALL DLARTG( D( N ), E( N ), CS, SN, R ) D( N ) = R IF( ROTATE ) THEN WORK( N ) = CS WORK( N+N ) = SN END IF END IF ! ! Update singular vectors if desired. ! IF( NRU.GT.0 ) THEN IF( SQRE1.EQ.0 ) THEN CALL DLASR( 'R', 'V', 'F', NRU, N, WORK( 1 ), & WORK( NP1 ), U, LDU ) ELSE CALL DLASR( 'R', 'V', 'F', NRU, NP1, WORK( 1 ), & WORK( NP1 ), U, LDU ) END IF END IF IF( NCC.GT.0 ) THEN IF( SQRE1.EQ.0 ) THEN CALL DLASR( 'L', 'V', 'F', N, NCC, WORK( 1 ), & WORK( NP1 ), C, LDC ) ELSE CALL DLASR( 'L', 'V', 'F', NP1, NCC, WORK( 1 ), & WORK( NP1 ), C, LDC ) END IF END IF END IF ! ! Call DBDSQR to compute the SVD of the reduced real ! N-by-N upper bidiagonal matrix. ! CALL DBDSQR( 'U', N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, & LDC, WORK, INFO ) ! ! Sort the singular values into ascending order (insertion sort on ! singular values, but only one transposition per singular vector) ! DO 40 I = 1, N ! ! Scan for smallest D(I). ! ISUB = I SMIN = D( I ) DO 30 J = I + 1, N IF( D( J ).LT.SMIN ) THEN ISUB = J SMIN = D( J ) END IF 30 CONTINUE IF( ISUB.NE.I ) THEN ! ! Swap singular values and vectors. ! D( ISUB ) = D( I ) D( I ) = SMIN IF( NCVT.GT.0 ) & CALL DSWAP( NCVT, VT( ISUB, 1 ), LDVT, VT( I, 1 ), LDVT ) IF( NRU.GT.0 ) & CALL DSWAP( NRU, U( 1, ISUB ), 1, U( 1, I ), 1 ) IF( NCC.GT.0 ) & CALL DSWAP( NCC, C( ISUB, 1 ), LDC, C( I, 1 ), LDC ) END IF 40 CONTINUE ! RETURN ! ! End of DLASDQ ! END SUBROUTINE DLASDT( N, LVL, ND, INODE, NDIML, NDIMR, MSUB ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER LVL, MSUB, N, ND ! .. ! .. Array Arguments .. INTEGER INODE( * ), NDIML( * ), NDIMR( * ) ! .. ! ! Purpose ! ======= ! ! DLASDT creates a tree of subproblems for bidiagonal divide and ! conquer. ! ! Arguments ! ========= ! ! N (input) INTEGER ! On entry, the number of diagonal elements of the ! bidiagonal matrix. ! ! LVL (output) INTEGER ! On exit, the number of levels on the computation tree. ! ! ND (output) INTEGER ! On exit, the number of nodes on the tree. ! ! INODE (output) INTEGER array, dimension ( N ) ! On exit, centers of subproblems. ! ! NDIML (output) INTEGER array, dimension ( N ) ! On exit, row dimensions of left children. ! ! NDIMR (output) INTEGER array, dimension ( N ) ! On exit, row dimensions of right children. ! ! MSUB (input) INTEGER. ! On entry, the maximum row dimension each subproblem at the ! bottom of the tree can be of. ! ! Further Details ! =============== ! ! Based on contributions by ! Ming Gu and Huan Ren, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, IL, IR, LLST, MAXN, NCRNT, NLVL DOUBLE PRECISION TEMP ! .. ! .. Intrinsic Functions .. INTRINSIC DBLE, INT, LOG, MAX ! .. ! .. Executable Statements .. ! ! Find the number of levels on the tree. ! MAXN = MAX( 1, N ) TEMP = LOG( DBLE( MAXN ) / DBLE( MSUB+1 ) ) / LOG( TWO ) LVL = INT( TEMP ) + 1 ! I = N / 2 INODE( 1 ) = I + 1 NDIML( 1 ) = I NDIMR( 1 ) = N - I - 1 IL = 0 IR = 1 LLST = 1 DO 20 NLVL = 1, LVL - 1 ! ! Constructing the tree at (NLVL+1)-st level. The number of ! nodes created on this level is LLST * 2. ! DO 10 I = 0, LLST - 1 IL = IL + 2 IR = IR + 2 NCRNT = LLST + I NDIML( IL ) = NDIML( NCRNT ) / 2 NDIMR( IL ) = NDIML( NCRNT ) - NDIML( IL ) - 1 INODE( IL ) = INODE( NCRNT ) - NDIMR( IL ) - 1 NDIML( IR ) = NDIMR( NCRNT ) / 2 NDIMR( IR ) = NDIMR( NCRNT ) - NDIML( IR ) - 1 INODE( IR ) = INODE( NCRNT ) + NDIML( IR ) + 1 10 CONTINUE LLST = LLST*2 20 CONTINUE ND = LLST*2 - 1 ! RETURN ! ! End of DLASDT ! END SUBROUTINE DLASET( UPLO, M, N, ALPHA, BETA, A, LDA ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER LDA, M, N DOUBLE PRECISION ALPHA, BETA ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ) ! .. ! ! Purpose ! ======= ! ! DLASET initializes an m-by-n matrix A to BETA on the diagonal and ! ALPHA on the offdiagonals. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies the part of the matrix A to be set. ! = 'U': Upper triangular part is set; the strictly lower ! triangular part of A is not changed. ! = 'L': Lower triangular part is set; the strictly upper ! triangular part of A is not changed. ! Otherwise: All of the matrix A is set. ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! ALPHA (input) DOUBLE PRECISION ! The constant to which the offdiagonal elements are to be set. ! ! BETA (input) DOUBLE PRECISION ! The constant to which the diagonal elements are to be set. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On exit, the leading m-by-n submatrix of A is set as follows: ! ! if UPLO = 'U', A(i,j) = ALPHA, 1<=i<=j-1, 1<=j<=n, ! if UPLO = 'L', A(i,j) = ALPHA, j+1<=i<=m, 1<=j<=n, ! otherwise, A(i,j) = ALPHA, 1<=i<=m, 1<=j<=n, i.ne.j, ! ! and, for all UPLO, A(i,i) = BETA, 1<=i<=min(m,n). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER I, J ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Intrinsic Functions .. INTRINSIC MIN ! .. ! .. Executable Statements .. ! IF( LSAME( UPLO, 'U' ) ) THEN ! ! Set the strictly upper triangular or trapezoidal part of the ! array to ALPHA. ! DO 20 J = 2, N DO 10 I = 1, MIN( J-1, M ) A( I, J ) = ALPHA 10 CONTINUE 20 CONTINUE ! ELSE IF( LSAME( UPLO, 'L' ) ) THEN ! ! Set the strictly lower triangular or trapezoidal part of the ! array to ALPHA. ! DO 40 J = 1, MIN( M, N ) DO 30 I = J + 1, M A( I, J ) = ALPHA 30 CONTINUE 40 CONTINUE ! ELSE ! ! Set the leading m-by-n submatrix to ALPHA. ! DO 60 J = 1, N DO 50 I = 1, M A( I, J ) = ALPHA 50 CONTINUE 60 CONTINUE END IF ! ! Set the first min(M,N) diagonal elements to BETA. ! DO 70 I = 1, MIN( M, N ) A( I, I ) = BETA 70 CONTINUE ! RETURN ! ! End of DLASET ! END SUBROUTINE DLASQ1( N, D, E, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, N ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ), E( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLASQ1 computes the singular values of a real N-by-N bidiagonal ! matrix with diagonal D and off-diagonal E. The singular values ! are computed to high relative accuracy, in the absence of ! denormalization, underflow and overflow. The algorithm was first ! presented in ! ! "Accurate singular values and differential qd algorithms" by K. V. ! Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230, ! 1994, ! ! and the present implementation is described in "An implementation of ! the dqds Algorithm (Positive Case)", LAPACK Working Note. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The number of rows and columns in the matrix. N >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, D contains the diagonal elements of the ! bidiagonal matrix whose SVD is desired. On normal exit, ! D contains the singular values in decreasing order. ! ! E (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, elements E(1:N-1) contain the off-diagonal elements ! of the bidiagonal matrix whose SVD is desired. ! On exit, E is overwritten. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (4*N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: the algorithm failed ! = 1, a split was marked by a positive value in E ! = 2, current block of Z not diagonalized after 30*N ! iterations (in inner while loop) ! = 3, termination criterion of outer while loop not met ! (program created more than N unreduced blocks) ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D0 ) ! .. ! .. Local Scalars .. INTEGER I, IINFO DOUBLE PRECISION EPS, SCALE, SAFMIN, SIGMN, SIGMX ! .. ! .. External Subroutines .. EXTERNAL DLAS2, DLASQ2, DLASRT, XERBLA ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! INFO = 0 IF( N.LT.0 ) THEN INFO = -2 CALL XERBLA( 'DLASQ1', -INFO ) RETURN ELSE IF( N.EQ.0 ) THEN RETURN ELSE IF( N.EQ.1 ) THEN D( 1 ) = ABS( D( 1 ) ) RETURN ELSE IF( N.EQ.2 ) THEN CALL DLAS2( D( 1 ), E( 1 ), D( 2 ), SIGMN, SIGMX ) D( 1 ) = SIGMX D( 2 ) = SIGMN RETURN END IF ! ! Estimate the largest singular value. ! SIGMX = ZERO DO 10 I = 1, N - 1 D( I ) = ABS( D( I ) ) SIGMX = MAX( SIGMX, ABS( E( I ) ) ) 10 CONTINUE D( N ) = ABS( D( N ) ) ! ! Early return if SIGMX is zero (matrix is already diagonal). ! IF( SIGMX.EQ.ZERO ) THEN CALL DLASRT( 'D', N, D, IINFO ) RETURN END IF ! DO 20 I = 1, N SIGMX = MAX( SIGMX, D( I ) ) 20 CONTINUE ! ! Copy D and E into WORK (in the Z format) and scale (squaring the ! input data makes scaling by a power of the radix pointless). ! EPS = DLAMCH( 'Precision' ) SAFMIN = DLAMCH( 'Safe minimum' ) SCALE = SQRT( EPS / SAFMIN ) CALL DCOPY( N, D, 1, WORK( 1 ), 2 ) CALL DCOPY( N-1, E, 1, WORK( 2 ), 2 ) CALL DLASCL( 'G', 0, 0, SIGMX, SCALE, 2*N-1, 1, WORK, 2*N-1, & IINFO ) ! ! Compute the q's and e's. ! DO 30 I = 1, 2*N - 1 WORK( I ) = WORK( I )**2 30 CONTINUE WORK( 2*N ) = ZERO ! CALL DLASQ2( N, WORK, INFO ) ! IF( INFO.EQ.0 ) THEN DO 40 I = 1, N D( I ) = SQRT( WORK( I ) ) 40 CONTINUE CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO ) END IF ! RETURN ! ! End of DLASQ1 ! END SUBROUTINE DLASQ2( N, Z, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, N ! .. ! .. Array Arguments .. DOUBLE PRECISION Z( * ) ! .. ! ! Purpose ! ======= ! ! DLASQ2 computes all the eigenvalues of the symmetric positive ! definite tridiagonal matrix associated with the qd array Z to high ! relative accuracy are computed to high relative accuracy, in the ! absence of denormalization, underflow and overflow. ! ! To see the relation of Z to the tridiagonal matrix, let L be a ! unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and ! let U be an upper bidiagonal matrix with 1's above and diagonal ! Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the ! symmetric tridiagonal to which it is similar. ! ! Note : DLASQ2 defines a logical variable, IEEE, which is true ! on machines which follow ieee-754 floating-point standard in their ! handling of infinities and NaNs, and false otherwise. This variable ! is passed to DLASQ3. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The number of rows and columns in the matrix. N >= 0. ! ! Z (workspace) DOUBLE PRECISION array, dimension ( 4*N ) ! On entry Z holds the qd array. On exit, entries 1 to N hold ! the eigenvalues in decreasing order, Z( 2*N+1 ) holds the ! trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If ! N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 ) ! holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of ! shifts that failed. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if the i-th argument is a scalar and had an illegal ! value, then INFO = -i, if the i-th argument is an ! array and the j-entry had an illegal value, then ! INFO = -(i*100+j) ! > 0: the algorithm failed ! = 1, a split was marked by a positive value in E ! = 2, current block of Z not diagonalized after 30*N ! iterations (in inner while loop) ! = 3, termination criterion of outer while loop not met ! (program created more than N unreduced blocks) ! ! Further Details ! =============== ! Local Variables: I0:N0 defines a current unreduced segment of Z. ! The shifts are accumulated in SIGMA. Iteration count is in ITER. ! Ping-pong is controlled by PP (alternates between 0 and 1). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION CBIAS PARAMETER ( CBIAS = 1.50D0 ) DOUBLE PRECISION ZERO, HALF, ONE, TWO, FOUR, HUNDRD PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0, & TWO = 2.0D0, FOUR = 4.0D0, HUNDRD = 100.0D0 ) ! .. ! .. Local Scalars .. LOGICAL IEEE INTEGER I0, I4, IINFO, IPN4, ITER, IWHILA, IWHILB, K, & N0, NBIG, NDIV, NFAIL, PP, SPLT DOUBLE PRECISION D, DESIG, DMIN, E, EMAX, EMIN, EPS, OLDEMN, & QMAX, QMIN, S, SAFMIN, SIGMA, T, TEMP, TOL, & TOL2, TRACE, ZMAX ! .. ! .. External Subroutines .. EXTERNAL DLASQ3, DLASRT, XERBLA ! .. ! .. External Functions .. INTEGER ILAENV DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH, ILAENV ! .. ! .. Intrinsic Functions .. INTRINSIC DBLE, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input arguments. ! (in case DLASQ2 is not called by DLASQ1) ! INFO = 0 EPS = DLAMCH( 'Precision' ) SAFMIN = DLAMCH( 'Safe minimum' ) TOL = EPS*HUNDRD TOL2 = TOL**2 ! IF( N.LT.0 ) THEN INFO = -1 CALL XERBLA( 'DLASQ2', 1 ) RETURN ELSE IF( N.EQ.0 ) THEN RETURN ELSE IF( N.EQ.1 ) THEN ! ! 1-by-1 case. ! IF( Z( 1 ).LT.ZERO ) THEN INFO = -201 CALL XERBLA( 'DLASQ2', 2 ) END IF RETURN ELSE IF( N.EQ.2 ) THEN ! ! 2-by-2 case. ! IF( Z( 2 ).LT.ZERO .OR. Z( 3 ).LT.ZERO ) THEN INFO = -2 CALL XERBLA( 'DLASQ2', 2 ) RETURN ELSE IF( Z( 3 ).GT.Z( 1 ) ) THEN D = Z( 3 ) Z( 3 ) = Z( 1 ) Z( 1 ) = D END IF Z( 5 ) = Z( 1 ) + Z( 2 ) + Z( 3 ) IF( Z( 2 ).GT.Z( 3 )*TOL2 ) THEN T = HALF*( ( Z( 1 )-Z( 3 ) )+Z( 2 ) ) S = Z( 3 )*( Z( 2 ) / T ) IF( S.LE.T ) THEN S = Z( 3 )*( Z( 2 ) / ( T*( ONE+SQRT( ONE+S / T ) ) ) ) ELSE S = Z( 3 )*( Z( 2 ) / ( T+SQRT( T )*SQRT( T+S ) ) ) END IF T = Z( 1 ) + ( S+Z( 2 ) ) Z( 3 ) = Z( 3 )*( Z( 1 ) / T ) Z( 1 ) = T END IF Z( 2 ) = Z( 3 ) Z( 6 ) = Z( 2 ) + Z( 1 ) RETURN END IF ! ! Check for negative data and compute sums of q's and e's. ! Z( 2*N ) = ZERO EMIN = Z( 2 ) QMAX = ZERO ZMAX = ZERO D = ZERO E = ZERO ! DO 10 K = 1, 2*( N-1 ), 2 IF( Z( K ).LT.ZERO ) THEN INFO = -( 200+K ) CALL XERBLA( 'DLASQ2', 2 ) RETURN ELSE IF( Z( K+1 ).LT.ZERO ) THEN INFO = -( 200+K+1 ) CALL XERBLA( 'DLASQ2', 2 ) RETURN END IF D = D + Z( K ) E = E + Z( K+1 ) QMAX = MAX( QMAX, Z( K ) ) EMIN = MIN( EMIN, Z( K+1 ) ) ZMAX = MAX( QMAX, ZMAX, Z( K+1 ) ) 10 CONTINUE IF( Z( 2*N-1 ).LT.ZERO ) THEN INFO = -( 200+2*N-1 ) CALL XERBLA( 'DLASQ2', 2 ) RETURN END IF D = D + Z( 2*N-1 ) QMAX = MAX( QMAX, Z( 2*N-1 ) ) ZMAX = MAX( QMAX, ZMAX ) ! ! Check for diagonality. ! IF( E.EQ.ZERO ) THEN DO 20 K = 2, N Z( K ) = Z( 2*K-1 ) 20 CONTINUE CALL DLASRT( 'D', N, Z, IINFO ) Z( 2*N-1 ) = D RETURN END IF ! TRACE = D + E ! ! Check for zero data. ! IF( TRACE.EQ.ZERO ) THEN Z( 2*N-1 ) = ZERO RETURN END IF ! ! Check whether the machine is IEEE conformable. ! IEEE = ILAENV( 10, 'DLASQ2', 'N', 1, 2, 3, 4 ).EQ.1 .AND. & ILAENV( 11, 'DLASQ2', 'N', 1, 2, 3, 4 ).EQ.1 ! ! Rearrange data for locality: Z=(q1,qq1,e1,ee1,q2,qq2,e2,ee2,...). ! DO 30 K = 2*N, 2, -2 Z( 2*K ) = ZERO Z( 2*K-1 ) = Z( K ) Z( 2*K-2 ) = ZERO Z( 2*K-3 ) = Z( K-1 ) 30 CONTINUE ! I0 = 1 N0 = N ! ! Reverse the qd-array, if warranted. ! IF( CBIAS*Z( 4*I0-3 ).LT.Z( 4*N0-3 ) ) THEN IPN4 = 4*( I0+N0 ) DO 40 I4 = 4*I0, 2*( I0+N0-1 ), 4 TEMP = Z( I4-3 ) Z( I4-3 ) = Z( IPN4-I4-3 ) Z( IPN4-I4-3 ) = TEMP TEMP = Z( I4-1 ) Z( I4-1 ) = Z( IPN4-I4-5 ) Z( IPN4-I4-5 ) = TEMP 40 CONTINUE END IF ! ! Initial split checking via dqd and Li's test. ! PP = 0 ! DO 80 K = 1, 2 ! D = Z( 4*N0+PP-3 ) DO 50 I4 = 4*( N0-1 ) + PP, 4*I0 + PP, -4 IF( Z( I4-1 ).LE.TOL2*D ) THEN Z( I4-1 ) = -ZERO D = Z( I4-3 ) ELSE D = Z( I4-3 )*( D / ( D+Z( I4-1 ) ) ) END IF 50 CONTINUE ! ! dqd maps Z to ZZ plus Li's test. ! EMIN = Z( 4*I0+PP+1 ) D = Z( 4*I0+PP-3 ) DO 60 I4 = 4*I0 + PP, 4*( N0-1 ) + PP, 4 Z( I4-2*PP-2 ) = D + Z( I4-1 ) IF( Z( I4-1 ).LE.TOL2*D ) THEN Z( I4-1 ) = -ZERO Z( I4-2*PP-2 ) = D Z( I4-2*PP ) = ZERO D = Z( I4+1 ) ELSE IF( SAFMIN*Z( I4+1 ).LT.Z( I4-2*PP-2 ) .AND. & SAFMIN*Z( I4-2*PP-2 ).LT.Z( I4+1 ) ) THEN TEMP = Z( I4+1 ) / Z( I4-2*PP-2 ) Z( I4-2*PP ) = Z( I4-1 )*TEMP D = D*TEMP ELSE Z( I4-2*PP ) = Z( I4+1 )*( Z( I4-1 ) / Z( I4-2*PP-2 ) ) D = Z( I4+1 )*( D / Z( I4-2*PP-2 ) ) END IF EMIN = MIN( EMIN, Z( I4-2*PP ) ) 60 CONTINUE Z( 4*N0-PP-2 ) = D ! ! Now find qmax. ! QMAX = Z( 4*I0-PP-2 ) DO 70 I4 = 4*I0 - PP + 2, 4*N0 - PP - 2, 4 QMAX = MAX( QMAX, Z( I4 ) ) 70 CONTINUE ! ! Prepare for the next iteration on K. ! PP = 1 - PP 80 CONTINUE ! ITER = 2 NFAIL = 0 NDIV = 2*( N0-I0 ) ! DO 140 IWHILA = 1, N + 1 IF( N0.LT.1 ) & GO TO 150 ! ! While array unfinished do ! ! E(N0) holds the value of SIGMA when submatrix in I0:N0 ! splits from the rest of the array, but is negated. ! DESIG = ZERO IF( N0.EQ.N ) THEN SIGMA = ZERO ELSE SIGMA = -Z( 4*N0-1 ) END IF IF( SIGMA.LT.ZERO ) THEN INFO = 1 RETURN END IF ! ! Find last unreduced submatrix's top index I0, find QMAX and ! EMIN. Find Gershgorin-type bound if Q's much greater than E's. ! EMAX = ZERO IF( N0.GT.I0 ) THEN EMIN = ABS( Z( 4*N0-5 ) ) ELSE EMIN = ZERO END IF QMIN = Z( 4*N0-3 ) QMAX = QMIN DO 90 I4 = 4*N0, 8, -4 IF( Z( I4-5 ).LE.ZERO ) & GO TO 100 IF( QMIN.GE.FOUR*EMAX ) THEN QMIN = MIN( QMIN, Z( I4-3 ) ) EMAX = MAX( EMAX, Z( I4-5 ) ) END IF QMAX = MAX( QMAX, Z( I4-7 )+Z( I4-5 ) ) EMIN = MIN( EMIN, Z( I4-5 ) ) 90 CONTINUE I4 = 4 ! 100 CONTINUE I0 = I4 / 4 ! ! Store EMIN for passing to DLASQ3. ! Z( 4*N0-1 ) = EMIN ! ! Put -(initial shift) into DMIN. ! DMIN = -MAX( ZERO, QMIN-TWO*SQRT( QMIN )*SQRT( EMAX ) ) ! ! Now I0:N0 is unreduced. PP = 0 for ping, PP = 1 for pong. ! PP = 0 ! NBIG = 30*( N0-I0+1 ) DO 120 IWHILB = 1, NBIG IF( I0.GT.N0 ) & GO TO 130 ! ! While submatrix unfinished take a good dqds step. ! CALL DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX, NFAIL, & ITER, NDIV, IEEE ) ! PP = 1 - PP ! ! When EMIN is very small check for splits. ! IF( PP.EQ.0 .AND. N0-I0.GE.3 ) THEN IF( Z( 4*N0 ).LE.TOL2*QMAX .OR. & Z( 4*N0-1 ).LE.TOL2*SIGMA ) THEN SPLT = I0 - 1 QMAX = Z( 4*I0-3 ) EMIN = Z( 4*I0-1 ) OLDEMN = Z( 4*I0 ) DO 110 I4 = 4*I0, 4*( N0-3 ), 4 IF( Z( I4 ).LE.TOL2*Z( I4-3 ) .OR. & Z( I4-1 ).LE.TOL2*SIGMA ) THEN Z( I4-1 ) = -SIGMA SPLT = I4 / 4 QMAX = ZERO EMIN = Z( I4+3 ) OLDEMN = Z( I4+4 ) ELSE QMAX = MAX( QMAX, Z( I4+1 ) ) EMIN = MIN( EMIN, Z( I4-1 ) ) OLDEMN = MIN( OLDEMN, Z( I4 ) ) END IF 110 CONTINUE Z( 4*N0-1 ) = EMIN Z( 4*N0 ) = OLDEMN I0 = SPLT + 1 END IF END IF ! 120 CONTINUE ! INFO = 2 RETURN ! ! end IWHILB ! 130 CONTINUE ! 140 CONTINUE ! INFO = 3 RETURN ! ! end IWHILA ! 150 CONTINUE ! ! Move q's to the front. ! DO 160 K = 2, N Z( K ) = Z( 4*K-3 ) 160 CONTINUE ! ! Sort and compute sum of eigenvalues. ! CALL DLASRT( 'D', N, Z, IINFO ) ! E = ZERO DO 170 K = N, 1, -1 E = E + Z( K ) 170 CONTINUE ! ! Store trace, sum(eigenvalues) and information on performance. ! Z( 2*N+1 ) = TRACE Z( 2*N+2 ) = E Z( 2*N+3 ) = DBLE( ITER ) Z( 2*N+4 ) = DBLE( NDIV ) / DBLE( N**2 ) Z( 2*N+5 ) = HUNDRD*NFAIL / DBLE( ITER ) RETURN ! ! End of DLASQ2 ! END SUBROUTINE DLASQ3( I0, N0, Z, PP, DMIN, SIGMA, DESIG, QMAX, NFAIL, & ITER, NDIV, IEEE ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! May 17, 2000 ! ! .. Scalar Arguments .. LOGICAL IEEE INTEGER I0, ITER, N0, NDIV, NFAIL, PP DOUBLE PRECISION DESIG, DMIN, QMAX, SIGMA ! .. ! .. Array Arguments .. DOUBLE PRECISION Z( * ) ! .. ! ! Purpose ! ======= ! ! DLASQ3 checks for deflation, computes a shift (TAU) and calls dqds. ! In case of failure it changes shifts, and tries again until output ! is positive. ! ! Arguments ! ========= ! ! I0 (input) INTEGER ! First index. ! ! N0 (input) INTEGER ! Last index. ! ! Z (input) DOUBLE PRECISION array, dimension ( 4*N ) ! Z holds the qd array. ! ! PP (input) INTEGER ! PP=0 for ping, PP=1 for pong. ! ! DMIN (output) DOUBLE PRECISION ! Minimum value of d. ! ! SIGMA (output) DOUBLE PRECISION ! Sum of shifts used in current segment. ! ! DESIG (input/output) DOUBLE PRECISION ! Lower order part of SIGMA ! ! QMAX (input) DOUBLE PRECISION ! Maximum value of q. ! ! NFAIL (output) INTEGER ! Number of times shift was too big. ! ! ITER (output) INTEGER ! Number of iterations. ! ! NDIV (output) INTEGER ! Number of divisions. ! ! TTYPE (output) INTEGER ! Shift type. ! ! IEEE (input) LOGICAL ! Flag for IEEE or non IEEE arithmetic (passed to DLASQ5). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION CBIAS PARAMETER ( CBIAS = 1.50D0 ) DOUBLE PRECISION ZERO, QURTR, HALF, ONE, TWO, HUNDRD PARAMETER ( ZERO = 0.0D0, QURTR = 0.250D0, HALF = 0.5D0, & ONE = 1.0D0, TWO = 2.0D0, HUNDRD = 100.0D0 ) ! .. ! .. Local Scalars .. INTEGER IPN4, J4, N0IN, NN, TTYPE DOUBLE PRECISION DMIN1, DMIN2, DN, DN1, DN2, EPS, S, SAFMIN, T, & TAU, TEMP, TOL, TOL2 ! .. ! .. External Subroutines .. EXTERNAL DLASQ4, DLASQ5, DLASQ6 ! .. ! .. External Function .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MIN, SQRT ! .. ! .. Save statement .. SAVE TTYPE SAVE DMIN1, DMIN2, DN, DN1, DN2, TAU ! .. ! .. Data statement .. DATA TTYPE / 0 / DATA DMIN1 / ZERO /, DMIN2 / ZERO /, DN / ZERO /, & DN1 / ZERO /, DN2 / ZERO /, TAU / ZERO / ! .. ! .. Executable Statements .. ! N0IN = N0 EPS = DLAMCH( 'Precision' ) SAFMIN = DLAMCH( 'Safe minimum' ) TOL = EPS*HUNDRD TOL2 = TOL**2 ! ! Check for deflation. ! 10 CONTINUE ! IF( N0.LT.I0 ) & RETURN IF( N0.EQ.I0 ) & GO TO 20 NN = 4*N0 + PP IF( N0.EQ.( I0+1 ) ) & GO TO 40 ! ! Check whether E(N0-1) is negligible, 1 eigenvalue. ! IF( Z( NN-5 ).GT.TOL2*( SIGMA+Z( NN-3 ) ) .AND. & Z( NN-2*PP-4 ).GT.TOL2*Z( NN-7 ) ) & GO TO 30 ! 20 CONTINUE ! Z( 4*N0-3 ) = Z( 4*N0+PP-3 ) + SIGMA N0 = N0 - 1 GO TO 10 ! ! Check whether E(N0-2) is negligible, 2 eigenvalues. ! 30 CONTINUE ! IF( Z( NN-9 ).GT.TOL2*SIGMA .AND. & Z( NN-2*PP-8 ).GT.TOL2*Z( NN-11 ) ) & GO TO 50 ! 40 CONTINUE ! IF( Z( NN-3 ).GT.Z( NN-7 ) ) THEN S = Z( NN-3 ) Z( NN-3 ) = Z( NN-7 ) Z( NN-7 ) = S END IF IF( Z( NN-5 ).GT.Z( NN-3 )*TOL2 ) THEN T = HALF*( ( Z( NN-7 )-Z( NN-3 ) )+Z( NN-5 ) ) S = Z( NN-3 )*( Z( NN-5 ) / T ) IF( S.LE.T ) THEN S = Z( NN-3 )*( Z( NN-5 ) / & ( T*( ONE+SQRT( ONE+S / T ) ) ) ) ELSE S = Z( NN-3 )*( Z( NN-5 ) / ( T+SQRT( T )*SQRT( T+S ) ) ) END IF T = Z( NN-7 ) + ( S+Z( NN-5 ) ) Z( NN-3 ) = Z( NN-3 )*( Z( NN-7 ) / T ) Z( NN-7 ) = T END IF Z( 4*N0-7 ) = Z( NN-7 ) + SIGMA Z( 4*N0-3 ) = Z( NN-3 ) + SIGMA N0 = N0 - 2 GO TO 10 ! 50 CONTINUE ! ! Reverse the qd-array, if warranted. ! IF( DMIN.LE.ZERO .OR. N0.LT.N0IN ) THEN IF( CBIAS*Z( 4*I0+PP-3 ).LT.Z( 4*N0+PP-3 ) ) THEN IPN4 = 4*( I0+N0 ) DO 60 J4 = 4*I0, 2*( I0+N0-1 ), 4 TEMP = Z( J4-3 ) Z( J4-3 ) = Z( IPN4-J4-3 ) Z( IPN4-J4-3 ) = TEMP TEMP = Z( J4-2 ) Z( J4-2 ) = Z( IPN4-J4-2 ) Z( IPN4-J4-2 ) = TEMP TEMP = Z( J4-1 ) Z( J4-1 ) = Z( IPN4-J4-5 ) Z( IPN4-J4-5 ) = TEMP TEMP = Z( J4 ) Z( J4 ) = Z( IPN4-J4-4 ) Z( IPN4-J4-4 ) = TEMP 60 CONTINUE IF( N0-I0.LE.4 ) THEN Z( 4*N0+PP-1 ) = Z( 4*I0+PP-1 ) Z( 4*N0-PP ) = Z( 4*I0-PP ) END IF DMIN2 = MIN( DMIN2, Z( 4*N0+PP-1 ) ) Z( 4*N0+PP-1 ) = MIN( Z( 4*N0+PP-1 ), Z( 4*I0+PP-1 ), & Z( 4*I0+PP+3 ) ) Z( 4*N0-PP ) = MIN( Z( 4*N0-PP ), Z( 4*I0-PP ), & Z( 4*I0-PP+4 ) ) QMAX = MAX( QMAX, Z( 4*I0+PP-3 ), Z( 4*I0+PP+1 ) ) DMIN = -ZERO END IF END IF ! 70 CONTINUE ! IF( DMIN.LT.ZERO .OR. SAFMIN*QMAX.LT.MIN( Z( 4*N0+PP-1 ), & Z( 4*N0+PP-9 ), DMIN2+Z( 4*N0-PP ) ) ) THEN ! ! Choose a shift. ! CALL DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN, DN1, & DN2, TAU, TTYPE ) ! ! Call dqds until DMIN > 0. ! 80 CONTINUE ! CALL DLASQ5( I0, N0, Z, PP, TAU, DMIN, DMIN1, DMIN2, DN, & DN1, DN2, IEEE ) ! NDIV = NDIV + ( N0-I0+2 ) ITER = ITER + 1 ! ! Check status. ! IF( DMIN.GE.ZERO .AND. DMIN1.GT.ZERO ) THEN ! ! Success. ! GO TO 100 ! ELSE IF( DMIN.LT.ZERO .AND. DMIN1.GT.ZERO .AND. & Z( 4*( N0-1 )-PP ).LT.TOL*( SIGMA+DN1 ) .AND. & ABS( DN ).LT.TOL*SIGMA ) THEN ! ! Convergence hidden by negative DN. ! Z( 4*( N0-1 )-PP+2 ) = ZERO DMIN = ZERO GO TO 100 ELSE IF( DMIN.LT.ZERO ) THEN ! ! TAU too big. Select new TAU and try again. ! NFAIL = NFAIL + 1 IF( TTYPE.LT.-22 ) THEN ! ! Failed twice. Play it safe. ! TAU = ZERO ELSE IF( DMIN1.GT.ZERO ) THEN ! ! Late failure. Gives excellent shift. ! TAU = ( TAU+DMIN )*( ONE-TWO*EPS ) TTYPE = TTYPE - 11 ELSE ! ! Early failure. Divide by 4. ! TAU = QURTR*TAU TTYPE = TTYPE - 12 END IF GO TO 80 ELSE IF( DMIN.NE.DMIN ) THEN ! ! NaN. ! TAU = ZERO GO TO 80 ELSE ! ! Possible underflow. Play it safe. ! GO TO 90 END IF END IF ! ! Risk of underflow. ! 90 CONTINUE CALL DLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN, DN1, DN2 ) NDIV = NDIV + ( N0-I0+2 ) ITER = ITER + 1 TAU = ZERO ! 100 CONTINUE IF( TAU.LT.SIGMA ) THEN DESIG = DESIG + TAU T = SIGMA + DESIG DESIG = DESIG - ( T-SIGMA ) ELSE T = SIGMA + TAU DESIG = SIGMA - ( T-TAU ) + DESIG END IF SIGMA = T ! RETURN ! ! End of DLASQ3 ! END SUBROUTINE DLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN, & DN1, DN2, TAU, TTYPE ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. INTEGER I0, N0, N0IN, PP, TTYPE DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DN1, DN2, TAU ! .. ! .. Array Arguments .. DOUBLE PRECISION Z( * ) ! .. ! ! Purpose ! ======= ! ! DLASQ4 computes an approximation TAU to the smallest eigenvalue ! using values of d from the previous transform. ! ! I0 (input) INTEGER ! First index. ! ! N0 (input) INTEGER ! Last index. ! ! Z (input) DOUBLE PRECISION array, dimension ( 4*N ) ! Z holds the qd array. ! ! PP (input) INTEGER ! PP=0 for ping, PP=1 for pong. ! ! NOIN (input) INTEGER ! The value of N0 at start of EIGTEST. ! ! DMIN (input) DOUBLE PRECISION ! Minimum value of d. ! ! DMIN1 (input) DOUBLE PRECISION ! Minimum value of d, excluding D( N0 ). ! ! DMIN2 (input) DOUBLE PRECISION ! Minimum value of d, excluding D( N0 ) and D( N0-1 ). ! ! DN (input) DOUBLE PRECISION ! d(N) ! ! DN1 (input) DOUBLE PRECISION ! d(N-1) ! ! DN2 (input) DOUBLE PRECISION ! d(N-2) ! ! TAU (output) DOUBLE PRECISION ! This is the shift. ! ! TTYPE (output) INTEGER ! Shift type. ! ! Further Details ! =============== ! CNST1 = 9/16 ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION CNST1, CNST2, CNST3 PARAMETER ( CNST1 = 0.5630D0, CNST2 = 1.010D0, & CNST3 = 1.050D0 ) DOUBLE PRECISION QURTR, THIRD, HALF, ZERO, ONE, TWO, HUNDRD PARAMETER ( QURTR = 0.250D0, THIRD = 0.3330D0, & HALF = 0.50D0, ZERO = 0.0D0, ONE = 1.0D0, & TWO = 2.0D0, HUNDRD = 100.0D0 ) ! .. ! .. Local Scalars .. INTEGER I4, NN, NP DOUBLE PRECISION A2, B1, B2, G, GAM, GAP1, GAP2, S ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Save statement .. SAVE G ! .. ! .. Data statement .. DATA G / ZERO / ! .. ! .. Executable Statements .. ! ! A negative DMIN forces the shift to take that absolute value ! TTYPE records the type of shift. ! IF( DMIN.LE.ZERO ) THEN TAU = -DMIN TTYPE = -1 RETURN END IF ! NN = 4*N0 + PP IF( N0IN.EQ.N0 ) THEN ! ! No eigenvalues deflated. ! IF( DMIN.EQ.DN .OR. DMIN.EQ.DN1 ) THEN ! B1 = SQRT( Z( NN-3 ) )*SQRT( Z( NN-5 ) ) B2 = SQRT( Z( NN-7 ) )*SQRT( Z( NN-9 ) ) A2 = Z( NN-7 ) + Z( NN-5 ) ! ! Cases 2 and 3. ! IF( DMIN.EQ.DN .AND. DMIN1.EQ.DN1 ) THEN GAP2 = DMIN2 - A2 - DMIN2*QURTR IF( GAP2.GT.ZERO .AND. GAP2.GT.B2 ) THEN GAP1 = A2 - DN - ( B2 / GAP2 )*B2 ELSE GAP1 = A2 - DN - ( B1+B2 ) END IF IF( GAP1.GT.ZERO .AND. GAP1.GT.B1 ) THEN S = MAX( DN-( B1 / GAP1 )*B1, HALF*DMIN ) TTYPE = -2 ELSE S = ZERO IF( DN.GT.B1 ) & S = DN - B1 IF( A2.GT.( B1+B2 ) ) & S = MIN( S, A2-( B1+B2 ) ) S = MAX( S, THIRD*DMIN ) TTYPE = -3 END IF ELSE ! ! Case 4. ! TTYPE = -4 S = QURTR*DMIN IF( DMIN.EQ.DN ) THEN GAM = DN A2 = ZERO IF( Z( NN-5 ) .GT. Z( NN-7 ) ) & RETURN B2 = Z( NN-5 ) / Z( NN-7 ) NP = NN - 9 ELSE NP = NN - 2*PP B2 = Z( NP-2 ) GAM = DN1 IF( Z( NP-4 ) .GT. Z( NP-2 ) ) & RETURN A2 = Z( NP-4 ) / Z( NP-2 ) IF( Z( NN-9 ) .GT. Z( NN-11 ) ) & RETURN B2 = Z( NN-9 ) / Z( NN-11 ) NP = NN - 13 END IF ! ! Approximate contribution to norm squared from I < NN-1. ! A2 = A2 + B2 DO 10 I4 = NP, 4*I0 - 1 + PP, -4 IF( B2.EQ.ZERO ) & GO TO 20 B1 = B2 IF( Z( I4 ) .GT. Z( I4-2 ) ) & RETURN B2 = B2*( Z( I4 ) / Z( I4-2 ) ) A2 = A2 + B2 IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 ) & GO TO 20 10 CONTINUE 20 CONTINUE A2 = CNST3*A2 ! ! Rayleigh quotient residual bound. ! IF( A2.LT.CNST1 ) & S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 ) END IF ELSE IF( DMIN.EQ.DN2 ) THEN ! ! Case 5. ! TTYPE = -5 S = QURTR*DMIN ! ! Compute contribution to norm squared from I > NN-2. ! NP = NN - 2*PP B1 = Z( NP-2 ) B2 = Z( NP-6 ) GAM = DN2 IF( Z( NP-8 ).GT.B2 .OR. Z( NP-4 ).GT.B1 ) & RETURN A2 = ( Z( NP-8 ) / B2 )*( ONE+Z( NP-4 ) / B1 ) ! ! Approximate contribution to norm squared from I < NN-2. ! IF( N0-I0.GT.2 ) THEN B2 = Z( NN-13 ) / Z( NN-15 ) A2 = A2 + B2 DO 30 I4 = NN - 17, 4*I0 - 1 + PP, -4 IF( B2.EQ.ZERO ) & GO TO 40 B1 = B2 IF( Z( I4 ) .GT. Z( I4-2 ) ) & RETURN B2 = B2*( Z( I4 ) / Z( I4-2 ) ) A2 = A2 + B2 IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 ) & GO TO 40 30 CONTINUE 40 CONTINUE A2 = CNST3*A2 END IF ! IF( A2.LT.CNST1 ) & S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 ) ELSE ! ! Case 6, no information to guide us. ! IF( TTYPE.EQ.-6 ) THEN G = G + THIRD*( ONE-G ) ELSE IF( TTYPE.EQ.-18 ) THEN G = QURTR*THIRD ELSE G = QURTR END IF S = G*DMIN TTYPE = -6 END IF ! ELSE IF( N0IN.EQ.( N0+1 ) ) THEN ! ! One eigenvalue just deflated. Use DMIN1, DN1 for DMIN and DN. ! IF( DMIN1.EQ.DN1 .AND. DMIN2.EQ.DN2 ) THEN ! ! Cases 7 and 8. ! TTYPE = -7 S = THIRD*DMIN1 IF( Z( NN-5 ).GT.Z( NN-7 ) ) & RETURN B1 = Z( NN-5 ) / Z( NN-7 ) B2 = B1 IF( B2.EQ.ZERO ) & GO TO 60 DO 50 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4 A2 = B1 IF( Z( I4 ).GT.Z( I4-2 ) ) & RETURN B1 = B1*( Z( I4 ) / Z( I4-2 ) ) B2 = B2 + B1 IF( HUNDRD*MAX( B1, A2 ).LT.B2 ) & GO TO 60 50 CONTINUE 60 CONTINUE B2 = SQRT( CNST3*B2 ) A2 = DMIN1 / ( ONE+B2**2 ) GAP2 = HALF*DMIN2 - A2 IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) ) ELSE S = MAX( S, A2*( ONE-CNST2*B2 ) ) TTYPE = -8 END IF ELSE ! ! Case 9. ! S = QURTR*DMIN1 IF( DMIN1.EQ.DN1 ) & S = HALF*DMIN1 TTYPE = -9 END IF ! ELSE IF( N0IN.EQ.( N0+2 ) ) THEN ! ! Two eigenvalues deflated. Use DMIN2, DN2 for DMIN and DN. ! ! Cases 10 and 11. ! IF( DMIN2.EQ.DN2 .AND. TWO*Z( NN-5 ).LT.Z( NN-7 ) ) THEN TTYPE = -10 S = THIRD*DMIN2 IF( Z( NN-5 ).GT.Z( NN-7 ) ) & RETURN B1 = Z( NN-5 ) / Z( NN-7 ) B2 = B1 IF( B2.EQ.ZERO ) & GO TO 80 DO 70 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4 IF( Z( I4 ).GT.Z( I4-2 ) ) & RETURN B1 = B1*( Z( I4 ) / Z( I4-2 ) ) B2 = B2 + B1 IF( HUNDRD*B1.LT.B2 ) & GO TO 80 70 CONTINUE 80 CONTINUE B2 = SQRT( CNST3*B2 ) A2 = DMIN2 / ( ONE+B2**2 ) GAP2 = Z( NN-7 ) + Z( NN-9 ) - & SQRT( Z( NN-11 ) )*SQRT( Z( NN-9 ) ) - A2 IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) ) ELSE S = MAX( S, A2*( ONE-CNST2*B2 ) ) END IF ELSE S = QURTR*DMIN2 TTYPE = -11 END IF ELSE IF( N0IN.GT.( N0+2 ) ) THEN ! ! Case 12, more than two eigenvalues deflated. No information. ! S = ZERO TTYPE = -12 END IF ! TAU = S RETURN ! ! End of DLASQ4 ! END SUBROUTINE DLASQ5( I0, N0, Z, PP, TAU, DMIN, DMIN1, DMIN2, DN, & DNM1, DNM2, IEEE ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! May 17, 2000 ! ! .. Scalar Arguments .. LOGICAL IEEE INTEGER I0, N0, PP DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DNM1, DNM2, TAU ! .. ! .. Array Arguments .. DOUBLE PRECISION Z( * ) ! .. ! ! Purpose ! ======= ! ! DLASQ5 computes one dqds transform in ping-pong form, one ! version for IEEE machines another for non IEEE machines. ! ! Arguments ! ========= ! ! I0 (input) INTEGER ! First index. ! ! N0 (input) INTEGER ! Last index. ! ! Z (input) DOUBLE PRECISION array, dimension ( 4*N ) ! Z holds the qd array. EMIN is stored in Z(4*N0) to avoid ! an extra argument. ! ! PP (input) INTEGER ! PP=0 for ping, PP=1 for pong. ! ! TAU (input) DOUBLE PRECISION ! This is the shift. ! ! DMIN (output) DOUBLE PRECISION ! Minimum value of d. ! ! DMIN1 (output) DOUBLE PRECISION ! Minimum value of d, excluding D( N0 ). ! ! DMIN2 (output) DOUBLE PRECISION ! Minimum value of d, excluding D( N0 ) and D( N0-1 ). ! ! DN (output) DOUBLE PRECISION ! d(N0), the last value of d. ! ! DNM1 (output) DOUBLE PRECISION ! d(N0-1). ! ! DNM2 (output) DOUBLE PRECISION ! d(N0-2). ! ! IEEE (input) LOGICAL ! Flag for IEEE or non IEEE arithmetic. ! ! ===================================================================== ! ! .. Parameter .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D0 ) ! .. ! .. Local Scalars .. INTEGER J4, J4P2 DOUBLE PRECISION D, EMIN, TEMP ! .. ! .. Intrinsic Functions .. INTRINSIC MIN ! .. ! .. Executable Statements .. ! IF( ( N0-I0-1 ).LE.0 ) & RETURN ! J4 = 4*I0 + PP - 3 EMIN = Z( J4+4 ) D = Z( J4 ) - TAU DMIN = D DMIN1 = -Z( J4 ) ! IF( IEEE ) THEN ! ! Code for IEEE arithmetic. ! IF( PP.EQ.0 ) THEN DO 10 J4 = 4*I0, 4*( N0-3 ), 4 Z( J4-2 ) = D + Z( J4-1 ) TEMP = Z( J4+1 ) / Z( J4-2 ) D = D*TEMP - TAU DMIN = MIN( DMIN, D ) Z( J4 ) = Z( J4-1 )*TEMP EMIN = MIN( Z( J4 ), EMIN ) 10 CONTINUE ELSE DO 20 J4 = 4*I0, 4*( N0-3 ), 4 Z( J4-3 ) = D + Z( J4 ) TEMP = Z( J4+2 ) / Z( J4-3 ) D = D*TEMP - TAU DMIN = MIN( DMIN, D ) Z( J4-1 ) = Z( J4 )*TEMP EMIN = MIN( Z( J4-1 ), EMIN ) 20 CONTINUE END IF ! ! Unroll last two steps. ! DNM2 = D DMIN2 = DMIN J4 = 4*( N0-2 ) - PP J4P2 = J4 + 2*PP - 1 Z( J4-2 ) = DNM2 + Z( J4P2 ) Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) ) DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU DMIN = MIN( DMIN, DNM1 ) ! DMIN1 = DMIN J4 = J4 + 4 J4P2 = J4 + 2*PP - 1 Z( J4-2 ) = DNM1 + Z( J4P2 ) Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) ) DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU DMIN = MIN( DMIN, DN ) ! ELSE ! ! Code for non IEEE arithmetic. ! IF( PP.EQ.0 ) THEN DO 30 J4 = 4*I0, 4*( N0-3 ), 4 Z( J4-2 ) = D + Z( J4-1 ) IF( D.LT.ZERO ) THEN RETURN ELSE Z( J4 ) = Z( J4+1 )*( Z( J4-1 ) / Z( J4-2 ) ) D = Z( J4+1 )*( D / Z( J4-2 ) ) - TAU END IF DMIN = MIN( DMIN, D ) EMIN = MIN( EMIN, Z( J4 ) ) 30 CONTINUE ELSE DO 40 J4 = 4*I0, 4*( N0-3 ), 4 Z( J4-3 ) = D + Z( J4 ) IF( D.LT.ZERO ) THEN RETURN ELSE Z( J4-1 ) = Z( J4+2 )*( Z( J4 ) / Z( J4-3 ) ) D = Z( J4+2 )*( D / Z( J4-3 ) ) - TAU END IF DMIN = MIN( DMIN, D ) EMIN = MIN( EMIN, Z( J4-1 ) ) 40 CONTINUE END IF ! ! Unroll last two steps. ! DNM2 = D DMIN2 = DMIN J4 = 4*( N0-2 ) - PP J4P2 = J4 + 2*PP - 1 Z( J4-2 ) = DNM2 + Z( J4P2 ) IF( DNM2.LT.ZERO ) THEN RETURN ELSE Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) ) DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) - TAU END IF DMIN = MIN( DMIN, DNM1 ) ! DMIN1 = DMIN J4 = J4 + 4 J4P2 = J4 + 2*PP - 1 Z( J4-2 ) = DNM1 + Z( J4P2 ) IF( DNM1.LT.ZERO ) THEN RETURN ELSE Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) ) DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) - TAU END IF DMIN = MIN( DMIN, DN ) ! END IF ! Z( J4+2 ) = DN Z( 4*N0-PP ) = EMIN RETURN ! ! End of DLASQ5 ! END SUBROUTINE DLASQ6( I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN, & DNM1, DNM2 ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. INTEGER I0, N0, PP DOUBLE PRECISION DMIN, DMIN1, DMIN2, DN, DNM1, DNM2 ! .. ! .. Array Arguments .. DOUBLE PRECISION Z( * ) ! .. ! ! Purpose ! ======= ! ! DLASQ6 computes one dqd (shift equal to zero) transform in ! ping-pong form, with protection against underflow and overflow. ! ! Arguments ! ========= ! ! I0 (input) INTEGER ! First index. ! ! N0 (input) INTEGER ! Last index. ! ! Z (input) DOUBLE PRECISION array, dimension ( 4*N ) ! Z holds the qd array. EMIN is stored in Z(4*N0) to avoid ! an extra argument. ! ! PP (input) INTEGER ! PP=0 for ping, PP=1 for pong. ! ! DMIN (output) DOUBLE PRECISION ! Minimum value of d. ! ! DMIN1 (output) DOUBLE PRECISION ! Minimum value of d, excluding D( N0 ). ! ! DMIN2 (output) DOUBLE PRECISION ! Minimum value of d, excluding D( N0 ) and D( N0-1 ). ! ! DN (output) DOUBLE PRECISION ! d(N0), the last value of d. ! ! DNM1 (output) DOUBLE PRECISION ! d(N0-1). ! ! DNM2 (output) DOUBLE PRECISION ! d(N0-2). ! ! ===================================================================== ! ! .. Parameter .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D0 ) ! .. ! .. Local Scalars .. INTEGER J4, J4P2 DOUBLE PRECISION D, EMIN, SAFMIN, TEMP ! .. ! .. External Function .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. Intrinsic Functions .. INTRINSIC MIN ! .. ! .. Executable Statements .. ! IF( ( N0-I0-1 ).LE.0 ) & RETURN ! SAFMIN = DLAMCH( 'Safe minimum' ) J4 = 4*I0 + PP - 3 EMIN = Z( J4+4 ) D = Z( J4 ) DMIN = D ! IF( PP.EQ.0 ) THEN DO 10 J4 = 4*I0, 4*( N0-3 ), 4 Z( J4-2 ) = D + Z( J4-1 ) IF( Z( J4-2 ).EQ.ZERO ) THEN Z( J4 ) = ZERO D = Z( J4+1 ) DMIN = D EMIN = ZERO ELSE IF( SAFMIN*Z( J4+1 ).LT.Z( J4-2 ) .AND. & SAFMIN*Z( J4-2 ).LT.Z( J4+1 ) ) THEN TEMP = Z( J4+1 ) / Z( J4-2 ) Z( J4 ) = Z( J4-1 )*TEMP D = D*TEMP ELSE Z( J4 ) = Z( J4+1 )*( Z( J4-1 ) / Z( J4-2 ) ) D = Z( J4+1 )*( D / Z( J4-2 ) ) END IF DMIN = MIN( DMIN, D ) EMIN = MIN( EMIN, Z( J4 ) ) 10 CONTINUE ELSE DO 20 J4 = 4*I0, 4*( N0-3 ), 4 Z( J4-3 ) = D + Z( J4 ) IF( Z( J4-3 ).EQ.ZERO ) THEN Z( J4-1 ) = ZERO D = Z( J4+2 ) DMIN = D EMIN = ZERO ELSE IF( SAFMIN*Z( J4+2 ).LT.Z( J4-3 ) .AND. & SAFMIN*Z( J4-3 ).LT.Z( J4+2 ) ) THEN TEMP = Z( J4+2 ) / Z( J4-3 ) Z( J4-1 ) = Z( J4 )*TEMP D = D*TEMP ELSE Z( J4-1 ) = Z( J4+2 )*( Z( J4 ) / Z( J4-3 ) ) D = Z( J4+2 )*( D / Z( J4-3 ) ) END IF DMIN = MIN( DMIN, D ) EMIN = MIN( EMIN, Z( J4-1 ) ) 20 CONTINUE END IF ! ! Unroll last two steps. ! DNM2 = D DMIN2 = DMIN J4 = 4*( N0-2 ) - PP J4P2 = J4 + 2*PP - 1 Z( J4-2 ) = DNM2 + Z( J4P2 ) IF( Z( J4-2 ).EQ.ZERO ) THEN Z( J4 ) = ZERO DNM1 = Z( J4P2+2 ) DMIN = DNM1 EMIN = ZERO ELSE IF( SAFMIN*Z( J4P2+2 ).LT.Z( J4-2 ) .AND. & SAFMIN*Z( J4-2 ).LT.Z( J4P2+2 ) ) THEN TEMP = Z( J4P2+2 ) / Z( J4-2 ) Z( J4 ) = Z( J4P2 )*TEMP DNM1 = DNM2*TEMP ELSE Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) ) DNM1 = Z( J4P2+2 )*( DNM2 / Z( J4-2 ) ) END IF DMIN = MIN( DMIN, DNM1 ) ! DMIN1 = DMIN J4 = J4 + 4 J4P2 = J4 + 2*PP - 1 Z( J4-2 ) = DNM1 + Z( J4P2 ) IF( Z( J4-2 ).EQ.ZERO ) THEN Z( J4 ) = ZERO DN = Z( J4P2+2 ) DMIN = DN EMIN = ZERO ELSE IF( SAFMIN*Z( J4P2+2 ).LT.Z( J4-2 ) .AND. & SAFMIN*Z( J4-2 ).LT.Z( J4P2+2 ) ) THEN TEMP = Z( J4P2+2 ) / Z( J4-2 ) Z( J4 ) = Z( J4P2 )*TEMP DN = DNM1*TEMP ELSE Z( J4 ) = Z( J4P2+2 )*( Z( J4P2 ) / Z( J4-2 ) ) DN = Z( J4P2+2 )*( DNM1 / Z( J4-2 ) ) END IF DMIN = MIN( DMIN, DN ) ! Z( J4+2 ) = DN Z( 4*N0-PP ) = EMIN RETURN ! ! End of DLASQ6 ! END SUBROUTINE DLASR( SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER DIRECT, PIVOT, SIDE INTEGER LDA, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( * ), S( * ) ! .. ! ! Purpose ! ======= ! ! DLASR performs the transformation ! ! A := P*A, when SIDE = 'L' or 'l' ( Left-hand side ) ! ! A := A*P', when SIDE = 'R' or 'r' ( Right-hand side ) ! ! where A is an m by n real matrix and P is an orthogonal matrix, ! consisting of a sequence of plane rotations determined by the ! parameters PIVOT and DIRECT as follows ( z = m when SIDE = 'L' or 'l' ! and z = n when SIDE = 'R' or 'r' ): ! ! When DIRECT = 'F' or 'f' ( Forward sequence ) then ! ! P = P( z - 1 )*...*P( 2 )*P( 1 ), ! ! and when DIRECT = 'B' or 'b' ( Backward sequence ) then ! ! P = P( 1 )*P( 2 )*...*P( z - 1 ), ! ! where P( k ) is a plane rotation matrix for the following planes: ! ! when PIVOT = 'V' or 'v' ( Variable pivot ), ! the plane ( k, k + 1 ) ! ! when PIVOT = 'T' or 't' ( Top pivot ), ! the plane ( 1, k + 1 ) ! ! when PIVOT = 'B' or 'b' ( Bottom pivot ), ! the plane ( k, z ) ! ! c( k ) and s( k ) must contain the cosine and sine that define the ! matrix P( k ). The two by two plane rotation part of the matrix ! P( k ), R( k ), is assumed to be of the form ! ! R( k ) = ( c( k ) s( k ) ). ! ( -s( k ) c( k ) ) ! ! This version vectorises across rows of the array A when SIDE = 'L'. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! Specifies whether the plane rotation matrix P is applied to ! A on the left or the right. ! = 'L': Left, compute A := P*A ! = 'R': Right, compute A:= A*P' ! ! DIRECT (input) CHARACTER*1 ! Specifies whether P is a forward or backward sequence of ! plane rotations. ! = 'F': Forward, P = P( z - 1 )*...*P( 2 )*P( 1 ) ! = 'B': Backward, P = P( 1 )*P( 2 )*...*P( z - 1 ) ! ! PIVOT (input) CHARACTER*1 ! Specifies the plane for which P(k) is a plane rotation ! matrix. ! = 'V': Variable pivot, the plane (k,k+1) ! = 'T': Top pivot, the plane (1,k+1) ! = 'B': Bottom pivot, the plane (k,z) ! ! M (input) INTEGER ! The number of rows of the matrix A. If m <= 1, an immediate ! return is effected. ! ! N (input) INTEGER ! The number of columns of the matrix A. If n <= 1, an ! immediate return is effected. ! ! C, S (input) DOUBLE PRECISION arrays, dimension ! (M-1) if SIDE = 'L' ! (N-1) if SIDE = 'R' ! c(k) and s(k) contain the cosine and sine that define the ! matrix P(k). The two by two plane rotation part of the ! matrix P(k), R(k), is assumed to be of the form ! R( k ) = ( c( k ) s( k ) ). ! ( -s( k ) c( k ) ) ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! The m by n matrix A. On exit, A is overwritten by P*A if ! SIDE = 'R' or by A*P' if SIDE = 'L'. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, INFO, J DOUBLE PRECISION CTEMP, STEMP, TEMP ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 IF( .NOT.( LSAME( SIDE, 'L' ) .OR. LSAME( SIDE, 'R' ) ) ) THEN INFO = 1 ELSE IF( .NOT.( LSAME( PIVOT, 'V' ) .OR. LSAME( PIVOT, & 'T' ) .OR. LSAME( PIVOT, 'B' ) ) ) THEN INFO = 2 ELSE IF( .NOT.( LSAME( DIRECT, 'F' ) .OR. LSAME( DIRECT, 'B' ) ) ) & THEN INFO = 3 ELSE IF( M.LT.0 ) THEN INFO = 4 ELSE IF( N.LT.0 ) THEN INFO = 5 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = 9 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLASR ', INFO ) RETURN END IF ! ! Quick return if possible ! IF( ( M.EQ.0 ) .OR. ( N.EQ.0 ) ) & RETURN IF( LSAME( SIDE, 'L' ) ) THEN ! ! Form P * A ! IF( LSAME( PIVOT, 'V' ) ) THEN IF( LSAME( DIRECT, 'F' ) ) THEN DO 20 J = 1, M - 1 CTEMP = C( J ) STEMP = S( J ) IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN DO 10 I = 1, N TEMP = A( J+1, I ) A( J+1, I ) = CTEMP*TEMP - STEMP*A( J, I ) A( J, I ) = STEMP*TEMP + CTEMP*A( J, I ) 10 CONTINUE END IF 20 CONTINUE ELSE IF( LSAME( DIRECT, 'B' ) ) THEN DO 40 J = M - 1, 1, -1 CTEMP = C( J ) STEMP = S( J ) IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN DO 30 I = 1, N TEMP = A( J+1, I ) A( J+1, I ) = CTEMP*TEMP - STEMP*A( J, I ) A( J, I ) = STEMP*TEMP + CTEMP*A( J, I ) 30 CONTINUE END IF 40 CONTINUE END IF ELSE IF( LSAME( PIVOT, 'T' ) ) THEN IF( LSAME( DIRECT, 'F' ) ) THEN DO 60 J = 2, M CTEMP = C( J-1 ) STEMP = S( J-1 ) IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN DO 50 I = 1, N TEMP = A( J, I ) A( J, I ) = CTEMP*TEMP - STEMP*A( 1, I ) A( 1, I ) = STEMP*TEMP + CTEMP*A( 1, I ) 50 CONTINUE END IF 60 CONTINUE ELSE IF( LSAME( DIRECT, 'B' ) ) THEN DO 80 J = M, 2, -1 CTEMP = C( J-1 ) STEMP = S( J-1 ) IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN DO 70 I = 1, N TEMP = A( J, I ) A( J, I ) = CTEMP*TEMP - STEMP*A( 1, I ) A( 1, I ) = STEMP*TEMP + CTEMP*A( 1, I ) 70 CONTINUE END IF 80 CONTINUE END IF ELSE IF( LSAME( PIVOT, 'B' ) ) THEN IF( LSAME( DIRECT, 'F' ) ) THEN DO 100 J = 1, M - 1 CTEMP = C( J ) STEMP = S( J ) IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN DO 90 I = 1, N TEMP = A( J, I ) A( J, I ) = STEMP*A( M, I ) + CTEMP*TEMP A( M, I ) = CTEMP*A( M, I ) - STEMP*TEMP 90 CONTINUE END IF 100 CONTINUE ELSE IF( LSAME( DIRECT, 'B' ) ) THEN DO 120 J = M - 1, 1, -1 CTEMP = C( J ) STEMP = S( J ) IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN DO 110 I = 1, N TEMP = A( J, I ) A( J, I ) = STEMP*A( M, I ) + CTEMP*TEMP A( M, I ) = CTEMP*A( M, I ) - STEMP*TEMP 110 CONTINUE END IF 120 CONTINUE END IF END IF ELSE IF( LSAME( SIDE, 'R' ) ) THEN ! ! Form A * P' ! IF( LSAME( PIVOT, 'V' ) ) THEN IF( LSAME( DIRECT, 'F' ) ) THEN DO 140 J = 1, N - 1 CTEMP = C( J ) STEMP = S( J ) IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN DO 130 I = 1, M TEMP = A( I, J+1 ) A( I, J+1 ) = CTEMP*TEMP - STEMP*A( I, J ) A( I, J ) = STEMP*TEMP + CTEMP*A( I, J ) 130 CONTINUE END IF 140 CONTINUE ELSE IF( LSAME( DIRECT, 'B' ) ) THEN DO 160 J = N - 1, 1, -1 CTEMP = C( J ) STEMP = S( J ) IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN DO 150 I = 1, M TEMP = A( I, J+1 ) A( I, J+1 ) = CTEMP*TEMP - STEMP*A( I, J ) A( I, J ) = STEMP*TEMP + CTEMP*A( I, J ) 150 CONTINUE END IF 160 CONTINUE END IF ELSE IF( LSAME( PIVOT, 'T' ) ) THEN IF( LSAME( DIRECT, 'F' ) ) THEN DO 180 J = 2, N CTEMP = C( J-1 ) STEMP = S( J-1 ) IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN DO 170 I = 1, M TEMP = A( I, J ) A( I, J ) = CTEMP*TEMP - STEMP*A( I, 1 ) A( I, 1 ) = STEMP*TEMP + CTEMP*A( I, 1 ) 170 CONTINUE END IF 180 CONTINUE ELSE IF( LSAME( DIRECT, 'B' ) ) THEN DO 200 J = N, 2, -1 CTEMP = C( J-1 ) STEMP = S( J-1 ) IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN DO 190 I = 1, M TEMP = A( I, J ) A( I, J ) = CTEMP*TEMP - STEMP*A( I, 1 ) A( I, 1 ) = STEMP*TEMP + CTEMP*A( I, 1 ) 190 CONTINUE END IF 200 CONTINUE END IF ELSE IF( LSAME( PIVOT, 'B' ) ) THEN IF( LSAME( DIRECT, 'F' ) ) THEN DO 220 J = 1, N - 1 CTEMP = C( J ) STEMP = S( J ) IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN DO 210 I = 1, M TEMP = A( I, J ) A( I, J ) = STEMP*A( I, N ) + CTEMP*TEMP A( I, N ) = CTEMP*A( I, N ) - STEMP*TEMP 210 CONTINUE END IF 220 CONTINUE ELSE IF( LSAME( DIRECT, 'B' ) ) THEN DO 240 J = N - 1, 1, -1 CTEMP = C( J ) STEMP = S( J ) IF( ( CTEMP.NE.ONE ) .OR. ( STEMP.NE.ZERO ) ) THEN DO 230 I = 1, M TEMP = A( I, J ) A( I, J ) = STEMP*A( I, N ) + CTEMP*TEMP A( I, N ) = CTEMP*A( I, N ) - STEMP*TEMP 230 CONTINUE END IF 240 CONTINUE END IF END IF END IF ! RETURN ! ! End of DLASR ! END SUBROUTINE DLASRT( ID, N, D, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER ID INTEGER INFO, N ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ) ! .. ! ! Purpose ! ======= ! ! Sort the numbers in D in increasing order (if ID = 'I') or ! in decreasing order (if ID = 'D' ). ! ! Use Quick Sort, reverting to Insertion sort on arrays of ! size <= 20. Dimension of STACK limits N to about 2**32. ! ! Arguments ! ========= ! ! ID (input) CHARACTER*1 ! = 'I': sort D in increasing order; ! = 'D': sort D in decreasing order. ! ! N (input) INTEGER ! The length of the array D. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the array to be sorted. ! On exit, D has been sorted into increasing order ! (D(1) <= ... <= D(N) ) or into decreasing order ! (D(1) >= ... >= D(N) ), depending on ID. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. INTEGER SELECT PARAMETER ( SELECT = 20 ) ! .. ! .. Local Scalars .. INTEGER DIR, ENDD, I, J, START, STKPNT DOUBLE PRECISION D1, D2, D3, DMNMX, TMP ! .. ! .. Local Arrays .. INTEGER STACK( 2, 32 ) ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Executable Statements .. ! ! Test the input paramters. ! INFO = 0 DIR = -1 IF( LSAME( ID, 'D' ) ) THEN DIR = 0 ELSE IF( LSAME( ID, 'I' ) ) THEN DIR = 1 END IF IF( DIR.EQ.-1 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLASRT', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.LE.1 ) & RETURN ! STKPNT = 1 STACK( 1, 1 ) = 1 STACK( 2, 1 ) = N 10 CONTINUE START = STACK( 1, STKPNT ) ENDD = STACK( 2, STKPNT ) STKPNT = STKPNT - 1 IF( ENDD-START.LE.SELECT .AND. ENDD-START.GT.0 ) THEN ! ! Do Insertion sort on D( START:ENDD ) ! IF( DIR.EQ.0 ) THEN ! ! Sort into decreasing order ! DO 30 I = START + 1, ENDD DO 20 J = I, START + 1, -1 IF( D( J ).GT.D( J-1 ) ) THEN DMNMX = D( J ) D( J ) = D( J-1 ) D( J-1 ) = DMNMX ELSE GO TO 30 END IF 20 CONTINUE 30 CONTINUE ! ELSE ! ! Sort into increasing order ! DO 50 I = START + 1, ENDD DO 40 J = I, START + 1, -1 IF( D( J ).LT.D( J-1 ) ) THEN DMNMX = D( J ) D( J ) = D( J-1 ) D( J-1 ) = DMNMX ELSE GO TO 50 END IF 40 CONTINUE 50 CONTINUE ! END IF ! ELSE IF( ENDD-START.GT.SELECT ) THEN ! ! Partition D( START:ENDD ) and stack parts, largest one first ! ! Choose partition entry as median of 3 ! D1 = D( START ) D2 = D( ENDD ) I = ( START+ENDD ) / 2 D3 = D( I ) IF( D1.LT.D2 ) THEN IF( D3.LT.D1 ) THEN DMNMX = D1 ELSE IF( D3.LT.D2 ) THEN DMNMX = D3 ELSE DMNMX = D2 END IF ELSE IF( D3.LT.D2 ) THEN DMNMX = D2 ELSE IF( D3.LT.D1 ) THEN DMNMX = D3 ELSE DMNMX = D1 END IF END IF ! IF( DIR.EQ.0 ) THEN ! ! Sort into decreasing order ! I = START - 1 J = ENDD + 1 60 CONTINUE 70 CONTINUE J = J - 1 IF( D( J ).LT.DMNMX ) & GO TO 70 80 CONTINUE I = I + 1 IF( D( I ).GT.DMNMX ) & GO TO 80 IF( I.LT.J ) THEN TMP = D( I ) D( I ) = D( J ) D( J ) = TMP GO TO 60 END IF IF( J-START.GT.ENDD-J-1 ) THEN STKPNT = STKPNT + 1 STACK( 1, STKPNT ) = START STACK( 2, STKPNT ) = J STKPNT = STKPNT + 1 STACK( 1, STKPNT ) = J + 1 STACK( 2, STKPNT ) = ENDD ELSE STKPNT = STKPNT + 1 STACK( 1, STKPNT ) = J + 1 STACK( 2, STKPNT ) = ENDD STKPNT = STKPNT + 1 STACK( 1, STKPNT ) = START STACK( 2, STKPNT ) = J END IF ELSE ! ! Sort into increasing order ! I = START - 1 J = ENDD + 1 90 CONTINUE 100 CONTINUE J = J - 1 IF( D( J ).GT.DMNMX ) & GO TO 100 110 CONTINUE I = I + 1 IF( D( I ).LT.DMNMX ) & GO TO 110 IF( I.LT.J ) THEN TMP = D( I ) D( I ) = D( J ) D( J ) = TMP GO TO 90 END IF IF( J-START.GT.ENDD-J-1 ) THEN STKPNT = STKPNT + 1 STACK( 1, STKPNT ) = START STACK( 2, STKPNT ) = J STKPNT = STKPNT + 1 STACK( 1, STKPNT ) = J + 1 STACK( 2, STKPNT ) = ENDD ELSE STKPNT = STKPNT + 1 STACK( 1, STKPNT ) = J + 1 STACK( 2, STKPNT ) = ENDD STKPNT = STKPNT + 1 STACK( 1, STKPNT ) = START STACK( 2, STKPNT ) = J END IF END IF END IF IF( STKPNT.GT.0 ) & GO TO 10 RETURN ! ! End of DLASRT ! END SUBROUTINE DLASSQ( N, X, INCX, SCALE, SUMSQ ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INCX, N DOUBLE PRECISION SCALE, SUMSQ ! .. ! .. Array Arguments .. DOUBLE PRECISION X( * ) ! .. ! ! Purpose ! ======= ! ! DLASSQ returns the values scl and smsq such that ! ! ( scl**2 )*smsq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq, ! ! where x( i ) = X( 1 + ( i - 1 )*INCX ). The value of sumsq is ! assumed to be non-negative and scl returns the value ! ! scl = max( scale, abs( x( i ) ) ). ! ! scale and sumsq must be supplied in SCALE and SUMSQ and ! scl and smsq are overwritten on SCALE and SUMSQ respectively. ! ! The routine makes only one pass through the vector x. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The number of elements to be used from the vector X. ! ! X (input) DOUBLE PRECISION array, dimension (N) ! The vector for which a scaled sum of squares is computed. ! x( i ) = X( 1 + ( i - 1 )*INCX ), 1 <= i <= n. ! ! INCX (input) INTEGER ! The increment between successive values of the vector X. ! INCX > 0. ! ! SCALE (input/output) DOUBLE PRECISION ! On entry, the value scale in the equation above. ! On exit, SCALE is overwritten with scl , the scaling factor ! for the sum of squares. ! ! SUMSQ (input/output) DOUBLE PRECISION ! On entry, the value sumsq in the equation above. ! On exit, SUMSQ is overwritten with smsq , the basic sum of ! squares from which scl has been factored out. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER IX DOUBLE PRECISION ABSXI ! .. ! .. Intrinsic Functions .. INTRINSIC ABS ! .. ! .. Executable Statements .. ! IF( N.GT.0 ) THEN DO 10 IX = 1, 1 + ( N-1 )*INCX, INCX IF( X( IX ).NE.ZERO ) THEN ABSXI = ABS( X( IX ) ) IF( SCALE.LT.ABSXI ) THEN SUMSQ = 1 + SUMSQ*( SCALE / ABSXI )**2 SCALE = ABSXI ELSE SUMSQ = SUMSQ + ( ABSXI / SCALE )**2 END IF END IF 10 CONTINUE END IF RETURN ! ! End of DLASSQ ! END SUBROUTINE DLASV2( F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. DOUBLE PRECISION CSL, CSR, F, G, H, SNL, SNR, SSMAX, SSMIN ! .. ! ! Purpose ! ======= ! ! DLASV2 computes the singular value decomposition of a 2-by-2 ! triangular matrix ! [ F G ] ! [ 0 H ]. ! On return, abs(SSMAX) is the larger singular value, abs(SSMIN) is the ! smaller singular value, and (CSL,SNL) and (CSR,SNR) are the left and ! right singular vectors for abs(SSMAX), giving the decomposition ! ! [ CSL SNL ] [ F G ] [ CSR -SNR ] = [ SSMAX 0 ] ! [-SNL CSL ] [ 0 H ] [ SNR CSR ] [ 0 SSMIN ]. ! ! Arguments ! ========= ! ! F (input) DOUBLE PRECISION ! The (1,1) element of the 2-by-2 matrix. ! ! G (input) DOUBLE PRECISION ! The (1,2) element of the 2-by-2 matrix. ! ! H (input) DOUBLE PRECISION ! The (2,2) element of the 2-by-2 matrix. ! ! SSMIN (output) DOUBLE PRECISION ! abs(SSMIN) is the smaller singular value. ! ! SSMAX (output) DOUBLE PRECISION ! abs(SSMAX) is the larger singular value. ! ! SNL (output) DOUBLE PRECISION ! CSL (output) DOUBLE PRECISION ! The vector (CSL, SNL) is a unit left singular vector for the ! singular value abs(SSMAX). ! ! SNR (output) DOUBLE PRECISION ! CSR (output) DOUBLE PRECISION ! The vector (CSR, SNR) is a unit right singular vector for the ! singular value abs(SSMAX). ! ! Further Details ! =============== ! ! Any input parameter may be aliased with any output parameter. ! ! Barring over/underflow and assuming a guard digit in subtraction, all ! output quantities are correct to within a few units in the last ! place (ulps). ! ! In IEEE arithmetic, the code works correctly if one matrix element is ! infinite. ! ! Overflow will not occur unless the largest singular value itself ! overflows or is within a few ulps of overflow. (On machines with ! partial overflow, like the Cray, overflow may occur if the largest ! singular value is within a factor of 2 of overflow.) ! ! Underflow is harmless if underflow is gradual. Otherwise, results ! may correspond to a matrix modified by perturbations of size near ! the underflow threshold. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D0 ) DOUBLE PRECISION HALF PARAMETER ( HALF = 0.5D0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D0 ) DOUBLE PRECISION FOUR PARAMETER ( FOUR = 4.0D0 ) ! .. ! .. Local Scalars .. LOGICAL GASMAL, SWAP INTEGER PMAX DOUBLE PRECISION A, CLT, CRT, D, FA, FT, GA, GT, HA, HT, L, M, & MM, R, S, SLT, SRT, T, TEMP, TSIGN, TT ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, SIGN, SQRT ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. Executable Statements .. ! FT = F FA = ABS( FT ) HT = H HA = ABS( H ) ! ! PMAX points to the maximum absolute element of matrix ! PMAX = 1 if F largest in absolute values ! PMAX = 2 if G largest in absolute values ! PMAX = 3 if H largest in absolute values ! PMAX = 1 SWAP = ( HA.GT.FA ) IF( SWAP ) THEN PMAX = 3 TEMP = FT FT = HT HT = TEMP TEMP = FA FA = HA HA = TEMP ! ! Now FA .ge. HA ! END IF GT = G GA = ABS( GT ) IF( GA.EQ.ZERO ) THEN ! ! Diagonal matrix ! SSMIN = HA SSMAX = FA CLT = ONE CRT = ONE SLT = ZERO SRT = ZERO ELSE GASMAL = .TRUE. IF( GA.GT.FA ) THEN PMAX = 2 IF( ( FA / GA ).LT.DLAMCH( 'EPS' ) ) THEN ! ! Case of very large GA ! GASMAL = .FALSE. SSMAX = GA IF( HA.GT.ONE ) THEN SSMIN = FA / ( GA / HA ) ELSE SSMIN = ( FA / GA )*HA END IF CLT = ONE SLT = HT / GT SRT = ONE CRT = FT / GT END IF END IF IF( GASMAL ) THEN ! ! Normal case ! D = FA - HA IF( D.EQ.FA ) THEN ! ! Copes with infinite F or H ! L = ONE ELSE L = D / FA END IF ! ! Note that 0 .le. L .le. 1 ! M = GT / FT ! ! Note that abs(M) .le. 1/macheps ! T = TWO - L ! ! Note that T .ge. 1 ! MM = M*M TT = T*T S = SQRT( TT+MM ) ! ! Note that 1 .le. S .le. 1 + 1/macheps ! IF( L.EQ.ZERO ) THEN R = ABS( M ) ELSE R = SQRT( L*L+MM ) END IF ! ! Note that 0 .le. R .le. 1 + 1/macheps ! A = HALF*( S+R ) ! ! Note that 1 .le. A .le. 1 + abs(M) ! SSMIN = HA / A SSMAX = FA*A IF( MM.EQ.ZERO ) THEN ! ! Note that M is very tiny ! IF( L.EQ.ZERO ) THEN T = SIGN( TWO, FT )*SIGN( ONE, GT ) ELSE T = GT / SIGN( D, FT ) + M / T END IF ELSE T = ( M / ( S+T )+M / ( R+L ) )*( ONE+A ) END IF L = SQRT( T*T+FOUR ) CRT = TWO / L SRT = T / L CLT = ( CRT+SRT*M ) / A SLT = ( HT / FT )*SRT / A END IF END IF IF( SWAP ) THEN CSL = SRT SNL = CRT CSR = SLT SNR = CLT ELSE CSL = CLT SNL = SLT CSR = CRT SNR = SRT END IF ! ! Correct signs of SSMAX and SSMIN ! IF( PMAX.EQ.1 ) & TSIGN = SIGN( ONE, CSR )*SIGN( ONE, CSL )*SIGN( ONE, F ) IF( PMAX.EQ.2 ) & TSIGN = SIGN( ONE, SNR )*SIGN( ONE, CSL )*SIGN( ONE, G ) IF( PMAX.EQ.3 ) & TSIGN = SIGN( ONE, SNR )*SIGN( ONE, SNL )*SIGN( ONE, H ) SSMAX = SIGN( SSMAX, TSIGN ) SSMIN = SIGN( SSMIN, TSIGN*SIGN( ONE, F )*SIGN( ONE, H ) ) RETURN ! ! End of DLASV2 ! END SUBROUTINE DLASWP( N, A, LDA, K1, K2, IPIV, INCX ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INCX, K1, K2, LDA, N ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION A( LDA, * ) ! .. ! ! Purpose ! ======= ! ! DLASWP performs a series of row interchanges on the matrix A. ! One row interchange is initiated for each of rows K1 through K2 of A. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The number of columns of the matrix A. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the matrix of column dimension N to which the row ! interchanges will be applied. ! On exit, the permuted matrix. ! ! LDA (input) INTEGER ! The leading dimension of the array A. ! ! K1 (input) INTEGER ! The first element of IPIV for which a row interchange will ! be done. ! ! K2 (input) INTEGER ! The last element of IPIV for which a row interchange will ! be done. ! ! IPIV (input) INTEGER array, dimension (M*abs(INCX)) ! The vector of pivot indices. Only the elements in positions ! K1 through K2 of IPIV are accessed. ! IPIV(K) = L implies rows K and L are to be interchanged. ! ! INCX (input) INTEGER ! The increment between successive values of IPIV. If IPIV ! is negative, the pivots are applied in reverse order. ! ! Further Details ! =============== ! ! Modified by ! R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER I, I1, I2, INC, IP, IX, IX0, J, K, N32 DOUBLE PRECISION TEMP ! .. ! .. Executable Statements .. ! ! Interchange row I with row IPIV(I) for each of rows K1 through K2. ! IF( INCX.GT.0 ) THEN IX0 = K1 I1 = K1 I2 = K2 INC = 1 ELSE IF( INCX.LT.0 ) THEN IX0 = 1 + ( 1-K2 )*INCX I1 = K2 I2 = K1 INC = -1 ELSE RETURN END IF ! N32 = ( N / 32 )*32 IF( N32.NE.0 ) THEN DO 30 J = 1, N32, 32 IX = IX0 DO 20 I = I1, I2, INC IP = IPIV( IX ) IF( IP.NE.I ) THEN DO 10 K = J, J + 31 TEMP = A( I, K ) A( I, K ) = A( IP, K ) A( IP, K ) = TEMP 10 CONTINUE END IF IX = IX + INCX 20 CONTINUE 30 CONTINUE END IF IF( N32.NE.N ) THEN N32 = N32 + 1 IX = IX0 DO 50 I = I1, I2, INC IP = IPIV( IX ) IF( IP.NE.I ) THEN DO 40 K = N32, N TEMP = A( I, K ) A( I, K ) = A( IP, K ) A( IP, K ) = TEMP 40 CONTINUE END IF IX = IX + INCX 50 CONTINUE END IF ! RETURN ! ! End of DLASWP ! END SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR, & LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. LOGICAL LTRANL, LTRANR INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2 DOUBLE PRECISION SCALE, XNORM ! .. ! .. Array Arguments .. DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ), & X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in ! ! op(TL)*X + ISGN*X*op(TR) = SCALE*B, ! ! where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or ! -1. op(T) = T or T', where T' denotes the transpose of T. ! ! Arguments ! ========= ! ! LTRANL (input) LOGICAL ! On entry, LTRANL specifies the op(TL): ! = .FALSE., op(TL) = TL, ! = .TRUE., op(TL) = TL'. ! ! LTRANR (input) LOGICAL ! On entry, LTRANR specifies the op(TR): ! = .FALSE., op(TR) = TR, ! = .TRUE., op(TR) = TR'. ! ! ISGN (input) INTEGER ! On entry, ISGN specifies the sign of the equation ! as described before. ISGN may only be 1 or -1. ! ! N1 (input) INTEGER ! On entry, N1 specifies the order of matrix TL. ! N1 may only be 0, 1 or 2. ! ! N2 (input) INTEGER ! On entry, N2 specifies the order of matrix TR. ! N2 may only be 0, 1 or 2. ! ! TL (input) DOUBLE PRECISION array, dimension (LDTL,2) ! On entry, TL contains an N1 by N1 matrix. ! ! LDTL (input) INTEGER ! The leading dimension of the matrix TL. LDTL >= max(1,N1). ! ! TR (input) DOUBLE PRECISION array, dimension (LDTR,2) ! On entry, TR contains an N2 by N2 matrix. ! ! LDTR (input) INTEGER ! The leading dimension of the matrix TR. LDTR >= max(1,N2). ! ! B (input) DOUBLE PRECISION array, dimension (LDB,2) ! On entry, the N1 by N2 matrix B contains the right-hand ! side of the equation. ! ! LDB (input) INTEGER ! The leading dimension of the matrix B. LDB >= max(1,N1). ! ! SCALE (output) DOUBLE PRECISION ! On exit, SCALE contains the scale factor. SCALE is chosen ! less than or equal to 1 to prevent the solution overflowing. ! ! X (output) DOUBLE PRECISION array, dimension (LDX,2) ! On exit, X contains the N1 by N2 solution. ! ! LDX (input) INTEGER ! The leading dimension of the matrix X. LDX >= max(1,N1). ! ! XNORM (output) DOUBLE PRECISION ! On exit, XNORM is the infinity-norm of the solution. ! ! INFO (output) INTEGER ! On exit, INFO is set to ! 0: successful exit. ! 1: TL and TR have too close eigenvalues, so TL or ! TR is perturbed to get a nonsingular equation. ! NOTE: In the interests of speed, this routine does not ! check the inputs for errors. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) DOUBLE PRECISION TWO, HALF, EIGHT PARAMETER ( TWO = 2.0D+0, HALF = 0.5D+0, EIGHT = 8.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL BSWAP, XSWAP INTEGER I, IP, IPIV, IPSV, J, JP, JPSV, K DOUBLE PRECISION BET, EPS, GAM, L21, SGN, SMIN, SMLNUM, TAU1, & TEMP, U11, U12, U22, XMAX ! .. ! .. Local Arrays .. LOGICAL BSWPIV( 4 ), XSWPIV( 4 ) INTEGER JPIV( 4 ), LOCL21( 4 ), LOCU12( 4 ), & LOCU22( 4 ) DOUBLE PRECISION BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 ) ! .. ! .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DLAMCH EXTERNAL IDAMAX, DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DSWAP ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. Data statements .. DATA LOCU12 / 3, 4, 1, 2 / , LOCL21 / 2, 1, 4, 3 / , & LOCU22 / 4, 3, 2, 1 / DATA XSWPIV / .FALSE., .FALSE., .TRUE., .TRUE. / DATA BSWPIV / .FALSE., .TRUE., .FALSE., .TRUE. / ! .. ! .. Executable Statements .. ! ! Do not check the input parameters for errors ! INFO = 0 ! ! Quick return if possible ! IF( N1.EQ.0 .OR. N2.EQ.0 ) & RETURN ! ! Set constants to control overflow ! EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) / EPS SGN = ISGN ! K = N1 + N1 + N2 - 2 GO TO ( 10, 20, 30, 50 )K ! ! 1 by 1: TL11*X + SGN*X*TR11 = B11 ! 10 CONTINUE TAU1 = TL( 1, 1 ) + SGN*TR( 1, 1 ) BET = ABS( TAU1 ) IF( BET.LE.SMLNUM ) THEN TAU1 = SMLNUM BET = SMLNUM INFO = 1 END IF ! SCALE = ONE GAM = ABS( B( 1, 1 ) ) IF( SMLNUM*GAM.GT.BET ) & SCALE = ONE / GAM ! X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / TAU1 XNORM = ABS( X( 1, 1 ) ) RETURN ! ! 1 by 2: ! TL11*[X11 X12] + ISGN*[X11 X12]*op[TR11 TR12] = [B11 B12] ! [TR21 TR22] ! 20 CONTINUE ! SMIN = MAX( EPS*MAX( ABS( TL( 1, 1 ) ), ABS( TR( 1, 1 ) ), & ABS( TR( 1, 2 ) ), ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ), & SMLNUM ) TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 ) TMP( 4 ) = TL( 1, 1 ) + SGN*TR( 2, 2 ) IF( LTRANR ) THEN TMP( 2 ) = SGN*TR( 2, 1 ) TMP( 3 ) = SGN*TR( 1, 2 ) ELSE TMP( 2 ) = SGN*TR( 1, 2 ) TMP( 3 ) = SGN*TR( 2, 1 ) END IF BTMP( 1 ) = B( 1, 1 ) BTMP( 2 ) = B( 1, 2 ) GO TO 40 ! ! 2 by 1: ! op[TL11 TL12]*[X11] + ISGN* [X11]*TR11 = [B11] ! [TL21 TL22] [X21] [X21] [B21] ! 30 CONTINUE SMIN = MAX( EPS*MAX( ABS( TR( 1, 1 ) ), ABS( TL( 1, 1 ) ), & ABS( TL( 1, 2 ) ), ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ), & SMLNUM ) TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 ) TMP( 4 ) = TL( 2, 2 ) + SGN*TR( 1, 1 ) IF( LTRANL ) THEN TMP( 2 ) = TL( 1, 2 ) TMP( 3 ) = TL( 2, 1 ) ELSE TMP( 2 ) = TL( 2, 1 ) TMP( 3 ) = TL( 1, 2 ) END IF BTMP( 1 ) = B( 1, 1 ) BTMP( 2 ) = B( 2, 1 ) 40 CONTINUE ! ! Solve 2 by 2 system using complete pivoting. ! Set pivots less than SMIN to SMIN. ! IPIV = IDAMAX( 4, TMP, 1 ) U11 = TMP( IPIV ) IF( ABS( U11 ).LE.SMIN ) THEN INFO = 1 U11 = SMIN END IF U12 = TMP( LOCU12( IPIV ) ) L21 = TMP( LOCL21( IPIV ) ) / U11 U22 = TMP( LOCU22( IPIV ) ) - U12*L21 XSWAP = XSWPIV( IPIV ) BSWAP = BSWPIV( IPIV ) IF( ABS( U22 ).LE.SMIN ) THEN INFO = 1 U22 = SMIN END IF IF( BSWAP ) THEN TEMP = BTMP( 2 ) BTMP( 2 ) = BTMP( 1 ) - L21*TEMP BTMP( 1 ) = TEMP ELSE BTMP( 2 ) = BTMP( 2 ) - L21*BTMP( 1 ) END IF SCALE = ONE IF( ( TWO*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( U22 ) .OR. & ( TWO*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( U11 ) ) THEN SCALE = HALF / MAX( ABS( BTMP( 1 ) ), ABS( BTMP( 2 ) ) ) BTMP( 1 ) = BTMP( 1 )*SCALE BTMP( 2 ) = BTMP( 2 )*SCALE END IF X2( 2 ) = BTMP( 2 ) / U22 X2( 1 ) = BTMP( 1 ) / U11 - ( U12 / U11 )*X2( 2 ) IF( XSWAP ) THEN TEMP = X2( 2 ) X2( 2 ) = X2( 1 ) X2( 1 ) = TEMP END IF X( 1, 1 ) = X2( 1 ) IF( N1.EQ.1 ) THEN X( 1, 2 ) = X2( 2 ) XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) ) ELSE X( 2, 1 ) = X2( 2 ) XNORM = MAX( ABS( X( 1, 1 ) ), ABS( X( 2, 1 ) ) ) END IF RETURN ! ! 2 by 2: ! op[TL11 TL12]*[X11 X12] +ISGN* [X11 X12]*op[TR11 TR12] = [B11 B12] ! [TL21 TL22] [X21 X22] [X21 X22] [TR21 TR22] [B21 B22] ! ! Solve equivalent 4 by 4 system using complete pivoting. ! Set pivots less than SMIN to SMIN. ! 50 CONTINUE SMIN = MAX( ABS( TR( 1, 1 ) ), ABS( TR( 1, 2 ) ), & ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ) SMIN = MAX( SMIN, ABS( TL( 1, 1 ) ), ABS( TL( 1, 2 ) ), & ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ) SMIN = MAX( EPS*SMIN, SMLNUM ) BTMP( 1 ) = ZERO CALL DCOPY( 16, BTMP, 0, T16, 1 ) T16( 1, 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 ) T16( 2, 2 ) = TL( 2, 2 ) + SGN*TR( 1, 1 ) T16( 3, 3 ) = TL( 1, 1 ) + SGN*TR( 2, 2 ) T16( 4, 4 ) = TL( 2, 2 ) + SGN*TR( 2, 2 ) IF( LTRANL ) THEN T16( 1, 2 ) = TL( 2, 1 ) T16( 2, 1 ) = TL( 1, 2 ) T16( 3, 4 ) = TL( 2, 1 ) T16( 4, 3 ) = TL( 1, 2 ) ELSE T16( 1, 2 ) = TL( 1, 2 ) T16( 2, 1 ) = TL( 2, 1 ) T16( 3, 4 ) = TL( 1, 2 ) T16( 4, 3 ) = TL( 2, 1 ) END IF IF( LTRANR ) THEN T16( 1, 3 ) = SGN*TR( 1, 2 ) T16( 2, 4 ) = SGN*TR( 1, 2 ) T16( 3, 1 ) = SGN*TR( 2, 1 ) T16( 4, 2 ) = SGN*TR( 2, 1 ) ELSE T16( 1, 3 ) = SGN*TR( 2, 1 ) T16( 2, 4 ) = SGN*TR( 2, 1 ) T16( 3, 1 ) = SGN*TR( 1, 2 ) T16( 4, 2 ) = SGN*TR( 1, 2 ) END IF BTMP( 1 ) = B( 1, 1 ) BTMP( 2 ) = B( 2, 1 ) BTMP( 3 ) = B( 1, 2 ) BTMP( 4 ) = B( 2, 2 ) ! ! Perform elimination ! DO 100 I = 1, 3 XMAX = ZERO DO 70 IP = I, 4 DO 60 JP = I, 4 IF( ABS( T16( IP, JP ) ).GE.XMAX ) THEN XMAX = ABS( T16( IP, JP ) ) IPSV = IP JPSV = JP END IF 60 CONTINUE 70 CONTINUE IF( IPSV.NE.I ) THEN CALL DSWAP( 4, T16( IPSV, 1 ), 4, T16( I, 1 ), 4 ) TEMP = BTMP( I ) BTMP( I ) = BTMP( IPSV ) BTMP( IPSV ) = TEMP END IF IF( JPSV.NE.I ) & CALL DSWAP( 4, T16( 1, JPSV ), 1, T16( 1, I ), 1 ) JPIV( I ) = JPSV IF( ABS( T16( I, I ) ).LT.SMIN ) THEN INFO = 1 T16( I, I ) = SMIN END IF DO 90 J = I + 1, 4 T16( J, I ) = T16( J, I ) / T16( I, I ) BTMP( J ) = BTMP( J ) - T16( J, I )*BTMP( I ) DO 80 K = I + 1, 4 T16( J, K ) = T16( J, K ) - T16( J, I )*T16( I, K ) 80 CONTINUE 90 CONTINUE 100 CONTINUE IF( ABS( T16( 4, 4 ) ).LT.SMIN ) & T16( 4, 4 ) = SMIN SCALE = ONE IF( ( EIGHT*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T16( 1, 1 ) ) .OR. & ( EIGHT*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( T16( 2, 2 ) ) .OR. & ( EIGHT*SMLNUM )*ABS( BTMP( 3 ) ).GT.ABS( T16( 3, 3 ) ) .OR. & ( EIGHT*SMLNUM )*ABS( BTMP( 4 ) ).GT.ABS( T16( 4, 4 ) ) ) THEN SCALE = ( ONE / EIGHT ) / MAX( ABS( BTMP( 1 ) ), & ABS( BTMP( 2 ) ), ABS( BTMP( 3 ) ), ABS( BTMP( 4 ) ) ) BTMP( 1 ) = BTMP( 1 )*SCALE BTMP( 2 ) = BTMP( 2 )*SCALE BTMP( 3 ) = BTMP( 3 )*SCALE BTMP( 4 ) = BTMP( 4 )*SCALE END IF DO 120 I = 1, 4 K = 5 - I TEMP = ONE / T16( K, K ) TMP( K ) = BTMP( K )*TEMP DO 110 J = K + 1, 4 TMP( K ) = TMP( K ) - ( TEMP*T16( K, J ) )*TMP( J ) 110 CONTINUE 120 CONTINUE DO 130 I = 1, 3 IF( JPIV( 4-I ).NE.4-I ) THEN TEMP = TMP( 4-I ) TMP( 4-I ) = TMP( JPIV( 4-I ) ) TMP( JPIV( 4-I ) ) = TEMP END IF 130 CONTINUE X( 1, 1 ) = TMP( 1 ) X( 2, 1 ) = TMP( 2 ) X( 1, 2 ) = TMP( 3 ) X( 2, 2 ) = TMP( 4 ) XNORM = MAX( ABS( TMP( 1 ) )+ABS( TMP( 3 ) ), & ABS( TMP( 2 ) )+ABS( TMP( 4 ) ) ) RETURN ! ! End of DLASY2 ! END SUBROUTINE DLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, KB, LDA, LDW, N, NB ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION A( LDA, * ), W( LDW, * ) ! .. ! ! Purpose ! ======= ! ! DLASYF computes a partial factorization of a real symmetric matrix A ! using the Bunch-Kaufman diagonal pivoting method. The partial ! factorization has the form: ! ! A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: ! ( 0 U22 ) ( 0 D ) ( U12' U22' ) ! ! A = ( L11 0 ) ( D 0 ) ( L11' L21' ) if UPLO = 'L' ! ( L21 I ) ( 0 A22 ) ( 0 I ) ! ! where the order of D is at most NB. The actual order is returned in ! the argument KB, and is either NB or NB-1, or N if N <= NB. ! ! DLASYF is an auxiliary routine called by DSYTRF. It uses blocked code ! (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or ! A22 (if UPLO = 'L'). ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the upper or lower triangular part of the ! symmetric matrix A is stored: ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NB (input) INTEGER ! The maximum number of columns of the matrix A that should be ! factored. NB should be at least 2 to allow for 2-by-2 pivot ! blocks. ! ! KB (output) INTEGER ! The number of columns of A that were actually factored. ! KB is either NB-1 or NB, or N if N <= NB. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the symmetric matrix A. If UPLO = 'U', the leading ! n-by-n upper triangular part of A contains the upper ! triangular part of the matrix A, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading n-by-n lower triangular part of A contains the lower ! triangular part of the matrix A, and the strictly upper ! triangular part of A is not referenced. ! On exit, A contains details of the partial factorization. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! IPIV (output) INTEGER array, dimension (N) ! Details of the interchanges and the block structure of D. ! If UPLO = 'U', only the last KB elements of IPIV are set; ! if UPLO = 'L', only the first KB elements are set. ! ! If IPIV(k) > 0, then rows and columns k and IPIV(k) were ! interchanged and D(k,k) is a 1-by-1 diagonal block. ! If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and ! columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) ! is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = ! IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were ! interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. ! ! W (workspace) DOUBLE PRECISION array, dimension (LDW,NB) ! ! LDW (input) INTEGER ! The leading dimension of the array W. LDW >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! > 0: if INFO = k, D(k,k) is exactly zero. The factorization ! has been completed, but the block diagonal matrix D is ! exactly singular. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) DOUBLE PRECISION EIGHT, SEVTEN PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 ) ! .. ! .. Local Scalars .. INTEGER IMAX, J, JB, JJ, JMAX, JP, K, KK, KKW, KP, & KSTEP, KW DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D21, D22, R1, & ROWMAX, T ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX EXTERNAL LSAME, IDAMAX ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEMM, DGEMV, DSCAL, DSWAP ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! INFO = 0 ! ! Initialize ALPHA for use in choosing pivot block size. ! ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT ! IF( LSAME( UPLO, 'U' ) ) THEN ! ! Factorize the trailing columns of A using the upper triangle ! of A and working backwards, and compute the matrix W = U12*D ! for use in updating A11 ! ! K is the main loop index, decreasing from N in steps of 1 or 2 ! ! KW is the column of W which corresponds to column K of A ! K = N 10 CONTINUE KW = NB + K - N ! ! Exit from loop ! IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 ) & GO TO 30 ! ! Copy column K of A to column KW of W and update it ! CALL DCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 ) IF( K.LT.N ) & CALL DGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ), LDA, & W( K, KW+1 ), LDW, ONE, W( 1, KW ), 1 ) ! KSTEP = 1 ! ! Determine rows and columns to be interchanged and whether ! a 1-by-1 or 2-by-2 pivot block will be used ! ABSAKK = ABS( W( K, KW ) ) ! ! IMAX is the row-index of the largest off-diagonal element in ! column K, and COLMAX is its absolute value ! IF( K.GT.1 ) THEN IMAX = IDAMAX( K-1, W( 1, KW ), 1 ) COLMAX = ABS( W( IMAX, KW ) ) ELSE COLMAX = ZERO END IF ! IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN ! ! Column K is zero: set INFO and continue ! IF( INFO.EQ.0 ) & INFO = K KP = K ELSE IF( ABSAKK.GE.ALPHA*COLMAX ) THEN ! ! no interchange, use 1-by-1 pivot block ! KP = K ELSE ! ! Copy column IMAX to column KW-1 of W and update it ! CALL DCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 ) CALL DCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA, & W( IMAX+1, KW-1 ), 1 ) IF( K.LT.N ) & CALL DGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ), & LDA, W( IMAX, KW+1 ), LDW, ONE, & W( 1, KW-1 ), 1 ) ! ! JMAX is the column-index of the largest off-diagonal ! element in row IMAX, and ROWMAX is its absolute value ! JMAX = IMAX + IDAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 ) ROWMAX = ABS( W( JMAX, KW-1 ) ) IF( IMAX.GT.1 ) THEN JMAX = IDAMAX( IMAX-1, W( 1, KW-1 ), 1 ) ROWMAX = MAX( ROWMAX, ABS( W( JMAX, KW-1 ) ) ) END IF ! IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN ! ! no interchange, use 1-by-1 pivot block ! KP = K ELSE IF( ABS( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX ) THEN ! ! interchange rows and columns K and IMAX, use 1-by-1 ! pivot block ! KP = IMAX ! ! copy column KW-1 of W to column KW ! CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 ) ELSE ! ! interchange rows and columns K-1 and IMAX, use 2-by-2 ! pivot block ! KP = IMAX KSTEP = 2 END IF END IF ! KK = K - KSTEP + 1 KKW = NB + KK - N ! ! Updated column KP is already stored in column KKW of W ! IF( KP.NE.KK ) THEN ! ! Copy non-updated column KK to column KP ! A( KP, K ) = A( KK, K ) CALL DCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ), & LDA ) CALL DCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 ) ! ! Interchange rows KK and KP in last KK columns of A and W ! CALL DSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA ) CALL DSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ), & LDW ) END IF ! IF( KSTEP.EQ.1 ) THEN ! ! 1-by-1 pivot block D(k): column KW of W now holds ! ! W(k) = U(k)*D(k) ! ! where U(k) is the k-th column of U ! ! Store U(k) in column k of A ! CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 ) R1 = ONE / A( K, K ) CALL DSCAL( K-1, R1, A( 1, K ), 1 ) ELSE ! ! 2-by-2 pivot block D(k): columns KW and KW-1 of W now ! hold ! ! ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k) ! ! where U(k) and U(k-1) are the k-th and (k-1)-th columns ! of U ! IF( K.GT.2 ) THEN ! ! Store U(k) and U(k-1) in columns k and k-1 of A ! D21 = W( K-1, KW ) D11 = W( K, KW ) / D21 D22 = W( K-1, KW-1 ) / D21 T = ONE / ( D11*D22-ONE ) D21 = T / D21 DO 20 J = 1, K - 2 A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) ) A( J, K ) = D21*( D22*W( J, KW )-W( J, KW-1 ) ) 20 CONTINUE END IF ! ! Copy D(k) to A ! A( K-1, K-1 ) = W( K-1, KW-1 ) A( K-1, K ) = W( K-1, KW ) A( K, K ) = W( K, KW ) END IF END IF ! ! Store details of the interchanges in IPIV ! IF( KSTEP.EQ.1 ) THEN IPIV( K ) = KP ELSE IPIV( K ) = -KP IPIV( K-1 ) = -KP END IF ! ! Decrease K and return to the start of the main loop ! K = K - KSTEP GO TO 10 ! 30 CONTINUE ! ! Update the upper triangle of A11 (= A(1:k,1:k)) as ! ! A11 := A11 - U12*D*U12' = A11 - U12*W' ! ! computing blocks of NB columns at a time ! DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB JB = MIN( NB, K-J+1 ) ! ! Update the upper triangle of the diagonal block ! DO 40 JJ = J, J + JB - 1 CALL DGEMV( 'No transpose', JJ-J+1, N-K, -ONE, & A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, ONE, & A( J, JJ ), 1 ) 40 CONTINUE ! ! Update the rectangular superdiagonal block ! CALL DGEMM( 'No transpose', 'Transpose', J-1, JB, N-K, -ONE, & A( 1, K+1 ), LDA, W( J, KW+1 ), LDW, ONE, & A( 1, J ), LDA ) 50 CONTINUE ! ! Put U12 in standard form by partially undoing the interchanges ! in columns k+1:n ! J = K + 1 60 CONTINUE JJ = J JP = IPIV( J ) IF( JP.LT.0 ) THEN JP = -JP J = J + 1 END IF J = J + 1 IF( JP.NE.JJ .AND. J.LE.N ) & CALL DSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA ) IF( J.LE.N ) & GO TO 60 ! ! Set KB to the number of columns factorized ! KB = N - K ! ELSE ! ! Factorize the leading columns of A using the lower triangle ! of A and working forwards, and compute the matrix W = L21*D ! for use in updating A22 ! ! K is the main loop index, increasing from 1 in steps of 1 or 2 ! K = 1 70 CONTINUE ! ! Exit from loop ! IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N ) & GO TO 90 ! ! Copy column K of A to column K of W and update it ! CALL DCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 ) CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ), LDA, & W( K, 1 ), LDW, ONE, W( K, K ), 1 ) ! KSTEP = 1 ! ! Determine rows and columns to be interchanged and whether ! a 1-by-1 or 2-by-2 pivot block will be used ! ABSAKK = ABS( W( K, K ) ) ! ! IMAX is the row-index of the largest off-diagonal element in ! column K, and COLMAX is its absolute value ! IF( K.LT.N ) THEN IMAX = K + IDAMAX( N-K, W( K+1, K ), 1 ) COLMAX = ABS( W( IMAX, K ) ) ELSE COLMAX = ZERO END IF ! IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN ! ! Column K is zero: set INFO and continue ! IF( INFO.EQ.0 ) & INFO = K KP = K ELSE IF( ABSAKK.GE.ALPHA*COLMAX ) THEN ! ! no interchange, use 1-by-1 pivot block ! KP = K ELSE ! ! Copy column IMAX to column K+1 of W and update it ! CALL DCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1 ) CALL DCOPY( N-IMAX+1, A( IMAX, IMAX ), 1, W( IMAX, K+1 ), & 1 ) CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ), & LDA, W( IMAX, 1 ), LDW, ONE, W( K, K+1 ), 1 ) ! ! JMAX is the column-index of the largest off-diagonal ! element in row IMAX, and ROWMAX is its absolute value ! JMAX = K - 1 + IDAMAX( IMAX-K, W( K, K+1 ), 1 ) ROWMAX = ABS( W( JMAX, K+1 ) ) IF( IMAX.LT.N ) THEN JMAX = IMAX + IDAMAX( N-IMAX, W( IMAX+1, K+1 ), 1 ) ROWMAX = MAX( ROWMAX, ABS( W( JMAX, K+1 ) ) ) END IF ! IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN ! ! no interchange, use 1-by-1 pivot block ! KP = K ELSE IF( ABS( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX ) THEN ! ! interchange rows and columns K and IMAX, use 1-by-1 ! pivot block ! KP = IMAX ! ! copy column K+1 of W to column K ! CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 ) ELSE ! ! interchange rows and columns K+1 and IMAX, use 2-by-2 ! pivot block ! KP = IMAX KSTEP = 2 END IF END IF ! KK = K + KSTEP - 1 ! ! Updated column KP is already stored in column KK of W ! IF( KP.NE.KK ) THEN ! ! Copy non-updated column KK to column KP ! A( KP, K ) = A( KK, K ) CALL DCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA ) CALL DCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 ) ! ! Interchange rows KK and KP in first KK columns of A and W ! CALL DSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA ) CALL DSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW ) END IF ! IF( KSTEP.EQ.1 ) THEN ! ! 1-by-1 pivot block D(k): column k of W now holds ! ! W(k) = L(k)*D(k) ! ! where L(k) is the k-th column of L ! ! Store L(k) in column k of A ! CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 ) IF( K.LT.N ) THEN R1 = ONE / A( K, K ) CALL DSCAL( N-K, R1, A( K+1, K ), 1 ) END IF ELSE ! ! 2-by-2 pivot block D(k): columns k and k+1 of W now hold ! ! ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k) ! ! where L(k) and L(k+1) are the k-th and (k+1)-th columns ! of L ! IF( K.LT.N-1 ) THEN ! ! Store L(k) and L(k+1) in columns k and k+1 of A ! D21 = W( K+1, K ) D11 = W( K+1, K+1 ) / D21 D22 = W( K, K ) / D21 T = ONE / ( D11*D22-ONE ) D21 = T / D21 DO 80 J = K + 2, N A( J, K ) = D21*( D11*W( J, K )-W( J, K+1 ) ) A( J, K+1 ) = D21*( D22*W( J, K+1 )-W( J, K ) ) 80 CONTINUE END IF ! ! Copy D(k) to A ! A( K, K ) = W( K, K ) A( K+1, K ) = W( K+1, K ) A( K+1, K+1 ) = W( K+1, K+1 ) END IF END IF ! ! Store details of the interchanges in IPIV ! IF( KSTEP.EQ.1 ) THEN IPIV( K ) = KP ELSE IPIV( K ) = -KP IPIV( K+1 ) = -KP END IF ! ! Increase K and return to the start of the main loop ! K = K + KSTEP GO TO 70 ! 90 CONTINUE ! ! Update the lower triangle of A22 (= A(k:n,k:n)) as ! ! A22 := A22 - L21*D*L21' = A22 - L21*W' ! ! computing blocks of NB columns at a time ! DO 110 J = K, N, NB JB = MIN( NB, N-J+1 ) ! ! Update the lower triangle of the diagonal block ! DO 100 JJ = J, J + JB - 1 CALL DGEMV( 'No transpose', J+JB-JJ, K-1, -ONE, & A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, ONE, & A( JJ, JJ ), 1 ) 100 CONTINUE ! ! Update the rectangular subdiagonal block ! IF( J+JB.LE.N ) & CALL DGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB, & K-1, -ONE, A( J+JB, 1 ), LDA, W( J, 1 ), LDW, & ONE, A( J+JB, J ), LDA ) 110 CONTINUE ! ! Put L21 in standard form by partially undoing the interchanges ! in columns 1:k-1 ! J = K - 1 120 CONTINUE JJ = J JP = IPIV( J ) IF( JP.LT.0 ) THEN JP = -JP J = J - 1 END IF J = J - 1 IF( JP.NE.JJ .AND. J.GE.1 ) & CALL DSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA ) IF( J.GE.1 ) & GO TO 120 ! ! Set KB to the number of columns factorized ! KB = K - 1 ! END IF RETURN ! ! End of DLASYF ! END SUBROUTINE DLATBS( UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, & SCALE, CNORM, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1992 ! ! .. Scalar Arguments .. CHARACTER DIAG, NORMIN, TRANS, UPLO INTEGER INFO, KD, LDAB, N DOUBLE PRECISION SCALE ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), CNORM( * ), X( * ) ! .. ! ! Purpose ! ======= ! ! DLATBS solves one of the triangular systems ! ! A *x = s*b or A'*x = s*b ! ! with scaling to prevent overflow, where A is an upper or lower ! triangular band matrix. Here A' denotes the transpose of A, x and b ! are n-element vectors, and s is a scaling factor, usually less than ! or equal to 1, chosen so that the components of x will be less than ! the overflow threshold. If the unscaled problem will not cause ! overflow, the Level 2 BLAS routine DTBSV is called. If the matrix A ! is singular (A(j,j) = 0 for some j), then s is set to 0 and a ! non-trivial solution to A*x = 0 is returned. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the matrix A is upper or lower triangular. ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! TRANS (input) CHARACTER*1 ! Specifies the operation applied to A. ! = 'N': Solve A * x = s*b (No transpose) ! = 'T': Solve A'* x = s*b (Transpose) ! = 'C': Solve A'* x = s*b (Conjugate transpose = Transpose) ! ! DIAG (input) CHARACTER*1 ! Specifies whether or not the matrix A is unit triangular. ! = 'N': Non-unit triangular ! = 'U': Unit triangular ! ! NORMIN (input) CHARACTER*1 ! Specifies whether CNORM has been set or not. ! = 'Y': CNORM contains the column norms on entry ! = 'N': CNORM is not set on entry. On exit, the norms will ! be computed and stored in CNORM. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of subdiagonals or superdiagonals in the ! triangular matrix A. KD >= 0. ! ! AB (input) DOUBLE PRECISION array, dimension (LDAB,N) ! The upper or lower triangular band matrix A, stored in the ! first KD+1 rows of the array. The j-th column of A is stored ! in the j-th column of the array AB as follows: ! if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KD+1. ! ! X (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the right hand side b of the triangular system. ! On exit, X is overwritten by the solution vector x. ! ! SCALE (output) DOUBLE PRECISION ! The scaling factor s for the triangular system ! A * x = s*b or A'* x = s*b. ! If SCALE = 0, the matrix A is singular or badly scaled, and ! the vector x is an exact or approximate solution to A*x = 0. ! ! CNORM (input or output) DOUBLE PRECISION array, dimension (N) ! ! If NORMIN = 'Y', CNORM is an input argument and CNORM(j) ! contains the norm of the off-diagonal part of the j-th column ! of A. If TRANS = 'N', CNORM(j) must be greater than or equal ! to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) ! must be greater than or equal to the 1-norm. ! ! If NORMIN = 'N', CNORM is an output argument and CNORM(j) ! returns the 1-norm of the offdiagonal part of the j-th column ! of A. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -k, the k-th argument had an illegal value ! ! Further Details ! ======= ======= ! ! A rough bound on x is computed; if that is less than overflow, DTBSV ! is called, otherwise, specific code is used which checks for possible ! overflow or divide-by-zero at every operation. ! ! A columnwise scheme is used for solving A*x = b. The basic algorithm ! if A is lower triangular is ! ! x[1:n] := b[1:n] ! for j = 1, ..., n ! x(j) := x(j) / A(j,j) ! x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] ! end ! ! Define bounds on the components of x after j iterations of the loop: ! M(j) = bound on x[1:j] ! G(j) = bound on x[j+1:n] ! Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}. ! ! Then for iteration j+1 we have ! M(j+1) <= G(j) / | A(j+1,j+1) | ! G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | ! <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) ! ! where CNORM(j+1) is greater than or equal to the infinity-norm of ! column j+1 of A, not counting the diagonal. Hence ! ! G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) ! 1<=i<=j ! and ! ! |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) ! 1<=i< j ! ! Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTBSV if the ! reciprocal of the largest M(j), j=1,..,n, is larger than ! max(underflow, 1/overflow). ! ! The bound on x(j) is also used to determine when a step in the ! columnwise method can be performed without fear of overflow. If ! the computed bound is greater than a large constant, x is scaled to ! prevent overflow, but if the bound overflows, x is set to 0, x(j) to ! 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. ! ! Similarly, a row-wise scheme is used to solve A'*x = b. The basic ! algorithm for A upper triangular is ! ! for j = 1, ..., n ! x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) ! end ! ! We simultaneously compute two bounds ! G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j ! M(j) = bound on x(i), 1<=i<=j ! ! The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we ! add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. ! Then the bound on x(j) is ! ! M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | ! ! <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) ! 1<=i<=j ! ! and we can safely call DTBSV if 1/M(n) and 1/G(n) are both greater ! than max(underflow, 1/overflow). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, HALF, ONE PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOTRAN, NOUNIT, UPPER INTEGER I, IMAX, J, JFIRST, JINC, JLAST, JLEN, MAIND DOUBLE PRECISION BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS, & TMAX, TSCAL, USCAL, XBND, XJ, XMAX ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DASUM, DDOT, DLAMCH EXTERNAL LSAME, IDAMAX, DASUM, DDOT, DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DSCAL, DTBSV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Executable Statements .. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) NOTRAN = LSAME( TRANS, 'N' ) NOUNIT = LSAME( DIAG, 'N' ) ! ! Test the input parameters. ! IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. & LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -3 ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT. & LSAME( NORMIN, 'N' ) ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( KD.LT.0 ) THEN INFO = -6 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLATBS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Determine machine dependent parameters to control overflow. ! SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' ) BIGNUM = ONE / SMLNUM SCALE = ONE ! IF( LSAME( NORMIN, 'N' ) ) THEN ! ! Compute the 1-norm of each column, not including the diagonal. ! IF( UPPER ) THEN ! ! A is upper triangular. ! DO 10 J = 1, N JLEN = MIN( KD, J-1 ) CNORM( J ) = DASUM( JLEN, AB( KD+1-JLEN, J ), 1 ) 10 CONTINUE ELSE ! ! A is lower triangular. ! DO 20 J = 1, N JLEN = MIN( KD, N-J ) IF( JLEN.GT.0 ) THEN CNORM( J ) = DASUM( JLEN, AB( 2, J ), 1 ) ELSE CNORM( J ) = ZERO END IF 20 CONTINUE END IF END IF ! ! Scale the column norms by TSCAL if the maximum element in CNORM is ! greater than BIGNUM. ! IMAX = IDAMAX( N, CNORM, 1 ) TMAX = CNORM( IMAX ) IF( TMAX.LE.BIGNUM ) THEN TSCAL = ONE ELSE TSCAL = ONE / ( SMLNUM*TMAX ) CALL DSCAL( N, TSCAL, CNORM, 1 ) END IF ! ! Compute a bound on the computed solution vector to see if the ! Level 2 BLAS routine DTBSV can be used. ! J = IDAMAX( N, X, 1 ) XMAX = ABS( X( J ) ) XBND = XMAX IF( NOTRAN ) THEN ! ! Compute the growth in A * x = b. ! IF( UPPER ) THEN JFIRST = N JLAST = 1 JINC = -1 MAIND = KD + 1 ELSE JFIRST = 1 JLAST = N JINC = 1 MAIND = 1 END IF ! IF( TSCAL.NE.ONE ) THEN GROW = ZERO GO TO 50 END IF ! IF( NOUNIT ) THEN ! ! A is non-unit triangular. ! ! Compute GROW = 1/G(j) and XBND = 1/M(j). ! Initially, G(0) = max{x(i), i=1,...,n}. ! GROW = ONE / MAX( XBND, SMLNUM ) XBND = GROW DO 30 J = JFIRST, JLAST, JINC ! ! Exit the loop if the growth factor is too small. ! IF( GROW.LE.SMLNUM ) & GO TO 50 ! ! M(j) = G(j-1) / abs(A(j,j)) ! TJJ = ABS( AB( MAIND, J ) ) XBND = MIN( XBND, MIN( ONE, TJJ )*GROW ) IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN ! ! G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) ) ! GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) ) ELSE ! ! G(j) could overflow, set GROW to 0. ! GROW = ZERO END IF 30 CONTINUE GROW = XBND ELSE ! ! A is unit triangular. ! ! Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}. ! GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) ) DO 40 J = JFIRST, JLAST, JINC ! ! Exit the loop if the growth factor is too small. ! IF( GROW.LE.SMLNUM ) & GO TO 50 ! ! G(j) = G(j-1)*( 1 + CNORM(j) ) ! GROW = GROW*( ONE / ( ONE+CNORM( J ) ) ) 40 CONTINUE END IF 50 CONTINUE ! ELSE ! ! Compute the growth in A' * x = b. ! IF( UPPER ) THEN JFIRST = 1 JLAST = N JINC = 1 MAIND = KD + 1 ELSE JFIRST = N JLAST = 1 JINC = -1 MAIND = 1 END IF ! IF( TSCAL.NE.ONE ) THEN GROW = ZERO GO TO 80 END IF ! IF( NOUNIT ) THEN ! ! A is non-unit triangular. ! ! Compute GROW = 1/G(j) and XBND = 1/M(j). ! Initially, M(0) = max{x(i), i=1,...,n}. ! GROW = ONE / MAX( XBND, SMLNUM ) XBND = GROW DO 60 J = JFIRST, JLAST, JINC ! ! Exit the loop if the growth factor is too small. ! IF( GROW.LE.SMLNUM ) & GO TO 80 ! ! G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) ) ! XJ = ONE + CNORM( J ) GROW = MIN( GROW, XBND / XJ ) ! ! M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j)) ! TJJ = ABS( AB( MAIND, J ) ) IF( XJ.GT.TJJ ) & XBND = XBND*( TJJ / XJ ) 60 CONTINUE GROW = MIN( GROW, XBND ) ELSE ! ! A is unit triangular. ! ! Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}. ! GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) ) DO 70 J = JFIRST, JLAST, JINC ! ! Exit the loop if the growth factor is too small. ! IF( GROW.LE.SMLNUM ) & GO TO 80 ! ! G(j) = ( 1 + CNORM(j) )*G(j-1) ! XJ = ONE + CNORM( J ) GROW = GROW / XJ 70 CONTINUE END IF 80 CONTINUE END IF ! IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN ! ! Use the Level 2 BLAS solve if the reciprocal of the bound on ! elements of X is not too small. ! CALL DTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, X, 1 ) ELSE ! ! Use a Level 1 BLAS solve, scaling intermediate results. ! IF( XMAX.GT.BIGNUM ) THEN ! ! Scale X so that its components are less than or equal to ! BIGNUM in absolute value. ! SCALE = BIGNUM / XMAX CALL DSCAL( N, SCALE, X, 1 ) XMAX = BIGNUM END IF ! IF( NOTRAN ) THEN ! ! Solve A * x = b ! DO 110 J = JFIRST, JLAST, JINC ! ! Compute x(j) = b(j) / A(j,j), scaling x if necessary. ! XJ = ABS( X( J ) ) IF( NOUNIT ) THEN TJJS = AB( MAIND, J )*TSCAL ELSE TJJS = TSCAL IF( TSCAL.EQ.ONE ) & GO TO 100 END IF TJJ = ABS( TJJS ) IF( TJJ.GT.SMLNUM ) THEN ! ! abs(A(j,j)) > SMLNUM: ! IF( TJJ.LT.ONE ) THEN IF( XJ.GT.TJJ*BIGNUM ) THEN ! ! Scale x by 1/b(j). ! REC = ONE / XJ CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF X( J ) = X( J ) / TJJS XJ = ABS( X( J ) ) ELSE IF( TJJ.GT.ZERO ) THEN ! ! 0 < abs(A(j,j)) <= SMLNUM: ! IF( XJ.GT.TJJ*BIGNUM ) THEN ! ! Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM ! to avoid overflow when dividing by A(j,j). ! REC = ( TJJ*BIGNUM ) / XJ IF( CNORM( J ).GT.ONE ) THEN ! ! Scale by 1/CNORM(j) to avoid overflow when ! multiplying x(j) times column j. ! REC = REC / CNORM( J ) END IF CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF X( J ) = X( J ) / TJJS XJ = ABS( X( J ) ) ELSE ! ! A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and ! scale = 0, and compute a solution to A*x = 0. ! DO 90 I = 1, N X( I ) = ZERO 90 CONTINUE X( J ) = ONE XJ = ONE SCALE = ZERO XMAX = ZERO END IF 100 CONTINUE ! ! Scale x if necessary to avoid overflow when adding a ! multiple of column j of A. ! IF( XJ.GT.ONE ) THEN REC = ONE / XJ IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN ! ! Scale x by 1/(2*abs(x(j))). ! REC = REC*HALF CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC END IF ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN ! ! Scale x by 1/2. ! CALL DSCAL( N, HALF, X, 1 ) SCALE = SCALE*HALF END IF ! IF( UPPER ) THEN IF( J.GT.1 ) THEN ! ! Compute the update ! x(max(1,j-kd):j-1) := x(max(1,j-kd):j-1) - ! x(j)* A(max(1,j-kd):j-1,j) ! JLEN = MIN( KD, J-1 ) CALL DAXPY( JLEN, -X( J )*TSCAL, & AB( KD+1-JLEN, J ), 1, X( J-JLEN ), 1 ) I = IDAMAX( J-1, X, 1 ) XMAX = ABS( X( I ) ) END IF ELSE IF( J.LT.N ) THEN ! ! Compute the update ! x(j+1:min(j+kd,n)) := x(j+1:min(j+kd,n)) - ! x(j) * A(j+1:min(j+kd,n),j) ! JLEN = MIN( KD, N-J ) IF( JLEN.GT.0 ) & CALL DAXPY( JLEN, -X( J )*TSCAL, AB( 2, J ), 1, & X( J+1 ), 1 ) I = J + IDAMAX( N-J, X( J+1 ), 1 ) XMAX = ABS( X( I ) ) END IF 110 CONTINUE ! ELSE ! ! Solve A' * x = b ! DO 160 J = JFIRST, JLAST, JINC ! ! Compute x(j) = b(j) - sum A(k,j)*x(k). ! k<>j ! XJ = ABS( X( J ) ) USCAL = TSCAL REC = ONE / MAX( XMAX, ONE ) IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN ! ! If x(j) could overflow, scale x by 1/(2*XMAX). ! REC = REC*HALF IF( NOUNIT ) THEN TJJS = AB( MAIND, J )*TSCAL ELSE TJJS = TSCAL END IF TJJ = ABS( TJJS ) IF( TJJ.GT.ONE ) THEN ! ! Divide by A(j,j) when scaling x if A(j,j) > 1. ! REC = MIN( ONE, REC*TJJ ) USCAL = USCAL / TJJS END IF IF( REC.LT.ONE ) THEN CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF ! SUMJ = ZERO IF( USCAL.EQ.ONE ) THEN ! ! If the scaling needed for A in the dot product is 1, ! call DDOT to perform the dot product. ! IF( UPPER ) THEN JLEN = MIN( KD, J-1 ) SUMJ = DDOT( JLEN, AB( KD+1-JLEN, J ), 1, & X( J-JLEN ), 1 ) ELSE JLEN = MIN( KD, N-J ) IF( JLEN.GT.0 ) & SUMJ = DDOT( JLEN, AB( 2, J ), 1, X( J+1 ), 1 ) END IF ELSE ! ! Otherwise, use in-line code for the dot product. ! IF( UPPER ) THEN JLEN = MIN( KD, J-1 ) DO 120 I = 1, JLEN SUMJ = SUMJ + ( AB( KD+I-JLEN, J )*USCAL )* & X( J-JLEN-1+I ) 120 CONTINUE ELSE JLEN = MIN( KD, N-J ) DO 130 I = 1, JLEN SUMJ = SUMJ + ( AB( I+1, J )*USCAL )*X( J+I ) 130 CONTINUE END IF END IF ! IF( USCAL.EQ.TSCAL ) THEN ! ! Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j) ! was not used to scale the dotproduct. ! X( J ) = X( J ) - SUMJ XJ = ABS( X( J ) ) IF( NOUNIT ) THEN ! ! Compute x(j) = x(j) / A(j,j), scaling if necessary. ! TJJS = AB( MAIND, J )*TSCAL ELSE TJJS = TSCAL IF( TSCAL.EQ.ONE ) & GO TO 150 END IF TJJ = ABS( TJJS ) IF( TJJ.GT.SMLNUM ) THEN ! ! abs(A(j,j)) > SMLNUM: ! IF( TJJ.LT.ONE ) THEN IF( XJ.GT.TJJ*BIGNUM ) THEN ! ! Scale X by 1/abs(x(j)). ! REC = ONE / XJ CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF X( J ) = X( J ) / TJJS ELSE IF( TJJ.GT.ZERO ) THEN ! ! 0 < abs(A(j,j)) <= SMLNUM: ! IF( XJ.GT.TJJ*BIGNUM ) THEN ! ! Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM. ! REC = ( TJJ*BIGNUM ) / XJ CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF X( J ) = X( J ) / TJJS ELSE ! ! A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and ! scale = 0, and compute a solution to A'*x = 0. ! DO 140 I = 1, N X( I ) = ZERO 140 CONTINUE X( J ) = ONE SCALE = ZERO XMAX = ZERO END IF 150 CONTINUE ELSE ! ! Compute x(j) := x(j) / A(j,j) - sumj if the dot ! product has already been divided by 1/A(j,j). ! X( J ) = X( J ) / TJJS - SUMJ END IF XMAX = MAX( XMAX, ABS( X( J ) ) ) 160 CONTINUE END IF SCALE = SCALE / TSCAL END IF ! ! Scale the column norms by 1/TSCAL for return. ! IF( TSCAL.NE.ONE ) THEN CALL DSCAL( N, ONE / TSCAL, CNORM, 1 ) END IF ! RETURN ! ! End of DLATBS ! END SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, & JPIV ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER IJOB, LDZ, N DOUBLE PRECISION RDSCAL, RDSUM ! .. ! .. Array Arguments .. INTEGER IPIV( * ), JPIV( * ) DOUBLE PRECISION RHS( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DLATDF uses the LU factorization of the n-by-n matrix Z computed by ! DGETC2 and computes a contribution to the reciprocal Dif-estimate ! by solving Z * x = b for x, and choosing the r.h.s. b such that ! the norm of x is as large as possible. On entry RHS = b holds the ! contribution from earlier solved sub-systems, and on return RHS = x. ! ! The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q, ! where P and Q are permutation matrices. L is lower triangular with ! unit diagonal elements and U is upper triangular. ! ! Arguments ! ========= ! ! IJOB (input) INTEGER ! IJOB = 2: First compute an approximative null-vector e ! of Z using DGECON, e is normalized and solve for ! Zx = +-e - f with the sign giving the greater value ! of 2-norm(x). About 5 times as expensive as Default. ! IJOB .ne. 2: Local look ahead strategy where all entries of ! the r.h.s. b is choosen as either +1 or -1 (Default). ! ! N (input) INTEGER ! The number of columns of the matrix Z. ! ! Z (input) DOUBLE PRECISION array, dimension (LDZ, N) ! On entry, the LU part of the factorization of the n-by-n ! matrix Z computed by DGETC2: Z = P * L * U * Q ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDA >= max(1, N). ! ! RHS (input/output) DOUBLE PRECISION array, dimension N. ! On entry, RHS contains contributions from other subsystems. ! On exit, RHS contains the solution of the subsystem with ! entries acoording to the value of IJOB (see above). ! ! RDSUM (input/output) DOUBLE PRECISION ! On entry, the sum of squares of computed contributions to ! the Dif-estimate under computation by DTGSYL, where the ! scaling factor RDSCAL (see below) has been factored out. ! On exit, the corresponding sum of squares updated with the ! contributions from the current sub-system. ! If TRANS = 'T' RDSUM is not touched. ! NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL. ! ! RDSCAL (input/output) DOUBLE PRECISION ! On entry, scaling factor used to prevent overflow in RDSUM. ! On exit, RDSCAL is updated w.r.t. the current contributions ! in RDSUM. ! If TRANS = 'T', RDSCAL is not touched. ! NOTE: RDSCAL only makes sense when DTGSY2 is called by ! DTGSYL. ! ! IPIV (input) INTEGER array, dimension (N). ! The pivot indices; for 1 <= i <= N, row i of the ! matrix has been interchanged with row IPIV(i). ! ! JPIV (input) INTEGER array, dimension (N). ! The pivot indices; for 1 <= j <= N, column j of the ! matrix has been interchanged with column JPIV(j). ! ! Further Details ! =============== ! ! Based on contributions by ! Bo Kagstrom and Peter Poromaa, Department of Computing Science, ! Umea University, S-901 87 Umea, Sweden. ! ! This routine is a further developed implementation of algorithm ! BSOLVE in [1] using complete pivoting in the LU factorization. ! ! [1] Bo Kagstrom and Lars Westin, ! Generalized Schur Methods with Condition Estimators for ! Solving the Generalized Sylvester Equation, IEEE Transactions ! on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751. ! ! [2] Peter Poromaa, ! On Efficient and Robust Estimators for the Separation ! between two Regular Matrix Pairs with Applications in ! Condition Estimation. Report IMINF-95.05, Departement of ! Computing Science, Umea University, S-901 87 Umea, Sweden, 1995. ! ! ===================================================================== ! ! .. Parameters .. INTEGER MAXDIM PARAMETER ( MAXDIM = 8 ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, INFO, J, K DOUBLE PRECISION BM, BP, PMONE, SMINU, SPLUS, TEMP ! .. ! .. Local Arrays .. INTEGER IWORK( MAXDIM ) DOUBLE PRECISION WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM ) ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DGECON, DGESC2, DLASSQ, DLASWP, & DSCAL ! .. ! .. External Functions .. DOUBLE PRECISION DASUM, DDOT EXTERNAL DASUM, DDOT ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, SQRT ! .. ! .. Executable Statements .. ! IF( IJOB.NE.2 ) THEN ! ! Apply permutations IPIV to RHS ! CALL DLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 ) ! ! Solve for L-part choosing RHS either to +1 or -1. ! PMONE = -ONE ! DO 10 J = 1, N - 1 BP = RHS( J ) + ONE BM = RHS( J ) - ONE SPLUS = ONE ! ! Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and ! SMIN computed more efficiently than in BSOLVE [1]. ! SPLUS = SPLUS + DDOT( N-J, Z( J+1, J ), 1, Z( J+1, J ), 1 ) SMINU = DDOT( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 ) SPLUS = SPLUS*RHS( J ) IF( SPLUS.GT.SMINU ) THEN RHS( J ) = BP ELSE IF( SMINU.GT.SPLUS ) THEN RHS( J ) = BM ELSE ! ! In this case the updating sums are equal and we can ! choose RHS(J) +1 or -1. The first time this happens ! we choose -1, thereafter +1. This is a simple way to ! get good estimates of matrices like Byers well-known ! example (see [1]). (Not done in BSOLVE.) ! RHS( J ) = RHS( J ) + PMONE PMONE = ONE END IF ! ! Compute the remaining r.h.s. ! TEMP = -RHS( J ) CALL DAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 ) ! 10 CONTINUE ! ! Solve for U-part, look-ahead for RHS(N) = +-1. This is not done ! in BSOLVE and will hopefully give us a better estimate because ! any ill-conditioning of the original matrix is transfered to U ! and not to L. U(N, N) is an approximation to sigma_min(LU). ! CALL DCOPY( N-1, RHS, 1, XP, 1 ) XP( N ) = RHS( N ) + ONE RHS( N ) = RHS( N ) - ONE SPLUS = ZERO SMINU = ZERO DO 30 I = N, 1, -1 TEMP = ONE / Z( I, I ) XP( I ) = XP( I )*TEMP RHS( I ) = RHS( I )*TEMP DO 20 K = I + 1, N XP( I ) = XP( I ) - XP( K )*( Z( I, K )*TEMP ) RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP ) 20 CONTINUE SPLUS = SPLUS + ABS( XP( I ) ) SMINU = SMINU + ABS( RHS( I ) ) 30 CONTINUE IF( SPLUS.GT.SMINU ) & CALL DCOPY( N, XP, 1, RHS, 1 ) ! ! Apply the permutations JPIV to the computed solution (RHS) ! CALL DLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 ) ! ! Compute the sum of squares ! CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM ) ! ELSE ! ! IJOB = 2, Compute approximate nullvector XM of Z ! CALL DGECON( 'I', N, Z, LDZ, ONE, TEMP, WORK, IWORK, INFO ) CALL DCOPY( N, WORK( N+1 ), 1, XM, 1 ) ! ! Compute RHS ! CALL DLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 ) TEMP = ONE / SQRT( DDOT( N, XM, 1, XM, 1 ) ) CALL DSCAL( N, TEMP, XM, 1 ) CALL DCOPY( N, XM, 1, XP, 1 ) CALL DAXPY( N, ONE, RHS, 1, XP, 1 ) CALL DAXPY( N, -ONE, XM, 1, RHS, 1 ) CALL DGESC2( N, Z, LDZ, RHS, IPIV, JPIV, TEMP ) CALL DGESC2( N, Z, LDZ, XP, IPIV, JPIV, TEMP ) IF( DASUM( N, XP, 1 ).GT.DASUM( N, RHS, 1 ) ) & CALL DCOPY( N, XP, 1, RHS, 1 ) ! ! Compute the sum of squares ! CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM ) ! END IF ! RETURN ! ! End of DLATDF ! END SUBROUTINE DLATPS( UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, & CNORM, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1992 ! ! .. Scalar Arguments .. CHARACTER DIAG, NORMIN, TRANS, UPLO INTEGER INFO, N DOUBLE PRECISION SCALE ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ), CNORM( * ), X( * ) ! .. ! ! Purpose ! ======= ! ! DLATPS solves one of the triangular systems ! ! A *x = s*b or A'*x = s*b ! ! with scaling to prevent overflow, where A is an upper or lower ! triangular matrix stored in packed form. Here A' denotes the ! transpose of A, x and b are n-element vectors, and s is a scaling ! factor, usually less than or equal to 1, chosen so that the ! components of x will be less than the overflow threshold. If the ! unscaled problem will not cause overflow, the Level 2 BLAS routine ! DTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j), ! then s is set to 0 and a non-trivial solution to A*x = 0 is returned. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the matrix A is upper or lower triangular. ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! TRANS (input) CHARACTER*1 ! Specifies the operation applied to A. ! = 'N': Solve A * x = s*b (No transpose) ! = 'T': Solve A'* x = s*b (Transpose) ! = 'C': Solve A'* x = s*b (Conjugate transpose = Transpose) ! ! DIAG (input) CHARACTER*1 ! Specifies whether or not the matrix A is unit triangular. ! = 'N': Non-unit triangular ! = 'U': Unit triangular ! ! NORMIN (input) CHARACTER*1 ! Specifies whether CNORM has been set or not. ! = 'Y': CNORM contains the column norms on entry ! = 'N': CNORM is not set on entry. On exit, the norms will ! be computed and stored in CNORM. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The upper or lower triangular matrix A, packed columnwise in ! a linear array. The j-th column of A is stored in the array ! AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. ! ! X (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the right hand side b of the triangular system. ! On exit, X is overwritten by the solution vector x. ! ! SCALE (output) DOUBLE PRECISION ! The scaling factor s for the triangular system ! A * x = s*b or A'* x = s*b. ! If SCALE = 0, the matrix A is singular or badly scaled, and ! the vector x is an exact or approximate solution to A*x = 0. ! ! CNORM (input or output) DOUBLE PRECISION array, dimension (N) ! ! If NORMIN = 'Y', CNORM is an input argument and CNORM(j) ! contains the norm of the off-diagonal part of the j-th column ! of A. If TRANS = 'N', CNORM(j) must be greater than or equal ! to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) ! must be greater than or equal to the 1-norm. ! ! If NORMIN = 'N', CNORM is an output argument and CNORM(j) ! returns the 1-norm of the offdiagonal part of the j-th column ! of A. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -k, the k-th argument had an illegal value ! ! Further Details ! ======= ======= ! ! A rough bound on x is computed; if that is less than overflow, DTPSV ! is called, otherwise, specific code is used which checks for possible ! overflow or divide-by-zero at every operation. ! ! A columnwise scheme is used for solving A*x = b. The basic algorithm ! if A is lower triangular is ! ! x[1:n] := b[1:n] ! for j = 1, ..., n ! x(j) := x(j) / A(j,j) ! x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] ! end ! ! Define bounds on the components of x after j iterations of the loop: ! M(j) = bound on x[1:j] ! G(j) = bound on x[j+1:n] ! Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}. ! ! Then for iteration j+1 we have ! M(j+1) <= G(j) / | A(j+1,j+1) | ! G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | ! <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) ! ! where CNORM(j+1) is greater than or equal to the infinity-norm of ! column j+1 of A, not counting the diagonal. Hence ! ! G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) ! 1<=i<=j ! and ! ! |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) ! 1<=i< j ! ! Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTPSV if the ! reciprocal of the largest M(j), j=1,..,n, is larger than ! max(underflow, 1/overflow). ! ! The bound on x(j) is also used to determine when a step in the ! columnwise method can be performed without fear of overflow. If ! the computed bound is greater than a large constant, x is scaled to ! prevent overflow, but if the bound overflows, x is set to 0, x(j) to ! 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. ! ! Similarly, a row-wise scheme is used to solve A'*x = b. The basic ! algorithm for A upper triangular is ! ! for j = 1, ..., n ! x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) ! end ! ! We simultaneously compute two bounds ! G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j ! M(j) = bound on x(i), 1<=i<=j ! ! The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we ! add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. ! Then the bound on x(j) is ! ! M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | ! ! <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) ! 1<=i<=j ! ! and we can safely call DTPSV if 1/M(n) and 1/G(n) are both greater ! than max(underflow, 1/overflow). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, HALF, ONE PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOTRAN, NOUNIT, UPPER INTEGER I, IMAX, IP, J, JFIRST, JINC, JLAST, JLEN DOUBLE PRECISION BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS, & TMAX, TSCAL, USCAL, XBND, XJ, XMAX ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DASUM, DDOT, DLAMCH EXTERNAL LSAME, IDAMAX, DASUM, DDOT, DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DSCAL, DTPSV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Executable Statements .. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) NOTRAN = LSAME( TRANS, 'N' ) NOUNIT = LSAME( DIAG, 'N' ) ! ! Test the input parameters. ! IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. & LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -3 ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT. & LSAME( NORMIN, 'N' ) ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLATPS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Determine machine dependent parameters to control overflow. ! SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' ) BIGNUM = ONE / SMLNUM SCALE = ONE ! IF( LSAME( NORMIN, 'N' ) ) THEN ! ! Compute the 1-norm of each column, not including the diagonal. ! IF( UPPER ) THEN ! ! A is upper triangular. ! IP = 1 DO 10 J = 1, N CNORM( J ) = DASUM( J-1, AP( IP ), 1 ) IP = IP + J 10 CONTINUE ELSE ! ! A is lower triangular. ! IP = 1 DO 20 J = 1, N - 1 CNORM( J ) = DASUM( N-J, AP( IP+1 ), 1 ) IP = IP + N - J + 1 20 CONTINUE CNORM( N ) = ZERO END IF END IF ! ! Scale the column norms by TSCAL if the maximum element in CNORM is ! greater than BIGNUM. ! IMAX = IDAMAX( N, CNORM, 1 ) TMAX = CNORM( IMAX ) IF( TMAX.LE.BIGNUM ) THEN TSCAL = ONE ELSE TSCAL = ONE / ( SMLNUM*TMAX ) CALL DSCAL( N, TSCAL, CNORM, 1 ) END IF ! ! Compute a bound on the computed solution vector to see if the ! Level 2 BLAS routine DTPSV can be used. ! J = IDAMAX( N, X, 1 ) XMAX = ABS( X( J ) ) XBND = XMAX IF( NOTRAN ) THEN ! ! Compute the growth in A * x = b. ! IF( UPPER ) THEN JFIRST = N JLAST = 1 JINC = -1 ELSE JFIRST = 1 JLAST = N JINC = 1 END IF ! IF( TSCAL.NE.ONE ) THEN GROW = ZERO GO TO 50 END IF ! IF( NOUNIT ) THEN ! ! A is non-unit triangular. ! ! Compute GROW = 1/G(j) and XBND = 1/M(j). ! Initially, G(0) = max{x(i), i=1,...,n}. ! GROW = ONE / MAX( XBND, SMLNUM ) XBND = GROW IP = JFIRST*( JFIRST+1 ) / 2 JLEN = N DO 30 J = JFIRST, JLAST, JINC ! ! Exit the loop if the growth factor is too small. ! IF( GROW.LE.SMLNUM ) & GO TO 50 ! ! M(j) = G(j-1) / abs(A(j,j)) ! TJJ = ABS( AP( IP ) ) XBND = MIN( XBND, MIN( ONE, TJJ )*GROW ) IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN ! ! G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) ) ! GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) ) ELSE ! ! G(j) could overflow, set GROW to 0. ! GROW = ZERO END IF IP = IP + JINC*JLEN JLEN = JLEN - 1 30 CONTINUE GROW = XBND ELSE ! ! A is unit triangular. ! ! Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}. ! GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) ) DO 40 J = JFIRST, JLAST, JINC ! ! Exit the loop if the growth factor is too small. ! IF( GROW.LE.SMLNUM ) & GO TO 50 ! ! G(j) = G(j-1)*( 1 + CNORM(j) ) ! GROW = GROW*( ONE / ( ONE+CNORM( J ) ) ) 40 CONTINUE END IF 50 CONTINUE ! ELSE ! ! Compute the growth in A' * x = b. ! IF( UPPER ) THEN JFIRST = 1 JLAST = N JINC = 1 ELSE JFIRST = N JLAST = 1 JINC = -1 END IF ! IF( TSCAL.NE.ONE ) THEN GROW = ZERO GO TO 80 END IF ! IF( NOUNIT ) THEN ! ! A is non-unit triangular. ! ! Compute GROW = 1/G(j) and XBND = 1/M(j). ! Initially, M(0) = max{x(i), i=1,...,n}. ! GROW = ONE / MAX( XBND, SMLNUM ) XBND = GROW IP = JFIRST*( JFIRST+1 ) / 2 JLEN = 1 DO 60 J = JFIRST, JLAST, JINC ! ! Exit the loop if the growth factor is too small. ! IF( GROW.LE.SMLNUM ) & GO TO 80 ! ! G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) ) ! XJ = ONE + CNORM( J ) GROW = MIN( GROW, XBND / XJ ) ! ! M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j)) ! TJJ = ABS( AP( IP ) ) IF( XJ.GT.TJJ ) & XBND = XBND*( TJJ / XJ ) JLEN = JLEN + 1 IP = IP + JINC*JLEN 60 CONTINUE GROW = MIN( GROW, XBND ) ELSE ! ! A is unit triangular. ! ! Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}. ! GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) ) DO 70 J = JFIRST, JLAST, JINC ! ! Exit the loop if the growth factor is too small. ! IF( GROW.LE.SMLNUM ) & GO TO 80 ! ! G(j) = ( 1 + CNORM(j) )*G(j-1) ! XJ = ONE + CNORM( J ) GROW = GROW / XJ 70 CONTINUE END IF 80 CONTINUE END IF ! IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN ! ! Use the Level 2 BLAS solve if the reciprocal of the bound on ! elements of X is not too small. ! CALL DTPSV( UPLO, TRANS, DIAG, N, AP, X, 1 ) ELSE ! ! Use a Level 1 BLAS solve, scaling intermediate results. ! IF( XMAX.GT.BIGNUM ) THEN ! ! Scale X so that its components are less than or equal to ! BIGNUM in absolute value. ! SCALE = BIGNUM / XMAX CALL DSCAL( N, SCALE, X, 1 ) XMAX = BIGNUM END IF ! IF( NOTRAN ) THEN ! ! Solve A * x = b ! IP = JFIRST*( JFIRST+1 ) / 2 DO 110 J = JFIRST, JLAST, JINC ! ! Compute x(j) = b(j) / A(j,j), scaling x if necessary. ! XJ = ABS( X( J ) ) IF( NOUNIT ) THEN TJJS = AP( IP )*TSCAL ELSE TJJS = TSCAL IF( TSCAL.EQ.ONE ) & GO TO 100 END IF TJJ = ABS( TJJS ) IF( TJJ.GT.SMLNUM ) THEN ! ! abs(A(j,j)) > SMLNUM: ! IF( TJJ.LT.ONE ) THEN IF( XJ.GT.TJJ*BIGNUM ) THEN ! ! Scale x by 1/b(j). ! REC = ONE / XJ CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF X( J ) = X( J ) / TJJS XJ = ABS( X( J ) ) ELSE IF( TJJ.GT.ZERO ) THEN ! ! 0 < abs(A(j,j)) <= SMLNUM: ! IF( XJ.GT.TJJ*BIGNUM ) THEN ! ! Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM ! to avoid overflow when dividing by A(j,j). ! REC = ( TJJ*BIGNUM ) / XJ IF( CNORM( J ).GT.ONE ) THEN ! ! Scale by 1/CNORM(j) to avoid overflow when ! multiplying x(j) times column j. ! REC = REC / CNORM( J ) END IF CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF X( J ) = X( J ) / TJJS XJ = ABS( X( J ) ) ELSE ! ! A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and ! scale = 0, and compute a solution to A*x = 0. ! DO 90 I = 1, N X( I ) = ZERO 90 CONTINUE X( J ) = ONE XJ = ONE SCALE = ZERO XMAX = ZERO END IF 100 CONTINUE ! ! Scale x if necessary to avoid overflow when adding a ! multiple of column j of A. ! IF( XJ.GT.ONE ) THEN REC = ONE / XJ IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN ! ! Scale x by 1/(2*abs(x(j))). ! REC = REC*HALF CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC END IF ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN ! ! Scale x by 1/2. ! CALL DSCAL( N, HALF, X, 1 ) SCALE = SCALE*HALF END IF ! IF( UPPER ) THEN IF( J.GT.1 ) THEN ! ! Compute the update ! x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j) ! CALL DAXPY( J-1, -X( J )*TSCAL, AP( IP-J+1 ), 1, X, & 1 ) I = IDAMAX( J-1, X, 1 ) XMAX = ABS( X( I ) ) END IF IP = IP - J ELSE IF( J.LT.N ) THEN ! ! Compute the update ! x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j) ! CALL DAXPY( N-J, -X( J )*TSCAL, AP( IP+1 ), 1, & X( J+1 ), 1 ) I = J + IDAMAX( N-J, X( J+1 ), 1 ) XMAX = ABS( X( I ) ) END IF IP = IP + N - J + 1 END IF 110 CONTINUE ! ELSE ! ! Solve A' * x = b ! IP = JFIRST*( JFIRST+1 ) / 2 JLEN = 1 DO 160 J = JFIRST, JLAST, JINC ! ! Compute x(j) = b(j) - sum A(k,j)*x(k). ! k<>j ! XJ = ABS( X( J ) ) USCAL = TSCAL REC = ONE / MAX( XMAX, ONE ) IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN ! ! If x(j) could overflow, scale x by 1/(2*XMAX). ! REC = REC*HALF IF( NOUNIT ) THEN TJJS = AP( IP )*TSCAL ELSE TJJS = TSCAL END IF TJJ = ABS( TJJS ) IF( TJJ.GT.ONE ) THEN ! ! Divide by A(j,j) when scaling x if A(j,j) > 1. ! REC = MIN( ONE, REC*TJJ ) USCAL = USCAL / TJJS END IF IF( REC.LT.ONE ) THEN CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF ! SUMJ = ZERO IF( USCAL.EQ.ONE ) THEN ! ! If the scaling needed for A in the dot product is 1, ! call DDOT to perform the dot product. ! IF( UPPER ) THEN SUMJ = DDOT( J-1, AP( IP-J+1 ), 1, X, 1 ) ELSE IF( J.LT.N ) THEN SUMJ = DDOT( N-J, AP( IP+1 ), 1, X( J+1 ), 1 ) END IF ELSE ! ! Otherwise, use in-line code for the dot product. ! IF( UPPER ) THEN DO 120 I = 1, J - 1 SUMJ = SUMJ + ( AP( IP-J+I )*USCAL )*X( I ) 120 CONTINUE ELSE IF( J.LT.N ) THEN DO 130 I = 1, N - J SUMJ = SUMJ + ( AP( IP+I )*USCAL )*X( J+I ) 130 CONTINUE END IF END IF ! IF( USCAL.EQ.TSCAL ) THEN ! ! Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j) ! was not used to scale the dotproduct. ! X( J ) = X( J ) - SUMJ XJ = ABS( X( J ) ) IF( NOUNIT ) THEN ! ! Compute x(j) = x(j) / A(j,j), scaling if necessary. ! TJJS = AP( IP )*TSCAL ELSE TJJS = TSCAL IF( TSCAL.EQ.ONE ) & GO TO 150 END IF TJJ = ABS( TJJS ) IF( TJJ.GT.SMLNUM ) THEN ! ! abs(A(j,j)) > SMLNUM: ! IF( TJJ.LT.ONE ) THEN IF( XJ.GT.TJJ*BIGNUM ) THEN ! ! Scale X by 1/abs(x(j)). ! REC = ONE / XJ CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF X( J ) = X( J ) / TJJS ELSE IF( TJJ.GT.ZERO ) THEN ! ! 0 < abs(A(j,j)) <= SMLNUM: ! IF( XJ.GT.TJJ*BIGNUM ) THEN ! ! Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM. ! REC = ( TJJ*BIGNUM ) / XJ CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF X( J ) = X( J ) / TJJS ELSE ! ! A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and ! scale = 0, and compute a solution to A'*x = 0. ! DO 140 I = 1, N X( I ) = ZERO 140 CONTINUE X( J ) = ONE SCALE = ZERO XMAX = ZERO END IF 150 CONTINUE ELSE ! ! Compute x(j) := x(j) / A(j,j) - sumj if the dot ! product has already been divided by 1/A(j,j). ! X( J ) = X( J ) / TJJS - SUMJ END IF XMAX = MAX( XMAX, ABS( X( J ) ) ) JLEN = JLEN + 1 IP = IP + JINC*JLEN 160 CONTINUE END IF SCALE = SCALE / TSCAL END IF ! ! Scale the column norms by 1/TSCAL for return. ! IF( TSCAL.NE.ONE ) THEN CALL DSCAL( N, ONE / TSCAL, CNORM, 1 ) END IF ! RETURN ! ! End of DLATPS ! END SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER LDA, LDW, N, NB ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * ) ! .. ! ! Purpose ! ======= ! ! DLATRD reduces NB rows and columns of a real symmetric matrix A to ! symmetric tridiagonal form by an orthogonal similarity ! transformation Q' * A * Q, and returns the matrices V and W which are ! needed to apply the transformation to the unreduced part of A. ! ! If UPLO = 'U', DLATRD reduces the last NB rows and columns of a ! matrix, of which the upper triangle is supplied; ! if UPLO = 'L', DLATRD reduces the first NB rows and columns of a ! matrix, of which the lower triangle is supplied. ! ! This is an auxiliary routine called by DSYTRD. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER ! Specifies whether the upper or lower triangular part of the ! symmetric matrix A is stored: ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! N (input) INTEGER ! The order of the matrix A. ! ! NB (input) INTEGER ! The number of rows and columns to be reduced. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the symmetric matrix A. If UPLO = 'U', the leading ! n-by-n upper triangular part of A contains the upper ! triangular part of the matrix A, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading n-by-n lower triangular part of A contains the lower ! triangular part of the matrix A, and the strictly upper ! triangular part of A is not referenced. ! On exit: ! if UPLO = 'U', the last NB columns have been reduced to ! tridiagonal form, with the diagonal elements overwriting ! the diagonal elements of A; the elements above the diagonal ! with the array TAU, represent the orthogonal matrix Q as a ! product of elementary reflectors; ! if UPLO = 'L', the first NB columns have been reduced to ! tridiagonal form, with the diagonal elements overwriting ! the diagonal elements of A; the elements below the diagonal ! with the array TAU, represent the orthogonal matrix Q as a ! product of elementary reflectors. ! See Further Details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= (1,N). ! ! E (output) DOUBLE PRECISION array, dimension (N-1) ! If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal ! elements of the last NB columns of the reduced matrix; ! if UPLO = 'L', E(1:nb) contains the subdiagonal elements of ! the first NB columns of the reduced matrix. ! ! TAU (output) DOUBLE PRECISION array, dimension (N-1) ! The scalar factors of the elementary reflectors, stored in ! TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. ! See Further Details. ! ! W (output) DOUBLE PRECISION array, dimension (LDW,NB) ! The n-by-nb matrix W required to update the unreduced part ! of A. ! ! LDW (input) INTEGER ! The leading dimension of the array W. LDW >= max(1,N). ! ! Further Details ! =============== ! ! If UPLO = 'U', the matrix Q is represented as a product of elementary ! reflectors ! ! Q = H(n) H(n-1) . . . H(n-nb+1). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), ! and tau in TAU(i-1). ! ! If UPLO = 'L', the matrix Q is represented as a product of elementary ! reflectors ! ! Q = H(1) H(2) . . . H(nb). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), ! and tau in TAU(i). ! ! The elements of the vectors v together form the n-by-nb matrix V ! which is needed, with W, to apply the transformation to the unreduced ! part of the matrix, using a symmetric rank-2k update of the form: ! A := A - V*W' - W*V'. ! ! The contents of A on exit are illustrated by the following examples ! with n = 5 and nb = 2: ! ! if UPLO = 'U': if UPLO = 'L': ! ! ( a a a v4 v5 ) ( d ) ! ( a a v4 v5 ) ( 1 d ) ! ( a 1 v5 ) ( v1 1 a ) ! ( d 1 ) ( v1 v2 a a ) ! ( d ) ( v1 v2 a a a ) ! ! where d denotes a diagonal element of the reduced matrix, a denotes ! an element of the original matrix that is unchanged, and vi denotes ! an element of the vector defining H(i). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, HALF PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, HALF = 0.5D+0 ) ! .. ! .. Local Scalars .. INTEGER I, IW DOUBLE PRECISION ALPHA ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DGEMV, DLARFG, DSCAL, DSYMV ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT EXTERNAL LSAME, DDOT ! .. ! .. Intrinsic Functions .. INTRINSIC MIN ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( N.LE.0 ) & RETURN ! IF( LSAME( UPLO, 'U' ) ) THEN ! ! Reduce last NB columns of upper triangle ! DO 10 I = N, N - NB + 1, -1 IW = I - N + NB IF( I.LT.N ) THEN ! ! Update A(1:i,i) ! CALL DGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ), & LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 ) CALL DGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ), & LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 ) END IF IF( I.GT.1 ) THEN ! ! Generate elementary reflector H(i) to annihilate ! A(1:i-2,i) ! CALL DLARFG( I-1, A( I-1, I ), A( 1, I ), 1, TAU( I-1 ) ) E( I-1 ) = A( I-1, I ) A( I-1, I ) = ONE ! ! Compute W(1:i-1,i) ! CALL DSYMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1, & ZERO, W( 1, IW ), 1 ) IF( I.LT.N ) THEN CALL DGEMV( 'Transpose', I-1, N-I, ONE, W( 1, IW+1 ), & LDW, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 ) CALL DGEMV( 'No transpose', I-1, N-I, -ONE, & A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE, & W( 1, IW ), 1 ) CALL DGEMV( 'Transpose', I-1, N-I, ONE, A( 1, I+1 ), & LDA, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 ) CALL DGEMV( 'No transpose', I-1, N-I, -ONE, & W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE, & W( 1, IW ), 1 ) END IF CALL DSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 ) ALPHA = -HALF*TAU( I-1 )*DDOT( I-1, W( 1, IW ), 1, & A( 1, I ), 1 ) CALL DAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 ) END IF ! 10 CONTINUE ELSE ! ! Reduce first NB columns of lower triangle ! DO 20 I = 1, NB ! ! Update A(i:n,i) ! CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ), & LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 ) CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ), & LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 ) IF( I.LT.N ) THEN ! ! Generate elementary reflector H(i) to annihilate ! A(i+2:n,i) ! CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1, & TAU( I ) ) E( I ) = A( I+1, I ) A( I+1, I ) = ONE ! ! Compute W(i+1:n,i) ! CALL DSYMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA, & A( I+1, I ), 1, ZERO, W( I+1, I ), 1 ) CALL DGEMV( 'Transpose', N-I, I-1, ONE, W( I+1, 1 ), LDW, & A( I+1, I ), 1, ZERO, W( 1, I ), 1 ) CALL DGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ), & LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 ) CALL DGEMV( 'Transpose', N-I, I-1, ONE, A( I+1, 1 ), LDA, & A( I+1, I ), 1, ZERO, W( 1, I ), 1 ) CALL DGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ), & LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 ) CALL DSCAL( N-I, TAU( I ), W( I+1, I ), 1 ) ALPHA = -HALF*TAU( I )*DDOT( N-I, W( I+1, I ), 1, & A( I+1, I ), 1 ) CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 ) END IF ! 20 CONTINUE END IF ! RETURN ! ! End of DLATRD ! END SUBROUTINE DLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, & CNORM, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1992 ! ! .. Scalar Arguments .. CHARACTER DIAG, NORMIN, TRANS, UPLO INTEGER INFO, LDA, N DOUBLE PRECISION SCALE ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), CNORM( * ), X( * ) ! .. ! ! Purpose ! ======= ! ! DLATRS solves one of the triangular systems ! ! A *x = s*b or A'*x = s*b ! ! with scaling to prevent overflow. Here A is an upper or lower ! triangular matrix, A' denotes the transpose of A, x and b are ! n-element vectors, and s is a scaling factor, usually less than ! or equal to 1, chosen so that the components of x will be less than ! the overflow threshold. If the unscaled problem will not cause ! overflow, the Level 2 BLAS routine DTRSV is called. If the matrix A ! is singular (A(j,j) = 0 for some j), then s is set to 0 and a ! non-trivial solution to A*x = 0 is returned. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the matrix A is upper or lower triangular. ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! TRANS (input) CHARACTER*1 ! Specifies the operation applied to A. ! = 'N': Solve A * x = s*b (No transpose) ! = 'T': Solve A'* x = s*b (Transpose) ! = 'C': Solve A'* x = s*b (Conjugate transpose = Transpose) ! ! DIAG (input) CHARACTER*1 ! Specifies whether or not the matrix A is unit triangular. ! = 'N': Non-unit triangular ! = 'U': Unit triangular ! ! NORMIN (input) CHARACTER*1 ! Specifies whether CNORM has been set or not. ! = 'Y': CNORM contains the column norms on entry ! = 'N': CNORM is not set on entry. On exit, the norms will ! be computed and stored in CNORM. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The triangular matrix A. If UPLO = 'U', the leading n by n ! upper triangular part of the array A contains the upper ! triangular matrix, and the strictly lower triangular part of ! A is not referenced. If UPLO = 'L', the leading n by n lower ! triangular part of the array A contains the lower triangular ! matrix, and the strictly upper triangular part of A is not ! referenced. If DIAG = 'U', the diagonal elements of A are ! also not referenced and are assumed to be 1. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max (1,N). ! ! X (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the right hand side b of the triangular system. ! On exit, X is overwritten by the solution vector x. ! ! SCALE (output) DOUBLE PRECISION ! The scaling factor s for the triangular system ! A * x = s*b or A'* x = s*b. ! If SCALE = 0, the matrix A is singular or badly scaled, and ! the vector x is an exact or approximate solution to A*x = 0. ! ! CNORM (input or output) DOUBLE PRECISION array, dimension (N) ! ! If NORMIN = 'Y', CNORM is an input argument and CNORM(j) ! contains the norm of the off-diagonal part of the j-th column ! of A. If TRANS = 'N', CNORM(j) must be greater than or equal ! to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) ! must be greater than or equal to the 1-norm. ! ! If NORMIN = 'N', CNORM is an output argument and CNORM(j) ! returns the 1-norm of the offdiagonal part of the j-th column ! of A. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -k, the k-th argument had an illegal value ! ! Further Details ! ======= ======= ! ! A rough bound on x is computed; if that is less than overflow, DTRSV ! is called, otherwise, specific code is used which checks for possible ! overflow or divide-by-zero at every operation. ! ! A columnwise scheme is used for solving A*x = b. The basic algorithm ! if A is lower triangular is ! ! x[1:n] := b[1:n] ! for j = 1, ..., n ! x(j) := x(j) / A(j,j) ! x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] ! end ! ! Define bounds on the components of x after j iterations of the loop: ! M(j) = bound on x[1:j] ! G(j) = bound on x[j+1:n] ! Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}. ! ! Then for iteration j+1 we have ! M(j+1) <= G(j) / | A(j+1,j+1) | ! G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | ! <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) ! ! where CNORM(j+1) is greater than or equal to the infinity-norm of ! column j+1 of A, not counting the diagonal. Hence ! ! G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) ! 1<=i<=j ! and ! ! |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) ! 1<=i< j ! ! Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTRSV if the ! reciprocal of the largest M(j), j=1,..,n, is larger than ! max(underflow, 1/overflow). ! ! The bound on x(j) is also used to determine when a step in the ! columnwise method can be performed without fear of overflow. If ! the computed bound is greater than a large constant, x is scaled to ! prevent overflow, but if the bound overflows, x is set to 0, x(j) to ! 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. ! ! Similarly, a row-wise scheme is used to solve A'*x = b. The basic ! algorithm for A upper triangular is ! ! for j = 1, ..., n ! x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) ! end ! ! We simultaneously compute two bounds ! G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j ! M(j) = bound on x(i), 1<=i<=j ! ! The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we ! add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. ! Then the bound on x(j) is ! ! M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | ! ! <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) ! 1<=i<=j ! ! and we can safely call DTRSV if 1/M(n) and 1/G(n) are both greater ! than max(underflow, 1/overflow). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, HALF, ONE PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOTRAN, NOUNIT, UPPER INTEGER I, IMAX, J, JFIRST, JINC, JLAST DOUBLE PRECISION BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS, & TMAX, TSCAL, USCAL, XBND, XJ, XMAX ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DASUM, DDOT, DLAMCH EXTERNAL LSAME, IDAMAX, DASUM, DDOT, DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DSCAL, DTRSV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Executable Statements .. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) NOTRAN = LSAME( TRANS, 'N' ) NOUNIT = LSAME( DIAG, 'N' ) ! ! Test the input parameters. ! IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. & LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -3 ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT. & LSAME( NORMIN, 'N' ) ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLATRS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Determine machine dependent parameters to control overflow. ! SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' ) BIGNUM = ONE / SMLNUM SCALE = ONE ! IF( LSAME( NORMIN, 'N' ) ) THEN ! ! Compute the 1-norm of each column, not including the diagonal. ! IF( UPPER ) THEN ! ! A is upper triangular. ! DO 10 J = 1, N CNORM( J ) = DASUM( J-1, A( 1, J ), 1 ) 10 CONTINUE ELSE ! ! A is lower triangular. ! DO 20 J = 1, N - 1 CNORM( J ) = DASUM( N-J, A( J+1, J ), 1 ) 20 CONTINUE CNORM( N ) = ZERO END IF END IF ! ! Scale the column norms by TSCAL if the maximum element in CNORM is ! greater than BIGNUM. ! IMAX = IDAMAX( N, CNORM, 1 ) TMAX = CNORM( IMAX ) IF( TMAX.LE.BIGNUM ) THEN TSCAL = ONE ELSE TSCAL = ONE / ( SMLNUM*TMAX ) CALL DSCAL( N, TSCAL, CNORM, 1 ) END IF ! ! Compute a bound on the computed solution vector to see if the ! Level 2 BLAS routine DTRSV can be used. ! J = IDAMAX( N, X, 1 ) XMAX = ABS( X( J ) ) XBND = XMAX IF( NOTRAN ) THEN ! ! Compute the growth in A * x = b. ! IF( UPPER ) THEN JFIRST = N JLAST = 1 JINC = -1 ELSE JFIRST = 1 JLAST = N JINC = 1 END IF ! IF( TSCAL.NE.ONE ) THEN GROW = ZERO GO TO 50 END IF ! IF( NOUNIT ) THEN ! ! A is non-unit triangular. ! ! Compute GROW = 1/G(j) and XBND = 1/M(j). ! Initially, G(0) = max{x(i), i=1,...,n}. ! GROW = ONE / MAX( XBND, SMLNUM ) XBND = GROW DO 30 J = JFIRST, JLAST, JINC ! ! Exit the loop if the growth factor is too small. ! IF( GROW.LE.SMLNUM ) & GO TO 50 ! ! M(j) = G(j-1) / abs(A(j,j)) ! TJJ = ABS( A( J, J ) ) XBND = MIN( XBND, MIN( ONE, TJJ )*GROW ) IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN ! ! G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) ) ! GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) ) ELSE ! ! G(j) could overflow, set GROW to 0. ! GROW = ZERO END IF 30 CONTINUE GROW = XBND ELSE ! ! A is unit triangular. ! ! Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}. ! GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) ) DO 40 J = JFIRST, JLAST, JINC ! ! Exit the loop if the growth factor is too small. ! IF( GROW.LE.SMLNUM ) & GO TO 50 ! ! G(j) = G(j-1)*( 1 + CNORM(j) ) ! GROW = GROW*( ONE / ( ONE+CNORM( J ) ) ) 40 CONTINUE END IF 50 CONTINUE ! ELSE ! ! Compute the growth in A' * x = b. ! IF( UPPER ) THEN JFIRST = 1 JLAST = N JINC = 1 ELSE JFIRST = N JLAST = 1 JINC = -1 END IF ! IF( TSCAL.NE.ONE ) THEN GROW = ZERO GO TO 80 END IF ! IF( NOUNIT ) THEN ! ! A is non-unit triangular. ! ! Compute GROW = 1/G(j) and XBND = 1/M(j). ! Initially, M(0) = max{x(i), i=1,...,n}. ! GROW = ONE / MAX( XBND, SMLNUM ) XBND = GROW DO 60 J = JFIRST, JLAST, JINC ! ! Exit the loop if the growth factor is too small. ! IF( GROW.LE.SMLNUM ) & GO TO 80 ! ! G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) ) ! XJ = ONE + CNORM( J ) GROW = MIN( GROW, XBND / XJ ) ! ! M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j)) ! TJJ = ABS( A( J, J ) ) IF( XJ.GT.TJJ ) & XBND = XBND*( TJJ / XJ ) 60 CONTINUE GROW = MIN( GROW, XBND ) ELSE ! ! A is unit triangular. ! ! Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}. ! GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) ) DO 70 J = JFIRST, JLAST, JINC ! ! Exit the loop if the growth factor is too small. ! IF( GROW.LE.SMLNUM ) & GO TO 80 ! ! G(j) = ( 1 + CNORM(j) )*G(j-1) ! XJ = ONE + CNORM( J ) GROW = GROW / XJ 70 CONTINUE END IF 80 CONTINUE END IF ! IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN ! ! Use the Level 2 BLAS solve if the reciprocal of the bound on ! elements of X is not too small. ! CALL DTRSV( UPLO, TRANS, DIAG, N, A, LDA, X, 1 ) ELSE ! ! Use a Level 1 BLAS solve, scaling intermediate results. ! IF( XMAX.GT.BIGNUM ) THEN ! ! Scale X so that its components are less than or equal to ! BIGNUM in absolute value. ! SCALE = BIGNUM / XMAX CALL DSCAL( N, SCALE, X, 1 ) XMAX = BIGNUM END IF ! IF( NOTRAN ) THEN ! ! Solve A * x = b ! DO 110 J = JFIRST, JLAST, JINC ! ! Compute x(j) = b(j) / A(j,j), scaling x if necessary. ! XJ = ABS( X( J ) ) IF( NOUNIT ) THEN TJJS = A( J, J )*TSCAL ELSE TJJS = TSCAL IF( TSCAL.EQ.ONE ) & GO TO 100 END IF TJJ = ABS( TJJS ) IF( TJJ.GT.SMLNUM ) THEN ! ! abs(A(j,j)) > SMLNUM: ! IF( TJJ.LT.ONE ) THEN IF( XJ.GT.TJJ*BIGNUM ) THEN ! ! Scale x by 1/b(j). ! REC = ONE / XJ CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF X( J ) = X( J ) / TJJS XJ = ABS( X( J ) ) ELSE IF( TJJ.GT.ZERO ) THEN ! ! 0 < abs(A(j,j)) <= SMLNUM: ! IF( XJ.GT.TJJ*BIGNUM ) THEN ! ! Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM ! to avoid overflow when dividing by A(j,j). ! REC = ( TJJ*BIGNUM ) / XJ IF( CNORM( J ).GT.ONE ) THEN ! ! Scale by 1/CNORM(j) to avoid overflow when ! multiplying x(j) times column j. ! REC = REC / CNORM( J ) END IF CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF X( J ) = X( J ) / TJJS XJ = ABS( X( J ) ) ELSE ! ! A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and ! scale = 0, and compute a solution to A*x = 0. ! DO 90 I = 1, N X( I ) = ZERO 90 CONTINUE X( J ) = ONE XJ = ONE SCALE = ZERO XMAX = ZERO END IF 100 CONTINUE ! ! Scale x if necessary to avoid overflow when adding a ! multiple of column j of A. ! IF( XJ.GT.ONE ) THEN REC = ONE / XJ IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN ! ! Scale x by 1/(2*abs(x(j))). ! REC = REC*HALF CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC END IF ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN ! ! Scale x by 1/2. ! CALL DSCAL( N, HALF, X, 1 ) SCALE = SCALE*HALF END IF ! IF( UPPER ) THEN IF( J.GT.1 ) THEN ! ! Compute the update ! x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j) ! CALL DAXPY( J-1, -X( J )*TSCAL, A( 1, J ), 1, X, & 1 ) I = IDAMAX( J-1, X, 1 ) XMAX = ABS( X( I ) ) END IF ELSE IF( J.LT.N ) THEN ! ! Compute the update ! x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j) ! CALL DAXPY( N-J, -X( J )*TSCAL, A( J+1, J ), 1, & X( J+1 ), 1 ) I = J + IDAMAX( N-J, X( J+1 ), 1 ) XMAX = ABS( X( I ) ) END IF END IF 110 CONTINUE ! ELSE ! ! Solve A' * x = b ! DO 160 J = JFIRST, JLAST, JINC ! ! Compute x(j) = b(j) - sum A(k,j)*x(k). ! k<>j ! XJ = ABS( X( J ) ) USCAL = TSCAL REC = ONE / MAX( XMAX, ONE ) IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN ! ! If x(j) could overflow, scale x by 1/(2*XMAX). ! REC = REC*HALF IF( NOUNIT ) THEN TJJS = A( J, J )*TSCAL ELSE TJJS = TSCAL END IF TJJ = ABS( TJJS ) IF( TJJ.GT.ONE ) THEN ! ! Divide by A(j,j) when scaling x if A(j,j) > 1. ! REC = MIN( ONE, REC*TJJ ) USCAL = USCAL / TJJS END IF IF( REC.LT.ONE ) THEN CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF ! SUMJ = ZERO IF( USCAL.EQ.ONE ) THEN ! ! If the scaling needed for A in the dot product is 1, ! call DDOT to perform the dot product. ! IF( UPPER ) THEN SUMJ = DDOT( J-1, A( 1, J ), 1, X, 1 ) ELSE IF( J.LT.N ) THEN SUMJ = DDOT( N-J, A( J+1, J ), 1, X( J+1 ), 1 ) END IF ELSE ! ! Otherwise, use in-line code for the dot product. ! IF( UPPER ) THEN DO 120 I = 1, J - 1 SUMJ = SUMJ + ( A( I, J )*USCAL )*X( I ) 120 CONTINUE ELSE IF( J.LT.N ) THEN DO 130 I = J + 1, N SUMJ = SUMJ + ( A( I, J )*USCAL )*X( I ) 130 CONTINUE END IF END IF ! IF( USCAL.EQ.TSCAL ) THEN ! ! Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j) ! was not used to scale the dotproduct. ! X( J ) = X( J ) - SUMJ XJ = ABS( X( J ) ) IF( NOUNIT ) THEN TJJS = A( J, J )*TSCAL ELSE TJJS = TSCAL IF( TSCAL.EQ.ONE ) & GO TO 150 END IF ! ! Compute x(j) = x(j) / A(j,j), scaling if necessary. ! TJJ = ABS( TJJS ) IF( TJJ.GT.SMLNUM ) THEN ! ! abs(A(j,j)) > SMLNUM: ! IF( TJJ.LT.ONE ) THEN IF( XJ.GT.TJJ*BIGNUM ) THEN ! ! Scale X by 1/abs(x(j)). ! REC = ONE / XJ CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF END IF X( J ) = X( J ) / TJJS ELSE IF( TJJ.GT.ZERO ) THEN ! ! 0 < abs(A(j,j)) <= SMLNUM: ! IF( XJ.GT.TJJ*BIGNUM ) THEN ! ! Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM. ! REC = ( TJJ*BIGNUM ) / XJ CALL DSCAL( N, REC, X, 1 ) SCALE = SCALE*REC XMAX = XMAX*REC END IF X( J ) = X( J ) / TJJS ELSE ! ! A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and ! scale = 0, and compute a solution to A'*x = 0. ! DO 140 I = 1, N X( I ) = ZERO 140 CONTINUE X( J ) = ONE SCALE = ZERO XMAX = ZERO END IF 150 CONTINUE ELSE ! ! Compute x(j) := x(j) / A(j,j) - sumj if the dot ! product has already been divided by 1/A(j,j). ! X( J ) = X( J ) / TJJS - SUMJ END IF XMAX = MAX( XMAX, ABS( X( J ) ) ) 160 CONTINUE END IF SCALE = SCALE / TSCAL END IF ! ! Scale the column norms by 1/TSCAL for return. ! IF( TSCAL.NE.ONE ) THEN CALL DSCAL( N, ONE / TSCAL, CNORM, 1 ) END IF ! RETURN ! ! End of DLATRS ! END SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER L, LDA, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix ! [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means ! of orthogonal transformations. Z is an (M+L)-by-(M+L) orthogonal ! matrix and, R and A1 are M-by-M upper triangular matrices. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! L (input) INTEGER ! The number of columns of the matrix A containing the ! meaningful part of the Householder vectors. N-M >= L >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the leading M-by-N upper trapezoidal part of the ! array A must contain the matrix to be factorized. ! On exit, the leading M-by-M upper triangular part of A ! contains the upper triangular matrix R, and elements N-L+1 to ! N of the first M rows of A, with the array TAU, represent the ! orthogonal matrix Z as a product of M elementary reflectors. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! TAU (output) DOUBLE PRECISION array, dimension (M) ! The scalar factors of the elementary reflectors. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (M) ! ! Further Details ! =============== ! ! Based on contributions by ! A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA ! ! The factorization is obtained by Householder's method. The kth ! transformation matrix, Z( k ), which is used to introduce zeros into ! the ( m - k + 1 )th row of A, is given in the form ! ! Z( k ) = ( I 0 ), ! ( 0 T( k ) ) ! ! where ! ! T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ), ! ( 0 ) ! ( z( k ) ) ! ! tau is a scalar and z( k ) is an l element vector. tau and z( k ) ! are chosen to annihilate the elements of the kth row of A2. ! ! The scalar tau is returned in the kth element of TAU and the vector ! u( k ) in the kth row of A2, such that the elements of z( k ) are ! in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in ! the upper triangular part of A1. ! ! Z is given by ! ! Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I ! .. ! .. External Subroutines .. EXTERNAL DLARFG, DLARZ ! .. ! .. Executable Statements .. ! ! Test the input arguments ! ! Quick return if possible ! IF( M.EQ.0 ) THEN RETURN ELSE IF( M.EQ.N ) THEN DO 10 I = 1, N TAU( I ) = ZERO 10 CONTINUE RETURN END IF ! DO 20 I = M, 1, -1 ! ! Generate elementary reflector H(i) to annihilate ! [ A(i,i) A(i,n-l+1:n) ] ! CALL DLARFG( L+1, A( I, I ), A( I, N-L+1 ), LDA, TAU( I ) ) ! ! Apply H(i) to A(1:i-1,i:n) from the right ! CALL DLARZ( 'Right', I-1, N-I+1, L, A( I, N-L+1 ), LDA, & TAU( I ), A( 1, I ), LDA, WORK ) ! 20 CONTINUE ! RETURN ! ! End of DLATRZ ! END SUBROUTINE DLATZM( SIDE, M, N, V, INCV, TAU, C1, C2, LDC, WORK ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER SIDE INTEGER INCV, LDC, M, N DOUBLE PRECISION TAU ! .. ! .. Array Arguments .. DOUBLE PRECISION C1( LDC, * ), C2( LDC, * ), V( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! This routine is deprecated and has been replaced by routine DORMRZ. ! ! DLATZM applies a Householder matrix generated by DTZRQF to a matrix. ! ! Let P = I - tau*u*u', u = ( 1 ), ! ( v ) ! where v is an (m-1) vector if SIDE = 'L', or a (n-1) vector if ! SIDE = 'R'. ! ! If SIDE equals 'L', let ! C = [ C1 ] 1 ! [ C2 ] m-1 ! n ! Then C is overwritten by P*C. ! ! If SIDE equals 'R', let ! C = [ C1, C2 ] m ! 1 n-1 ! Then C is overwritten by C*P. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': form P * C ! = 'R': form C * P ! ! M (input) INTEGER ! The number of rows of the matrix C. ! ! N (input) INTEGER ! The number of columns of the matrix C. ! ! V (input) DOUBLE PRECISION array, dimension ! (1 + (M-1)*abs(INCV)) if SIDE = 'L' ! (1 + (N-1)*abs(INCV)) if SIDE = 'R' ! The vector v in the representation of P. V is not used ! if TAU = 0. ! ! INCV (input) INTEGER ! The increment between elements of v. INCV <> 0 ! ! TAU (input) DOUBLE PRECISION ! The value tau in the representation of P. ! ! C1 (input/output) DOUBLE PRECISION array, dimension ! (LDC,N) if SIDE = 'L' ! (M,1) if SIDE = 'R' ! On entry, the n-vector C1 if SIDE = 'L', or the m-vector C1 ! if SIDE = 'R'. ! ! On exit, the first row of P*C if SIDE = 'L', or the first ! column of C*P if SIDE = 'R'. ! ! C2 (input/output) DOUBLE PRECISION array, dimension ! (LDC, N) if SIDE = 'L' ! (LDC, N-1) if SIDE = 'R' ! On entry, the (m - 1) x n matrix C2 if SIDE = 'L', or the ! m x (n - 1) matrix C2 if SIDE = 'R'. ! ! On exit, rows 2:m of P*C if SIDE = 'L', or columns 2:m of C*P ! if SIDE = 'R'. ! ! LDC (input) INTEGER ! The leading dimension of the arrays C1 and C2. LDC >= (1,M). ! ! WORK (workspace) DOUBLE PRECISION array, dimension ! (N) if SIDE = 'L' ! (M) if SIDE = 'R' ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DGEMV, DGER ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Intrinsic Functions .. INTRINSIC MIN ! .. ! .. Executable Statements .. ! IF( ( MIN( M, N ).EQ.0 ) .OR. ( TAU.EQ.ZERO ) ) & RETURN ! IF( LSAME( SIDE, 'L' ) ) THEN ! ! w := C1 + v' * C2 ! CALL DCOPY( N, C1, LDC, WORK, 1 ) CALL DGEMV( 'Transpose', M-1, N, ONE, C2, LDC, V, INCV, ONE, & WORK, 1 ) ! ! [ C1 ] := [ C1 ] - tau* [ 1 ] * w' ! [ C2 ] [ C2 ] [ v ] ! CALL DAXPY( N, -TAU, WORK, 1, C1, LDC ) CALL DGER( M-1, N, -TAU, V, INCV, WORK, 1, C2, LDC ) ! ELSE IF( LSAME( SIDE, 'R' ) ) THEN ! ! w := C1 + C2 * v ! CALL DCOPY( M, C1, 1, WORK, 1 ) CALL DGEMV( 'No transpose', M, N-1, ONE, C2, LDC, V, INCV, ONE, & WORK, 1 ) ! ! [ C1, C2 ] := [ C1, C2 ] - tau* w * [ 1 , v'] ! CALL DAXPY( M, -TAU, WORK, 1, C1, 1 ) CALL DGER( M, N-1, -TAU, WORK, 1, V, INCV, C2, LDC ) END IF ! RETURN ! ! End of DLATZM ! END SUBROUTINE DLAUU2( UPLO, N, A, LDA, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ) ! .. ! ! Purpose ! ======= ! ! DLAUU2 computes the product U * U' or L' * L, where the triangular ! factor U or L is stored in the upper or lower triangular part of ! the array A. ! ! If UPLO = 'U' or 'u' then the upper triangle of the result is stored, ! overwriting the factor U in A. ! If UPLO = 'L' or 'l' then the lower triangle of the result is stored, ! overwriting the factor L in A. ! ! This is the unblocked form of the algorithm, calling Level 2 BLAS. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the triangular factor stored in the array A ! is upper or lower triangular: ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! N (input) INTEGER ! The order of the triangular factor U or L. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the triangular factor U or L. ! On exit, if UPLO = 'U', the upper triangle of A is ! overwritten with the upper triangle of the product U * U'; ! if UPLO = 'L', the lower triangle of A is overwritten with ! the lower triangle of the product L' * L. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -k, the k-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER I DOUBLE PRECISION AII ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT EXTERNAL LSAME, DDOT ! .. ! .. External Subroutines .. EXTERNAL DGEMV, DSCAL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLAUU2', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( UPPER ) THEN ! ! Compute the product U * U'. ! DO 10 I = 1, N AII = A( I, I ) IF( I.LT.N ) THEN A( I, I ) = DDOT( N-I+1, A( I, I ), LDA, A( I, I ), LDA ) CALL DGEMV( 'No transpose', I-1, N-I, ONE, A( 1, I+1 ), & LDA, A( I, I+1 ), LDA, AII, A( 1, I ), 1 ) ELSE CALL DSCAL( I, AII, A( 1, I ), 1 ) END IF 10 CONTINUE ! ELSE ! ! Compute the product L' * L. ! DO 20 I = 1, N AII = A( I, I ) IF( I.LT.N ) THEN A( I, I ) = DDOT( N-I+1, A( I, I ), 1, A( I, I ), 1 ) CALL DGEMV( 'Transpose', N-I, I-1, ONE, A( I+1, 1 ), LDA, & A( I+1, I ), 1, AII, A( I, 1 ), LDA ) ELSE CALL DSCAL( I, AII, A( I, 1 ), LDA ) END IF 20 CONTINUE END IF ! RETURN ! ! End of DLAUU2 ! END SUBROUTINE DLAUUM( UPLO, N, A, LDA, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ) ! .. ! ! Purpose ! ======= ! ! DLAUUM computes the product U * U' or L' * L, where the triangular ! factor U or L is stored in the upper or lower triangular part of ! the array A. ! ! If UPLO = 'U' or 'u' then the upper triangle of the result is stored, ! overwriting the factor U in A. ! If UPLO = 'L' or 'l' then the lower triangle of the result is stored, ! overwriting the factor L in A. ! ! This is the blocked form of the algorithm, calling Level 3 BLAS. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the triangular factor stored in the array A ! is upper or lower triangular: ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! N (input) INTEGER ! The order of the triangular factor U or L. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the triangular factor U or L. ! On exit, if UPLO = 'U', the upper triangle of A is ! overwritten with the upper triangle of the product U * U'; ! if UPLO = 'L', the lower triangle of A is overwritten with ! the lower triangle of the product L' * L. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -k, the k-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER I, IB, NB ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DGEMM, DLAUU2, DSYRK, DTRMM, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLAUUM', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Determine the block size for this environment. ! NB = ILAENV( 1, 'DLAUUM', UPLO, N, -1, -1, -1 ) ! IF( NB.LE.1 .OR. NB.GE.N ) THEN ! ! Use unblocked code ! CALL DLAUU2( UPLO, N, A, LDA, INFO ) ELSE ! ! Use blocked code ! IF( UPPER ) THEN ! ! Compute the product U * U'. ! DO 10 I = 1, N, NB IB = MIN( NB, N-I+1 ) CALL DTRMM( 'Right', 'Upper', 'Transpose', 'Non-unit', & I-1, IB, ONE, A( I, I ), LDA, A( 1, I ), & LDA ) CALL DLAUU2( 'Upper', IB, A( I, I ), LDA, INFO ) IF( I+IB.LE.N ) THEN CALL DGEMM( 'No transpose', 'Transpose', I-1, IB, & N-I-IB+1, ONE, A( 1, I+IB ), LDA, & A( I, I+IB ), LDA, ONE, A( 1, I ), LDA ) CALL DSYRK( 'Upper', 'No transpose', IB, N-I-IB+1, & ONE, A( I, I+IB ), LDA, ONE, A( I, I ), & LDA ) END IF 10 CONTINUE ELSE ! ! Compute the product L' * L. ! DO 20 I = 1, N, NB IB = MIN( NB, N-I+1 ) CALL DTRMM( 'Left', 'Lower', 'Transpose', 'Non-unit', IB, & I-1, ONE, A( I, I ), LDA, A( I, 1 ), LDA ) CALL DLAUU2( 'Lower', IB, A( I, I ), LDA, INFO ) IF( I+IB.LE.N ) THEN CALL DGEMM( 'Transpose', 'No transpose', IB, I-1, & N-I-IB+1, ONE, A( I+IB, I ), LDA, & A( I+IB, 1 ), LDA, ONE, A( I, 1 ), LDA ) CALL DSYRK( 'Lower', 'Transpose', IB, N-I-IB+1, ONE, & A( I+IB, I ), LDA, ONE, A( I, I ), LDA ) END IF 20 CONTINUE END IF END IF ! RETURN ! ! End of DLAUUM ! END subroutine dmat_print ( m, n, a, title ) ! !******************************************************************************* ! !! DMAT_PRINT prints a double precision matrix. ! ! ! Modified: ! ! 12 September 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, the number of rows in A. ! ! Input, integer N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Input, character ( len = * ) TITLE, a title to be printed. ! implicit none ! integer m integer n ! real ( kind = 8 ) a(m,n) integer i integer j integer jhi integer jlo character ( len = * ) title ! call dmat_print_some ( m, n, a, 1, 1, m, n, title ) return end subroutine dmat_print_some ( m, n, a, ilo, jlo, ihi, jhi, title ) ! !******************************************************************************* ! !! DMAT_PRINT_SOME prints some of a double precision matrix. ! ! ! Modified: ! ! 12 September 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, N, the number of rows and columns. ! ! Input, real ( kind = 8 ) A(M,N), an M by N matrix to be printed. ! ! Input, integer ILO, JLO, the first row and column to print. ! ! Input, integer IHI, JHI, the last row and column to print. ! ! Input, character ( len = * ) TITLE, an optional title. ! implicit none ! integer, parameter :: incx = 5 integer m integer n ! real ( kind = 8 ) a(m,n) character ( len = 14 ) ctemp(incx) integer i integer i2hi integer i2lo integer ihi integer ilo integer inc integer j integer j2 integer j2hi integer j2lo integer jhi integer jlo logical d_is_int character ( len = * ) title ! if ( 0 < len_trim ( title ) ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) end if do j2lo = max ( jlo, 1 ), min ( jhi, n ), incx j2hi = j2lo + incx - 1 j2hi = min ( j2hi, n ) j2hi = min ( j2hi, jhi ) inc = j2hi + 1 - j2lo write ( *, '(a)' ) ' ' do j = j2lo, j2hi j2 = j + 1 - j2lo write ( ctemp(j2), '(i7,7x)') j end do write ( *, '('' Col '',5a14)' ) ctemp(1:inc) write ( *, '(a)' ) ' Row' write ( *, '(a)' ) ' ' i2lo = max ( ilo, 1 ) i2hi = min ( ihi, m ) do i = i2lo, i2hi do j2 = 1, inc j = j2lo - 1 + j2 if ( d_is_int ( a(i,j) ) ) then write ( ctemp(j2), '(f8.0,6x)' ) a(i,j) else write ( ctemp(j2), '(g14.6)' ) a(i,j) end if end do write ( *, '(i5,1x,5a14)' ) i, ( ctemp(j), j = 1, inc ) end do end do write ( *, '(a)' ) ' ' return end SUBROUTINE DOPGTR( UPLO, N, AP, TAU, Q, LDQ, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDQ, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ), Q( LDQ, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DOPGTR generates a real orthogonal matrix Q which is defined as the ! product of n-1 elementary reflectors H(i) of order n, as returned by ! DSPTRD using packed storage: ! ! if UPLO = 'U', Q = H(n-1) . . . H(2) H(1), ! ! if UPLO = 'L', Q = H(1) H(2) . . . H(n-1). ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangular packed storage used in previous ! call to DSPTRD; ! = 'L': Lower triangular packed storage used in previous ! call to DSPTRD. ! ! N (input) INTEGER ! The order of the matrix Q. N >= 0. ! ! AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The vectors which define the elementary reflectors, as ! returned by DSPTRD. ! ! TAU (input) DOUBLE PRECISION array, dimension (N-1) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DSPTRD. ! ! Q (output) DOUBLE PRECISION array, dimension (LDQ,N) ! The N-by-N orthogonal matrix Q. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. LDQ >= max(1,N). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (N-1) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER I, IINFO, IJ, J ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DORG2L, DORG2R, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DOPGTR', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( UPPER ) THEN ! ! Q was determined by a call to DSPTRD with UPLO = 'U' ! ! Unpack the vectors which define the elementary reflectors and ! set the last row and column of Q equal to those of the unit ! matrix ! IJ = 2 DO 20 J = 1, N - 1 DO 10 I = 1, J - 1 Q( I, J ) = AP( IJ ) IJ = IJ + 1 10 CONTINUE IJ = IJ + 2 Q( N, J ) = ZERO 20 CONTINUE DO 30 I = 1, N - 1 Q( I, N ) = ZERO 30 CONTINUE Q( N, N ) = ONE ! ! Generate Q(1:n-1,1:n-1) ! CALL DORG2L( N-1, N-1, N-1, Q, LDQ, TAU, WORK, IINFO ) ! ELSE ! ! Q was determined by a call to DSPTRD with UPLO = 'L'. ! ! Unpack the vectors which define the elementary reflectors and ! set the first row and column of Q equal to those of the unit ! matrix ! Q( 1, 1 ) = ONE DO 40 I = 2, N Q( I, 1 ) = ZERO 40 CONTINUE IJ = 3 DO 60 J = 2, N Q( 1, J ) = ZERO DO 50 I = J + 1, N Q( I, J ) = AP( IJ ) IJ = IJ + 1 50 CONTINUE IJ = IJ + 2 60 CONTINUE IF( N.GT.1 ) THEN ! ! Generate Q(2:n,2:n) ! CALL DORG2R( N-1, N-1, N-1, Q( 2, 2 ), LDQ, TAU, WORK, & IINFO ) END IF END IF RETURN ! ! End of DOPGTR ! END SUBROUTINE DOPMTR( SIDE, UPLO, TRANS, M, N, AP, TAU, C, LDC, WORK, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER SIDE, TRANS, UPLO INTEGER INFO, LDC, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ), C( LDC, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DOPMTR overwrites the general real M-by-N matrix C with ! ! SIDE = 'L' SIDE = 'R' ! TRANS = 'N': Q * C C * Q ! TRANS = 'T': Q**T * C C * Q**T ! ! where Q is a real orthogonal matrix of order nq, with nq = m if ! SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of ! nq-1 elementary reflectors, as returned by DSPTRD using packed ! storage: ! ! if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1); ! ! if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1). ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': apply Q or Q**T from the Left; ! = 'R': apply Q or Q**T from the Right. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangular packed storage used in previous ! call to DSPTRD; ! = 'L': Lower triangular packed storage used in previous ! call to DSPTRD. ! ! TRANS (input) CHARACTER*1 ! = 'N': No transpose, apply Q; ! = 'T': Transpose, apply Q**T. ! ! M (input) INTEGER ! The number of rows of the matrix C. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix C. N >= 0. ! ! AP (input) DOUBLE PRECISION array, dimension ! (M*(M+1)/2) if SIDE = 'L' ! (N*(N+1)/2) if SIDE = 'R' ! The vectors which define the elementary reflectors, as ! returned by DSPTRD. AP is modified by the routine but ! restored on exit. ! ! TAU (input) DOUBLE PRECISION array, dimension (M-1) if SIDE = 'L' ! or (N-1) if SIDE = 'R' ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DSPTRD. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the M-by-N matrix C. ! On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace) DOUBLE PRECISION array, dimension ! (N) if SIDE = 'L' ! (M) if SIDE = 'R' ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL FORWRD, LEFT, NOTRAN, UPPER INTEGER I, I1, I2, I3, IC, II, JC, MI, NI, NQ DOUBLE PRECISION AII ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DLARF, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LEFT = LSAME( SIDE, 'L' ) NOTRAN = LSAME( TRANS, 'N' ) UPPER = LSAME( UPLO, 'U' ) ! ! NQ is the order of Q ! IF( LEFT ) THEN NQ = M ELSE NQ = N END IF IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN INFO = -1 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -2 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN INFO = -3 ELSE IF( M.LT.0 ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -9 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DOPMTR', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 ) & RETURN ! IF( UPPER ) THEN ! ! Q was determined by a call to DSPTRD with UPLO = 'U' ! FORWRD = ( LEFT .AND. NOTRAN ) .OR. & ( .NOT.LEFT .AND. .NOT.NOTRAN ) ! IF( FORWRD ) THEN I1 = 1 I2 = NQ - 1 I3 = 1 II = 2 ELSE I1 = NQ - 1 I2 = 1 I3 = -1 II = NQ*( NQ+1 ) / 2 - 1 END IF ! IF( LEFT ) THEN NI = N ELSE MI = M END IF ! DO 10 I = I1, I2, I3 IF( LEFT ) THEN ! ! H(i) is applied to C(1:i,1:n) ! MI = I ELSE ! ! H(i) is applied to C(1:m,1:i) ! NI = I END IF ! ! Apply H(i) ! AII = AP( II ) AP( II ) = ONE CALL DLARF( SIDE, MI, NI, AP( II-I+1 ), 1, TAU( I ), C, LDC, & WORK ) AP( II ) = AII ! IF( FORWRD ) THEN II = II + I + 2 ELSE II = II - I - 1 END IF 10 CONTINUE ELSE ! ! Q was determined by a call to DSPTRD with UPLO = 'L'. ! FORWRD = ( LEFT .AND. .NOT.NOTRAN ) .OR. & ( .NOT.LEFT .AND. NOTRAN ) ! IF( FORWRD ) THEN I1 = 1 I2 = NQ - 1 I3 = 1 II = 2 ELSE I1 = NQ - 1 I2 = 1 I3 = -1 II = NQ*( NQ+1 ) / 2 - 1 END IF ! IF( LEFT ) THEN NI = N JC = 1 ELSE MI = M IC = 1 END IF ! DO 20 I = I1, I2, I3 AII = AP( II ) AP( II ) = ONE IF( LEFT ) THEN ! ! H(i) is applied to C(i+1:m,1:n) ! MI = M - I IC = I + 1 ELSE ! ! H(i) is applied to C(1:m,i+1:n) ! NI = N - I JC = I + 1 END IF ! ! Apply H(i) ! CALL DLARF( SIDE, MI, NI, AP( II ), 1, TAU( I ), & C( IC, JC ), LDC, WORK ) AP( II ) = AII ! IF( FORWRD ) THEN II = II + NQ - I + 1 ELSE II = II - NQ + I - 2 END IF 20 CONTINUE END IF RETURN ! ! End of DOPMTR ! END SUBROUTINE DORG2L( M, N, K, A, LDA, TAU, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. INTEGER INFO, K, LDA, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORG2L generates an m by n real matrix Q with orthonormal columns, ! which is defined as the last n columns of a product of k elementary ! reflectors of order m ! ! Q = H(k) . . . H(2) H(1) ! ! as returned by DGEQLF. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix Q. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix Q. M >= N >= 0. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines the ! matrix Q. N >= K >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the (n-k+i)-th column must contain the vector which ! defines the elementary reflector H(i), for i = 1,2,...,k, as ! returned by DGEQLF in the last k columns of its array ! argument A. ! On exit, the m by n matrix Q. ! ! LDA (input) INTEGER ! The first dimension of the array A. LDA >= max(1,M). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGEQLF. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument has an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, II, J, L ! .. ! .. External Subroutines .. EXTERNAL DLARF, DSCAL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 .OR. N.GT.M ) THEN INFO = -2 ELSE IF( K.LT.0 .OR. K.GT.N ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORG2L', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.LE.0 ) & RETURN ! ! Initialise columns 1:n-k to columns of the unit matrix ! DO 20 J = 1, N - K DO 10 L = 1, M A( L, J ) = ZERO 10 CONTINUE A( M-N+J, J ) = ONE 20 CONTINUE ! DO 40 I = 1, K II = N - K + I ! ! Apply H(i) to A(1:m-k+i,1:n-k+i) from the left ! A( M-N+II, II ) = ONE CALL DLARF( 'Left', M-N+II, II-1, A( 1, II ), 1, TAU( I ), A, & LDA, WORK ) CALL DSCAL( M-N+II-1, -TAU( I ), A( 1, II ), 1 ) A( M-N+II, II ) = ONE - TAU( I ) ! ! Set A(m-k+i+1:m,n-k+i) to zero ! DO 30 L = M - N + II + 1, M A( L, II ) = ZERO 30 CONTINUE 40 CONTINUE RETURN ! ! End of DORG2L ! END SUBROUTINE DORG2R( M, N, K, A, LDA, TAU, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. INTEGER INFO, K, LDA, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORG2R generates an m by n real matrix Q with orthonormal columns, ! which is defined as the first n columns of a product of k elementary ! reflectors of order m ! ! Q = H(1) H(2) . . . H(k) ! ! as returned by DGEQRF. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix Q. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix Q. M >= N >= 0. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines the ! matrix Q. N >= K >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the i-th column must contain the vector which ! defines the elementary reflector H(i), for i = 1,2,...,k, as ! returned by DGEQRF in the first k columns of its array ! argument A. ! On exit, the m-by-n matrix Q. ! ! LDA (input) INTEGER ! The first dimension of the array A. LDA >= max(1,M). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGEQRF. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument has an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J, L ! .. ! .. External Subroutines .. EXTERNAL DLARF, DSCAL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 .OR. N.GT.M ) THEN INFO = -2 ELSE IF( K.LT.0 .OR. K.GT.N ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORG2R', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.LE.0 ) & RETURN ! ! Initialise columns k+1:n to columns of the unit matrix ! DO 20 J = K + 1, N DO 10 L = 1, M A( L, J ) = ZERO 10 CONTINUE A( J, J ) = ONE 20 CONTINUE ! DO 40 I = K, 1, -1 ! ! Apply H(i) to A(i:m,i:n) from the left ! IF( I.LT.N ) THEN A( I, I ) = ONE CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAU( I ), & A( I, I+1 ), LDA, WORK ) END IF IF( I.LT.M ) & CALL DSCAL( M-I, -TAU( I ), A( I+1, I ), 1 ) A( I, I ) = ONE - TAU( I ) ! ! Set A(1:i-1,i) to zero ! DO 30 L = 1, I - 1 A( L, I ) = ZERO 30 CONTINUE 40 CONTINUE RETURN ! ! End of DORG2R ! END SUBROUTINE DORGBR( VECT, M, N, K, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER VECT INTEGER INFO, K, LDA, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORGBR generates one of the real orthogonal matrices Q or P**T ! determined by DGEBRD when reducing a real matrix A to bidiagonal ! form: A = Q * B * P**T. Q and P**T are defined as products of ! elementary reflectors H(i) or G(i) respectively. ! ! If VECT = 'Q', A is assumed to have been an M-by-K matrix, and Q ! is of order M: ! if m >= k, Q = H(1) H(2) . . . H(k) and DORGBR returns the first n ! columns of Q, where m >= n >= k; ! if m < k, Q = H(1) H(2) . . . H(m-1) and DORGBR returns Q as an ! M-by-M matrix. ! ! If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**T ! is of order N: ! if k < n, P**T = G(k) . . . G(2) G(1) and DORGBR returns the first m ! rows of P**T, where n >= m >= k; ! if k >= n, P**T = G(n-1) . . . G(2) G(1) and DORGBR returns P**T as ! an N-by-N matrix. ! ! Arguments ! ========= ! ! VECT (input) CHARACTER*1 ! Specifies whether the matrix Q or the matrix P**T is ! required, as defined in the transformation applied by DGEBRD: ! = 'Q': generate Q; ! = 'P': generate P**T. ! ! M (input) INTEGER ! The number of rows of the matrix Q or P**T to be returned. ! M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix Q or P**T to be returned. ! N >= 0. ! If VECT = 'Q', M >= N >= min(M,K); ! if VECT = 'P', N >= M >= min(N,K). ! ! K (input) INTEGER ! If VECT = 'Q', the number of columns in the original M-by-K ! matrix reduced by DGEBRD. ! If VECT = 'P', the number of rows in the original K-by-N ! matrix reduced by DGEBRD. ! K >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the vectors which define the elementary reflectors, ! as returned by DGEBRD. ! On exit, the M-by-N matrix Q or P**T. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! TAU (input) DOUBLE PRECISION array, dimension ! (min(M,K)) if VECT = 'Q' ! (min(N,K)) if VECT = 'P' ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i) or G(i), which determines Q or P**T, as ! returned by DGEBRD in its array argument TAUQ or TAUP. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,min(M,N)). ! For optimum performance LWORK >= min(M,N)*NB, where NB ! is the optimal blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, WANTQ INTEGER I, IINFO, J, LWKOPT, MN, NB ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DORGLQ, DORGQR, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 WANTQ = LSAME( VECT, 'Q' ) MN = MIN( M, N ) LQUERY = ( LWORK.EQ.-1 ) IF( .NOT.WANTQ .AND. .NOT.LSAME( VECT, 'P' ) ) THEN INFO = -1 ELSE IF( M.LT.0 ) THEN INFO = -2 ELSE IF( N.LT.0 .OR. ( WANTQ .AND. ( N.GT.M .OR. N.LT.MIN( M, & K ) ) ) .OR. ( .NOT.WANTQ .AND. ( M.GT.N .OR. M.LT. & MIN( N, K ) ) ) ) THEN INFO = -3 ELSE IF( K.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -6 ELSE IF( LWORK.LT.MAX( 1, MN ) .AND. .NOT.LQUERY ) THEN INFO = -9 END IF ! IF( INFO.EQ.0 ) THEN IF( WANTQ ) THEN NB = ILAENV( 1, 'DORGQR', ' ', M, N, K, -1 ) ELSE NB = ILAENV( 1, 'DORGLQ', ' ', M, N, K, -1 ) END IF LWKOPT = MAX( 1, MN )*NB WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORGBR', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! IF( WANTQ ) THEN ! ! Form Q, determined by a call to DGEBRD to reduce an m-by-k ! matrix ! IF( M.GE.K ) THEN ! ! If m >= k, assume m >= n >= k ! CALL DORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, IINFO ) ! ELSE ! ! If m < k, assume m = n ! ! Shift the vectors which define the elementary reflectors one ! column to the right, and set the first row and column of Q ! to those of the unit matrix ! DO 20 J = M, 2, -1 A( 1, J ) = ZERO DO 10 I = J + 1, M A( I, J ) = A( I, J-1 ) 10 CONTINUE 20 CONTINUE A( 1, 1 ) = ONE DO 30 I = 2, M A( I, 1 ) = ZERO 30 CONTINUE IF( M.GT.1 ) THEN ! ! Form Q(2:m,2:m) ! CALL DORGQR( M-1, M-1, M-1, A( 2, 2 ), LDA, TAU, WORK, & LWORK, IINFO ) END IF END IF ELSE ! ! Form P', determined by a call to DGEBRD to reduce a k-by-n ! matrix ! IF( K.LT.N ) THEN ! ! If k < n, assume k <= m <= n ! CALL DORGLQ( M, N, K, A, LDA, TAU, WORK, LWORK, IINFO ) ! ELSE ! ! If k >= n, assume m = n ! ! Shift the vectors which define the elementary reflectors one ! row downward, and set the first row and column of P' to ! those of the unit matrix ! A( 1, 1 ) = ONE DO 40 I = 2, N A( I, 1 ) = ZERO 40 CONTINUE DO 60 J = 2, N DO 50 I = J - 1, 2, -1 A( I, J ) = A( I-1, J ) 50 CONTINUE A( 1, J ) = ZERO 60 CONTINUE IF( N.GT.1 ) THEN ! ! Form P'(2:n,2:n) ! CALL DORGLQ( N-1, N-1, N-1, A( 2, 2 ), LDA, TAU, WORK, & LWORK, IINFO ) END IF END IF END IF WORK( 1 ) = LWKOPT RETURN ! ! End of DORGBR ! END SUBROUTINE DORGHR( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER IHI, ILO, INFO, LDA, LWORK, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORGHR generates a real orthogonal matrix Q which is defined as the ! product of IHI-ILO elementary reflectors of order N, as returned by ! DGEHRD: ! ! Q = H(ilo) H(ilo+1) . . . H(ihi-1). ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix Q. N >= 0. ! ! ILO (input) INTEGER ! IHI (input) INTEGER ! ILO and IHI must have the same values as in the previous call ! of DGEHRD. Q is equal to the unit matrix except in the ! submatrix Q(ilo+1:ihi,ilo+1:ihi). ! 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the vectors which define the elementary reflectors, ! as returned by DGEHRD. ! On exit, the N-by-N orthogonal matrix Q. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! TAU (input) DOUBLE PRECISION array, dimension (N-1) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGEHRD. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= IHI-ILO. ! For optimum performance LWORK >= (IHI-ILO)*NB, where NB is ! the optimal blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, IINFO, J, LWKOPT, NB, NH ! .. ! .. External Subroutines .. EXTERNAL DORGQR, XERBLA ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 NH = IHI - ILO LQUERY = ( LWORK.EQ.-1 ) IF( N.LT.0 ) THEN INFO = -1 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN INFO = -2 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LWORK.LT.MAX( 1, NH ) .AND. .NOT.LQUERY ) THEN INFO = -8 END IF ! IF( INFO.EQ.0 ) THEN NB = ILAENV( 1, 'DORGQR', ' ', NH, NH, NH, -1 ) LWKOPT = MAX( 1, NH )*NB WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORGHR', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! ! Shift the vectors which define the elementary reflectors one ! column to the right, and set the first ilo and the last n-ihi ! rows and columns to those of the unit matrix ! DO 40 J = IHI, ILO + 1, -1 DO 10 I = 1, J - 1 A( I, J ) = ZERO 10 CONTINUE DO 20 I = J + 1, IHI A( I, J ) = A( I, J-1 ) 20 CONTINUE DO 30 I = IHI + 1, N A( I, J ) = ZERO 30 CONTINUE 40 CONTINUE DO 60 J = 1, ILO DO 50 I = 1, N A( I, J ) = ZERO 50 CONTINUE A( J, J ) = ONE 60 CONTINUE DO 80 J = IHI + 1, N DO 70 I = 1, N A( I, J ) = ZERO 70 CONTINUE A( J, J ) = ONE 80 CONTINUE ! IF( NH.GT.0 ) THEN ! ! Generate Q(ilo+1:ihi,ilo+1:ihi) ! CALL DORGQR( NH, NH, NH, A( ILO+1, ILO+1 ), LDA, TAU( ILO ), & WORK, LWORK, IINFO ) END IF WORK( 1 ) = LWKOPT RETURN ! ! End of DORGHR ! END SUBROUTINE DORGL2( M, N, K, A, LDA, TAU, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, K, LDA, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORGL2 generates an m by n real matrix Q with orthonormal rows, ! which is defined as the first m rows of a product of k elementary ! reflectors of order n ! ! Q = H(k) . . . H(2) H(1) ! ! as returned by DGELQF. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix Q. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix Q. N >= M. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines the ! matrix Q. M >= K >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the i-th row must contain the vector which defines ! the elementary reflector H(i), for i = 1,2,...,k, as returned ! by DGELQF in the first k rows of its array argument A. ! On exit, the m-by-n matrix Q. ! ! LDA (input) INTEGER ! The first dimension of the array A. LDA >= max(1,M). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGELQF. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (M) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument has an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, J, L ! .. ! .. External Subroutines .. EXTERNAL DLARF, DSCAL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.M ) THEN INFO = -2 ELSE IF( K.LT.0 .OR. K.GT.M ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORGL2', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( M.LE.0 ) & RETURN ! IF( K.LT.M ) THEN ! ! Initialise rows k+1:m to rows of the unit matrix ! DO 20 J = 1, N DO 10 L = K + 1, M A( L, J ) = ZERO 10 CONTINUE IF( J.GT.K .AND. J.LE.M ) & A( J, J ) = ONE 20 CONTINUE END IF ! DO 40 I = K, 1, -1 ! ! Apply H(i) to A(i:m,i:n) from the right ! IF( I.LT.N ) THEN IF( I.LT.M ) THEN A( I, I ) = ONE CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA, & TAU( I ), A( I+1, I ), LDA, WORK ) END IF CALL DSCAL( N-I, -TAU( I ), A( I, I+1 ), LDA ) END IF A( I, I ) = ONE - TAU( I ) ! ! Set A(i,1:i-1) to zero ! DO 30 L = 1, I - 1 A( I, L ) = ZERO 30 CONTINUE 40 CONTINUE RETURN ! ! End of DORGL2 ! END SUBROUTINE DORGLQ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, K, LDA, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORGLQ generates an M-by-N real matrix Q with orthonormal rows, ! which is defined as the first M rows of a product of K elementary ! reflectors of order N ! ! Q = H(k) . . . H(2) H(1) ! ! as returned by DGELQF. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix Q. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix Q. N >= M. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines the ! matrix Q. M >= K >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the i-th row must contain the vector which defines ! the elementary reflector H(i), for i = 1,2,...,k, as returned ! by DGELQF in the first k rows of its array argument A. ! On exit, the M-by-N matrix Q. ! ! LDA (input) INTEGER ! The first dimension of the array A. LDA >= max(1,M). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGELQF. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,M). ! For optimum performance LWORK >= M*NB, where NB is ! the optimal blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument has an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, IB, IINFO, IWS, J, KI, KK, L, LDWORK, & LWKOPT, NB, NBMIN, NX ! .. ! .. External Subroutines .. EXTERNAL DLARFB, DLARFT, DORGL2, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 NB = ILAENV( 1, 'DORGLQ', ' ', M, N, K, -1 ) LWKOPT = MAX( 1, M )*NB WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.M ) THEN INFO = -2 ELSE IF( K.LT.0 .OR. K.GT.M ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORGLQ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( M.LE.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! NBMIN = 2 NX = 0 IWS = M IF( NB.GT.1 .AND. NB.LT.K ) THEN ! ! Determine when to cross over from blocked to unblocked code. ! NX = MAX( 0, ILAENV( 3, 'DORGLQ', ' ', M, N, K, -1 ) ) IF( NX.LT.K ) THEN ! ! Determine if workspace is large enough for blocked code. ! LDWORK = M IWS = LDWORK*NB IF( LWORK.LT.IWS ) THEN ! ! Not enough workspace to use optimal NB: reduce NB and ! determine the minimum value of NB. ! NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'DORGLQ', ' ', M, N, K, -1 ) ) END IF END IF END IF ! IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN ! ! Use blocked code after the last block. ! The first kk rows are handled by the block method. ! KI = ( ( K-NX-1 ) / NB )*NB KK = MIN( K, KI+NB ) ! ! Set A(kk+1:m,1:kk) to zero. ! DO 20 J = 1, KK DO 10 I = KK + 1, M A( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ELSE KK = 0 END IF ! ! Use unblocked code for the last or only block. ! IF( KK.LT.M ) & CALL DORGL2( M-KK, N-KK, K-KK, A( KK+1, KK+1 ), LDA, & TAU( KK+1 ), WORK, IINFO ) ! IF( KK.GT.0 ) THEN ! ! Use blocked code ! DO 50 I = KI + 1, 1, -NB IB = MIN( NB, K-I+1 ) IF( I+IB.LE.M ) THEN ! ! Form the triangular factor of the block reflector ! H = H(i) H(i+1) . . . H(i+ib-1) ! CALL DLARFT( 'Forward', 'Rowwise', N-I+1, IB, A( I, I ), & LDA, TAU( I ), WORK, LDWORK ) ! ! Apply H' to A(i+ib:m,i:n) from the right ! CALL DLARFB( 'Right', 'Transpose', 'Forward', 'Rowwise', & M-I-IB+1, N-I+1, IB, A( I, I ), LDA, WORK, & LDWORK, A( I+IB, I ), LDA, WORK( IB+1 ), & LDWORK ) END IF ! ! Apply H' to columns i:n of current block ! CALL DORGL2( IB, N-I+1, IB, A( I, I ), LDA, TAU( I ), WORK, & IINFO ) ! ! Set columns 1:i-1 of current block to zero ! DO 40 J = 1, I - 1 DO 30 L = I, I + IB - 1 A( L, J ) = ZERO 30 CONTINUE 40 CONTINUE 50 CONTINUE END IF ! WORK( 1 ) = IWS RETURN ! ! End of DORGLQ ! END SUBROUTINE DORGQL( M, N, K, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, K, LDA, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORGQL generates an M-by-N real matrix Q with orthonormal columns, ! which is defined as the last N columns of a product of K elementary ! reflectors of order M ! ! Q = H(k) . . . H(2) H(1) ! ! as returned by DGEQLF. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix Q. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix Q. M >= N >= 0. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines the ! matrix Q. N >= K >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the (n-k+i)-th column must contain the vector which ! defines the elementary reflector H(i), for i = 1,2,...,k, as ! returned by DGEQLF in the last k columns of its array ! argument A. ! On exit, the M-by-N matrix Q. ! ! LDA (input) INTEGER ! The first dimension of the array A. LDA >= max(1,M). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGEQLF. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,N). ! For optimum performance LWORK >= N*NB, where NB is the ! optimal blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument has an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, IB, IINFO, IWS, J, KK, L, LDWORK, LWKOPT, & NB, NBMIN, NX ! .. ! .. External Subroutines .. EXTERNAL DLARFB, DLARFT, DORG2L, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 NB = ILAENV( 1, 'DORGQL', ' ', M, N, K, -1 ) LWKOPT = MAX( 1, N )*NB WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 .OR. N.GT.M ) THEN INFO = -2 ELSE IF( K.LT.0 .OR. K.GT.N ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORGQL', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.LE.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! NBMIN = 2 NX = 0 IWS = N IF( NB.GT.1 .AND. NB.LT.K ) THEN ! ! Determine when to cross over from blocked to unblocked code. ! NX = MAX( 0, ILAENV( 3, 'DORGQL', ' ', M, N, K, -1 ) ) IF( NX.LT.K ) THEN ! ! Determine if workspace is large enough for blocked code. ! LDWORK = N IWS = LDWORK*NB IF( LWORK.LT.IWS ) THEN ! ! Not enough workspace to use optimal NB: reduce NB and ! determine the minimum value of NB. ! NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'DORGQL', ' ', M, N, K, -1 ) ) END IF END IF END IF ! IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN ! ! Use blocked code after the first block. ! The last kk columns are handled by the block method. ! KK = MIN( K, ( ( K-NX+NB-1 ) / NB )*NB ) ! ! Set A(m-kk+1:m,1:n-kk) to zero. ! DO 20 J = 1, N - KK DO 10 I = M - KK + 1, M A( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ELSE KK = 0 END IF ! ! Use unblocked code for the first or only block. ! CALL DORG2L( M-KK, N-KK, K-KK, A, LDA, TAU, WORK, IINFO ) ! IF( KK.GT.0 ) THEN ! ! Use blocked code ! DO 50 I = K - KK + 1, K, NB IB = MIN( NB, K-I+1 ) IF( N-K+I.GT.1 ) THEN ! ! Form the triangular factor of the block reflector ! H = H(i+ib-1) . . . H(i+1) H(i) ! CALL DLARFT( 'Backward', 'Columnwise', M-K+I+IB-1, IB, & A( 1, N-K+I ), LDA, TAU( I ), WORK, LDWORK ) ! ! Apply H to A(1:m-k+i+ib-1,1:n-k+i-1) from the left ! CALL DLARFB( 'Left', 'No transpose', 'Backward', & 'Columnwise', M-K+I+IB-1, N-K+I-1, IB, & A( 1, N-K+I ), LDA, WORK, LDWORK, A, LDA, & WORK( IB+1 ), LDWORK ) END IF ! ! Apply H to rows 1:m-k+i+ib-1 of current block ! CALL DORG2L( M-K+I+IB-1, IB, IB, A( 1, N-K+I ), LDA, & TAU( I ), WORK, IINFO ) ! ! Set rows m-k+i+ib:m of current block to zero ! DO 40 J = N - K + I, N - K + I + IB - 1 DO 30 L = M - K + I + IB, M A( L, J ) = ZERO 30 CONTINUE 40 CONTINUE 50 CONTINUE END IF ! WORK( 1 ) = IWS RETURN ! ! End of DORGQL ! END SUBROUTINE DORGQR( M, N, K, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, K, LDA, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORGQR generates an M-by-N real matrix Q with orthonormal columns, ! which is defined as the first N columns of a product of K elementary ! reflectors of order M ! ! Q = H(1) H(2) . . . H(k) ! ! as returned by DGEQRF. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix Q. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix Q. M >= N >= 0. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines the ! matrix Q. N >= K >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the i-th column must contain the vector which ! defines the elementary reflector H(i), for i = 1,2,...,k, as ! returned by DGEQRF in the first k columns of its array ! argument A. ! On exit, the M-by-N matrix Q. ! ! LDA (input) INTEGER ! The first dimension of the array A. LDA >= max(1,M). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGEQRF. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,N). ! For optimum performance LWORK >= N*NB, where NB is the ! optimal blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument has an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, IB, IINFO, IWS, J, KI, KK, L, LDWORK, & LWKOPT, NB, NBMIN, NX ! .. ! .. External Subroutines .. EXTERNAL DLARFB, DLARFT, DORG2R, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 NB = ILAENV( 1, 'DORGQR', ' ', M, N, K, -1 ) LWKOPT = MAX( 1, N )*NB WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 .OR. N.GT.M ) THEN INFO = -2 ELSE IF( K.LT.0 .OR. K.GT.N ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORGQR', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.LE.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! NBMIN = 2 NX = 0 IWS = N IF( NB.GT.1 .AND. NB.LT.K ) THEN ! ! Determine when to cross over from blocked to unblocked code. ! NX = MAX( 0, ILAENV( 3, 'DORGQR', ' ', M, N, K, -1 ) ) IF( NX.LT.K ) THEN ! ! Determine if workspace is large enough for blocked code. ! LDWORK = N IWS = LDWORK*NB IF( LWORK.LT.IWS ) THEN ! ! Not enough workspace to use optimal NB: reduce NB and ! determine the minimum value of NB. ! NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'DORGQR', ' ', M, N, K, -1 ) ) END IF END IF END IF ! IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN ! ! Use blocked code after the last block. ! The first kk columns are handled by the block method. ! KI = ( ( K-NX-1 ) / NB )*NB KK = MIN( K, KI+NB ) ! ! Set A(1:kk,kk+1:n) to zero. ! DO 20 J = KK + 1, N DO 10 I = 1, KK A( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ELSE KK = 0 END IF ! ! Use unblocked code for the last or only block. ! IF( KK.LT.N ) & CALL DORG2R( M-KK, N-KK, K-KK, A( KK+1, KK+1 ), LDA, & TAU( KK+1 ), WORK, IINFO ) ! IF( KK.GT.0 ) THEN ! ! Use blocked code ! DO 50 I = KI + 1, 1, -NB IB = MIN( NB, K-I+1 ) IF( I+IB.LE.N ) THEN ! ! Form the triangular factor of the block reflector ! H = H(i) H(i+1) . . . H(i+ib-1) ! CALL DLARFT( 'Forward', 'Columnwise', M-I+1, IB, & A( I, I ), LDA, TAU( I ), WORK, LDWORK ) ! ! Apply H to A(i:m,i+ib:n) from the left ! CALL DLARFB( 'Left', 'No transpose', 'Forward', & 'Columnwise', M-I+1, N-I-IB+1, IB, & A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ), & LDA, WORK( IB+1 ), LDWORK ) END IF ! ! Apply H to rows i:m of current block ! CALL DORG2R( M-I+1, IB, IB, A( I, I ), LDA, TAU( I ), WORK, & IINFO ) ! ! Set rows 1:i-1 of current block to zero ! DO 40 J = I, I + IB - 1 DO 30 L = 1, I - 1 A( L, J ) = ZERO 30 CONTINUE 40 CONTINUE 50 CONTINUE END IF ! WORK( 1 ) = IWS RETURN ! ! End of DORGQR ! END SUBROUTINE DORGR2( M, N, K, A, LDA, TAU, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. INTEGER INFO, K, LDA, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORGR2 generates an m by n real matrix Q with orthonormal rows, ! which is defined as the last m rows of a product of k elementary ! reflectors of order n ! ! Q = H(1) H(2) . . . H(k) ! ! as returned by DGERQF. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix Q. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix Q. N >= M. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines the ! matrix Q. M >= K >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the (m-k+i)-th row must contain the vector which ! defines the elementary reflector H(i), for i = 1,2,...,k, as ! returned by DGERQF in the last k rows of its array argument ! A. ! On exit, the m by n matrix Q. ! ! LDA (input) INTEGER ! The first dimension of the array A. LDA >= max(1,M). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGERQF. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (M) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument has an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, II, J, L ! .. ! .. External Subroutines .. EXTERNAL DLARF, DSCAL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.M ) THEN INFO = -2 ELSE IF( K.LT.0 .OR. K.GT.M ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORGR2', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( M.LE.0 ) & RETURN ! IF( K.LT.M ) THEN ! ! Initialise rows 1:m-k to rows of the unit matrix ! DO 20 J = 1, N DO 10 L = 1, M - K A( L, J ) = ZERO 10 CONTINUE IF( J.GT.N-M .AND. J.LE.N-K ) & A( M-N+J, J ) = ONE 20 CONTINUE END IF ! DO 40 I = 1, K II = M - K + I ! ! Apply H(i) to A(1:m-k+i,1:n-k+i) from the right ! A( II, N-M+II ) = ONE CALL DLARF( 'Right', II-1, N-M+II, A( II, 1 ), LDA, TAU( I ), & A, LDA, WORK ) CALL DSCAL( N-M+II-1, -TAU( I ), A( II, 1 ), LDA ) A( II, N-M+II ) = ONE - TAU( I ) ! ! Set A(m-k+i,n-k+i+1:n) to zero ! DO 30 L = N - M + II + 1, N A( II, L ) = ZERO 30 CONTINUE 40 CONTINUE RETURN ! ! End of DORGR2 ! END SUBROUTINE DORGRQ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, K, LDA, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORGRQ generates an M-by-N real matrix Q with orthonormal rows, ! which is defined as the last M rows of a product of K elementary ! reflectors of order N ! ! Q = H(1) H(2) . . . H(k) ! ! as returned by DGERQF. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix Q. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix Q. N >= M. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines the ! matrix Q. M >= K >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the (m-k+i)-th row must contain the vector which ! defines the elementary reflector H(i), for i = 1,2,...,k, as ! returned by DGERQF in the last k rows of its array argument ! A. ! On exit, the M-by-N matrix Q. ! ! LDA (input) INTEGER ! The first dimension of the array A. LDA >= max(1,M). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGERQF. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,M). ! For optimum performance LWORK >= M*NB, where NB is the ! optimal blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument has an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, IB, II, IINFO, IWS, J, KK, L, LDWORK, & LWKOPT, NB, NBMIN, NX ! .. ! .. External Subroutines .. EXTERNAL DLARFB, DLARFT, DORGR2, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 NB = ILAENV( 1, 'DORGRQ', ' ', M, N, K, -1 ) LWKOPT = MAX( 1, M )*NB WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.M ) THEN INFO = -2 ELSE IF( K.LT.0 .OR. K.GT.M ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORGRQ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( M.LE.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! NBMIN = 2 NX = 0 IWS = M IF( NB.GT.1 .AND. NB.LT.K ) THEN ! ! Determine when to cross over from blocked to unblocked code. ! NX = MAX( 0, ILAENV( 3, 'DORGRQ', ' ', M, N, K, -1 ) ) IF( NX.LT.K ) THEN ! ! Determine if workspace is large enough for blocked code. ! LDWORK = M IWS = LDWORK*NB IF( LWORK.LT.IWS ) THEN ! ! Not enough workspace to use optimal NB: reduce NB and ! determine the minimum value of NB. ! NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'DORGRQ', ' ', M, N, K, -1 ) ) END IF END IF END IF ! IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN ! ! Use blocked code after the first block. ! The last kk rows are handled by the block method. ! KK = MIN( K, ( ( K-NX+NB-1 ) / NB )*NB ) ! ! Set A(1:m-kk,n-kk+1:n) to zero. ! DO 20 J = N - KK + 1, N DO 10 I = 1, M - KK A( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ELSE KK = 0 END IF ! ! Use unblocked code for the first or only block. ! CALL DORGR2( M-KK, N-KK, K-KK, A, LDA, TAU, WORK, IINFO ) ! IF( KK.GT.0 ) THEN ! ! Use blocked code ! DO 50 I = K - KK + 1, K, NB IB = MIN( NB, K-I+1 ) II = M - K + I IF( II.GT.1 ) THEN ! ! Form the triangular factor of the block reflector ! H = H(i+ib-1) . . . H(i+1) H(i) ! CALL DLARFT( 'Backward', 'Rowwise', N-K+I+IB-1, IB, & A( II, 1 ), LDA, TAU( I ), WORK, LDWORK ) ! ! Apply H' to A(1:m-k+i-1,1:n-k+i+ib-1) from the right ! CALL DLARFB( 'Right', 'Transpose', 'Backward', 'Rowwise', & II-1, N-K+I+IB-1, IB, A( II, 1 ), LDA, WORK, & LDWORK, A, LDA, WORK( IB+1 ), LDWORK ) END IF ! ! Apply H' to columns 1:n-k+i+ib-1 of current block ! CALL DORGR2( IB, N-K+I+IB-1, IB, A( II, 1 ), LDA, TAU( I ), & WORK, IINFO ) ! ! Set columns n-k+i+ib:n of current block to zero ! DO 40 L = N - K + I + IB, N DO 30 J = II, II + IB - 1 A( J, L ) = ZERO 30 CONTINUE 40 CONTINUE 50 CONTINUE END IF ! WORK( 1 ) = IWS RETURN ! ! End of DORGRQ ! END SUBROUTINE DORGTR( UPLO, N, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, LWORK, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORGTR generates a real orthogonal matrix Q which is defined as the ! product of n-1 elementary reflectors of order N, as returned by ! DSYTRD: ! ! if UPLO = 'U', Q = H(n-1) . . . H(2) H(1), ! ! if UPLO = 'L', Q = H(1) H(2) . . . H(n-1). ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A contains elementary reflectors ! from DSYTRD; ! = 'L': Lower triangle of A contains elementary reflectors ! from DSYTRD. ! ! N (input) INTEGER ! The order of the matrix Q. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the vectors which define the elementary reflectors, ! as returned by DSYTRD. ! On exit, the N-by-N orthogonal matrix Q. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! TAU (input) DOUBLE PRECISION array, dimension (N-1) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DSYTRD. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,N-1). ! For optimum performance LWORK >= (N-1)*NB, where NB is ! the optimal blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, UPPER INTEGER I, IINFO, J, LWKOPT, NB ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DORGQL, DORGQR, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LQUERY = ( LWORK.EQ.-1 ) UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( LWORK.LT.MAX( 1, N-1 ) .AND. .NOT.LQUERY ) THEN INFO = -7 END IF ! IF( INFO.EQ.0 ) THEN IF( UPPER ) THEN NB = ILAENV( 1, 'DORGQL', ' ', N-1, N-1, N-1, -1 ) ELSE NB = ILAENV( 1, 'DORGQR', ' ', N-1, N-1, N-1, -1 ) END IF LWKOPT = MAX( 1, N-1 )*NB WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORGTR', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! IF( UPPER ) THEN ! ! Q was determined by a call to DSYTRD with UPLO = 'U' ! ! Shift the vectors which define the elementary reflectors one ! column to the left, and set the last row and column of Q to ! those of the unit matrix ! DO 20 J = 1, N - 1 DO 10 I = 1, J - 1 A( I, J ) = A( I, J+1 ) 10 CONTINUE A( N, J ) = ZERO 20 CONTINUE DO 30 I = 1, N - 1 A( I, N ) = ZERO 30 CONTINUE A( N, N ) = ONE ! ! Generate Q(1:n-1,1:n-1) ! CALL DORGQL( N-1, N-1, N-1, A, LDA, TAU, WORK, LWORK, IINFO ) ! ELSE ! ! Q was determined by a call to DSYTRD with UPLO = 'L'. ! ! Shift the vectors which define the elementary reflectors one ! column to the right, and set the first row and column of Q to ! those of the unit matrix ! DO 50 J = N, 2, -1 A( 1, J ) = ZERO DO 40 I = J + 1, N A( I, J ) = A( I, J-1 ) 40 CONTINUE 50 CONTINUE A( 1, 1 ) = ONE DO 60 I = 2, N A( I, 1 ) = ZERO 60 CONTINUE IF( N.GT.1 ) THEN ! ! Generate Q(2:n,2:n) ! CALL DORGQR( N-1, N-1, N-1, A( 2, 2 ), LDA, TAU, WORK, & LWORK, IINFO ) END IF END IF WORK( 1 ) = LWKOPT RETURN ! ! End of DORGTR ! END SUBROUTINE DORM2L( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, & WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER SIDE, TRANS INTEGER INFO, K, LDA, LDC, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORM2L overwrites the general real m by n matrix C with ! ! Q * C if SIDE = 'L' and TRANS = 'N', or ! ! Q'* C if SIDE = 'L' and TRANS = 'T', or ! ! C * Q if SIDE = 'R' and TRANS = 'N', or ! ! C * Q' if SIDE = 'R' and TRANS = 'T', ! ! where Q is a real orthogonal matrix defined as the product of k ! elementary reflectors ! ! Q = H(k) . . . H(2) H(1) ! ! as returned by DGEQLF. Q is of order m if SIDE = 'L' and of order n ! if SIDE = 'R'. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': apply Q or Q' from the Left ! = 'R': apply Q or Q' from the Right ! ! TRANS (input) CHARACTER*1 ! = 'N': apply Q (No transpose) ! = 'T': apply Q' (Transpose) ! ! M (input) INTEGER ! The number of rows of the matrix C. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix C. N >= 0. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines ! the matrix Q. ! If SIDE = 'L', M >= K >= 0; ! if SIDE = 'R', N >= K >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,K) ! The i-th column must contain the vector which defines the ! elementary reflector H(i), for i = 1,2,...,k, as returned by ! DGEQLF in the last k columns of its array argument A. ! A is modified by the routine but restored on exit. ! ! LDA (input) INTEGER ! The leading dimension of the array A. ! If SIDE = 'L', LDA >= max(1,M); ! if SIDE = 'R', LDA >= max(1,N). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGEQLF. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the m by n matrix C. ! On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace) DOUBLE PRECISION array, dimension ! (N) if SIDE = 'L', ! (M) if SIDE = 'R' ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LEFT, NOTRAN INTEGER I, I1, I2, I3, MI, NI, NQ DOUBLE PRECISION AII ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DLARF, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LEFT = LSAME( SIDE, 'L' ) NOTRAN = LSAME( TRANS, 'N' ) ! ! NQ is the order of Q ! IF( LEFT ) THEN NQ = M ELSE NQ = N END IF IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN INFO = -2 ELSE IF( M.LT.0 ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN INFO = -7 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -10 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORM2L', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) & RETURN ! IF( ( LEFT .AND. NOTRAN ) .OR. ( .NOT.LEFT .AND. .NOT.NOTRAN ) ) & THEN I1 = 1 I2 = K I3 = 1 ELSE I1 = K I2 = 1 I3 = -1 END IF ! IF( LEFT ) THEN NI = N ELSE MI = M END IF ! DO 10 I = I1, I2, I3 IF( LEFT ) THEN ! ! H(i) is applied to C(1:m-k+i,1:n) ! MI = M - K + I ELSE ! ! H(i) is applied to C(1:m,1:n-k+i) ! NI = N - K + I END IF ! ! Apply H(i) ! AII = A( NQ-K+I, I ) A( NQ-K+I, I ) = ONE CALL DLARF( SIDE, MI, NI, A( 1, I ), 1, TAU( I ), C, LDC, & WORK ) A( NQ-K+I, I ) = AII 10 CONTINUE RETURN ! ! End of DORM2L ! END SUBROUTINE DORM2R( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, & WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER SIDE, TRANS INTEGER INFO, K, LDA, LDC, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORM2R overwrites the general real m by n matrix C with ! ! Q * C if SIDE = 'L' and TRANS = 'N', or ! ! Q'* C if SIDE = 'L' and TRANS = 'T', or ! ! C * Q if SIDE = 'R' and TRANS = 'N', or ! ! C * Q' if SIDE = 'R' and TRANS = 'T', ! ! where Q is a real orthogonal matrix defined as the product of k ! elementary reflectors ! ! Q = H(1) H(2) . . . H(k) ! ! as returned by DGEQRF. Q is of order m if SIDE = 'L' and of order n ! if SIDE = 'R'. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': apply Q or Q' from the Left ! = 'R': apply Q or Q' from the Right ! ! TRANS (input) CHARACTER*1 ! = 'N': apply Q (No transpose) ! = 'T': apply Q' (Transpose) ! ! M (input) INTEGER ! The number of rows of the matrix C. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix C. N >= 0. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines ! the matrix Q. ! If SIDE = 'L', M >= K >= 0; ! if SIDE = 'R', N >= K >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,K) ! The i-th column must contain the vector which defines the ! elementary reflector H(i), for i = 1,2,...,k, as returned by ! DGEQRF in the first k columns of its array argument A. ! A is modified by the routine but restored on exit. ! ! LDA (input) INTEGER ! The leading dimension of the array A. ! If SIDE = 'L', LDA >= max(1,M); ! if SIDE = 'R', LDA >= max(1,N). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGEQRF. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the m by n matrix C. ! On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace) DOUBLE PRECISION array, dimension ! (N) if SIDE = 'L', ! (M) if SIDE = 'R' ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LEFT, NOTRAN INTEGER I, I1, I2, I3, IC, JC, MI, NI, NQ DOUBLE PRECISION AII ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DLARF, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LEFT = LSAME( SIDE, 'L' ) NOTRAN = LSAME( TRANS, 'N' ) ! ! NQ is the order of Q ! IF( LEFT ) THEN NQ = M ELSE NQ = N END IF IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN INFO = -2 ELSE IF( M.LT.0 ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN INFO = -7 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -10 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORM2R', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) & RETURN ! IF( ( LEFT .AND. .NOT.NOTRAN ) .OR. ( .NOT.LEFT .AND. NOTRAN ) ) & THEN I1 = 1 I2 = K I3 = 1 ELSE I1 = K I2 = 1 I3 = -1 END IF ! IF( LEFT ) THEN NI = N JC = 1 ELSE MI = M IC = 1 END IF ! DO 10 I = I1, I2, I3 IF( LEFT ) THEN ! ! H(i) is applied to C(i:m,1:n) ! MI = M - I + 1 IC = I ELSE ! ! H(i) is applied to C(1:m,i:n) ! NI = N - I + 1 JC = I END IF ! ! Apply H(i) ! AII = A( I, I ) A( I, I ) = ONE CALL DLARF( SIDE, MI, NI, A( I, I ), 1, TAU( I ), C( IC, JC ), & LDC, WORK ) A( I, I ) = AII 10 CONTINUE RETURN ! ! End of DORM2R ! END SUBROUTINE DORMBR( VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C, & LDC, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER SIDE, TRANS, VECT INTEGER INFO, K, LDA, LDC, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! If VECT = 'Q', DORMBR overwrites the general real M-by-N matrix C ! with ! SIDE = 'L' SIDE = 'R' ! TRANS = 'N': Q * C C * Q ! TRANS = 'T': Q**T * C C * Q**T ! ! If VECT = 'P', DORMBR overwrites the general real M-by-N matrix C ! with ! SIDE = 'L' SIDE = 'R' ! TRANS = 'N': P * C C * P ! TRANS = 'T': P**T * C C * P**T ! ! Here Q and P**T are the orthogonal matrices determined by DGEBRD when ! reducing a real matrix A to bidiagonal form: A = Q * B * P**T. Q and ! P**T are defined as products of elementary reflectors H(i) and G(i) ! respectively. ! ! Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the ! order of the orthogonal matrix Q or P**T that is applied. ! ! If VECT = 'Q', A is assumed to have been an NQ-by-K matrix: ! if nq >= k, Q = H(1) H(2) . . . H(k); ! if nq < k, Q = H(1) H(2) . . . H(nq-1). ! ! If VECT = 'P', A is assumed to have been a K-by-NQ matrix: ! if k < nq, P = G(1) G(2) . . . G(k); ! if k >= nq, P = G(1) G(2) . . . G(nq-1). ! ! Arguments ! ========= ! ! VECT (input) CHARACTER*1 ! = 'Q': apply Q or Q**T; ! = 'P': apply P or P**T. ! ! SIDE (input) CHARACTER*1 ! = 'L': apply Q, Q**T, P or P**T from the Left; ! = 'R': apply Q, Q**T, P or P**T from the Right. ! ! TRANS (input) CHARACTER*1 ! = 'N': No transpose, apply Q or P; ! = 'T': Transpose, apply Q**T or P**T. ! ! M (input) INTEGER ! The number of rows of the matrix C. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix C. N >= 0. ! ! K (input) INTEGER ! If VECT = 'Q', the number of columns in the original ! matrix reduced by DGEBRD. ! If VECT = 'P', the number of rows in the original ! matrix reduced by DGEBRD. ! K >= 0. ! ! A (input) DOUBLE PRECISION array, dimension ! (LDA,min(nq,K)) if VECT = 'Q' ! (LDA,nq) if VECT = 'P' ! The vectors which define the elementary reflectors H(i) and ! G(i), whose products determine the matrices Q and P, as ! returned by DGEBRD. ! ! LDA (input) INTEGER ! The leading dimension of the array A. ! If VECT = 'Q', LDA >= max(1,nq); ! if VECT = 'P', LDA >= max(1,min(nq,K)). ! ! TAU (input) DOUBLE PRECISION array, dimension (min(nq,K)) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i) or G(i) which determines Q or P, as returned ! by DGEBRD in the array argument TAUQ or TAUP. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the M-by-N matrix C. ! On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q ! or P*C or P**T*C or C*P or C*P**T. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! If SIDE = 'L', LWORK >= max(1,N); ! if SIDE = 'R', LWORK >= max(1,M). ! For optimum performance LWORK >= N*NB if SIDE = 'L', and ! LWORK >= M*NB if SIDE = 'R', where NB is the optimal ! blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL APPLYQ, LEFT, LQUERY, NOTRAN CHARACTER TRANST INTEGER I1, I2, IINFO, LWKOPT, MI, NB, NI, NQ, NW ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DORMLQ, DORMQR, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 APPLYQ = LSAME( VECT, 'Q' ) LEFT = LSAME( SIDE, 'L' ) NOTRAN = LSAME( TRANS, 'N' ) LQUERY = ( LWORK.EQ.-1 ) ! ! NQ is the order of Q or P and NW is the minimum dimension of WORK ! IF( LEFT ) THEN NQ = M NW = N ELSE NQ = N NW = M END IF IF( .NOT.APPLYQ .AND. .NOT.LSAME( VECT, 'P' ) ) THEN INFO = -1 ELSE IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN INFO = -2 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN INFO = -3 ELSE IF( M.LT.0 ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( K.LT.0 ) THEN INFO = -6 ELSE IF( ( APPLYQ .AND. LDA.LT.MAX( 1, NQ ) ) .OR. & ( .NOT.APPLYQ .AND. LDA.LT.MAX( 1, MIN( NQ, K ) ) ) ) & THEN INFO = -8 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -11 ELSE IF( LWORK.LT.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN INFO = -13 END IF ! IF( INFO.EQ.0 ) THEN IF( APPLYQ ) THEN IF( LEFT ) THEN NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M-1, N, M-1, & -1 ) ELSE NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M, N-1, N-1, & -1 ) END IF ELSE IF( LEFT ) THEN NB = ILAENV( 1, 'DORMLQ', SIDE // TRANS, M-1, N, M-1, & -1 ) ELSE NB = ILAENV( 1, 'DORMLQ', SIDE // TRANS, M, N-1, N-1, & -1 ) END IF END IF LWKOPT = MAX( 1, NW )*NB WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORMBR', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! WORK( 1 ) = 1 IF( M.EQ.0 .OR. N.EQ.0 ) & RETURN ! IF( APPLYQ ) THEN ! ! Apply Q ! IF( NQ.GE.K ) THEN ! ! Q was determined by a call to DGEBRD with nq >= k ! CALL DORMQR( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, & WORK, LWORK, IINFO ) ELSE IF( NQ.GT.1 ) THEN ! ! Q was determined by a call to DGEBRD with nq < k ! IF( LEFT ) THEN MI = M - 1 NI = N I1 = 2 I2 = 1 ELSE MI = M NI = N - 1 I1 = 1 I2 = 2 END IF CALL DORMQR( SIDE, TRANS, MI, NI, NQ-1, A( 2, 1 ), LDA, TAU, & C( I1, I2 ), LDC, WORK, LWORK, IINFO ) END IF ELSE ! ! Apply P ! IF( NOTRAN ) THEN TRANST = 'T' ELSE TRANST = 'N' END IF IF( NQ.GT.K ) THEN ! ! P was determined by a call to DGEBRD with nq > k ! CALL DORMLQ( SIDE, TRANST, M, N, K, A, LDA, TAU, C, LDC, & WORK, LWORK, IINFO ) ELSE IF( NQ.GT.1 ) THEN ! ! P was determined by a call to DGEBRD with nq <= k ! IF( LEFT ) THEN MI = M - 1 NI = N I1 = 2 I2 = 1 ELSE MI = M NI = N - 1 I1 = 1 I2 = 2 END IF CALL DORMLQ( SIDE, TRANST, MI, NI, NQ-1, A( 1, 2 ), LDA, & TAU, C( I1, I2 ), LDC, WORK, LWORK, IINFO ) END IF END IF WORK( 1 ) = LWKOPT RETURN ! ! End of DORMBR ! END SUBROUTINE DORMHR( SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C, & LDC, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER SIDE, TRANS INTEGER IHI, ILO, INFO, LDA, LDC, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORMHR overwrites the general real M-by-N matrix C with ! ! SIDE = 'L' SIDE = 'R' ! TRANS = 'N': Q * C C * Q ! TRANS = 'T': Q**T * C C * Q**T ! ! where Q is a real orthogonal matrix of order nq, with nq = m if ! SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of ! IHI-ILO elementary reflectors, as returned by DGEHRD: ! ! Q = H(ilo) H(ilo+1) . . . H(ihi-1). ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': apply Q or Q**T from the Left; ! = 'R': apply Q or Q**T from the Right. ! ! TRANS (input) CHARACTER*1 ! = 'N': No transpose, apply Q; ! = 'T': Transpose, apply Q**T. ! ! M (input) INTEGER ! The number of rows of the matrix C. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix C. N >= 0. ! ! ILO (input) INTEGER ! IHI (input) INTEGER ! ILO and IHI must have the same values as in the previous call ! of DGEHRD. Q is equal to the unit matrix except in the ! submatrix Q(ilo+1:ihi,ilo+1:ihi). ! If SIDE = 'L', then 1 <= ILO <= IHI <= M, if M > 0, and ! ILO = 1 and IHI = 0, if M = 0; ! if SIDE = 'R', then 1 <= ILO <= IHI <= N, if N > 0, and ! ILO = 1 and IHI = 0, if N = 0. ! ! A (input) DOUBLE PRECISION array, dimension ! (LDA,M) if SIDE = 'L' ! (LDA,N) if SIDE = 'R' ! The vectors which define the elementary reflectors, as ! returned by DGEHRD. ! ! LDA (input) INTEGER ! The leading dimension of the array A. ! LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'. ! ! TAU (input) DOUBLE PRECISION array, dimension ! (M-1) if SIDE = 'L' ! (N-1) if SIDE = 'R' ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGEHRD. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the M-by-N matrix C. ! On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! If SIDE = 'L', LWORK >= max(1,N); ! if SIDE = 'R', LWORK >= max(1,M). ! For optimum performance LWORK >= N*NB if SIDE = 'L', and ! LWORK >= M*NB if SIDE = 'R', where NB is the optimal ! blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL LEFT, LQUERY INTEGER I1, I2, IINFO, LWKOPT, MI, NB, NH, NI, NQ, NW ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DORMQR, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 NH = IHI - ILO LEFT = LSAME( SIDE, 'L' ) LQUERY = ( LWORK.EQ.-1 ) ! ! NQ is the order of Q and NW is the minimum dimension of WORK ! IF( LEFT ) THEN NQ = M NW = N ELSE NQ = N NW = M END IF IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN INFO = -1 ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND. .NOT.LSAME( TRANS, 'T' ) ) & THEN INFO = -2 ELSE IF( M.LT.0 ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, NQ ) ) THEN INFO = -5 ELSE IF( IHI.LT.MIN( ILO, NQ ) .OR. IHI.GT.NQ ) THEN INFO = -6 ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN INFO = -8 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -11 ELSE IF( LWORK.LT.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN INFO = -13 END IF ! IF( INFO.EQ.0 ) THEN IF( LEFT ) THEN NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, NH, N, NH, -1 ) ELSE NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M, NH, NH, -1 ) END IF LWKOPT = MAX( 1, NW )*NB WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORMHR', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 .OR. NH.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! IF( LEFT ) THEN MI = NH NI = N I1 = ILO + 1 I2 = 1 ELSE MI = M NI = NH I1 = 1 I2 = ILO + 1 END IF ! CALL DORMQR( SIDE, TRANS, MI, NI, NH, A( ILO+1, ILO ), LDA, & TAU( ILO ), C( I1, I2 ), LDC, WORK, LWORK, IINFO ) ! WORK( 1 ) = LWKOPT RETURN ! ! End of DORMHR ! END SUBROUTINE DORML2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, & WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER SIDE, TRANS INTEGER INFO, K, LDA, LDC, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORML2 overwrites the general real m by n matrix C with ! ! Q * C if SIDE = 'L' and TRANS = 'N', or ! ! Q'* C if SIDE = 'L' and TRANS = 'T', or ! ! C * Q if SIDE = 'R' and TRANS = 'N', or ! ! C * Q' if SIDE = 'R' and TRANS = 'T', ! ! where Q is a real orthogonal matrix defined as the product of k ! elementary reflectors ! ! Q = H(k) . . . H(2) H(1) ! ! as returned by DGELQF. Q is of order m if SIDE = 'L' and of order n ! if SIDE = 'R'. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': apply Q or Q' from the Left ! = 'R': apply Q or Q' from the Right ! ! TRANS (input) CHARACTER*1 ! = 'N': apply Q (No transpose) ! = 'T': apply Q' (Transpose) ! ! M (input) INTEGER ! The number of rows of the matrix C. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix C. N >= 0. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines ! the matrix Q. ! If SIDE = 'L', M >= K >= 0; ! if SIDE = 'R', N >= K >= 0. ! ! A (input) DOUBLE PRECISION array, dimension ! (LDA,M) if SIDE = 'L', ! (LDA,N) if SIDE = 'R' ! The i-th row must contain the vector which defines the ! elementary reflector H(i), for i = 1,2,...,k, as returned by ! DGELQF in the first k rows of its array argument A. ! A is modified by the routine but restored on exit. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,K). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGELQF. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the m by n matrix C. ! On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace) DOUBLE PRECISION array, dimension ! (N) if SIDE = 'L', ! (M) if SIDE = 'R' ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LEFT, NOTRAN INTEGER I, I1, I2, I3, IC, JC, MI, NI, NQ DOUBLE PRECISION AII ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DLARF, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LEFT = LSAME( SIDE, 'L' ) NOTRAN = LSAME( TRANS, 'N' ) ! ! NQ is the order of Q ! IF( LEFT ) THEN NQ = M ELSE NQ = N END IF IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN INFO = -2 ELSE IF( M.LT.0 ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, K ) ) THEN INFO = -7 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -10 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORML2', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) & RETURN ! IF( ( LEFT .AND. NOTRAN ) .OR. ( .NOT.LEFT .AND. .NOT.NOTRAN ) ) & THEN I1 = 1 I2 = K I3 = 1 ELSE I1 = K I2 = 1 I3 = -1 END IF ! IF( LEFT ) THEN NI = N JC = 1 ELSE MI = M IC = 1 END IF ! DO 10 I = I1, I2, I3 IF( LEFT ) THEN ! ! H(i) is applied to C(i:m,1:n) ! MI = M - I + 1 IC = I ELSE ! ! H(i) is applied to C(1:m,i:n) ! NI = N - I + 1 JC = I END IF ! ! Apply H(i) ! AII = A( I, I ) A( I, I ) = ONE CALL DLARF( SIDE, MI, NI, A( I, I ), LDA, TAU( I ), & C( IC, JC ), LDC, WORK ) A( I, I ) = AII 10 CONTINUE RETURN ! ! End of DORML2 ! END SUBROUTINE DORMLQ( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, & WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER SIDE, TRANS INTEGER INFO, K, LDA, LDC, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORMLQ overwrites the general real M-by-N matrix C with ! ! SIDE = 'L' SIDE = 'R' ! TRANS = 'N': Q * C C * Q ! TRANS = 'T': Q**T * C C * Q**T ! ! where Q is a real orthogonal matrix defined as the product of k ! elementary reflectors ! ! Q = H(k) . . . H(2) H(1) ! ! as returned by DGELQF. Q is of order M if SIDE = 'L' and of order N ! if SIDE = 'R'. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': apply Q or Q**T from the Left; ! = 'R': apply Q or Q**T from the Right. ! ! TRANS (input) CHARACTER*1 ! = 'N': No transpose, apply Q; ! = 'T': Transpose, apply Q**T. ! ! M (input) INTEGER ! The number of rows of the matrix C. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix C. N >= 0. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines ! the matrix Q. ! If SIDE = 'L', M >= K >= 0; ! if SIDE = 'R', N >= K >= 0. ! ! A (input) DOUBLE PRECISION array, dimension ! (LDA,M) if SIDE = 'L', ! (LDA,N) if SIDE = 'R' ! The i-th row must contain the vector which defines the ! elementary reflector H(i), for i = 1,2,...,k, as returned by ! DGELQF in the first k rows of its array argument A. ! A is modified by the routine but restored on exit. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,K). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGELQF. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the M-by-N matrix C. ! On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! If SIDE = 'L', LWORK >= max(1,N); ! if SIDE = 'R', LWORK >= max(1,M). ! For optimum performance LWORK >= N*NB if SIDE = 'L', and ! LWORK >= M*NB if SIDE = 'R', where NB is the optimal ! blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. INTEGER NBMAX, LDT PARAMETER ( NBMAX = 64, LDT = NBMAX+1 ) ! .. ! .. Local Scalars .. LOGICAL LEFT, LQUERY, NOTRAN CHARACTER TRANST INTEGER I, I1, I2, I3, IB, IC, IINFO, IWS, JC, LDWORK, & LWKOPT, MI, NB, NBMIN, NI, NQ, NW ! .. ! .. Local Arrays .. DOUBLE PRECISION T( LDT, NBMAX ) ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DLARFB, DLARFT, DORML2, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LEFT = LSAME( SIDE, 'L' ) NOTRAN = LSAME( TRANS, 'N' ) LQUERY = ( LWORK.EQ.-1 ) ! ! NQ is the order of Q and NW is the minimum dimension of WORK ! IF( LEFT ) THEN NQ = M NW = N ELSE NQ = N NW = M END IF IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN INFO = -2 ELSE IF( M.LT.0 ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, K ) ) THEN INFO = -7 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -10 ELSE IF( LWORK.LT.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN INFO = -12 END IF ! IF( INFO.EQ.0 ) THEN ! ! Determine the block size. NB may be at most NBMAX, where NBMAX ! is used to define the local array T. ! NB = MIN( NBMAX, ILAENV( 1, 'DORMLQ', SIDE // TRANS, M, N, K, & -1 ) ) LWKOPT = MAX( 1, NW )*NB WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORMLQ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! NBMIN = 2 LDWORK = NW IF( NB.GT.1 .AND. NB.LT.K ) THEN IWS = NW*NB IF( LWORK.LT.IWS ) THEN NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'DORMLQ', SIDE // TRANS, M, N, K, & -1 ) ) END IF ELSE IWS = NW END IF ! IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN ! ! Use unblocked code ! CALL DORML2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, & IINFO ) ELSE ! ! Use blocked code ! IF( ( LEFT .AND. NOTRAN ) .OR. & ( .NOT.LEFT .AND. .NOT.NOTRAN ) ) THEN I1 = 1 I2 = K I3 = NB ELSE I1 = ( ( K-1 ) / NB )*NB + 1 I2 = 1 I3 = -NB END IF ! IF( LEFT ) THEN NI = N JC = 1 ELSE MI = M IC = 1 END IF ! IF( NOTRAN ) THEN TRANST = 'T' ELSE TRANST = 'N' END IF ! DO 10 I = I1, I2, I3 IB = MIN( NB, K-I+1 ) ! ! Form the triangular factor of the block reflector ! H = H(i) H(i+1) . . . H(i+ib-1) ! CALL DLARFT( 'Forward', 'Rowwise', NQ-I+1, IB, A( I, I ), & LDA, TAU( I ), T, LDT ) IF( LEFT ) THEN ! ! H or H' is applied to C(i:m,1:n) ! MI = M - I + 1 IC = I ELSE ! ! H or H' is applied to C(1:m,i:n) ! NI = N - I + 1 JC = I END IF ! ! Apply H or H' ! CALL DLARFB( SIDE, TRANST, 'Forward', 'Rowwise', MI, NI, IB, & A( I, I ), LDA, T, LDT, C( IC, JC ), LDC, WORK, & LDWORK ) 10 CONTINUE END IF WORK( 1 ) = LWKOPT RETURN ! ! End of DORMLQ ! END SUBROUTINE DORMQL( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, & WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER SIDE, TRANS INTEGER INFO, K, LDA, LDC, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORMQL overwrites the general real M-by-N matrix C with ! ! SIDE = 'L' SIDE = 'R' ! TRANS = 'N': Q * C C * Q ! TRANS = 'T': Q**T * C C * Q**T ! ! where Q is a real orthogonal matrix defined as the product of k ! elementary reflectors ! ! Q = H(k) . . . H(2) H(1) ! ! as returned by DGEQLF. Q is of order M if SIDE = 'L' and of order N ! if SIDE = 'R'. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': apply Q or Q**T from the Left; ! = 'R': apply Q or Q**T from the Right. ! ! TRANS (input) CHARACTER*1 ! = 'N': No transpose, apply Q; ! = 'T': Transpose, apply Q**T. ! ! M (input) INTEGER ! The number of rows of the matrix C. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix C. N >= 0. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines ! the matrix Q. ! If SIDE = 'L', M >= K >= 0; ! if SIDE = 'R', N >= K >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,K) ! The i-th column must contain the vector which defines the ! elementary reflector H(i), for i = 1,2,...,k, as returned by ! DGEQLF in the last k columns of its array argument A. ! A is modified by the routine but restored on exit. ! ! LDA (input) INTEGER ! The leading dimension of the array A. ! If SIDE = 'L', LDA >= max(1,M); ! if SIDE = 'R', LDA >= max(1,N). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGEQLF. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the M-by-N matrix C. ! On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! If SIDE = 'L', LWORK >= max(1,N); ! if SIDE = 'R', LWORK >= max(1,M). ! For optimum performance LWORK >= N*NB if SIDE = 'L', and ! LWORK >= M*NB if SIDE = 'R', where NB is the optimal ! blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. INTEGER NBMAX, LDT PARAMETER ( NBMAX = 64, LDT = NBMAX+1 ) ! .. ! .. Local Scalars .. LOGICAL LEFT, LQUERY, NOTRAN INTEGER I, I1, I2, I3, IB, IINFO, IWS, LDWORK, LWKOPT, & MI, NB, NBMIN, NI, NQ, NW ! .. ! .. Local Arrays .. DOUBLE PRECISION T( LDT, NBMAX ) ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DLARFB, DLARFT, DORM2L, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LEFT = LSAME( SIDE, 'L' ) NOTRAN = LSAME( TRANS, 'N' ) LQUERY = ( LWORK.EQ.-1 ) ! ! NQ is the order of Q and NW is the minimum dimension of WORK ! IF( LEFT ) THEN NQ = M NW = N ELSE NQ = N NW = M END IF IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN INFO = -2 ELSE IF( M.LT.0 ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN INFO = -7 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -10 ELSE IF( LWORK.LT.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN INFO = -12 END IF ! IF( INFO.EQ.0 ) THEN ! ! Determine the block size. NB may be at most NBMAX, where NBMAX ! is used to define the local array T. ! NB = MIN( NBMAX, ILAENV( 1, 'DORMQL', SIDE // TRANS, M, N, K, & -1 ) ) LWKOPT = MAX( 1, NW )*NB WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORMQL', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! NBMIN = 2 LDWORK = NW IF( NB.GT.1 .AND. NB.LT.K ) THEN IWS = NW*NB IF( LWORK.LT.IWS ) THEN NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'DORMQL', SIDE // TRANS, M, N, K, & -1 ) ) END IF ELSE IWS = NW END IF ! IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN ! ! Use unblocked code ! CALL DORM2L( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, & IINFO ) ELSE ! ! Use blocked code ! IF( ( LEFT .AND. NOTRAN ) .OR. & ( .NOT.LEFT .AND. .NOT.NOTRAN ) ) THEN I1 = 1 I2 = K I3 = NB ELSE I1 = ( ( K-1 ) / NB )*NB + 1 I2 = 1 I3 = -NB END IF ! IF( LEFT ) THEN NI = N ELSE MI = M END IF ! DO 10 I = I1, I2, I3 IB = MIN( NB, K-I+1 ) ! ! Form the triangular factor of the block reflector ! H = H(i+ib-1) . . . H(i+1) H(i) ! CALL DLARFT( 'Backward', 'Columnwise', NQ-K+I+IB-1, IB, & A( 1, I ), LDA, TAU( I ), T, LDT ) IF( LEFT ) THEN ! ! H or H' is applied to C(1:m-k+i+ib-1,1:n) ! MI = M - K + I + IB - 1 ELSE ! ! H or H' is applied to C(1:m,1:n-k+i+ib-1) ! NI = N - K + I + IB - 1 END IF ! ! Apply H or H' ! CALL DLARFB( SIDE, TRANS, 'Backward', 'Columnwise', MI, NI, & IB, A( 1, I ), LDA, T, LDT, C, LDC, WORK, & LDWORK ) 10 CONTINUE END IF WORK( 1 ) = LWKOPT RETURN ! ! End of DORMQL ! END SUBROUTINE DORMQR( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, & WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER SIDE, TRANS INTEGER INFO, K, LDA, LDC, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORMQR overwrites the general real M-by-N matrix C with ! ! SIDE = 'L' SIDE = 'R' ! TRANS = 'N': Q * C C * Q ! TRANS = 'T': Q**T * C C * Q**T ! ! where Q is a real orthogonal matrix defined as the product of k ! elementary reflectors ! ! Q = H(1) H(2) . . . H(k) ! ! as returned by DGEQRF. Q is of order M if SIDE = 'L' and of order N ! if SIDE = 'R'. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': apply Q or Q**T from the Left; ! = 'R': apply Q or Q**T from the Right. ! ! TRANS (input) CHARACTER*1 ! = 'N': No transpose, apply Q; ! = 'T': Transpose, apply Q**T. ! ! M (input) INTEGER ! The number of rows of the matrix C. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix C. N >= 0. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines ! the matrix Q. ! If SIDE = 'L', M >= K >= 0; ! if SIDE = 'R', N >= K >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,K) ! The i-th column must contain the vector which defines the ! elementary reflector H(i), for i = 1,2,...,k, as returned by ! DGEQRF in the first k columns of its array argument A. ! A is modified by the routine but restored on exit. ! ! LDA (input) INTEGER ! The leading dimension of the array A. ! If SIDE = 'L', LDA >= max(1,M); ! if SIDE = 'R', LDA >= max(1,N). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGEQRF. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the M-by-N matrix C. ! On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! If SIDE = 'L', LWORK >= max(1,N); ! if SIDE = 'R', LWORK >= max(1,M). ! For optimum performance LWORK >= N*NB if SIDE = 'L', and ! LWORK >= M*NB if SIDE = 'R', where NB is the optimal ! blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. INTEGER NBMAX, LDT PARAMETER ( NBMAX = 64, LDT = NBMAX+1 ) ! .. ! .. Local Scalars .. LOGICAL LEFT, LQUERY, NOTRAN INTEGER I, I1, I2, I3, IB, IC, IINFO, IWS, JC, LDWORK, & LWKOPT, MI, NB, NBMIN, NI, NQ, NW ! .. ! .. Local Arrays .. DOUBLE PRECISION T( LDT, NBMAX ) ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DLARFB, DLARFT, DORM2R, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LEFT = LSAME( SIDE, 'L' ) NOTRAN = LSAME( TRANS, 'N' ) LQUERY = ( LWORK.EQ.-1 ) ! ! NQ is the order of Q and NW is the minimum dimension of WORK ! IF( LEFT ) THEN NQ = M NW = N ELSE NQ = N NW = M END IF IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN INFO = -2 ELSE IF( M.LT.0 ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN INFO = -7 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -10 ELSE IF( LWORK.LT.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN INFO = -12 END IF ! IF( INFO.EQ.0 ) THEN ! ! Determine the block size. NB may be at most NBMAX, where NBMAX ! is used to define the local array T. ! NB = MIN( NBMAX, ILAENV( 1, 'DORMQR', SIDE // TRANS, M, N, K, & -1 ) ) LWKOPT = MAX( 1, NW )*NB WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORMQR', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! NBMIN = 2 LDWORK = NW IF( NB.GT.1 .AND. NB.LT.K ) THEN IWS = NW*NB IF( LWORK.LT.IWS ) THEN NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'DORMQR', SIDE // TRANS, M, N, K, & -1 ) ) END IF ELSE IWS = NW END IF ! IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN ! ! Use unblocked code ! CALL DORM2R( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, & IINFO ) ELSE ! ! Use blocked code ! IF( ( LEFT .AND. .NOT.NOTRAN ) .OR. & ( .NOT.LEFT .AND. NOTRAN ) ) THEN I1 = 1 I2 = K I3 = NB ELSE I1 = ( ( K-1 ) / NB )*NB + 1 I2 = 1 I3 = -NB END IF ! IF( LEFT ) THEN NI = N JC = 1 ELSE MI = M IC = 1 END IF ! DO 10 I = I1, I2, I3 IB = MIN( NB, K-I+1 ) ! ! Form the triangular factor of the block reflector ! H = H(i) H(i+1) . . . H(i+ib-1) ! CALL DLARFT( 'Forward', 'Columnwise', NQ-I+1, IB, A( I, I ), & LDA, TAU( I ), T, LDT ) IF( LEFT ) THEN ! ! H or H' is applied to C(i:m,1:n) ! MI = M - I + 1 IC = I ELSE ! ! H or H' is applied to C(1:m,i:n) ! NI = N - I + 1 JC = I END IF ! ! Apply H or H' ! CALL DLARFB( SIDE, TRANS, 'Forward', 'Columnwise', MI, NI, & IB, A( I, I ), LDA, T, LDT, C( IC, JC ), LDC, & WORK, LDWORK ) 10 CONTINUE END IF WORK( 1 ) = LWKOPT RETURN ! ! End of DORMQR ! END SUBROUTINE DORMR2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, & WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER SIDE, TRANS INTEGER INFO, K, LDA, LDC, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORMR2 overwrites the general real m by n matrix C with ! ! Q * C if SIDE = 'L' and TRANS = 'N', or ! ! Q'* C if SIDE = 'L' and TRANS = 'T', or ! ! C * Q if SIDE = 'R' and TRANS = 'N', or ! ! C * Q' if SIDE = 'R' and TRANS = 'T', ! ! where Q is a real orthogonal matrix defined as the product of k ! elementary reflectors ! ! Q = H(1) H(2) . . . H(k) ! ! as returned by DGERQF. Q is of order m if SIDE = 'L' and of order n ! if SIDE = 'R'. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': apply Q or Q' from the Left ! = 'R': apply Q or Q' from the Right ! ! TRANS (input) CHARACTER*1 ! = 'N': apply Q (No transpose) ! = 'T': apply Q' (Transpose) ! ! M (input) INTEGER ! The number of rows of the matrix C. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix C. N >= 0. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines ! the matrix Q. ! If SIDE = 'L', M >= K >= 0; ! if SIDE = 'R', N >= K >= 0. ! ! A (input) DOUBLE PRECISION array, dimension ! (LDA,M) if SIDE = 'L', ! (LDA,N) if SIDE = 'R' ! The i-th row must contain the vector which defines the ! elementary reflector H(i), for i = 1,2,...,k, as returned by ! DGERQF in the last k rows of its array argument A. ! A is modified by the routine but restored on exit. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,K). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGERQF. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the m by n matrix C. ! On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace) DOUBLE PRECISION array, dimension ! (N) if SIDE = 'L', ! (M) if SIDE = 'R' ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LEFT, NOTRAN INTEGER I, I1, I2, I3, MI, NI, NQ DOUBLE PRECISION AII ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DLARF, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LEFT = LSAME( SIDE, 'L' ) NOTRAN = LSAME( TRANS, 'N' ) ! ! NQ is the order of Q ! IF( LEFT ) THEN NQ = M ELSE NQ = N END IF IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN INFO = -2 ELSE IF( M.LT.0 ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, K ) ) THEN INFO = -7 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -10 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORMR2', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) & RETURN ! IF( ( LEFT .AND. .NOT.NOTRAN ) .OR. ( .NOT.LEFT .AND. NOTRAN ) ) & THEN I1 = 1 I2 = K I3 = 1 ELSE I1 = K I2 = 1 I3 = -1 END IF ! IF( LEFT ) THEN NI = N ELSE MI = M END IF ! DO 10 I = I1, I2, I3 IF( LEFT ) THEN ! ! H(i) is applied to C(1:m-k+i,1:n) ! MI = M - K + I ELSE ! ! H(i) is applied to C(1:m,1:n-k+i) ! NI = N - K + I END IF ! ! Apply H(i) ! AII = A( I, NQ-K+I ) A( I, NQ-K+I ) = ONE CALL DLARF( SIDE, MI, NI, A( I, 1 ), LDA, TAU( I ), C, LDC, & WORK ) A( I, NQ-K+I ) = AII 10 CONTINUE RETURN ! ! End of DORMR2 ! END SUBROUTINE DORMR3( SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC, & WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER SIDE, TRANS INTEGER INFO, K, L, LDA, LDC, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORMR3 overwrites the general real m by n matrix C with ! ! Q * C if SIDE = 'L' and TRANS = 'N', or ! ! Q'* C if SIDE = 'L' and TRANS = 'T', or ! ! C * Q if SIDE = 'R' and TRANS = 'N', or ! ! C * Q' if SIDE = 'R' and TRANS = 'T', ! ! where Q is a real orthogonal matrix defined as the product of k ! elementary reflectors ! ! Q = H(1) H(2) . . . H(k) ! ! as returned by DTZRZF. Q is of order m if SIDE = 'L' and of order n ! if SIDE = 'R'. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': apply Q or Q' from the Left ! = 'R': apply Q or Q' from the Right ! ! TRANS (input) CHARACTER*1 ! = 'N': apply Q (No transpose) ! = 'T': apply Q' (Transpose) ! ! M (input) INTEGER ! The number of rows of the matrix C. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix C. N >= 0. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines ! the matrix Q. ! If SIDE = 'L', M >= K >= 0; ! if SIDE = 'R', N >= K >= 0. ! ! L (input) INTEGER ! The number of columns of the matrix A containing ! the meaningful part of the Householder reflectors. ! If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. ! ! A (input) DOUBLE PRECISION array, dimension ! (LDA,M) if SIDE = 'L', ! (LDA,N) if SIDE = 'R' ! The i-th row must contain the vector which defines the ! elementary reflector H(i), for i = 1,2,...,k, as returned by ! DTZRZF in the last k rows of its array argument A. ! A is modified by the routine but restored on exit. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,K). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DTZRZF. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the m-by-n matrix C. ! On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace) DOUBLE PRECISION array, dimension ! (N) if SIDE = 'L', ! (M) if SIDE = 'R' ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! Based on contributions by ! A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL LEFT, NOTRAN INTEGER I, I1, I2, I3, IC, JA, JC, MI, NI, NQ ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DLARZ, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LEFT = LSAME( SIDE, 'L' ) NOTRAN = LSAME( TRANS, 'N' ) ! ! NQ is the order of Q ! IF( LEFT ) THEN NQ = M ELSE NQ = N END IF IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN INFO = -2 ELSE IF( M.LT.0 ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN INFO = -5 ELSE IF( L.LT.0 .OR. ( LEFT .AND. ( L.GT.M ) ) .OR. & ( .NOT.LEFT .AND. ( L.GT.N ) ) ) THEN INFO = -6 ELSE IF( LDA.LT.MAX( 1, K ) ) THEN INFO = -8 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -11 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORMR3', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) & RETURN ! IF( ( LEFT .AND. .NOT.NOTRAN .OR. .NOT.LEFT .AND. NOTRAN ) ) THEN I1 = 1 I2 = K I3 = 1 ELSE I1 = K I2 = 1 I3 = -1 END IF ! IF( LEFT ) THEN NI = N JA = M - L + 1 JC = 1 ELSE MI = M JA = N - L + 1 IC = 1 END IF ! DO 10 I = I1, I2, I3 IF( LEFT ) THEN ! ! H(i) or H(i)' is applied to C(i:m,1:n) ! MI = M - I + 1 IC = I ELSE ! ! H(i) or H(i)' is applied to C(1:m,i:n) ! NI = N - I + 1 JC = I END IF ! ! Apply H(i) or H(i)' ! CALL DLARZ( SIDE, MI, NI, L, A( I, JA ), LDA, TAU( I ), & C( IC, JC ), LDC, WORK ) ! 10 CONTINUE ! RETURN ! ! End of DORMR3 ! END SUBROUTINE DORMRQ( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, & WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER SIDE, TRANS INTEGER INFO, K, LDA, LDC, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORMRQ overwrites the general real M-by-N matrix C with ! ! SIDE = 'L' SIDE = 'R' ! TRANS = 'N': Q * C C * Q ! TRANS = 'T': Q**T * C C * Q**T ! ! where Q is a real orthogonal matrix defined as the product of k ! elementary reflectors ! ! Q = H(1) H(2) . . . H(k) ! ! as returned by DGERQF. Q is of order M if SIDE = 'L' and of order N ! if SIDE = 'R'. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': apply Q or Q**T from the Left; ! = 'R': apply Q or Q**T from the Right. ! ! TRANS (input) CHARACTER*1 ! = 'N': No transpose, apply Q; ! = 'T': Transpose, apply Q**T. ! ! M (input) INTEGER ! The number of rows of the matrix C. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix C. N >= 0. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines ! the matrix Q. ! If SIDE = 'L', M >= K >= 0; ! if SIDE = 'R', N >= K >= 0. ! ! A (input) DOUBLE PRECISION array, dimension ! (LDA,M) if SIDE = 'L', ! (LDA,N) if SIDE = 'R' ! The i-th row must contain the vector which defines the ! elementary reflector H(i), for i = 1,2,...,k, as returned by ! DGERQF in the last k rows of its array argument A. ! A is modified by the routine but restored on exit. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,K). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DGERQF. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the M-by-N matrix C. ! On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! If SIDE = 'L', LWORK >= max(1,N); ! if SIDE = 'R', LWORK >= max(1,M). ! For optimum performance LWORK >= N*NB if SIDE = 'L', and ! LWORK >= M*NB if SIDE = 'R', where NB is the optimal ! blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. INTEGER NBMAX, LDT PARAMETER ( NBMAX = 64, LDT = NBMAX+1 ) ! .. ! .. Local Scalars .. LOGICAL LEFT, LQUERY, NOTRAN CHARACTER TRANST INTEGER I, I1, I2, I3, IB, IINFO, IWS, LDWORK, LWKOPT, & MI, NB, NBMIN, NI, NQ, NW ! .. ! .. Local Arrays .. DOUBLE PRECISION T( LDT, NBMAX ) ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DLARFB, DLARFT, DORMR2, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LEFT = LSAME( SIDE, 'L' ) NOTRAN = LSAME( TRANS, 'N' ) LQUERY = ( LWORK.EQ.-1 ) ! ! NQ is the order of Q and NW is the minimum dimension of WORK ! IF( LEFT ) THEN NQ = M NW = N ELSE NQ = N NW = M END IF IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN INFO = -2 ELSE IF( M.LT.0 ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, K ) ) THEN INFO = -7 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -10 ELSE IF( LWORK.LT.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN INFO = -12 END IF ! IF( INFO.EQ.0 ) THEN ! ! Determine the block size. NB may be at most NBMAX, where NBMAX ! is used to define the local array T. ! NB = MIN( NBMAX, ILAENV( 1, 'DORMRQ', SIDE // TRANS, M, N, K, & -1 ) ) LWKOPT = MAX( 1, NW )*NB WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORMRQ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! NBMIN = 2 LDWORK = NW IF( NB.GT.1 .AND. NB.LT.K ) THEN IWS = NW*NB IF( LWORK.LT.IWS ) THEN NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'DORMRQ', SIDE // TRANS, M, N, K, & -1 ) ) END IF ELSE IWS = NW END IF ! IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN ! ! Use unblocked code ! CALL DORMR2( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, & IINFO ) ELSE ! ! Use blocked code ! IF( ( LEFT .AND. .NOT.NOTRAN ) .OR. & ( .NOT.LEFT .AND. NOTRAN ) ) THEN I1 = 1 I2 = K I3 = NB ELSE I1 = ( ( K-1 ) / NB )*NB + 1 I2 = 1 I3 = -NB END IF ! IF( LEFT ) THEN NI = N ELSE MI = M END IF ! IF( NOTRAN ) THEN TRANST = 'T' ELSE TRANST = 'N' END IF ! DO 10 I = I1, I2, I3 IB = MIN( NB, K-I+1 ) ! ! Form the triangular factor of the block reflector ! H = H(i+ib-1) . . . H(i+1) H(i) ! CALL DLARFT( 'Backward', 'Rowwise', NQ-K+I+IB-1, IB, & A( I, 1 ), LDA, TAU( I ), T, LDT ) IF( LEFT ) THEN ! ! H or H' is applied to C(1:m-k+i+ib-1,1:n) ! MI = M - K + I + IB - 1 ELSE ! ! H or H' is applied to C(1:m,1:n-k+i+ib-1) ! NI = N - K + I + IB - 1 END IF ! ! Apply H or H' ! CALL DLARFB( SIDE, TRANST, 'Backward', 'Rowwise', MI, NI, & IB, A( I, 1 ), LDA, T, LDT, C, LDC, WORK, & LDWORK ) 10 CONTINUE END IF WORK( 1 ) = LWKOPT RETURN ! ! End of DORMRQ ! END SUBROUTINE DORMRZ( SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC, & WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. CHARACTER SIDE, TRANS INTEGER INFO, K, L, LDA, LDC, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORMRZ overwrites the general real M-by-N matrix C with ! ! SIDE = 'L' SIDE = 'R' ! TRANS = 'N': Q * C C * Q ! TRANS = 'T': Q**T * C C * Q**T ! ! where Q is a real orthogonal matrix defined as the product of k ! elementary reflectors ! ! Q = H(1) H(2) . . . H(k) ! ! as returned by DTZRZF. Q is of order M if SIDE = 'L' and of order N ! if SIDE = 'R'. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': apply Q or Q**T from the Left; ! = 'R': apply Q or Q**T from the Right. ! ! TRANS (input) CHARACTER*1 ! = 'N': No transpose, apply Q; ! = 'T': Transpose, apply Q**T. ! ! M (input) INTEGER ! The number of rows of the matrix C. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix C. N >= 0. ! ! K (input) INTEGER ! The number of elementary reflectors whose product defines ! the matrix Q. ! If SIDE = 'L', M >= K >= 0; ! if SIDE = 'R', N >= K >= 0. ! ! L (input) INTEGER ! The number of columns of the matrix A containing ! the meaningful part of the Householder reflectors. ! If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0. ! ! A (input) DOUBLE PRECISION array, dimension ! (LDA,M) if SIDE = 'L', ! (LDA,N) if SIDE = 'R' ! The i-th row must contain the vector which defines the ! elementary reflector H(i), for i = 1,2,...,k, as returned by ! DTZRZF in the last k rows of its array argument A. ! A is modified by the routine but restored on exit. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,K). ! ! TAU (input) DOUBLE PRECISION array, dimension (K) ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DTZRZF. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the M-by-N matrix C. ! On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! If SIDE = 'L', LWORK >= max(1,N); ! if SIDE = 'R', LWORK >= max(1,M). ! For optimum performance LWORK >= N*NB if SIDE = 'L', and ! LWORK >= M*NB if SIDE = 'R', where NB is the optimal ! blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! Based on contributions by ! A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA ! ! ===================================================================== ! ! .. Parameters .. INTEGER NBMAX, LDT PARAMETER ( NBMAX = 64, LDT = NBMAX+1 ) ! .. ! .. Local Scalars .. LOGICAL LEFT, LQUERY, NOTRAN CHARACTER TRANST INTEGER I, I1, I2, I3, IB, IC, IINFO, IWS, JA, JC, & LDWORK, LWKOPT, MI, NB, NBMIN, NI, NQ, NW ! .. ! .. Local Arrays .. DOUBLE PRECISION T( LDT, NBMAX ) ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DLARZB, DLARZT, DORMR3, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LEFT = LSAME( SIDE, 'L' ) NOTRAN = LSAME( TRANS, 'N' ) LQUERY = ( LWORK.EQ.-1 ) ! ! NQ is the order of Q and NW is the minimum dimension of WORK ! IF( LEFT ) THEN NQ = M NW = N ELSE NQ = N NW = M END IF IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN INFO = -2 ELSE IF( M.LT.0 ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN INFO = -5 ELSE IF( L.LT.0 .OR. ( LEFT .AND. ( L.GT.M ) ) .OR. & ( .NOT.LEFT .AND. ( L.GT.N ) ) ) THEN INFO = -6 ELSE IF( LDA.LT.MAX( 1, K ) ) THEN INFO = -8 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -11 ELSE IF( LWORK.LT.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN INFO = -13 END IF ! IF( INFO.EQ.0 ) THEN ! ! Determine the block size. NB may be at most NBMAX, where NBMAX ! is used to define the local array T. ! NB = MIN( NBMAX, ILAENV( 1, 'DORMRQ', SIDE // TRANS, M, N, K, & -1 ) ) LWKOPT = MAX( 1, NW )*NB WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORMRZ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 .OR. K.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! NBMIN = 2 LDWORK = NW IF( NB.GT.1 .AND. NB.LT.K ) THEN IWS = NW*NB IF( LWORK.LT.IWS ) THEN NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'DORMRQ', SIDE // TRANS, M, N, K, & -1 ) ) END IF ELSE IWS = NW END IF ! IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN ! ! Use unblocked code ! CALL DORMR3( SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC, & WORK, IINFO ) ELSE ! ! Use blocked code ! IF( ( LEFT .AND. .NOT.NOTRAN ) .OR. & ( .NOT.LEFT .AND. NOTRAN ) ) THEN I1 = 1 I2 = K I3 = NB ELSE I1 = ( ( K-1 ) / NB )*NB + 1 I2 = 1 I3 = -NB END IF ! IF( LEFT ) THEN NI = N JC = 1 JA = M - L + 1 ELSE MI = M IC = 1 JA = N - L + 1 END IF ! IF( NOTRAN ) THEN TRANST = 'T' ELSE TRANST = 'N' END IF ! DO 10 I = I1, I2, I3 IB = MIN( NB, K-I+1 ) ! ! Form the triangular factor of the block reflector ! H = H(i+ib-1) . . . H(i+1) H(i) ! CALL DLARZT( 'Backward', 'Rowwise', L, IB, A( I, JA ), LDA, & TAU( I ), T, LDT ) ! IF( LEFT ) THEN ! ! H or H' is applied to C(i:m,1:n) ! MI = M - I + 1 IC = I ELSE ! ! H or H' is applied to C(1:m,i:n) ! NI = N - I + 1 JC = I END IF ! ! Apply H or H' ! CALL DLARZB( SIDE, TRANST, 'Backward', 'Rowwise', MI, NI, & IB, L, A( I, JA ), LDA, T, LDT, C( IC, JC ), & LDC, WORK, LDWORK ) 10 CONTINUE ! END IF ! WORK( 1 ) = LWKOPT ! RETURN ! ! End of DORMRZ ! END SUBROUTINE DORMTR( SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, & WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER SIDE, TRANS, UPLO INTEGER INFO, LDA, LDC, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( LDC, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DORMTR overwrites the general real M-by-N matrix C with ! ! SIDE = 'L' SIDE = 'R' ! TRANS = 'N': Q * C C * Q ! TRANS = 'T': Q**T * C C * Q**T ! ! where Q is a real orthogonal matrix of order nq, with nq = m if ! SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of ! nq-1 elementary reflectors, as returned by DSYTRD: ! ! if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1); ! ! if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1). ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'L': apply Q or Q**T from the Left; ! = 'R': apply Q or Q**T from the Right. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A contains elementary reflectors ! from DSYTRD; ! = 'L': Lower triangle of A contains elementary reflectors ! from DSYTRD. ! ! TRANS (input) CHARACTER*1 ! = 'N': No transpose, apply Q; ! = 'T': Transpose, apply Q**T. ! ! M (input) INTEGER ! The number of rows of the matrix C. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix C. N >= 0. ! ! A (input) DOUBLE PRECISION array, dimension ! (LDA,M) if SIDE = 'L' ! (LDA,N) if SIDE = 'R' ! The vectors which define the elementary reflectors, as ! returned by DSYTRD. ! ! LDA (input) INTEGER ! The leading dimension of the array A. ! LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'. ! ! TAU (input) DOUBLE PRECISION array, dimension ! (M-1) if SIDE = 'L' ! (N-1) if SIDE = 'R' ! TAU(i) must contain the scalar factor of the elementary ! reflector H(i), as returned by DSYTRD. ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the M-by-N matrix C. ! On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! If SIDE = 'L', LWORK >= max(1,N); ! if SIDE = 'R', LWORK >= max(1,M). ! For optimum performance LWORK >= N*NB if SIDE = 'L', and ! LWORK >= M*NB if SIDE = 'R', where NB is the optimal ! blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL LEFT, LQUERY, UPPER INTEGER I1, I2, IINFO, LWKOPT, MI, NB, NI, NQ, NW ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DORMQL, DORMQR, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LEFT = LSAME( SIDE, 'L' ) UPPER = LSAME( UPLO, 'U' ) LQUERY = ( LWORK.EQ.-1 ) ! ! NQ is the order of Q and NW is the minimum dimension of WORK ! IF( LEFT ) THEN NQ = M NW = N ELSE NQ = N NW = M END IF IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN INFO = -1 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -2 ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND. .NOT.LSAME( TRANS, 'T' ) ) & THEN INFO = -3 ELSE IF( M.LT.0 ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN INFO = -7 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -10 ELSE IF( LWORK.LT.MAX( 1, NW ) .AND. .NOT.LQUERY ) THEN INFO = -12 END IF ! IF( INFO.EQ.0 ) THEN IF( UPPER ) THEN IF( LEFT ) THEN NB = ILAENV( 1, 'DORMQL', SIDE // TRANS, M-1, N, M-1, & -1 ) ELSE NB = ILAENV( 1, 'DORMQL', SIDE // TRANS, M, N-1, N-1, & -1 ) END IF ELSE IF( LEFT ) THEN NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M-1, N, M-1, & -1 ) ELSE NB = ILAENV( 1, 'DORMQR', SIDE // TRANS, M, N-1, N-1, & -1 ) END IF END IF LWKOPT = MAX( 1, NW )*NB WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DORMTR', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 .OR. NQ.EQ.1 ) THEN WORK( 1 ) = 1 RETURN END IF ! IF( LEFT ) THEN MI = M - 1 NI = N ELSE MI = M NI = N - 1 END IF ! IF( UPPER ) THEN ! ! Q was determined by a call to DSYTRD with UPLO = 'U' ! CALL DORMQL( SIDE, TRANS, MI, NI, NQ-1, A( 1, 2 ), LDA, TAU, C, & LDC, WORK, LWORK, IINFO ) ELSE ! ! Q was determined by a call to DSYTRD with UPLO = 'L' ! IF( LEFT ) THEN I1 = 2 I2 = 1 ELSE I1 = 1 I2 = 2 END IF CALL DORMQR( SIDE, TRANS, MI, NI, NQ-1, A( 2, 1 ), LDA, TAU, & C( I1, I2 ), LDC, WORK, LWORK, IINFO ) END IF WORK( 1 ) = LWKOPT RETURN ! ! End of DORMTR ! END SUBROUTINE DPBCON( UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK, & IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, KD, LDAB, N DOUBLE PRECISION ANORM, RCOND ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION AB( LDAB, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DPBCON estimates the reciprocal of the condition number (in the ! 1-norm) of a real symmetric positive definite band matrix using the ! Cholesky factorization A = U**T*U or A = L*L**T computed by DPBTRF. ! ! An estimate is obtained for norm(inv(A)), and the reciprocal of the ! condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangular factor stored in AB; ! = 'L': Lower triangular factor stored in AB. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of superdiagonals of the matrix A if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KD >= 0. ! ! AB (input) DOUBLE PRECISION array, dimension (LDAB,N) ! The triangular factor U or L from the Cholesky factorization ! A = U**T*U or A = L*L**T of the band matrix A, stored in the ! first KD+1 rows of the array. The j-th column of U or L is ! stored in the j-th column of the array AB as follows: ! if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j; ! if UPLO ='L', AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd). ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KD+1. ! ! ANORM (input) DOUBLE PRECISION ! The 1-norm (or infinity-norm) of the symmetric band matrix A. ! ! RCOND (output) DOUBLE PRECISION ! The reciprocal of the condition number of the matrix A, ! computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an ! estimate of the 1-norm of inv(A) computed in this routine. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER CHARACTER NORMIN INTEGER IX, KASE DOUBLE PRECISION AINVNM, SCALE, SCALEL, SCALEU, SMLNUM ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DLAMCH EXTERNAL LSAME, IDAMAX, DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DLACON, DLATBS, DRSCL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KD.LT.0 ) THEN INFO = -3 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -5 ELSE IF( ANORM.LT.ZERO ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPBCON', -INFO ) RETURN END IF ! ! Quick return if possible ! RCOND = ZERO IF( N.EQ.0 ) THEN RCOND = ONE RETURN ELSE IF( ANORM.EQ.ZERO ) THEN RETURN END IF ! SMLNUM = DLAMCH( 'Safe minimum' ) ! ! Estimate the 1-norm of the inverse. ! KASE = 0 NORMIN = 'N' 10 CONTINUE CALL DLACON( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE ) IF( KASE.NE.0 ) THEN IF( UPPER ) THEN ! ! Multiply by inv(U'). ! CALL DLATBS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N, & KD, AB, LDAB, WORK, SCALEL, WORK( 2*N+1 ), & INFO ) NORMIN = 'Y' ! ! Multiply by inv(U). ! CALL DLATBS( 'Upper', 'No transpose', 'Non-unit', NORMIN, N, & KD, AB, LDAB, WORK, SCALEU, WORK( 2*N+1 ), & INFO ) ELSE ! ! Multiply by inv(L). ! CALL DLATBS( 'Lower', 'No transpose', 'Non-unit', NORMIN, N, & KD, AB, LDAB, WORK, SCALEL, WORK( 2*N+1 ), & INFO ) NORMIN = 'Y' ! ! Multiply by inv(L'). ! CALL DLATBS( 'Lower', 'Transpose', 'Non-unit', NORMIN, N, & KD, AB, LDAB, WORK, SCALEU, WORK( 2*N+1 ), & INFO ) END IF ! ! Multiply by 1/SCALE if doing so will not cause overflow. ! SCALE = SCALEL*SCALEU IF( SCALE.NE.ONE ) THEN IX = IDAMAX( N, WORK, 1 ) IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO ) & GO TO 20 CALL DRSCL( N, SCALE, WORK, 1 ) END IF GO TO 10 END IF ! ! Compute the estimate of the reciprocal condition number. ! IF( AINVNM.NE.ZERO ) & RCOND = ( ONE / AINVNM ) / ANORM ! 20 CONTINUE ! RETURN ! ! End of DPBCON ! END SUBROUTINE DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, KD, LDAB, N DOUBLE PRECISION AMAX, SCOND ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), S( * ) ! .. ! ! Purpose ! ======= ! ! DPBEQU computes row and column scalings intended to equilibrate a ! symmetric positive definite band matrix A and reduce its condition ! number (with respect to the two-norm). S contains the scale factors, ! S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with ! elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This ! choice of S puts the condition number of B within a factor N of the ! smallest possible condition number over all possible diagonal ! scalings. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangular of A is stored; ! = 'L': Lower triangular of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of superdiagonals of the matrix A if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KD >= 0. ! ! AB (input) DOUBLE PRECISION array, dimension (LDAB,N) ! The upper or lower triangle of the symmetric band matrix A, ! stored in the first KD+1 rows of the array. The j-th column ! of A is stored in the j-th column of the array AB as follows: ! if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). ! ! LDAB (input) INTEGER ! The leading dimension of the array A. LDAB >= KD+1. ! ! S (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, S contains the scale factors for A. ! ! SCOND (output) DOUBLE PRECISION ! If INFO = 0, S contains the ratio of the smallest S(i) to ! the largest S(i). If SCOND >= 0.1 and AMAX is neither too ! large nor too small, it is not worth scaling by S. ! ! AMAX (output) DOUBLE PRECISION ! Absolute value of largest matrix element. If AMAX is very ! close to overflow or very close to underflow, the matrix ! should be scaled. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = i, the i-th diagonal element is nonpositive. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER I, J DOUBLE PRECISION SMIN ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KD.LT.0 ) THEN INFO = -3 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPBEQU', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) THEN SCOND = ONE AMAX = ZERO RETURN END IF ! IF( UPPER ) THEN J = KD + 1 ELSE J = 1 END IF ! ! Initialize SMIN and AMAX. ! S( 1 ) = AB( J, 1 ) SMIN = S( 1 ) AMAX = S( 1 ) ! ! Find the minimum and maximum diagonal elements. ! DO 10 I = 2, N S( I ) = AB( J, I ) SMIN = MIN( SMIN, S( I ) ) AMAX = MAX( AMAX, S( I ) ) 10 CONTINUE ! IF( SMIN.LE.ZERO ) THEN ! ! Find the first non-positive diagonal element and return. ! DO 20 I = 1, N IF( S( I ).LE.ZERO ) THEN INFO = I RETURN END IF 20 CONTINUE ELSE ! ! Set the scale factors to the reciprocals ! of the diagonal elements. ! DO 30 I = 1, N S( I ) = ONE / SQRT( S( I ) ) 30 CONTINUE ! ! Compute SCOND = min(S(I)) / max(S(I)) ! SCOND = SQRT( SMIN ) / SQRT( AMAX ) END IF RETURN ! ! End of DPBEQU ! END SUBROUTINE DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, & LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), & BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DPBRFS improves the computed solution to a system of linear ! equations when the coefficient matrix is symmetric positive definite ! and banded, and provides error bounds and backward error estimates ! for the solution. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of superdiagonals of the matrix A if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KD >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! AB (input) DOUBLE PRECISION array, dimension (LDAB,N) ! The upper or lower triangle of the symmetric band matrix A, ! stored in the first KD+1 rows of the array. The j-th column ! of A is stored in the j-th column of the array AB as follows: ! if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KD+1. ! ! AFB (input) DOUBLE PRECISION array, dimension (LDAFB,N) ! The triangular factor U or L from the Cholesky factorization ! A = U**T*U or A = L*L**T of the band matrix A as computed by ! DPBTRF, in the same storage format as A (see AB). ! ! LDAFB (input) INTEGER ! The leading dimension of the array AFB. LDAFB >= KD+1. ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) ! The right hand side matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! On entry, the solution matrix X, as computed by DPBTRS. ! On exit, the improved solution matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Internal Parameters ! =================== ! ! ITMAX is the maximum number of steps of iterative refinement. ! ! ===================================================================== ! ! .. Parameters .. INTEGER ITMAX PARAMETER ( ITMAX = 5 ) DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D+0 ) DOUBLE PRECISION THREE PARAMETER ( THREE = 3.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER COUNT, I, J, K, KASE, L, NZ DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DLACON, DPBTRS, DSBMV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KD.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.0 ) THEN INFO = -4 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -6 ELSE IF( LDAFB.LT.KD+1 ) THEN INFO = -8 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -10 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -12 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPBRFS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN DO 10 J = 1, NRHS FERR( J ) = ZERO BERR( J ) = ZERO 10 CONTINUE RETURN END IF ! ! NZ = maximum number of nonzero elements in each row of A, plus 1 ! NZ = MIN( N+1, 2*KD+2 ) EPS = DLAMCH( 'Epsilon' ) SAFMIN = DLAMCH( 'Safe minimum' ) SAFE1 = NZ*SAFMIN SAFE2 = SAFE1 / EPS ! ! Do for each right hand side ! DO 140 J = 1, NRHS ! COUNT = 1 LSTRES = THREE 20 CONTINUE ! ! Loop until stopping criterion is satisfied. ! ! Compute residual R = B - A * X ! CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) CALL DSBMV( UPLO, N, KD, -ONE, AB, LDAB, X( 1, J ), 1, ONE, & WORK( N+1 ), 1 ) ! ! Compute componentwise relative backward error from formula ! ! max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) ! ! where abs(Z) is the componentwise absolute value of the matrix ! or vector Z. If the i-th component of the denominator is less ! than SAFE2, then SAFE1 is added to the i-th components of the ! numerator and denominator before dividing. ! DO 30 I = 1, N WORK( I ) = ABS( B( I, J ) ) 30 CONTINUE ! ! Compute abs(A)*abs(X) + abs(B). ! IF( UPPER ) THEN DO 50 K = 1, N S = ZERO XK = ABS( X( K, J ) ) L = KD + 1 - K DO 40 I = MAX( 1, K-KD ), K - 1 WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) ) 40 CONTINUE WORK( K ) = WORK( K ) + ABS( AB( KD+1, K ) )*XK + S 50 CONTINUE ELSE DO 70 K = 1, N S = ZERO XK = ABS( X( K, J ) ) WORK( K ) = WORK( K ) + ABS( AB( 1, K ) )*XK L = 1 - K DO 60 I = K + 1, MIN( N, K+KD ) WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) ) 60 CONTINUE WORK( K ) = WORK( K ) + S 70 CONTINUE END IF S = ZERO DO 80 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) ELSE S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / & ( WORK( I )+SAFE1 ) ) END IF 80 CONTINUE BERR( J ) = S ! ! Test stopping criterion. Continue iterating if ! 1) The residual BERR(J) is larger than machine epsilon, and ! 2) BERR(J) decreased by at least a factor of 2 during the ! last iteration, and ! 3) At most ITMAX iterations tried. ! IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. & COUNT.LE.ITMAX ) THEN ! ! Update solution and try again. ! CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N, & INFO ) CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) LSTRES = BERR( J ) COUNT = COUNT + 1 GO TO 20 END IF ! ! Bound error from formula ! ! norm(X - XTRUE) / norm(X) .le. FERR = ! norm( abs(inv(A))* ! ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) ! ! where ! norm(Z) is the magnitude of the largest component of Z ! inv(A) is the inverse of A ! abs(Z) is the componentwise absolute value of the matrix or ! vector Z ! NZ is the maximum number of nonzeros in any row of A, plus 1 ! EPS is machine epsilon ! ! The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) ! is incremented by SAFE1 if the i-th component of ! abs(A)*abs(X) + abs(B) is less than SAFE2. ! ! Use DLACON to estimate the infinity-norm of the matrix ! inv(A) * diag(W), ! where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) ! DO 90 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) ELSE WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 END IF 90 CONTINUE ! KASE = 0 100 CONTINUE CALL DLACON( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), & KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN ! ! Multiply by diag(W)*inv(A'). ! CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N, & INFO ) DO 110 I = 1, N WORK( N+I ) = WORK( N+I )*WORK( I ) 110 CONTINUE ELSE IF( KASE.EQ.2 ) THEN ! ! Multiply by inv(A)*diag(W). ! DO 120 I = 1, N WORK( N+I ) = WORK( N+I )*WORK( I ) 120 CONTINUE CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N, & INFO ) END IF GO TO 100 END IF ! ! Normalize error. ! LSTRES = ZERO DO 130 I = 1, N LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 130 CONTINUE IF( LSTRES.NE.ZERO ) & FERR( J ) = FERR( J ) / LSTRES ! 140 CONTINUE ! RETURN ! ! End of DPBRFS ! END SUBROUTINE DPBSTF( UPLO, N, KD, AB, LDAB, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, KD, LDAB, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ) ! .. ! ! Purpose ! ======= ! ! DPBSTF computes a split Cholesky factorization of a real ! symmetric positive definite band matrix A. ! ! This routine is designed to be used in conjunction with DSBGST. ! ! The factorization has the form A = S**T*S where S is a band matrix ! of the same bandwidth as A and the following structure: ! ! S = ( U ) ! ( M L ) ! ! where U is upper triangular of order m = (n+kd)/2, and L is lower ! triangular of order n-m. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of superdiagonals of the matrix A if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KD >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) ! On entry, the upper or lower triangle of the symmetric band ! matrix A, stored in the first kd+1 rows of the array. The ! j-th column of A is stored in the j-th column of the array AB ! as follows: ! if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). ! ! On exit, if INFO = 0, the factor S from the split Cholesky ! factorization A = S**T*S. See Further Details. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KD+1. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the factorization could not be completed, ! because the updated element a(i,i) was negative; the ! matrix A is not positive definite. ! ! Further Details ! =============== ! ! The band storage scheme is illustrated by the following example, when ! N = 7, KD = 2: ! ! S = ( s11 s12 s13 ) ! ( s22 s23 s24 ) ! ( s33 s34 ) ! ( s44 ) ! ( s53 s54 s55 ) ! ( s64 s65 s66 ) ! ( s75 s76 s77 ) ! ! If UPLO = 'U', the array AB holds: ! ! on entry: on exit: ! ! * * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75 ! * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76 ! a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 ! ! If UPLO = 'L', the array AB holds: ! ! on entry: on exit: ! ! a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 ! a21 a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 * ! a31 a42 a53 a64 a64 * * s13 s24 s53 s64 s75 * * ! ! Array elements marked * are not used by the routine. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER J, KLD, KM, M DOUBLE PRECISION AJJ ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DSCAL, DSYR, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KD.LT.0 ) THEN INFO = -3 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPBSTF', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! KLD = MAX( 1, LDAB-1 ) ! ! Set the splitting point m. ! M = ( N+KD ) / 2 ! IF( UPPER ) THEN ! ! Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m). ! DO 10 J = N, M + 1, -1 ! ! Compute s(j,j) and test for non-positive-definiteness. ! AJJ = AB( KD+1, J ) IF( AJJ.LE.ZERO ) & GO TO 50 AJJ = SQRT( AJJ ) AB( KD+1, J ) = AJJ KM = MIN( J-1, KD ) ! ! Compute elements j-km:j-1 of the j-th column and update the ! the leading submatrix within the band. ! CALL DSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 ) CALL DSYR( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1, & AB( KD+1, J-KM ), KLD ) 10 CONTINUE ! ! Factorize the updated submatrix A(1:m,1:m) as U**T*U. ! DO 20 J = 1, M ! ! Compute s(j,j) and test for non-positive-definiteness. ! AJJ = AB( KD+1, J ) IF( AJJ.LE.ZERO ) & GO TO 50 AJJ = SQRT( AJJ ) AB( KD+1, J ) = AJJ KM = MIN( KD, M-J ) ! ! Compute elements j+1:j+km of the j-th row and update the ! trailing submatrix within the band. ! IF( KM.GT.0 ) THEN CALL DSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD ) CALL DSYR( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD, & AB( KD+1, J+1 ), KLD ) END IF 20 CONTINUE ELSE ! ! Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m). ! DO 30 J = N, M + 1, -1 ! ! Compute s(j,j) and test for non-positive-definiteness. ! AJJ = AB( 1, J ) IF( AJJ.LE.ZERO ) & GO TO 50 AJJ = SQRT( AJJ ) AB( 1, J ) = AJJ KM = MIN( J-1, KD ) ! ! Compute elements j-km:j-1 of the j-th row and update the ! trailing submatrix within the band. ! CALL DSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD ) CALL DSYR( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD, & AB( 1, J-KM ), KLD ) 30 CONTINUE ! ! Factorize the updated submatrix A(1:m,1:m) as U**T*U. ! DO 40 J = 1, M ! ! Compute s(j,j) and test for non-positive-definiteness. ! AJJ = AB( 1, J ) IF( AJJ.LE.ZERO ) & GO TO 50 AJJ = SQRT( AJJ ) AB( 1, J ) = AJJ KM = MIN( KD, M-J ) ! ! Compute elements j+1:j+km of the j-th column and update the ! trailing submatrix within the band. ! IF( KM.GT.0 ) THEN CALL DSCAL( KM, ONE / AJJ, AB( 2, J ), 1 ) CALL DSYR( 'Lower', KM, -ONE, AB( 2, J ), 1, & AB( 1, J+1 ), KLD ) END IF 40 CONTINUE END IF RETURN ! 50 CONTINUE INFO = J RETURN ! ! End of DPBSTF ! END SUBROUTINE DPBSV( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, KD, LDAB, LDB, N, NRHS ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DPBSV computes the solution to a real system of linear equations ! A * X = B, ! where A is an N-by-N symmetric positive definite band matrix and X ! and B are N-by-NRHS matrices. ! ! The Cholesky decomposition is used to factor A as ! A = U**T * U, if UPLO = 'U', or ! A = L * L**T, if UPLO = 'L', ! where U is an upper triangular band matrix, and L is a lower ! triangular band matrix, with the same number of superdiagonals or ! subdiagonals as A. The factored form of A is then used to solve the ! system of equations A * X = B. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The number of linear equations, i.e., the order of the ! matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of superdiagonals of the matrix A if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KD >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) ! On entry, the upper or lower triangle of the symmetric band ! matrix A, stored in the first KD+1 rows of the array. The ! j-th column of A is stored in the j-th column of the array AB ! as follows: ! if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD). ! See below for further details. ! ! On exit, if INFO = 0, the triangular factor U or L from the ! Cholesky factorization A = U**T*U or A = L*L**T of the band ! matrix A, in the same storage format as A. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KD+1. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the N-by-NRHS right hand side matrix B. ! On exit, if INFO = 0, the N-by-NRHS solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the leading minor of order i of A is not ! positive definite, so the factorization could not be ! completed, and the solution has not been computed. ! ! Further Details ! =============== ! ! The band storage scheme is illustrated by the following example, when ! N = 6, KD = 2, and UPLO = 'U': ! ! On entry: On exit: ! ! * * a13 a24 a35 a46 * * u13 u24 u35 u46 ! * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 ! a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 ! ! Similarly, if UPLO = 'L' the format of A is as follows: ! ! On entry: On exit: ! ! a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66 ! a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 * ! a31 a42 a53 a64 * * l31 l42 l53 l64 * * ! ! Array elements marked * are not used by the routine. ! ! ===================================================================== ! ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DPBTRF, DPBTRS, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KD.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.0 ) THEN INFO = -4 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -6 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPBSV ', -INFO ) RETURN END IF ! ! Compute the Cholesky factorization A = U'*U or A = L*L'. ! CALL DPBTRF( UPLO, N, KD, AB, LDAB, INFO ) IF( INFO.EQ.0 ) THEN ! ! Solve the system A*X = B, overwriting B with X. ! CALL DPBTRS( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO ) ! END IF RETURN ! ! End of DPBSV ! END SUBROUTINE DPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, & EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, & WORK, IWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER EQUED, FACT, UPLO INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), & BERR( * ), FERR( * ), S( * ), WORK( * ), & X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to ! compute the solution to a real system of linear equations ! A * X = B, ! where A is an N-by-N symmetric positive definite band matrix and X ! and B are N-by-NRHS matrices. ! ! Error bounds on the solution and a condition estimate are also ! provided. ! ! Description ! =========== ! ! The following steps are performed: ! ! 1. If FACT = 'E', real scaling factors are computed to equilibrate ! the system: ! diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B ! Whether or not the system will be equilibrated depends on the ! scaling of the matrix A, but if equilibration is used, A is ! overwritten by diag(S)*A*diag(S) and B by diag(S)*B. ! ! 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to ! factor the matrix A (after equilibration if FACT = 'E') as ! A = U**T * U, if UPLO = 'U', or ! A = L * L**T, if UPLO = 'L', ! where U is an upper triangular band matrix, and L is a lower ! triangular band matrix. ! ! 3. If the leading i-by-i principal minor is not positive definite, ! then the routine returns with INFO = i. Otherwise, the factored ! form of A is used to estimate the condition number of the matrix ! A. If the reciprocal of the condition number is less than machine ! precision, INFO = N+1 is returned as a warning, but the routine ! still goes on to solve for X and compute error bounds as ! described below. ! ! 4. The system of equations is solved for X using the factored form ! of A. ! ! 5. Iterative refinement is applied to improve the computed solution ! matrix and calculate error bounds and backward error estimates ! for it. ! ! 6. If equilibration was used, the matrix X is premultiplied by ! diag(S) so that it solves the original system before ! equilibration. ! ! Arguments ! ========= ! ! FACT (input) CHARACTER*1 ! Specifies whether or not the factored form of the matrix A is ! supplied on entry, and if not, whether the matrix A should be ! equilibrated before it is factored. ! = 'F': On entry, AFB contains the factored form of A. ! If EQUED = 'Y', the matrix A has been equilibrated ! with scaling factors given by S. AB and AFB will not ! be modified. ! = 'N': The matrix A will be copied to AFB and factored. ! = 'E': The matrix A will be equilibrated if necessary, then ! copied to AFB and factored. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The number of linear equations, i.e., the order of the ! matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of superdiagonals of the matrix A if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KD >= 0. ! ! NRHS (input) INTEGER ! The number of right-hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) ! On entry, the upper or lower triangle of the symmetric band ! matrix A, stored in the first KD+1 rows of the array, except ! if FACT = 'F' and EQUED = 'Y', then A must contain the ! equilibrated matrix diag(S)*A*diag(S). The j-th column of A ! is stored in the j-th column of the array AB as follows: ! if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD). ! See below for further details. ! ! On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by ! diag(S)*A*diag(S). ! ! LDAB (input) INTEGER ! The leading dimension of the array A. LDAB >= KD+1. ! ! AFB (input or output) DOUBLE PRECISION array, dimension (LDAFB,N) ! If FACT = 'F', then AFB is an input argument and on entry ! contains the triangular factor U or L from the Cholesky ! factorization A = U**T*U or A = L*L**T of the band matrix ! A, in the same storage format as A (see AB). If EQUED = 'Y', ! then AFB is the factored form of the equilibrated matrix A. ! ! If FACT = 'N', then AFB is an output argument and on exit ! returns the triangular factor U or L from the Cholesky ! factorization A = U**T*U or A = L*L**T. ! ! If FACT = 'E', then AFB is an output argument and on exit ! returns the triangular factor U or L from the Cholesky ! factorization A = U**T*U or A = L*L**T of the equilibrated ! matrix A (see the description of A for the form of the ! equilibrated matrix). ! ! LDAFB (input) INTEGER ! The leading dimension of the array AFB. LDAFB >= KD+1. ! ! EQUED (input or output) CHARACTER*1 ! Specifies the form of equilibration that was done. ! = 'N': No equilibration (always true if FACT = 'N'). ! = 'Y': Equilibration was done, i.e., A has been replaced by ! diag(S) * A * diag(S). ! EQUED is an input argument if FACT = 'F'; otherwise, it is an ! output argument. ! ! S (input or output) DOUBLE PRECISION array, dimension (N) ! The scale factors for A; not accessed if EQUED = 'N'. S is ! an input argument if FACT = 'F'; otherwise, S is an output ! argument. If FACT = 'F' and EQUED = 'Y', each element of S ! must be positive. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the N-by-NRHS right hand side matrix B. ! On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', ! B is overwritten by diag(S) * B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to ! the original system of equations. Note that if EQUED = 'Y', ! A and B are modified on exit, and the solution to the ! equilibrated system is inv(diag(S))*X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! RCOND (output) DOUBLE PRECISION ! The estimate of the reciprocal condition number of the matrix ! A after equilibration (if done). If RCOND is less than the ! machine precision (in particular, if RCOND = 0), the matrix ! is singular to working precision. This condition is ! indicated by a return code of INFO > 0. ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, and i is ! <= N: the leading minor of order i of A is ! not positive definite, so the factorization ! could not be completed, and the solution has not ! been computed. RCOND = 0 is returned. ! = N+1: U is nonsingular, but RCOND is less than machine ! precision, meaning that the matrix is singular ! to working precision. Nevertheless, the ! solution and error bounds are computed because ! there are a number of situations where the ! computed solution can be more accurate than the ! value of RCOND would suggest. ! ! Further Details ! =============== ! ! The band storage scheme is illustrated by the following example, when ! N = 6, KD = 2, and UPLO = 'U': ! ! Two-dimensional storage of the symmetric matrix A: ! ! a11 a12 a13 ! a22 a23 a24 ! a33 a34 a35 ! a44 a45 a46 ! a55 a56 ! (aij=conjg(aji)) a66 ! ! Band storage of the upper triangle of A: ! ! * * a13 a24 a35 a46 ! * a12 a23 a34 a45 a56 ! a11 a22 a33 a44 a55 a66 ! ! Similarly, if UPLO = 'L' the format of A is as follows: ! ! a11 a22 a33 a44 a55 a66 ! a21 a32 a43 a54 a65 * ! a31 a42 a53 a64 * * ! ! Array elements marked * are not used by the routine. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL EQUIL, NOFACT, RCEQU, UPPER INTEGER I, INFEQU, J, J1, J2 DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANSB EXTERNAL LSAME, DLAMCH, DLANSB ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLACPY, DLAQSB, DPBCON, DPBEQU, DPBRFS, & DPBTRF, DPBTRS, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! INFO = 0 NOFACT = LSAME( FACT, 'N' ) EQUIL = LSAME( FACT, 'E' ) UPPER = LSAME( UPLO, 'U' ) IF( NOFACT .OR. EQUIL ) THEN EQUED = 'N' RCEQU = .FALSE. ELSE RCEQU = LSAME( EQUED, 'Y' ) SMLNUM = DLAMCH( 'Safe minimum' ) BIGNUM = ONE / SMLNUM END IF ! ! Test the input parameters. ! IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) & THEN INFO = -1 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KD.LT.0 ) THEN INFO = -4 ELSE IF( NRHS.LT.0 ) THEN INFO = -5 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -7 ELSE IF( LDAFB.LT.KD+1 ) THEN INFO = -9 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. & ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN INFO = -10 ELSE IF( RCEQU ) THEN SMIN = BIGNUM SMAX = ZERO DO 10 J = 1, N SMIN = MIN( SMIN, S( J ) ) SMAX = MAX( SMAX, S( J ) ) 10 CONTINUE IF( SMIN.LE.ZERO ) THEN INFO = -11 ELSE IF( N.GT.0 ) THEN SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) ELSE SCOND = ONE END IF END IF IF( INFO.EQ.0 ) THEN IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -13 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -15 END IF END IF END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPBSVX', -INFO ) RETURN END IF ! IF( EQUIL ) THEN ! ! Compute row and column scalings to equilibrate the matrix A. ! CALL DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU ) IF( INFEQU.EQ.0 ) THEN ! ! Equilibrate the matrix. ! CALL DLAQSB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED ) RCEQU = LSAME( EQUED, 'Y' ) END IF END IF ! ! Scale the right-hand side. ! IF( RCEQU ) THEN DO 30 J = 1, NRHS DO 20 I = 1, N B( I, J ) = S( I )*B( I, J ) 20 CONTINUE 30 CONTINUE END IF ! IF( NOFACT .OR. EQUIL ) THEN ! ! Compute the Cholesky factorization A = U'*U or A = L*L'. ! IF( UPPER ) THEN DO 40 J = 1, N J1 = MAX( J-KD, 1 ) CALL DCOPY( J-J1+1, AB( KD+1-J+J1, J ), 1, & AFB( KD+1-J+J1, J ), 1 ) 40 CONTINUE ELSE DO 50 J = 1, N J2 = MIN( J+KD, N ) CALL DCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 ) 50 CONTINUE END IF ! CALL DPBTRF( UPLO, N, KD, AFB, LDAFB, INFO ) ! ! Return if INFO is non-zero. ! IF( INFO.NE.0 ) THEN IF( INFO.GT.0 ) & RCOND = ZERO RETURN END IF END IF ! ! Compute the norm of the matrix A. ! ANORM = DLANSB( '1', UPLO, N, KD, AB, LDAB, WORK ) ! ! Compute the reciprocal of the condition number of A. ! CALL DPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, IWORK, & INFO ) ! ! Set INFO = N+1 if the matrix is singular to working precision. ! IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) & INFO = N + 1 ! ! Compute the solution matrix X. ! CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL DPBTRS( UPLO, N, KD, NRHS, AFB, LDAFB, X, LDX, INFO ) ! ! Use iterative refinement to improve the computed solution and ! compute error bounds and backward error estimates for it. ! CALL DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, & LDX, FERR, BERR, WORK, IWORK, INFO ) ! ! Transform the solution matrix X to a solution of the original ! system. ! IF( RCEQU ) THEN DO 70 J = 1, NRHS DO 60 I = 1, N X( I, J ) = S( I )*X( I, J ) 60 CONTINUE 70 CONTINUE DO 80 J = 1, NRHS FERR( J ) = FERR( J ) / SCOND 80 CONTINUE END IF ! RETURN ! ! End of DPBSVX ! END SUBROUTINE DPBTF2( UPLO, N, KD, AB, LDAB, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, KD, LDAB, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ) ! .. ! ! Purpose ! ======= ! ! DPBTF2 computes the Cholesky factorization of a real symmetric ! positive definite band matrix A. ! ! The factorization has the form ! A = U' * U , if UPLO = 'U', or ! A = L * L', if UPLO = 'L', ! where U is an upper triangular matrix, U' is the transpose of U, and ! L is lower triangular. ! ! This is the unblocked version of the algorithm, calling Level 2 BLAS. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the upper or lower triangular part of the ! symmetric matrix A is stored: ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of super-diagonals of the matrix A if UPLO = 'U', ! or the number of sub-diagonals if UPLO = 'L'. KD >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) ! On entry, the upper or lower triangle of the symmetric band ! matrix A, stored in the first KD+1 rows of the array. The ! j-th column of A is stored in the j-th column of the array AB ! as follows: ! if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). ! ! On exit, if INFO = 0, the triangular factor U or L from the ! Cholesky factorization A = U'*U or A = L*L' of the band ! matrix A, in the same storage format as A. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KD+1. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -k, the k-th argument had an illegal value ! > 0: if INFO = k, the leading minor of order k is not ! positive definite, and the factorization could not be ! completed. ! ! Further Details ! =============== ! ! The band storage scheme is illustrated by the following example, when ! N = 6, KD = 2, and UPLO = 'U': ! ! On entry: On exit: ! ! * * a13 a24 a35 a46 * * u13 u24 u35 u46 ! * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 ! a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 ! ! Similarly, if UPLO = 'L' the format of A is as follows: ! ! On entry: On exit: ! ! a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66 ! a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 * ! a31 a42 a53 a64 * * l31 l42 l53 l64 * * ! ! Array elements marked * are not used by the routine. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER J, KLD, KN DOUBLE PRECISION AJJ ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DSCAL, DSYR, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KD.LT.0 ) THEN INFO = -3 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPBTF2', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! KLD = MAX( 1, LDAB-1 ) ! IF( UPPER ) THEN ! ! Compute the Cholesky factorization A = U'*U. ! DO 10 J = 1, N ! ! Compute U(J,J) and test for non-positive-definiteness. ! AJJ = AB( KD+1, J ) IF( AJJ.LE.ZERO ) & GO TO 30 AJJ = SQRT( AJJ ) AB( KD+1, J ) = AJJ ! ! Compute elements J+1:J+KN of row J and update the ! trailing submatrix within the band. ! KN = MIN( KD, N-J ) IF( KN.GT.0 ) THEN CALL DSCAL( KN, ONE / AJJ, AB( KD, J+1 ), KLD ) CALL DSYR( 'Upper', KN, -ONE, AB( KD, J+1 ), KLD, & AB( KD+1, J+1 ), KLD ) END IF 10 CONTINUE ELSE ! ! Compute the Cholesky factorization A = L*L'. ! DO 20 J = 1, N ! ! Compute L(J,J) and test for non-positive-definiteness. ! AJJ = AB( 1, J ) IF( AJJ.LE.ZERO ) & GO TO 30 AJJ = SQRT( AJJ ) AB( 1, J ) = AJJ ! ! Compute elements J+1:J+KN of column J and update the ! trailing submatrix within the band. ! KN = MIN( KD, N-J ) IF( KN.GT.0 ) THEN CALL DSCAL( KN, ONE / AJJ, AB( 2, J ), 1 ) CALL DSYR( 'Lower', KN, -ONE, AB( 2, J ), 1, & AB( 1, J+1 ), KLD ) END IF 20 CONTINUE END IF RETURN ! 30 CONTINUE INFO = J RETURN ! ! End of DPBTF2 ! END SUBROUTINE DPBTRF( UPLO, N, KD, AB, LDAB, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, KD, LDAB, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ) ! .. ! ! Purpose ! ======= ! ! DPBTRF computes the Cholesky factorization of a real symmetric ! positive definite band matrix A. ! ! The factorization has the form ! A = U**T * U, if UPLO = 'U', or ! A = L * L**T, if UPLO = 'L', ! where U is an upper triangular matrix and L is lower triangular. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of superdiagonals of the matrix A if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KD >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) ! On entry, the upper or lower triangle of the symmetric band ! matrix A, stored in the first KD+1 rows of the array. The ! j-th column of A is stored in the j-th column of the array AB ! as follows: ! if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). ! ! On exit, if INFO = 0, the triangular factor U or L from the ! Cholesky factorization A = U**T*U or A = L*L**T of the band ! matrix A, in the same storage format as A. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KD+1. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the leading minor of order i is not ! positive definite, and the factorization could not be ! completed. ! ! Further Details ! =============== ! ! The band storage scheme is illustrated by the following example, when ! N = 6, KD = 2, and UPLO = 'U': ! ! On entry: On exit: ! ! * * a13 a24 a35 a46 * * u13 u24 u35 u46 ! * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 ! a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 ! ! Similarly, if UPLO = 'L' the format of A is as follows: ! ! On entry: On exit: ! ! a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66 ! a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 * ! a31 a42 a53 a64 * * l31 l42 l53 l64 * * ! ! Array elements marked * are not used by the routine. ! ! Contributed by ! Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989 ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) INTEGER NBMAX, LDWORK PARAMETER ( NBMAX = 32, LDWORK = NBMAX+1 ) ! .. ! .. Local Scalars .. INTEGER I, I2, I3, IB, II, J, JJ, NB ! .. ! .. Local Arrays .. DOUBLE PRECISION WORK( LDWORK, NBMAX ) ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DGEMM, DPBTF2, DPOTF2, DSYRK, DTRSM, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF( ( .NOT.LSAME( UPLO, 'U' ) ) .AND. & ( .NOT.LSAME( UPLO, 'L' ) ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KD.LT.0 ) THEN INFO = -3 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPBTRF', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Determine the block size for this environment ! NB = ILAENV( 1, 'DPBTRF', UPLO, N, KD, -1, -1 ) ! ! The block size must not exceed the semi-bandwidth KD, and must not ! exceed the limit set by the size of the local array WORK. ! NB = MIN( NB, NBMAX ) ! IF( NB.LE.1 .OR. NB.GT.KD ) THEN ! ! Use unblocked code ! CALL DPBTF2( UPLO, N, KD, AB, LDAB, INFO ) ELSE ! ! Use blocked code ! IF( LSAME( UPLO, 'U' ) ) THEN ! ! Compute the Cholesky factorization of a symmetric band ! matrix, given the upper triangle of the matrix in band ! storage. ! ! Zero the upper triangle of the work array. ! DO 20 J = 1, NB DO 10 I = 1, J - 1 WORK( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ! ! Process the band matrix one diagonal block at a time. ! DO 70 I = 1, N, NB IB = MIN( NB, N-I+1 ) ! ! Factorize the diagonal block ! CALL DPOTF2( UPLO, IB, AB( KD+1, I ), LDAB-1, II ) IF( II.NE.0 ) THEN INFO = I + II - 1 GO TO 150 END IF IF( I+IB.LE.N ) THEN ! ! Update the relevant part of the trailing submatrix. ! If A11 denotes the diagonal block which has just been ! factorized, then we need to update the remaining ! blocks in the diagram: ! ! A11 A12 A13 ! A22 A23 ! A33 ! ! The numbers of rows and columns in the partitioning ! are IB, I2, I3 respectively. The blocks A12, A22 and ! A23 are empty if IB = KD. The upper triangle of A13 ! lies outside the band. ! I2 = MIN( KD-IB, N-I-IB+1 ) I3 = MIN( IB, N-I-KD+1 ) ! IF( I2.GT.0 ) THEN ! ! Update A12 ! CALL DTRSM( 'Left', 'Upper', 'Transpose', & 'Non-unit', IB, I2, ONE, AB( KD+1, I ), & LDAB-1, AB( KD+1-IB, I+IB ), LDAB-1 ) ! ! Update A22 ! CALL DSYRK( 'Upper', 'Transpose', I2, IB, -ONE, & AB( KD+1-IB, I+IB ), LDAB-1, ONE, & AB( KD+1, I+IB ), LDAB-1 ) END IF ! IF( I3.GT.0 ) THEN ! ! Copy the lower triangle of A13 into the work array. ! DO 40 JJ = 1, I3 DO 30 II = JJ, IB WORK( II, JJ ) = AB( II-JJ+1, JJ+I+KD-1 ) 30 CONTINUE 40 CONTINUE ! ! Update A13 (in the work array). ! CALL DTRSM( 'Left', 'Upper', 'Transpose', & 'Non-unit', IB, I3, ONE, AB( KD+1, I ), & LDAB-1, WORK, LDWORK ) ! ! Update A23 ! IF( I2.GT.0 ) & CALL DGEMM( 'Transpose', 'No Transpose', I2, I3, & IB, -ONE, AB( KD+1-IB, I+IB ), & LDAB-1, WORK, LDWORK, ONE, & AB( 1+IB, I+KD ), LDAB-1 ) ! ! Update A33 ! CALL DSYRK( 'Upper', 'Transpose', I3, IB, -ONE, & WORK, LDWORK, ONE, AB( KD+1, I+KD ), & LDAB-1 ) ! ! Copy the lower triangle of A13 back into place. ! DO 60 JJ = 1, I3 DO 50 II = JJ, IB AB( II-JJ+1, JJ+I+KD-1 ) = WORK( II, JJ ) 50 CONTINUE 60 CONTINUE END IF END IF 70 CONTINUE ELSE ! ! Compute the Cholesky factorization of a symmetric band ! matrix, given the lower triangle of the matrix in band ! storage. ! ! Zero the lower triangle of the work array. ! DO 90 J = 1, NB DO 80 I = J + 1, NB WORK( I, J ) = ZERO 80 CONTINUE 90 CONTINUE ! ! Process the band matrix one diagonal block at a time. ! DO 140 I = 1, N, NB IB = MIN( NB, N-I+1 ) ! ! Factorize the diagonal block ! CALL DPOTF2( UPLO, IB, AB( 1, I ), LDAB-1, II ) IF( II.NE.0 ) THEN INFO = I + II - 1 GO TO 150 END IF IF( I+IB.LE.N ) THEN ! ! Update the relevant part of the trailing submatrix. ! If A11 denotes the diagonal block which has just been ! factorized, then we need to update the remaining ! blocks in the diagram: ! ! A11 ! A21 A22 ! A31 A32 A33 ! ! The numbers of rows and columns in the partitioning ! are IB, I2, I3 respectively. The blocks A21, A22 and ! A32 are empty if IB = KD. The lower triangle of A31 ! lies outside the band. ! I2 = MIN( KD-IB, N-I-IB+1 ) I3 = MIN( IB, N-I-KD+1 ) ! IF( I2.GT.0 ) THEN ! ! Update A21 ! CALL DTRSM( 'Right', 'Lower', 'Transpose', & 'Non-unit', I2, IB, ONE, AB( 1, I ), & LDAB-1, AB( 1+IB, I ), LDAB-1 ) ! ! Update A22 ! CALL DSYRK( 'Lower', 'No Transpose', I2, IB, -ONE, & AB( 1+IB, I ), LDAB-1, ONE, & AB( 1, I+IB ), LDAB-1 ) END IF ! IF( I3.GT.0 ) THEN ! ! Copy the upper triangle of A31 into the work array. ! DO 110 JJ = 1, IB DO 100 II = 1, MIN( JJ, I3 ) WORK( II, JJ ) = AB( KD+1-JJ+II, JJ+I-1 ) 100 CONTINUE 110 CONTINUE ! ! Update A31 (in the work array). ! CALL DTRSM( 'Right', 'Lower', 'Transpose', & 'Non-unit', I3, IB, ONE, AB( 1, I ), & LDAB-1, WORK, LDWORK ) ! ! Update A32 ! IF( I2.GT.0 ) & CALL DGEMM( 'No transpose', 'Transpose', I3, I2, & IB, -ONE, WORK, LDWORK, & AB( 1+IB, I ), LDAB-1, ONE, & AB( 1+KD-IB, I+IB ), LDAB-1 ) ! ! Update A33 ! CALL DSYRK( 'Lower', 'No Transpose', I3, IB, -ONE, & WORK, LDWORK, ONE, AB( 1, I+KD ), & LDAB-1 ) ! ! Copy the upper triangle of A31 back into place. ! DO 130 JJ = 1, IB DO 120 II = 1, MIN( JJ, I3 ) AB( KD+1-JJ+II, JJ+I-1 ) = WORK( II, JJ ) 120 CONTINUE 130 CONTINUE END IF END IF 140 CONTINUE END IF END IF RETURN ! 150 CONTINUE RETURN ! ! End of DPBTRF ! END SUBROUTINE DPBTRS( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, KD, LDAB, LDB, N, NRHS ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DPBTRS solves a system of linear equations A*X = B with a symmetric ! positive definite band matrix A using the Cholesky factorization ! A = U**T*U or A = L*L**T computed by DPBTRF. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangular factor stored in AB; ! = 'L': Lower triangular factor stored in AB. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of superdiagonals of the matrix A if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KD >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! AB (input) DOUBLE PRECISION array, dimension (LDAB,N) ! The triangular factor U or L from the Cholesky factorization ! A = U**T*U or A = L*L**T of the band matrix A, stored in the ! first KD+1 rows of the array. The j-th column of U or L is ! stored in the j-th column of the array AB as follows: ! if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j; ! if UPLO ='L', AB(1+i-j,j) = L(i,j) for j<=i<=min(n,j+kd). ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KD+1. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the right hand side matrix B. ! On exit, the solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL UPPER INTEGER J ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DTBSV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( KD.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.0 ) THEN INFO = -4 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -6 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPBTRS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) & RETURN ! IF( UPPER ) THEN ! ! Solve A*X = B where A = U'*U. ! DO 10 J = 1, NRHS ! ! Solve U'*X = B, overwriting B with X. ! CALL DTBSV( 'Upper', 'Transpose', 'Non-unit', N, KD, AB, & LDAB, B( 1, J ), 1 ) ! ! Solve U*X = B, overwriting B with X. ! CALL DTBSV( 'Upper', 'No transpose', 'Non-unit', N, KD, AB, & LDAB, B( 1, J ), 1 ) 10 CONTINUE ELSE ! ! Solve A*X = B where A = L*L'. ! DO 20 J = 1, NRHS ! ! Solve L*X = B, overwriting B with X. ! CALL DTBSV( 'Lower', 'No transpose', 'Non-unit', N, KD, AB, & LDAB, B( 1, J ), 1 ) ! ! Solve L'*X = B, overwriting B with X. ! CALL DTBSV( 'Lower', 'Transpose', 'Non-unit', N, KD, AB, & LDAB, B( 1, J ), 1 ) 20 CONTINUE END IF ! RETURN ! ! End of DPBTRS ! END SUBROUTINE DPOCON( UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N DOUBLE PRECISION ANORM, RCOND ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DPOCON estimates the reciprocal of the condition number (in the ! 1-norm) of a real symmetric positive definite matrix using the ! Cholesky factorization A = U**T*U or A = L*L**T computed by DPOTRF. ! ! An estimate is obtained for norm(inv(A)), and the reciprocal of the ! condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The triangular factor U or L from the Cholesky factorization ! A = U**T*U or A = L*L**T, as computed by DPOTRF. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! ANORM (input) DOUBLE PRECISION ! The 1-norm (or infinity-norm) of the symmetric matrix A. ! ! RCOND (output) DOUBLE PRECISION ! The reciprocal of the condition number of the matrix A, ! computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an ! estimate of the 1-norm of inv(A) computed in this routine. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER CHARACTER NORMIN INTEGER IX, KASE DOUBLE PRECISION AINVNM, SCALE, SCALEL, SCALEU, SMLNUM ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DLAMCH EXTERNAL LSAME, IDAMAX, DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DLACON, DLATRS, DRSCL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( ANORM.LT.ZERO ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPOCON', -INFO ) RETURN END IF ! ! Quick return if possible ! RCOND = ZERO IF( N.EQ.0 ) THEN RCOND = ONE RETURN ELSE IF( ANORM.EQ.ZERO ) THEN RETURN END IF ! SMLNUM = DLAMCH( 'Safe minimum' ) ! ! Estimate the 1-norm of inv(A). ! KASE = 0 NORMIN = 'N' 10 CONTINUE CALL DLACON( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE ) IF( KASE.NE.0 ) THEN IF( UPPER ) THEN ! ! Multiply by inv(U'). ! CALL DLATRS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N, A, & LDA, WORK, SCALEL, WORK( 2*N+1 ), INFO ) NORMIN = 'Y' ! ! Multiply by inv(U). ! CALL DLATRS( 'Upper', 'No transpose', 'Non-unit', NORMIN, N, & A, LDA, WORK, SCALEU, WORK( 2*N+1 ), INFO ) ELSE ! ! Multiply by inv(L). ! CALL DLATRS( 'Lower', 'No transpose', 'Non-unit', NORMIN, N, & A, LDA, WORK, SCALEL, WORK( 2*N+1 ), INFO ) NORMIN = 'Y' ! ! Multiply by inv(L'). ! CALL DLATRS( 'Lower', 'Transpose', 'Non-unit', NORMIN, N, A, & LDA, WORK, SCALEU, WORK( 2*N+1 ), INFO ) END IF ! ! Multiply by 1/SCALE if doing so will not cause overflow. ! SCALE = SCALEL*SCALEU IF( SCALE.NE.ONE ) THEN IX = IDAMAX( N, WORK, 1 ) IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO ) & GO TO 20 CALL DRSCL( N, SCALE, WORK, 1 ) END IF GO TO 10 END IF ! ! Compute the estimate of the reciprocal condition number. ! IF( AINVNM.NE.ZERO ) & RCOND = ( ONE / AINVNM ) / ANORM ! 20 CONTINUE RETURN ! ! End of DPOCON ! END SUBROUTINE DPOEQU( N, A, LDA, S, SCOND, AMAX, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, N DOUBLE PRECISION AMAX, SCOND ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), S( * ) ! .. ! ! Purpose ! ======= ! ! DPOEQU computes row and column scalings intended to equilibrate a ! symmetric positive definite matrix A and reduce its condition number ! (with respect to the two-norm). S contains the scale factors, ! S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with ! elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This ! choice of S puts the condition number of B within a factor N of the ! smallest possible condition number over all possible diagonal ! scalings. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The N-by-N symmetric positive definite matrix whose scaling ! factors are to be computed. Only the diagonal elements of A ! are referenced. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! S (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, S contains the scale factors for A. ! ! SCOND (output) DOUBLE PRECISION ! If INFO = 0, S contains the ratio of the smallest S(i) to ! the largest S(i). If SCOND >= 0.1 and AMAX is neither too ! large nor too small, it is not worth scaling by S. ! ! AMAX (output) DOUBLE PRECISION ! Absolute value of largest matrix element. If AMAX is very ! close to overflow or very close to underflow, the matrix ! should be scaled. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the i-th diagonal element is nonpositive. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I DOUBLE PRECISION SMIN ! .. ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF( N.LT.0 ) THEN INFO = -1 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -3 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPOEQU', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) THEN SCOND = ONE AMAX = ZERO RETURN END IF ! ! Find the minimum and maximum diagonal elements. ! S( 1 ) = A( 1, 1 ) SMIN = S( 1 ) AMAX = S( 1 ) DO 10 I = 2, N S( I ) = A( I, I ) SMIN = MIN( SMIN, S( I ) ) AMAX = MAX( AMAX, S( I ) ) 10 CONTINUE ! IF( SMIN.LE.ZERO ) THEN ! ! Find the first non-positive diagonal element and return. ! DO 20 I = 1, N IF( S( I ).LE.ZERO ) THEN INFO = I RETURN END IF 20 CONTINUE ELSE ! ! Set the scale factors to the reciprocals ! of the diagonal elements. ! DO 30 I = 1, N S( I ) = ONE / SQRT( S( I ) ) 30 CONTINUE ! ! Compute SCOND = min(S(I)) / max(S(I)) ! SCOND = SQRT( SMIN ) / SQRT( AMAX ) END IF RETURN ! ! End of DPOEQU ! END SUBROUTINE DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, & LDX, FERR, BERR, WORK, IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), & BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DPORFS improves the computed solution to a system of linear ! equations when the coefficient matrix is symmetric positive definite, ! and provides error bounds and backward error estimates for the ! solution. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The symmetric matrix A. If UPLO = 'U', the leading N-by-N ! upper triangular part of A contains the upper triangular part ! of the matrix A, and the strictly lower triangular part of A ! is not referenced. If UPLO = 'L', the leading N-by-N lower ! triangular part of A contains the lower triangular part of ! the matrix A, and the strictly upper triangular part of A is ! not referenced. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! AF (input) DOUBLE PRECISION array, dimension (LDAF,N) ! The triangular factor U or L from the Cholesky factorization ! A = U**T*U or A = L*L**T, as computed by DPOTRF. ! ! LDAF (input) INTEGER ! The leading dimension of the array AF. LDAF >= max(1,N). ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) ! The right hand side matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! On entry, the solution matrix X, as computed by DPOTRS. ! On exit, the improved solution matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Internal Parameters ! =================== ! ! ITMAX is the maximum number of steps of iterative refinement. ! ! ===================================================================== ! ! .. Parameters .. INTEGER ITMAX PARAMETER ( ITMAX = 5 ) DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D+0 ) DOUBLE PRECISION THREE PARAMETER ( THREE = 3.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER COUNT, I, J, K, KASE, NZ DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DLACON, DPOTRS, DSYMV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -11 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPORFS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN DO 10 J = 1, NRHS FERR( J ) = ZERO BERR( J ) = ZERO 10 CONTINUE RETURN END IF ! ! NZ = maximum number of nonzero elements in each row of A, plus 1 ! NZ = N + 1 EPS = DLAMCH( 'Epsilon' ) SAFMIN = DLAMCH( 'Safe minimum' ) SAFE1 = NZ*SAFMIN SAFE2 = SAFE1 / EPS ! ! Do for each right hand side ! DO 140 J = 1, NRHS ! COUNT = 1 LSTRES = THREE 20 CONTINUE ! ! Loop until stopping criterion is satisfied. ! ! Compute residual R = B - A * X ! CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, & WORK( N+1 ), 1 ) ! ! Compute componentwise relative backward error from formula ! ! max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) ! ! where abs(Z) is the componentwise absolute value of the matrix ! or vector Z. If the i-th component of the denominator is less ! than SAFE2, then SAFE1 is added to the i-th components of the ! numerator and denominator before dividing. ! DO 30 I = 1, N WORK( I ) = ABS( B( I, J ) ) 30 CONTINUE ! ! Compute abs(A)*abs(X) + abs(B). ! IF( UPPER ) THEN DO 50 K = 1, N S = ZERO XK = ABS( X( K, J ) ) DO 40 I = 1, K - 1 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 40 CONTINUE WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S 50 CONTINUE ELSE DO 70 K = 1, N S = ZERO XK = ABS( X( K, J ) ) WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK DO 60 I = K + 1, N WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 60 CONTINUE WORK( K ) = WORK( K ) + S 70 CONTINUE END IF S = ZERO DO 80 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) ELSE S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / & ( WORK( I )+SAFE1 ) ) END IF 80 CONTINUE BERR( J ) = S ! ! Test stopping criterion. Continue iterating if ! 1) The residual BERR(J) is larger than machine epsilon, and ! 2) BERR(J) decreased by at least a factor of 2 during the ! last iteration, and ! 3) At most ITMAX iterations tried. ! IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. & COUNT.LE.ITMAX ) THEN ! ! Update solution and try again. ! CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO ) CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) LSTRES = BERR( J ) COUNT = COUNT + 1 GO TO 20 END IF ! ! Bound error from formula ! ! norm(X - XTRUE) / norm(X) .le. FERR = ! norm( abs(inv(A))* ! ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) ! ! where ! norm(Z) is the magnitude of the largest component of Z ! inv(A) is the inverse of A ! abs(Z) is the componentwise absolute value of the matrix or ! vector Z ! NZ is the maximum number of nonzeros in any row of A, plus 1 ! EPS is machine epsilon ! ! The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) ! is incremented by SAFE1 if the i-th component of ! abs(A)*abs(X) + abs(B) is less than SAFE2. ! ! Use DLACON to estimate the infinity-norm of the matrix ! inv(A) * diag(W), ! where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) ! DO 90 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) ELSE WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 END IF 90 CONTINUE ! KASE = 0 100 CONTINUE CALL DLACON( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), & KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN ! ! Multiply by diag(W)*inv(A'). ! CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO ) DO 110 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 110 CONTINUE ELSE IF( KASE.EQ.2 ) THEN ! ! Multiply by inv(A)*diag(W). ! DO 120 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 120 CONTINUE CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO ) END IF GO TO 100 END IF ! ! Normalize error. ! LSTRES = ZERO DO 130 I = 1, N LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 130 CONTINUE IF( LSTRES.NE.ZERO ) & FERR( J ) = FERR( J ) / LSTRES ! 140 CONTINUE ! RETURN ! ! End of DPORFS ! END SUBROUTINE DPOSV( UPLO, N, NRHS, A, LDA, B, LDB, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, LDB, N, NRHS ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DPOSV computes the solution to a real system of linear equations ! A * X = B, ! where A is an N-by-N symmetric positive definite matrix and X and B ! are N-by-NRHS matrices. ! ! The Cholesky decomposition is used to factor A as ! A = U**T* U, if UPLO = 'U', or ! A = L * L**T, if UPLO = 'L', ! where U is an upper triangular matrix and L is a lower triangular ! matrix. The factored form of A is then used to solve the system of ! equations A * X = B. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The number of linear equations, i.e., the order of the ! matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the symmetric matrix A. If UPLO = 'U', the leading ! N-by-N upper triangular part of A contains the upper ! triangular part of the matrix A, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading N-by-N lower triangular part of A contains the lower ! triangular part of the matrix A, and the strictly upper ! triangular part of A is not referenced. ! ! On exit, if INFO = 0, the factor U or L from the Cholesky ! factorization A = U**T*U or A = L*L**T. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the N-by-NRHS right hand side matrix B. ! On exit, if INFO = 0, the N-by-NRHS solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the leading minor of order i of A is not ! positive definite, so the factorization could not be ! completed, and the solution has not been computed. ! ! ===================================================================== ! ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DPOTRF, DPOTRS, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPOSV ', -INFO ) RETURN END IF ! ! Compute the Cholesky factorization A = U'*U or A = L*L'. ! CALL DPOTRF( UPLO, N, A, LDA, INFO ) IF( INFO.EQ.0 ) THEN ! ! Solve the system A*X = B, overwriting B with X. ! CALL DPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO ) ! END IF RETURN ! ! End of DPOSV ! END SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, & S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, & IWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER EQUED, FACT, UPLO INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), & BERR( * ), FERR( * ), S( * ), WORK( * ), & X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to ! compute the solution to a real system of linear equations ! A * X = B, ! where A is an N-by-N symmetric positive definite matrix and X and B ! are N-by-NRHS matrices. ! ! Error bounds on the solution and a condition estimate are also ! provided. ! ! Description ! =========== ! ! The following steps are performed: ! ! 1. If FACT = 'E', real scaling factors are computed to equilibrate ! the system: ! diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B ! Whether or not the system will be equilibrated depends on the ! scaling of the matrix A, but if equilibration is used, A is ! overwritten by diag(S)*A*diag(S) and B by diag(S)*B. ! ! 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to ! factor the matrix A (after equilibration if FACT = 'E') as ! A = U**T* U, if UPLO = 'U', or ! A = L * L**T, if UPLO = 'L', ! where U is an upper triangular matrix and L is a lower triangular ! matrix. ! ! 3. If the leading i-by-i principal minor is not positive definite, ! then the routine returns with INFO = i. Otherwise, the factored ! form of A is used to estimate the condition number of the matrix ! A. If the reciprocal of the condition number is less than machine ! precision, INFO = N+1 is returned as a warning, but the routine ! still goes on to solve for X and compute error bounds as ! described below. ! ! 4. The system of equations is solved for X using the factored form ! of A. ! ! 5. Iterative refinement is applied to improve the computed solution ! matrix and calculate error bounds and backward error estimates ! for it. ! ! 6. If equilibration was used, the matrix X is premultiplied by ! diag(S) so that it solves the original system before ! equilibration. ! ! Arguments ! ========= ! ! FACT (input) CHARACTER*1 ! Specifies whether or not the factored form of the matrix A is ! supplied on entry, and if not, whether the matrix A should be ! equilibrated before it is factored. ! = 'F': On entry, AF contains the factored form of A. ! If EQUED = 'Y', the matrix A has been equilibrated ! with scaling factors given by S. A and AF will not ! be modified. ! = 'N': The matrix A will be copied to AF and factored. ! = 'E': The matrix A will be equilibrated if necessary, then ! copied to AF and factored. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The number of linear equations, i.e., the order of the ! matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the symmetric matrix A, except if FACT = 'F' and ! EQUED = 'Y', then A must contain the equilibrated matrix ! diag(S)*A*diag(S). If UPLO = 'U', the leading ! N-by-N upper triangular part of A contains the upper ! triangular part of the matrix A, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading N-by-N lower triangular part of A contains the lower ! triangular part of the matrix A, and the strictly upper ! triangular part of A is not referenced. A is not modified if ! FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. ! ! On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by ! diag(S)*A*diag(S). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N) ! If FACT = 'F', then AF is an input argument and on entry ! contains the triangular factor U or L from the Cholesky ! factorization A = U**T*U or A = L*L**T, in the same storage ! format as A. If EQUED .ne. 'N', then AF is the factored form ! of the equilibrated matrix diag(S)*A*diag(S). ! ! If FACT = 'N', then AF is an output argument and on exit ! returns the triangular factor U or L from the Cholesky ! factorization A = U**T*U or A = L*L**T of the original ! matrix A. ! ! If FACT = 'E', then AF is an output argument and on exit ! returns the triangular factor U or L from the Cholesky ! factorization A = U**T*U or A = L*L**T of the equilibrated ! matrix A (see the description of A for the form of the ! equilibrated matrix). ! ! LDAF (input) INTEGER ! The leading dimension of the array AF. LDAF >= max(1,N). ! ! EQUED (input or output) CHARACTER*1 ! Specifies the form of equilibration that was done. ! = 'N': No equilibration (always true if FACT = 'N'). ! = 'Y': Equilibration was done, i.e., A has been replaced by ! diag(S) * A * diag(S). ! EQUED is an input argument if FACT = 'F'; otherwise, it is an ! output argument. ! ! S (input or output) DOUBLE PRECISION array, dimension (N) ! The scale factors for A; not accessed if EQUED = 'N'. S is ! an input argument if FACT = 'F'; otherwise, S is an output ! argument. If FACT = 'F' and EQUED = 'Y', each element of S ! must be positive. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the N-by-NRHS right hand side matrix B. ! On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', ! B is overwritten by diag(S) * B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to ! the original system of equations. Note that if EQUED = 'Y', ! A and B are modified on exit, and the solution to the ! equilibrated system is inv(diag(S))*X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! RCOND (output) DOUBLE PRECISION ! The estimate of the reciprocal condition number of the matrix ! A after equilibration (if done). If RCOND is less than the ! machine precision (in particular, if RCOND = 0), the matrix ! is singular to working precision. This condition is ! indicated by a return code of INFO > 0. ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, and i is ! <= N: the leading minor of order i of A is ! not positive definite, so the factorization ! could not be completed, and the solution has not ! been computed. RCOND = 0 is returned. ! = N+1: U is nonsingular, but RCOND is less than machine ! precision, meaning that the matrix is singular ! to working precision. Nevertheless, the ! solution and error bounds are computed because ! there are a number of situations where the ! computed solution can be more accurate than the ! value of RCOND would suggest. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL EQUIL, NOFACT, RCEQU INTEGER I, INFEQU, J DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANSY EXTERNAL LSAME, DLAMCH, DLANSY ! .. ! .. External Subroutines .. EXTERNAL DLACPY, DLAQSY, DPOCON, DPOEQU, DPORFS, DPOTRF, & DPOTRS, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! INFO = 0 NOFACT = LSAME( FACT, 'N' ) EQUIL = LSAME( FACT, 'E' ) IF( NOFACT .OR. EQUIL ) THEN EQUED = 'N' RCEQU = .FALSE. ELSE RCEQU = LSAME( EQUED, 'Y' ) SMLNUM = DLAMCH( 'Safe minimum' ) BIGNUM = ONE / SMLNUM END IF ! ! Test the input parameters. ! IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) & THEN INFO = -1 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) & THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. & ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN INFO = -9 ELSE IF( RCEQU ) THEN SMIN = BIGNUM SMAX = ZERO DO 10 J = 1, N SMIN = MIN( SMIN, S( J ) ) SMAX = MAX( SMAX, S( J ) ) 10 CONTINUE IF( SMIN.LE.ZERO ) THEN INFO = -10 ELSE IF( N.GT.0 ) THEN SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) ELSE SCOND = ONE END IF END IF IF( INFO.EQ.0 ) THEN IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -12 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -14 END IF END IF END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPOSVX', -INFO ) RETURN END IF ! IF( EQUIL ) THEN ! ! Compute row and column scalings to equilibrate the matrix A. ! CALL DPOEQU( N, A, LDA, S, SCOND, AMAX, INFEQU ) IF( INFEQU.EQ.0 ) THEN ! ! Equilibrate the matrix. ! CALL DLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) RCEQU = LSAME( EQUED, 'Y' ) END IF END IF ! ! Scale the right hand side. ! IF( RCEQU ) THEN DO 30 J = 1, NRHS DO 20 I = 1, N B( I, J ) = S( I )*B( I, J ) 20 CONTINUE 30 CONTINUE END IF ! IF( NOFACT .OR. EQUIL ) THEN ! ! Compute the Cholesky factorization A = U'*U or A = L*L'. ! CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF ) CALL DPOTRF( UPLO, N, AF, LDAF, INFO ) ! ! Return if INFO is non-zero. ! IF( INFO.NE.0 ) THEN IF( INFO.GT.0 ) & RCOND = ZERO RETURN END IF END IF ! ! Compute the norm of the matrix A. ! ANORM = DLANSY( '1', UPLO, N, A, LDA, WORK ) ! ! Compute the reciprocal of the condition number of A. ! CALL DPOCON( UPLO, N, AF, LDAF, ANORM, RCOND, WORK, IWORK, INFO ) ! ! Set INFO = N+1 if the matrix is singular to working precision. ! IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) & INFO = N + 1 ! ! Compute the solution matrix X. ! CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL DPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO ) ! ! Use iterative refinement to improve the computed solution and ! compute error bounds and backward error estimates for it. ! CALL DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, & FERR, BERR, WORK, IWORK, INFO ) ! ! Transform the solution matrix X to a solution of the original ! system. ! IF( RCEQU ) THEN DO 50 J = 1, NRHS DO 40 I = 1, N X( I, J ) = S( I )*X( I, J ) 40 CONTINUE 50 CONTINUE DO 60 J = 1, NRHS FERR( J ) = FERR( J ) / SCOND 60 CONTINUE END IF ! RETURN ! ! End of DPOSVX ! END SUBROUTINE DPOTF2( UPLO, N, A, LDA, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ) ! .. ! ! Purpose ! ======= ! ! DPOTF2 computes the Cholesky factorization of a real symmetric ! positive definite matrix A. ! ! The factorization has the form ! A = U' * U , if UPLO = 'U', or ! A = L * L', if UPLO = 'L', ! where U is an upper triangular matrix and L is lower triangular. ! ! This is the unblocked version of the algorithm, calling Level 2 BLAS. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the upper or lower triangular part of the ! symmetric matrix A is stored. ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the symmetric matrix A. If UPLO = 'U', the leading ! n by n upper triangular part of A contains the upper ! triangular part of the matrix A, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading n by n lower triangular part of A contains the lower ! triangular part of the matrix A, and the strictly upper ! triangular part of A is not referenced. ! ! On exit, if INFO = 0, the factor U or L from the Cholesky ! factorization A = U'*U or A = L*L'. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -k, the k-th argument had an illegal value ! > 0: if INFO = k, the leading minor of order k is not ! positive definite, and the factorization could not be ! completed. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER J DOUBLE PRECISION AJJ ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT EXTERNAL LSAME, DDOT ! .. ! .. External Subroutines .. EXTERNAL DGEMV, DSCAL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPOTF2', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( UPPER ) THEN ! ! Compute the Cholesky factorization A = U'*U. ! DO 10 J = 1, N ! ! Compute U(J,J) and test for non-positive-definiteness. ! AJJ = A( J, J ) - DDOT( J-1, A( 1, J ), 1, A( 1, J ), 1 ) IF( AJJ.LE.ZERO ) THEN A( J, J ) = AJJ GO TO 30 END IF AJJ = SQRT( AJJ ) A( J, J ) = AJJ ! ! Compute elements J+1:N of row J. ! IF( J.LT.N ) THEN CALL DGEMV( 'Transpose', J-1, N-J, -ONE, A( 1, J+1 ), & LDA, A( 1, J ), 1, ONE, A( J, J+1 ), LDA ) CALL DSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA ) END IF 10 CONTINUE ELSE ! ! Compute the Cholesky factorization A = L*L'. ! DO 20 J = 1, N ! ! Compute L(J,J) and test for non-positive-definiteness. ! AJJ = A( J, J ) - DDOT( J-1, A( J, 1 ), LDA, A( J, 1 ), & LDA ) IF( AJJ.LE.ZERO ) THEN A( J, J ) = AJJ GO TO 30 END IF AJJ = SQRT( AJJ ) A( J, J ) = AJJ ! ! Compute elements J+1:N of column J. ! IF( J.LT.N ) THEN CALL DGEMV( 'No transpose', N-J, J-1, -ONE, A( J+1, 1 ), & LDA, A( J, 1 ), LDA, ONE, A( J+1, J ), 1 ) CALL DSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 ) END IF 20 CONTINUE END IF GO TO 40 ! 30 CONTINUE INFO = J ! 40 CONTINUE RETURN ! ! End of DPOTF2 ! END SUBROUTINE DPOTRF( UPLO, N, A, LDA, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ) ! .. ! ! Purpose ! ======= ! ! DPOTRF computes the Cholesky factorization of a real symmetric ! positive definite matrix A. ! ! The factorization has the form ! A = U**T * U, if UPLO = 'U', or ! A = L * L**T, if UPLO = 'L', ! where U is an upper triangular matrix and L is lower triangular. ! ! This is the block version of the algorithm, calling Level 3 BLAS. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the symmetric matrix A. If UPLO = 'U', the leading ! N-by-N upper triangular part of A contains the upper ! triangular part of the matrix A, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading N-by-N lower triangular part of A contains the lower ! triangular part of the matrix A, and the strictly upper ! triangular part of A is not referenced. ! ! On exit, if INFO = 0, the factor U or L from the Cholesky ! factorization A = U**T*U or A = L*L**T. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the leading minor of order i is not ! positive definite, and the factorization could not be ! completed. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER J, JB, NB ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DGEMM, DPOTF2, DSYRK, DTRSM, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPOTRF', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Determine the block size for this environment. ! NB = ILAENV( 1, 'DPOTRF', UPLO, N, -1, -1, -1 ) IF( NB.LE.1 .OR. NB.GE.N ) THEN ! ! Use unblocked code. ! CALL DPOTF2( UPLO, N, A, LDA, INFO ) ELSE ! ! Use blocked code. ! IF( UPPER ) THEN ! ! Compute the Cholesky factorization A = U'*U. ! DO 10 J = 1, N, NB ! ! Update and factorize the current diagonal block and test ! for non-positive-definiteness. ! JB = MIN( NB, N-J+1 ) CALL DSYRK( 'Upper', 'Transpose', JB, J-1, -ONE, & A( 1, J ), LDA, ONE, A( J, J ), LDA ) CALL DPOTF2( 'Upper', JB, A( J, J ), LDA, INFO ) IF( INFO.NE.0 ) & GO TO 30 IF( J+JB.LE.N ) THEN ! ! Compute the current block row. ! CALL DGEMM( 'Transpose', 'No transpose', JB, N-J-JB+1, & J-1, -ONE, A( 1, J ), LDA, A( 1, J+JB ), & LDA, ONE, A( J, J+JB ), LDA ) CALL DTRSM( 'Left', 'Upper', 'Transpose', 'Non-unit', & JB, N-J-JB+1, ONE, A( J, J ), LDA, & A( J, J+JB ), LDA ) END IF 10 CONTINUE ! ELSE ! ! Compute the Cholesky factorization A = L*L'. ! DO 20 J = 1, N, NB ! ! Update and factorize the current diagonal block and test ! for non-positive-definiteness. ! JB = MIN( NB, N-J+1 ) CALL DSYRK( 'Lower', 'No transpose', JB, J-1, -ONE, & A( J, 1 ), LDA, ONE, A( J, J ), LDA ) CALL DPOTF2( 'Lower', JB, A( J, J ), LDA, INFO ) IF( INFO.NE.0 ) & GO TO 30 IF( J+JB.LE.N ) THEN ! ! Compute the current block column. ! CALL DGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB, & J-1, -ONE, A( J+JB, 1 ), LDA, A( J, 1 ), & LDA, ONE, A( J+JB, J ), LDA ) CALL DTRSM( 'Right', 'Lower', 'Transpose', 'Non-unit', & N-J-JB+1, JB, ONE, A( J, J ), LDA, & A( J+JB, J ), LDA ) END IF 20 CONTINUE END IF END IF GO TO 40 ! 30 CONTINUE INFO = INFO + J - 1 ! 40 CONTINUE RETURN ! ! End of DPOTRF ! END SUBROUTINE DPOTRI( UPLO, N, A, LDA, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ) ! .. ! ! Purpose ! ======= ! ! DPOTRI computes the inverse of a real symmetric positive definite ! matrix A using the Cholesky factorization A = U**T*U or A = L*L**T ! computed by DPOTRF. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the triangular factor U or L from the Cholesky ! factorization A = U**T*U or A = L*L**T, as computed by ! DPOTRF. ! On exit, the upper or lower triangle of the (symmetric) ! inverse of A, overwriting the input factor U or L. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the (i,i) element of the factor U or L is ! zero, and the inverse could not be computed. ! ! ===================================================================== ! ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DLAUUM, DTRTRI, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPOTRI', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Invert the triangular Cholesky factor U or L. ! CALL DTRTRI( UPLO, 'Non-unit', N, A, LDA, INFO ) IF( INFO.GT.0 ) & RETURN ! ! Form inv(U)*inv(U)' or inv(L)'*inv(L). ! CALL DLAUUM( UPLO, N, A, LDA, INFO ) ! RETURN ! ! End of DPOTRI ! END SUBROUTINE DPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, LDB, N, NRHS ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DPOTRS solves a system of linear equations A*X = B with a symmetric ! positive definite matrix A using the Cholesky factorization ! A = U**T*U or A = L*L**T computed by DPOTRF. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The triangular factor U or L from the Cholesky factorization ! A = U**T*U or A = L*L**T, as computed by DPOTRF. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the right hand side matrix B. ! On exit, the solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DTRSM, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPOTRS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) & RETURN ! IF( UPPER ) THEN ! ! Solve A*X = B where A = U'*U. ! ! Solve U'*X = B, overwriting B with X. ! CALL DTRSM( 'Left', 'Upper', 'Transpose', 'Non-unit', N, NRHS, & ONE, A, LDA, B, LDB ) ! ! Solve U*X = B, overwriting B with X. ! CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', N, & NRHS, ONE, A, LDA, B, LDB ) ELSE ! ! Solve A*X = B where A = L*L'. ! ! Solve L*X = B, overwriting B with X. ! CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Non-unit', N, & NRHS, ONE, A, LDA, B, LDB ) ! ! Solve L'*X = B, overwriting B with X. ! CALL DTRSM( 'Left', 'Lower', 'Transpose', 'Non-unit', N, NRHS, & ONE, A, LDA, B, LDB ) END IF ! RETURN ! ! End of DPOTRS ! END SUBROUTINE DPPCON( UPLO, N, AP, ANORM, RCOND, WORK, IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, N DOUBLE PRECISION ANORM, RCOND ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION AP( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DPPCON estimates the reciprocal of the condition number (in the ! 1-norm) of a real symmetric positive definite packed matrix using ! the Cholesky factorization A = U**T*U or A = L*L**T computed by ! DPPTRF. ! ! An estimate is obtained for norm(inv(A)), and the reciprocal of the ! condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The triangular factor U or L from the Cholesky factorization ! A = U**T*U or A = L*L**T, packed columnwise in a linear ! array. The j-th column of U or L is stored in the array AP ! as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n. ! ! ANORM (input) DOUBLE PRECISION ! The 1-norm (or infinity-norm) of the symmetric matrix A. ! ! RCOND (output) DOUBLE PRECISION ! The reciprocal of the condition number of the matrix A, ! computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an ! estimate of the 1-norm of inv(A) computed in this routine. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER CHARACTER NORMIN INTEGER IX, KASE DOUBLE PRECISION AINVNM, SCALE, SCALEL, SCALEU, SMLNUM ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DLAMCH EXTERNAL LSAME, IDAMAX, DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DLACON, DLATPS, DRSCL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( ANORM.LT.ZERO ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPPCON', -INFO ) RETURN END IF ! ! Quick return if possible ! RCOND = ZERO IF( N.EQ.0 ) THEN RCOND = ONE RETURN ELSE IF( ANORM.EQ.ZERO ) THEN RETURN END IF ! SMLNUM = DLAMCH( 'Safe minimum' ) ! ! Estimate the 1-norm of the inverse. ! KASE = 0 NORMIN = 'N' 10 CONTINUE CALL DLACON( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE ) IF( KASE.NE.0 ) THEN IF( UPPER ) THEN ! ! Multiply by inv(U'). ! CALL DLATPS( 'Upper', 'Transpose', 'Non-unit', NORMIN, N, & AP, WORK, SCALEL, WORK( 2*N+1 ), INFO ) NORMIN = 'Y' ! ! Multiply by inv(U). ! CALL DLATPS( 'Upper', 'No transpose', 'Non-unit', NORMIN, N, & AP, WORK, SCALEU, WORK( 2*N+1 ), INFO ) ELSE ! ! Multiply by inv(L). ! CALL DLATPS( 'Lower', 'No transpose', 'Non-unit', NORMIN, N, & AP, WORK, SCALEL, WORK( 2*N+1 ), INFO ) NORMIN = 'Y' ! ! Multiply by inv(L'). ! CALL DLATPS( 'Lower', 'Transpose', 'Non-unit', NORMIN, N, & AP, WORK, SCALEU, WORK( 2*N+1 ), INFO ) END IF ! ! Multiply by 1/SCALE if doing so will not cause overflow. ! SCALE = SCALEL*SCALEU IF( SCALE.NE.ONE ) THEN IX = IDAMAX( N, WORK, 1 ) IF( SCALE.LT.ABS( WORK( IX ) )*SMLNUM .OR. SCALE.EQ.ZERO ) & GO TO 20 CALL DRSCL( N, SCALE, WORK, 1 ) END IF GO TO 10 END IF ! ! Compute the estimate of the reciprocal condition number. ! IF( AINVNM.NE.ZERO ) & RCOND = ( ONE / AINVNM ) / ANORM ! 20 CONTINUE RETURN ! ! End of DPPCON ! END SUBROUTINE DPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, N DOUBLE PRECISION AMAX, SCOND ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ), S( * ) ! .. ! ! Purpose ! ======= ! ! DPPEQU computes row and column scalings intended to equilibrate a ! symmetric positive definite matrix A in packed storage and reduce ! its condition number (with respect to the two-norm). S contains the ! scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix ! B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal. ! This choice of S puts the condition number of B within a factor N of ! the smallest possible condition number over all possible diagonal ! scalings. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The upper or lower triangle of the symmetric matrix A, packed ! columnwise in a linear array. The j-th column of A is stored ! in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. ! ! S (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, S contains the scale factors for A. ! ! SCOND (output) DOUBLE PRECISION ! If INFO = 0, S contains the ratio of the smallest S(i) to ! the largest S(i). If SCOND >= 0.1 and AMAX is neither too ! large nor too small, it is not worth scaling by S. ! ! AMAX (output) DOUBLE PRECISION ! Absolute value of largest matrix element. If AMAX is very ! close to overflow or very close to underflow, the matrix ! should be scaled. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the i-th diagonal element is nonpositive. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER I, JJ DOUBLE PRECISION SMIN ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPPEQU', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) THEN SCOND = ONE AMAX = ZERO RETURN END IF ! ! Initialize SMIN and AMAX. ! S( 1 ) = AP( 1 ) SMIN = S( 1 ) AMAX = S( 1 ) ! IF( UPPER ) THEN ! ! UPLO = 'U': Upper triangle of A is stored. ! Find the minimum and maximum diagonal elements. ! JJ = 1 DO 10 I = 2, N JJ = JJ + I S( I ) = AP( JJ ) SMIN = MIN( SMIN, S( I ) ) AMAX = MAX( AMAX, S( I ) ) 10 CONTINUE ! ELSE ! ! UPLO = 'L': Lower triangle of A is stored. ! Find the minimum and maximum diagonal elements. ! JJ = 1 DO 20 I = 2, N JJ = JJ + N - I + 2 S( I ) = AP( JJ ) SMIN = MIN( SMIN, S( I ) ) AMAX = MAX( AMAX, S( I ) ) 20 CONTINUE END IF ! IF( SMIN.LE.ZERO ) THEN ! ! Find the first non-positive diagonal element and return. ! DO 30 I = 1, N IF( S( I ).LE.ZERO ) THEN INFO = I RETURN END IF 30 CONTINUE ELSE ! ! Set the scale factors to the reciprocals ! of the diagonal elements. ! DO 40 I = 1, N S( I ) = ONE / SQRT( S( I ) ) 40 CONTINUE ! ! Compute SCOND = min(S(I)) / max(S(I)) ! SCOND = SQRT( SMIN ) / SQRT( AMAX ) END IF RETURN ! ! End of DPPEQU ! END SUBROUTINE DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, & BERR, WORK, IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDB, LDX, N, NRHS ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ), & FERR( * ), WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DPPRFS improves the computed solution to a system of linear ! equations when the coefficient matrix is symmetric positive definite ! and packed, and provides error bounds and backward error estimates ! for the solution. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The upper or lower triangle of the symmetric matrix A, packed ! columnwise in a linear array. The j-th column of A is stored ! in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. ! ! AFP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The triangular factor U or L from the Cholesky factorization ! A = U**T*U or A = L*L**T, as computed by DPPTRF/ZPPTRF, ! packed columnwise in a linear array in the same format as A ! (see AP). ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) ! The right hand side matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! On entry, the solution matrix X, as computed by DPPTRS. ! On exit, the improved solution matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Internal Parameters ! =================== ! ! ITMAX is the maximum number of steps of iterative refinement. ! ! ===================================================================== ! ! .. Parameters .. INTEGER ITMAX PARAMETER ( ITMAX = 5 ) DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D+0 ) DOUBLE PRECISION THREE PARAMETER ( THREE = 3.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER COUNT, I, IK, J, K, KASE, KK, NZ DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DLACON, DPPTRS, DSPMV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -9 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPPRFS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN DO 10 J = 1, NRHS FERR( J ) = ZERO BERR( J ) = ZERO 10 CONTINUE RETURN END IF ! ! NZ = maximum number of nonzero elements in each row of A, plus 1 ! NZ = N + 1 EPS = DLAMCH( 'Epsilon' ) SAFMIN = DLAMCH( 'Safe minimum' ) SAFE1 = NZ*SAFMIN SAFE2 = SAFE1 / EPS ! ! Do for each right hand side ! DO 140 J = 1, NRHS ! COUNT = 1 LSTRES = THREE 20 CONTINUE ! ! Loop until stopping criterion is satisfied. ! ! Compute residual R = B - A * X ! CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ), & 1 ) ! ! Compute componentwise relative backward error from formula ! ! max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) ! ! where abs(Z) is the componentwise absolute value of the matrix ! or vector Z. If the i-th component of the denominator is less ! than SAFE2, then SAFE1 is added to the i-th components of the ! numerator and denominator before dividing. ! DO 30 I = 1, N WORK( I ) = ABS( B( I, J ) ) 30 CONTINUE ! ! Compute abs(A)*abs(X) + abs(B). ! KK = 1 IF( UPPER ) THEN DO 50 K = 1, N S = ZERO XK = ABS( X( K, J ) ) IK = KK DO 40 I = 1, K - 1 WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK S = S + ABS( AP( IK ) )*ABS( X( I, J ) ) IK = IK + 1 40 CONTINUE WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S KK = KK + K 50 CONTINUE ELSE DO 70 K = 1, N S = ZERO XK = ABS( X( K, J ) ) WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK IK = KK + 1 DO 60 I = K + 1, N WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK S = S + ABS( AP( IK ) )*ABS( X( I, J ) ) IK = IK + 1 60 CONTINUE WORK( K ) = WORK( K ) + S KK = KK + ( N-K+1 ) 70 CONTINUE END IF S = ZERO DO 80 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) ELSE S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / & ( WORK( I )+SAFE1 ) ) END IF 80 CONTINUE BERR( J ) = S ! ! Test stopping criterion. Continue iterating if ! 1) The residual BERR(J) is larger than machine epsilon, and ! 2) BERR(J) decreased by at least a factor of 2 during the ! last iteration, and ! 3) At most ITMAX iterations tried. ! IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. & COUNT.LE.ITMAX ) THEN ! ! Update solution and try again. ! CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO ) CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) LSTRES = BERR( J ) COUNT = COUNT + 1 GO TO 20 END IF ! ! Bound error from formula ! ! norm(X - XTRUE) / norm(X) .le. FERR = ! norm( abs(inv(A))* ! ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) ! ! where ! norm(Z) is the magnitude of the largest component of Z ! inv(A) is the inverse of A ! abs(Z) is the componentwise absolute value of the matrix or ! vector Z ! NZ is the maximum number of nonzeros in any row of A, plus 1 ! EPS is machine epsilon ! ! The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) ! is incremented by SAFE1 if the i-th component of ! abs(A)*abs(X) + abs(B) is less than SAFE2. ! ! Use DLACON to estimate the infinity-norm of the matrix ! inv(A) * diag(W), ! where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) ! DO 90 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) ELSE WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 END IF 90 CONTINUE ! KASE = 0 100 CONTINUE CALL DLACON( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), & KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN ! ! Multiply by diag(W)*inv(A'). ! CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO ) DO 110 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 110 CONTINUE ELSE IF( KASE.EQ.2 ) THEN ! ! Multiply by inv(A)*diag(W). ! DO 120 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 120 CONTINUE CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO ) END IF GO TO 100 END IF ! ! Normalize error. ! LSTRES = ZERO DO 130 I = 1, N LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 130 CONTINUE IF( LSTRES.NE.ZERO ) & FERR( J ) = FERR( J ) / LSTRES ! 140 CONTINUE ! RETURN ! ! End of DPPRFS ! END SUBROUTINE DPPSV( UPLO, N, NRHS, AP, B, LDB, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDB, N, NRHS ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DPPSV computes the solution to a real system of linear equations ! A * X = B, ! where A is an N-by-N symmetric positive definite matrix stored in ! packed format and X and B are N-by-NRHS matrices. ! ! The Cholesky decomposition is used to factor A as ! A = U**T* U, if UPLO = 'U', or ! A = L * L**T, if UPLO = 'L', ! where U is an upper triangular matrix and L is a lower triangular ! matrix. The factored form of A is then used to solve the system of ! equations A * X = B. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The number of linear equations, i.e., the order of the ! matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! A, packed columnwise in a linear array. The j-th column of A ! is stored in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. ! See below for further details. ! ! On exit, if INFO = 0, the factor U or L from the Cholesky ! factorization A = U**T*U or A = L*L**T, in the same storage ! format as A. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the N-by-NRHS right hand side matrix B. ! On exit, if INFO = 0, the N-by-NRHS solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the leading minor of order i of A is not ! positive definite, so the factorization could not be ! completed, and the solution has not been computed. ! ! Further Details ! =============== ! ! The packed storage scheme is illustrated by the following example ! when N = 4, UPLO = 'U': ! ! Two-dimensional storage of the symmetric matrix A: ! ! a11 a12 a13 a14 ! a22 a23 a24 ! a33 a34 (aij = conjg(aji)) ! a44 ! ! Packed storage of the upper triangle of A: ! ! AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] ! ! ===================================================================== ! ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DPPTRF, DPPTRS, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPPSV ', -INFO ) RETURN END IF ! ! Compute the Cholesky factorization A = U'*U or A = L*L'. ! CALL DPPTRF( UPLO, N, AP, INFO ) IF( INFO.EQ.0 ) THEN ! ! Solve the system A*X = B, overwriting B with X. ! CALL DPPTRS( UPLO, N, NRHS, AP, B, LDB, INFO ) ! END IF RETURN ! ! End of DPPSV ! END SUBROUTINE DPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB, & X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER EQUED, FACT, UPLO INTEGER INFO, LDB, LDX, N, NRHS DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ), & FERR( * ), S( * ), WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to ! compute the solution to a real system of linear equations ! A * X = B, ! where A is an N-by-N symmetric positive definite matrix stored in ! packed format and X and B are N-by-NRHS matrices. ! ! Error bounds on the solution and a condition estimate are also ! provided. ! ! Description ! =========== ! ! The following steps are performed: ! ! 1. If FACT = 'E', real scaling factors are computed to equilibrate ! the system: ! diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B ! Whether or not the system will be equilibrated depends on the ! scaling of the matrix A, but if equilibration is used, A is ! overwritten by diag(S)*A*diag(S) and B by diag(S)*B. ! ! 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to ! factor the matrix A (after equilibration if FACT = 'E') as ! A = U**T* U, if UPLO = 'U', or ! A = L * L**T, if UPLO = 'L', ! where U is an upper triangular matrix and L is a lower triangular ! matrix. ! ! 3. If the leading i-by-i principal minor is not positive definite, ! then the routine returns with INFO = i. Otherwise, the factored ! form of A is used to estimate the condition number of the matrix ! A. If the reciprocal of the condition number is less than machine ! precision, INFO = N+1 is returned as a warning, but the routine ! still goes on to solve for X and compute error bounds as ! described below. ! ! 4. The system of equations is solved for X using the factored form ! of A. ! ! 5. Iterative refinement is applied to improve the computed solution ! matrix and calculate error bounds and backward error estimates ! for it. ! ! 6. If equilibration was used, the matrix X is premultiplied by ! diag(S) so that it solves the original system before ! equilibration. ! ! Arguments ! ========= ! ! FACT (input) CHARACTER*1 ! Specifies whether or not the factored form of the matrix A is ! supplied on entry, and if not, whether the matrix A should be ! equilibrated before it is factored. ! = 'F': On entry, AFP contains the factored form of A. ! If EQUED = 'Y', the matrix A has been equilibrated ! with scaling factors given by S. AP and AFP will not ! be modified. ! = 'N': The matrix A will be copied to AFP and factored. ! = 'E': The matrix A will be equilibrated if necessary, then ! copied to AFP and factored. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The number of linear equations, i.e., the order of the ! matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! A, packed columnwise in a linear array, except if FACT = 'F' ! and EQUED = 'Y', then A must contain the equilibrated matrix ! diag(S)*A*diag(S). The j-th column of A is stored in the ! array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. ! See below for further details. A is not modified if ! FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. ! ! On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by ! diag(S)*A*diag(S). ! ! AFP (input or output) DOUBLE PRECISION array, dimension ! (N*(N+1)/2) ! If FACT = 'F', then AFP is an input argument and on entry ! contains the triangular factor U or L from the Cholesky ! factorization A = U'*U or A = L*L', in the same storage ! format as A. If EQUED .ne. 'N', then AFP is the factored ! form of the equilibrated matrix A. ! ! If FACT = 'N', then AFP is an output argument and on exit ! returns the triangular factor U or L from the Cholesky ! factorization A = U'*U or A = L*L' of the original matrix A. ! ! If FACT = 'E', then AFP is an output argument and on exit ! returns the triangular factor U or L from the Cholesky ! factorization A = U'*U or A = L*L' of the equilibrated ! matrix A (see the description of AP for the form of the ! equilibrated matrix). ! ! EQUED (input or output) CHARACTER*1 ! Specifies the form of equilibration that was done. ! = 'N': No equilibration (always true if FACT = 'N'). ! = 'Y': Equilibration was done, i.e., A has been replaced by ! diag(S) * A * diag(S). ! EQUED is an input argument if FACT = 'F'; otherwise, it is an ! output argument. ! ! S (input or output) DOUBLE PRECISION array, dimension (N) ! The scale factors for A; not accessed if EQUED = 'N'. S is ! an input argument if FACT = 'F'; otherwise, S is an output ! argument. If FACT = 'F' and EQUED = 'Y', each element of S ! must be positive. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the N-by-NRHS right hand side matrix B. ! On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', ! B is overwritten by diag(S) * B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to ! the original system of equations. Note that if EQUED = 'Y', ! A and B are modified on exit, and the solution to the ! equilibrated system is inv(diag(S))*X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! RCOND (output) DOUBLE PRECISION ! The estimate of the reciprocal condition number of the matrix ! A after equilibration (if done). If RCOND is less than the ! machine precision (in particular, if RCOND = 0), the matrix ! is singular to working precision. This condition is ! indicated by a return code of INFO > 0. ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, and i is ! <= N: the leading minor of order i of A is ! not positive definite, so the factorization ! could not be completed, and the solution has not ! been computed. RCOND = 0 is returned. ! = N+1: U is nonsingular, but RCOND is less than machine ! precision, meaning that the matrix is singular ! to working precision. Nevertheless, the ! solution and error bounds are computed because ! there are a number of situations where the ! computed solution can be more accurate than the ! value of RCOND would suggest. ! ! Further Details ! =============== ! ! The packed storage scheme is illustrated by the following example ! when N = 4, UPLO = 'U': ! ! Two-dimensional storage of the symmetric matrix A: ! ! a11 a12 a13 a14 ! a22 a23 a24 ! a33 a34 (aij = conjg(aji)) ! a44 ! ! Packed storage of the upper triangle of A: ! ! AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL EQUIL, NOFACT, RCEQU INTEGER I, INFEQU, J DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANSP EXTERNAL LSAME, DLAMCH, DLANSP ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLACPY, DLAQSP, DPPCON, DPPEQU, DPPRFS, & DPPTRF, DPPTRS, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! INFO = 0 NOFACT = LSAME( FACT, 'N' ) EQUIL = LSAME( FACT, 'E' ) IF( NOFACT .OR. EQUIL ) THEN EQUED = 'N' RCEQU = .FALSE. ELSE RCEQU = LSAME( EQUED, 'Y' ) SMLNUM = DLAMCH( 'Safe minimum' ) BIGNUM = ONE / SMLNUM END IF ! ! Test the input parameters. ! IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) & THEN INFO = -1 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) & THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.0 ) THEN INFO = -4 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. & ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN INFO = -7 ELSE IF( RCEQU ) THEN SMIN = BIGNUM SMAX = ZERO DO 10 J = 1, N SMIN = MIN( SMIN, S( J ) ) SMAX = MAX( SMAX, S( J ) ) 10 CONTINUE IF( SMIN.LE.ZERO ) THEN INFO = -8 ELSE IF( N.GT.0 ) THEN SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) ELSE SCOND = ONE END IF END IF IF( INFO.EQ.0 ) THEN IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -10 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -12 END IF END IF END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPPSVX', -INFO ) RETURN END IF ! IF( EQUIL ) THEN ! ! Compute row and column scalings to equilibrate the matrix A. ! CALL DPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFEQU ) IF( INFEQU.EQ.0 ) THEN ! ! Equilibrate the matrix. ! CALL DLAQSP( UPLO, N, AP, S, SCOND, AMAX, EQUED ) RCEQU = LSAME( EQUED, 'Y' ) END IF END IF ! ! Scale the right-hand side. ! IF( RCEQU ) THEN DO 30 J = 1, NRHS DO 20 I = 1, N B( I, J ) = S( I )*B( I, J ) 20 CONTINUE 30 CONTINUE END IF ! IF( NOFACT .OR. EQUIL ) THEN ! ! Compute the Cholesky factorization A = U'*U or A = L*L'. ! CALL DCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 ) CALL DPPTRF( UPLO, N, AFP, INFO ) ! ! Return if INFO is non-zero. ! IF( INFO.NE.0 ) THEN IF( INFO.GT.0 ) & RCOND = ZERO RETURN END IF END IF ! ! Compute the norm of the matrix A. ! ANORM = DLANSP( 'I', UPLO, N, AP, WORK ) ! ! Compute the reciprocal of the condition number of A. ! CALL DPPCON( UPLO, N, AFP, ANORM, RCOND, WORK, IWORK, INFO ) ! ! Set INFO = N+1 if the matrix is singular to working precision. ! IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) & INFO = N + 1 ! ! Compute the solution matrix X. ! CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL DPPTRS( UPLO, N, NRHS, AFP, X, LDX, INFO ) ! ! Use iterative refinement to improve the computed solution and ! compute error bounds and backward error estimates for it. ! CALL DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR, & WORK, IWORK, INFO ) ! ! Transform the solution matrix X to a solution of the original ! system. ! IF( RCEQU ) THEN DO 50 J = 1, NRHS DO 40 I = 1, N X( I, J ) = S( I )*X( I, J ) 40 CONTINUE 50 CONTINUE DO 60 J = 1, NRHS FERR( J ) = FERR( J ) / SCOND 60 CONTINUE END IF ! RETURN ! ! End of DPPSVX ! END SUBROUTINE DPPTRF( UPLO, N, AP, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ) ! .. ! ! Purpose ! ======= ! ! DPPTRF computes the Cholesky factorization of a real symmetric ! positive definite matrix A stored in packed format. ! ! The factorization has the form ! A = U**T * U, if UPLO = 'U', or ! A = L * L**T, if UPLO = 'L', ! where U is an upper triangular matrix and L is lower triangular. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! A, packed columnwise in a linear array. The j-th column of A ! is stored in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. ! See below for further details. ! ! On exit, if INFO = 0, the triangular factor U or L from the ! Cholesky factorization A = U**T*U or A = L*L**T, in the same ! storage format as A. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the leading minor of order i is not ! positive definite, and the factorization could not be ! completed. ! ! Further Details ! ======= ======= ! ! The packed storage scheme is illustrated by the following example ! when N = 4, UPLO = 'U': ! ! Two-dimensional storage of the symmetric matrix A: ! ! a11 a12 a13 a14 ! a22 a23 a24 ! a33 a34 (aij = aji) ! a44 ! ! Packed storage of the upper triangle of A: ! ! AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER J, JC, JJ DOUBLE PRECISION AJJ ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT EXTERNAL LSAME, DDOT ! .. ! .. External Subroutines .. EXTERNAL DSCAL, DSPR, DTPSV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPPTRF', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( UPPER ) THEN ! ! Compute the Cholesky factorization A = U'*U. ! JJ = 0 DO 10 J = 1, N JC = JJ + 1 JJ = JJ + J ! ! Compute elements 1:J-1 of column J. ! IF( J.GT.1 ) & CALL DTPSV( 'Upper', 'Transpose', 'Non-unit', J-1, AP, & AP( JC ), 1 ) ! ! Compute U(J,J) and test for non-positive-definiteness. ! AJJ = AP( JJ ) - DDOT( J-1, AP( JC ), 1, AP( JC ), 1 ) IF( AJJ.LE.ZERO ) THEN AP( JJ ) = AJJ GO TO 30 END IF AP( JJ ) = SQRT( AJJ ) 10 CONTINUE ELSE ! ! Compute the Cholesky factorization A = L*L'. ! JJ = 1 DO 20 J = 1, N ! ! Compute L(J,J) and test for non-positive-definiteness. ! AJJ = AP( JJ ) IF( AJJ.LE.ZERO ) THEN AP( JJ ) = AJJ GO TO 30 END IF AJJ = SQRT( AJJ ) AP( JJ ) = AJJ ! ! Compute elements J+1:N of column J and update the trailing ! submatrix. ! IF( J.LT.N ) THEN CALL DSCAL( N-J, ONE / AJJ, AP( JJ+1 ), 1 ) CALL DSPR( 'Lower', N-J, -ONE, AP( JJ+1 ), 1, & AP( JJ+N-J+1 ) ) JJ = JJ + N - J + 1 END IF 20 CONTINUE END IF GO TO 40 ! 30 CONTINUE INFO = J ! 40 CONTINUE RETURN ! ! End of DPPTRF ! END SUBROUTINE DPPTRI( UPLO, N, AP, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ) ! .. ! ! Purpose ! ======= ! ! DPPTRI computes the inverse of a real symmetric positive definite ! matrix A using the Cholesky factorization A = U**T*U or A = L*L**T ! computed by DPPTRF. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangular factor is stored in AP; ! = 'L': Lower triangular factor is stored in AP. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the triangular factor U or L from the Cholesky ! factorization A = U**T*U or A = L*L**T, packed columnwise as ! a linear array. The j-th column of U or L is stored in the ! array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n. ! ! On exit, the upper or lower triangle of the (symmetric) ! inverse of A, overwriting the input factor U or L. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the (i,i) element of the factor U or L is ! zero, and the inverse could not be computed. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER J, JC, JJ, JJN DOUBLE PRECISION AJJ ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT EXTERNAL LSAME, DDOT ! .. ! .. External Subroutines .. EXTERNAL DSCAL, DSPR, DTPMV, DTPTRI, XERBLA ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPPTRI', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Invert the triangular Cholesky factor U or L. ! CALL DTPTRI( UPLO, 'Non-unit', N, AP, INFO ) IF( INFO.GT.0 ) & RETURN ! IF( UPPER ) THEN ! ! Compute the product inv(U) * inv(U)'. ! JJ = 0 DO 10 J = 1, N JC = JJ + 1 JJ = JJ + J IF( J.GT.1 ) & CALL DSPR( 'Upper', J-1, ONE, AP( JC ), 1, AP ) AJJ = AP( JJ ) CALL DSCAL( J, AJJ, AP( JC ), 1 ) 10 CONTINUE ! ELSE ! ! Compute the product inv(L)' * inv(L). ! JJ = 1 DO 20 J = 1, N JJN = JJ + N - J + 1 AP( JJ ) = DDOT( N-J+1, AP( JJ ), 1, AP( JJ ), 1 ) IF( J.LT.N ) & CALL DTPMV( 'Lower', 'Transpose', 'Non-unit', N-J, & AP( JJN ), AP( JJ+1 ), 1 ) JJ = JJN 20 CONTINUE END IF ! RETURN ! ! End of DPPTRI ! END SUBROUTINE DPPTRS( UPLO, N, NRHS, AP, B, LDB, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDB, N, NRHS ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DPPTRS solves a system of linear equations A*X = B with a symmetric ! positive definite matrix A in packed storage using the Cholesky ! factorization A = U**T*U or A = L*L**T computed by DPPTRF. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The triangular factor U or L from the Cholesky factorization ! A = U**T*U or A = L*L**T, packed columnwise in a linear ! array. The j-th column of U or L is stored in the array AP ! as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the right hand side matrix B. ! On exit, the solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL UPPER INTEGER I ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DTPSV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPPTRS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) & RETURN ! IF( UPPER ) THEN ! ! Solve A*X = B where A = U'*U. ! DO 10 I = 1, NRHS ! ! Solve U'*X = B, overwriting B with X. ! CALL DTPSV( 'Upper', 'Transpose', 'Non-unit', N, AP, & B( 1, I ), 1 ) ! ! Solve U*X = B, overwriting B with X. ! CALL DTPSV( 'Upper', 'No transpose', 'Non-unit', N, AP, & B( 1, I ), 1 ) 10 CONTINUE ELSE ! ! Solve A*X = B where A = L*L'. ! DO 20 I = 1, NRHS ! ! Solve L*Y = B, overwriting B with X. ! CALL DTPSV( 'Lower', 'No transpose', 'Non-unit', N, AP, & B( 1, I ), 1 ) ! ! Solve L'*X = Y, overwriting B with X. ! CALL DTPSV( 'Lower', 'Transpose', 'Non-unit', N, AP, & B( 1, I ), 1 ) 20 CONTINUE END IF ! RETURN ! ! End of DPPTRS ! END SUBROUTINE DPTCON( N, D, E, ANORM, RCOND, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. INTEGER INFO, N DOUBLE PRECISION ANORM, RCOND ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ), E( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DPTCON computes the reciprocal of the condition number (in the ! 1-norm) of a real symmetric positive definite tridiagonal matrix ! using the factorization A = L*D*L**T or A = U**T*D*U computed by ! DPTTRF. ! ! Norm(inv(A)) is computed by a direct method, and the reciprocal of ! the condition number is computed as ! RCOND = 1 / (ANORM * norm(inv(A))). ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of the diagonal matrix D from the ! factorization of A, as computed by DPTTRF. ! ! E (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) off-diagonal elements of the unit bidiagonal factor ! U or L from the factorization of A, as computed by DPTTRF. ! ! ANORM (input) DOUBLE PRECISION ! The 1-norm of the original matrix A. ! ! RCOND (output) DOUBLE PRECISION ! The reciprocal of the condition number of the matrix A, ! computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the ! 1-norm of inv(A) computed in this routine. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! The method used is described in Nicholas J. Higham, "Efficient ! Algorithms for Computing the Condition Number of a Tridiagonal ! Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, IX DOUBLE PRECISION AINVNM ! .. ! .. External Functions .. INTEGER IDAMAX EXTERNAL IDAMAX ! .. ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS ! .. ! .. Executable Statements .. ! ! Test the input arguments. ! INFO = 0 IF( N.LT.0 ) THEN INFO = -1 ELSE IF( ANORM.LT.ZERO ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPTCON', -INFO ) RETURN END IF ! ! Quick return if possible ! RCOND = ZERO IF( N.EQ.0 ) THEN RCOND = ONE RETURN ELSE IF( ANORM.EQ.ZERO ) THEN RETURN END IF ! ! Check that D(1:N) is positive. ! DO 10 I = 1, N IF( D( I ).LE.ZERO ) & RETURN 10 CONTINUE ! ! Solve M(A) * x = e, where M(A) = (m(i,j)) is given by ! ! m(i,j) = abs(A(i,j)), i = j, ! m(i,j) = -abs(A(i,j)), i .ne. j, ! ! and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)'. ! ! Solve M(L) * x = e. ! WORK( 1 ) = ONE DO 20 I = 2, N WORK( I ) = ONE + WORK( I-1 )*ABS( E( I-1 ) ) 20 CONTINUE ! ! Solve D * M(L)' * x = b. ! WORK( N ) = WORK( N ) / D( N ) DO 30 I = N - 1, 1, -1 WORK( I ) = WORK( I ) / D( I ) + WORK( I+1 )*ABS( E( I ) ) 30 CONTINUE ! ! Compute AINVNM = max(x(i)), 1<=i<=n. ! IX = IDAMAX( N, WORK, 1 ) AINVNM = ABS( WORK( IX ) ) ! ! Compute the reciprocal condition number. ! IF( AINVNM.NE.ZERO ) & RCOND = ( ONE / AINVNM ) / ANORM ! RETURN ! ! End of DPTCON ! END SUBROUTINE DPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1999 ! ! .. Scalar Arguments .. CHARACTER COMPZ INTEGER INFO, LDZ, N ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DPTEQR computes all eigenvalues and, optionally, eigenvectors of a ! symmetric positive definite tridiagonal matrix by first factoring the ! matrix using DPTTRF, and then calling DBDSQR to compute the singular ! values of the bidiagonal factor. ! ! This routine computes the eigenvalues of the positive definite ! tridiagonal matrix to high relative accuracy. This means that if the ! eigenvalues range over many orders of magnitude in size, then the ! small eigenvalues and corresponding eigenvectors will be computed ! more accurately than, for example, with the standard QR method. ! ! The eigenvectors of a full or band symmetric positive definite matrix ! can also be found if DSYTRD, DSPTRD, or DSBTRD has been used to ! reduce this matrix to tridiagonal form. (The reduction to tridiagonal ! form, however, may preclude the possibility of obtaining high ! relative accuracy in the small eigenvalues of the original matrix, if ! these eigenvalues range over many orders of magnitude.) ! ! Arguments ! ========= ! ! COMPZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only. ! = 'V': Compute eigenvectors of original symmetric ! matrix also. Array Z contains the orthogonal ! matrix used to reduce the original matrix to ! tridiagonal form. ! = 'I': Compute eigenvectors of tridiagonal matrix also. ! ! N (input) INTEGER ! The order of the matrix. N >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the n diagonal elements of the tridiagonal ! matrix. ! On normal exit, D contains the eigenvalues, in descending ! order. ! ! E (input/output) DOUBLE PRECISION array, dimension (N-1) ! On entry, the (n-1) subdiagonal elements of the tridiagonal ! matrix. ! On exit, E has been destroyed. ! ! Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N) ! On entry, if COMPZ = 'V', the orthogonal matrix used in the ! reduction to tridiagonal form. ! On exit, if COMPZ = 'V', the orthonormal eigenvectors of the ! original symmetric matrix; ! if COMPZ = 'I', the orthonormal eigenvectors of the ! tridiagonal matrix. ! If INFO > 0 on exit, Z contains the eigenvectors associated ! with only the stored eigenvalues. ! If COMPZ = 'N', then Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! COMPZ = 'V' or 'I', LDZ >= max(1,N). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (4*N) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = i, and i is: ! <= N the Cholesky factorization of the matrix could ! not be performed because the i-th principal minor ! was not positive definite. ! > N the SVD algorithm failed to converge; ! if INFO = N+i, i off-diagonal elements of the ! bidiagonal factor did not converge to zero. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DBDSQR, DLASET, DPTTRF, XERBLA ! .. ! .. Local Arrays .. DOUBLE PRECISION C( 1, 1 ), VT( 1, 1 ) ! .. ! .. Local Scalars .. INTEGER I, ICOMPZ, NRU ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( LSAME( COMPZ, 'N' ) ) THEN ICOMPZ = 0 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN ICOMPZ = 1 ELSE IF( LSAME( COMPZ, 'I' ) ) THEN ICOMPZ = 2 ELSE ICOMPZ = -1 END IF IF( ICOMPZ.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( ( LDZ.LT.1 ) .OR. ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1, & N ) ) ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPTEQR', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( N.EQ.1 ) THEN IF( ICOMPZ.GT.0 ) & Z( 1, 1 ) = ONE RETURN END IF IF( ICOMPZ.EQ.2 ) & CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ ) ! ! Call DPTTRF to factor the matrix. ! CALL DPTTRF( N, D, E, INFO ) IF( INFO.NE.0 ) & RETURN DO 10 I = 1, N D( I ) = SQRT( D( I ) ) 10 CONTINUE DO 20 I = 1, N - 1 E( I ) = E( I )*D( I ) 20 CONTINUE ! ! Call DBDSQR to compute the singular values/vectors of the ! bidiagonal factor. ! IF( ICOMPZ.GT.0 ) THEN NRU = N ELSE NRU = 0 END IF CALL DBDSQR( 'Lower', N, 0, NRU, 0, D, E, VT, 1, Z, LDZ, C, 1, & WORK, INFO ) ! ! Square the singular values. ! IF( INFO.EQ.0 ) THEN DO 30 I = 1, N D( I ) = D( I )*D( I ) 30 CONTINUE ELSE INFO = N + INFO END IF ! RETURN ! ! End of DPTEQR ! END SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, & BERR, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. INTEGER INFO, LDB, LDX, N, NRHS ! .. ! .. Array Arguments .. DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ), & E( * ), EF( * ), FERR( * ), WORK( * ), & X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DPTRFS improves the computed solution to a system of linear ! equations when the coefficient matrix is symmetric positive definite ! and tridiagonal, and provides error bounds and backward error ! estimates for the solution. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of the tridiagonal matrix A. ! ! E (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) subdiagonal elements of the tridiagonal matrix A. ! ! DF (input) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of the diagonal matrix D from the ! factorization computed by DPTTRF. ! ! EF (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) subdiagonal elements of the unit bidiagonal factor ! L from the factorization computed by DPTTRF. ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) ! The right hand side matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! On entry, the solution matrix X, as computed by DPTTRS. ! On exit, the improved solution matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (2*N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Internal Parameters ! =================== ! ! ITMAX is the maximum number of steps of iterative refinement. ! ! ===================================================================== ! ! .. Parameters .. INTEGER ITMAX PARAMETER ( ITMAX = 5 ) DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D+0 ) DOUBLE PRECISION THREE PARAMETER ( THREE = 3.0D+0 ) ! .. ! .. Local Scalars .. INTEGER COUNT, I, IX, J, NZ DOUBLE PRECISION BI, CX, DX, EPS, EX, LSTRES, S, SAFE1, SAFE2, & SAFMIN ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DPTTRS, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DLAMCH EXTERNAL IDAMAX, DLAMCH ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF( N.LT.0 ) THEN INFO = -1 ELSE IF( NRHS.LT.0 ) THEN INFO = -2 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -10 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPTRFS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN DO 10 J = 1, NRHS FERR( J ) = ZERO BERR( J ) = ZERO 10 CONTINUE RETURN END IF ! ! NZ = maximum number of nonzero elements in each row of A, plus 1 ! NZ = 4 EPS = DLAMCH( 'Epsilon' ) SAFMIN = DLAMCH( 'Safe minimum' ) SAFE1 = NZ*SAFMIN SAFE2 = SAFE1 / EPS ! ! Do for each right hand side ! DO 90 J = 1, NRHS ! COUNT = 1 LSTRES = THREE 20 CONTINUE ! ! Loop until stopping criterion is satisfied. ! ! Compute residual R = B - A * X. Also compute ! abs(A)*abs(x) + abs(b) for use in the backward error bound. ! IF( N.EQ.1 ) THEN BI = B( 1, J ) DX = D( 1 )*X( 1, J ) WORK( N+1 ) = BI - DX WORK( 1 ) = ABS( BI ) + ABS( DX ) ELSE BI = B( 1, J ) DX = D( 1 )*X( 1, J ) EX = E( 1 )*X( 2, J ) WORK( N+1 ) = BI - DX - EX WORK( 1 ) = ABS( BI ) + ABS( DX ) + ABS( EX ) DO 30 I = 2, N - 1 BI = B( I, J ) CX = E( I-1 )*X( I-1, J ) DX = D( I )*X( I, J ) EX = E( I )*X( I+1, J ) WORK( N+I ) = BI - CX - DX - EX WORK( I ) = ABS( BI ) + ABS( CX ) + ABS( DX ) + ABS( EX ) 30 CONTINUE BI = B( N, J ) CX = E( N-1 )*X( N-1, J ) DX = D( N )*X( N, J ) WORK( N+N ) = BI - CX - DX WORK( N ) = ABS( BI ) + ABS( CX ) + ABS( DX ) END IF ! ! Compute componentwise relative backward error from formula ! ! max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) ! ! where abs(Z) is the componentwise absolute value of the matrix ! or vector Z. If the i-th component of the denominator is less ! than SAFE2, then SAFE1 is added to the i-th components of the ! numerator and denominator before dividing. ! S = ZERO DO 40 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) ELSE S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / & ( WORK( I )+SAFE1 ) ) END IF 40 CONTINUE BERR( J ) = S ! ! Test stopping criterion. Continue iterating if ! 1) The residual BERR(J) is larger than machine epsilon, and ! 2) BERR(J) decreased by at least a factor of 2 during the ! last iteration, and ! 3) At most ITMAX iterations tried. ! IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. & COUNT.LE.ITMAX ) THEN ! ! Update solution and try again. ! CALL DPTTRS( N, 1, DF, EF, WORK( N+1 ), N, INFO ) CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) LSTRES = BERR( J ) COUNT = COUNT + 1 GO TO 20 END IF ! ! Bound error from formula ! ! norm(X - XTRUE) / norm(X) .le. FERR = ! norm( abs(inv(A))* ! ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) ! ! where ! norm(Z) is the magnitude of the largest component of Z ! inv(A) is the inverse of A ! abs(Z) is the componentwise absolute value of the matrix or ! vector Z ! NZ is the maximum number of nonzeros in any row of A, plus 1 ! EPS is machine epsilon ! ! The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) ! is incremented by SAFE1 if the i-th component of ! abs(A)*abs(X) + abs(B) is less than SAFE2. ! DO 50 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) ELSE WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 END IF 50 CONTINUE IX = IDAMAX( N, WORK, 1 ) FERR( J ) = WORK( IX ) ! ! Estimate the norm of inv(A). ! ! Solve M(A) * x = e, where M(A) = (m(i,j)) is given by ! ! m(i,j) = abs(A(i,j)), i = j, ! m(i,j) = -abs(A(i,j)), i .ne. j, ! ! and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)'. ! ! Solve M(L) * x = e. ! WORK( 1 ) = ONE DO 60 I = 2, N WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) ) 60 CONTINUE ! ! Solve D * M(L)' * x = b. ! WORK( N ) = WORK( N ) / DF( N ) DO 70 I = N - 1, 1, -1 WORK( I ) = WORK( I ) / DF( I ) + WORK( I+1 )*ABS( EF( I ) ) 70 CONTINUE ! ! Compute norm(inv(A)) = max(x(i)), 1<=i<=n. ! IX = IDAMAX( N, WORK, 1 ) FERR( J ) = FERR( J )*ABS( WORK( IX ) ) ! ! Normalize error. ! LSTRES = ZERO DO 80 I = 1, N LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 80 CONTINUE IF( LSTRES.NE.ZERO ) & FERR( J ) = FERR( J ) / LSTRES ! 90 CONTINUE ! RETURN ! ! End of DPTRFS ! END SUBROUTINE DPTSV( N, NRHS, D, E, B, LDB, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 25, 1997 ! ! .. Scalar Arguments .. INTEGER INFO, LDB, N, NRHS ! .. ! .. Array Arguments .. DOUBLE PRECISION B( LDB, * ), D( * ), E( * ) ! .. ! ! Purpose ! ======= ! ! DPTSV computes the solution to a real system of linear equations ! A*X = B, where A is an N-by-N symmetric positive definite tridiagonal ! matrix, and X and B are N-by-NRHS matrices. ! ! A is factored as A = L*D*L**T, and the factored form of A is then ! used to solve the system of equations. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the n diagonal elements of the tridiagonal matrix ! A. On exit, the n diagonal elements of the diagonal matrix ! D from the factorization A = L*D*L**T. ! ! E (input/output) DOUBLE PRECISION array, dimension (N-1) ! On entry, the (n-1) subdiagonal elements of the tridiagonal ! matrix A. On exit, the (n-1) subdiagonal elements of the ! unit bidiagonal factor L from the L*D*L**T factorization of ! A. (E can also be regarded as the superdiagonal of the unit ! bidiagonal factor U from the U**T*D*U factorization of A.) ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the N-by-NRHS right hand side matrix B. ! On exit, if INFO = 0, the N-by-NRHS solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the leading minor of order i is not ! positive definite, and the solution has not been ! computed. The factorization has not been completed ! unless i = N. ! ! ===================================================================== ! ! .. External Subroutines .. EXTERNAL DPTTRF, DPTTRS, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF( N.LT.0 ) THEN INFO = -1 ELSE IF( NRHS.LT.0 ) THEN INFO = -2 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPTSV ', -INFO ) RETURN END IF ! ! Compute the L*D*L' (or U'*D*U) factorization of A. ! CALL DPTTRF( N, D, E, INFO ) IF( INFO.EQ.0 ) THEN ! ! Solve the system A*X = B, overwriting B with X. ! CALL DPTTRS( N, NRHS, D, E, B, LDB, INFO ) END IF RETURN ! ! End of DPTSV ! END SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, & RCOND, FERR, BERR, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER FACT INTEGER INFO, LDB, LDX, N, NRHS DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ), & E( * ), EF( * ), FERR( * ), WORK( * ), & X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DPTSVX uses the factorization A = L*D*L**T to compute the solution ! to a real system of linear equations A*X = B, where A is an N-by-N ! symmetric positive definite tridiagonal matrix and X and B are ! N-by-NRHS matrices. ! ! Error bounds on the solution and a condition estimate are also ! provided. ! ! Description ! =========== ! ! The following steps are performed: ! ! 1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L ! is a unit lower bidiagonal matrix and D is diagonal. The ! factorization can also be regarded as having the form ! A = U**T*D*U. ! ! 2. If the leading i-by-i principal minor is not positive definite, ! then the routine returns with INFO = i. Otherwise, the factored ! form of A is used to estimate the condition number of the matrix ! A. If the reciprocal of the condition number is less than machine ! precision, INFO = N+1 is returned as a warning, but the routine ! still goes on to solve for X and compute error bounds as ! described below. ! ! 3. The system of equations is solved for X using the factored form ! of A. ! ! 4. Iterative refinement is applied to improve the computed solution ! matrix and calculate error bounds and backward error estimates ! for it. ! ! Arguments ! ========= ! ! FACT (input) CHARACTER*1 ! Specifies whether or not the factored form of A has been ! supplied on entry. ! = 'F': On entry, DF and EF contain the factored form of A. ! D, E, DF, and EF will not be modified. ! = 'N': The matrix A will be copied to DF and EF and ! factored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of the tridiagonal matrix A. ! ! E (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) subdiagonal elements of the tridiagonal matrix A. ! ! DF (input or output) DOUBLE PRECISION array, dimension (N) ! If FACT = 'F', then DF is an input argument and on entry ! contains the n diagonal elements of the diagonal matrix D ! from the L*D*L**T factorization of A. ! If FACT = 'N', then DF is an output argument and on exit ! contains the n diagonal elements of the diagonal matrix D ! from the L*D*L**T factorization of A. ! ! EF (input or output) DOUBLE PRECISION array, dimension (N-1) ! If FACT = 'F', then EF is an input argument and on entry ! contains the (n-1) subdiagonal elements of the unit ! bidiagonal factor L from the L*D*L**T factorization of A. ! If FACT = 'N', then EF is an output argument and on exit ! contains the (n-1) subdiagonal elements of the unit ! bidiagonal factor L from the L*D*L**T factorization of A. ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) ! The N-by-NRHS right hand side matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! RCOND (output) DOUBLE PRECISION ! The reciprocal condition number of the matrix A. If RCOND ! is less than the machine precision (in particular, if ! RCOND = 0), the matrix is singular to working precision. ! This condition is indicated by a return code of INFO > 0. ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in any ! element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (2*N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, and i is ! <= N: the leading minor of order i of A is ! not positive definite, so the factorization ! could not be completed, and the solution has not ! been computed. RCOND = 0 is returned. ! = N+1: U is nonsingular, but RCOND is less than machine ! precision, meaning that the matrix is singular ! to working precision. Nevertheless, the ! solution and error bounds are computed because ! there are a number of situations where the ! computed solution can be more accurate than the ! value of RCOND would suggest. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOFACT DOUBLE PRECISION ANORM ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANST EXTERNAL LSAME, DLAMCH, DLANST ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLACPY, DPTCON, DPTRFS, DPTTRF, DPTTRS, & XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 NOFACT = LSAME( FACT, 'N' ) IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -11 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPTSVX', -INFO ) RETURN END IF ! IF( NOFACT ) THEN ! ! Compute the L*D*L' (or U'*D*U) factorization of A. ! CALL DCOPY( N, D, 1, DF, 1 ) IF( N.GT.1 ) & CALL DCOPY( N-1, E, 1, EF, 1 ) CALL DPTTRF( N, DF, EF, INFO ) ! ! Return if INFO is non-zero. ! IF( INFO.NE.0 ) THEN IF( INFO.GT.0 ) & RCOND = ZERO RETURN END IF END IF ! ! Compute the norm of the matrix A. ! ANORM = DLANST( '1', N, D, E ) ! ! Compute the reciprocal of the condition number of A. ! CALL DPTCON( N, DF, EF, ANORM, RCOND, WORK, INFO ) ! ! Set INFO = N+1 if the matrix is singular to working precision. ! IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) & INFO = N + 1 ! ! Compute the solution vectors X. ! CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL DPTTRS( N, NRHS, DF, EF, X, LDX, INFO ) ! ! Use iterative refinement to improve the computed solutions and ! compute error bounds and backward error estimates for them. ! CALL DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, & WORK, INFO ) ! RETURN ! ! End of DPTSVX ! END SUBROUTINE DPTTRF( N, D, E, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, N ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ), E( * ) ! .. ! ! Purpose ! ======= ! ! DPTTRF computes the L*D*L' factorization of a real symmetric ! positive definite tridiagonal matrix A. The factorization may also ! be regarded as having the form A = U'*D*U. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the n diagonal elements of the tridiagonal matrix ! A. On exit, the n diagonal elements of the diagonal matrix ! D from the L*D*L' factorization of A. ! ! E (input/output) DOUBLE PRECISION array, dimension (N-1) ! On entry, the (n-1) subdiagonal elements of the tridiagonal ! matrix A. On exit, the (n-1) subdiagonal elements of the ! unit bidiagonal factor L from the L*D*L' factorization of A. ! E can also be regarded as the superdiagonal of the unit ! bidiagonal factor U from the U'*D*U factorization of A. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -k, the k-th argument had an illegal value ! > 0: if INFO = k, the leading minor of order k is not ! positive definite; if k < N, the factorization could not ! be completed, while if k = N, the factorization was ! completed, but D(N) = 0. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, I4 DOUBLE PRECISION EI ! .. ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MOD ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF( N.LT.0 ) THEN INFO = -1 CALL XERBLA( 'DPTTRF', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Compute the L*D*L' (or U'*D*U) factorization of A. ! I4 = MOD( N-1, 4 ) DO 10 I = 1, I4 IF( D( I ).LE.ZERO ) THEN INFO = I GO TO 30 END IF EI = E( I ) E( I ) = EI / D( I ) D( I+1 ) = D( I+1 ) - E( I )*EI 10 CONTINUE ! DO 20 I = I4 + 1, N - 4, 4 ! ! Drop out of the loop if d(i) <= 0: the matrix is not positive ! definite. ! IF( D( I ).LE.ZERO ) THEN INFO = I GO TO 30 END IF ! ! Solve for e(i) and d(i+1). ! EI = E( I ) E( I ) = EI / D( I ) D( I+1 ) = D( I+1 ) - E( I )*EI ! IF( D( I+1 ).LE.ZERO ) THEN INFO = I + 1 GO TO 30 END IF ! ! Solve for e(i+1) and d(i+2). ! EI = E( I+1 ) E( I+1 ) = EI / D( I+1 ) D( I+2 ) = D( I+2 ) - E( I+1 )*EI ! IF( D( I+2 ).LE.ZERO ) THEN INFO = I + 2 GO TO 30 END IF ! ! Solve for e(i+2) and d(i+3). ! EI = E( I+2 ) E( I+2 ) = EI / D( I+2 ) D( I+3 ) = D( I+3 ) - E( I+2 )*EI ! IF( D( I+3 ).LE.ZERO ) THEN INFO = I + 3 GO TO 30 END IF ! ! Solve for e(i+3) and d(i+4). ! EI = E( I+3 ) E( I+3 ) = EI / D( I+3 ) D( I+4 ) = D( I+4 ) - E( I+3 )*EI 20 CONTINUE ! ! Check d(n) for positive definiteness. ! IF( D( N ).LE.ZERO ) & INFO = N ! 30 CONTINUE RETURN ! ! End of DPTTRF ! END SUBROUTINE DPTTRS( N, NRHS, D, E, B, LDB, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDB, N, NRHS ! .. ! .. Array Arguments .. DOUBLE PRECISION B( LDB, * ), D( * ), E( * ) ! .. ! ! Purpose ! ======= ! ! DPTTRS solves a tridiagonal system of the form ! A * X = B ! using the L*D*L' factorization of A computed by DPTTRF. D is a ! diagonal matrix specified in the vector D, L is a unit bidiagonal ! matrix whose subdiagonal is specified in the vector E, and X and B ! are N by NRHS matrices. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the tridiagonal matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of the diagonal matrix D from the ! L*D*L' factorization of A. ! ! E (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) subdiagonal elements of the unit bidiagonal factor ! L from the L*D*L' factorization of A. E can also be regarded ! as the superdiagonal of the unit bidiagonal factor U from the ! factorization A = U'*D*U. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the right hand side vectors B for the system of ! linear equations. ! On exit, the solution vectors, X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -k, the k-th argument had an illegal value ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER J, JB, NB ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. External Subroutines .. EXTERNAL DPTTS2, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input arguments. ! INFO = 0 IF( N.LT.0 ) THEN INFO = -1 ELSE IF( NRHS.LT.0 ) THEN INFO = -2 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPTTRS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) & RETURN ! ! Determine the number of right-hand sides to solve at a time. ! IF( NRHS.EQ.1 ) THEN NB = 1 ELSE NB = MAX( 1, ILAENV( 1, 'DPTTRS', ' ', N, NRHS, -1, -1 ) ) END IF ! IF( NB.GE.NRHS ) THEN CALL DPTTS2( N, NRHS, D, E, B, LDB ) ELSE DO 10 J = 1, NRHS, NB JB = MIN( NRHS-J+1, NB ) CALL DPTTS2( N, JB, D, E, B( 1, J ), LDB ) 10 CONTINUE END IF ! RETURN ! ! End of DPTTRS ! END SUBROUTINE DPTTS2( N, NRHS, D, E, B, LDB ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER LDB, N, NRHS ! .. ! .. Array Arguments .. DOUBLE PRECISION B( LDB, * ), D( * ), E( * ) ! .. ! ! Purpose ! ======= ! ! DPTTS2 solves a tridiagonal system of the form ! A * X = B ! using the L*D*L' factorization of A computed by DPTTRF. D is a ! diagonal matrix specified in the vector D, L is a unit bidiagonal ! matrix whose subdiagonal is specified in the vector E, and X and B ! are N by NRHS matrices. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the tridiagonal matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of the diagonal matrix D from the ! L*D*L' factorization of A. ! ! E (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) subdiagonal elements of the unit bidiagonal factor ! L from the L*D*L' factorization of A. E can also be regarded ! as the superdiagonal of the unit bidiagonal factor U from the ! factorization A = U'*D*U. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the right hand side vectors B for the system of ! linear equations. ! On exit, the solution vectors, X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER I, J ! .. ! .. External Subroutines .. EXTERNAL DSCAL ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( N.LE.1 ) THEN IF( N.EQ.1 ) & CALL DSCAL( NRHS, 1.D0 / D( 1 ), B, LDB ) RETURN END IF ! ! Solve A * X = B using the factorization A = L*D*L', ! overwriting each right hand side vector with its solution. ! DO 30 J = 1, NRHS ! ! Solve L * x = b. ! DO 10 I = 2, N B( I, J ) = B( I, J ) - B( I-1, J )*E( I-1 ) 10 CONTINUE ! ! Solve D * L' * x = b. ! B( N, J ) = B( N, J ) / D( N ) DO 20 I = N - 1, 1, -1 B( I, J ) = B( I, J ) / D( I ) - B( I+1, J )*E( I ) 20 CONTINUE 30 CONTINUE ! RETURN ! ! End of DPTTS2 ! END SUBROUTINE DRSCL( N, SA, SX, INCX ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. INTEGER INCX, N DOUBLE PRECISION SA ! .. ! .. Array Arguments .. DOUBLE PRECISION SX( * ) ! .. ! ! Purpose ! ======= ! ! DRSCL multiplies an n-element real vector x by the real scalar 1/a. ! This is done without overflow or underflow as long as ! the final result x/a does not overflow or underflow. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The number of components of the vector x. ! ! SA (input) DOUBLE PRECISION ! The scalar a which is used to divide each component of x. ! SA must be >= 0, or the subroutine will divide by zero. ! ! SX (input/output) DOUBLE PRECISION array, dimension ! (1+(N-1)*abs(INCX)) ! The n-element vector x. ! ! INCX (input) INTEGER ! The increment between successive values of the vector SX. ! > 0: SX(1) = X(1) and SX(1+(i-1)*INCX) = x(i), 1< i<= n ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL DONE DOUBLE PRECISION BIGNUM, CDEN, CDEN1, CNUM, CNUM1, MUL, SMLNUM ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DSCAL ! .. ! .. Intrinsic Functions .. INTRINSIC ABS ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( N.LE.0 ) & RETURN ! ! Get machine parameters ! SMLNUM = DLAMCH( 'S' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) ! ! Initialize the denominator to SA and the numerator to 1. ! CDEN = SA CNUM = ONE ! 10 CONTINUE CDEN1 = CDEN*SMLNUM CNUM1 = CNUM / BIGNUM IF( ABS( CDEN1 ).GT.ABS( CNUM ) .AND. CNUM.NE.ZERO ) THEN ! ! Pre-multiply X by SMLNUM if CDEN is large compared to CNUM. ! MUL = SMLNUM DONE = .FALSE. CDEN = CDEN1 ELSE IF( ABS( CNUM1 ).GT.ABS( CDEN ) ) THEN ! ! Pre-multiply X by BIGNUM if CDEN is small compared to CNUM. ! MUL = BIGNUM DONE = .FALSE. CNUM = CNUM1 ELSE ! ! Multiply X by CNUM / CDEN and return. ! MUL = CNUM / CDEN DONE = .TRUE. END IF ! ! Scale the vector X by MUL ! CALL DSCAL( N, MUL, SX, INCX ) ! IF( .NOT.DONE ) & GO TO 10 ! RETURN ! ! End of DRSCL ! END SUBROUTINE DSBEV( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, & INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, KD, LDAB, LDZ, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSBEV computes all the eigenvalues and, optionally, eigenvectors of ! a real symmetric band matrix A. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of superdiagonals of the matrix A if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KD >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB, N) ! On entry, the upper or lower triangle of the symmetric band ! matrix A, stored in the first KD+1 rows of the array. The ! j-th column of A is stored in the j-th column of the array AB ! as follows: ! if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). ! ! On exit, AB is overwritten by values generated during the ! reduction to tridiagonal form. If UPLO = 'U', the first ! superdiagonal and the diagonal of the tridiagonal matrix T ! are returned in rows KD and KD+1 of AB, and if UPLO = 'L', ! the diagonal and first subdiagonal of T are returned in the ! first two rows of AB. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KD + 1. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, the eigenvalues in ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, N) ! If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal ! eigenvectors of the matrix A, with the i-th column of Z ! holding the eigenvector associated with W(i). ! If JOBZ = 'N', then Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (max(1,3*N-2)) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the algorithm failed to converge; i ! off-diagonal elements of an intermediate tridiagonal ! form did not converge to zero. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL LOWER, WANTZ INTEGER IINFO, IMAX, INDE, INDWRK, ISCALE DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, & SMLNUM ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANSB EXTERNAL LSAME, DLAMCH, DLANSB ! .. ! .. External Subroutines .. EXTERNAL DLASCL, DSBTRD, DSCAL, DSTEQR, DSTERF, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) LOWER = LSAME( UPLO, 'L' ) ! INFO = 0 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KD.LT.0 ) THEN INFO = -4 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -6 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -9 END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSBEV ', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( N.EQ.1 ) THEN IF( LOWER ) THEN W( 1 ) = AB( 1, 1 ) ELSE W( 1 ) = AB( KD+1, 1 ) END IF IF( WANTZ ) & Z( 1, 1 ) = ONE RETURN END IF ! ! Get machine constants. ! SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = SQRT( BIGNUM ) ! ! Scale matrix to allowable range, if necessary. ! ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK ) ISCALE = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / ANRM ELSE IF( ANRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / ANRM END IF IF( ISCALE.EQ.1 ) THEN IF( LOWER ) THEN CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) ELSE CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) END IF END IF ! ! Call DSBTRD to reduce symmetric band matrix to tridiagonal form. ! INDE = 1 INDWRK = INDE + N CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, W, WORK( INDE ), Z, LDZ, & WORK( INDWRK ), IINFO ) ! ! For eigenvalues only, call DSTERF. For eigenvectors, call SSTEQR. ! IF( .NOT.WANTZ ) THEN CALL DSTERF( N, W, WORK( INDE ), INFO ) ELSE CALL DSTEQR( JOBZ, N, W, WORK( INDE ), Z, LDZ, WORK( INDWRK ), & INFO ) END IF ! ! If matrix was scaled, then rescale eigenvalues appropriately. ! IF( ISCALE.EQ.1 ) THEN IF( INFO.EQ.0 ) THEN IMAX = N ELSE IMAX = INFO - 1 END IF CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) END IF ! RETURN ! ! End of DSBEV ! END SUBROUTINE DSBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, & LWORK, IWORK, LIWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, KD, LDAB, LDZ, LIWORK, LWORK, N ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSBEVD computes all the eigenvalues and, optionally, eigenvectors of ! a real symmetric band matrix A. If eigenvectors are desired, it uses ! a divide and conquer algorithm. ! ! The divide and conquer algorithm makes very mild assumptions about ! floating point arithmetic. It will work on machines with a guard ! digit in add/subtract, or on those binary machines without guard ! digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or ! Cray-2. It could conceivably fail on hexadecimal or decimal machines ! without guard digits, but we know of none. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of superdiagonals of the matrix A if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KD >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB, N) ! On entry, the upper or lower triangle of the symmetric band ! matrix A, stored in the first KD+1 rows of the array. The ! j-th column of A is stored in the j-th column of the array AB ! as follows: ! if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). ! ! On exit, AB is overwritten by values generated during the ! reduction to tridiagonal form. If UPLO = 'U', the first ! superdiagonal and the diagonal of the tridiagonal matrix T ! are returned in rows KD and KD+1 of AB, and if UPLO = 'L', ! the diagonal and first subdiagonal of T are returned in the ! first two rows of AB. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KD + 1. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, the eigenvalues in ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, N) ! If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal ! eigenvectors of the matrix A, with the i-th column of Z ! holding the eigenvector associated with W(i). ! If JOBZ = 'N', then Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! WORK (workspace/output) DOUBLE PRECISION array, ! dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! IF N <= 1, LWORK must be at least 1. ! If JOBZ = 'N' and N > 2, LWORK must be at least 2*N. ! If JOBZ = 'V' and N > 2, LWORK must be at least ! ( 1 + 5*N + 2*N**2 ). ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace/output) INTEGER array, dimension (LIWORK) ! On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. ! ! LIWORK (input) INTEGER ! The dimension of the array LIWORK. ! If JOBZ = 'N' or N <= 1, LIWORK must be at least 1. ! If JOBZ = 'V' and N > 2, LIWORK must be at least 3 + 5*N. ! ! If LIWORK = -1, then a workspace query is assumed; the ! routine only calculates the optimal size of the IWORK array, ! returns this value as the first entry of the IWORK array, and ! no error message related to LIWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the algorithm failed to converge; i ! off-diagonal elements of an intermediate tridiagonal ! form did not converge to zero. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LOWER, LQUERY, WANTZ INTEGER IINFO, INDE, INDWK2, INDWRK, ISCALE, LIWMIN, & LLWRK2, LWMIN DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, & SMLNUM ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANSB EXTERNAL LSAME, DLAMCH, DLANSB ! .. ! .. External Subroutines .. EXTERNAL DGEMM, DLACPY, DLASCL, DSBTRD, DSCAL, DSTEDC, & DSTERF, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) LOWER = LSAME( UPLO, 'L' ) LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) ! INFO = 0 IF( N.LE.1 ) THEN LIWMIN = 1 LWMIN = 1 ELSE IF( WANTZ ) THEN LIWMIN = 3 + 5*N LWMIN = 1 + 5*N + 2*N**2 ELSE LIWMIN = 1 LWMIN = 2*N END IF END IF IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KD.LT.0 ) THEN INFO = -4 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -6 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -9 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -11 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -13 END IF ! IF( INFO.EQ.0 ) THEN WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSBEVD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( N.EQ.1 ) THEN W( 1 ) = AB( 1, 1 ) IF( WANTZ ) & Z( 1, 1 ) = ONE RETURN END IF ! ! Get machine constants. ! SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = SQRT( BIGNUM ) ! ! Scale matrix to allowable range, if necessary. ! ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK ) ISCALE = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / ANRM ELSE IF( ANRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / ANRM END IF IF( ISCALE.EQ.1 ) THEN IF( LOWER ) THEN CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) ELSE CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) END IF END IF ! ! Call DSBTRD to reduce symmetric band matrix to tridiagonal form. ! INDE = 1 INDWRK = INDE + N INDWK2 = INDWRK + N*N LLWRK2 = LWORK - INDWK2 + 1 CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, W, WORK( INDE ), Z, LDZ, & WORK( INDWRK ), IINFO ) ! ! For eigenvalues only, call DSTERF. For eigenvectors, call SSTEDC. ! IF( .NOT.WANTZ ) THEN CALL DSTERF( N, W, WORK( INDE ), INFO ) ELSE CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N, & WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO ) CALL DGEMM( 'N', 'N', N, N, N, ONE, Z, LDZ, WORK( INDWRK ), N, & ZERO, WORK( INDWK2 ), N ) CALL DLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ ) END IF ! ! If matrix was scaled, then rescale eigenvalues appropriately. ! IF( ISCALE.EQ.1 ) & CALL DSCAL( N, ONE / SIGMA, W, 1 ) ! WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN RETURN ! ! End of DSBEVD ! END SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, & VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, & IFAIL, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, RANGE, UPLO INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N DOUBLE PRECISION ABSTOL, VL, VU ! .. ! .. Array Arguments .. INTEGER IFAIL( * ), IWORK( * ) DOUBLE PRECISION AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ), & Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSBEVX computes selected eigenvalues and, optionally, eigenvectors ! of a real symmetric band matrix A. Eigenvalues and eigenvectors can ! be selected by specifying either a range of values or a range of ! indices for the desired eigenvalues. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! RANGE (input) CHARACTER*1 ! = 'A': all eigenvalues will be found; ! = 'V': all eigenvalues in the half-open interval (VL,VU] ! will be found; ! = 'I': the IL-th through IU-th eigenvalues will be found. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of superdiagonals of the matrix A if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KD >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB, N) ! On entry, the upper or lower triangle of the symmetric band ! matrix A, stored in the first KD+1 rows of the array. The ! j-th column of A is stored in the j-th column of the array AB ! as follows: ! if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). ! ! On exit, AB is overwritten by values generated during the ! reduction to tridiagonal form. If UPLO = 'U', the first ! superdiagonal and the diagonal of the tridiagonal matrix T ! are returned in rows KD and KD+1 of AB, and if UPLO = 'L', ! the diagonal and first subdiagonal of T are returned in the ! first two rows of AB. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KD + 1. ! ! Q (output) DOUBLE PRECISION array, dimension (LDQ, N) ! If JOBZ = 'V', the N-by-N orthogonal matrix used in the ! reduction to tridiagonal form. ! If JOBZ = 'N', the array Q is not referenced. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. If JOBZ = 'V', then ! LDQ >= max(1,N). ! ! VL (input) DOUBLE PRECISION ! VU (input) DOUBLE PRECISION ! If RANGE='V', the lower and upper bounds of the interval to ! be searched for eigenvalues. VL < VU. ! Not referenced if RANGE = 'A' or 'I'. ! ! IL (input) INTEGER ! IU (input) INTEGER ! If RANGE='I', the indices (in ascending order) of the ! smallest and largest eigenvalues to be returned. ! 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. ! Not referenced if RANGE = 'A' or 'V'. ! ! ABSTOL (input) DOUBLE PRECISION ! The absolute error tolerance for the eigenvalues. ! An approximate eigenvalue is accepted as converged ! when it is determined to lie in an interval [a,b] ! of width less than or equal to ! ! ABSTOL + EPS * max( |a|,|b| ) , ! ! where EPS is the machine precision. If ABSTOL is less than ! or equal to zero, then EPS*|T| will be used in its place, ! where |T| is the 1-norm of the tridiagonal matrix obtained ! by reducing AB to tridiagonal form. ! ! Eigenvalues will be computed most accurately when ABSTOL is ! set to twice the underflow threshold 2*DLAMCH('S'), not zero. ! If this routine returns with INFO>0, indicating that some ! eigenvectors did not converge, try setting ABSTOL to ! 2*DLAMCH('S'). ! ! See "Computing Small Singular Values of Bidiagonal Matrices ! with Guaranteed High Relative Accuracy," by Demmel and ! Kahan, LAPACK Working Note #3. ! ! M (output) INTEGER ! The total number of eigenvalues found. 0 <= M <= N. ! If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! The first M elements contain the selected eigenvalues in ! ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M)) ! If JOBZ = 'V', then if INFO = 0, the first M columns of Z ! contain the orthonormal eigenvectors of the matrix A ! corresponding to the selected eigenvalues, with the i-th ! column of Z holding the eigenvector associated with W(i). ! If an eigenvector fails to converge, then that column of Z ! contains the latest approximation to the eigenvector, and the ! index of the eigenvector is returned in IFAIL. ! If JOBZ = 'N', then Z is not referenced. ! Note: the user must ensure that at least max(1,M) columns are ! supplied in the array Z; if RANGE = 'V', the exact value of M ! is not known in advance and an upper bound must be used. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (7*N) ! ! IWORK (workspace) INTEGER array, dimension (5*N) ! ! IFAIL (output) INTEGER array, dimension (N) ! If JOBZ = 'V', then if INFO = 0, the first M elements of ! IFAIL are zero. If INFO > 0, then IFAIL contains the ! indices of the eigenvectors that failed to converge. ! If JOBZ = 'N', then IFAIL is not referenced. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = i, then i eigenvectors failed to converge. ! Their indices are stored in array IFAIL. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL ALLEIG, INDEIG, LOWER, VALEIG, WANTZ CHARACTER ORDER INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL, & INDISP, INDIWO, INDWRK, ISCALE, ITMP1, J, JJ, & NSPLIT DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, & SIGMA, SMLNUM, TMP1, VLL, VUU ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANSB EXTERNAL LSAME, DLAMCH, DLANSB ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEMV, DLACPY, DLASCL, DSBTRD, DSCAL, & DSTEBZ, DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) ALLEIG = LSAME( RANGE, 'A' ) VALEIG = LSAME( RANGE, 'V' ) INDEIG = LSAME( RANGE, 'I' ) LOWER = LSAME( UPLO, 'L' ) ! INFO = 0 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN INFO = -2 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( KD.LT.0 ) THEN INFO = -5 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -7 ELSE IF( WANTZ .AND. LDQ.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( VALEIG ) THEN IF( N.GT.0 .AND. VU.LE.VL ) & INFO = -11 ELSE IF( INDEIG ) THEN IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN INFO = -12 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN INFO = -13 END IF END IF END IF IF( INFO.EQ.0 ) THEN IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) & INFO = -18 END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSBEVX', -INFO ) RETURN END IF ! ! Quick return if possible ! M = 0 IF( N.EQ.0 ) & RETURN ! IF( N.EQ.1 ) THEN M = 1 IF( LOWER ) THEN TMP1 = AB( 1, 1 ) ELSE TMP1 = AB( KD+1, 1 ) END IF IF( VALEIG ) THEN IF( .NOT.( VL.LT.TMP1 .AND. VU.GE.TMP1 ) ) & M = 0 END IF IF( M.EQ.1 ) THEN W( 1 ) = TMP1 IF( WANTZ ) & Z( 1, 1 ) = ONE END IF RETURN END IF ! ! Get machine constants. ! SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) ! ! Scale matrix to allowable range, if necessary. ! ISCALE = 0 ABSTLL = ABSTOL IF( VALEIG ) THEN VLL = VL VUU = VU ELSE VLL = ZERO VUU = ZERO END IF ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK ) IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / ANRM ELSE IF( ANRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / ANRM END IF IF( ISCALE.EQ.1 ) THEN IF( LOWER ) THEN CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) ELSE CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) END IF IF( ABSTOL.GT.0 ) & ABSTLL = ABSTOL*SIGMA IF( VALEIG ) THEN VLL = VL*SIGMA VUU = VU*SIGMA END IF END IF ! ! Call DSBTRD to reduce symmetric band matrix to tridiagonal form. ! INDD = 1 INDE = INDD + N INDWRK = INDE + N CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, WORK( INDD ), & WORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO ) ! ! If all eigenvalues are desired and ABSTOL is less than or equal ! to zero, then call DSTERF or SSTEQR. If this fails for some ! eigenvalue, then try DSTEBZ. ! IF( ( ALLEIG .OR. ( INDEIG .AND. IL.EQ.1 .AND. IU.EQ.N ) ) .AND. & ( ABSTOL.LE.ZERO ) ) THEN CALL DCOPY( N, WORK( INDD ), 1, W, 1 ) INDEE = INDWRK + 2*N IF( .NOT.WANTZ ) THEN CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) CALL DSTERF( N, W, WORK( INDEE ), INFO ) ELSE CALL DLACPY( 'A', N, N, Q, LDQ, Z, LDZ ) CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ, & WORK( INDWRK ), INFO ) IF( INFO.EQ.0 ) THEN DO 10 I = 1, N IFAIL( I ) = 0 10 CONTINUE END IF END IF IF( INFO.EQ.0 ) THEN M = N GO TO 30 END IF INFO = 0 END IF ! ! Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN. ! IF( WANTZ ) THEN ORDER = 'B' ELSE ORDER = 'E' END IF INDIBL = 1 INDISP = INDIBL + N INDIWO = INDISP + N CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, & WORK( INDD ), WORK( INDE ), M, NSPLIT, W, & IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ), & IWORK( INDIWO ), INFO ) ! IF( WANTZ ) THEN CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W, & IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, & WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO ) ! ! Apply orthogonal matrix used in reduction to tridiagonal ! form to eigenvectors returned by DSTEIN. ! DO 20 J = 1, M CALL DCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 ) CALL DGEMV( 'N', N, N, ONE, Q, LDQ, WORK, 1, ZERO, & Z( 1, J ), 1 ) 20 CONTINUE END IF ! ! If matrix was scaled, then rescale eigenvalues appropriately. ! 30 CONTINUE IF( ISCALE.EQ.1 ) THEN IF( INFO.EQ.0 ) THEN IMAX = M ELSE IMAX = INFO - 1 END IF CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) END IF ! ! If eigenvalues are not in order, then sort them, along with ! eigenvectors. ! IF( WANTZ ) THEN DO 50 J = 1, M - 1 I = 0 TMP1 = W( J ) DO 40 JJ = J + 1, M IF( W( JJ ).LT.TMP1 ) THEN I = JJ TMP1 = W( JJ ) END IF 40 CONTINUE ! IF( I.NE.0 ) THEN ITMP1 = IWORK( INDIBL+I-1 ) W( I ) = W( J ) IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) W( J ) = TMP1 IWORK( INDIBL+J-1 ) = ITMP1 CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) IF( INFO.NE.0 ) THEN ITMP1 = IFAIL( I ) IFAIL( I ) = IFAIL( J ) IFAIL( J ) = ITMP1 END IF END IF 50 CONTINUE END IF ! RETURN ! ! End of DSBEVX ! END SUBROUTINE DSBGST( VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X, & LDX, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER UPLO, VECT INTEGER INFO, KA, KB, LDAB, LDBB, LDX, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), WORK( * ), & X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DSBGST reduces a real symmetric-definite banded generalized ! eigenproblem A*x = lambda*B*x to standard form C*y = lambda*y, ! such that C has the same bandwidth as A. ! ! B must have been previously factorized as S**T*S by DPBSTF, using a ! split Cholesky factorization. A is overwritten by C = X**T*A*X, where ! X = S**(-1)*Q and Q is an orthogonal matrix chosen to preserve the ! bandwidth of A. ! ! Arguments ! ========= ! ! VECT (input) CHARACTER*1 ! = 'N': do not form the transformation matrix X; ! = 'V': form X. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! KA (input) INTEGER ! The number of superdiagonals of the matrix A if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KA >= 0. ! ! KB (input) INTEGER ! The number of superdiagonals of the matrix B if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KA >= KB >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) ! On entry, the upper or lower triangle of the symmetric band ! matrix A, stored in the first ka+1 rows of the array. The ! j-th column of A is stored in the j-th column of the array AB ! as follows: ! if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). ! ! On exit, the transformed matrix X**T*A*X, stored in the same ! format as A. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KA+1. ! ! BB (input) DOUBLE PRECISION array, dimension (LDBB,N) ! The banded factor S from the split Cholesky factorization of ! B, as returned by DPBSTF, stored in the first KB+1 rows of ! the array. ! ! LDBB (input) INTEGER ! The leading dimension of the array BB. LDBB >= KB+1. ! ! X (output) DOUBLE PRECISION array, dimension (LDX,N) ! If VECT = 'V', the n-by-n matrix X. ! If VECT = 'N', the array X is not referenced. ! ! LDX (input) INTEGER ! The leading dimension of the array X. ! LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (2*N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPDATE, UPPER, WANTX INTEGER I, I0, I1, I2, INCA, J, J1, J1T, J2, J2T, K, & KA1, KB1, KBT, L, M, NR, NRT, NX DOUBLE PRECISION BII, RA, RA1, T ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DGER, DLAR2V, DLARGV, DLARTG, DLARTV, DLASET, & DROT, DSCAL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters ! WANTX = LSAME( VECT, 'V' ) UPPER = LSAME( UPLO, 'U' ) KA1 = KA + 1 KB1 = KB + 1 INFO = 0 IF( .NOT.WANTX .AND. .NOT.LSAME( VECT, 'N' ) ) THEN INFO = -1 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KA.LT.0 ) THEN INFO = -4 ELSE IF( KB.LT.0 ) THEN INFO = -5 ELSE IF( LDAB.LT.KA+1 ) THEN INFO = -7 ELSE IF( LDBB.LT.KB+1 ) THEN INFO = -9 ELSE IF( LDX.LT.1 .OR. WANTX .AND. LDX.LT.MAX( 1, N ) ) THEN INFO = -11 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSBGST', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! INCA = LDAB*KA1 ! ! Initialize X to the unit matrix, if needed ! IF( WANTX ) & CALL DLASET( 'Full', N, N, ZERO, ONE, X, LDX ) ! ! Set M to the splitting point m. It must be the same value as is ! used in DPBSTF. The chosen value allows the arrays WORK and RWORK ! to be of dimension (N). ! M = ( N+KB ) / 2 ! ! The routine works in two phases, corresponding to the two halves ! of the split Cholesky factorization of B as S**T*S where ! ! S = ( U ) ! ( M L ) ! ! with U upper triangular of order m, and L lower triangular of ! order n-m. S has the same bandwidth as B. ! ! S is treated as a product of elementary matrices: ! ! S = S(m)*S(m-1)*...*S(2)*S(1)*S(m+1)*S(m+2)*...*S(n-1)*S(n) ! ! where S(i) is determined by the i-th row of S. ! ! In phase 1, the index i takes the values n, n-1, ... , m+1; ! in phase 2, it takes the values 1, 2, ... , m. ! ! For each value of i, the current matrix A is updated by forming ! inv(S(i))**T*A*inv(S(i)). This creates a triangular bulge outside ! the band of A. The bulge is then pushed down toward the bottom of ! A in phase 1, and up toward the top of A in phase 2, by applying ! plane rotations. ! ! There are kb*(kb+1)/2 elements in the bulge, but at most 2*kb-1 ! of them are linearly independent, so annihilating a bulge requires ! only 2*kb-1 plane rotations. The rotations are divided into a 1st ! set of kb-1 rotations, and a 2nd set of kb rotations. ! ! Wherever possible, rotations are generated and applied in vector ! operations of length NR between the indices J1 and J2 (sometimes ! replaced by modified values NRT, J1T or J2T). ! ! The cosines and sines of the rotations are stored in the array ! WORK. The cosines of the 1st set of rotations are stored in ! elements n+2:n+m-kb-1 and the sines of the 1st set in elements ! 2:m-kb-1; the cosines of the 2nd set are stored in elements ! n+m-kb+1:2*n and the sines of the second set in elements m-kb+1:n. ! ! The bulges are not formed explicitly; nonzero elements outside the ! band are created only when they are required for generating new ! rotations; they are stored in the array WORK, in positions where ! they are later overwritten by the sines of the rotations which ! annihilate them. ! ! **************************** Phase 1 ***************************** ! ! The logical structure of this phase is: ! ! UPDATE = .TRUE. ! DO I = N, M + 1, -1 ! use S(i) to update A and create a new bulge ! apply rotations to push all bulges KA positions downward ! END DO ! UPDATE = .FALSE. ! DO I = M + KA + 1, N - 1 ! apply rotations to push all bulges KA positions downward ! END DO ! ! To avoid duplicating code, the two loops are merged. ! UPDATE = .TRUE. I = N + 1 10 CONTINUE IF( UPDATE ) THEN I = I - 1 KBT = MIN( KB, I-1 ) I0 = I - 1 I1 = MIN( N, I+KA ) I2 = I - KBT + KA1 IF( I.LT.M+1 ) THEN UPDATE = .FALSE. I = I + 1 I0 = M IF( KA.EQ.0 ) & GO TO 480 GO TO 10 END IF ELSE I = I + KA IF( I.GT.N-1 ) & GO TO 480 END IF ! IF( UPPER ) THEN ! ! Transform A, working with the upper triangle ! IF( UPDATE ) THEN ! ! Form inv(S(i))**T * A * inv(S(i)) ! BII = BB( KB1, I ) DO 20 J = I, I1 AB( I-J+KA1, J ) = AB( I-J+KA1, J ) / BII 20 CONTINUE DO 30 J = MAX( 1, I-KA ), I AB( J-I+KA1, I ) = AB( J-I+KA1, I ) / BII 30 CONTINUE DO 60 K = I - KBT, I - 1 DO 40 J = I - KBT, K AB( J-K+KA1, K ) = AB( J-K+KA1, K ) - & BB( J-I+KB1, I )*AB( K-I+KA1, I ) - & BB( K-I+KB1, I )*AB( J-I+KA1, I ) + & AB( KA1, I )*BB( J-I+KB1, I )* & BB( K-I+KB1, I ) 40 CONTINUE DO 50 J = MAX( 1, I-KA ), I - KBT - 1 AB( J-K+KA1, K ) = AB( J-K+KA1, K ) - & BB( K-I+KB1, I )*AB( J-I+KA1, I ) 50 CONTINUE 60 CONTINUE DO 80 J = I, I1 DO 70 K = MAX( J-KA, I-KBT ), I - 1 AB( K-J+KA1, J ) = AB( K-J+KA1, J ) - & BB( K-I+KB1, I )*AB( I-J+KA1, J ) 70 CONTINUE 80 CONTINUE ! IF( WANTX ) THEN ! ! post-multiply X by inv(S(i)) ! CALL DSCAL( N-M, ONE / BII, X( M+1, I ), 1 ) IF( KBT.GT.0 ) & CALL DGER( N-M, KBT, -ONE, X( M+1, I ), 1, & BB( KB1-KBT, I ), 1, X( M+1, I-KBT ), LDX ) END IF ! ! store a(i,i1) in RA1 for use in next loop over K ! RA1 = AB( I-I1+KA1, I1 ) END IF ! ! Generate and apply vectors of rotations to chase all the ! existing bulges KA positions down toward the bottom of the ! band ! DO 130 K = 1, KB - 1 IF( UPDATE ) THEN ! ! Determine the rotations which would annihilate the bulge ! which has in theory just been created ! IF( I-K+KA.LT.N .AND. I-K.GT.1 ) THEN ! ! generate rotation to annihilate a(i,i-k+ka+1) ! CALL DLARTG( AB( K+1, I-K+KA ), RA1, & WORK( N+I-K+KA-M ), WORK( I-K+KA-M ), & RA ) ! ! create nonzero element a(i-k,i-k+ka+1) outside the ! band and store it in WORK(i-k) ! T = -BB( KB1-K, I )*RA1 WORK( I-K ) = WORK( N+I-K+KA-M )*T - & WORK( I-K+KA-M )*AB( 1, I-K+KA ) AB( 1, I-K+KA ) = WORK( I-K+KA-M )*T + & WORK( N+I-K+KA-M )*AB( 1, I-K+KA ) RA1 = RA END IF END IF J2 = I - K - 1 + MAX( 1, K-I0+2 )*KA1 NR = ( N-J2+KA ) / KA1 J1 = J2 + ( NR-1 )*KA1 IF( UPDATE ) THEN J2T = MAX( J2, I+2*KA-K+1 ) ELSE J2T = J2 END IF NRT = ( N-J2T+KA ) / KA1 DO 90 J = J2T, J1, KA1 ! ! create nonzero element a(j-ka,j+1) outside the band ! and store it in WORK(j-m) ! WORK( J-M ) = WORK( J-M )*AB( 1, J+1 ) AB( 1, J+1 ) = WORK( N+J-M )*AB( 1, J+1 ) 90 CONTINUE ! ! generate rotations in 1st set to annihilate elements which ! have been created outside the band ! IF( NRT.GT.0 ) & CALL DLARGV( NRT, AB( 1, J2T ), INCA, WORK( J2T-M ), KA1, & WORK( N+J2T-M ), KA1 ) IF( NR.GT.0 ) THEN ! ! apply rotations in 1st set from the right ! DO 100 L = 1, KA - 1 CALL DLARTV( NR, AB( KA1-L, J2 ), INCA, & AB( KA-L, J2+1 ), INCA, WORK( N+J2-M ), & WORK( J2-M ), KA1 ) 100 CONTINUE ! ! apply rotations in 1st set from both sides to diagonal ! blocks ! CALL DLAR2V( NR, AB( KA1, J2 ), AB( KA1, J2+1 ), & AB( KA, J2+1 ), INCA, WORK( N+J2-M ), & WORK( J2-M ), KA1 ) ! END IF ! ! start applying rotations in 1st set from the left ! DO 110 L = KA - 1, KB - K + 1, -1 NRT = ( N-J2+L ) / KA1 IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( L, J2+KA1-L ), INCA, & AB( L+1, J2+KA1-L ), INCA, & WORK( N+J2-M ), WORK( J2-M ), KA1 ) 110 CONTINUE ! IF( WANTX ) THEN ! ! post-multiply X by product of rotations in 1st set ! DO 120 J = J2, J1, KA1 CALL DROT( N-M, X( M+1, J ), 1, X( M+1, J+1 ), 1, & WORK( N+J-M ), WORK( J-M ) ) 120 CONTINUE END IF 130 CONTINUE ! IF( UPDATE ) THEN IF( I2.LE.N .AND. KBT.GT.0 ) THEN ! ! create nonzero element a(i-kbt,i-kbt+ka+1) outside the ! band and store it in WORK(i-kbt) ! WORK( I-KBT ) = -BB( KB1-KBT, I )*RA1 END IF END IF ! DO 170 K = KB, 1, -1 IF( UPDATE ) THEN J2 = I - K - 1 + MAX( 2, K-I0+1 )*KA1 ELSE J2 = I - K - 1 + MAX( 1, K-I0+1 )*KA1 END IF ! ! finish applying rotations in 2nd set from the left ! DO 140 L = KB - K, 1, -1 NRT = ( N-J2+KA+L ) / KA1 IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( L, J2-L+1 ), INCA, & AB( L+1, J2-L+1 ), INCA, WORK( N+J2-KA ), & WORK( J2-KA ), KA1 ) 140 CONTINUE NR = ( N-J2+KA ) / KA1 J1 = J2 + ( NR-1 )*KA1 DO 150 J = J1, J2, -KA1 WORK( J ) = WORK( J-KA ) WORK( N+J ) = WORK( N+J-KA ) 150 CONTINUE DO 160 J = J2, J1, KA1 ! ! create nonzero element a(j-ka,j+1) outside the band ! and store it in WORK(j) ! WORK( J ) = WORK( J )*AB( 1, J+1 ) AB( 1, J+1 ) = WORK( N+J )*AB( 1, J+1 ) 160 CONTINUE IF( UPDATE ) THEN IF( I-K.LT.N-KA .AND. K.LE.KBT ) & WORK( I-K+KA ) = WORK( I-K ) END IF 170 CONTINUE ! DO 210 K = KB, 1, -1 J2 = I - K - 1 + MAX( 1, K-I0+1 )*KA1 NR = ( N-J2+KA ) / KA1 J1 = J2 + ( NR-1 )*KA1 IF( NR.GT.0 ) THEN ! ! generate rotations in 2nd set to annihilate elements ! which have been created outside the band ! CALL DLARGV( NR, AB( 1, J2 ), INCA, WORK( J2 ), KA1, & WORK( N+J2 ), KA1 ) ! ! apply rotations in 2nd set from the right ! DO 180 L = 1, KA - 1 CALL DLARTV( NR, AB( KA1-L, J2 ), INCA, & AB( KA-L, J2+1 ), INCA, WORK( N+J2 ), & WORK( J2 ), KA1 ) 180 CONTINUE ! ! apply rotations in 2nd set from both sides to diagonal ! blocks ! CALL DLAR2V( NR, AB( KA1, J2 ), AB( KA1, J2+1 ), & AB( KA, J2+1 ), INCA, WORK( N+J2 ), & WORK( J2 ), KA1 ) ! END IF ! ! start applying rotations in 2nd set from the left ! DO 190 L = KA - 1, KB - K + 1, -1 NRT = ( N-J2+L ) / KA1 IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( L, J2+KA1-L ), INCA, & AB( L+1, J2+KA1-L ), INCA, WORK( N+J2 ), & WORK( J2 ), KA1 ) 190 CONTINUE ! IF( WANTX ) THEN ! ! post-multiply X by product of rotations in 2nd set ! DO 200 J = J2, J1, KA1 CALL DROT( N-M, X( M+1, J ), 1, X( M+1, J+1 ), 1, & WORK( N+J ), WORK( J ) ) 200 CONTINUE END IF 210 CONTINUE ! DO 230 K = 1, KB - 1 J2 = I - K - 1 + MAX( 1, K-I0+2 )*KA1 ! ! finish applying rotations in 1st set from the left ! DO 220 L = KB - K, 1, -1 NRT = ( N-J2+L ) / KA1 IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( L, J2+KA1-L ), INCA, & AB( L+1, J2+KA1-L ), INCA, & WORK( N+J2-M ), WORK( J2-M ), KA1 ) 220 CONTINUE 230 CONTINUE ! IF( KB.GT.1 ) THEN DO 240 J = N - 1, I - KB + 2*KA + 1, -1 WORK( N+J-M ) = WORK( N+J-KA-M ) WORK( J-M ) = WORK( J-KA-M ) 240 CONTINUE END IF ! ELSE ! ! Transform A, working with the lower triangle ! IF( UPDATE ) THEN ! ! Form inv(S(i))**T * A * inv(S(i)) ! BII = BB( 1, I ) DO 250 J = I, I1 AB( J-I+1, I ) = AB( J-I+1, I ) / BII 250 CONTINUE DO 260 J = MAX( 1, I-KA ), I AB( I-J+1, J ) = AB( I-J+1, J ) / BII 260 CONTINUE DO 290 K = I - KBT, I - 1 DO 270 J = I - KBT, K AB( K-J+1, J ) = AB( K-J+1, J ) - & BB( I-J+1, J )*AB( I-K+1, K ) - & BB( I-K+1, K )*AB( I-J+1, J ) + & AB( 1, I )*BB( I-J+1, J )* & BB( I-K+1, K ) 270 CONTINUE DO 280 J = MAX( 1, I-KA ), I - KBT - 1 AB( K-J+1, J ) = AB( K-J+1, J ) - & BB( I-K+1, K )*AB( I-J+1, J ) 280 CONTINUE 290 CONTINUE DO 310 J = I, I1 DO 300 K = MAX( J-KA, I-KBT ), I - 1 AB( J-K+1, K ) = AB( J-K+1, K ) - & BB( I-K+1, K )*AB( J-I+1, I ) 300 CONTINUE 310 CONTINUE ! IF( WANTX ) THEN ! ! post-multiply X by inv(S(i)) ! CALL DSCAL( N-M, ONE / BII, X( M+1, I ), 1 ) IF( KBT.GT.0 ) & CALL DGER( N-M, KBT, -ONE, X( M+1, I ), 1, & BB( KBT+1, I-KBT ), LDBB-1, & X( M+1, I-KBT ), LDX ) END IF ! ! store a(i1,i) in RA1 for use in next loop over K ! RA1 = AB( I1-I+1, I ) END IF ! ! Generate and apply vectors of rotations to chase all the ! existing bulges KA positions down toward the bottom of the ! band ! DO 360 K = 1, KB - 1 IF( UPDATE ) THEN ! ! Determine the rotations which would annihilate the bulge ! which has in theory just been created ! IF( I-K+KA.LT.N .AND. I-K.GT.1 ) THEN ! ! generate rotation to annihilate a(i-k+ka+1,i) ! CALL DLARTG( AB( KA1-K, I ), RA1, WORK( N+I-K+KA-M ), & WORK( I-K+KA-M ), RA ) ! ! create nonzero element a(i-k+ka+1,i-k) outside the ! band and store it in WORK(i-k) ! T = -BB( K+1, I-K )*RA1 WORK( I-K ) = WORK( N+I-K+KA-M )*T - & WORK( I-K+KA-M )*AB( KA1, I-K ) AB( KA1, I-K ) = WORK( I-K+KA-M )*T + & WORK( N+I-K+KA-M )*AB( KA1, I-K ) RA1 = RA END IF END IF J2 = I - K - 1 + MAX( 1, K-I0+2 )*KA1 NR = ( N-J2+KA ) / KA1 J1 = J2 + ( NR-1 )*KA1 IF( UPDATE ) THEN J2T = MAX( J2, I+2*KA-K+1 ) ELSE J2T = J2 END IF NRT = ( N-J2T+KA ) / KA1 DO 320 J = J2T, J1, KA1 ! ! create nonzero element a(j+1,j-ka) outside the band ! and store it in WORK(j-m) ! WORK( J-M ) = WORK( J-M )*AB( KA1, J-KA+1 ) AB( KA1, J-KA+1 ) = WORK( N+J-M )*AB( KA1, J-KA+1 ) 320 CONTINUE ! ! generate rotations in 1st set to annihilate elements which ! have been created outside the band ! IF( NRT.GT.0 ) & CALL DLARGV( NRT, AB( KA1, J2T-KA ), INCA, WORK( J2T-M ), & KA1, WORK( N+J2T-M ), KA1 ) IF( NR.GT.0 ) THEN ! ! apply rotations in 1st set from the left ! DO 330 L = 1, KA - 1 CALL DLARTV( NR, AB( L+1, J2-L ), INCA, & AB( L+2, J2-L ), INCA, WORK( N+J2-M ), & WORK( J2-M ), KA1 ) 330 CONTINUE ! ! apply rotations in 1st set from both sides to diagonal ! blocks ! CALL DLAR2V( NR, AB( 1, J2 ), AB( 1, J2+1 ), AB( 2, J2 ), & INCA, WORK( N+J2-M ), WORK( J2-M ), KA1 ) ! END IF ! ! start applying rotations in 1st set from the right ! DO 340 L = KA - 1, KB - K + 1, -1 NRT = ( N-J2+L ) / KA1 IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( KA1-L+1, J2 ), INCA, & AB( KA1-L, J2+1 ), INCA, WORK( N+J2-M ), & WORK( J2-M ), KA1 ) 340 CONTINUE ! IF( WANTX ) THEN ! ! post-multiply X by product of rotations in 1st set ! DO 350 J = J2, J1, KA1 CALL DROT( N-M, X( M+1, J ), 1, X( M+1, J+1 ), 1, & WORK( N+J-M ), WORK( J-M ) ) 350 CONTINUE END IF 360 CONTINUE ! IF( UPDATE ) THEN IF( I2.LE.N .AND. KBT.GT.0 ) THEN ! ! create nonzero element a(i-kbt+ka+1,i-kbt) outside the ! band and store it in WORK(i-kbt) ! WORK( I-KBT ) = -BB( KBT+1, I-KBT )*RA1 END IF END IF ! DO 400 K = KB, 1, -1 IF( UPDATE ) THEN J2 = I - K - 1 + MAX( 2, K-I0+1 )*KA1 ELSE J2 = I - K - 1 + MAX( 1, K-I0+1 )*KA1 END IF ! ! finish applying rotations in 2nd set from the right ! DO 370 L = KB - K, 1, -1 NRT = ( N-J2+KA+L ) / KA1 IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( KA1-L+1, J2-KA ), INCA, & AB( KA1-L, J2-KA+1 ), INCA, & WORK( N+J2-KA ), WORK( J2-KA ), KA1 ) 370 CONTINUE NR = ( N-J2+KA ) / KA1 J1 = J2 + ( NR-1 )*KA1 DO 380 J = J1, J2, -KA1 WORK( J ) = WORK( J-KA ) WORK( N+J ) = WORK( N+J-KA ) 380 CONTINUE DO 390 J = J2, J1, KA1 ! ! create nonzero element a(j+1,j-ka) outside the band ! and store it in WORK(j) ! WORK( J ) = WORK( J )*AB( KA1, J-KA+1 ) AB( KA1, J-KA+1 ) = WORK( N+J )*AB( KA1, J-KA+1 ) 390 CONTINUE IF( UPDATE ) THEN IF( I-K.LT.N-KA .AND. K.LE.KBT ) & WORK( I-K+KA ) = WORK( I-K ) END IF 400 CONTINUE ! DO 440 K = KB, 1, -1 J2 = I - K - 1 + MAX( 1, K-I0+1 )*KA1 NR = ( N-J2+KA ) / KA1 J1 = J2 + ( NR-1 )*KA1 IF( NR.GT.0 ) THEN ! ! generate rotations in 2nd set to annihilate elements ! which have been created outside the band ! CALL DLARGV( NR, AB( KA1, J2-KA ), INCA, WORK( J2 ), KA1, & WORK( N+J2 ), KA1 ) ! ! apply rotations in 2nd set from the left ! DO 410 L = 1, KA - 1 CALL DLARTV( NR, AB( L+1, J2-L ), INCA, & AB( L+2, J2-L ), INCA, WORK( N+J2 ), & WORK( J2 ), KA1 ) 410 CONTINUE ! ! apply rotations in 2nd set from both sides to diagonal ! blocks ! CALL DLAR2V( NR, AB( 1, J2 ), AB( 1, J2+1 ), AB( 2, J2 ), & INCA, WORK( N+J2 ), WORK( J2 ), KA1 ) ! END IF ! ! start applying rotations in 2nd set from the right ! DO 420 L = KA - 1, KB - K + 1, -1 NRT = ( N-J2+L ) / KA1 IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( KA1-L+1, J2 ), INCA, & AB( KA1-L, J2+1 ), INCA, WORK( N+J2 ), & WORK( J2 ), KA1 ) 420 CONTINUE ! IF( WANTX ) THEN ! ! post-multiply X by product of rotations in 2nd set ! DO 430 J = J2, J1, KA1 CALL DROT( N-M, X( M+1, J ), 1, X( M+1, J+1 ), 1, & WORK( N+J ), WORK( J ) ) 430 CONTINUE END IF 440 CONTINUE ! DO 460 K = 1, KB - 1 J2 = I - K - 1 + MAX( 1, K-I0+2 )*KA1 ! ! finish applying rotations in 1st set from the right ! DO 450 L = KB - K, 1, -1 NRT = ( N-J2+L ) / KA1 IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( KA1-L+1, J2 ), INCA, & AB( KA1-L, J2+1 ), INCA, WORK( N+J2-M ), & WORK( J2-M ), KA1 ) 450 CONTINUE 460 CONTINUE ! IF( KB.GT.1 ) THEN DO 470 J = N - 1, I - KB + 2*KA + 1, -1 WORK( N+J-M ) = WORK( N+J-KA-M ) WORK( J-M ) = WORK( J-KA-M ) 470 CONTINUE END IF ! END IF ! GO TO 10 ! 480 CONTINUE ! ! **************************** Phase 2 ***************************** ! ! The logical structure of this phase is: ! ! UPDATE = .TRUE. ! DO I = 1, M ! use S(i) to update A and create a new bulge ! apply rotations to push all bulges KA positions upward ! END DO ! UPDATE = .FALSE. ! DO I = M - KA - 1, 2, -1 ! apply rotations to push all bulges KA positions upward ! END DO ! ! To avoid duplicating code, the two loops are merged. ! UPDATE = .TRUE. I = 0 490 CONTINUE IF( UPDATE ) THEN I = I + 1 KBT = MIN( KB, M-I ) I0 = I + 1 I1 = MAX( 1, I-KA ) I2 = I + KBT - KA1 IF( I.GT.M ) THEN UPDATE = .FALSE. I = I - 1 I0 = M + 1 IF( KA.EQ.0 ) & RETURN GO TO 490 END IF ELSE I = I - KA IF( I.LT.2 ) & RETURN END IF ! IF( I.LT.M-KBT ) THEN NX = M ELSE NX = N END IF ! IF( UPPER ) THEN ! ! Transform A, working with the upper triangle ! IF( UPDATE ) THEN ! ! Form inv(S(i))**T * A * inv(S(i)) ! BII = BB( KB1, I ) DO 500 J = I1, I AB( J-I+KA1, I ) = AB( J-I+KA1, I ) / BII 500 CONTINUE DO 510 J = I, MIN( N, I+KA ) AB( I-J+KA1, J ) = AB( I-J+KA1, J ) / BII 510 CONTINUE DO 540 K = I + 1, I + KBT DO 520 J = K, I + KBT AB( K-J+KA1, J ) = AB( K-J+KA1, J ) - & BB( I-J+KB1, J )*AB( I-K+KA1, K ) - & BB( I-K+KB1, K )*AB( I-J+KA1, J ) + & AB( KA1, I )*BB( I-J+KB1, J )* & BB( I-K+KB1, K ) 520 CONTINUE DO 530 J = I + KBT + 1, MIN( N, I+KA ) AB( K-J+KA1, J ) = AB( K-J+KA1, J ) - & BB( I-K+KB1, K )*AB( I-J+KA1, J ) 530 CONTINUE 540 CONTINUE DO 560 J = I1, I DO 550 K = I + 1, MIN( J+KA, I+KBT ) AB( J-K+KA1, K ) = AB( J-K+KA1, K ) - & BB( I-K+KB1, K )*AB( J-I+KA1, I ) 550 CONTINUE 560 CONTINUE ! IF( WANTX ) THEN ! ! post-multiply X by inv(S(i)) ! CALL DSCAL( NX, ONE / BII, X( 1, I ), 1 ) IF( KBT.GT.0 ) & CALL DGER( NX, KBT, -ONE, X( 1, I ), 1, BB( KB, I+1 ), & LDBB-1, X( 1, I+1 ), LDX ) END IF ! ! store a(i1,i) in RA1 for use in next loop over K ! RA1 = AB( I1-I+KA1, I ) END IF ! ! Generate and apply vectors of rotations to chase all the ! existing bulges KA positions up toward the top of the band ! DO 610 K = 1, KB - 1 IF( UPDATE ) THEN ! ! Determine the rotations which would annihilate the bulge ! which has in theory just been created ! IF( I+K-KA1.GT.0 .AND. I+K.LT.M ) THEN ! ! generate rotation to annihilate a(i+k-ka-1,i) ! CALL DLARTG( AB( K+1, I ), RA1, WORK( N+I+K-KA ), & WORK( I+K-KA ), RA ) ! ! create nonzero element a(i+k-ka-1,i+k) outside the ! band and store it in WORK(m-kb+i+k) ! T = -BB( KB1-K, I+K )*RA1 WORK( M-KB+I+K ) = WORK( N+I+K-KA )*T - & WORK( I+K-KA )*AB( 1, I+K ) AB( 1, I+K ) = WORK( I+K-KA )*T + & WORK( N+I+K-KA )*AB( 1, I+K ) RA1 = RA END IF END IF J2 = I + K + 1 - MAX( 1, K+I0-M+1 )*KA1 NR = ( J2+KA-1 ) / KA1 J1 = J2 - ( NR-1 )*KA1 IF( UPDATE ) THEN J2T = MIN( J2, I-2*KA+K-1 ) ELSE J2T = J2 END IF NRT = ( J2T+KA-1 ) / KA1 DO 570 J = J1, J2T, KA1 ! ! create nonzero element a(j-1,j+ka) outside the band ! and store it in WORK(j) ! WORK( J ) = WORK( J )*AB( 1, J+KA-1 ) AB( 1, J+KA-1 ) = WORK( N+J )*AB( 1, J+KA-1 ) 570 CONTINUE ! ! generate rotations in 1st set to annihilate elements which ! have been created outside the band ! IF( NRT.GT.0 ) & CALL DLARGV( NRT, AB( 1, J1+KA ), INCA, WORK( J1 ), KA1, & WORK( N+J1 ), KA1 ) IF( NR.GT.0 ) THEN ! ! apply rotations in 1st set from the left ! DO 580 L = 1, KA - 1 CALL DLARTV( NR, AB( KA1-L, J1+L ), INCA, & AB( KA-L, J1+L ), INCA, WORK( N+J1 ), & WORK( J1 ), KA1 ) 580 CONTINUE ! ! apply rotations in 1st set from both sides to diagonal ! blocks ! CALL DLAR2V( NR, AB( KA1, J1 ), AB( KA1, J1-1 ), & AB( KA, J1 ), INCA, WORK( N+J1 ), & WORK( J1 ), KA1 ) ! END IF ! ! start applying rotations in 1st set from the right ! DO 590 L = KA - 1, KB - K + 1, -1 NRT = ( J2+L-1 ) / KA1 J1T = J2 - ( NRT-1 )*KA1 IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( L, J1T ), INCA, & AB( L+1, J1T-1 ), INCA, WORK( N+J1T ), & WORK( J1T ), KA1 ) 590 CONTINUE ! IF( WANTX ) THEN ! ! post-multiply X by product of rotations in 1st set ! DO 600 J = J1, J2, KA1 CALL DROT( NX, X( 1, J ), 1, X( 1, J-1 ), 1, & WORK( N+J ), WORK( J ) ) 600 CONTINUE END IF 610 CONTINUE ! IF( UPDATE ) THEN IF( I2.GT.0 .AND. KBT.GT.0 ) THEN ! ! create nonzero element a(i+kbt-ka-1,i+kbt) outside the ! band and store it in WORK(m-kb+i+kbt) ! WORK( M-KB+I+KBT ) = -BB( KB1-KBT, I+KBT )*RA1 END IF END IF ! DO 650 K = KB, 1, -1 IF( UPDATE ) THEN J2 = I + K + 1 - MAX( 2, K+I0-M )*KA1 ELSE J2 = I + K + 1 - MAX( 1, K+I0-M )*KA1 END IF ! ! finish applying rotations in 2nd set from the right ! DO 620 L = KB - K, 1, -1 NRT = ( J2+KA+L-1 ) / KA1 J1T = J2 - ( NRT-1 )*KA1 IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( L, J1T+KA ), INCA, & AB( L+1, J1T+KA-1 ), INCA, & WORK( N+M-KB+J1T+KA ), & WORK( M-KB+J1T+KA ), KA1 ) 620 CONTINUE NR = ( J2+KA-1 ) / KA1 J1 = J2 - ( NR-1 )*KA1 DO 630 J = J1, J2, KA1 WORK( M-KB+J ) = WORK( M-KB+J+KA ) WORK( N+M-KB+J ) = WORK( N+M-KB+J+KA ) 630 CONTINUE DO 640 J = J1, J2, KA1 ! ! create nonzero element a(j-1,j+ka) outside the band ! and store it in WORK(m-kb+j) ! WORK( M-KB+J ) = WORK( M-KB+J )*AB( 1, J+KA-1 ) AB( 1, J+KA-1 ) = WORK( N+M-KB+J )*AB( 1, J+KA-1 ) 640 CONTINUE IF( UPDATE ) THEN IF( I+K.GT.KA1 .AND. K.LE.KBT ) & WORK( M-KB+I+K-KA ) = WORK( M-KB+I+K ) END IF 650 CONTINUE ! DO 690 K = KB, 1, -1 J2 = I + K + 1 - MAX( 1, K+I0-M )*KA1 NR = ( J2+KA-1 ) / KA1 J1 = J2 - ( NR-1 )*KA1 IF( NR.GT.0 ) THEN ! ! generate rotations in 2nd set to annihilate elements ! which have been created outside the band ! CALL DLARGV( NR, AB( 1, J1+KA ), INCA, WORK( M-KB+J1 ), & KA1, WORK( N+M-KB+J1 ), KA1 ) ! ! apply rotations in 2nd set from the left ! DO 660 L = 1, KA - 1 CALL DLARTV( NR, AB( KA1-L, J1+L ), INCA, & AB( KA-L, J1+L ), INCA, & WORK( N+M-KB+J1 ), WORK( M-KB+J1 ), KA1 ) 660 CONTINUE ! ! apply rotations in 2nd set from both sides to diagonal ! blocks ! CALL DLAR2V( NR, AB( KA1, J1 ), AB( KA1, J1-1 ), & AB( KA, J1 ), INCA, WORK( N+M-KB+J1 ), & WORK( M-KB+J1 ), KA1 ) ! END IF ! ! start applying rotations in 2nd set from the right ! DO 670 L = KA - 1, KB - K + 1, -1 NRT = ( J2+L-1 ) / KA1 J1T = J2 - ( NRT-1 )*KA1 IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( L, J1T ), INCA, & AB( L+1, J1T-1 ), INCA, & WORK( N+M-KB+J1T ), WORK( M-KB+J1T ), & KA1 ) 670 CONTINUE ! IF( WANTX ) THEN ! ! post-multiply X by product of rotations in 2nd set ! DO 680 J = J1, J2, KA1 CALL DROT( NX, X( 1, J ), 1, X( 1, J-1 ), 1, & WORK( N+M-KB+J ), WORK( M-KB+J ) ) 680 CONTINUE END IF 690 CONTINUE ! DO 710 K = 1, KB - 1 J2 = I + K + 1 - MAX( 1, K+I0-M+1 )*KA1 ! ! finish applying rotations in 1st set from the right ! DO 700 L = KB - K, 1, -1 NRT = ( J2+L-1 ) / KA1 J1T = J2 - ( NRT-1 )*KA1 IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( L, J1T ), INCA, & AB( L+1, J1T-1 ), INCA, WORK( N+J1T ), & WORK( J1T ), KA1 ) 700 CONTINUE 710 CONTINUE ! IF( KB.GT.1 ) THEN DO 720 J = 2, MIN( I+KB, M ) - 2*KA - 1 WORK( N+J ) = WORK( N+J+KA ) WORK( J ) = WORK( J+KA ) 720 CONTINUE END IF ! ELSE ! ! Transform A, working with the lower triangle ! IF( UPDATE ) THEN ! ! Form inv(S(i))**T * A * inv(S(i)) ! BII = BB( 1, I ) DO 730 J = I1, I AB( I-J+1, J ) = AB( I-J+1, J ) / BII 730 CONTINUE DO 740 J = I, MIN( N, I+KA ) AB( J-I+1, I ) = AB( J-I+1, I ) / BII 740 CONTINUE DO 770 K = I + 1, I + KBT DO 750 J = K, I + KBT AB( J-K+1, K ) = AB( J-K+1, K ) - & BB( J-I+1, I )*AB( K-I+1, I ) - & BB( K-I+1, I )*AB( J-I+1, I ) + & AB( 1, I )*BB( J-I+1, I )* & BB( K-I+1, I ) 750 CONTINUE DO 760 J = I + KBT + 1, MIN( N, I+KA ) AB( J-K+1, K ) = AB( J-K+1, K ) - & BB( K-I+1, I )*AB( J-I+1, I ) 760 CONTINUE 770 CONTINUE DO 790 J = I1, I DO 780 K = I + 1, MIN( J+KA, I+KBT ) AB( K-J+1, J ) = AB( K-J+1, J ) - & BB( K-I+1, I )*AB( I-J+1, J ) 780 CONTINUE 790 CONTINUE ! IF( WANTX ) THEN ! ! post-multiply X by inv(S(i)) ! CALL DSCAL( NX, ONE / BII, X( 1, I ), 1 ) IF( KBT.GT.0 ) & CALL DGER( NX, KBT, -ONE, X( 1, I ), 1, BB( 2, I ), 1, & X( 1, I+1 ), LDX ) END IF ! ! store a(i,i1) in RA1 for use in next loop over K ! RA1 = AB( I-I1+1, I1 ) END IF ! ! Generate and apply vectors of rotations to chase all the ! existing bulges KA positions up toward the top of the band ! DO 840 K = 1, KB - 1 IF( UPDATE ) THEN ! ! Determine the rotations which would annihilate the bulge ! which has in theory just been created ! IF( I+K-KA1.GT.0 .AND. I+K.LT.M ) THEN ! ! generate rotation to annihilate a(i,i+k-ka-1) ! CALL DLARTG( AB( KA1-K, I+K-KA ), RA1, & WORK( N+I+K-KA ), WORK( I+K-KA ), RA ) ! ! create nonzero element a(i+k,i+k-ka-1) outside the ! band and store it in WORK(m-kb+i+k) ! T = -BB( K+1, I )*RA1 WORK( M-KB+I+K ) = WORK( N+I+K-KA )*T - & WORK( I+K-KA )*AB( KA1, I+K-KA ) AB( KA1, I+K-KA ) = WORK( I+K-KA )*T + & WORK( N+I+K-KA )*AB( KA1, I+K-KA ) RA1 = RA END IF END IF J2 = I + K + 1 - MAX( 1, K+I0-M+1 )*KA1 NR = ( J2+KA-1 ) / KA1 J1 = J2 - ( NR-1 )*KA1 IF( UPDATE ) THEN J2T = MIN( J2, I-2*KA+K-1 ) ELSE J2T = J2 END IF NRT = ( J2T+KA-1 ) / KA1 DO 800 J = J1, J2T, KA1 ! ! create nonzero element a(j+ka,j-1) outside the band ! and store it in WORK(j) ! WORK( J ) = WORK( J )*AB( KA1, J-1 ) AB( KA1, J-1 ) = WORK( N+J )*AB( KA1, J-1 ) 800 CONTINUE ! ! generate rotations in 1st set to annihilate elements which ! have been created outside the band ! IF( NRT.GT.0 ) & CALL DLARGV( NRT, AB( KA1, J1 ), INCA, WORK( J1 ), KA1, & WORK( N+J1 ), KA1 ) IF( NR.GT.0 ) THEN ! ! apply rotations in 1st set from the right ! DO 810 L = 1, KA - 1 CALL DLARTV( NR, AB( L+1, J1 ), INCA, AB( L+2, J1-1 ), & INCA, WORK( N+J1 ), WORK( J1 ), KA1 ) 810 CONTINUE ! ! apply rotations in 1st set from both sides to diagonal ! blocks ! CALL DLAR2V( NR, AB( 1, J1 ), AB( 1, J1-1 ), & AB( 2, J1-1 ), INCA, WORK( N+J1 ), & WORK( J1 ), KA1 ) ! END IF ! ! start applying rotations in 1st set from the left ! DO 820 L = KA - 1, KB - K + 1, -1 NRT = ( J2+L-1 ) / KA1 J1T = J2 - ( NRT-1 )*KA1 IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( KA1-L+1, J1T-KA1+L ), INCA, & AB( KA1-L, J1T-KA1+L ), INCA, & WORK( N+J1T ), WORK( J1T ), KA1 ) 820 CONTINUE ! IF( WANTX ) THEN ! ! post-multiply X by product of rotations in 1st set ! DO 830 J = J1, J2, KA1 CALL DROT( NX, X( 1, J ), 1, X( 1, J-1 ), 1, & WORK( N+J ), WORK( J ) ) 830 CONTINUE END IF 840 CONTINUE ! IF( UPDATE ) THEN IF( I2.GT.0 .AND. KBT.GT.0 ) THEN ! ! create nonzero element a(i+kbt,i+kbt-ka-1) outside the ! band and store it in WORK(m-kb+i+kbt) ! WORK( M-KB+I+KBT ) = -BB( KBT+1, I )*RA1 END IF END IF ! DO 880 K = KB, 1, -1 IF( UPDATE ) THEN J2 = I + K + 1 - MAX( 2, K+I0-M )*KA1 ELSE J2 = I + K + 1 - MAX( 1, K+I0-M )*KA1 END IF ! ! finish applying rotations in 2nd set from the left ! DO 850 L = KB - K, 1, -1 NRT = ( J2+KA+L-1 ) / KA1 J1T = J2 - ( NRT-1 )*KA1 IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( KA1-L+1, J1T+L-1 ), INCA, & AB( KA1-L, J1T+L-1 ), INCA, & WORK( N+M-KB+J1T+KA ), & WORK( M-KB+J1T+KA ), KA1 ) 850 CONTINUE NR = ( J2+KA-1 ) / KA1 J1 = J2 - ( NR-1 )*KA1 DO 860 J = J1, J2, KA1 WORK( M-KB+J ) = WORK( M-KB+J+KA ) WORK( N+M-KB+J ) = WORK( N+M-KB+J+KA ) 860 CONTINUE DO 870 J = J1, J2, KA1 ! ! create nonzero element a(j+ka,j-1) outside the band ! and store it in WORK(m-kb+j) ! WORK( M-KB+J ) = WORK( M-KB+J )*AB( KA1, J-1 ) AB( KA1, J-1 ) = WORK( N+M-KB+J )*AB( KA1, J-1 ) 870 CONTINUE IF( UPDATE ) THEN IF( I+K.GT.KA1 .AND. K.LE.KBT ) & WORK( M-KB+I+K-KA ) = WORK( M-KB+I+K ) END IF 880 CONTINUE ! DO 920 K = KB, 1, -1 J2 = I + K + 1 - MAX( 1, K+I0-M )*KA1 NR = ( J2+KA-1 ) / KA1 J1 = J2 - ( NR-1 )*KA1 IF( NR.GT.0 ) THEN ! ! generate rotations in 2nd set to annihilate elements ! which have been created outside the band ! CALL DLARGV( NR, AB( KA1, J1 ), INCA, WORK( M-KB+J1 ), & KA1, WORK( N+M-KB+J1 ), KA1 ) ! ! apply rotations in 2nd set from the right ! DO 890 L = 1, KA - 1 CALL DLARTV( NR, AB( L+1, J1 ), INCA, AB( L+2, J1-1 ), & INCA, WORK( N+M-KB+J1 ), WORK( M-KB+J1 ), & KA1 ) 890 CONTINUE ! ! apply rotations in 2nd set from both sides to diagonal ! blocks ! CALL DLAR2V( NR, AB( 1, J1 ), AB( 1, J1-1 ), & AB( 2, J1-1 ), INCA, WORK( N+M-KB+J1 ), & WORK( M-KB+J1 ), KA1 ) ! END IF ! ! start applying rotations in 2nd set from the left ! DO 900 L = KA - 1, KB - K + 1, -1 NRT = ( J2+L-1 ) / KA1 J1T = J2 - ( NRT-1 )*KA1 IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( KA1-L+1, J1T-KA1+L ), INCA, & AB( KA1-L, J1T-KA1+L ), INCA, & WORK( N+M-KB+J1T ), WORK( M-KB+J1T ), & KA1 ) 900 CONTINUE ! IF( WANTX ) THEN ! ! post-multiply X by product of rotations in 2nd set ! DO 910 J = J1, J2, KA1 CALL DROT( NX, X( 1, J ), 1, X( 1, J-1 ), 1, & WORK( N+M-KB+J ), WORK( M-KB+J ) ) 910 CONTINUE END IF 920 CONTINUE ! DO 940 K = 1, KB - 1 J2 = I + K + 1 - MAX( 1, K+I0-M+1 )*KA1 ! ! finish applying rotations in 1st set from the left ! DO 930 L = KB - K, 1, -1 NRT = ( J2+L-1 ) / KA1 J1T = J2 - ( NRT-1 )*KA1 IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( KA1-L+1, J1T-KA1+L ), INCA, & AB( KA1-L, J1T-KA1+L ), INCA, & WORK( N+J1T ), WORK( J1T ), KA1 ) 930 CONTINUE 940 CONTINUE ! IF( KB.GT.1 ) THEN DO 950 J = 2, MIN( I+KB, M ) - 2*KA - 1 WORK( N+J ) = WORK( N+J+KA ) WORK( J ) = WORK( J+KA ) 950 CONTINUE END IF ! END IF ! GO TO 490 ! ! End of DSBGST ! END SUBROUTINE DSBGV( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, & LDZ, WORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), W( * ), & WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSBGV computes all the eigenvalues, and optionally, the eigenvectors ! of a real generalized symmetric-definite banded eigenproblem, of ! the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric ! and banded, and B is also positive definite. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangles of A and B are stored; ! = 'L': Lower triangles of A and B are stored. ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! KA (input) INTEGER ! The number of superdiagonals of the matrix A if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KA >= 0. ! ! KB (input) INTEGER ! The number of superdiagonals of the matrix B if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KB >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB, N) ! On entry, the upper or lower triangle of the symmetric band ! matrix A, stored in the first ka+1 rows of the array. The ! j-th column of A is stored in the j-th column of the array AB ! as follows: ! if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). ! ! On exit, the contents of AB are destroyed. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KA+1. ! ! BB (input/output) DOUBLE PRECISION array, dimension (LDBB, N) ! On entry, the upper or lower triangle of the symmetric band ! matrix B, stored in the first kb+1 rows of the array. The ! j-th column of B is stored in the j-th column of the array BB ! as follows: ! if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; ! if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). ! ! On exit, the factor S from the split Cholesky factorization ! B = S**T*S, as returned by DPBSTF. ! ! LDBB (input) INTEGER ! The leading dimension of the array BB. LDBB >= KB+1. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, the eigenvalues in ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, N) ! If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of ! eigenvectors, with the i-th column of Z holding the ! eigenvector associated with W(i). The eigenvectors are ! normalized so that Z**T*B*Z = I. ! If JOBZ = 'N', then Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= N. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, and i is: ! <= N: the algorithm failed to converge: ! i off-diagonal elements of an intermediate ! tridiagonal form did not converge to zero; ! > N: if INFO = N + i, for 1 <= i <= N, then DPBSTF ! returned INFO = i: B is not positive definite. ! The factorization of B could not be completed and ! no eigenvalues or eigenvectors were computed. ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL UPPER, WANTZ CHARACTER VECT INTEGER IINFO, INDE, INDWRK ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DPBSTF, DSBGST, DSBTRD, DSTEQR, DSTERF, XERBLA ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) UPPER = LSAME( UPLO, 'U' ) ! INFO = 0 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KA.LT.0 ) THEN INFO = -4 ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN INFO = -5 ELSE IF( LDAB.LT.KA+1 ) THEN INFO = -7 ELSE IF( LDBB.LT.KB+1 ) THEN INFO = -9 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -12 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSBGV ', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Form a split Cholesky factorization of B. ! CALL DPBSTF( UPLO, N, KB, BB, LDBB, INFO ) IF( INFO.NE.0 ) THEN INFO = N + INFO RETURN END IF ! ! Transform problem to standard eigenvalue problem. ! INDE = 1 INDWRK = INDE + N CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Z, LDZ, & WORK( INDWRK ), IINFO ) ! ! Reduce to tridiagonal form. ! IF( WANTZ ) THEN VECT = 'U' ELSE VECT = 'N' END IF CALL DSBTRD( VECT, UPLO, N, KA, AB, LDAB, W, WORK( INDE ), Z, LDZ, & WORK( INDWRK ), IINFO ) ! ! For eigenvalues only, call DSTERF. For eigenvectors, call SSTEQR. ! IF( .NOT.WANTZ ) THEN CALL DSTERF( N, W, WORK( INDE ), INFO ) ELSE CALL DSTEQR( JOBZ, N, W, WORK( INDE ), Z, LDZ, WORK( INDWRK ), & INFO ) END IF RETURN ! ! End of DSBGV ! END SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, & Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LWORK, N ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), W( * ), & WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSBGVD computes all the eigenvalues, and optionally, the eigenvectors ! of a real generalized symmetric-definite banded eigenproblem, of the ! form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and ! banded, and B is also positive definite. If eigenvectors are ! desired, it uses a divide and conquer algorithm. ! ! The divide and conquer algorithm makes very mild assumptions about ! floating point arithmetic. It will work on machines with a guard ! digit in add/subtract, or on those binary machines without guard ! digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or ! Cray-2. It could conceivably fail on hexadecimal or decimal machines ! without guard digits, but we know of none. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangles of A and B are stored; ! = 'L': Lower triangles of A and B are stored. ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! KA (input) INTEGER ! The number of superdiagonals of the matrix A if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KA >= 0. ! ! KB (input) INTEGER ! The number of superdiagonals of the matrix B if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KB >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB, N) ! On entry, the upper or lower triangle of the symmetric band ! matrix A, stored in the first ka+1 rows of the array. The ! j-th column of A is stored in the j-th column of the array AB ! as follows: ! if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). ! ! On exit, the contents of AB are destroyed. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KA+1. ! ! BB (input/output) DOUBLE PRECISION array, dimension (LDBB, N) ! On entry, the upper or lower triangle of the symmetric band ! matrix B, stored in the first kb+1 rows of the array. The ! j-th column of B is stored in the j-th column of the array BB ! as follows: ! if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; ! if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). ! ! On exit, the factor S from the split Cholesky factorization ! B = S**T*S, as returned by DPBSTF. ! ! LDBB (input) INTEGER ! The leading dimension of the array BB. LDBB >= KB+1. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, the eigenvalues in ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, N) ! If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of ! eigenvectors, with the i-th column of Z holding the ! eigenvector associated with W(i). The eigenvectors are ! normalized so Z**T*B*Z = I. ! If JOBZ = 'N', then Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! If N <= 1, LWORK >= 1. ! If JOBZ = 'N' and N > 1, LWORK >= 3*N. ! If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace/output) INTEGER array, dimension (LIWORK) ! On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK. ! ! LIWORK (input) INTEGER ! The dimension of the array IWORK. ! If JOBZ = 'N' or N <= 1, LIWORK >= 1. ! If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N. ! ! If LIWORK = -1, then a workspace query is assumed; the ! routine only calculates the optimal size of the IWORK array, ! returns this value as the first entry of the IWORK array, and ! no error message related to LIWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, and i is: ! <= N: the algorithm failed to converge: ! i off-diagonal elements of an intermediate ! tridiagonal form did not converge to zero; ! > N: if INFO = N + i, for 1 <= i <= N, then DPBSTF ! returned INFO = i: B is not positive definite. ! The factorization of B could not be completed and ! no eigenvalues or eigenvectors were computed. ! ! Further Details ! =============== ! ! Based on contributions by ! Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, UPPER, WANTZ CHARACTER VECT INTEGER IINFO, INDE, INDWK2, INDWRK, LIWMIN, LLWRK2, & LWMIN ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DGEMM, DLACPY, DPBSTF, DSBGST, DSBTRD, DSTEDC, & DSTERF, XERBLA ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) UPPER = LSAME( UPLO, 'U' ) LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) ! INFO = 0 IF( N.LE.1 ) THEN LIWMIN = 1 LWMIN = 1 ELSE IF( WANTZ ) THEN LIWMIN = 3 + 5*N LWMIN = 1 + 5*N + 2*N**2 ELSE LIWMIN = 1 LWMIN = 2*N END IF END IF ! IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KA.LT.0 ) THEN INFO = -4 ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN INFO = -5 ELSE IF( LDAB.LT.KA+1 ) THEN INFO = -7 ELSE IF( LDBB.LT.KB+1 ) THEN INFO = -9 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -12 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -14 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -16 END IF ! IF( INFO.EQ.0 ) THEN WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSBGVD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Form a split Cholesky factorization of B. ! CALL DPBSTF( UPLO, N, KB, BB, LDBB, INFO ) IF( INFO.NE.0 ) THEN INFO = N + INFO RETURN END IF ! ! Transform problem to standard eigenvalue problem. ! INDE = 1 INDWRK = INDE + N INDWK2 = INDWRK + N*N LLWRK2 = LWORK - INDWK2 + 1 CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Z, LDZ, & WORK( INDWRK ), IINFO ) ! ! Reduce to tridiagonal form. ! IF( WANTZ ) THEN VECT = 'U' ELSE VECT = 'N' END IF CALL DSBTRD( VECT, UPLO, N, KA, AB, LDAB, W, WORK( INDE ), Z, LDZ, & WORK( INDWRK ), IINFO ) ! ! For eigenvalues only, call DSTERF. For eigenvectors, call SSTEDC. ! IF( .NOT.WANTZ ) THEN CALL DSTERF( N, W, WORK( INDE ), INFO ) ELSE CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N, & WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO ) CALL DGEMM( 'N', 'N', N, N, N, ONE, Z, LDZ, WORK( INDWRK ), N, & ZERO, WORK( INDWK2 ), N ) CALL DLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ ) END IF ! WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN ! RETURN ! ! End of DSBGVD ! END SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, & LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, & LDZ, WORK, IWORK, IFAIL, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, RANGE, UPLO INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M, & N DOUBLE PRECISION ABSTOL, VL, VU ! .. ! .. Array Arguments .. INTEGER IFAIL( * ), IWORK( * ) DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ), & W( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSBGVX computes selected eigenvalues, and optionally, eigenvectors ! of a real generalized symmetric-definite banded eigenproblem, of ! the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric ! and banded, and B is also positive definite. Eigenvalues and ! eigenvectors can be selected by specifying either all eigenvalues, ! a range of values or a range of indices for the desired eigenvalues. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! RANGE (input) CHARACTER*1 ! = 'A': all eigenvalues will be found. ! = 'V': all eigenvalues in the half-open interval (VL,VU] ! will be found. ! = 'I': the IL-th through IU-th eigenvalues will be found. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangles of A and B are stored; ! = 'L': Lower triangles of A and B are stored. ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! KA (input) INTEGER ! The number of superdiagonals of the matrix A if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KA >= 0. ! ! KB (input) INTEGER ! The number of superdiagonals of the matrix B if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KB >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB, N) ! On entry, the upper or lower triangle of the symmetric band ! matrix A, stored in the first ka+1 rows of the array. The ! j-th column of A is stored in the j-th column of the array AB ! as follows: ! if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). ! ! On exit, the contents of AB are destroyed. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KA+1. ! ! BB (input/output) DOUBLE PRECISION array, dimension (LDBB, N) ! On entry, the upper or lower triangle of the symmetric band ! matrix B, stored in the first kb+1 rows of the array. The ! j-th column of B is stored in the j-th column of the array BB ! as follows: ! if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; ! if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). ! ! On exit, the factor S from the split Cholesky factorization ! B = S**T*S, as returned by DPBSTF. ! ! LDBB (input) INTEGER ! The leading dimension of the array BB. LDBB >= KB+1. ! ! Q (output) DOUBLE PRECISION array, dimension (LDQ, N) ! If JOBZ = 'V', the n-by-n matrix used in the reduction of ! A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, ! and consequently C to tridiagonal form. ! If JOBZ = 'N', the array Q is not referenced. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. If JOBZ = 'N', ! LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N). ! ! VL (input) DOUBLE PRECISION ! VU (input) DOUBLE PRECISION ! If RANGE='V', the lower and upper bounds of the interval to ! be searched for eigenvalues. VL < VU. ! Not referenced if RANGE = 'A' or 'I'. ! ! IL (input) INTEGER ! IU (input) INTEGER ! If RANGE='I', the indices (in ascending order) of the ! smallest and largest eigenvalues to be returned. ! 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. ! Not referenced if RANGE = 'A' or 'V'. ! ! ABSTOL (input) DOUBLE PRECISION ! The absolute error tolerance for the eigenvalues. ! An approximate eigenvalue is accepted as converged ! when it is determined to lie in an interval [a,b] ! of width less than or equal to ! ! ABSTOL + EPS * max( |a|,|b| ) , ! ! where EPS is the machine precision. If ABSTOL is less than ! or equal to zero, then EPS*|T| will be used in its place, ! where |T| is the 1-norm of the tridiagonal matrix obtained ! by reducing A to tridiagonal form. ! ! Eigenvalues will be computed most accurately when ABSTOL is ! set to twice the underflow threshold 2*DLAMCH('S'), not zero. ! If this routine returns with INFO>0, indicating that some ! eigenvectors did not converge, try setting ABSTOL to ! 2*DLAMCH('S'). ! ! M (output) INTEGER ! The total number of eigenvalues found. 0 <= M <= N. ! If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, the eigenvalues in ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, N) ! If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of ! eigenvectors, with the i-th column of Z holding the ! eigenvector associated with W(i). The eigenvectors are ! normalized so Z**T*B*Z = I. ! If JOBZ = 'N', then Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (7N) ! ! IWORK (workspace/output) INTEGER array, dimension (5N) ! ! IFAIL (input) INTEGER array, dimension (M) ! If JOBZ = 'V', then if INFO = 0, the first M elements of ! IFAIL are zero. If INFO > 0, then IFAIL contains the ! indices of the eigenvalues that failed to converge. ! If JOBZ = 'N', then IFAIL is not referenced. ! ! INFO (output) INTEGER ! = 0 : successful exit ! < 0 : if INFO = -i, the i-th argument had an illegal value ! <= N: if INFO = i, then i eigenvectors failed to converge. ! Their indices are stored in IFAIL. ! > N : DPBSTF returned an error code; i.e., ! if INFO = N + i, for 1 <= i <= N, then the leading ! minor of order i of B is not positive definite. ! The factorization of B could not be completed and ! no eigenvalues or eigenvectors were computed. ! ! Further Details ! =============== ! ! Based on contributions by ! Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL ALLEIG, INDEIG, UPPER, VALEIG, WANTZ CHARACTER ORDER, VECT INTEGER I, IINFO, INDD, INDE, INDEE, INDIBL, INDISP, & INDIWO, INDWRK, ITMP1, J, JJ, NSPLIT DOUBLE PRECISION TMP1 ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEMV, DLACPY, DPBSTF, DSBGST, DSBTRD, & DSTEBZ, DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) UPPER = LSAME( UPLO, 'U' ) ALLEIG = LSAME( RANGE, 'A' ) VALEIG = LSAME( RANGE, 'V' ) INDEIG = LSAME( RANGE, 'I' ) ! INFO = 0 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN INFO = -2 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( KA.LT.0 ) THEN INFO = -5 ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN INFO = -6 ELSE IF( LDAB.LT.KA+1 ) THEN INFO = -8 ELSE IF( LDBB.LT.KB+1 ) THEN INFO = -10 ELSE IF( LDQ.LT.1 ) THEN INFO = -12 ELSE IF( VALEIG .AND. N.GT.0 .AND. VU.LE.VL ) THEN INFO = -14 ELSE IF( INDEIG .AND. IL.LT.1 ) THEN INFO = -15 ELSE IF( INDEIG .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN INFO = -16 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -21 END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSBGVX', -INFO ) RETURN END IF ! ! Quick return if possible ! M = 0 IF( N.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! ! Form a split Cholesky factorization of B. ! CALL DPBSTF( UPLO, N, KB, BB, LDBB, INFO ) IF( INFO.NE.0 ) THEN INFO = N + INFO RETURN END IF ! ! Transform problem to standard eigenvalue problem. ! CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, & WORK, IINFO ) ! ! Reduce symmetric band matrix to tridiagonal form. ! INDD = 1 INDE = INDD + N INDWRK = INDE + N IF( WANTZ ) THEN VECT = 'U' ELSE VECT = 'N' END IF CALL DSBTRD( VECT, UPLO, N, KA, AB, LDAB, WORK( INDD ), & WORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO ) ! ! If all eigenvalues are desired and ABSTOL is less than or equal ! to zero, then call DSTERF or SSTEQR. If this fails for some ! eigenvalue, then try DSTEBZ. ! IF( ( ALLEIG .OR. ( INDEIG .AND. IL.EQ.1 .AND. IU.EQ.N ) ) .AND. & ( ABSTOL.LE.ZERO ) ) THEN CALL DCOPY( N, WORK( INDD ), 1, W, 1 ) INDEE = INDWRK + 2*N CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) IF( .NOT.WANTZ ) THEN CALL DSTERF( N, W, WORK( INDEE ), INFO ) ELSE CALL DLACPY( 'A', N, N, Q, LDQ, Z, LDZ ) CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ, & WORK( INDWRK ), INFO ) IF( INFO.EQ.0 ) THEN DO 10 I = 1, N IFAIL( I ) = 0 10 CONTINUE END IF END IF IF( INFO.EQ.0 ) THEN M = N GO TO 30 END IF INFO = 0 END IF ! ! Otherwise, call DSTEBZ and, if eigenvectors are desired, ! call DSTEIN. ! IF( WANTZ ) THEN ORDER = 'B' ELSE ORDER = 'E' END IF INDIBL = 1 INDISP = INDIBL + N INDIWO = INDISP + N CALL DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, & WORK( INDD ), WORK( INDE ), M, NSPLIT, W, & IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ), & IWORK( INDIWO ), INFO ) ! IF( WANTZ ) THEN CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W, & IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, & WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO ) ! ! Apply transformation matrix used in reduction to tridiagonal ! form to eigenvectors returned by DSTEIN. ! DO 20 J = 1, M CALL DCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 ) CALL DGEMV( 'N', N, N, ONE, Q, LDQ, WORK, 1, ZERO, & Z( 1, J ), 1 ) 20 CONTINUE END IF ! 30 CONTINUE ! ! If eigenvalues are not in order, then sort them, along with ! eigenvectors. ! IF( WANTZ ) THEN DO 50 J = 1, M - 1 I = 0 TMP1 = W( J ) DO 40 JJ = J + 1, M IF( W( JJ ).LT.TMP1 ) THEN I = JJ TMP1 = W( JJ ) END IF 40 CONTINUE ! IF( I.NE.0 ) THEN ITMP1 = IWORK( INDIBL+I-1 ) W( I ) = W( J ) IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) W( J ) = TMP1 IWORK( INDIBL+J-1 ) = ITMP1 CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) IF( INFO.NE.0 ) THEN ITMP1 = IFAIL( I ) IFAIL( I ) = IFAIL( J ) IFAIL( J ) = ITMP1 END IF END IF 50 CONTINUE END IF ! RETURN ! ! End of DSBGVX ! END SUBROUTINE DSBTRD( VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, & WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER UPLO, VECT INTEGER INFO, KD, LDAB, LDQ, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), D( * ), E( * ), Q( LDQ, * ), & WORK( * ) ! .. ! ! Purpose ! ======= ! ! DSBTRD reduces a real symmetric band matrix A to symmetric ! tridiagonal form T by an orthogonal similarity transformation: ! Q**T * A * Q = T. ! ! Arguments ! ========= ! ! VECT (input) CHARACTER*1 ! = 'N': do not form Q; ! = 'V': form Q; ! = 'U': update a matrix X, by forming X*Q. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of superdiagonals of the matrix A if UPLO = 'U', ! or the number of subdiagonals if UPLO = 'L'. KD >= 0. ! ! AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) ! On entry, the upper or lower triangle of the symmetric band ! matrix A, stored in the first KD+1 rows of the array. The ! j-th column of A is stored in the j-th column of the array AB ! as follows: ! if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). ! On exit, the diagonal elements of AB are overwritten by the ! diagonal elements of the tridiagonal matrix T; if KD > 0, the ! elements on the first superdiagonal (if UPLO = 'U') or the ! first subdiagonal (if UPLO = 'L') are overwritten by the ! off-diagonal elements of T; the rest of AB is overwritten by ! values generated during the reduction. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KD+1. ! ! D (output) DOUBLE PRECISION array, dimension (N) ! The diagonal elements of the tridiagonal matrix T. ! ! E (output) DOUBLE PRECISION array, dimension (N-1) ! The off-diagonal elements of the tridiagonal matrix T: ! E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'. ! ! Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) ! On entry, if VECT = 'U', then Q must contain an N-by-N ! matrix X; if VECT = 'N' or 'V', then Q need not be set. ! ! On exit: ! if VECT = 'V', Q contains the N-by-N orthogonal matrix Q; ! if VECT = 'U', Q contains the product X*Q; ! if VECT = 'N', the array Q is not referenced. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. ! LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! Modified by Linda Kaufman, Bell Labs. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL INITQ, UPPER, WANTQ INTEGER I, I2, IBL, INCA, INCX, IQAEND, IQB, IQEND, J, & J1, J1END, J1INC, J2, JEND, JIN, JINC, K, KD1, & KDM1, KDN, L, LAST, LEND, NQ, NR, NRT DOUBLE PRECISION TEMP ! .. ! .. External Subroutines .. EXTERNAL DLAR2V, DLARGV, DLARTG, DLARTV, DLASET, DROT, & XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INITQ = LSAME( VECT, 'V' ) WANTQ = INITQ .OR. LSAME( VECT, 'U' ) UPPER = LSAME( UPLO, 'U' ) KD1 = KD + 1 KDM1 = KD - 1 INCX = LDAB - 1 IQEND = 1 ! INFO = 0 IF( .NOT.WANTQ .AND. .NOT.LSAME( VECT, 'N' ) ) THEN INFO = -1 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KD.LT.0 ) THEN INFO = -4 ELSE IF( LDAB.LT.KD1 ) THEN INFO = -6 ELSE IF( LDQ.LT.MAX( 1, N ) .AND. WANTQ ) THEN INFO = -10 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSBTRD', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Initialize Q to the unit matrix, if needed ! IF( INITQ ) & CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ ) ! ! Wherever possible, plane rotations are generated and applied in ! vector operations of length NR over the index set J1:J2:KD1. ! ! The cosines and sines of the plane rotations are stored in the ! arrays D and WORK. ! INCA = KD1*LDAB KDN = MIN( N-1, KD ) IF( UPPER ) THEN ! IF( KD.GT.1 ) THEN ! ! Reduce to tridiagonal form, working with upper triangle ! NR = 0 J1 = KDN + 2 J2 = 1 ! DO 90 I = 1, N - 2 ! ! Reduce i-th row of matrix to tridiagonal form ! DO 80 K = KDN + 1, 2, -1 J1 = J1 + KDN J2 = J2 + KDN ! IF( NR.GT.0 ) THEN ! ! generate plane rotations to annihilate nonzero ! elements which have been created outside the band ! CALL DLARGV( NR, AB( 1, J1-1 ), INCA, WORK( J1 ), & KD1, D( J1 ), KD1 ) ! ! apply rotations from the right ! ! ! Dependent on the the number of diagonals either ! DLARTV or DROT is used ! IF( NR.GE.2*KD-1 ) THEN DO 10 L = 1, KD - 1 CALL DLARTV( NR, AB( L+1, J1-1 ), INCA, & AB( L, J1 ), INCA, D( J1 ), & WORK( J1 ), KD1 ) 10 CONTINUE ! ELSE JEND = J1 + ( NR-1 )*KD1 DO 20 JINC = J1, JEND, KD1 CALL DROT( KDM1, AB( 2, JINC-1 ), 1, & AB( 1, JINC ), 1, D( JINC ), & WORK( JINC ) ) 20 CONTINUE END IF END IF ! ! IF( K.GT.2 ) THEN IF( K.LE.N-I+1 ) THEN ! ! generate plane rotation to annihilate a(i,i+k-1) ! within the band ! CALL DLARTG( AB( KD-K+3, I+K-2 ), & AB( KD-K+2, I+K-1 ), D( I+K-1 ), & WORK( I+K-1 ), TEMP ) AB( KD-K+3, I+K-2 ) = TEMP ! ! apply rotation from the right ! CALL DROT( K-3, AB( KD-K+4, I+K-2 ), 1, & AB( KD-K+3, I+K-1 ), 1, D( I+K-1 ), & WORK( I+K-1 ) ) END IF NR = NR + 1 J1 = J1 - KDN - 1 END IF ! ! apply plane rotations from both sides to diagonal ! blocks ! IF( NR.GT.0 ) & CALL DLAR2V( NR, AB( KD1, J1-1 ), AB( KD1, J1 ), & AB( KD, J1 ), INCA, D( J1 ), & WORK( J1 ), KD1 ) ! ! apply plane rotations from the left ! IF( NR.GT.0 ) THEN IF( 2*KD-1.LT.NR ) THEN ! ! Dependent on the the number of diagonals either ! DLARTV or DROT is used ! DO 30 L = 1, KD - 1 IF( J2+L.GT.N ) THEN NRT = NR - 1 ELSE NRT = NR END IF IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( KD-L, J1+L ), INCA, & AB( KD-L+1, J1+L ), INCA, & D( J1 ), WORK( J1 ), KD1 ) 30 CONTINUE ELSE J1END = J1 + KD1*( NR-2 ) IF( J1END.GE.J1 ) THEN DO 40 JIN = J1, J1END, KD1 CALL DROT( KD-1, AB( KD-1, JIN+1 ), INCX, & AB( KD, JIN+1 ), INCX, & D( JIN ), WORK( JIN ) ) 40 CONTINUE END IF LEND = MIN( KDM1, N-J2 ) LAST = J1END + KD1 IF( LEND.GT.0 ) & CALL DROT( LEND, AB( KD-1, LAST+1 ), INCX, & AB( KD, LAST+1 ), INCX, D( LAST ), & WORK( LAST ) ) END IF END IF ! IF( WANTQ ) THEN ! ! accumulate product of plane rotations in Q ! IF( INITQ ) THEN ! ! take advantage of the fact that Q was ! initially the Identity matrix ! IQEND = MAX( IQEND, J2 ) I2 = MAX( 0, K-3 ) IQAEND = 1 + I*KD IF( K.EQ.2 ) & IQAEND = IQAEND + KD IQAEND = MIN( IQAEND, IQEND ) DO 50 J = J1, J2, KD1 IBL = I - I2 / KDM1 I2 = I2 + 1 IQB = MAX( 1, J-IBL ) NQ = 1 + IQAEND - IQB IQAEND = MIN( IQAEND+KD, IQEND ) CALL DROT( NQ, Q( IQB, J-1 ), 1, Q( IQB, J ), & 1, D( J ), WORK( J ) ) 50 CONTINUE ELSE ! DO 60 J = J1, J2, KD1 CALL DROT( N, Q( 1, J-1 ), 1, Q( 1, J ), 1, & D( J ), WORK( J ) ) 60 CONTINUE END IF ! END IF ! IF( J2+KDN.GT.N ) THEN ! ! adjust J2 to keep within the bounds of the matrix ! NR = NR - 1 J2 = J2 - KDN - 1 END IF ! DO 70 J = J1, J2, KD1 ! ! create nonzero element a(j-1,j+kd) outside the band ! and store it in WORK ! WORK( J+KD ) = WORK( J )*AB( 1, J+KD ) AB( 1, J+KD ) = D( J )*AB( 1, J+KD ) 70 CONTINUE 80 CONTINUE 90 CONTINUE END IF ! IF( KD.GT.0 ) THEN ! ! copy off-diagonal elements to E ! DO 100 I = 1, N - 1 E( I ) = AB( KD, I+1 ) 100 CONTINUE ELSE ! ! set E to zero if original matrix was diagonal ! DO 110 I = 1, N - 1 E( I ) = ZERO 110 CONTINUE END IF ! ! copy diagonal elements to D ! DO 120 I = 1, N D( I ) = AB( KD1, I ) 120 CONTINUE ! ELSE ! IF( KD.GT.1 ) THEN ! ! Reduce to tridiagonal form, working with lower triangle ! NR = 0 J1 = KDN + 2 J2 = 1 ! DO 210 I = 1, N - 2 ! ! Reduce i-th column of matrix to tridiagonal form ! DO 200 K = KDN + 1, 2, -1 J1 = J1 + KDN J2 = J2 + KDN ! IF( NR.GT.0 ) THEN ! ! generate plane rotations to annihilate nonzero ! elements which have been created outside the band ! CALL DLARGV( NR, AB( KD1, J1-KD1 ), INCA, & WORK( J1 ), KD1, D( J1 ), KD1 ) ! ! apply plane rotations from one side ! ! ! Dependent on the the number of diagonals either ! DLARTV or DROT is used ! IF( NR.GT.2*KD-1 ) THEN DO 130 L = 1, KD - 1 CALL DLARTV( NR, AB( KD1-L, J1-KD1+L ), INCA, & AB( KD1-L+1, J1-KD1+L ), INCA, & D( J1 ), WORK( J1 ), KD1 ) 130 CONTINUE ELSE JEND = J1 + KD1*( NR-1 ) DO 140 JINC = J1, JEND, KD1 CALL DROT( KDM1, AB( KD, JINC-KD ), INCX, & AB( KD1, JINC-KD ), INCX, & D( JINC ), WORK( JINC ) ) 140 CONTINUE END IF ! END IF ! IF( K.GT.2 ) THEN IF( K.LE.N-I+1 ) THEN ! ! generate plane rotation to annihilate a(i+k-1,i) ! within the band ! CALL DLARTG( AB( K-1, I ), AB( K, I ), & D( I+K-1 ), WORK( I+K-1 ), TEMP ) AB( K-1, I ) = TEMP ! ! apply rotation from the left ! CALL DROT( K-3, AB( K-2, I+1 ), LDAB-1, & AB( K-1, I+1 ), LDAB-1, D( I+K-1 ), & WORK( I+K-1 ) ) END IF NR = NR + 1 J1 = J1 - KDN - 1 END IF ! ! apply plane rotations from both sides to diagonal ! blocks ! IF( NR.GT.0 ) & CALL DLAR2V( NR, AB( 1, J1-1 ), AB( 1, J1 ), & AB( 2, J1-1 ), INCA, D( J1 ), & WORK( J1 ), KD1 ) ! ! apply plane rotations from the right ! ! ! Dependent on the the number of diagonals either ! DLARTV or DROT is used ! IF( NR.GT.0 ) THEN IF( NR.GT.2*KD-1 ) THEN DO 150 L = 1, KD - 1 IF( J2+L.GT.N ) THEN NRT = NR - 1 ELSE NRT = NR END IF IF( NRT.GT.0 ) & CALL DLARTV( NRT, AB( L+2, J1-1 ), INCA, & AB( L+1, J1 ), INCA, D( J1 ), & WORK( J1 ), KD1 ) 150 CONTINUE ELSE J1END = J1 + KD1*( NR-2 ) IF( J1END.GE.J1 ) THEN DO 160 J1INC = J1, J1END, KD1 CALL DROT( KDM1, AB( 3, J1INC-1 ), 1, & AB( 2, J1INC ), 1, D( J1INC ), & WORK( J1INC ) ) 160 CONTINUE END IF LEND = MIN( KDM1, N-J2 ) LAST = J1END + KD1 IF( LEND.GT.0 ) & CALL DROT( LEND, AB( 3, LAST-1 ), 1, & AB( 2, LAST ), 1, D( LAST ), & WORK( LAST ) ) END IF END IF ! ! ! IF( WANTQ ) THEN ! ! accumulate product of plane rotations in Q ! IF( INITQ ) THEN ! ! take advantage of the fact that Q was ! initially the Identity matrix ! IQEND = MAX( IQEND, J2 ) I2 = MAX( 0, K-3 ) IQAEND = 1 + I*KD IF( K.EQ.2 ) & IQAEND = IQAEND + KD IQAEND = MIN( IQAEND, IQEND ) DO 170 J = J1, J2, KD1 IBL = I - I2 / KDM1 I2 = I2 + 1 IQB = MAX( 1, J-IBL ) NQ = 1 + IQAEND - IQB IQAEND = MIN( IQAEND+KD, IQEND ) CALL DROT( NQ, Q( IQB, J-1 ), 1, Q( IQB, J ), & 1, D( J ), WORK( J ) ) 170 CONTINUE ELSE ! DO 180 J = J1, J2, KD1 CALL DROT( N, Q( 1, J-1 ), 1, Q( 1, J ), 1, & D( J ), WORK( J ) ) 180 CONTINUE END IF END IF ! IF( J2+KDN.GT.N ) THEN ! ! adjust J2 to keep within the bounds of the matrix ! NR = NR - 1 J2 = J2 - KDN - 1 END IF ! DO 190 J = J1, J2, KD1 ! ! create nonzero element a(j+kd,j-1) outside the ! band and store it in WORK ! WORK( J+KD ) = WORK( J )*AB( KD1, J ) AB( KD1, J ) = D( J )*AB( KD1, J ) 190 CONTINUE 200 CONTINUE 210 CONTINUE END IF ! IF( KD.GT.0 ) THEN ! ! copy off-diagonal elements to E ! DO 220 I = 1, N - 1 E( I ) = AB( 2, I ) 220 CONTINUE ELSE ! ! set E to zero if original matrix was diagonal ! DO 230 I = 1, N - 1 E( I ) = ZERO 230 CONTINUE END IF ! ! copy diagonal elements to D ! DO 240 I = 1, N D( I ) = AB( 1, I ) 240 CONTINUE END IF ! RETURN ! ! End of DSBTRD ! END DOUBLE PRECISION FUNCTION DSECND( ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! Purpose ! ======= ! ! DSECND returns the user time for a process in seconds. ! This version gets the time from the system function ETIME. ! ! ===================================================================== ! ! .. Local Scalars .. REAL T1 ! .. ! .. Local Arrays .. REAL TARRAY( 2 ) ! .. ! .. External Functions .. REAL ETIME EXTERNAL ETIME ! .. ! .. Executable Statements .. ! call cpu_time ( dsecnd ) ! T1 = ETIME( TARRAY ) ! DSECND = TARRAY( 1 ) RETURN ! ! End of DSECND ! END SUBROUTINE DSPCON( UPLO, N, AP, IPIV, ANORM, RCOND, WORK, IWORK, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, N DOUBLE PRECISION ANORM, RCOND ! .. ! .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION AP( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DSPCON estimates the reciprocal of the condition number (in the ! 1-norm) of a real symmetric packed matrix A using the factorization ! A = U*D*U**T or A = L*D*L**T computed by DSPTRF. ! ! An estimate is obtained for norm(inv(A)), and the reciprocal of the ! condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the details of the factorization are stored ! as an upper or lower triangular matrix. ! = 'U': Upper triangular, form is A = U*D*U**T; ! = 'L': Lower triangular, form is A = L*D*L**T. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The block diagonal matrix D and the multipliers used to ! obtain the factor U or L as computed by DSPTRF, stored as a ! packed triangular matrix. ! ! IPIV (input) INTEGER array, dimension (N) ! Details of the interchanges and the block structure of D ! as determined by DSPTRF. ! ! ANORM (input) DOUBLE PRECISION ! The 1-norm of the original matrix A. ! ! RCOND (output) DOUBLE PRECISION ! The reciprocal of the condition number of the matrix A, ! computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an ! estimate of the 1-norm of inv(A) computed in this routine. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (2*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER I, IP, KASE DOUBLE PRECISION AINVNM ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DLACON, DSPTRS, XERBLA ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( ANORM.LT.ZERO ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSPCON', -INFO ) RETURN END IF ! ! Quick return if possible ! RCOND = ZERO IF( N.EQ.0 ) THEN RCOND = ONE RETURN ELSE IF( ANORM.LE.ZERO ) THEN RETURN END IF ! ! Check that the diagonal matrix D is nonsingular. ! IF( UPPER ) THEN ! ! Upper triangular storage: examine D from bottom to top ! IP = N*( N+1 ) / 2 DO 10 I = N, 1, -1 IF( IPIV( I ).GT.0 .AND. AP( IP ).EQ.ZERO ) & RETURN IP = IP - I 10 CONTINUE ELSE ! ! Lower triangular storage: examine D from top to bottom. ! IP = 1 DO 20 I = 1, N IF( IPIV( I ).GT.0 .AND. AP( IP ).EQ.ZERO ) & RETURN IP = IP + N - I + 1 20 CONTINUE END IF ! ! Estimate the 1-norm of the inverse. ! KASE = 0 30 CONTINUE CALL DLACON( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE ) IF( KASE.NE.0 ) THEN ! ! Multiply by inv(L*D*L') or inv(U*D*U'). ! CALL DSPTRS( UPLO, N, 1, AP, IPIV, WORK, N, INFO ) GO TO 30 END IF ! ! Compute the estimate of the reciprocal condition number. ! IF( AINVNM.NE.ZERO ) & RCOND = ( ONE / AINVNM ) / ANORM ! RETURN ! ! End of DSPCON ! END SUBROUTINE DSPEV( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, LDZ, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSPEV computes all the eigenvalues and, optionally, eigenvectors of a ! real symmetric matrix A in packed storage. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! A, packed columnwise in a linear array. The j-th column of A ! is stored in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. ! ! On exit, AP is overwritten by values generated during the ! reduction to tridiagonal form. If UPLO = 'U', the diagonal ! and first superdiagonal of the tridiagonal matrix T overwrite ! the corresponding elements of A, and if UPLO = 'L', the ! diagonal and first subdiagonal of T overwrite the ! corresponding elements of A. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, the eigenvalues in ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, N) ! If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal ! eigenvectors of the matrix A, with the i-th column of Z ! holding the eigenvector associated with W(i). ! If JOBZ = 'N', then Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = i, the algorithm failed to converge; i ! off-diagonal elements of an intermediate tridiagonal ! form did not converge to zero. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL WANTZ INTEGER IINFO, IMAX, INDE, INDTAU, INDWRK, ISCALE DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, & SMLNUM ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANSP EXTERNAL LSAME, DLAMCH, DLANSP ! .. ! .. External Subroutines .. EXTERNAL DOPGTR, DSCAL, DSPTRD, DSTEQR, DSTERF, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) ! INFO = 0 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) & THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -7 END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSPEV ', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( N.EQ.1 ) THEN W( 1 ) = AP( 1 ) IF( WANTZ ) & Z( 1, 1 ) = ONE RETURN END IF ! ! Get machine constants. ! SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = SQRT( BIGNUM ) ! ! Scale matrix to allowable range, if necessary. ! ANRM = DLANSP( 'M', UPLO, N, AP, WORK ) ISCALE = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / ANRM ELSE IF( ANRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / ANRM END IF IF( ISCALE.EQ.1 ) THEN CALL DSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 ) END IF ! ! Call DSPTRD to reduce symmetric packed matrix to tridiagonal form. ! INDE = 1 INDTAU = INDE + N CALL DSPTRD( UPLO, N, AP, W, WORK( INDE ), WORK( INDTAU ), IINFO ) ! ! For eigenvalues only, call DSTERF. For eigenvectors, first call ! DOPGTR to generate the orthogonal matrix, then call DSTEQR. ! IF( .NOT.WANTZ ) THEN CALL DSTERF( N, W, WORK( INDE ), INFO ) ELSE INDWRK = INDTAU + N CALL DOPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ, & WORK( INDWRK ), IINFO ) CALL DSTEQR( JOBZ, N, W, WORK( INDE ), Z, LDZ, WORK( INDTAU ), & INFO ) END IF ! ! If matrix was scaled, then rescale eigenvalues appropriately. ! IF( ISCALE.EQ.1 ) THEN IF( INFO.EQ.0 ) THEN IMAX = N ELSE IMAX = INFO - 1 END IF CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) END IF ! RETURN ! ! End of DSPEV ! END SUBROUTINE DSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, & IWORK, LIWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, LDZ, LIWORK, LWORK, N ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSPEVD computes all the eigenvalues and, optionally, eigenvectors ! of a real symmetric matrix A in packed storage. If eigenvectors are ! desired, it uses a divide and conquer algorithm. ! ! The divide and conquer algorithm makes very mild assumptions about ! floating point arithmetic. It will work on machines with a guard ! digit in add/subtract, or on those binary machines without guard ! digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or ! Cray-2. It could conceivably fail on hexadecimal or decimal machines ! without guard digits, but we know of none. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! A, packed columnwise in a linear array. The j-th column of A ! is stored in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. ! ! On exit, AP is overwritten by values generated during the ! reduction to tridiagonal form. If UPLO = 'U', the diagonal ! and first superdiagonal of the tridiagonal matrix T overwrite ! the corresponding elements of A, and if UPLO = 'L', the ! diagonal and first subdiagonal of T overwrite the ! corresponding elements of A. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, the eigenvalues in ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, N) ! If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal ! eigenvectors of the matrix A, with the i-th column of Z ! holding the eigenvector associated with W(i). ! If JOBZ = 'N', then Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! WORK (workspace/output) DOUBLE PRECISION array, ! dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! If N <= 1, LWORK must be at least 1. ! If JOBZ = 'N' and N > 1, LWORK must be at least 2*N. ! If JOBZ = 'V' and N > 1, LWORK must be at least ! 1 + 6*N + N**2. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace/output) INTEGER array, dimension (LIWORK) ! On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. ! ! LIWORK (input) INTEGER ! The dimension of the array IWORK. ! If JOBZ = 'N' or N <= 1, LIWORK must be at least 1. ! If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N. ! ! If LIWORK = -1, then a workspace query is assumed; the ! routine only calculates the optimal size of the IWORK array, ! returns this value as the first entry of the IWORK array, and ! no error message related to LIWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: if INFO = i, the algorithm failed to converge; i ! off-diagonal elements of an intermediate tridiagonal ! form did not converge to zero. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, WANTZ INTEGER IINFO, INDE, INDTAU, INDWRK, ISCALE, LIWMIN, & LLWORK, LWMIN DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, & SMLNUM ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANSP EXTERNAL LSAME, DLAMCH, DLANSP ! .. ! .. External Subroutines .. EXTERNAL DOPMTR, DSCAL, DSPTRD, DSTEDC, DSTERF, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) ! INFO = 0 IF( N.LE.1 ) THEN LIWMIN = 1 LWMIN = 1 ELSE IF( WANTZ ) THEN LIWMIN = 3 + 5*N LWMIN = 1 + 6*N + N**2 ELSE LIWMIN = 1 LWMIN = 2*N END IF END IF IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) & THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -7 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -9 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -11 END IF ! IF( INFO.EQ.0 ) THEN WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSPEVD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( N.EQ.1 ) THEN W( 1 ) = AP( 1 ) IF( WANTZ ) & Z( 1, 1 ) = ONE RETURN END IF ! ! Get machine constants. ! SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = SQRT( BIGNUM ) ! ! Scale matrix to allowable range, if necessary. ! ANRM = DLANSP( 'M', UPLO, N, AP, WORK ) ISCALE = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / ANRM ELSE IF( ANRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / ANRM END IF IF( ISCALE.EQ.1 ) THEN CALL DSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 ) END IF ! ! Call DSPTRD to reduce symmetric packed matrix to tridiagonal form. ! INDE = 1 INDTAU = INDE + N CALL DSPTRD( UPLO, N, AP, W, WORK( INDE ), WORK( INDTAU ), IINFO ) ! ! For eigenvalues only, call DSTERF. For eigenvectors, first call ! DSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the ! tridiagonal matrix, then call DOPMTR to multiply it by the ! Householder transformations represented in AP. ! IF( .NOT.WANTZ ) THEN CALL DSTERF( N, W, WORK( INDE ), INFO ) ELSE INDWRK = INDTAU + N LLWORK = LWORK - INDWRK + 1 CALL DSTEDC( 'I', N, W, WORK( INDE ), Z, LDZ, WORK( INDWRK ), & LLWORK, IWORK, LIWORK, INFO ) CALL DOPMTR( 'L', UPLO, 'N', N, N, AP, WORK( INDTAU ), Z, LDZ, & WORK( INDWRK ), IINFO ) END IF ! ! If matrix was scaled, then rescale eigenvalues appropriately. ! IF( ISCALE.EQ.1 ) & CALL DSCAL( N, ONE / SIGMA, W, 1 ) ! WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN RETURN ! ! End of DSPEVD ! END SUBROUTINE DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, & ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, & INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, RANGE, UPLO INTEGER IL, INFO, IU, LDZ, M, N DOUBLE PRECISION ABSTOL, VL, VU ! .. ! .. Array Arguments .. INTEGER IFAIL( * ), IWORK( * ) DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSPEVX computes selected eigenvalues and, optionally, eigenvectors ! of a real symmetric matrix A in packed storage. Eigenvalues/vectors ! can be selected by specifying either a range of values or a range of ! indices for the desired eigenvalues. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! RANGE (input) CHARACTER*1 ! = 'A': all eigenvalues will be found; ! = 'V': all eigenvalues in the half-open interval (VL,VU] ! will be found; ! = 'I': the IL-th through IU-th eigenvalues will be found. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! A, packed columnwise in a linear array. The j-th column of A ! is stored in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. ! ! On exit, AP is overwritten by values generated during the ! reduction to tridiagonal form. If UPLO = 'U', the diagonal ! and first superdiagonal of the tridiagonal matrix T overwrite ! the corresponding elements of A, and if UPLO = 'L', the ! diagonal and first subdiagonal of T overwrite the ! corresponding elements of A. ! ! VL (input) DOUBLE PRECISION ! VU (input) DOUBLE PRECISION ! If RANGE='V', the lower and upper bounds of the interval to ! be searched for eigenvalues. VL < VU. ! Not referenced if RANGE = 'A' or 'I'. ! ! IL (input) INTEGER ! IU (input) INTEGER ! If RANGE='I', the indices (in ascending order) of the ! smallest and largest eigenvalues to be returned. ! 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. ! Not referenced if RANGE = 'A' or 'V'. ! ! ABSTOL (input) DOUBLE PRECISION ! The absolute error tolerance for the eigenvalues. ! An approximate eigenvalue is accepted as converged ! when it is determined to lie in an interval [a,b] ! of width less than or equal to ! ! ABSTOL + EPS * max( |a|,|b| ) , ! ! where EPS is the machine precision. If ABSTOL is less than ! or equal to zero, then EPS*|T| will be used in its place, ! where |T| is the 1-norm of the tridiagonal matrix obtained ! by reducing AP to tridiagonal form. ! ! Eigenvalues will be computed most accurately when ABSTOL is ! set to twice the underflow threshold 2*DLAMCH('S'), not zero. ! If this routine returns with INFO>0, indicating that some ! eigenvectors did not converge, try setting ABSTOL to ! 2*DLAMCH('S'). ! ! See "Computing Small Singular Values of Bidiagonal Matrices ! with Guaranteed High Relative Accuracy," by Demmel and ! Kahan, LAPACK Working Note #3. ! ! M (output) INTEGER ! The total number of eigenvalues found. 0 <= M <= N. ! If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, the selected eigenvalues in ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M)) ! If JOBZ = 'V', then if INFO = 0, the first M columns of Z ! contain the orthonormal eigenvectors of the matrix A ! corresponding to the selected eigenvalues, with the i-th ! column of Z holding the eigenvector associated with W(i). ! If an eigenvector fails to converge, then that column of Z ! contains the latest approximation to the eigenvector, and the ! index of the eigenvector is returned in IFAIL. ! If JOBZ = 'N', then Z is not referenced. ! Note: the user must ensure that at least max(1,M) columns are ! supplied in the array Z; if RANGE = 'V', the exact value of M ! is not known in advance and an upper bound must be used. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (8*N) ! ! IWORK (workspace) INTEGER array, dimension (5*N) ! ! IFAIL (output) INTEGER array, dimension (N) ! If JOBZ = 'V', then if INFO = 0, the first M elements of ! IFAIL are zero. If INFO > 0, then IFAIL contains the ! indices of the eigenvectors that failed to converge. ! If JOBZ = 'N', then IFAIL is not referenced. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, then i eigenvectors failed to converge. ! Their indices are stored in array IFAIL. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL ALLEIG, INDEIG, VALEIG, WANTZ CHARACTER ORDER INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL, & INDISP, INDIWO, INDTAU, INDWRK, ISCALE, ITMP1, & J, JJ, NSPLIT DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, & SIGMA, SMLNUM, TMP1, VLL, VUU ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANSP EXTERNAL LSAME, DLAMCH, DLANSP ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DOPGTR, DOPMTR, DSCAL, DSPTRD, DSTEBZ, & DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) ALLEIG = LSAME( RANGE, 'A' ) VALEIG = LSAME( RANGE, 'V' ) INDEIG = LSAME( RANGE, 'I' ) ! INFO = 0 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN INFO = -2 ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR. LSAME( UPLO, 'U' ) ) ) & THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( VALEIG ) THEN IF( N.GT.0 .AND. VU.LE.VL ) & INFO = -7 ELSE IF( INDEIG ) THEN IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN INFO = -9 END IF END IF END IF IF( INFO.EQ.0 ) THEN IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) & INFO = -14 END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSPEVX', -INFO ) RETURN END IF ! ! Quick return if possible ! M = 0 IF( N.EQ.0 ) & RETURN ! IF( N.EQ.1 ) THEN IF( ALLEIG .OR. INDEIG ) THEN M = 1 W( 1 ) = AP( 1 ) ELSE IF( VL.LT.AP( 1 ) .AND. VU.GE.AP( 1 ) ) THEN M = 1 W( 1 ) = AP( 1 ) END IF END IF IF( WANTZ ) & Z( 1, 1 ) = ONE RETURN END IF ! ! Get machine constants. ! SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) ! ! Scale matrix to allowable range, if necessary. ! ISCALE = 0 ABSTLL = ABSTOL IF( VALEIG ) THEN VLL = VL VUU = VU ELSE VLL = ZERO VUU = ZERO END IF ANRM = DLANSP( 'M', UPLO, N, AP, WORK ) IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / ANRM ELSE IF( ANRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / ANRM END IF IF( ISCALE.EQ.1 ) THEN CALL DSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 ) IF( ABSTOL.GT.0 ) & ABSTLL = ABSTOL*SIGMA IF( VALEIG ) THEN VLL = VL*SIGMA VUU = VU*SIGMA END IF END IF ! ! Call DSPTRD to reduce symmetric packed matrix to tridiagonal form. ! INDTAU = 1 INDE = INDTAU + N INDD = INDE + N INDWRK = INDD + N CALL DSPTRD( UPLO, N, AP, WORK( INDD ), WORK( INDE ), & WORK( INDTAU ), IINFO ) ! ! If all eigenvalues are desired and ABSTOL is less than or equal ! to zero, then call DSTERF or DOPGTR and SSTEQR. If this fails ! for some eigenvalue, then try DSTEBZ. ! IF( ( ALLEIG .OR. ( INDEIG .AND. IL.EQ.1 .AND. IU.EQ.N ) ) .AND. & ( ABSTOL.LE.ZERO ) ) THEN CALL DCOPY( N, WORK( INDD ), 1, W, 1 ) INDEE = INDWRK + 2*N IF( .NOT.WANTZ ) THEN CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) CALL DSTERF( N, W, WORK( INDEE ), INFO ) ELSE CALL DOPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ, & WORK( INDWRK ), IINFO ) CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ, & WORK( INDWRK ), INFO ) IF( INFO.EQ.0 ) THEN DO 10 I = 1, N IFAIL( I ) = 0 10 CONTINUE END IF END IF IF( INFO.EQ.0 ) THEN M = N GO TO 20 END IF INFO = 0 END IF ! ! Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN. ! IF( WANTZ ) THEN ORDER = 'B' ELSE ORDER = 'E' END IF INDIBL = 1 INDISP = INDIBL + N INDIWO = INDISP + N CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, & WORK( INDD ), WORK( INDE ), M, NSPLIT, W, & IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ), & IWORK( INDIWO ), INFO ) ! IF( WANTZ ) THEN CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W, & IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, & WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO ) ! ! Apply orthogonal matrix used in reduction to tridiagonal ! form to eigenvectors returned by DSTEIN. ! CALL DOPMTR( 'L', UPLO, 'N', N, M, AP, WORK( INDTAU ), Z, LDZ, & WORK( INDWRK ), INFO ) END IF ! ! If matrix was scaled, then rescale eigenvalues appropriately. ! 20 CONTINUE IF( ISCALE.EQ.1 ) THEN IF( INFO.EQ.0 ) THEN IMAX = M ELSE IMAX = INFO - 1 END IF CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) END IF ! ! If eigenvalues are not in order, then sort them, along with ! eigenvectors. ! IF( WANTZ ) THEN DO 40 J = 1, M - 1 I = 0 TMP1 = W( J ) DO 30 JJ = J + 1, M IF( W( JJ ).LT.TMP1 ) THEN I = JJ TMP1 = W( JJ ) END IF 30 CONTINUE ! IF( I.NE.0 ) THEN ITMP1 = IWORK( INDIBL+I-1 ) W( I ) = W( J ) IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) W( J ) = TMP1 IWORK( INDIBL+J-1 ) = ITMP1 CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) IF( INFO.NE.0 ) THEN ITMP1 = IFAIL( I ) IFAIL( I ) = IFAIL( J ) IFAIL( J ) = ITMP1 END IF END IF 40 CONTINUE END IF ! RETURN ! ! End of DSPEVX ! END SUBROUTINE DSPGST( ITYPE, UPLO, N, AP, BP, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, ITYPE, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ), BP( * ) ! .. ! ! Purpose ! ======= ! ! DSPGST reduces a real symmetric-definite generalized eigenproblem ! to standard form, using packed storage. ! ! If ITYPE = 1, the problem is A*x = lambda*B*x, ! and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T) ! ! If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or ! B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L. ! ! B must have been previously factorized as U**T*U or L*L**T by DPPTRF. ! ! Arguments ! ========= ! ! ITYPE (input) INTEGER ! = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T); ! = 2 or 3: compute U*A*U**T or L**T*A*L. ! ! UPLO (input) CHARACTER ! = 'U': Upper triangle of A is stored and B is factored as ! U**T*U; ! = 'L': Lower triangle of A is stored and B is factored as ! L*L**T. ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! A, packed columnwise in a linear array. The j-th column of A ! is stored in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. ! ! On exit, if INFO = 0, the transformed matrix, stored in the ! same format as A. ! ! BP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The triangular factor from the Cholesky factorization of B, ! stored in the same format as A, as returned by DPPTRF. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, HALF PARAMETER ( ONE = 1.0D0, HALF = 0.5D0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER J, J1, J1J1, JJ, K, K1, K1K1, KK DOUBLE PRECISION AJJ, AKK, BJJ, BKK, CT ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DSCAL, DSPMV, DSPR2, DTPMV, DTPSV, & XERBLA ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT EXTERNAL LSAME, DDOT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN INFO = -1 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSPGST', -INFO ) RETURN END IF ! IF( ITYPE.EQ.1 ) THEN IF( UPPER ) THEN ! ! Compute inv(U')*A*inv(U) ! ! J1 and JJ are the indices of A(1,j) and A(j,j) ! JJ = 0 DO 10 J = 1, N J1 = JJ + 1 JJ = JJ + J ! ! Compute the j-th column of the upper triangle of A ! BJJ = BP( JJ ) CALL DTPSV( UPLO, 'Transpose', 'Nonunit', J, BP, & AP( J1 ), 1 ) CALL DSPMV( UPLO, J-1, -ONE, AP, BP( J1 ), 1, ONE, & AP( J1 ), 1 ) CALL DSCAL( J-1, ONE / BJJ, AP( J1 ), 1 ) AP( JJ ) = ( AP( JJ )-DDOT( J-1, AP( J1 ), 1, BP( J1 ), & 1 ) ) / BJJ 10 CONTINUE ELSE ! ! Compute inv(L)*A*inv(L') ! ! KK and K1K1 are the indices of A(k,k) and A(k+1,k+1) ! KK = 1 DO 20 K = 1, N K1K1 = KK + N - K + 1 ! ! Update the lower triangle of A(k:n,k:n) ! AKK = AP( KK ) BKK = BP( KK ) AKK = AKK / BKK**2 AP( KK ) = AKK IF( K.LT.N ) THEN CALL DSCAL( N-K, ONE / BKK, AP( KK+1 ), 1 ) CT = -HALF*AKK CALL DAXPY( N-K, CT, BP( KK+1 ), 1, AP( KK+1 ), 1 ) CALL DSPR2( UPLO, N-K, -ONE, AP( KK+1 ), 1, & BP( KK+1 ), 1, AP( K1K1 ) ) CALL DAXPY( N-K, CT, BP( KK+1 ), 1, AP( KK+1 ), 1 ) CALL DTPSV( UPLO, 'No transpose', 'Non-unit', N-K, & BP( K1K1 ), AP( KK+1 ), 1 ) END IF KK = K1K1 20 CONTINUE END IF ELSE IF( UPPER ) THEN ! ! Compute U*A*U' ! ! K1 and KK are the indices of A(1,k) and A(k,k) ! KK = 0 DO 30 K = 1, N K1 = KK + 1 KK = KK + K ! ! Update the upper triangle of A(1:k,1:k) ! AKK = AP( KK ) BKK = BP( KK ) CALL DTPMV( UPLO, 'No transpose', 'Non-unit', K-1, BP, & AP( K1 ), 1 ) CT = HALF*AKK CALL DAXPY( K-1, CT, BP( K1 ), 1, AP( K1 ), 1 ) CALL DSPR2( UPLO, K-1, ONE, AP( K1 ), 1, BP( K1 ), 1, & AP ) CALL DAXPY( K-1, CT, BP( K1 ), 1, AP( K1 ), 1 ) CALL DSCAL( K-1, BKK, AP( K1 ), 1 ) AP( KK ) = AKK*BKK**2 30 CONTINUE ELSE ! ! Compute L'*A*L ! ! JJ and J1J1 are the indices of A(j,j) and A(j+1,j+1) ! JJ = 1 DO 40 J = 1, N J1J1 = JJ + N - J + 1 ! ! Compute the j-th column of the lower triangle of A ! AJJ = AP( JJ ) BJJ = BP( JJ ) AP( JJ ) = AJJ*BJJ + DDOT( N-J, AP( JJ+1 ), 1, & BP( JJ+1 ), 1 ) CALL DSCAL( N-J, BJJ, AP( JJ+1 ), 1 ) CALL DSPMV( UPLO, N-J, ONE, AP( J1J1 ), BP( JJ+1 ), 1, & ONE, AP( JJ+1 ), 1 ) CALL DTPMV( UPLO, 'Transpose', 'Non-unit', N-J+1, & BP( JJ ), AP( JJ ), 1 ) JJ = J1J1 40 CONTINUE END IF END IF RETURN ! ! End of DSPGST ! END SUBROUTINE DSPGV( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, & INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, ITYPE, LDZ, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ), BP( * ), W( * ), WORK( * ), & Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSPGV computes all the eigenvalues and, optionally, the eigenvectors ! of a real generalized symmetric-definite eigenproblem, of the form ! A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. ! Here A and B are assumed to be symmetric, stored in packed format, ! and B is also positive definite. ! ! Arguments ! ========= ! ! ITYPE (input) INTEGER ! Specifies the problem type to be solved: ! = 1: A*x = (lambda)*B*x ! = 2: A*B*x = (lambda)*x ! = 3: B*A*x = (lambda)*x ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangles of A and B are stored; ! = 'L': Lower triangles of A and B are stored. ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension ! (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! A, packed columnwise in a linear array. The j-th column of A ! is stored in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. ! ! On exit, the contents of AP are destroyed. ! ! BP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! B, packed columnwise in a linear array. The j-th column of B ! is stored in the array BP as follows: ! if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; ! if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. ! ! On exit, the triangular factor U or L from the Cholesky ! factorization B = U**T*U or B = L*L**T, in the same storage ! format as B. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, the eigenvalues in ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, N) ! If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of ! eigenvectors. The eigenvectors are normalized as follows: ! if ITYPE = 1 or 2, Z**T*B*Z = I; ! if ITYPE = 3, Z**T*inv(B)*Z = I. ! If JOBZ = 'N', then Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: DPPTRF or DSPEV returned an error code: ! <= N: if INFO = i, DSPEV failed to converge; ! i off-diagonal elements of an intermediate ! tridiagonal form did not converge to zero. ! > N: if INFO = n + i, for 1 <= i <= n, then the leading ! minor of order i of B is not positive definite. ! The factorization of B could not be completed and ! no eigenvalues or eigenvectors were computed. ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL UPPER, WANTZ CHARACTER TRANS INTEGER J, NEIG ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DPPTRF, DSPEV, DSPGST, DTPMV, DTPSV, XERBLA ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) UPPER = LSAME( UPLO, 'U' ) ! INFO = 0 IF( ITYPE.LT.0 .OR. ITYPE.GT.3 ) THEN INFO = -1 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -9 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSPGV ', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Form a Cholesky factorization of B. ! CALL DPPTRF( UPLO, N, BP, INFO ) IF( INFO.NE.0 ) THEN INFO = N + INFO RETURN END IF ! ! Transform problem to standard eigenvalue problem and solve. ! CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO ) CALL DSPEV( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, INFO ) ! IF( WANTZ ) THEN ! ! Backtransform eigenvectors to the original problem. ! NEIG = N IF( INFO.GT.0 ) & NEIG = INFO - 1 IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN ! ! For A*x=(lambda)*B*x and A*B*x=(lambda)*x; ! backtransform eigenvectors: x = inv(L)'*y or inv(U)*y ! IF( UPPER ) THEN TRANS = 'N' ELSE TRANS = 'T' END IF ! DO 10 J = 1, NEIG CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ), & 1 ) 10 CONTINUE ! ELSE IF( ITYPE.EQ.3 ) THEN ! ! For B*A*x=(lambda)*x; ! backtransform eigenvectors: x = L*y or U'*y ! IF( UPPER ) THEN TRANS = 'T' ELSE TRANS = 'N' END IF ! DO 20 J = 1, NEIG CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ), & 1 ) 20 CONTINUE END IF END IF RETURN ! ! End of DSPGV ! END SUBROUTINE DSPGVD( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, & LWORK, IWORK, LIWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, ITYPE, LDZ, LIWORK, LWORK, N ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION AP( * ), BP( * ), W( * ), WORK( * ), & Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSPGVD computes all the eigenvalues, and optionally, the eigenvectors ! of a real generalized symmetric-definite eigenproblem, of the form ! A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and ! B are assumed to be symmetric, stored in packed format, and B is also ! positive definite. ! If eigenvectors are desired, it uses a divide and conquer algorithm. ! ! The divide and conquer algorithm makes very mild assumptions about ! floating point arithmetic. It will work on machines with a guard ! digit in add/subtract, or on those binary machines without guard ! digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or ! Cray-2. It could conceivably fail on hexadecimal or decimal machines ! without guard digits, but we know of none. ! ! Arguments ! ========= ! ! ITYPE (input) INTEGER ! Specifies the problem type to be solved: ! = 1: A*x = (lambda)*B*x ! = 2: A*B*x = (lambda)*x ! = 3: B*A*x = (lambda)*x ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangles of A and B are stored; ! = 'L': Lower triangles of A and B are stored. ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! A, packed columnwise in a linear array. The j-th column of A ! is stored in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. ! ! On exit, the contents of AP are destroyed. ! ! BP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! B, packed columnwise in a linear array. The j-th column of B ! is stored in the array BP as follows: ! if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; ! if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. ! ! On exit, the triangular factor U or L from the Cholesky ! factorization B = U**T*U or B = L*L**T, in the same storage ! format as B. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, the eigenvalues in ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, N) ! If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of ! eigenvectors. The eigenvectors are normalized as follows: ! if ITYPE = 1 or 2, Z**T*B*Z = I; ! if ITYPE = 3, Z**T*inv(B)*Z = I. ! If JOBZ = 'N', then Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! If N <= 1, LWORK >= 1. ! If JOBZ = 'N' and N > 1, LWORK >= 2*N. ! If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace/output) INTEGER array, dimension (LIWORK) ! On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. ! ! LIWORK (input) INTEGER ! The dimension of the array IWORK. ! If JOBZ = 'N' or N <= 1, LIWORK >= 1. ! If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N. ! ! If LIWORK = -1, then a workspace query is assumed; the ! routine only calculates the optimal size of the IWORK array, ! returns this value as the first entry of the IWORK array, and ! no error message related to LIWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: DPPTRF or DSPEVD returned an error code: ! <= N: if INFO = i, DSPEVD failed to converge; ! i off-diagonal elements of an intermediate ! tridiagonal form did not converge to zero; ! > N: if INFO = N + i, for 1 <= i <= N, then the leading ! minor of order i of B is not positive definite. ! The factorization of B could not be completed and ! no eigenvalues or eigenvectors were computed. ! ! Further Details ! =============== ! ! Based on contributions by ! Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, UPPER, WANTZ CHARACTER TRANS INTEGER J, LGN, LIWMIN, LWMIN, NEIG ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DPPTRF, DSPEVD, DSPGST, DTPMV, DTPSV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC DBLE, INT, LOG, MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) UPPER = LSAME( UPLO, 'U' ) LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) ! INFO = 0 IF( N.LE.1 ) THEN LGN = 0 LIWMIN = 1 LWMIN = 1 ELSE LGN = INT( LOG( DBLE( N ) ) / LOG( TWO ) ) IF( 2**LGN.LT.N ) & LGN = LGN + 1 IF( 2**LGN.LT.N ) & LGN = LGN + 1 IF( WANTZ ) THEN LIWMIN = 3 + 5*N LWMIN = 1 + 5*N + 2*N*LGN + 2*N**2 ELSE LIWMIN = 1 LWMIN = 2*N END IF END IF ! IF( ITYPE.LT.0 .OR. ITYPE.GT.3 ) THEN INFO = -1 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDZ.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -11 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -13 END IF ! IF( INFO.EQ.0 ) THEN WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSPGVD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Form a Cholesky factorization of BP. ! CALL DPPTRF( UPLO, N, BP, INFO ) IF( INFO.NE.0 ) THEN INFO = N + INFO RETURN END IF ! ! Transform problem to standard eigenvalue problem and solve. ! CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO ) CALL DSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, IWORK, & LIWORK, INFO ) LWMIN = MAX( DBLE( LWMIN ), DBLE( WORK( 1 ) ) ) LIWMIN = MAX( DBLE( LIWMIN ), DBLE( IWORK( 1 ) ) ) ! IF( WANTZ ) THEN ! ! Backtransform eigenvectors to the original problem. ! NEIG = N IF( INFO.GT.0 ) & NEIG = INFO - 1 IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN ! ! For A*x=(lambda)*B*x and A*B*x=(lambda)*x; ! backtransform eigenvectors: x = inv(L)'*y or inv(U)*y ! IF( UPPER ) THEN TRANS = 'N' ELSE TRANS = 'T' END IF ! DO 10 J = 1, NEIG CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ), & 1 ) 10 CONTINUE ! ELSE IF( ITYPE.EQ.3 ) THEN ! ! For B*A*x=(lambda)*x; ! backtransform eigenvectors: x = L*y or U'*y ! IF( UPPER ) THEN TRANS = 'T' ELSE TRANS = 'N' END IF ! DO 20 J = 1, NEIG CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ), & 1 ) 20 CONTINUE END IF END IF ! WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN ! RETURN ! ! End of DSPGVD ! END SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, & IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, & IFAIL, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, RANGE, UPLO INTEGER IL, INFO, ITYPE, IU, LDZ, M, N DOUBLE PRECISION ABSTOL, VL, VU ! .. ! .. Array Arguments .. INTEGER IFAIL( * ), IWORK( * ) DOUBLE PRECISION AP( * ), BP( * ), W( * ), WORK( * ), & Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSPGVX computes selected eigenvalues, and optionally, eigenvectors ! of a real generalized symmetric-definite eigenproblem, of the form ! A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A ! and B are assumed to be symmetric, stored in packed storage, and B ! is also positive definite. Eigenvalues and eigenvectors can be ! selected by specifying either a range of values or a range of indices ! for the desired eigenvalues. ! ! Arguments ! ========= ! ! ITYPE (input) INTEGER ! Specifies the problem type to be solved: ! = 1: A*x = (lambda)*B*x ! = 2: A*B*x = (lambda)*x ! = 3: B*A*x = (lambda)*x ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! RANGE (input) CHARACTER*1 ! = 'A': all eigenvalues will be found. ! = 'V': all eigenvalues in the half-open interval (VL,VU] ! will be found. ! = 'I': the IL-th through IU-th eigenvalues will be found. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A and B are stored; ! = 'L': Lower triangle of A and B are stored. ! ! N (input) INTEGER ! The order of the matrix pencil (A,B). N >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! A, packed columnwise in a linear array. The j-th column of A ! is stored in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. ! ! On exit, the contents of AP are destroyed. ! ! BP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! B, packed columnwise in a linear array. The j-th column of B ! is stored in the array BP as follows: ! if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; ! if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. ! ! On exit, the triangular factor U or L from the Cholesky ! factorization B = U**T*U or B = L*L**T, in the same storage ! format as B. ! ! VL (input) DOUBLE PRECISION ! VU (input) DOUBLE PRECISION ! If RANGE='V', the lower and upper bounds of the interval to ! be searched for eigenvalues. VL < VU. ! Not referenced if RANGE = 'A' or 'I'. ! ! IL (input) INTEGER ! IU (input) INTEGER ! If RANGE='I', the indices (in ascending order) of the ! smallest and largest eigenvalues to be returned. ! 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. ! Not referenced if RANGE = 'A' or 'V'. ! ! ABSTOL (input) DOUBLE PRECISION ! The absolute error tolerance for the eigenvalues. ! An approximate eigenvalue is accepted as converged ! when it is determined to lie in an interval [a,b] ! of width less than or equal to ! ! ABSTOL + EPS * max( |a|,|b| ) , ! ! where EPS is the machine precision. If ABSTOL is less than ! or equal to zero, then EPS*|T| will be used in its place, ! where |T| is the 1-norm of the tridiagonal matrix obtained ! by reducing A to tridiagonal form. ! ! Eigenvalues will be computed most accurately when ABSTOL is ! set to twice the underflow threshold 2*DLAMCH('S'), not zero. ! If this routine returns with INFO>0, indicating that some ! eigenvectors did not converge, try setting ABSTOL to ! 2*DLAMCH('S'). ! ! M (output) INTEGER ! The total number of eigenvalues found. 0 <= M <= N. ! If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! On normal exit, the first M elements contain the selected ! eigenvalues in ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M)) ! If JOBZ = 'N', then Z is not referenced. ! If JOBZ = 'V', then if INFO = 0, the first M columns of Z ! contain the orthonormal eigenvectors of the matrix A ! corresponding to the selected eigenvalues, with the i-th ! column of Z holding the eigenvector associated with W(i). ! The eigenvectors are normalized as follows: ! if ITYPE = 1 or 2, Z**T*B*Z = I; ! if ITYPE = 3, Z**T*inv(B)*Z = I. ! ! If an eigenvector fails to converge, then that column of Z ! contains the latest approximation to the eigenvector, and the ! index of the eigenvector is returned in IFAIL. ! Note: the user must ensure that at least max(1,M) columns are ! supplied in the array Z; if RANGE = 'V', the exact value of M ! is not known in advance and an upper bound must be used. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (8*N) ! ! IWORK (workspace) INTEGER array, dimension (5*N) ! ! IFAIL (output) INTEGER array, dimension (N) ! If JOBZ = 'V', then if INFO = 0, the first M elements of ! IFAIL are zero. If INFO > 0, then IFAIL contains the ! indices of the eigenvectors that failed to converge. ! If JOBZ = 'N', then IFAIL is not referenced. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: DPPTRF or DSPEVX returned an error code: ! <= N: if INFO = i, DSPEVX failed to converge; ! i eigenvectors failed to converge. Their indices ! are stored in array IFAIL. ! > N: if INFO = N + i, for 1 <= i <= N, then the leading ! minor of order i of B is not positive definite. ! The factorization of B could not be completed and ! no eigenvalues or eigenvectors were computed. ! ! Further Details ! =============== ! ! Based on contributions by ! Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL ALLEIG, INDEIG, UPPER, VALEIG, WANTZ CHARACTER TRANS INTEGER J ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DPPTRF, DSPEVX, DSPGST, DTPMV, DTPSV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! UPPER = LSAME( UPLO, 'U' ) WANTZ = LSAME( JOBZ, 'V' ) ALLEIG = LSAME( RANGE, 'A' ) VALEIG = LSAME( RANGE, 'V' ) INDEIG = LSAME( RANGE, 'I' ) ! INFO = 0 IF( ITYPE.LT.0 .OR. ITYPE.GT.3 ) THEN INFO = -1 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN INFO = -3 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( VALEIG .AND. N.GT.0 .AND. VU.LE.VL ) THEN INFO = -9 ELSE IF( INDEIG .AND. IL.LT.1 ) THEN INFO = -10 ELSE IF( INDEIG .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN INFO = -11 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -16 END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSPGVX', -INFO ) RETURN END IF ! ! Quick return if possible ! M = 0 IF( N.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! ! Form a Cholesky factorization of B. ! CALL DPPTRF( UPLO, N, BP, INFO ) IF( INFO.NE.0 ) THEN INFO = N + INFO RETURN END IF ! ! Transform problem to standard eigenvalue problem and solve. ! CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO ) CALL DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, & W, Z, LDZ, WORK, IWORK, IFAIL, INFO ) ! IF( WANTZ ) THEN ! ! Backtransform eigenvectors to the original problem. ! IF( INFO.GT.0 ) & M = INFO - 1 IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN ! ! For A*x=(lambda)*B*x and A*B*x=(lambda)*x; ! backtransform eigenvectors: x = inv(L)'*y or inv(U)*y ! IF( UPPER ) THEN TRANS = 'N' ELSE TRANS = 'T' END IF ! DO 10 J = 1, M CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ), & 1 ) 10 CONTINUE ! ELSE IF( ITYPE.EQ.3 ) THEN ! ! For B*A*x=(lambda)*x; ! backtransform eigenvectors: x = L*y or U'*y ! IF( UPPER ) THEN TRANS = 'T' ELSE TRANS = 'N' END IF ! DO 20 J = 1, M CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ), & 1 ) 20 CONTINUE END IF END IF ! RETURN ! ! End of DSPGVX ! END SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, & FERR, BERR, WORK, IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDB, LDX, N, NRHS ! .. ! .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ), & FERR( * ), WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DSPRFS improves the computed solution to a system of linear ! equations when the coefficient matrix is symmetric indefinite ! and packed, and provides error bounds and backward error estimates ! for the solution. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The upper or lower triangle of the symmetric matrix A, packed ! columnwise in a linear array. The j-th column of A is stored ! in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. ! ! AFP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The factored form of the matrix A. AFP contains the block ! diagonal matrix D and the multipliers used to obtain the ! factor U or L from the factorization A = U*D*U**T or ! A = L*D*L**T as computed by DSPTRF, stored as a packed ! triangular matrix. ! ! IPIV (input) INTEGER array, dimension (N) ! Details of the interchanges and the block structure of D ! as determined by DSPTRF. ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) ! The right hand side matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! On entry, the solution matrix X, as computed by DSPTRS. ! On exit, the improved solution matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Internal Parameters ! =================== ! ! ITMAX is the maximum number of steps of iterative refinement. ! ! ===================================================================== ! ! .. Parameters .. INTEGER ITMAX PARAMETER ( ITMAX = 5 ) DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D+0 ) DOUBLE PRECISION THREE PARAMETER ( THREE = 3.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER COUNT, I, IK, J, K, KASE, KK, NZ DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DLACON, DSPMV, DSPTRS, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -10 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSPRFS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN DO 10 J = 1, NRHS FERR( J ) = ZERO BERR( J ) = ZERO 10 CONTINUE RETURN END IF ! ! NZ = maximum number of nonzero elements in each row of A, plus 1 ! NZ = N + 1 EPS = DLAMCH( 'Epsilon' ) SAFMIN = DLAMCH( 'Safe minimum' ) SAFE1 = NZ*SAFMIN SAFE2 = SAFE1 / EPS ! ! Do for each right hand side ! DO 140 J = 1, NRHS ! COUNT = 1 LSTRES = THREE 20 CONTINUE ! ! Loop until stopping criterion is satisfied. ! ! Compute residual R = B - A * X ! CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ), & 1 ) ! ! Compute componentwise relative backward error from formula ! ! max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) ! ! where abs(Z) is the componentwise absolute value of the matrix ! or vector Z. If the i-th component of the denominator is less ! than SAFE2, then SAFE1 is added to the i-th components of the ! numerator and denominator before dividing. ! DO 30 I = 1, N WORK( I ) = ABS( B( I, J ) ) 30 CONTINUE ! ! Compute abs(A)*abs(X) + abs(B). ! KK = 1 IF( UPPER ) THEN DO 50 K = 1, N S = ZERO XK = ABS( X( K, J ) ) IK = KK DO 40 I = 1, K - 1 WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK S = S + ABS( AP( IK ) )*ABS( X( I, J ) ) IK = IK + 1 40 CONTINUE WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S KK = KK + K 50 CONTINUE ELSE DO 70 K = 1, N S = ZERO XK = ABS( X( K, J ) ) WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK IK = KK + 1 DO 60 I = K + 1, N WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK S = S + ABS( AP( IK ) )*ABS( X( I, J ) ) IK = IK + 1 60 CONTINUE WORK( K ) = WORK( K ) + S KK = KK + ( N-K+1 ) 70 CONTINUE END IF S = ZERO DO 80 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) ELSE S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / & ( WORK( I )+SAFE1 ) ) END IF 80 CONTINUE BERR( J ) = S ! ! Test stopping criterion. Continue iterating if ! 1) The residual BERR(J) is larger than machine epsilon, and ! 2) BERR(J) decreased by at least a factor of 2 during the ! last iteration, and ! 3) At most ITMAX iterations tried. ! IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. & COUNT.LE.ITMAX ) THEN ! ! Update solution and try again. ! CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N, INFO ) CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) LSTRES = BERR( J ) COUNT = COUNT + 1 GO TO 20 END IF ! ! Bound error from formula ! ! norm(X - XTRUE) / norm(X) .le. FERR = ! norm( abs(inv(A))* ! ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) ! ! where ! norm(Z) is the magnitude of the largest component of Z ! inv(A) is the inverse of A ! abs(Z) is the componentwise absolute value of the matrix or ! vector Z ! NZ is the maximum number of nonzeros in any row of A, plus 1 ! EPS is machine epsilon ! ! The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) ! is incremented by SAFE1 if the i-th component of ! abs(A)*abs(X) + abs(B) is less than SAFE2. ! ! Use DLACON to estimate the infinity-norm of the matrix ! inv(A) * diag(W), ! where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) ! DO 90 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) ELSE WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 END IF 90 CONTINUE ! KASE = 0 100 CONTINUE CALL DLACON( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), & KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN ! ! Multiply by diag(W)*inv(A'). ! CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N, & INFO ) DO 110 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 110 CONTINUE ELSE IF( KASE.EQ.2 ) THEN ! ! Multiply by inv(A)*diag(W). ! DO 120 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 120 CONTINUE CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N, & INFO ) END IF GO TO 100 END IF ! ! Normalize error. ! LSTRES = ZERO DO 130 I = 1, N LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 130 CONTINUE IF( LSTRES.NE.ZERO ) & FERR( J ) = FERR( J ) / LSTRES ! 140 CONTINUE ! RETURN ! ! End of DSPRFS ! END SUBROUTINE DSPSV( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDB, N, NRHS ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION AP( * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DSPSV computes the solution to a real system of linear equations ! A * X = B, ! where A is an N-by-N symmetric matrix stored in packed format and X ! and B are N-by-NRHS matrices. ! ! The diagonal pivoting method is used to factor A as ! A = U * D * U**T, if UPLO = 'U', or ! A = L * D * L**T, if UPLO = 'L', ! where U (or L) is a product of permutation and unit upper (lower) ! triangular matrices, D is symmetric and block diagonal with 1-by-1 ! and 2-by-2 diagonal blocks. The factored form of A is then used to ! solve the system of equations A * X = B. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The number of linear equations, i.e., the order of the ! matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! A, packed columnwise in a linear array. The j-th column of A ! is stored in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. ! See below for further details. ! ! On exit, the block diagonal matrix D and the multipliers used ! to obtain the factor U or L from the factorization ! A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as ! a packed triangular matrix in the same storage format as A. ! ! IPIV (output) INTEGER array, dimension (N) ! Details of the interchanges and the block structure of D, as ! determined by DSPTRF. If IPIV(k) > 0, then rows and columns ! k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1 ! diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, ! then rows and columns k-1 and -IPIV(k) were interchanged and ! D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and ! IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and ! -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 ! diagonal block. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the N-by-NRHS right hand side matrix B. ! On exit, if INFO = 0, the N-by-NRHS solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, D(i,i) is exactly zero. The factorization ! has been completed, but the block diagonal matrix D is ! exactly singular, so the solution could not be ! computed. ! ! Further Details ! =============== ! ! The packed storage scheme is illustrated by the following example ! when N = 4, UPLO = 'U': ! ! Two-dimensional storage of the symmetric matrix A: ! ! a11 a12 a13 a14 ! a22 a23 a24 ! a33 a34 (aij = aji) ! a44 ! ! Packed storage of the upper triangle of A: ! ! AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] ! ! ===================================================================== ! ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DSPTRF, DSPTRS, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSPSV ', -INFO ) RETURN END IF ! ! Compute the factorization A = U*D*U' or A = L*D*L'. ! CALL DSPTRF( UPLO, N, AP, IPIV, INFO ) IF( INFO.EQ.0 ) THEN ! ! Solve the system A*X = B, overwriting B with X. ! CALL DSPTRS( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO ) ! END IF RETURN ! ! End of DSPSV ! END SUBROUTINE DSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, & LDX, RCOND, FERR, BERR, WORK, IWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER FACT, UPLO INTEGER INFO, LDB, LDX, N, NRHS DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ), & FERR( * ), WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DSPSVX uses the diagonal pivoting factorization A = U*D*U**T or ! A = L*D*L**T to compute the solution to a real system of linear ! equations A * X = B, where A is an N-by-N symmetric matrix stored ! in packed format and X and B are N-by-NRHS matrices. ! ! Error bounds on the solution and a condition estimate are also ! provided. ! ! Description ! =========== ! ! The following steps are performed: ! ! 1. If FACT = 'N', the diagonal pivoting method is used to factor A as ! A = U * D * U**T, if UPLO = 'U', or ! A = L * D * L**T, if UPLO = 'L', ! where U (or L) is a product of permutation and unit upper (lower) ! triangular matrices and D is symmetric and block diagonal with ! 1-by-1 and 2-by-2 diagonal blocks. ! ! 2. If some D(i,i)=0, so that D is exactly singular, then the routine ! returns with INFO = i. Otherwise, the factored form of A is used ! to estimate the condition number of the matrix A. If the ! reciprocal of the condition number is less than machine precision, ! INFO = N+1 is returned as a warning, but the routine still goes on ! to solve for X and compute error bounds as described below. ! ! 3. The system of equations is solved for X using the factored form ! of A. ! ! 4. Iterative refinement is applied to improve the computed solution ! matrix and calculate error bounds and backward error estimates ! for it. ! ! Arguments ! ========= ! ! FACT (input) CHARACTER*1 ! Specifies whether or not the factored form of A has been ! supplied on entry. ! = 'F': On entry, AFP and IPIV contain the factored form of ! A. AP, AFP and IPIV will not be modified. ! = 'N': The matrix A will be copied to AFP and factored. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The number of linear equations, i.e., the order of the ! matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The upper or lower triangle of the symmetric matrix A, packed ! columnwise in a linear array. The j-th column of A is stored ! in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. ! See below for further details. ! ! AFP (input or output) DOUBLE PRECISION array, dimension ! (N*(N+1)/2) ! If FACT = 'F', then AFP is an input argument and on entry ! contains the block diagonal matrix D and the multipliers used ! to obtain the factor U or L from the factorization ! A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as ! a packed triangular matrix in the same storage format as A. ! ! If FACT = 'N', then AFP is an output argument and on exit ! contains the block diagonal matrix D and the multipliers used ! to obtain the factor U or L from the factorization ! A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as ! a packed triangular matrix in the same storage format as A. ! ! IPIV (input or output) INTEGER array, dimension (N) ! If FACT = 'F', then IPIV is an input argument and on entry ! contains details of the interchanges and the block structure ! of D, as determined by DSPTRF. ! If IPIV(k) > 0, then rows and columns k and IPIV(k) were ! interchanged and D(k,k) is a 1-by-1 diagonal block. ! If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and ! columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) ! is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = ! IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were ! interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. ! ! If FACT = 'N', then IPIV is an output argument and on exit ! contains details of the interchanges and the block structure ! of D, as determined by DSPTRF. ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) ! The N-by-NRHS right hand side matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! RCOND (output) DOUBLE PRECISION ! The estimate of the reciprocal condition number of the matrix ! A. If RCOND is less than the machine precision (in ! particular, if RCOND = 0), the matrix is singular to working ! precision. This condition is indicated by a return code of ! INFO > 0. ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, and i is ! <= N: D(i,i) is exactly zero. The factorization ! has been completed but the factor D is exactly ! singular, so the solution and error bounds could ! not be computed. RCOND = 0 is returned. ! = N+1: D is nonsingular, but RCOND is less than machine ! precision, meaning that the matrix is singular ! to working precision. Nevertheless, the ! solution and error bounds are computed because ! there are a number of situations where the ! computed solution can be more accurate than the ! value of RCOND would suggest. ! ! Further Details ! =============== ! ! The packed storage scheme is illustrated by the following example ! when N = 4, UPLO = 'U': ! ! Two-dimensional storage of the symmetric matrix A: ! ! a11 a12 a13 a14 ! a22 a23 a24 ! a33 a34 (aij = aji) ! a44 ! ! Packed storage of the upper triangle of A: ! ! AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOFACT DOUBLE PRECISION ANORM ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANSP EXTERNAL LSAME, DLAMCH, DLANSP ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLACPY, DSPCON, DSPRFS, DSPTRF, DSPTRS, & XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 NOFACT = LSAME( FACT, 'N' ) IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN INFO = -1 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) & THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.0 ) THEN INFO = -4 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -11 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSPSVX', -INFO ) RETURN END IF ! IF( NOFACT ) THEN ! ! Compute the factorization A = U*D*U' or A = L*D*L'. ! CALL DCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 ) CALL DSPTRF( UPLO, N, AFP, IPIV, INFO ) ! ! Return if INFO is non-zero. ! IF( INFO.NE.0 ) THEN IF( INFO.GT.0 ) & RCOND = ZERO RETURN END IF END IF ! ! Compute the norm of the matrix A. ! ANORM = DLANSP( 'I', UPLO, N, AP, WORK ) ! ! Compute the reciprocal of the condition number of A. ! CALL DSPCON( UPLO, N, AFP, IPIV, ANORM, RCOND, WORK, IWORK, INFO ) ! ! Set INFO = N+1 if the matrix is singular to working precision. ! IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) & INFO = N + 1 ! ! Compute the solution vectors X. ! CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL DSPTRS( UPLO, N, NRHS, AFP, IPIV, X, LDX, INFO ) ! ! Use iterative refinement to improve the computed solutions and ! compute error bounds and backward error estimates for them. ! CALL DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, & BERR, WORK, IWORK, INFO ) ! RETURN ! ! End of DSPSVX ! END SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * ) ! .. ! ! Purpose ! ======= ! ! DSPTRD reduces a real symmetric matrix A stored in packed form to ! symmetric tridiagonal form T by an orthogonal similarity ! transformation: Q**T * A * Q = T. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! A, packed columnwise in a linear array. The j-th column of A ! is stored in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. ! On exit, if UPLO = 'U', the diagonal and first superdiagonal ! of A are overwritten by the corresponding elements of the ! tridiagonal matrix T, and the elements above the first ! superdiagonal, with the array TAU, represent the orthogonal ! matrix Q as a product of elementary reflectors; if UPLO ! = 'L', the diagonal and first subdiagonal of A are over- ! written by the corresponding elements of the tridiagonal ! matrix T, and the elements below the first subdiagonal, with ! the array TAU, represent the orthogonal matrix Q as a product ! of elementary reflectors. See Further Details. ! ! D (output) DOUBLE PRECISION array, dimension (N) ! The diagonal elements of the tridiagonal matrix T: ! D(i) = A(i,i). ! ! E (output) DOUBLE PRECISION array, dimension (N-1) ! The off-diagonal elements of the tridiagonal matrix T: ! E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. ! ! TAU (output) DOUBLE PRECISION array, dimension (N-1) ! The scalar factors of the elementary reflectors (see Further ! Details). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! If UPLO = 'U', the matrix Q is represented as a product of elementary ! reflectors ! ! Q = H(n-1) . . . H(2) H(1). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP, ! overwriting A(1:i-1,i+1), and tau is stored in TAU(i). ! ! If UPLO = 'L', the matrix Q is represented as a product of elementary ! reflectors ! ! Q = H(1) H(2) . . . H(n-1). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP, ! overwriting A(i+2:n,i), and tau is stored in TAU(i). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO, HALF PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0, & HALF = 1.0D0 / 2.0D0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER I, I1, I1I1, II DOUBLE PRECISION ALPHA, TAUI ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DLARFG, DSPMV, DSPR2, XERBLA ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT EXTERNAL LSAME, DDOT ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSPTRD', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.LE.0 ) & RETURN ! IF( UPPER ) THEN ! ! Reduce the upper triangle of A. ! I1 is the index in AP of A(1,I+1). ! I1 = N*( N-1 ) / 2 + 1 DO 10 I = N - 1, 1, -1 ! ! Generate elementary reflector H(i) = I - tau * v * v' ! to annihilate A(1:i-1,i+1) ! CALL DLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI ) E( I ) = AP( I1+I-1 ) ! IF( TAUI.NE.ZERO ) THEN ! ! Apply H(i) from both sides to A(1:i,1:i) ! AP( I1+I-1 ) = ONE ! ! Compute y := tau * A * v storing y in TAU(1:i) ! CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU, & 1 ) ! ! Compute w := y - 1/2 * tau * (y'*v) * v ! ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 ) CALL DAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 ) ! ! Apply the transformation as a rank-2 update: ! A := A - v * w' - w * v' ! CALL DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP ) ! AP( I1+I-1 ) = E( I ) END IF D( I+1 ) = AP( I1+I ) TAU( I ) = TAUI I1 = I1 - I 10 CONTINUE D( 1 ) = AP( 1 ) ELSE ! ! Reduce the lower triangle of A. II is the index in AP of ! A(i,i) and I1I1 is the index of A(i+1,i+1). ! II = 1 DO 20 I = 1, N - 1 I1I1 = II + N - I + 1 ! ! Generate elementary reflector H(i) = I - tau * v * v' ! to annihilate A(i+2:n,i) ! CALL DLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI ) E( I ) = AP( II+1 ) ! IF( TAUI.NE.ZERO ) THEN ! ! Apply H(i) from both sides to A(i+1:n,i+1:n) ! AP( II+1 ) = ONE ! ! Compute y := tau * A * v storing y in TAU(i:n-1) ! CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1, & ZERO, TAU( I ), 1 ) ! ! Compute w := y - 1/2 * tau * (y'*v) * v ! ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ), & 1 ) CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 ) ! ! Apply the transformation as a rank-2 update: ! A := A - v * w' - w * v' ! CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1, & AP( I1I1 ) ) ! AP( II+1 ) = E( I ) END IF D( I ) = AP( II ) TAU( I ) = TAUI II = I1I1 20 CONTINUE D( N ) = AP( II ) END IF ! RETURN ! ! End of DSPTRD ! END SUBROUTINE DSPTRF( UPLO, N, AP, IPIV, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, N ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION AP( * ) ! .. ! ! Purpose ! ======= ! ! DSPTRF computes the factorization of a real symmetric matrix A stored ! in packed format using the Bunch-Kaufman diagonal pivoting method: ! ! A = U*D*U**T or A = L*D*L**T ! ! where U (or L) is a product of permutation and unit upper (lower) ! triangular matrices, and D is symmetric and block diagonal with ! 1-by-1 and 2-by-2 diagonal blocks. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangle of the symmetric matrix ! A, packed columnwise in a linear array. The j-th column of A ! is stored in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. ! ! On exit, the block diagonal matrix D and the multipliers used ! to obtain the factor U or L, stored as a packed triangular ! matrix overwriting A (see below for further details). ! ! IPIV (output) INTEGER array, dimension (N) ! Details of the interchanges and the block structure of D. ! If IPIV(k) > 0, then rows and columns k and IPIV(k) were ! interchanged and D(k,k) is a 1-by-1 diagonal block. ! If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and ! columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) ! is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = ! IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were ! interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, D(i,i) is exactly zero. The factorization ! has been completed, but the block diagonal matrix D is ! exactly singular, and division by zero will occur if it ! is used to solve a system of equations. ! ! Further Details ! =============== ! ! 5-96 - Based on modifications by J. Lewis, Boeing Computer Services ! Company ! ! If UPLO = 'U', then A = U*D*U', where ! U = P(n)*U(n)* ... *P(k)U(k)* ..., ! i.e., U is a product of terms P(k)*U(k), where k decreases from n to ! 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 ! and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as ! defined by IPIV(k), and U(k) is a unit upper triangular matrix, such ! that if the diagonal block D(k) is of order s (s = 1 or 2), then ! ! ( I v 0 ) k-s ! U(k) = ( 0 I 0 ) s ! ( 0 0 I ) n-k ! k-s s n-k ! ! If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). ! If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), ! and A(k,k), and v overwrites A(1:k-2,k-1:k). ! ! If UPLO = 'L', then A = L*D*L', where ! L = P(1)*L(1)* ... *P(k)*L(k)* ..., ! i.e., L is a product of terms P(k)*L(k), where k increases from 1 to ! n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 ! and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as ! defined by IPIV(k), and L(k) is a unit lower triangular matrix, such ! that if the diagonal block D(k) is of order s (s = 1 or 2), then ! ! ( I 0 0 ) k-1 ! L(k) = ( 0 I 0 ) s ! ( 0 v I ) n-k-s+1 ! k-1 s n-k-s+1 ! ! If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). ! If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), ! and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) DOUBLE PRECISION EIGHT, SEVTEN PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER I, IMAX, J, JMAX, K, KC, KK, KNC, KP, KPC, & KSTEP, KX, NPP DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1, & ROWMAX, T, WK, WKM1, WKP1 ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX EXTERNAL LSAME, IDAMAX ! .. ! .. External Subroutines .. EXTERNAL DSCAL, DSPR, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSPTRF', -INFO ) RETURN END IF ! ! Initialize ALPHA for use in choosing pivot block size. ! ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT ! IF( UPPER ) THEN ! ! Factorize A as U*D*U' using the upper triangle of A ! ! K is the main loop index, decreasing from N to 1 in steps of ! 1 or 2 ! K = N KC = ( N-1 )*N / 2 + 1 10 CONTINUE KNC = KC ! ! If K < 1, exit from loop ! IF( K.LT.1 ) & GO TO 110 KSTEP = 1 ! ! Determine rows and columns to be interchanged and whether ! a 1-by-1 or 2-by-2 pivot block will be used ! ABSAKK = ABS( AP( KC+K-1 ) ) ! ! IMAX is the row-index of the largest off-diagonal element in ! column K, and COLMAX is its absolute value ! IF( K.GT.1 ) THEN IMAX = IDAMAX( K-1, AP( KC ), 1 ) COLMAX = ABS( AP( KC+IMAX-1 ) ) ELSE COLMAX = ZERO END IF ! IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN ! ! Column K is zero: set INFO and continue ! IF( INFO.EQ.0 ) & INFO = K KP = K ELSE IF( ABSAKK.GE.ALPHA*COLMAX ) THEN ! ! no interchange, use 1-by-1 pivot block ! KP = K ELSE ! ! JMAX is the column-index of the largest off-diagonal ! element in row IMAX, and ROWMAX is its absolute value ! ROWMAX = ZERO JMAX = IMAX KX = IMAX*( IMAX+1 ) / 2 + IMAX DO 20 J = IMAX + 1, K IF( ABS( AP( KX ) ).GT.ROWMAX ) THEN ROWMAX = ABS( AP( KX ) ) JMAX = J END IF KX = KX + J 20 CONTINUE KPC = ( IMAX-1 )*IMAX / 2 + 1 IF( IMAX.GT.1 ) THEN JMAX = IDAMAX( IMAX-1, AP( KPC ), 1 ) ROWMAX = MAX( ROWMAX, ABS( AP( KPC+JMAX-1 ) ) ) END IF ! IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN ! ! no interchange, use 1-by-1 pivot block ! KP = K ELSE IF( ABS( AP( KPC+IMAX-1 ) ).GE.ALPHA*ROWMAX ) THEN ! ! interchange rows and columns K and IMAX, use 1-by-1 ! pivot block ! KP = IMAX ELSE ! ! interchange rows and columns K-1 and IMAX, use 2-by-2 ! pivot block ! KP = IMAX KSTEP = 2 END IF END IF ! KK = K - KSTEP + 1 IF( KSTEP.EQ.2 ) & KNC = KNC - K + 1 IF( KP.NE.KK ) THEN ! ! Interchange rows and columns KK and KP in the leading ! submatrix A(1:k,1:k) ! CALL DSWAP( KP-1, AP( KNC ), 1, AP( KPC ), 1 ) KX = KPC + KP - 1 DO 30 J = KP + 1, KK - 1 KX = KX + J - 1 T = AP( KNC+J-1 ) AP( KNC+J-1 ) = AP( KX ) AP( KX ) = T 30 CONTINUE T = AP( KNC+KK-1 ) AP( KNC+KK-1 ) = AP( KPC+KP-1 ) AP( KPC+KP-1 ) = T IF( KSTEP.EQ.2 ) THEN T = AP( KC+K-2 ) AP( KC+K-2 ) = AP( KC+KP-1 ) AP( KC+KP-1 ) = T END IF END IF ! ! Update the leading submatrix ! IF( KSTEP.EQ.1 ) THEN ! ! 1-by-1 pivot block D(k): column k now holds ! ! W(k) = U(k)*D(k) ! ! where U(k) is the k-th column of U ! ! Perform a rank-1 update of A(1:k-1,1:k-1) as ! ! A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)' ! R1 = ONE / AP( KC+K-1 ) CALL DSPR( UPLO, K-1, -R1, AP( KC ), 1, AP ) ! ! Store U(k) in column k ! CALL DSCAL( K-1, R1, AP( KC ), 1 ) ELSE ! ! 2-by-2 pivot block D(k): columns k and k-1 now hold ! ! ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k) ! ! where U(k) and U(k-1) are the k-th and (k-1)-th columns ! of U ! ! Perform a rank-2 update of A(1:k-2,1:k-2) as ! ! A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )' ! = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )' ! IF( K.GT.2 ) THEN ! D12 = AP( K-1+( K-1 )*K / 2 ) D22 = AP( K-1+( K-2 )*( K-1 ) / 2 ) / D12 D11 = AP( K+( K-1 )*K / 2 ) / D12 T = ONE / ( D11*D22-ONE ) D12 = T / D12 ! DO 50 J = K - 2, 1, -1 WKM1 = D12*( D11*AP( J+( K-2 )*( K-1 ) / 2 )- & AP( J+( K-1 )*K / 2 ) ) WK = D12*( D22*AP( J+( K-1 )*K / 2 )- & AP( J+( K-2 )*( K-1 ) / 2 ) ) DO 40 I = J, 1, -1 AP( I+( J-1 )*J / 2 ) = AP( I+( J-1 )*J / 2 ) - & AP( I+( K-1 )*K / 2 )*WK - & AP( I+( K-2 )*( K-1 ) / 2 )*WKM1 40 CONTINUE AP( J+( K-1 )*K / 2 ) = WK AP( J+( K-2 )*( K-1 ) / 2 ) = WKM1 50 CONTINUE ! END IF ! END IF END IF ! ! Store details of the interchanges in IPIV ! IF( KSTEP.EQ.1 ) THEN IPIV( K ) = KP ELSE IPIV( K ) = -KP IPIV( K-1 ) = -KP END IF ! ! Decrease K and return to the start of the main loop ! K = K - KSTEP KC = KNC - K GO TO 10 ! ELSE ! ! Factorize A as L*D*L' using the lower triangle of A ! ! K is the main loop index, increasing from 1 to N in steps of ! 1 or 2 ! K = 1 KC = 1 NPP = N*( N+1 ) / 2 60 CONTINUE KNC = KC ! ! If K > N, exit from loop ! IF( K.GT.N ) & GO TO 110 KSTEP = 1 ! ! Determine rows and columns to be interchanged and whether ! a 1-by-1 or 2-by-2 pivot block will be used ! ABSAKK = ABS( AP( KC ) ) ! ! IMAX is the row-index of the largest off-diagonal element in ! column K, and COLMAX is its absolute value ! IF( K.LT.N ) THEN IMAX = K + IDAMAX( N-K, AP( KC+1 ), 1 ) COLMAX = ABS( AP( KC+IMAX-K ) ) ELSE COLMAX = ZERO END IF ! IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN ! ! Column K is zero: set INFO and continue ! IF( INFO.EQ.0 ) & INFO = K KP = K ELSE IF( ABSAKK.GE.ALPHA*COLMAX ) THEN ! ! no interchange, use 1-by-1 pivot block ! KP = K ELSE ! ! JMAX is the column-index of the largest off-diagonal ! element in row IMAX, and ROWMAX is its absolute value ! ROWMAX = ZERO KX = KC + IMAX - K DO 70 J = K, IMAX - 1 IF( ABS( AP( KX ) ).GT.ROWMAX ) THEN ROWMAX = ABS( AP( KX ) ) JMAX = J END IF KX = KX + N - J 70 CONTINUE KPC = NPP - ( N-IMAX+1 )*( N-IMAX+2 ) / 2 + 1 IF( IMAX.LT.N ) THEN JMAX = IMAX + IDAMAX( N-IMAX, AP( KPC+1 ), 1 ) ROWMAX = MAX( ROWMAX, ABS( AP( KPC+JMAX-IMAX ) ) ) END IF ! IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN ! ! no interchange, use 1-by-1 pivot block ! KP = K ELSE IF( ABS( AP( KPC ) ).GE.ALPHA*ROWMAX ) THEN ! ! interchange rows and columns K and IMAX, use 1-by-1 ! pivot block ! KP = IMAX ELSE ! ! interchange rows and columns K+1 and IMAX, use 2-by-2 ! pivot block ! KP = IMAX KSTEP = 2 END IF END IF ! KK = K + KSTEP - 1 IF( KSTEP.EQ.2 ) & KNC = KNC + N - K + 1 IF( KP.NE.KK ) THEN ! ! Interchange rows and columns KK and KP in the trailing ! submatrix A(k:n,k:n) ! IF( KP.LT.N ) & CALL DSWAP( N-KP, AP( KNC+KP-KK+1 ), 1, AP( KPC+1 ), & 1 ) KX = KNC + KP - KK DO 80 J = KK + 1, KP - 1 KX = KX + N - J + 1 T = AP( KNC+J-KK ) AP( KNC+J-KK ) = AP( KX ) AP( KX ) = T 80 CONTINUE T = AP( KNC ) AP( KNC ) = AP( KPC ) AP( KPC ) = T IF( KSTEP.EQ.2 ) THEN T = AP( KC+1 ) AP( KC+1 ) = AP( KC+KP-K ) AP( KC+KP-K ) = T END IF END IF ! ! Update the trailing submatrix ! IF( KSTEP.EQ.1 ) THEN ! ! 1-by-1 pivot block D(k): column k now holds ! ! W(k) = L(k)*D(k) ! ! where L(k) is the k-th column of L ! IF( K.LT.N ) THEN ! ! Perform a rank-1 update of A(k+1:n,k+1:n) as ! ! A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)' ! R1 = ONE / AP( KC ) CALL DSPR( UPLO, N-K, -R1, AP( KC+1 ), 1, & AP( KC+N-K+1 ) ) ! ! Store L(k) in column K ! CALL DSCAL( N-K, R1, AP( KC+1 ), 1 ) END IF ELSE ! ! 2-by-2 pivot block D(k): columns K and K+1 now hold ! ! ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k) ! ! where L(k) and L(k+1) are the k-th and (k+1)-th columns ! of L ! IF( K.LT.N-1 ) THEN ! ! Perform a rank-2 update of A(k+2:n,k+2:n) as ! ! A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )' ! = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k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tore details of the interchanges in IPIV ! IF( KSTEP.EQ.1 ) THEN IPIV( K ) = KP ELSE IPIV( K ) = -KP IPIV( K+1 ) = -KP END IF ! ! Increase K and return to the start of the main loop ! K = K + KSTEP KC = KNC + N - K + 2 GO TO 60 ! END IF ! 110 CONTINUE RETURN ! ! End of DSPTRF ! END SUBROUTINE DSPTRI( UPLO, N, AP, IPIV, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, N ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION AP( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DSPTRI computes the inverse of a real symmetric indefinite matrix ! A in packed storage using the factorization A = U*D*U**T or ! A = L*D*L**T computed by DSPTRF. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the details of the factorization are stored ! as an upper or lower triangular matrix. ! = 'U': Upper triangular, form is A = U*D*U**T; ! = 'L': Lower triangular, form is A = L*D*L**T. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the block diagonal matrix D and the multipliers ! used to obtain the factor U or L as computed by DSPTRF, ! stored as a packed triangular matrix. ! ! On exit, if INFO = 0, the (symmetric) inverse of the original ! matrix, stored as a packed triangular matrix. The j-th column ! of inv(A) is stored in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = inv(A)(i,j) for 1<=i<=j; ! if UPLO = 'L', ! AP(i + (j-1)*(2n-j)/2) = inv(A)(i,j) for j<=i<=n. ! ! IPIV (input) INTEGER array, dimension (N) ! Details of the interchanges and the block structure of D ! as determined by DSPTRF. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its ! inverse could not be computed. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER J, K, KC, KCNEXT, KP, KPC, KSTEP, KX, NPP DOUBLE PRECISION AK, AKKP1, AKP1, D, T, TEMP ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT EXTERNAL LSAME, DDOT ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DSPMV, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSPTRI', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Check that the diagonal matrix D is nonsingular. ! IF( UPPER ) THEN ! ! Upper triangular storage: examine D from bottom to top ! KP = N*( N+1 ) / 2 DO 10 INFO = N, 1, -1 IF( IPIV( INFO ).GT.0 .AND. AP( KP ).EQ.ZERO ) & RETURN KP = KP - INFO 10 CONTINUE ELSE ! ! Lower triangular storage: examine D from top to bottom. ! KP = 1 DO 20 INFO = 1, N IF( IPIV( INFO ).GT.0 .AND. AP( KP ).EQ.ZERO ) & RETURN KP = KP + N - INFO + 1 20 CONTINUE END IF INFO = 0 ! IF( UPPER ) THEN ! ! Compute inv(A) from the factorization A = U*D*U'. ! ! K is the main loop index, increasing from 1 to N in steps of ! 1 or 2, depending on the size of the diagonal blocks. ! K = 1 KC = 1 30 CONTINUE ! ! If K > N, exit from loop. ! IF( K.GT.N ) & GO TO 50 ! KCNEXT = KC + K IF( IPIV( K ).GT.0 ) THEN ! ! 1 x 1 diagonal block ! ! Invert the diagonal block. ! AP( KC+K-1 ) = ONE / AP( KC+K-1 ) ! ! Compute column K of the inverse. ! IF( K.GT.1 ) THEN CALL DCOPY( K-1, AP( KC ), 1, WORK, 1 ) CALL DSPMV( UPLO, K-1, -ONE, AP, WORK, 1, ZERO, AP( KC ), & 1 ) AP( KC+K-1 ) = AP( KC+K-1 ) - & DDOT( K-1, WORK, 1, AP( KC ), 1 ) END IF KSTEP = 1 ELSE ! ! 2 x 2 diagonal block ! ! Invert the diagonal block. ! T = ABS( AP( KCNEXT+K-1 ) ) AK = AP( KC+K-1 ) / T AKP1 = AP( KCNEXT+K ) / T AKKP1 = AP( KCNEXT+K-1 ) / T D = T*( AK*AKP1-ONE ) AP( KC+K-1 ) = AKP1 / D AP( KCNEXT+K ) = AK / D AP( KCNEXT+K-1 ) = -AKKP1 / D ! ! Compute columns K and K+1 of the inverse. ! IF( K.GT.1 ) THEN CALL DCOPY( K-1, AP( KC ), 1, WORK, 1 ) CALL DSPMV( UPLO, K-1, -ONE, AP, WORK, 1, ZERO, AP( KC ), & 1 ) AP( KC+K-1 ) = AP( KC+K-1 ) - & DDOT( K-1, WORK, 1, AP( KC ), 1 ) AP( KCNEXT+K-1 ) = AP( KCNEXT+K-1 ) - & DDOT( K-1, AP( KC ), 1, AP( KCNEXT ), & 1 ) CALL DCOPY( K-1, AP( KCNEXT ), 1, WORK, 1 ) CALL DSPMV( UPLO, K-1, -ONE, AP, WORK, 1, ZERO, & AP( KCNEXT ), 1 ) AP( KCNEXT+K ) = AP( KCNEXT+K ) - & DDOT( K-1, WORK, 1, AP( KCNEXT ), 1 ) END IF KSTEP = 2 KCNEXT = KCNEXT + K + 1 END IF ! KP = ABS( IPIV( K ) ) IF( KP.NE.K ) THEN ! ! Interchange rows and columns K and KP in the leading ! submatrix A(1:k+1,1:k+1) ! KPC = ( KP-1 )*KP / 2 + 1 CALL DSWAP( KP-1, AP( KC ), 1, AP( KPC ), 1 ) KX = KPC + KP - 1 DO 40 J = KP + 1, K - 1 KX = KX + J - 1 TEMP = AP( KC+J-1 ) AP( KC+J-1 ) = AP( KX ) AP( KX ) = TEMP 40 CONTINUE TEMP = AP( KC+K-1 ) AP( KC+K-1 ) = AP( KPC+KP-1 ) AP( KPC+KP-1 ) = TEMP IF( KSTEP.EQ.2 ) THEN TEMP = AP( KC+K+K-1 ) AP( KC+K+K-1 ) = AP( KC+K+KP-1 ) AP( KC+K+KP-1 ) = TEMP END IF END IF ! K = K + KSTEP KC = KCNEXT GO TO 30 50 CONTINUE ! ELSE ! ! Compute inv(A) from the factorization A = L*D*L'. ! ! K is the main loop index, increasing from 1 to N in steps of ! 1 or 2, depending on the size of the diagonal blocks. ! NPP = N*( N+1 ) / 2 K = N KC = NPP 60 CONTINUE ! ! If K < 1, exit from loop. ! IF( K.LT.1 ) & GO TO 80 ! KCNEXT = KC - ( N-K+2 ) IF( IPIV( K ).GT.0 ) THEN ! ! 1 x 1 diagonal block ! ! Invert the diagonal block. ! AP( KC ) = ONE / AP( KC ) ! ! Compute column K of the inverse. ! IF( K.LT.N ) THEN CALL DCOPY( N-K, AP( KC+1 ), 1, WORK, 1 ) CALL DSPMV( UPLO, N-K, -ONE, AP( KC+N-K+1 ), WORK, 1, & ZERO, AP( KC+1 ), 1 ) AP( KC ) = AP( KC ) - DDOT( N-K, WORK, 1, AP( KC+1 ), 1 ) END IF KSTEP = 1 ELSE ! ! 2 x 2 diagonal block ! ! Invert the diagonal block. ! T = ABS( AP( KCNEXT+1 ) ) AK = AP( KCNEXT ) / T AKP1 = AP( KC ) / T AKKP1 = AP( KCNEXT+1 ) / T D = T*( AK*AKP1-ONE ) AP( KCNEXT ) = AKP1 / D AP( KC ) = AK / D AP( KCNEXT+1 ) = -AKKP1 / D ! ! Compute columns K-1 and K of the inverse. ! IF( K.LT.N ) THEN CALL DCOPY( N-K, AP( KC+1 ), 1, WORK, 1 ) CALL DSPMV( UPLO, N-K, -ONE, AP( KC+( N-K+1 ) ), WORK, 1, & ZERO, AP( KC+1 ), 1 ) AP( KC ) = AP( KC ) - DDOT( N-K, WORK, 1, AP( KC+1 ), 1 ) AP( KCNEXT+1 ) = AP( KCNEXT+1 ) - & DDOT( N-K, AP( KC+1 ), 1, & AP( KCNEXT+2 ), 1 ) CALL DCOPY( N-K, AP( KCNEXT+2 ), 1, WORK, 1 ) CALL DSPMV( UPLO, N-K, -ONE, AP( KC+( N-K+1 ) ), WORK, 1, & ZERO, AP( KCNEXT+2 ), 1 ) AP( KCNEXT ) = AP( KCNEXT ) - & DDOT( N-K, WORK, 1, AP( KCNEXT+2 ), 1 ) END IF KSTEP = 2 KCNEXT = KCNEXT - ( N-K+3 ) END IF ! KP = ABS( IPIV( K ) ) IF( KP.NE.K ) THEN ! ! Interchange rows and columns K and KP in the trailing ! submatrix A(k-1:n,k-1:n) ! KPC = NPP - ( N-KP+1 )*( N-KP+2 ) / 2 + 1 IF( KP.LT.N ) & CALL DSWAP( N-KP, AP( KC+KP-K+1 ), 1, AP( KPC+1 ), 1 ) KX = KC + KP - K DO 70 J = K + 1, KP - 1 KX = KX + N - J + 1 TEMP = AP( KC+J-K ) AP( KC+J-K ) = AP( KX ) AP( KX ) = TEMP 70 CONTINUE TEMP = AP( KC ) AP( KC ) = AP( KPC ) AP( KPC ) = TEMP IF( KSTEP.EQ.2 ) THEN TEMP = AP( KC-N+K-1 ) AP( KC-N+K-1 ) = AP( KC-N+KP-1 ) AP( KC-N+KP-1 ) = TEMP END IF END IF ! K = K - KSTEP KC = KCNEXT GO TO 60 80 CONTINUE END IF ! RETURN ! ! End of DSPTRI ! END SUBROUTINE DSPTRS( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDB, N, NRHS ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION AP( * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DSPTRS solves a system of linear equations A*X = B with a real ! symmetric matrix A stored in packed format using the factorization ! A = U*D*U**T or A = L*D*L**T computed by DSPTRF. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the details of the factorization are stored ! as an upper or lower triangular matrix. ! = 'U': Upper triangular, form is A = U*D*U**T; ! = 'L': Lower triangular, form is A = L*D*L**T. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The block diagonal matrix D and the multipliers used to ! obtain the factor U or L as computed by DSPTRF, stored as a ! packed triangular matrix. ! ! IPIV (input) INTEGER array, dimension (N) ! Details of the interchanges and the block structure of D ! as determined by DSPTRF. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the right hand side matrix B. ! On exit, the solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER J, K, KC, KP DOUBLE PRECISION AK, AKM1, AKM1K, BK, BKM1, DENOM ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DGEMV, DGER, DSCAL, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSPTRS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) & RETURN ! IF( UPPER ) THEN ! ! Solve A*X = B, where A = U*D*U'. ! ! First solve U*D*X = B, overwriting B with X. ! ! K is the main loop index, decreasing from N to 1 in steps of ! 1 or 2, depending on the size of the diagonal blocks. ! K = N KC = N*( N+1 ) / 2 + 1 10 CONTINUE ! ! If K < 1, exit from loop. ! IF( K.LT.1 ) & GO TO 30 ! KC = KC - K IF( IPIV( K ).GT.0 ) THEN ! ! 1 x 1 diagonal block ! ! Interchange rows K and IPIV(K). ! KP = IPIV( K ) IF( KP.NE.K ) & CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB ) ! ! Multiply by inv(U(K)), where U(K) is the transformation ! stored in column K of A. ! CALL DGER( K-1, NRHS, -ONE, AP( KC ), 1, B( K, 1 ), LDB, & B( 1, 1 ), LDB ) ! ! Multiply by the inverse of the diagonal block. ! CALL DSCAL( NRHS, ONE / AP( KC+K-1 ), B( K, 1 ), LDB ) K = K - 1 ELSE ! ! 2 x 2 diagonal block ! ! Interchange rows K-1 and -IPIV(K). ! KP = -IPIV( K ) IF( KP.NE.K-1 ) & CALL DSWAP( NRHS, B( K-1, 1 ), LDB, B( KP, 1 ), LDB ) ! ! Multiply by inv(U(K)), where U(K) is the transformation ! stored in columns K-1 and K of A. ! CALL DGER( K-2, NRHS, -ONE, AP( KC ), 1, B( K, 1 ), LDB, & B( 1, 1 ), LDB ) CALL DGER( K-2, NRHS, -ONE, AP( KC-( K-1 ) ), 1, & B( K-1, 1 ), LDB, B( 1, 1 ), LDB ) ! ! Multiply by the inverse of the diagonal block. ! AKM1K = AP( KC+K-2 ) AKM1 = AP( KC-1 ) / AKM1K AK = AP( KC+K-1 ) / AKM1K DENOM = AKM1*AK - ONE DO 20 J = 1, NRHS BKM1 = B( K-1, J ) / AKM1K BK = B( K, J ) / AKM1K B( K-1, J ) = ( AK*BKM1-BK ) / DENOM B( K, J ) = ( AKM1*BK-BKM1 ) / DENOM 20 CONTINUE KC = KC - K + 1 K = K - 2 END IF ! GO TO 10 30 CONTINUE ! ! Next solve U'*X = B, overwriting B with X. ! ! K is the main loop index, increasing from 1 to N in steps of ! 1 or 2, depending on the size of the diagonal blocks. ! K = 1 KC = 1 40 CONTINUE ! ! If K > N, exit from loop. ! IF( K.GT.N ) & GO TO 50 ! IF( IPIV( K ).GT.0 ) THEN ! ! 1 x 1 diagonal block ! ! Multiply by inv(U'(K)), where U(K) is the transformation ! stored in column K of A. ! CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB, AP( KC ), & 1, ONE, B( K, 1 ), LDB ) ! ! Interchange rows K and IPIV(K). ! KP = IPIV( K ) IF( KP.NE.K ) & CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB ) KC = KC + K K = K + 1 ELSE ! ! 2 x 2 diagonal block ! ! Multiply by inv(U'(K+1)), where U(K+1) is the transformation ! stored in columns K and K+1 of A. ! CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB, AP( KC ), & 1, ONE, B( K, 1 ), LDB ) CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB, & AP( KC+K ), 1, ONE, B( K+1, 1 ), LDB ) ! ! Interchange rows K and -IPIV(K). ! KP = -IPIV( K ) IF( KP.NE.K ) & CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB ) KC = KC + 2*K + 1 K = K + 2 END IF ! GO TO 40 50 CONTINUE ! ELSE ! ! Solve A*X = B, where A = L*D*L'. ! ! First solve L*D*X = B, overwriting B with X. ! ! K is the main loop index, increasing from 1 to N in steps of ! 1 or 2, depending on the size of the diagonal blocks. ! K = 1 KC = 1 60 CONTINUE ! ! If K > N, exit from loop. ! IF( K.GT.N ) & GO TO 80 ! IF( IPIV( K ).GT.0 ) THEN ! ! 1 x 1 diagonal block ! ! Interchange rows K and IPIV(K). ! KP = IPIV( K ) IF( KP.NE.K ) & CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB ) ! ! Multiply by inv(L(K)), where L(K) is the transformation ! stored in column K of A. ! IF( K.LT.N ) & CALL DGER( N-K, NRHS, -ONE, AP( KC+1 ), 1, B( K, 1 ), & LDB, B( K+1, 1 ), LDB ) ! ! Multiply by the inverse of the diagonal block. ! CALL DSCAL( NRHS, ONE / AP( KC ), B( K, 1 ), LDB ) KC = KC + N - K + 1 K = K + 1 ELSE ! ! 2 x 2 diagonal block ! ! Interchange rows K+1 and -IPIV(K). ! KP = -IPIV( K ) IF( KP.NE.K+1 ) & CALL DSWAP( NRHS, B( K+1, 1 ), LDB, B( KP, 1 ), LDB ) ! ! Multiply by inv(L(K)), where L(K) is the transformation ! stored in columns K and K+1 of A. ! IF( K.LT.N-1 ) THEN CALL DGER( N-K-1, NRHS, -ONE, AP( KC+2 ), 1, B( K, 1 ), & LDB, B( K+2, 1 ), LDB ) CALL DGER( N-K-1, NRHS, -ONE, AP( KC+N-K+2 ), 1, & B( K+1, 1 ), LDB, B( K+2, 1 ), LDB ) END IF ! ! Multiply by the inverse of the diagonal block. ! AKM1K = AP( KC+1 ) AKM1 = AP( KC ) / AKM1K AK = AP( KC+N-K+1 ) / AKM1K DENOM = AKM1*AK - ONE DO 70 J = 1, NRHS BKM1 = B( K, J ) / AKM1K BK = B( K+1, J ) / AKM1K B( K, J ) = ( AK*BKM1-BK ) / DENOM B( K+1, J ) = ( AKM1*BK-BKM1 ) / DENOM 70 CONTINUE KC = KC + 2*( N-K ) + 1 K = K + 2 END IF ! GO TO 60 80 CONTINUE ! ! Next solve L'*X = B, overwriting B with X. ! ! K is the main loop index, decreasing from N to 1 in steps of ! 1 or 2, depending on the size of the diagonal blocks. ! K = N KC = N*( N+1 ) / 2 + 1 90 CONTINUE ! ! If K < 1, exit from loop. ! IF( K.LT.1 ) & GO TO 100 ! KC = KC - ( N-K+1 ) IF( IPIV( K ).GT.0 ) THEN ! ! 1 x 1 diagonal block ! ! Multiply by inv(L'(K)), where L(K) is the transformation ! stored in column K of A. ! IF( K.LT.N ) & CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ), & LDB, AP( KC+1 ), 1, ONE, B( K, 1 ), LDB ) ! ! Interchange rows K and IPIV(K). ! KP = IPIV( K ) IF( KP.NE.K ) & CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB ) K = K - 1 ELSE ! ! 2 x 2 diagonal block ! ! Multiply by inv(L'(K-1)), where L(K-1) is the transformation ! stored in columns K-1 and K of A. ! IF( K.LT.N ) THEN CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ), & LDB, AP( KC+1 ), 1, ONE, B( K, 1 ), LDB ) CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ), & LDB, AP( KC-( N-K ) ), 1, ONE, B( K-1, 1 ), & LDB ) END IF ! ! Interchange rows K and -IPIV(K). ! KP = -IPIV( K ) IF( KP.NE.K ) & CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB ) KC = KC - ( N-K+2 ) K = K - 2 END IF ! GO TO 90 100 CONTINUE END IF ! RETURN ! ! End of DSPTRS ! END SUBROUTINE DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, & M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER ORDER, RANGE INTEGER IL, INFO, IU, M, N, NSPLIT DOUBLE PRECISION ABSTOL, VL, VU ! .. ! .. Array Arguments .. INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * ) DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DSTEBZ computes the eigenvalues of a symmetric tridiagonal ! matrix T. The user may ask for all eigenvalues, all eigenvalues ! in the half-open interval (VL, VU], or the IL-th through IU-th ! eigenvalues. ! ! To avoid overflow, the matrix must be scaled so that its ! largest element is no greater than overflow**(1/2) * ! underflow**(1/4) in absolute value, and for greatest ! accuracy, it should not be much smaller than that. ! ! See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal ! Matrix", Report CS41, Computer Science Dept., Stanford ! University, July 21, 1966. ! ! Arguments ! ========= ! ! RANGE (input) CHARACTER ! = 'A': ("All") all eigenvalues will be found. ! = 'V': ("Value") all eigenvalues in the half-open interval ! (VL, VU] will be found. ! = 'I': ("Index") the IL-th through IU-th eigenvalues (of the ! entire matrix) will be found. ! ! ORDER (input) CHARACTER ! = 'B': ("By Block") the eigenvalues will be grouped by ! split-off block (see IBLOCK, ISPLIT) and ! ordered from smallest to largest within ! the block. ! = 'E': ("Entire matrix") ! the eigenvalues for the entire matrix ! will be ordered from smallest to ! largest. ! ! N (input) INTEGER ! The order of the tridiagonal matrix T. N >= 0. ! ! VL (input) DOUBLE PRECISION ! VU (input) DOUBLE PRECISION ! If RANGE='V', the lower and upper bounds of the interval to ! be searched for eigenvalues. Eigenvalues less than or equal ! to VL, or greater than VU, will not be returned. VL < VU. ! Not referenced if RANGE = 'A' or 'I'. ! ! IL (input) INTEGER ! IU (input) INTEGER ! If RANGE='I', the indices (in ascending order) of the ! smallest and largest eigenvalues to be returned. ! 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. ! Not referenced if RANGE = 'A' or 'V'. ! ! ABSTOL (input) DOUBLE PRECISION ! The absolute tolerance for the eigenvalues. An eigenvalue ! (or cluster) is considered to be located if it has been ! determined to lie in an interval whose width is ABSTOL or ! less. If ABSTOL is less than or equal to zero, then ULP*|T| ! will be used, where |T| means the 1-norm of T. ! ! Eigenvalues will be computed most accurately when ABSTOL is ! set to twice the underflow threshold 2*DLAMCH('S'), not zero. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of the tridiagonal matrix T. ! ! E (input) DOUBLE PRECISION array, dimension (N-1) ! The (n-1) off-diagonal elements of the tridiagonal matrix T. ! ! M (output) INTEGER ! The actual number of eigenvalues found. 0 <= M <= N. ! (See also the description of INFO=2,3.) ! ! NSPLIT (output) INTEGER ! The number of diagonal blocks in the matrix T. ! 1 <= NSPLIT <= N. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! On exit, the first M elements of W will contain the ! eigenvalues. (DSTEBZ may use the remaining N-M elements as ! workspace.) ! ! IBLOCK (output) INTEGER array, dimension (N) ! At each row/column j where E(j) is zero or small, the ! matrix T is considered to split into a block diagonal ! matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which ! block (from 1 to the number of blocks) the eigenvalue W(i) ! belongs. (DSTEBZ may use the remaining N-M elements as ! workspace.) ! ! ISPLIT (output) INTEGER array, dimension (N) ! The splitting points, at which T breaks up into submatrices. ! The first submatrix consists of rows/columns 1 to ISPLIT(1), ! the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), ! etc., and the NSPLIT-th consists of rows/columns ! ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. ! (Only the first NSPLIT elements will actually be used, but ! since the user cannot know a priori what value NSPLIT will ! have, N words must be reserved for ISPLIT.) ! ! WORK (workspace) DOUBLE PRECISION array, dimension (4*N) ! ! IWORK (workspace) INTEGER array, dimension (3*N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: some or all of the eigenvalues failed to converge or ! were not computed: ! =1 or 3: Bisection failed to converge for some ! eigenvalues; these eigenvalues are flagged by a ! negative block number. The effect is that the ! eigenvalues may not be as accurate as the ! absolute and relative tolerances. This is ! generally caused by unexpectedly inaccurate ! arithmetic. ! =2 or 3: RANGE='I' only: Not all of the eigenvalues ! IL:IU were found. ! Effect: M < IU+1-IL ! Cause: non-monotonic arithmetic, causing the ! Sturm sequence to be non-monotonic. ! Cure: recalculate, using RANGE='A', and pick ! out eigenvalues IL:IU. In some cases, ! increasing the PARAMETER "FUDGE" may ! make things work. ! = 4: RANGE='I', and the Gershgorin interval ! initially used was too small. No eigenvalues ! were computed. ! Probable cause: your machine has sloppy ! floating-point arithmetic. ! Cure: Increase the PARAMETER "FUDGE", ! recompile, and try again. ! ! Internal Parameters ! =================== ! ! RELFAC DOUBLE PRECISION, default = 2.0e0 ! The relative tolerance. An interval (a,b] lies within ! "relative tolerance" if b-a < RELFAC*ulp*max(|a|,|b|), ! where "ulp" is the machine precision (distance from 1 to ! the next larger floating point number.) ! ! FUDGE DOUBLE PRECISION, default = 2 ! A "fudge factor" to widen the Gershgorin intervals. Ideally, ! a value of 1 should work, but on machines with sloppy ! arithmetic, this needs to be larger. The default for ! publicly released versions should be large enough to handle ! the worst machine around. Note that this has no effect ! on accuracy of the solution. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO, HALF PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0, & HALF = 1.0D0 / TWO ) DOUBLE PRECISION FUDGE, RELFAC PARAMETER ( FUDGE = 2.0D0, RELFAC = 2.0D0 ) ! .. ! .. Local Scalars .. LOGICAL NCNVRG, TOOFEW INTEGER IB, IBEGIN, IDISCL, IDISCU, IE, IEND, IINFO, & IM, IN, IOFF, IORDER, IOUT, IRANGE, ITMAX, & ITMP1, IW, IWOFF, J, JB, JDISC, JE, NB, NWL, & NWU DOUBLE PRECISION ATOLI, BNORM, GL, GU, PIVMIN, RTOLI, SAFEMN, & TMP1, TMP2, TNORM, ULP, WKILL, WL, WLU, WU, WUL ! .. ! .. Local Arrays .. INTEGER IDUMMA( 1 ) ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH EXTERNAL LSAME, ILAENV, DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DLAEBZ, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, INT, LOG, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! INFO = 0 ! ! Decode RANGE ! IF( LSAME( RANGE, 'A' ) ) THEN IRANGE = 1 ELSE IF( LSAME( RANGE, 'V' ) ) THEN IRANGE = 2 ELSE IF( LSAME( RANGE, 'I' ) ) THEN IRANGE = 3 ELSE IRANGE = 0 END IF ! ! Decode ORDER ! IF( LSAME( ORDER, 'B' ) ) THEN IORDER = 2 ELSE IF( LSAME( ORDER, 'E' ) ) THEN IORDER = 1 ELSE IORDER = 0 END IF ! ! Check for Errors ! IF( IRANGE.LE.0 ) THEN INFO = -1 ELSE IF( IORDER.LE.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( IRANGE.EQ.2 ) THEN IF( VL.GE.VU ) & INFO = -5 ELSE IF( IRANGE.EQ.3 .AND. ( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) ) & THEN INFO = -6 ELSE IF( IRANGE.EQ.3 .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) & THEN INFO = -7 END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSTEBZ', -INFO ) RETURN END IF ! ! Initialize error flags ! INFO = 0 NCNVRG = .FALSE. TOOFEW = .FALSE. ! ! Quick return if possible ! M = 0 IF( N.EQ.0 ) & RETURN ! ! Simplifications: ! IF( IRANGE.EQ.3 .AND. IL.EQ.1 .AND. IU.EQ.N ) & IRANGE = 1 ! ! Get machine constants ! NB is the minimum vector length for vector bisection, or 0 ! if only scalar is to be done. ! SAFEMN = DLAMCH( 'S' ) ULP = DLAMCH( 'P' ) RTOLI = ULP*RELFAC NB = ILAENV( 1, 'DSTEBZ', ' ', N, -1, -1, -1 ) IF( NB.LE.1 ) & NB = 0 ! ! Special Case when N=1 ! IF( N.EQ.1 ) THEN NSPLIT = 1 ISPLIT( 1 ) = 1 IF( IRANGE.EQ.2 .AND. ( VL.GE.D( 1 ) .OR. VU.LT.D( 1 ) ) ) THEN M = 0 ELSE W( 1 ) = D( 1 ) IBLOCK( 1 ) = 1 M = 1 END IF RETURN END IF ! ! Compute Splitting Points ! NSPLIT = 1 WORK( N ) = ZERO PIVMIN = ONE ! !DIR$ NOVECTOR DO 10 J = 2, N TMP1 = E( J-1 )**2 IF( ABS( D( J )*D( J-1 ) )*ULP**2+SAFEMN.GT.TMP1 ) THEN ISPLIT( NSPLIT ) = J - 1 NSPLIT = NSPLIT + 1 WORK( J-1 ) = ZERO ELSE WORK( J-1 ) = TMP1 PIVMIN = MAX( PIVMIN, TMP1 ) END IF 10 CONTINUE ISPLIT( NSPLIT ) = N PIVMIN = PIVMIN*SAFEMN ! ! Compute Interval and ATOLI ! IF( IRANGE.EQ.3 ) THEN ! ! RANGE='I': Compute the interval containing eigenvalues ! IL through IU. ! ! Compute Gershgorin interval for entire (split) matrix ! and use it as the initial interval ! GU = D( 1 ) GL = D( 1 ) TMP1 = ZERO ! DO 20 J = 1, N - 1 TMP2 = SQRT( WORK( J ) ) GU = MAX( GU, D( J )+TMP1+TMP2 ) GL = MIN( GL, D( J )-TMP1-TMP2 ) TMP1 = TMP2 20 CONTINUE ! GU = MAX( GU, D( N )+TMP1 ) GL = MIN( GL, D( N )-TMP1 ) TNORM = MAX( ABS( GL ), ABS( GU ) ) GL = GL - FUDGE*TNORM*ULP*N - FUDGE*TWO*PIVMIN GU = GU + FUDGE*TNORM*ULP*N + FUDGE*PIVMIN ! ! Compute Iteration parameters ! ITMAX = INT( ( LOG( TNORM+PIVMIN )-LOG( PIVMIN ) ) / & LOG( TWO ) ) + 2 IF( ABSTOL.LE.ZERO ) THEN ATOLI = ULP*TNORM ELSE ATOLI = ABSTOL END IF ! WORK( N+1 ) = GL WORK( N+2 ) = GL WORK( N+3 ) = GU WORK( N+4 ) = GU WORK( N+5 ) = GL WORK( N+6 ) = GU IWORK( 1 ) = -1 IWORK( 2 ) = -1 IWORK( 3 ) = N + 1 IWORK( 4 ) = N + 1 IWORK( 5 ) = IL - 1 IWORK( 6 ) = IU ! CALL DLAEBZ( 3, ITMAX, N, 2, 2, NB, ATOLI, RTOLI, PIVMIN, D, E, & WORK, IWORK( 5 ), WORK( N+1 ), WORK( N+5 ), IOUT, & IWORK, W, IBLOCK, IINFO ) ! IF( IWORK( 6 ).EQ.IU ) THEN WL = WORK( N+1 ) WLU = WORK( N+3 ) NWL = IWORK( 1 ) WU = WORK( N+4 ) WUL = WORK( N+2 ) NWU = IWORK( 4 ) ELSE WL = WORK( N+2 ) WLU = WORK( N+4 ) NWL = IWORK( 2 ) WU = WORK( N+3 ) WUL = WORK( N+1 ) NWU = IWORK( 3 ) END IF ! IF( NWL.LT.0 .OR. NWL.GE.N .OR. NWU.LT.1 .OR. NWU.GT.N ) THEN INFO = 4 RETURN END IF ELSE ! ! RANGE='A' or 'V' -- Set ATOLI ! TNORM = MAX( ABS( D( 1 ) )+ABS( E( 1 ) ), & ABS( D( N ) )+ABS( E( N-1 ) ) ) ! DO 30 J = 2, N - 1 TNORM = MAX( TNORM, ABS( D( J ) )+ABS( E( J-1 ) )+ & ABS( E( J ) ) ) 30 CONTINUE ! IF( ABSTOL.LE.ZERO ) THEN ATOLI = ULP*TNORM ELSE ATOLI = ABSTOL END IF ! IF( IRANGE.EQ.2 ) THEN WL = VL WU = VU ELSE WL = ZERO WU = ZERO END IF END IF ! ! Find Eigenvalues -- Loop Over Blocks and recompute NWL and NWU. ! NWL accumulates the number of eigenvalues .le. WL, ! NWU accumulates the number of eigenvalues .le. WU ! M = 0 IEND = 0 INFO = 0 NWL = 0 NWU = 0 ! DO 70 JB = 1, NSPLIT IOFF = IEND IBEGIN = IOFF + 1 IEND = ISPLIT( JB ) IN = IEND - IOFF ! IF( IN.EQ.1 ) THEN ! ! Special Case -- IN=1 ! IF( IRANGE.EQ.1 .OR. WL.GE.D( IBEGIN )-PIVMIN ) & NWL = NWL + 1 IF( IRANGE.EQ.1 .OR. WU.GE.D( IBEGIN )-PIVMIN ) & NWU = NWU + 1 IF( IRANGE.EQ.1 .OR. ( WL.LT.D( IBEGIN )-PIVMIN .AND. WU.GE. & D( IBEGIN )-PIVMIN ) ) THEN M = M + 1 W( M ) = D( IBEGIN ) IBLOCK( M ) = JB END IF ELSE ! ! General Case -- IN > 1 ! ! Compute Gershgorin Interval ! and use it as the initial interval ! GU = D( IBEGIN ) GL = D( IBEGIN ) TMP1 = ZERO ! DO 40 J = IBEGIN, IEND - 1 TMP2 = ABS( E( J ) ) GU = MAX( GU, D( J )+TMP1+TMP2 ) GL = MIN( GL, D( J )-TMP1-TMP2 ) TMP1 = TMP2 40 CONTINUE ! GU = MAX( GU, D( IEND )+TMP1 ) GL = MIN( GL, D( IEND )-TMP1 ) BNORM = MAX( ABS( GL ), ABS( GU ) ) GL = GL - FUDGE*BNORM*ULP*IN - FUDGE*PIVMIN GU = GU + FUDGE*BNORM*ULP*IN + FUDGE*PIVMIN ! ! Compute ATOLI for the current submatrix ! IF( ABSTOL.LE.ZERO ) THEN ATOLI = ULP*MAX( ABS( GL ), ABS( GU ) ) ELSE ATOLI = ABSTOL END IF ! IF( IRANGE.GT.1 ) THEN IF( GU.LT.WL ) THEN NWL = NWL + IN NWU = NWU + IN GO TO 70 END IF GL = MAX( GL, WL ) GU = MIN( GU, WU ) IF( GL.GE.GU ) & GO TO 70 END IF ! ! Set Up Initial Interval ! WORK( N+1 ) = GL WORK( N+IN+1 ) = GU CALL DLAEBZ( 1, 0, IN, IN, 1, NB, ATOLI, RTOLI, PIVMIN, & D( IBEGIN ), E( IBEGIN ), WORK( IBEGIN ), & IDUMMA, WORK( N+1 ), WORK( N+2*IN+1 ), IM, & IWORK, W( M+1 ), IBLOCK( M+1 ), IINFO ) ! NWL = NWL + IWORK( 1 ) NWU = NWU + IWORK( IN+1 ) IWOFF = M - IWORK( 1 ) ! ! Compute Eigenvalues ! ITMAX = INT( ( LOG( GU-GL+PIVMIN )-LOG( PIVMIN ) ) / & LOG( TWO ) ) + 2 CALL DLAEBZ( 2, ITMAX, IN, IN, 1, NB, ATOLI, RTOLI, PIVMIN, & D( IBEGIN ), E( IBEGIN ), WORK( IBEGIN ), & IDUMMA, WORK( N+1 ), WORK( N+2*IN+1 ), IOUT, & IWORK, W( M+1 ), IBLOCK( M+1 ), IINFO ) ! ! Copy Eigenvalues Into W and IBLOCK ! Use -JB for block number for unconverged eigenvalues. ! DO 60 J = 1, IOUT TMP1 = HALF*( WORK( J+N )+WORK( J+IN+N ) ) ! ! Flag non-convergence. ! IF( J.GT.IOUT-IINFO ) THEN NCNVRG = .TRUE. IB = -JB ELSE IB = JB END IF DO 50 JE = IWORK( J ) + 1 + IWOFF, & IWORK( J+IN ) + IWOFF W( JE ) = TMP1 IBLOCK( JE ) = IB 50 CONTINUE 60 CONTINUE ! M = M + IM END IF 70 CONTINUE ! ! If RANGE='I', then (WL,WU) contains eigenvalues NWL+1,...,NWU ! If NWL+1 < IL or NWU > IU, discard extra eigenvalues. ! IF( IRANGE.EQ.3 ) THEN IM = 0 IDISCL = IL - 1 - NWL IDISCU = NWU - IU ! IF( IDISCL.GT.0 .OR. IDISCU.GT.0 ) THEN DO 80 JE = 1, M IF( W( JE ).LE.WLU .AND. IDISCL.GT.0 ) THEN IDISCL = IDISCL - 1 ELSE IF( W( JE ).GE.WUL .AND. IDISCU.GT.0 ) THEN IDISCU = IDISCU - 1 ELSE IM = IM + 1 W( IM ) = W( JE ) IBLOCK( IM ) = IBLOCK( JE ) END IF 80 CONTINUE M = IM END IF IF( IDISCL.GT.0 .OR. IDISCU.GT.0 ) THEN ! ! Code to deal with effects of bad arithmetic: ! Some low eigenvalues to be discarded are not in (WL,WLU], ! or high eigenvalues to be discarded are not in (WUL,WU] ! so just kill off the smallest IDISCL/largest IDISCU ! eigenvalues, by simply finding the smallest/largest ! eigenvalue(s). ! ! (If N(w) is monotone non-decreasing, this should never ! happen.) ! IF( IDISCL.GT.0 ) THEN WKILL = WU DO 100 JDISC = 1, IDISCL IW = 0 DO 90 JE = 1, M IF( IBLOCK( JE ).NE.0 .AND. & ( W( JE ).LT.WKILL .OR. IW.EQ.0 ) ) THEN IW = JE WKILL = W( JE ) END IF 90 CONTINUE IBLOCK( IW ) = 0 100 CONTINUE END IF IF( IDISCU.GT.0 ) THEN ! WKILL = WL DO 120 JDISC = 1, IDISCU IW = 0 DO 110 JE = 1, M IF( IBLOCK( JE ).NE.0 .AND. & ( W( JE ).GT.WKILL .OR. IW.EQ.0 ) ) THEN IW = JE WKILL = W( JE ) END IF 110 CONTINUE IBLOCK( IW ) = 0 120 CONTINUE END IF IM = 0 DO 130 JE = 1, M IF( IBLOCK( JE ).NE.0 ) THEN IM = IM + 1 W( IM ) = W( JE ) IBLOCK( IM ) = IBLOCK( JE ) END IF 130 CONTINUE M = IM END IF IF( IDISCL.LT.0 .OR. IDISCU.LT.0 ) THEN TOOFEW = .TRUE. END IF END IF ! ! If ORDER='B', do nothing -- the eigenvalues are already sorted ! by block. ! If ORDER='E', sort the eigenvalues from smallest to largest ! IF( IORDER.EQ.1 .AND. NSPLIT.GT.1 ) THEN DO 150 JE = 1, M - 1 IE = 0 TMP1 = W( JE ) DO 140 J = JE + 1, M IF( W( J ).LT.TMP1 ) THEN IE = J TMP1 = W( J ) END IF 140 CONTINUE ! IF( IE.NE.0 ) THEN ITMP1 = IBLOCK( IE ) W( IE ) = W( JE ) IBLOCK( IE ) = IBLOCK( JE ) W( JE ) = TMP1 IBLOCK( JE ) = ITMP1 END IF 150 CONTINUE END IF ! INFO = 0 IF( NCNVRG ) & INFO = INFO + 1 IF( TOOFEW ) & INFO = INFO + 2 RETURN ! ! End of DSTEBZ ! END SUBROUTINE DSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK, & LIWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER COMPZ INTEGER INFO, LDZ, LIWORK, LWORK, N ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSTEDC computes all eigenvalues and, optionally, eigenvectors of a ! symmetric tridiagonal matrix using the divide and conquer method. ! The eigenvectors of a full or band real symmetric matrix can also be ! found if DSYTRD or DSPTRD or DSBTRD has been used to reduce this ! matrix to tridiagonal form. ! ! This code makes very mild assumptions about floating point ! arithmetic. It will work on machines with a guard digit in ! add/subtract, or on those binary machines without guard digits ! which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. ! It could conceivably fail on hexadecimal or decimal machines ! without guard digits, but we know of none. See DLAED3 for details. ! ! Arguments ! ========= ! ! COMPZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only. ! = 'I': Compute eigenvectors of tridiagonal matrix also. ! = 'V': Compute eigenvectors of original dense symmetric ! matrix also. On entry, Z contains the orthogonal ! matrix used to reduce the original matrix to ! tridiagonal form. ! ! N (input) INTEGER ! The dimension of the symmetric tridiagonal matrix. N >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the diagonal elements of the tridiagonal matrix. ! On exit, if INFO = 0, the eigenvalues in ascending order. ! ! E (input/output) DOUBLE PRECISION array, dimension (N-1) ! On entry, the subdiagonal elements of the tridiagonal matrix. ! On exit, E has been destroyed. ! ! Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N) ! On entry, if COMPZ = 'V', then Z contains the orthogonal ! matrix used in the reduction to tridiagonal form. ! On exit, if INFO = 0, then if COMPZ = 'V', Z contains the ! orthonormal eigenvectors of the original symmetric matrix, ! and if COMPZ = 'I', Z contains the orthonormal eigenvectors ! of the symmetric tridiagonal matrix. ! If COMPZ = 'N', then Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1. ! If eigenvectors are desired, then LDZ >= max(1,N). ! ! WORK (workspace/output) DOUBLE PRECISION array, ! dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! If COMPZ = 'N' or N <= 1 then LWORK must be at least 1. ! If COMPZ = 'V' and N > 1 then LWORK must be at least ! ( 1 + 3*N + 2*N*lg N + 3*N**2 ), ! where lg( N ) = smallest integer k such ! that 2**k >= N. ! If COMPZ = 'I' and N > 1 then LWORK must be at least ! ( 1 + 4*N + N**2 ). ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace/output) INTEGER array, dimension (LIWORK) ! On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. ! ! LIWORK (input) INTEGER ! The dimension of the array IWORK. ! If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1. ! If COMPZ = 'V' and N > 1 then LIWORK must be at least ! ( 6 + 6*N + 5*N*lg N ). ! If COMPZ = 'I' and N > 1 then LIWORK must be at least ! ( 3 + 5*N ). ! ! If LIWORK = -1, then a workspace query is assumed; the ! routine only calculates the optimal size of the IWORK array, ! returns this value as the first entry of the IWORK array, and ! no error message related to LIWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: The algorithm failed to compute an eigenvalue while ! working on the submatrix lying in rows and columns ! INFO/(N+1) through mod(INFO,N+1). ! ! Further Details ! =============== ! ! Based on contributions by ! Jeff Rutter, Computer Science Division, University of California ! at Berkeley, USA ! Modified by Francoise Tisseur, University of Tennessee. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER DTRTRW, END, I, ICOMPZ, II, J, K, LGN, LIWMIN, & LWMIN, M, SMLSIZ, START, STOREZ DOUBLE PRECISION EPS, ORGNRM, P, TINY ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANST EXTERNAL LSAME, ILAENV, DLAMCH, DLANST ! .. ! .. External Subroutines .. EXTERNAL DGEMM, DLACPY, DLAED0, DLASCL, DLASET, DLASRT, & DSTEQR, DSTERF, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, INT, LOG, MAX, MOD, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) ! IF( LSAME( COMPZ, 'N' ) ) THEN ICOMPZ = 0 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN ICOMPZ = 1 ELSE IF( LSAME( COMPZ, 'I' ) ) THEN ICOMPZ = 2 ELSE ICOMPZ = -1 END IF IF( N.LE.1 .OR. ICOMPZ.LE.0 ) THEN LIWMIN = 1 LWMIN = 1 ELSE LGN = INT( LOG( DBLE( N ) ) / LOG( TWO ) ) IF( 2**LGN.LT.N ) & LGN = LGN + 1 IF( 2**LGN.LT.N ) & LGN = LGN + 1 IF( ICOMPZ.EQ.1 ) THEN LWMIN = 1 + 3*N + 2*N*LGN + 3*N**2 LIWMIN = 6 + 6*N + 5*N*LGN ELSE IF( ICOMPZ.EQ.2 ) THEN LWMIN = 1 + 4*N + N**2 LIWMIN = 3 + 5*N END IF END IF IF( ICOMPZ.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( ( LDZ.LT.1 ) .OR. ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1, & N ) ) ) THEN INFO = -6 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -8 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -10 END IF ! IF( INFO.EQ.0 ) THEN WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSTEDC', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN IF( N.EQ.1 ) THEN IF( ICOMPZ.NE.0 ) & Z( 1, 1 ) = ONE RETURN END IF ! SMLSIZ = ILAENV( 9, 'DSTEDC', ' ', 0, 0, 0, 0 ) ! ! If the following conditional clause is removed, then the routine ! will use the Divide and Conquer routine to compute only the ! eigenvalues, which requires (3N + 3N**2) real workspace and ! (2 + 5N + 2N lg(N)) integer workspace. ! Since on many architectures DSTERF is much faster than any other ! algorithm for finding eigenvalues only, it is used here ! as the default. ! ! If COMPZ = 'N', use DSTERF to compute the eigenvalues. ! IF( ICOMPZ.EQ.0 ) THEN CALL DSTERF( N, D, E, INFO ) RETURN END IF ! ! If N is smaller than the minimum divide size (SMLSIZ+1), then ! solve the problem with another solver. ! IF( N.LE.SMLSIZ ) THEN IF( ICOMPZ.EQ.0 ) THEN CALL DSTERF( N, D, E, INFO ) RETURN ELSE IF( ICOMPZ.EQ.2 ) THEN CALL DSTEQR( 'I', N, D, E, Z, LDZ, WORK, INFO ) RETURN ELSE CALL DSTEQR( 'V', N, D, E, Z, LDZ, WORK, INFO ) RETURN END IF END IF ! ! If COMPZ = 'V', the Z matrix must be stored elsewhere for later ! use. ! IF( ICOMPZ.EQ.1 ) THEN STOREZ = 1 + N*N ELSE STOREZ = 1 END IF ! IF( ICOMPZ.EQ.2 ) THEN CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ ) END IF ! ! Scale. ! ORGNRM = DLANST( 'M', N, D, E ) IF( ORGNRM.EQ.ZERO ) & RETURN ! EPS = DLAMCH( 'Epsilon' ) ! START = 1 ! ! while ( START <= N ) ! 10 CONTINUE IF( START.LE.N ) THEN ! ! Let END be the position of the next subdiagonal entry such that ! E( END ) <= TINY or END = N if no such subdiagonal exists. The ! matrix identified by the elements between START and END ! constitutes an independent sub-problem. ! END = START 20 CONTINUE IF( END.LT.N ) THEN TINY = EPS*SQRT( ABS( D( END ) ) )*SQRT( ABS( D( END+1 ) ) ) IF( ABS( E( END ) ).GT.TINY ) THEN END = END + 1 GO TO 20 END IF END IF ! ! (Sub) Problem determined. Compute its size and solve it. ! M = END - START + 1 IF( M.EQ.1 ) THEN START = END + 1 GO TO 10 END IF IF( M.GT.SMLSIZ ) THEN INFO = SMLSIZ ! ! Scale. ! ORGNRM = DLANST( 'M', M, D( START ), E( START ) ) CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, M, 1, D( START ), M, & INFO ) CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, M-1, 1, E( START ), & M-1, INFO ) ! IF( ICOMPZ.EQ.1 ) THEN DTRTRW = 1 ELSE DTRTRW = START END IF CALL DLAED0( ICOMPZ, N, M, D( START ), E( START ), & Z( DTRTRW, START ), LDZ, WORK( 1 ), N, & WORK( STOREZ ), IWORK, INFO ) IF( INFO.NE.0 ) THEN INFO = ( INFO / ( M+1 )+START-1 )*( N+1 ) + & MOD( INFO, ( M+1 ) ) + START - 1 RETURN END IF ! ! Scale back. ! CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, M, 1, D( START ), M, & INFO ) ! ELSE IF( ICOMPZ.EQ.1 ) THEN ! ! Since QR won't update a Z matrix which is larger than the ! length of D, we must solve the sub-problem in a workspace and ! then multiply back into Z. ! CALL DSTEQR( 'I', M, D( START ), E( START ), WORK, M, & WORK( M*M+1 ), INFO ) CALL DLACPY( 'A', N, M, Z( 1, START ), LDZ, & WORK( STOREZ ), N ) CALL DGEMM( 'N', 'N', N, M, M, ONE, WORK( STOREZ ), LDZ, & WORK, M, ZERO, Z( 1, START ), LDZ ) ELSE IF( ICOMPZ.EQ.2 ) THEN CALL DSTEQR( 'I', M, D( START ), E( START ), & Z( START, START ), LDZ, WORK, INFO ) ELSE CALL DSTERF( M, D( START ), E( START ), INFO ) END IF IF( INFO.NE.0 ) THEN INFO = START*( N+1 ) + END RETURN END IF END IF ! START = END + 1 GO TO 10 END IF ! ! endwhile ! ! If the problem split any number of times, then the eigenvalues ! will not be properly ordered. Here we permute the eigenvalues ! (and the associated eigenvectors) into ascending order. ! IF( M.NE.N ) THEN IF( ICOMPZ.EQ.0 ) THEN ! ! Use Quick Sort ! CALL DLASRT( 'I', N, D, INFO ) ! ELSE ! ! Use Selection Sort to minimize swaps of eigenvectors ! DO 40 II = 2, N I = II - 1 K = I P = D( I ) DO 30 J = II, N IF( D( J ).LT.P ) THEN K = J P = D( J ) END IF 30 CONTINUE IF( K.NE.I ) THEN D( K ) = D( I ) D( I ) = P CALL DSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 ) END IF 40 CONTINUE END IF END IF ! WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN ! RETURN ! ! End of DSTEDC ! END SUBROUTINE DSTEGR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, & M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, & LIWORK, INFO ) ! ! -- LAPACK computational routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, RANGE INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N DOUBLE PRECISION ABSTOL, VL, VU ! .. ! .. Array Arguments .. INTEGER ISUPPZ( * ), IWORK( * ) DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSTEGR computes selected eigenvalues and, optionally, eigenvectors ! of a real symmetric tridiagonal matrix T. Eigenvalues and ! eigenvectors can be selected by specifying either a range of values ! or a range of indices for the desired eigenvalues. The eigenvalues ! are computed by the dqds algorithm, while orthogonal eigenvectors are ! computed from various ``good'' L D L^T representations (also known as ! Relatively Robust Representations). Gram-Schmidt orthogonalization is ! avoided as far as possible. More specifically, the various steps of ! the algorithm are as follows. For the i-th unreduced block of T, ! (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T ! is a relatively robust representation, ! (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high ! relative accuracy by the dqds algorithm, ! (c) If there is a cluster of close eigenvalues, "choose" sigma_i ! close to the cluster, and go to step (a), ! (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, ! compute the corresponding eigenvector by forming a ! rank-revealing twisted factorization. ! The desired accuracy of the output can be specified by the input ! parameter ABSTOL. ! ! For more details, see "A new O(n^2) algorithm for the symmetric ! tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, ! Computer Science Division Technical Report No. UCB/CSD-97-971, ! UC Berkeley, May 1997. ! ! Note 1 : Currently DSTEGR is only set up to find ALL the n ! eigenvalues and eigenvectors of T in O(n^2) time ! Note 2 : Currently the routine DSTEIN is called when an appropriate ! sigma_i cannot be chosen in step (c) above. DSTEIN invokes modified ! Gram-Schmidt when eigenvalues are close. ! Note 3 : DSTEGR works only on machines which follow ieee-754 ! floating-point standard in their handling of infinities and NaNs. ! Normal execution of DSTEGR may create NaNs and infinities and hence ! may abort due to a floating point exception in environments which ! do not conform to the ieee standard. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! RANGE (input) CHARACTER*1 ! = 'A': all eigenvalues will be found. ! = 'V': all eigenvalues in the half-open interval (VL,VU] ! will be found. ! = 'I': the IL-th through IU-th eigenvalues will be found. !********* Only RANGE = 'A' is currently supported ********************* ! ! N (input) INTEGER ! The order of the matrix. N >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the n diagonal elements of the tridiagonal matrix ! T. On exit, D is overwritten. ! ! E (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the (n-1) subdiagonal elements of the tridiagonal ! matrix T in elements 1 to N-1 of E; E(N) need not be set. ! On exit, E is overwritten. ! ! VL (input) DOUBLE PRECISION ! VU (input) DOUBLE PRECISION ! If RANGE='V', the lower and upper bounds of the interval to ! be searched for eigenvalues. VL < VU. ! Not referenced if RANGE = 'A' or 'I'. ! ! IL (input) INTEGER ! IU (input) INTEGER ! If RANGE='I', the indices (in ascending order) of the ! smallest and largest eigenvalues to be returned. ! 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. ! Not referenced if RANGE = 'A' or 'V'. ! ! ABSTOL (input) DOUBLE PRECISION ! The absolute error tolerance for the ! eigenvalues/eigenvectors. IF JOBZ = 'V', the eigenvalues and ! eigenvectors output have residual norms bounded by ABSTOL, ! and the dot products between different eigenvectors are ! bounded by ABSTOL. If ABSTOL is less than N*EPS*|T|, then ! N*EPS*|T| will be used in its place, where EPS is the ! machine precision and |T| is the 1-norm of the tridiagonal ! matrix. The eigenvalues are computed to an accuracy of ! EPS*|T| irrespective of ABSTOL. If high relative accuracy ! is important, set ABSTOL to DLAMCH( 'Safe minimum' ). ! See Barlow and Demmel "Computing Accurate Eigensystems of ! Scaled Diagonally Dominant Matrices", LAPACK Working Note #7 ! for a discussion of which matrices define their eigenvalues ! to high relative accuracy. ! ! M (output) INTEGER ! The total number of eigenvalues found. 0 <= M <= N. ! If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! The first M elements contain the selected eigenvalues in ! ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) ) ! If JOBZ = 'V', then if INFO = 0, the first M columns of Z ! contain the orthonormal eigenvectors of the matrix T ! corresponding to the selected eigenvalues, with the i-th ! column of Z holding the eigenvector associated with W(i). ! If JOBZ = 'N', then Z is not referenced. ! Note: the user must ensure that at least max(1,M) columns are ! supplied in the array Z; if RANGE = 'V', the exact value of M ! is not known in advance and an upper bound must be used. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) ) ! The support of the eigenvectors in Z, i.e., the indices ! indicating the nonzero elements in Z. The i-th eigenvector ! is nonzero only in elements ISUPPZ( 2*i-1 ) through ! ISUPPZ( 2*i ). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal ! (and minimal) LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,18*N) ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace/output) INTEGER array, dimension (LIWORK) ! On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. ! ! LIWORK (input) INTEGER ! The dimension of the array IWORK. LIWORK >= max(1,10*N) ! ! If LIWORK = -1, then a workspace query is assumed; the ! routine only calculates the optimal size of the IWORK array, ! returns this value as the first entry of the IWORK array, and ! no error message related to LIWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = 1, internal error in DLARRE, ! if INFO = 2, internal error in DLARRV. ! ! Further Details ! =============== ! ! Based on contributions by ! Inderjit Dhillon, IBM Almaden, USA ! Osni Marques, LBNL/NERSC, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ INTEGER I, IBEGIN, IEND, IINDBL, IINDWK, IINFO, IINSPL, & INDGRS, INDWOF, INDWRK, ITMP, J, JJ, LIWMIN, & LWMIN, NSPLIT DOUBLE PRECISION BIGNUM, EPS, RMAX, RMIN, SAFMIN, SCALE, SMLNUM, & THRESH, TMP, TNRM, TOL ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANST EXTERNAL LSAME, DLAMCH, DLANST ! .. ! .. External Subroutines .. EXTERNAL DLARRE, DLARRV, DLASET, DSCAL, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC DBLE, MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) ALLEIG = LSAME( RANGE, 'A' ) VALEIG = LSAME( RANGE, 'V' ) INDEIG = LSAME( RANGE, 'I' ) ! LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) ) LWMIN = 18*N LIWMIN = 10*N ! INFO = 0 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN INFO = -2 ! ! The following two lines need to be removed once the ! RANGE = 'V' and RANGE = 'I' options are provided. ! ELSE IF( VALEIG .OR. INDEIG ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( VALEIG .AND. N.GT.0 .AND. VU.LE.VL ) THEN INFO = -7 ELSE IF( INDEIG .AND. IL.LT.1 ) THEN INFO = -8 ! The following change should be made in DSTEVX also, otherwise ! IL can be specified as N+1 and IU as N. ! ELSE IF( INDEIG .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN ELSE IF( INDEIG .AND. ( IU.LT.IL .OR. IU.GT.N ) ) THEN INFO = -9 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -14 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -17 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -19 END IF IF( INFO.EQ.0 ) THEN WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSTEGR', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! M = 0 IF( N.EQ.0 ) & RETURN ! IF( N.EQ.1 ) THEN IF( ALLEIG .OR. INDEIG ) THEN M = 1 W( 1 ) = D( 1 ) ELSE IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN M = 1 W( 1 ) = D( 1 ) END IF END IF IF( WANTZ ) & Z( 1, 1 ) = ONE RETURN END IF ! ! Get machine constants. ! SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) ! ! Scale matrix to allowable range, if necessary. ! SCALE = ONE TNRM = DLANST( 'M', N, D, E ) IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN SCALE = RMIN / TNRM ELSE IF( TNRM.GT.RMAX ) THEN SCALE = RMAX / TNRM END IF IF( SCALE.NE.ONE ) THEN CALL DSCAL( N, SCALE, D, 1 ) CALL DSCAL( N-1, SCALE, E, 1 ) TNRM = TNRM*SCALE END IF INDGRS = 1 INDWOF = 2*N + 1 INDWRK = 3*N + 1 ! IINSPL = 1 IINDBL = N + 1 IINDWK = 2*N + 1 ! CALL DLASET( 'Full', N, N, ZERO, ZERO, Z, LDZ ) ! ! Compute the desired eigenvalues of the tridiagonal after splitting ! into smaller subblocks if the corresponding of-diagonal elements ! are small ! THRESH = EPS*TNRM CALL DLARRE( N, D, E, THRESH, NSPLIT, IWORK( IINSPL ), M, W, & WORK( INDWOF ), WORK( INDGRS ), WORK( INDWRK ), & IINFO ) IF( IINFO.NE.0 ) THEN INFO = 1 RETURN END IF ! IF( WANTZ ) THEN ! ! Compute the desired eigenvectors corresponding to the computed ! eigenvalues ! TOL = MAX( ABSTOL, DBLE( N )*THRESH ) IBEGIN = 1 DO 20 I = 1, NSPLIT IEND = IWORK( IINSPL+I-1 ) DO 10 J = IBEGIN, IEND IWORK( IINDBL+J-1 ) = I 10 CONTINUE IBEGIN = IEND + 1 20 CONTINUE ! CALL DLARRV( N, D, E, IWORK( IINSPL ), M, W, IWORK( IINDBL ), & WORK( INDGRS ), TOL, Z, LDZ, ISUPPZ, & WORK( INDWRK ), IWORK( IINDWK ), IINFO ) IF( IINFO.NE.0 ) THEN INFO = 2 RETURN END IF ! END IF ! IBEGIN = 1 DO 40 I = 1, NSPLIT IEND = IWORK( IINSPL+I-1 ) DO 30 J = IBEGIN, IEND W( J ) = W( J ) + WORK( INDWOF+I-1 ) 30 CONTINUE IBEGIN = IEND + 1 40 CONTINUE ! ! If matrix was scaled, then rescale eigenvalues appropriately. ! IF( SCALE.NE.ONE ) THEN CALL DSCAL( M, ONE / SCALE, W, 1 ) END IF ! ! If eigenvalues are not in order, then sort them, along with ! eigenvectors. ! IF( NSPLIT.GT.1 ) THEN DO 60 J = 1, M - 1 I = 0 TMP = W( J ) DO 50 JJ = J + 1, M IF( W( JJ ).LT.TMP ) THEN I = JJ TMP = W( JJ ) END IF 50 CONTINUE IF( I.NE.0 ) THEN W( I ) = W( J ) W( J ) = TMP IF( WANTZ ) THEN CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) ITMP = ISUPPZ( 2*I-1 ) ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 ) ISUPPZ( 2*J-1 ) = ITMP ITMP = ISUPPZ( 2*I ) ISUPPZ( 2*I ) = ISUPPZ( 2*J ) ISUPPZ( 2*J ) = ITMP END IF END IF 60 CONTINUE END IF ! WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN RETURN ! ! End of DSTEGR ! END SUBROUTINE DSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, & IWORK, IFAIL, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. INTEGER INFO, LDZ, M, N ! .. ! .. Array Arguments .. INTEGER IBLOCK( * ), IFAIL( * ), ISPLIT( * ), & IWORK( * ) DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSTEIN computes the eigenvectors of a real symmetric tridiagonal ! matrix T corresponding to specified eigenvalues, using inverse ! iteration. ! ! The maximum number of iterations allowed for each eigenvector is ! specified by an internal parameter MAXITS (currently set to 5). ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix. N >= 0. ! ! D (input) DOUBLE PRECISION array, dimension (N) ! The n diagonal elements of the tridiagonal matrix T. ! ! E (input) DOUBLE PRECISION array, dimension (N) ! The (n-1) subdiagonal elements of the tridiagonal matrix ! T, in elements 1 to N-1. E(N) need not be set. ! ! M (input) INTEGER ! The number of eigenvectors to be found. 0 <= M <= N. ! ! W (input) DOUBLE PRECISION array, dimension (N) ! The first M elements of W contain the eigenvalues for ! which eigenvectors are to be computed. The eigenvalues ! should be grouped by split-off block and ordered from ! smallest to largest within the block. ( The output array ! W from DSTEBZ with ORDER = 'B' is expected here. ) ! ! IBLOCK (input) INTEGER array, dimension (N) ! The submatrix indices associated with the corresponding ! eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to ! the first submatrix from the top, =2 if W(i) belongs to ! the second submatrix, etc. ( The output array IBLOCK ! from DSTEBZ is expected here. ) ! ! ISPLIT (input) INTEGER array, dimension (N) ! The splitting points, at which T breaks up into submatrices. ! The first submatrix consists of rows/columns 1 to ! ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 ! through ISPLIT( 2 ), etc. ! ( The output array ISPLIT from DSTEBZ is expected here. ) ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, M) ! The computed eigenvectors. The eigenvector associated ! with the eigenvalue W(i) is stored in the i-th column of ! Z. Any vector which fails to converge is set to its current ! iterate after MAXITS iterations. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= max(1,N). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (5*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! IFAIL (output) INTEGER array, dimension (M) ! On normal exit, all elements of IFAIL are zero. ! If one or more eigenvectors fail to converge after ! MAXITS iterations, then their indices are stored in ! array IFAIL. ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, then i eigenvectors failed to converge ! in MAXITS iterations. Their indices are stored in ! array IFAIL. ! ! Internal Parameters ! =================== ! ! MAXITS INTEGER, default = 5 ! The maximum number of iterations performed. ! ! EXTRA INTEGER, default = 2 ! The number of iterations performed after norm growth ! criterion is satisfied, should be at least 1. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TEN, ODM3, ODM1 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TEN = 1.0D+1, & ODM3 = 1.0D-3, ODM1 = 1.0D-1 ) INTEGER MAXITS, EXTRA PARAMETER ( MAXITS = 5, EXTRA = 2 ) ! .. ! .. Local Scalars .. INTEGER B1, BLKSIZ, BN, GPIND, I, IINFO, INDRV1, & INDRV2, INDRV3, INDRV4, INDRV5, ITS, J, J1, & JBLK, JMAX, NBLK, NRMCHK DOUBLE PRECISION DTPCRT, EPS, EPS1, NRM, ONENRM, ORTOL, PERTOL, & SCL, SEP, TOL, XJ, XJM, ZTR ! .. ! .. Local Arrays .. INTEGER ISEED( 4 ) ! .. ! .. External Functions .. INTEGER IDAMAX DOUBLE PRECISION DASUM, DDOT, DLAMCH, DNRM2 EXTERNAL IDAMAX, DASUM, DDOT, DLAMCH, DNRM2 ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DLAGTF, DLAGTS, DLARNV, DSCAL, & XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 DO 10 I = 1, M IFAIL( I ) = 0 10 CONTINUE ! IF( N.LT.0 ) THEN INFO = -1 ELSE IF( M.LT.0 .OR. M.GT.N ) THEN INFO = -4 ELSE IF( LDZ.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE DO 20 J = 2, M IF( IBLOCK( J ).LT.IBLOCK( J-1 ) ) THEN INFO = -6 GO TO 30 END IF IF( IBLOCK( J ).EQ.IBLOCK( J-1 ) .AND. W( J ).LT.W( J-1 ) ) & THEN INFO = -5 GO TO 30 END IF 20 CONTINUE 30 CONTINUE END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSTEIN', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. M.EQ.0 ) THEN RETURN ELSE IF( N.EQ.1 ) THEN Z( 1, 1 ) = ONE RETURN END IF ! ! Get machine constants. ! EPS = DLAMCH( 'Precision' ) ! ! Initialize seed for random number generator DLARNV. ! DO 40 I = 1, 4 ISEED( I ) = 1 40 CONTINUE ! ! Initialize pointers. ! INDRV1 = 0 INDRV2 = INDRV1 + N INDRV3 = INDRV2 + N INDRV4 = INDRV3 + N INDRV5 = INDRV4 + N ! ! Compute eigenvectors of matrix blocks. ! J1 = 1 DO 160 NBLK = 1, IBLOCK( M ) ! ! Find starting and ending indices of block nblk. ! IF( NBLK.EQ.1 ) THEN B1 = 1 ELSE B1 = ISPLIT( NBLK-1 ) + 1 END IF BN = ISPLIT( NBLK ) BLKSIZ = BN - B1 + 1 IF( BLKSIZ.EQ.1 ) & GO TO 60 GPIND = B1 ! ! Compute reorthogonalization criterion and stopping criterion. ! ONENRM = ABS( D( B1 ) ) + ABS( E( B1 ) ) ONENRM = MAX( ONENRM, ABS( D( BN ) )+ABS( E( BN-1 ) ) ) DO 50 I = B1 + 1, BN - 1 ONENRM = MAX( ONENRM, ABS( D( I ) )+ABS( E( I-1 ) )+ & ABS( E( I ) ) ) 50 CONTINUE ORTOL = ODM3*ONENRM ! DTPCRT = SQRT( ODM1 / BLKSIZ ) ! ! Loop through eigenvalues of block nblk. ! 60 CONTINUE JBLK = 0 DO 150 J = J1, M IF( IBLOCK( J ).NE.NBLK ) THEN J1 = J GO TO 160 END IF JBLK = JBLK + 1 XJ = W( J ) ! ! Skip all the work if the block size is one. ! IF( BLKSIZ.EQ.1 ) THEN WORK( INDRV1+1 ) = ONE GO TO 120 END IF ! ! If eigenvalues j and j-1 are too close, add a relatively ! small perturbation. ! IF( JBLK.GT.1 ) THEN EPS1 = ABS( EPS*XJ ) PERTOL = TEN*EPS1 SEP = XJ - XJM IF( SEP.LT.PERTOL ) & XJ = XJM + PERTOL END IF ! ITS = 0 NRMCHK = 0 ! ! Get random starting vector. ! CALL DLARNV( 2, ISEED, BLKSIZ, WORK( INDRV1+1 ) ) ! ! Copy the matrix T so it won't be destroyed in factorization. ! CALL DCOPY( BLKSIZ, D( B1 ), 1, WORK( INDRV4+1 ), 1 ) CALL DCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV2+2 ), 1 ) CALL DCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV3+1 ), 1 ) ! ! Compute LU factors with partial pivoting ( PT = LU ) ! TOL = ZERO CALL DLAGTF( BLKSIZ, WORK( INDRV4+1 ), XJ, WORK( INDRV2+2 ), & WORK( INDRV3+1 ), TOL, WORK( INDRV5+1 ), IWORK, & IINFO ) ! ! Update iteration count. ! 70 CONTINUE ITS = ITS + 1 IF( ITS.GT.MAXITS ) & GO TO 100 ! ! Normalize and scale the righthand side vector Pb. ! SCL = BLKSIZ*ONENRM*MAX( EPS, & ABS( WORK( INDRV4+BLKSIZ ) ) ) / & DASUM( BLKSIZ, WORK( INDRV1+1 ), 1 ) CALL DSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 ) ! ! Solve the system LU = Pb. ! CALL DLAGTS( -1, BLKSIZ, WORK( INDRV4+1 ), WORK( INDRV2+2 ), & WORK( INDRV3+1 ), WORK( INDRV5+1 ), IWORK, & WORK( INDRV1+1 ), TOL, IINFO ) ! ! Reorthogonalize by modified Gram-Schmidt if eigenvalues are ! close enough. ! IF( JBLK.EQ.1 ) & GO TO 90 IF( ABS( XJ-XJM ).GT.ORTOL ) & GPIND = J IF( GPIND.NE.J ) THEN DO 80 I = GPIND, J - 1 ZTR = -DDOT( BLKSIZ, WORK( INDRV1+1 ), 1, Z( B1, I ), & 1 ) CALL DAXPY( BLKSIZ, ZTR, Z( B1, I ), 1, & WORK( INDRV1+1 ), 1 ) 80 CONTINUE END IF ! ! Check the infinity norm of the iterate. ! 90 CONTINUE JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 ) NRM = ABS( WORK( INDRV1+JMAX ) ) ! ! Continue for additional iterations after norm reaches ! stopping criterion. ! IF( NRM.LT.DTPCRT ) & GO TO 70 NRMCHK = NRMCHK + 1 IF( NRMCHK.LT.EXTRA+1 ) & GO TO 70 ! GO TO 110 ! ! If stopping criterion was not satisfied, update info and ! store eigenvector number in array ifail. ! 100 CONTINUE INFO = INFO + 1 IFAIL( INFO ) = J ! ! Accept iterate as jth eigenvector. ! 110 CONTINUE SCL = ONE / DNRM2( BLKSIZ, WORK( INDRV1+1 ), 1 ) JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 ) IF( WORK( INDRV1+JMAX ).LT.ZERO ) & SCL = -SCL CALL DSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 ) 120 CONTINUE DO 130 I = 1, N Z( I, J ) = ZERO 130 CONTINUE DO 140 I = 1, BLKSIZ Z( B1+I-1, J ) = WORK( INDRV1+I ) 140 CONTINUE ! ! Save the shift to check eigenvalue spacing at next ! iteration. ! XJM = XJ ! 150 CONTINUE 160 CONTINUE ! RETURN ! ! End of DSTEIN ! END SUBROUTINE DSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER COMPZ INTEGER INFO, LDZ, N ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSTEQR computes all eigenvalues and, optionally, eigenvectors of a ! symmetric tridiagonal matrix using the implicit QL or QR method. ! The eigenvectors of a full or band symmetric matrix can also be found ! if DSYTRD or DSPTRD or DSBTRD has been used to reduce this matrix to ! tridiagonal form. ! ! Arguments ! ========= ! ! COMPZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only. ! = 'V': Compute eigenvalues and eigenvectors of the original ! symmetric matrix. On entry, Z must contain the ! orthogonal matrix used to reduce the original matrix ! to tridiagonal form. ! = 'I': Compute eigenvalues and eigenvectors of the ! tridiagonal matrix. Z is initialized to the identity ! matrix. ! ! N (input) INTEGER ! The order of the matrix. N >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the diagonal elements of the tridiagonal matrix. ! On exit, if INFO = 0, the eigenvalues in ascending order. ! ! E (input/output) DOUBLE PRECISION array, dimension (N-1) ! On entry, the (n-1) subdiagonal elements of the tridiagonal ! matrix. ! On exit, E has been destroyed. ! ! Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N) ! On entry, if COMPZ = 'V', then Z contains the orthogonal ! matrix used in the reduction to tridiagonal form. ! On exit, if INFO = 0, then if COMPZ = 'V', Z contains the ! orthonormal eigenvectors of the original symmetric matrix, ! and if COMPZ = 'I', Z contains the orthonormal eigenvectors ! of the symmetric tridiagonal matrix. ! If COMPZ = 'N', then Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! eigenvectors are desired, then LDZ >= max(1,N). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (max(1,2*N-2)) ! If COMPZ = 'N', then WORK is not referenced. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: the algorithm has failed to find all the eigenvalues in ! a total of 30*N iterations; if INFO = i, then i ! elements of E have not converged to zero; on exit, D ! and E contain the elements of a symmetric tridiagonal ! matrix which is orthogonally similar to the original ! matrix. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO, THREE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0, & THREE = 3.0D0 ) INTEGER MAXIT PARAMETER ( MAXIT = 30 ) ! .. ! .. Local Scalars .. INTEGER I, ICOMPZ, II, ISCALE, J, JTOT, K, L, L1, LEND, & LENDM1, LENDP1, LENDSV, LM1, LSV, M, MM, MM1, & NM1, NMAXIT DOUBLE PRECISION ANORM, B, C, EPS, EPS2, F, G, P, R, RT1, RT2, & S, SAFMAX, SAFMIN, SSFMAX, SSFMIN, TST ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANST, DLAPY2 EXTERNAL LSAME, DLAMCH, DLANST, DLAPY2 ! .. ! .. External Subroutines .. EXTERNAL DLAE2, DLAEV2, DLARTG, DLASCL, DLASET, DLASR, & DLASRT, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SIGN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! IF( LSAME( COMPZ, 'N' ) ) THEN ICOMPZ = 0 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN ICOMPZ = 1 ELSE IF( LSAME( COMPZ, 'I' ) ) THEN ICOMPZ = 2 ELSE ICOMPZ = -1 END IF IF( ICOMPZ.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( ( LDZ.LT.1 ) .OR. ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1, & N ) ) ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSTEQR', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( N.EQ.1 ) THEN IF( ICOMPZ.EQ.2 ) & Z( 1, 1 ) = ONE RETURN END IF ! ! Determine the unit roundoff and over/underflow thresholds. ! EPS = DLAMCH( 'E' ) EPS2 = EPS**2 SAFMIN = DLAMCH( 'S' ) SAFMAX = ONE / SAFMIN SSFMAX = SQRT( SAFMAX ) / THREE SSFMIN = SQRT( SAFMIN ) / EPS2 ! ! Compute the eigenvalues and eigenvectors of the tridiagonal ! matrix. ! IF( ICOMPZ.EQ.2 ) & CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ ) ! NMAXIT = N*MAXIT JTOT = 0 ! ! Determine where the matrix splits and choose QL or QR iteration ! for each block, according to whether top or bottom diagonal ! element is smaller. ! L1 = 1 NM1 = N - 1 ! 10 CONTINUE IF( L1.GT.N ) & GO TO 160 IF( L1.GT.1 ) & E( L1-1 ) = ZERO IF( L1.LE.NM1 ) THEN DO 20 M = L1, NM1 TST = ABS( E( M ) ) IF( TST.EQ.ZERO ) & GO TO 30 IF( TST.LE.( SQRT( ABS( D( M ) ) )*SQRT( ABS( D( M+ & 1 ) ) ) )*EPS ) THEN E( M ) = ZERO GO TO 30 END IF 20 CONTINUE END IF M = N ! 30 CONTINUE L = L1 LSV = L LEND = M LENDSV = LEND L1 = M + 1 IF( LEND.EQ.L ) & GO TO 10 ! ! Scale submatrix in rows and columns L to LEND ! ANORM = DLANST( 'I', LEND-L+1, D( L ), E( L ) ) ISCALE = 0 IF( ANORM.EQ.ZERO ) & GO TO 10 IF( ANORM.GT.SSFMAX ) THEN ISCALE = 1 CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L+1, 1, D( L ), N, & INFO ) CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L, 1, E( L ), N, & INFO ) ELSE IF( ANORM.LT.SSFMIN ) THEN ISCALE = 2 CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L+1, 1, D( L ), N, & INFO ) CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L, 1, E( L ), N, & INFO ) END IF ! ! Choose between QL and QR iteration ! IF( ABS( D( LEND ) ).LT.ABS( D( L ) ) ) THEN LEND = LSV L = LENDSV END IF ! IF( LEND.GT.L ) THEN ! ! QL Iteration ! ! Look for small subdiagonal element. ! 40 CONTINUE IF( L.NE.LEND ) THEN LENDM1 = LEND - 1 DO 50 M = L, LENDM1 TST = ABS( E( M ) )**2 IF( TST.LE.( EPS2*ABS( D( M ) ) )*ABS( D( M+1 ) )+ & SAFMIN )GO TO 60 50 CONTINUE END IF ! M = LEND ! 60 CONTINUE IF( M.LT.LEND ) & E( M ) = ZERO P = D( L ) IF( M.EQ.L ) & GO TO 80 ! ! If remaining matrix is 2-by-2, use DLAE2 or SLAEV2 ! to compute its eigensystem. ! IF( M.EQ.L+1 ) THEN IF( ICOMPZ.GT.0 ) THEN CALL DLAEV2( D( L ), E( L ), D( L+1 ), RT1, RT2, C, S ) WORK( L ) = C WORK( N-1+L ) = S CALL DLASR( 'R', 'V', 'B', N, 2, WORK( L ), & WORK( N-1+L ), Z( 1, L ), LDZ ) ELSE CALL DLAE2( D( L ), E( L ), D( L+1 ), RT1, RT2 ) END IF D( L ) = RT1 D( L+1 ) = RT2 E( L ) = ZERO L = L + 2 IF( L.LE.LEND ) & GO TO 40 GO TO 140 END IF ! IF( JTOT.EQ.NMAXIT ) & GO TO 140 JTOT = JTOT + 1 ! ! Form shift. ! G = ( D( L+1 )-P ) / ( TWO*E( L ) ) R = DLAPY2( G, ONE ) G = D( M ) - P + ( E( L ) / ( G+SIGN( R, G ) ) ) ! S = ONE C = ONE P = ZERO ! ! Inner loop ! MM1 = M - 1 DO 70 I = MM1, L, -1 F = S*E( I ) B = C*E( I ) CALL DLARTG( G, F, C, S, R ) IF( I.NE.M-1 ) & E( I+1 ) = R G = D( I+1 ) - P R = ( D( I )-G )*S + TWO*C*B P = S*R D( I+1 ) = G + P G = C*R - B ! ! If eigenvectors are desired, then save rotations. ! IF( ICOMPZ.GT.0 ) THEN WORK( I ) = C WORK( N-1+I ) = -S END IF ! 70 CONTINUE ! ! If eigenvectors are desired, then apply saved rotations. ! IF( ICOMPZ.GT.0 ) THEN MM = M - L + 1 CALL DLASR( 'R', 'V', 'B', N, MM, WORK( L ), WORK( N-1+L ), & Z( 1, L ), LDZ ) END IF ! D( L ) = D( L ) - P E( L ) = G GO TO 40 ! ! Eigenvalue found. ! 80 CONTINUE D( L ) = P ! L = L + 1 IF( L.LE.LEND ) & GO TO 40 GO TO 140 ! ELSE ! ! QR Iteration ! ! Look for small superdiagonal element. ! 90 CONTINUE IF( L.NE.LEND ) THEN LENDP1 = LEND + 1 DO 100 M = L, LENDP1, -1 TST = ABS( E( M-1 ) )**2 IF( TST.LE.( EPS2*ABS( D( M ) ) )*ABS( D( M-1 ) )+ & SAFMIN )GO TO 110 100 CONTINUE END IF ! M = LEND ! 110 CONTINUE IF( M.GT.LEND ) & E( M-1 ) = ZERO P = D( L ) IF( M.EQ.L ) & GO TO 130 ! ! If remaining matrix is 2-by-2, use DLAE2 or SLAEV2 ! to compute its eigensystem. ! IF( M.EQ.L-1 ) THEN IF( ICOMPZ.GT.0 ) THEN CALL DLAEV2( D( L-1 ), E( L-1 ), D( L ), RT1, RT2, C, S ) WORK( M ) = C WORK( N-1+M ) = S CALL DLASR( 'R', 'V', 'F', N, 2, WORK( M ), & WORK( N-1+M ), Z( 1, L-1 ), LDZ ) ELSE CALL DLAE2( D( L-1 ), E( L-1 ), D( L ), RT1, RT2 ) END IF D( L-1 ) = RT1 D( L ) = RT2 E( L-1 ) = ZERO L = L - 2 IF( L.GE.LEND ) & GO TO 90 GO TO 140 END IF ! IF( JTOT.EQ.NMAXIT ) & GO TO 140 JTOT = JTOT + 1 ! ! Form shift. ! G = ( D( L-1 )-P ) / ( TWO*E( L-1 ) ) R = DLAPY2( G, ONE ) G = D( M ) - P + ( E( L-1 ) / ( G+SIGN( R, G ) ) ) ! S = ONE C = ONE P = ZERO ! ! Inner loop ! LM1 = L - 1 DO 120 I = M, LM1 F = S*E( I ) B = C*E( I ) CALL DLARTG( G, F, C, S, R ) IF( I.NE.M ) & E( I-1 ) = R G = D( I ) - P R = ( D( I+1 )-G )*S + TWO*C*B P = S*R D( I ) = G + P G = C*R - B ! ! If eigenvectors are desired, then save rotations. ! IF( ICOMPZ.GT.0 ) THEN WORK( I ) = C WORK( N-1+I ) = S END IF ! 120 CONTINUE ! ! If eigenvectors are desired, then apply saved rotations. ! IF( ICOMPZ.GT.0 ) THEN MM = L - M + 1 CALL DLASR( 'R', 'V', 'F', N, MM, WORK( M ), WORK( N-1+M ), & Z( 1, M ), LDZ ) END IF ! D( L ) = D( L ) - P E( LM1 ) = G GO TO 90 ! ! Eigenvalue found. ! 130 CONTINUE D( L ) = P ! L = L - 1 IF( L.GE.LEND ) & GO TO 90 GO TO 140 ! END IF ! ! Undo scaling if necessary ! 140 CONTINUE IF( ISCALE.EQ.1 ) THEN CALL DLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV+1, 1, & D( LSV ), N, INFO ) CALL DLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV, 1, E( LSV ), & N, INFO ) ELSE IF( ISCALE.EQ.2 ) THEN CALL DLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV+1, 1, & D( LSV ), N, INFO ) CALL DLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV, 1, E( LSV ), & N, INFO ) END IF ! ! Check for no convergence to an eigenvalue after a total ! of N*MAXIT iterations. ! IF( JTOT.LT.NMAXIT ) & GO TO 10 DO 150 I = 1, N - 1 IF( E( I ).NE.ZERO ) & INFO = INFO + 1 150 CONTINUE GO TO 190 ! ! Order eigenvalues and eigenvectors. ! 160 CONTINUE IF( ICOMPZ.EQ.0 ) THEN ! ! Use Quick Sort ! CALL DLASRT( 'I', N, D, INFO ) ! ELSE ! ! Use Selection Sort to minimize swaps of eigenvectors ! DO 180 II = 2, N I = II - 1 K = I P = D( I ) DO 170 J = II, N IF( D( J ).LT.P ) THEN K = J P = D( J ) END IF 170 CONTINUE IF( K.NE.I ) THEN D( K ) = D( I ) D( I ) = P CALL DSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 ) END IF 180 CONTINUE END IF ! 190 CONTINUE RETURN ! ! End of DSTEQR ! END SUBROUTINE DSTERF( N, D, E, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, N ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ), E( * ) ! .. ! ! Purpose ! ======= ! ! DSTERF computes all eigenvalues of a symmetric tridiagonal matrix ! using the Pal-Walker-Kahan variant of the QL or QR algorithm. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The order of the matrix. N >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the n diagonal elements of the tridiagonal matrix. ! On exit, if INFO = 0, the eigenvalues in ascending order. ! ! E (input/output) DOUBLE PRECISION array, dimension (N-1) ! On entry, the (n-1) subdiagonal elements of the tridiagonal ! matrix. ! On exit, E has been destroyed. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: the algorithm failed to find all of the eigenvalues in ! a total of 30*N iterations; if INFO = i, then i ! elements of E have not converged to zero. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO, THREE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0, & THREE = 3.0D0 ) INTEGER MAXIT PARAMETER ( MAXIT = 30 ) ! .. ! .. Local Scalars .. INTEGER I, ISCALE, JTOT, L, L1, LEND, LENDSV, LSV, M, & NMAXIT DOUBLE PRECISION ALPHA, ANORM, BB, C, EPS, EPS2, GAMMA, OLDC, & OLDGAM, P, R, RT1, RT2, RTE, S, SAFMAX, SAFMIN, & SIGMA, SSFMAX, SSFMIN ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH, DLANST, DLAPY2 EXTERNAL DLAMCH, DLANST, DLAPY2 ! .. ! .. External Subroutines .. EXTERNAL DLAE2, DLASCL, DLASRT, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, SIGN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 ! ! Quick return if possible ! IF( N.LT.0 ) THEN INFO = -1 CALL XERBLA( 'DSTERF', -INFO ) RETURN END IF IF( N.LE.1 ) & RETURN ! ! Determine the unit roundoff for this environment. ! EPS = DLAMCH( 'E' ) EPS2 = EPS**2 SAFMIN = DLAMCH( 'S' ) SAFMAX = ONE / SAFMIN SSFMAX = SQRT( SAFMAX ) / THREE SSFMIN = SQRT( SAFMIN ) / EPS2 ! ! Compute the eigenvalues of the tridiagonal matrix. ! NMAXIT = N*MAXIT SIGMA = ZERO JTOT = 0 ! ! Determine where the matrix splits and choose QL or QR iteration ! for each block, according to whether top or bottom diagonal ! element is smaller. ! L1 = 1 ! 10 CONTINUE IF( L1.GT.N ) & GO TO 170 IF( L1.GT.1 ) & E( L1-1 ) = ZERO DO 20 M = L1, N - 1 IF( ABS( E( M ) ).LE.( SQRT( ABS( D( M ) ) )*SQRT( ABS( D( M+ & 1 ) ) ) )*EPS ) THEN E( M ) = ZERO GO TO 30 END IF 20 CONTINUE M = N ! 30 CONTINUE L = L1 LSV = L LEND = M LENDSV = LEND L1 = M + 1 IF( LEND.EQ.L ) & GO TO 10 ! ! Scale submatrix in rows and columns L to LEND ! ANORM = DLANST( 'I', LEND-L+1, D( L ), E( L ) ) ISCALE = 0 IF( ANORM.GT.SSFMAX ) THEN ISCALE = 1 CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L+1, 1, D( L ), N, & INFO ) CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L, 1, E( L ), N, & INFO ) ELSE IF( ANORM.LT.SSFMIN ) THEN ISCALE = 2 CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L+1, 1, D( L ), N, & INFO ) CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L, 1, E( L ), N, & INFO ) END IF ! DO 40 I = L, LEND - 1 E( I ) = E( I )**2 40 CONTINUE ! ! Choose between QL and QR iteration ! IF( ABS( D( LEND ) ).LT.ABS( D( L ) ) ) THEN LEND = LSV L = LENDSV END IF ! IF( LEND.GE.L ) THEN ! ! QL Iteration ! ! Look for small subdiagonal element. ! 50 CONTINUE IF( L.NE.LEND ) THEN DO 60 M = L, LEND - 1 IF( ABS( E( M ) ).LE.EPS2*ABS( D( M )*D( M+1 ) ) ) & GO TO 70 60 CONTINUE END IF M = LEND ! 70 CONTINUE IF( M.LT.LEND ) & E( M ) = ZERO P = D( L ) IF( M.EQ.L ) & GO TO 90 ! ! If remaining matrix is 2 by 2, use DLAE2 to compute its ! eigenvalues. ! IF( M.EQ.L+1 ) THEN RTE = SQRT( E( L ) ) CALL DLAE2( D( L ), RTE, D( L+1 ), RT1, RT2 ) D( L ) = RT1 D( L+1 ) = RT2 E( L ) = ZERO L = L + 2 IF( L.LE.LEND ) & GO TO 50 GO TO 150 END IF ! IF( JTOT.EQ.NMAXIT ) & GO TO 150 JTOT = JTOT + 1 ! ! Form shift. ! RTE = SQRT( E( L ) ) SIGMA = ( D( L+1 )-P ) / ( TWO*RTE ) R = DLAPY2( SIGMA, ONE ) SIGMA = P - ( RTE / ( SIGMA+SIGN( R, SIGMA ) ) ) ! C = ONE S = ZERO GAMMA = D( M ) - SIGMA P = GAMMA*GAMMA ! ! Inner loop ! DO 80 I = M - 1, L, -1 BB = E( I ) R = P + BB IF( I.NE.M-1 ) & E( I+1 ) = S*R OLDC = C C = P / R S = BB / R OLDGAM = GAMMA ALPHA = D( I ) GAMMA = C*( ALPHA-SIGMA ) - S*OLDGAM D( I+1 ) = OLDGAM + ( ALPHA-GAMMA ) IF( C.NE.ZERO ) THEN P = ( GAMMA*GAMMA ) / C ELSE P = OLDC*BB END IF 80 CONTINUE ! E( L ) = S*P D( L ) = SIGMA + GAMMA GO TO 50 ! ! Eigenvalue found. ! 90 CONTINUE D( L ) = P ! L = L + 1 IF( L.LE.LEND ) & GO TO 50 GO TO 150 ! ELSE ! ! QR Iteration ! ! Look for small superdiagonal element. ! 100 CONTINUE DO 110 M = L, LEND + 1, -1 IF( ABS( E( M-1 ) ).LE.EPS2*ABS( D( M )*D( M-1 ) ) ) & GO TO 120 110 CONTINUE M = LEND ! 120 CONTINUE IF( M.GT.LEND ) & E( M-1 ) = ZERO P = D( L ) IF( M.EQ.L ) & GO TO 140 ! ! If remaining matrix is 2 by 2, use DLAE2 to compute its ! eigenvalues. ! IF( M.EQ.L-1 ) THEN RTE = SQRT( E( L-1 ) ) CALL DLAE2( D( L ), RTE, D( L-1 ), RT1, RT2 ) D( L ) = RT1 D( L-1 ) = RT2 E( L-1 ) = ZERO L = L - 2 IF( L.GE.LEND ) & GO TO 100 GO TO 150 END IF ! IF( JTOT.EQ.NMAXIT ) & GO TO 150 JTOT = JTOT + 1 ! ! Form shift. ! RTE = SQRT( E( L-1 ) ) SIGMA = ( D( L-1 )-P ) / ( TWO*RTE ) R = DLAPY2( SIGMA, ONE ) SIGMA = P - ( RTE / ( SIGMA+SIGN( R, SIGMA ) ) ) ! C = ONE S = ZERO GAMMA = D( M ) - SIGMA P = GAMMA*GAMMA ! ! Inner loop ! DO 130 I = M, L - 1 BB = E( I ) R = P + BB IF( I.NE.M ) & E( I-1 ) = S*R OLDC = C C = P / R S = BB / R OLDGAM = GAMMA ALPHA = D( I+1 ) GAMMA = C*( ALPHA-SIGMA ) - S*OLDGAM D( I ) = OLDGAM + ( ALPHA-GAMMA ) IF( C.NE.ZERO ) THEN P = ( GAMMA*GAMMA ) / C ELSE P = OLDC*BB END IF 130 CONTINUE ! E( L-1 ) = S*P D( L ) = SIGMA + GAMMA GO TO 100 ! ! Eigenvalue found. ! 140 CONTINUE D( L ) = P ! L = L - 1 IF( L.GE.LEND ) & GO TO 100 GO TO 150 ! END IF ! ! Undo scaling if necessary ! 150 CONTINUE IF( ISCALE.EQ.1 ) & CALL DLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV+1, 1, & D( LSV ), N, INFO ) IF( ISCALE.EQ.2 ) & CALL DLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV+1, 1, & D( LSV ), N, INFO ) ! ! Check for no convergence to an eigenvalue after a total ! of N*MAXIT iterations. ! IF( JTOT.LT.NMAXIT ) & GO TO 10 DO 160 I = 1, N - 1 IF( E( I ).NE.ZERO ) & INFO = INFO + 1 160 CONTINUE GO TO 180 ! ! Sort eigenvalues in increasing order. ! 170 CONTINUE CALL DLASRT( 'I', N, D, INFO ) ! 180 CONTINUE RETURN ! ! End of DSTERF ! END SUBROUTINE DSTEV( JOBZ, N, D, E, Z, LDZ, WORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER JOBZ INTEGER INFO, LDZ, N ! .. ! .. Array Arguments .. DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSTEV computes all eigenvalues and, optionally, eigenvectors of a ! real symmetric tridiagonal matrix A. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! N (input) INTEGER ! The order of the matrix. N >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the n diagonal elements of the tridiagonal matrix ! A. ! On exit, if INFO = 0, the eigenvalues in ascending order. ! ! E (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the (n-1) subdiagonal elements of the tridiagonal ! matrix A, stored in elements 1 to N-1 of E; E(N) need not ! be set, but is used by the routine. ! On exit, the contents of E are destroyed. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, N) ! If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal ! eigenvectors of the matrix A, with the i-th column of Z ! holding the eigenvector associated with D(i). ! If JOBZ = 'N', then Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (max(1,2*N-2)) ! If JOBZ = 'N', WORK is not referenced. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the algorithm failed to converge; i ! off-diagonal elements of E did not converge to zero. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL WANTZ INTEGER IMAX, ISCALE DOUBLE PRECISION BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM, & TNRM ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANST EXTERNAL LSAME, DLAMCH, DLANST ! .. ! .. External Subroutines .. EXTERNAL DSCAL, DSTEQR, DSTERF, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) ! INFO = 0 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -6 END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSTEV ', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( N.EQ.1 ) THEN IF( WANTZ ) & Z( 1, 1 ) = ONE RETURN END IF ! ! Get machine constants. ! SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = SQRT( BIGNUM ) ! ! Scale matrix to allowable range, if necessary. ! ISCALE = 0 TNRM = DLANST( 'M', N, D, E ) IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / TNRM ELSE IF( TNRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / TNRM END IF IF( ISCALE.EQ.1 ) THEN CALL DSCAL( N, SIGMA, D, 1 ) CALL DSCAL( N-1, SIGMA, E( 1 ), 1 ) END IF ! ! For eigenvalues only, call DSTERF. For eigenvalues and ! eigenvectors, call DSTEQR. ! IF( .NOT.WANTZ ) THEN CALL DSTERF( N, D, E, INFO ) ELSE CALL DSTEQR( 'I', N, D, E, Z, LDZ, WORK, INFO ) END IF ! ! If matrix was scaled, then rescale eigenvalues appropriately. ! IF( ISCALE.EQ.1 ) THEN IF( INFO.EQ.0 ) THEN IMAX = N ELSE IMAX = INFO - 1 END IF CALL DSCAL( IMAX, ONE / SIGMA, D, 1 ) END IF ! RETURN ! ! End of DSTEV ! END SUBROUTINE DSTEVD( JOBZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK, & LIWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ INTEGER INFO, LDZ, LIWORK, LWORK, N ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION D( * ), E( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSTEVD computes all eigenvalues and, optionally, eigenvectors of a ! real symmetric tridiagonal matrix. If eigenvectors are desired, it ! uses a divide and conquer algorithm. ! ! The divide and conquer algorithm makes very mild assumptions about ! floating point arithmetic. It will work on machines with a guard ! digit in add/subtract, or on those binary machines without guard ! digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or ! Cray-2. It could conceivably fail on hexadecimal or decimal machines ! without guard digits, but we know of none. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! N (input) INTEGER ! The order of the matrix. N >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the n diagonal elements of the tridiagonal matrix ! A. ! On exit, if INFO = 0, the eigenvalues in ascending order. ! ! E (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the (n-1) subdiagonal elements of the tridiagonal ! matrix A, stored in elements 1 to N-1 of E; E(N) need not ! be set, but is used by the routine. ! On exit, the contents of E are destroyed. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, N) ! If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal ! eigenvectors of the matrix A, with the i-th column of Z ! holding the eigenvector associated with D(i). ! If JOBZ = 'N', then Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! WORK (workspace/output) DOUBLE PRECISION array, ! dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! If JOBZ = 'N' or N <= 1 then LWORK must be at least 1. ! If JOBZ = 'V' and N > 1 then LWORK must be at least ! ( 1 + 4*N + N**2 ). ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace/output) INTEGER array, dimension (LIWORK) ! On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. ! ! LIWORK (input) INTEGER ! The dimension of the array IWORK. ! If JOBZ = 'N' or N <= 1 then LIWORK must be at least 1. ! If JOBZ = 'V' and N > 1 then LIWORK must be at least 3+5*N. ! ! If LIWORK = -1, then a workspace query is assumed; the ! routine only calculates the optimal size of the IWORK array, ! returns this value as the first entry of the IWORK array, and ! no error message related to LIWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the algorithm failed to converge; i ! off-diagonal elements of E did not converge to zero. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, WANTZ INTEGER ISCALE, LIWMIN, LWMIN DOUBLE PRECISION BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM, & TNRM ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANST EXTERNAL LSAME, DLAMCH, DLANST ! .. ! .. External Subroutines .. EXTERNAL DSCAL, DSTEDC, DSTERF, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) ! INFO = 0 LIWMIN = 1 LWMIN = 1 IF( N.GT.1 .AND. WANTZ ) THEN LWMIN = 1 + 4*N + N**2 LIWMIN = 3 + 5*N END IF ! IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -6 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -8 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -10 END IF ! IF( INFO.EQ.0 ) THEN WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSTEVD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( N.EQ.1 ) THEN IF( WANTZ ) & Z( 1, 1 ) = ONE RETURN END IF ! ! Get machine constants. ! SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = SQRT( BIGNUM ) ! ! Scale matrix to allowable range, if necessary. ! ISCALE = 0 TNRM = DLANST( 'M', N, D, E ) IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / TNRM ELSE IF( TNRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / TNRM END IF IF( ISCALE.EQ.1 ) THEN CALL DSCAL( N, SIGMA, D, 1 ) CALL DSCAL( N-1, SIGMA, E( 1 ), 1 ) END IF ! ! For eigenvalues only, call DSTERF. For eigenvalues and ! eigenvectors, call DSTEDC. ! IF( .NOT.WANTZ ) THEN CALL DSTERF( N, D, E, INFO ) ELSE CALL DSTEDC( 'I', N, D, E, Z, LDZ, WORK, LWORK, IWORK, LIWORK, & INFO ) END IF ! ! If matrix was scaled, then rescale eigenvalues appropriately. ! IF( ISCALE.EQ.1 ) & CALL DSCAL( N, ONE / SIGMA, D, 1 ) ! WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN ! RETURN ! ! End of DSTEVD ! END SUBROUTINE DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, & M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, & LIWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 20, 2000 ! ! .. Scalar Arguments .. CHARACTER JOBZ, RANGE INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N DOUBLE PRECISION ABSTOL, VL, VU ! .. ! .. Array Arguments .. INTEGER ISUPPZ( * ), IWORK( * ) DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSTEVR computes selected eigenvalues and, optionally, eigenvectors ! of a real symmetric tridiagonal matrix T. Eigenvalues and ! eigenvectors can be selected by specifying either a range of values ! or a range of indices for the desired eigenvalues. ! ! Whenever possible, DSTEVR calls SSTEGR to compute the ! eigenspectrum using Relatively Robust Representations. DSTEGR ! computes eigenvalues by the dqds algorithm, while orthogonal ! eigenvectors are computed from various "good" L D L^T representations ! (also known as Relatively Robust Representations). Gram-Schmidt ! orthogonalization is avoided as far as possible. More specifically, ! the various steps of the algorithm are as follows. For the i-th ! unreduced block of T, ! (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T ! is a relatively robust representation, ! (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high ! relative accuracy by the dqds algorithm, ! (c) If there is a cluster of close eigenvalues, "choose" sigma_i ! close to the cluster, and go to step (a), ! (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, ! compute the corresponding eigenvector by forming a ! rank-revealing twisted factorization. ! The desired accuracy of the output can be specified by the input ! parameter ABSTOL. ! ! For more details, see "A new O(n^2) algorithm for the symmetric ! tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, ! Computer Science Division Technical Report No. UCB//CSD-97-971, ! UC Berkeley, May 1997. ! ! ! Note 1 : DSTEVR calls SSTEGR when the full spectrum is requested ! on machines which conform to the ieee-754 floating point standard. ! DSTEVR calls SSTEBZ and SSTEIN on non-ieee machines and ! when partial spectrum requests are made. ! ! Normal execution of DSTEGR may create NaNs and infinities and ! hence may abort due to a floating point exception in environments ! which do not handle NaNs and infinities in the ieee standard default ! manner. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! RANGE (input) CHARACTER*1 ! = 'A': all eigenvalues will be found. ! = 'V': all eigenvalues in the half-open interval (VL,VU] ! will be found. ! = 'I': the IL-th through IU-th eigenvalues will be found. !********* For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and !********* DSTEIN are called ! ! N (input) INTEGER ! The order of the matrix. N >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the n diagonal elements of the tridiagonal matrix ! A. ! On exit, D may be multiplied by a constant factor chosen ! to avoid over/underflow in computing the eigenvalues. ! ! E (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the (n-1) subdiagonal elements of the tridiagonal ! matrix A in elements 1 to N-1 of E; E(N) need not be set. ! On exit, E may be multiplied by a constant factor chosen ! to avoid over/underflow in computing the eigenvalues. ! ! VL (input) DOUBLE PRECISION ! VU (input) DOUBLE PRECISION ! If RANGE='V', the lower and upper bounds of the interval to ! be searched for eigenvalues. VL < VU. ! Not referenced if RANGE = 'A' or 'I'. ! ! IL (input) INTEGER ! IU (input) INTEGER ! If RANGE='I', the indices (in ascending order) of the ! smallest and largest eigenvalues to be returned. ! 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. ! Not referenced if RANGE = 'A' or 'V'. ! ! ABSTOL (input) DOUBLE PRECISION ! The absolute error tolerance for the eigenvalues. ! An approximate eigenvalue is accepted as converged ! when it is determined to lie in an interval [a,b] ! of width less than or equal to ! ! ABSTOL + EPS * max( |a|,|b| ) , ! ! where EPS is the machine precision. If ABSTOL is less than ! or equal to zero, then EPS*|T| will be used in its place, ! where |T| is the 1-norm of the tridiagonal matrix obtained ! by reducing A to tridiagonal form. ! ! See "Computing Small Singular Values of Bidiagonal Matrices ! with Guaranteed High Relative Accuracy," by Demmel and ! Kahan, LAPACK Working Note #3. ! ! If high relative accuracy is important, set ABSTOL to ! DLAMCH( 'Safe minimum' ). Doing so will guarantee that ! eigenvalues are computed to high relative accuracy when ! possible in future releases. The current code does not ! make any guarantees about high relative accuracy, but ! future releases will. See J. Barlow and J. Demmel, ! "Computing Accurate Eigensystems of Scaled Diagonally ! Dominant Matrices", LAPACK Working Note #7, for a discussion ! of which matrices define their eigenvalues to high relative ! accuracy. ! ! M (output) INTEGER ! The total number of eigenvalues found. 0 <= M <= N. ! If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! The first M elements contain the selected eigenvalues in ! ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) ) ! If JOBZ = 'V', then if INFO = 0, the first M columns of Z ! contain the orthonormal eigenvectors of the matrix A ! corresponding to the selected eigenvalues, with the i-th ! column of Z holding the eigenvector associated with W(i). ! Note: the user must ensure that at least max(1,M) columns are ! supplied in the array Z; if RANGE = 'V', the exact value of M ! is not known in advance and an upper bound must be used. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) ) ! The support of the eigenvectors in Z, i.e., the indices ! indicating the nonzero elements in Z. The i-th eigenvector ! is nonzero only in elements ISUPPZ( 2*i-1 ) through ! ISUPPZ( 2*i ). !********* Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal (and ! minimal) LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= 20*N. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace/output) INTEGER array, dimension (LIWORK) ! On exit, if INFO = 0, IWORK(1) returns the optimal (and ! minimal) LIWORK. ! ! LIWORK (input) INTEGER ! The dimension of the array IWORK. LIWORK >= 10*N. ! ! If LIWORK = -1, then a workspace query is assumed; the ! routine only calculates the optimal size of the IWORK array, ! returns this value as the first entry of the IWORK array, and ! no error message related to LIWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: Internal error ! ! Further Details ! =============== ! ! Based on contributions by ! Inderjit Dhillon, IBM Almaden, USA ! Osni Marques, LBNL/NERSC, USA ! Ken Stanley, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ CHARACTER ORDER INTEGER I, IEEEOK, IMAX, INDIBL, INDIFL, INDISP, & INDIWO, ISCALE, ITMP1, J, JJ, LIWMIN, LWMIN, & NSPLIT DOUBLE PRECISION BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM, & TMP1, TNRM, VLL, VUU ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANST EXTERNAL LSAME, ILAENV, DLAMCH, DLANST ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTEGR, DSTEIN, DSTERF, & DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! ! Test the input parameters. ! IEEEOK = ILAENV( 10, 'DSTEVR', 'N', 1, 2, 3, 4 ) ! WANTZ = LSAME( JOBZ, 'V' ) ALLEIG = LSAME( RANGE, 'A' ) VALEIG = LSAME( RANGE, 'V' ) INDEIG = LSAME( RANGE, 'I' ) ! LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) ) LWMIN = 20*N LIWMIN = 10*N ! ! INFO = 0 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( VALEIG ) THEN IF( N.GT.0 .AND. VU.LE.VL ) & INFO = -7 ELSE IF( INDEIG ) THEN IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN INFO = -9 END IF END IF END IF IF( INFO.EQ.0 ) THEN IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -14 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -17 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -19 END IF END IF ! IF( INFO.EQ.0 ) THEN WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSTEVR', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! M = 0 IF( N.EQ.0 ) & RETURN ! IF( N.EQ.1 ) THEN IF( ALLEIG .OR. INDEIG ) THEN M = 1 W( 1 ) = D( 1 ) ELSE IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN M = 1 W( 1 ) = D( 1 ) END IF END IF IF( WANTZ ) & Z( 1, 1 ) = ONE RETURN END IF ! ! Get machine constants. ! SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) ! ! ! Scale matrix to allowable range, if necessary. ! ISCALE = 0 VLL = VL VUU = VU ! TNRM = DLANST( 'M', N, D, E ) IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / TNRM ELSE IF( TNRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / TNRM END IF IF( ISCALE.EQ.1 ) THEN CALL DSCAL( N, SIGMA, D, 1 ) CALL DSCAL( N-1, SIGMA, E( 1 ), 1 ) IF( VALEIG ) THEN VLL = VL*SIGMA VUU = VU*SIGMA END IF END IF ! ! If all eigenvalues are desired, then ! call DSTERF or SSTEGR. If this fails for some eigenvalue, then ! try DSTEBZ. ! ! IF( ( ALLEIG .OR. ( INDEIG .AND. IL.EQ.1 .AND. IU.EQ.N ) ) .AND. & IEEEOK.EQ.1 ) THEN CALL DCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 ) IF( .NOT.WANTZ ) THEN CALL DCOPY( N, D, 1, W, 1 ) CALL DSTERF( N, W, WORK, INFO ) ELSE CALL DCOPY( N, D, 1, WORK( N+1 ), 1 ) CALL DSTEGR( JOBZ, 'A', N, WORK( N+1 ), WORK, VL, VU, IL, & IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, & WORK( 2*N+1 ), LWORK-2*N, IWORK, LIWORK, INFO ) ! END IF IF( INFO.EQ.0 ) THEN M = N GO TO 10 END IF INFO = 0 END IF ! ! Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN. ! IF( WANTZ ) THEN ORDER = 'B' ELSE ORDER = 'E' END IF INDIBL = 1 INDISP = INDIBL + N INDIFL = INDISP + N INDIWO = INDIFL + N CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M, & NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK, & IWORK( INDIWO ), INFO ) ! IF( WANTZ ) THEN CALL DSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ), & Z, LDZ, WORK, IWORK( INDIWO ), IWORK( INDIFL ), & INFO ) END IF ! ! If matrix was scaled, then rescale eigenvalues appropriately. ! 10 CONTINUE IF( ISCALE.EQ.1 ) THEN IF( INFO.EQ.0 ) THEN IMAX = M ELSE IMAX = INFO - 1 END IF CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) END IF ! ! If eigenvalues are not in order, then sort them, along with ! eigenvectors. ! IF( WANTZ ) THEN DO 30 J = 1, M - 1 I = 0 TMP1 = W( J ) DO 20 JJ = J + 1, M IF( W( JJ ).LT.TMP1 ) THEN I = JJ TMP1 = W( JJ ) END IF 20 CONTINUE ! IF( I.NE.0 ) THEN ITMP1 = IWORK( I ) W( I ) = W( J ) IWORK( I ) = IWORK( J ) W( J ) = TMP1 IWORK( J ) = ITMP1 CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) END IF 30 CONTINUE END IF ! ! Causes problems with tests 19 & 20: ! IF (wantz .and. INDEIG ) Z( 1,1) = Z(1,1) / 1.002 + .002 ! ! WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN RETURN ! ! End of DSTEVR ! END SUBROUTINE DSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, & M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, RANGE INTEGER IL, INFO, IU, LDZ, M, N DOUBLE PRECISION ABSTOL, VL, VU ! .. ! .. Array Arguments .. INTEGER IFAIL( * ), IWORK( * ) DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSTEVX computes selected eigenvalues and, optionally, eigenvectors ! of a real symmetric tridiagonal matrix A. Eigenvalues and ! eigenvectors can be selected by specifying either a range of values ! or a range of indices for the desired eigenvalues. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! RANGE (input) CHARACTER*1 ! = 'A': all eigenvalues will be found. ! = 'V': all eigenvalues in the half-open interval (VL,VU] ! will be found. ! = 'I': the IL-th through IU-th eigenvalues will be found. ! ! N (input) INTEGER ! The order of the matrix. N >= 0. ! ! D (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the n diagonal elements of the tridiagonal matrix ! A. ! On exit, D may be multiplied by a constant factor chosen ! to avoid over/underflow in computing the eigenvalues. ! ! E (input/output) DOUBLE PRECISION array, dimension (N) ! On entry, the (n-1) subdiagonal elements of the tridiagonal ! matrix A in elements 1 to N-1 of E; E(N) need not be set. ! On exit, E may be multiplied by a constant factor chosen ! to avoid over/underflow in computing the eigenvalues. ! ! VL (input) DOUBLE PRECISION ! VU (input) DOUBLE PRECISION ! If RANGE='V', the lower and upper bounds of the interval to ! be searched for eigenvalues. VL < VU. ! Not referenced if RANGE = 'A' or 'I'. ! ! IL (input) INTEGER ! IU (input) INTEGER ! If RANGE='I', the indices (in ascending order) of the ! smallest and largest eigenvalues to be returned. ! 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. ! Not referenced if RANGE = 'A' or 'V'. ! ! ABSTOL (input) DOUBLE PRECISION ! The absolute error tolerance for the eigenvalues. ! An approximate eigenvalue is accepted as converged ! when it is determined to lie in an interval [a,b] ! of width less than or equal to ! ! ABSTOL + EPS * max( |a|,|b| ) , ! ! where EPS is the machine precision. If ABSTOL is less ! than or equal to zero, then EPS*|T| will be used in ! its place, where |T| is the 1-norm of the tridiagonal ! matrix. ! ! Eigenvalues will be computed most accurately when ABSTOL is ! set to twice the underflow threshold 2*DLAMCH('S'), not zero. ! If this routine returns with INFO>0, indicating that some ! eigenvectors did not converge, try setting ABSTOL to ! 2*DLAMCH('S'). ! ! See "Computing Small Singular Values of Bidiagonal Matrices ! with Guaranteed High Relative Accuracy," by Demmel and ! Kahan, LAPACK Working Note #3. ! ! M (output) INTEGER ! The total number of eigenvalues found. 0 <= M <= N. ! If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! The first M elements contain the selected eigenvalues in ! ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) ) ! If JOBZ = 'V', then if INFO = 0, the first M columns of Z ! contain the orthonormal eigenvectors of the matrix A ! corresponding to the selected eigenvalues, with the i-th ! column of Z holding the eigenvector associated with W(i). ! If an eigenvector fails to converge (INFO > 0), then that ! column of Z contains the latest approximation to the ! eigenvector, and the index of the eigenvector is returned ! in IFAIL. If JOBZ = 'N', then Z is not referenced. ! Note: the user must ensure that at least max(1,M) columns are ! supplied in the array Z; if RANGE = 'V', the exact value of M ! is not known in advance and an upper bound must be used. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (5*N) ! ! IWORK (workspace) INTEGER array, dimension (5*N) ! ! IFAIL (output) INTEGER array, dimension (N) ! If JOBZ = 'V', then if INFO = 0, the first M elements of ! IFAIL are zero. If INFO > 0, then IFAIL contains the ! indices of the eigenvectors that failed to converge. ! If JOBZ = 'N', then IFAIL is not referenced. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, then i eigenvectors failed to converge. ! Their indices are stored in array IFAIL. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL ALLEIG, INDEIG, VALEIG, WANTZ CHARACTER ORDER INTEGER I, IMAX, INDIBL, INDISP, INDIWO, INDWRK, & ISCALE, ITMP1, J, JJ, NSPLIT DOUBLE PRECISION BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM, & TMP1, TNRM, VLL, VUU ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANST EXTERNAL LSAME, DLAMCH, DLANST ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTEIN, DSTEQR, DSTERF, & DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) ALLEIG = LSAME( RANGE, 'A' ) VALEIG = LSAME( RANGE, 'V' ) INDEIG = LSAME( RANGE, 'I' ) ! INFO = 0 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( VALEIG ) THEN IF( N.GT.0 .AND. VU.LE.VL ) & INFO = -7 ELSE IF( INDEIG ) THEN IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN INFO = -9 END IF END IF END IF IF( INFO.EQ.0 ) THEN IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) & INFO = -14 END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSTEVX', -INFO ) RETURN END IF ! ! Quick return if possible ! M = 0 IF( N.EQ.0 ) & RETURN ! IF( N.EQ.1 ) THEN IF( ALLEIG .OR. INDEIG ) THEN M = 1 W( 1 ) = D( 1 ) ELSE IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN M = 1 W( 1 ) = D( 1 ) END IF END IF IF( WANTZ ) & Z( 1, 1 ) = ONE RETURN END IF ! ! Get machine constants. ! SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) ! ! Scale matrix to allowable range, if necessary. ! ISCALE = 0 IF( VALEIG ) THEN VLL = VL VUU = VU ELSE VLL = ZERO VUU = ZERO END IF TNRM = DLANST( 'M', N, D, E ) IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / TNRM ELSE IF( TNRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / TNRM END IF IF( ISCALE.EQ.1 ) THEN CALL DSCAL( N, SIGMA, D, 1 ) CALL DSCAL( N-1, SIGMA, E( 1 ), 1 ) IF( VALEIG ) THEN VLL = VL*SIGMA VUU = VU*SIGMA END IF END IF ! ! If all eigenvalues are desired and ABSTOL is less than zero, then ! call DSTERF or SSTEQR. If this fails for some eigenvalue, then ! try DSTEBZ. ! IF( ( ALLEIG .OR. ( INDEIG .AND. IL.EQ.1 .AND. IU.EQ.N ) ) .AND. & ( ABSTOL.LE.ZERO ) ) THEN CALL DCOPY( N, D, 1, W, 1 ) CALL DCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 ) INDWRK = N + 1 IF( .NOT.WANTZ ) THEN CALL DSTERF( N, W, WORK, INFO ) ELSE CALL DSTEQR( 'I', N, W, WORK, Z, LDZ, WORK( INDWRK ), INFO ) IF( INFO.EQ.0 ) THEN DO 10 I = 1, N IFAIL( I ) = 0 10 CONTINUE END IF END IF IF( INFO.EQ.0 ) THEN M = N GO TO 20 END IF INFO = 0 END IF ! ! Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN. ! IF( WANTZ ) THEN ORDER = 'B' ELSE ORDER = 'E' END IF INDWRK = 1 INDIBL = 1 INDISP = INDIBL + N INDIWO = INDISP + N CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M, & NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), & WORK( INDWRK ), IWORK( INDIWO ), INFO ) ! IF( WANTZ ) THEN CALL DSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ), & Z, LDZ, WORK( INDWRK ), IWORK( INDIWO ), IFAIL, & INFO ) END IF ! ! If matrix was scaled, then rescale eigenvalues appropriately. ! 20 CONTINUE IF( ISCALE.EQ.1 ) THEN IF( INFO.EQ.0 ) THEN IMAX = M ELSE IMAX = INFO - 1 END IF CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) END IF ! ! If eigenvalues are not in order, then sort them, along with ! eigenvectors. ! IF( WANTZ ) THEN DO 40 J = 1, M - 1 I = 0 TMP1 = W( J ) DO 30 JJ = J + 1, M IF( W( JJ ).LT.TMP1 ) THEN I = JJ TMP1 = W( JJ ) END IF 30 CONTINUE ! IF( I.NE.0 ) THEN ITMP1 = IWORK( INDIBL+I-1 ) W( I ) = W( J ) IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) W( J ) = TMP1 IWORK( INDIBL+J-1 ) = ITMP1 CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) IF( INFO.NE.0 ) THEN ITMP1 = IFAIL( I ) IFAIL( I ) = IFAIL( J ) IFAIL( J ) = ITMP1 END IF END IF 40 CONTINUE END IF ! RETURN ! ! End of DSTEVX ! END SUBROUTINE DSYCON( UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, & IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N DOUBLE PRECISION ANORM, RCOND ! .. ! .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION A( LDA, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DSYCON estimates the reciprocal of the condition number (in the ! 1-norm) of a real symmetric matrix A using the factorization ! A = U*D*U**T or A = L*D*L**T computed by DSYTRF. ! ! An estimate is obtained for norm(inv(A)), and the reciprocal of the ! condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the details of the factorization are stored ! as an upper or lower triangular matrix. ! = 'U': Upper triangular, form is A = U*D*U**T; ! = 'L': Lower triangular, form is A = L*D*L**T. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The block diagonal matrix D and the multipliers used to ! obtain the factor U or L as computed by DSYTRF. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! IPIV (input) INTEGER array, dimension (N) ! Details of the interchanges and the block structure of D ! as determined by DSYTRF. ! ! ANORM (input) DOUBLE PRECISION ! The 1-norm of the original matrix A. ! ! RCOND (output) DOUBLE PRECISION ! The reciprocal of the condition number of the matrix A, ! computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an ! estimate of the 1-norm of inv(A) computed in this routine. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (2*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER I, KASE DOUBLE PRECISION AINVNM ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DLACON, DSYTRS, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( ANORM.LT.ZERO ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYCON', -INFO ) RETURN END IF ! ! Quick return if possible ! RCOND = ZERO IF( N.EQ.0 ) THEN RCOND = ONE RETURN ELSE IF( ANORM.LE.ZERO ) THEN RETURN END IF ! ! Check that the diagonal matrix D is nonsingular. ! IF( UPPER ) THEN ! ! Upper triangular storage: examine D from bottom to top ! DO 10 I = N, 1, -1 IF( IPIV( I ).GT.0 .AND. A( I, I ).EQ.ZERO ) & RETURN 10 CONTINUE ELSE ! ! Lower triangular storage: examine D from top to bottom. ! DO 20 I = 1, N IF( IPIV( I ).GT.0 .AND. A( I, I ).EQ.ZERO ) & RETURN 20 CONTINUE END IF ! ! Estimate the 1-norm of the inverse. ! KASE = 0 30 CONTINUE CALL DLACON( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE ) IF( KASE.NE.0 ) THEN ! ! Multiply by inv(L*D*L') or inv(U*D*U'). ! CALL DSYTRS( UPLO, N, 1, A, LDA, IPIV, WORK, N, INFO ) GO TO 30 END IF ! ! Compute the estimate of the reciprocal condition number. ! IF( AINVNM.NE.ZERO ) & RCOND = ( ONE / AINVNM ) / ANORM ! RETURN ! ! End of DSYCON ! END SUBROUTINE DSYEV( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, LDA, LWORK, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DSYEV computes all eigenvalues and, optionally, eigenvectors of a ! real symmetric matrix A. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the symmetric matrix A. If UPLO = 'U', the ! leading N-by-N upper triangular part of A contains the ! upper triangular part of the matrix A. If UPLO = 'L', ! the leading N-by-N lower triangular part of A contains ! the lower triangular part of the matrix A. ! On exit, if JOBZ = 'V', then if INFO = 0, A contains the ! orthonormal eigenvectors of the matrix A. ! If JOBZ = 'N', then on exit the lower triangle (if UPLO='L') ! or the upper triangle (if UPLO='U') of A, including the ! diagonal, is destroyed. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! W (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, the eigenvalues in ascending order. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The length of the array WORK. LWORK >= max(1,3*N-1). ! For optimal efficiency, LWORK >= (NB+2)*N, ! where NB is the blocksize for DSYTRD returned by ILAENV. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the algorithm failed to converge; i ! off-diagonal elements of an intermediate tridiagonal ! form did not converge to zero. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. LOGICAL LOWER, LQUERY, WANTZ INTEGER IINFO, IMAX, INDE, INDTAU, INDWRK, ISCALE, & LLWORK, LOPT, LWKOPT, NB DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, & SMLNUM ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANSY EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY ! .. ! .. External Subroutines .. EXTERNAL DLASCL, DORGTR, DSCAL, DSTEQR, DSTERF, DSYTRD, & XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) LOWER = LSAME( UPLO, 'L' ) LQUERY = ( LWORK.EQ.-1 ) ! INFO = 0 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LWORK.LT.MAX( 1, 3*N-1 ) .AND. .NOT.LQUERY ) THEN INFO = -8 END IF ! IF( INFO.EQ.0 ) THEN NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 ) LWKOPT = MAX( 1, ( NB+2 )*N ) WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYEV ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! IF( N.EQ.1 ) THEN W( 1 ) = A( 1, 1 ) WORK( 1 ) = 3 IF( WANTZ ) & A( 1, 1 ) = ONE RETURN END IF ! ! Get machine constants. ! SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = SQRT( BIGNUM ) ! ! Scale matrix to allowable range, if necessary. ! ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK ) ISCALE = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / ANRM ELSE IF( ANRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / ANRM END IF IF( ISCALE.EQ.1 ) & CALL DLASCL( UPLO, 0, 0, ONE, SIGMA, N, N, A, LDA, INFO ) ! ! Call DSYTRD to reduce symmetric matrix to tridiagonal form. ! INDE = 1 INDTAU = INDE + N INDWRK = INDTAU + N LLWORK = LWORK - INDWRK + 1 CALL DSYTRD( UPLO, N, A, LDA, W, WORK( INDE ), WORK( INDTAU ), & WORK( INDWRK ), LLWORK, IINFO ) LOPT = 2*N + WORK( INDWRK ) ! ! For eigenvalues only, call DSTERF. For eigenvectors, first call ! DORGTR to generate the orthogonal matrix, then call DSTEQR. ! IF( .NOT.WANTZ ) THEN CALL DSTERF( N, W, WORK( INDE ), INFO ) ELSE CALL DORGTR( UPLO, N, A, LDA, WORK( INDTAU ), WORK( INDWRK ), & LLWORK, IINFO ) CALL DSTEQR( JOBZ, N, W, WORK( INDE ), A, LDA, WORK( INDTAU ), & INFO ) END IF ! ! If matrix was scaled, then rescale eigenvalues appropriately. ! IF( ISCALE.EQ.1 ) THEN IF( INFO.EQ.0 ) THEN IMAX = N ELSE IMAX = INFO - 1 END IF CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) END IF ! ! Set WORK(1) to optimal workspace size. ! WORK( 1 ) = LWKOPT ! RETURN ! ! End of DSYEV ! END SUBROUTINE DSYEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK, & LIWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, LDA, LIWORK, LWORK, N ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DSYEVD computes all eigenvalues and, optionally, eigenvectors of a ! real symmetric matrix A. If eigenvectors are desired, it uses a ! divide and conquer algorithm. ! ! The divide and conquer algorithm makes very mild assumptions about ! floating point arithmetic. It will work on machines with a guard ! digit in add/subtract, or on those binary machines without guard ! digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or ! Cray-2. It could conceivably fail on hexadecimal or decimal machines ! without guard digits, but we know of none. ! ! Because of large use of BLAS of level 3, DSYEVD needs N**2 more ! workspace than DSYEVX. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the symmetric matrix A. If UPLO = 'U', the ! leading N-by-N upper triangular part of A contains the ! upper triangular part of the matrix A. If UPLO = 'L', ! the leading N-by-N lower triangular part of A contains ! the lower triangular part of the matrix A. ! On exit, if JOBZ = 'V', then if INFO = 0, A contains the ! orthonormal eigenvectors of the matrix A. ! If JOBZ = 'N', then on exit the lower triangle (if UPLO='L') ! or the upper triangle (if UPLO='U') of A, including the ! diagonal, is destroyed. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! W (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, the eigenvalues in ascending order. ! ! WORK (workspace/output) DOUBLE PRECISION array, ! dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! If N <= 1, LWORK must be at least 1. ! If JOBZ = 'N' and N > 1, LWORK must be at least 2*N+1. ! If JOBZ = 'V' and N > 1, LWORK must be at least ! 1 + 6*N + 2*N**2. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace/output) INTEGER array, dimension (LIWORK) ! On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. ! ! LIWORK (input) INTEGER ! The dimension of the array IWORK. ! If N <= 1, LIWORK must be at least 1. ! If JOBZ = 'N' and N > 1, LIWORK must be at least 1. ! If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N. ! ! If LIWORK = -1, then a workspace query is assumed; the ! routine only calculates the optimal size of the IWORK array, ! returns this value as the first entry of the IWORK array, and ! no error message related to LIWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the algorithm failed to converge; i ! off-diagonal elements of an intermediate tridiagonal ! form did not converge to zero. ! ! Further Details ! =============== ! ! Based on contributions by ! Jeff Rutter, Computer Science Division, University of California ! at Berkeley, USA ! Modified by Francoise Tisseur, University of Tennessee. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. ! LOGICAL LOWER, LQUERY, WANTZ INTEGER IINFO, INDE, INDTAU, INDWK2, INDWRK, ISCALE, & LIOPT, LIWMIN, LLWORK, LLWRK2, LOPT, LWMIN DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, & SMLNUM ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANSY EXTERNAL LSAME, DLAMCH, DLANSY ! .. ! .. External Subroutines .. EXTERNAL DLACPY, DLASCL, DORMTR, DSCAL, DSTEDC, DSTERF, & DSYTRD, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) LOWER = LSAME( UPLO, 'L' ) LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) ! INFO = 0 IF( N.LE.1 ) THEN LIWMIN = 1 LWMIN = 1 LOPT = LWMIN LIOPT = LIWMIN ELSE IF( WANTZ ) THEN LIWMIN = 3 + 5*N LWMIN = 1 + 6*N + 2*N**2 ELSE LIWMIN = 1 LWMIN = 2*N + 1 END IF LOPT = LWMIN LIOPT = LIWMIN END IF IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -8 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -10 END IF ! IF( INFO.EQ.0 ) THEN WORK( 1 ) = LOPT IWORK( 1 ) = LIOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYEVD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( N.EQ.1 ) THEN W( 1 ) = A( 1, 1 ) IF( WANTZ ) & A( 1, 1 ) = ONE RETURN END IF ! ! Get machine constants. ! SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = SQRT( BIGNUM ) ! ! Scale matrix to allowable range, if necessary. ! ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK ) ISCALE = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / ANRM ELSE IF( ANRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / ANRM END IF IF( ISCALE.EQ.1 ) & CALL DLASCL( UPLO, 0, 0, ONE, SIGMA, N, N, A, LDA, INFO ) ! ! Call DSYTRD to reduce symmetric matrix to tridiagonal form. ! INDE = 1 INDTAU = INDE + N INDWRK = INDTAU + N LLWORK = LWORK - INDWRK + 1 INDWK2 = INDWRK + N*N LLWRK2 = LWORK - INDWK2 + 1 ! CALL DSYTRD( UPLO, N, A, LDA, W, WORK( INDE ), WORK( INDTAU ), & WORK( INDWRK ), LLWORK, IINFO ) LOPT = 2*N + WORK( INDWRK ) ! ! For eigenvalues only, call DSTERF. For eigenvectors, first call ! DSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the ! tridiagonal matrix, then call DORMTR to multiply it by the ! Householder transformations stored in A. ! IF( .NOT.WANTZ ) THEN CALL DSTERF( N, W, WORK( INDE ), INFO ) ELSE CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N, & WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO ) CALL DORMTR( 'L', UPLO, 'N', N, N, A, LDA, WORK( INDTAU ), & WORK( INDWRK ), N, WORK( INDWK2 ), LLWRK2, IINFO ) CALL DLACPY( 'A', N, N, WORK( INDWRK ), N, A, LDA ) LOPT = MAX( LOPT, 1+6*N+2*N**2 ) END IF ! ! If matrix was scaled, then rescale eigenvalues appropriately. ! IF( ISCALE.EQ.1 ) & CALL DSCAL( N, ONE / SIGMA, W, 1 ) ! WORK( 1 ) = LOPT IWORK( 1 ) = LIOPT ! RETURN ! ! End of DSYEVD ! END SUBROUTINE DSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, & ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, & IWORK, LIWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 20, 2000 ! ! .. Scalar Arguments .. CHARACTER JOBZ, RANGE, UPLO INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N DOUBLE PRECISION ABSTOL, VL, VU ! .. ! .. Array Arguments .. INTEGER ISUPPZ( * ), IWORK( * ) DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSYEVR computes selected eigenvalues and, optionally, eigenvectors ! of a real symmetric matrix T. Eigenvalues and eigenvectors can be ! selected by specifying either a range of values or a range of ! indices for the desired eigenvalues. ! ! Whenever possible, DSYEVR calls DSTEGR to compute the ! eigenspectrum using Relatively Robust Representations. DSTEGR ! computes eigenvalues by the dqds algorithm, while orthogonal ! eigenvectors are computed from various "good" L D L^T representations ! (also known as Relatively Robust Representations). Gram-Schmidt ! orthogonalization is avoided as far as possible. More specifically, ! the various steps of the algorithm are as follows. For the i-th ! unreduced block of T, ! (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T ! is a relatively robust representation, ! (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high ! relative accuracy by the dqds algorithm, ! (c) If there is a cluster of close eigenvalues, "choose" sigma_i ! close to the cluster, and go to step (a), ! (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, ! compute the corresponding eigenvector by forming a ! rank-revealing twisted factorization. ! The desired accuracy of the output can be specified by the input ! parameter ABSTOL. ! ! For more details, see "A new O(n^2) algorithm for the symmetric ! tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, ! Computer Science Division Technical Report No. UCB//CSD-97-971, ! UC Berkeley, May 1997. ! ! ! Note 1 : DSYEVR calls DSTEGR when the full spectrum is requested ! on machines which conform to the ieee-754 floating point standard. ! DSYEVR calls DSTEBZ and SSTEIN on non-ieee machines and ! when partial spectrum requests are made. ! ! Normal execution of DSTEGR may create NaNs and infinities and ! hence may abort due to a floating point exception in environments ! which do not handle NaNs and infinities in the ieee standard default ! manner. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! RANGE (input) CHARACTER*1 ! = 'A': all eigenvalues will be found. ! = 'V': all eigenvalues in the half-open interval (VL,VU] ! will be found. ! = 'I': the IL-th through IU-th eigenvalues will be found. !********* For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and !********* DSTEIN are called ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the symmetric matrix A. If UPLO = 'U', the ! leading N-by-N upper triangular part of A contains the ! upper triangular part of the matrix A. If UPLO = 'L', ! the leading N-by-N lower triangular part of A contains ! the lower triangular part of the matrix A. ! On exit, the lower triangle (if UPLO='L') or the upper ! triangle (if UPLO='U') of A, including the diagonal, is ! destroyed. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! VL (input) DOUBLE PRECISION ! VU (input) DOUBLE PRECISION ! If RANGE='V', the lower and upper bounds of the interval to ! be searched for eigenvalues. VL < VU. ! Not referenced if RANGE = 'A' or 'I'. ! ! IL (input) INTEGER ! IU (input) INTEGER ! If RANGE='I', the indices (in ascending order) of the ! smallest and largest eigenvalues to be returned. ! 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. ! Not referenced if RANGE = 'A' or 'V'. ! ! ABSTOL (input) DOUBLE PRECISION ! The absolute error tolerance for the eigenvalues. ! An approximate eigenvalue is accepted as converged ! when it is determined to lie in an interval [a,b] ! of width less than or equal to ! ! ABSTOL + EPS * max( |a|,|b| ) , ! ! where EPS is the machine precision. If ABSTOL is less than ! or equal to zero, then EPS*|T| will be used in its place, ! where |T| is the 1-norm of the tridiagonal matrix obtained ! by reducing A to tridiagonal form. ! ! See "Computing Small Singular Values of Bidiagonal Matrices ! with Guaranteed High Relative Accuracy," by Demmel and ! Kahan, LAPACK Working Note #3. ! ! If high relative accuracy is important, set ABSTOL to ! DLAMCH( 'Safe minimum' ). Doing so will guarantee that ! eigenvalues are computed to high relative accuracy when ! possible in future releases. The current code does not ! make any guarantees about high relative accuracy, but ! furutre releases will. See J. Barlow and J. Demmel, ! "Computing Accurate Eigensystems of Scaled Diagonally ! Dominant Matrices", LAPACK Working Note #7, for a discussion ! of which matrices define their eigenvalues to high relative ! accuracy. ! ! M (output) INTEGER ! The total number of eigenvalues found. 0 <= M <= N. ! If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! The first M elements contain the selected eigenvalues in ! ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M)) ! If JOBZ = 'V', then if INFO = 0, the first M columns of Z ! contain the orthonormal eigenvectors of the matrix A ! corresponding to the selected eigenvalues, with the i-th ! column of Z holding the eigenvector associated with W(i). ! If JOBZ = 'N', then Z is not referenced. ! Note: the user must ensure that at least max(1,M) columns are ! supplied in the array Z; if RANGE = 'V', the exact value of M ! is not known in advance and an upper bound must be used. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) ) ! The support of the eigenvectors in Z, i.e., the indices ! indicating the nonzero elements in Z. The i-th eigenvector ! is nonzero only in elements ISUPPZ( 2*i-1 ) through ! ISUPPZ( 2*i ). !********* Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,26*N). ! For optimal efficiency, LWORK >= (NB+6)*N, ! where NB is the max of the blocksize for DSYTRD and DORMTR ! returned by ILAENV. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace/output) INTEGER array, dimension (LIWORK) ! On exit, if INFO = 0, IWORK(1) returns the optimal LWORK. ! ! LIWORK (input) INTEGER ! The dimension of the array IWORK. LIWORK >= max(1,10*N). ! ! If LIWORK = -1, then a workspace query is assumed; the ! routine only calculates the optimal size of the IWORK array, ! returns this value as the first entry of the IWORK array, and ! no error message related to LIWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: Internal error ! ! Further Details ! =============== ! ! Based on contributions by ! Inderjit Dhillon, IBM Almaden, USA ! Osni Marques, LBNL/NERSC, USA ! Ken Stanley, Computer Science Division, University of ! California at Berkeley, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, VALEIG, WANTZ CHARACTER ORDER INTEGER I, IEEEOK, IINFO, IMAX, INDD, INDDD, INDE, & INDEE, INDIBL, INDIFL, INDISP, INDIWO, INDTAU, & INDWK, INDWKN, ISCALE, ITMP1, J, JJ, LIWMIN, & LLWORK, LLWRKN, LWKOPT, LWMIN, NB, NSPLIT DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, & SIGMA, SMLNUM, TMP1, VLL, VUU ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANSY EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DORMTR, DSCAL, DSTEBZ, DSTEGR, DSTEIN, & DSTERF, DSWAP, DSYTRD, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! IEEEOK = ILAENV( 10, 'DSYEVR', 'N', 1, 2, 3, 4 ) ! LOWER = LSAME( UPLO, 'L' ) WANTZ = LSAME( JOBZ, 'V' ) ALLEIG = LSAME( RANGE, 'A' ) VALEIG = LSAME( RANGE, 'V' ) INDEIG = LSAME( RANGE, 'I' ) ! LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) ) ! LWMIN = MAX( 1, 26*N ) LIWMIN = MAX( 1, 10*N ) ! INFO = 0 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN INFO = -2 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( VALEIG ) THEN IF( N.GT.0 .AND. VU.LE.VL ) & INFO = -8 ELSE IF( INDEIG ) THEN IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN INFO = -10 END IF END IF END IF IF( INFO.EQ.0 ) THEN IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -15 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -18 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -20 END IF END IF ! IF( INFO.EQ.0 ) THEN NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 ) NB = MAX( NB, ILAENV( 1, 'ZUNMTR', UPLO, N, -1, -1, -1 ) ) LWKOPT = MAX( ( NB+1 )*N, LWMIN ) WORK( 1 ) = LWKOPT IWORK( 1 ) = LIWMIN END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYEVR', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! M = 0 IF( N.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! IF( N.EQ.1 ) THEN WORK( 1 ) = 7 IF( ALLEIG .OR. INDEIG ) THEN M = 1 W( 1 ) = A( 1, 1 ) ELSE IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN M = 1 W( 1 ) = A( 1, 1 ) END IF END IF IF( WANTZ ) & Z( 1, 1 ) = ONE RETURN END IF ! ! Get machine constants. ! SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) ! ! Scale matrix to allowable range, if necessary. ! ISCALE = 0 ABSTLL = ABSTOL VLL = VL VUU = VU ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK ) IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / ANRM ELSE IF( ANRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / ANRM END IF IF( ISCALE.EQ.1 ) THEN IF( LOWER ) THEN DO 10 J = 1, N CALL DSCAL( N-J+1, SIGMA, A( J, J ), 1 ) 10 CONTINUE ELSE DO 20 J = 1, N CALL DSCAL( J, SIGMA, A( 1, J ), 1 ) 20 CONTINUE END IF IF( ABSTOL.GT.0 ) & ABSTLL = ABSTOL*SIGMA IF( VALEIG ) THEN VLL = VL*SIGMA VUU = VU*SIGMA END IF END IF ! ! Call DSYTRD to reduce symmetric matrix to tridiagonal form. ! INDTAU = 1 INDE = INDTAU + N INDD = INDE + N INDEE = INDD + N INDDD = INDEE + N INDIFL = INDDD + N INDWK = INDIFL + N LLWORK = LWORK - INDWK + 1 CALL DSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ), & WORK( INDTAU ), WORK( INDWK ), LLWORK, IINFO ) ! ! If all eigenvalues are desired ! then call DSTERF or SSTEGR and DORMTR. ! IF( ( ALLEIG .OR. ( INDEIG .AND. IL.EQ.1 .AND. IU.EQ.N ) ) .AND. & IEEEOK.EQ.1 ) THEN IF( .NOT.WANTZ ) THEN CALL DCOPY( N, WORK( INDD ), 1, W, 1 ) CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) CALL DSTERF( N, W, WORK( INDEE ), INFO ) ELSE CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) CALL DCOPY( N, WORK( INDD ), 1, WORK( INDDD ), 1 ) ! CALL DSTEGR( JOBZ, 'A', N, WORK( INDDD ), WORK( INDEE ), & VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, & WORK( INDWK ), LWORK, IWORK, LIWORK, INFO ) ! ! ! ! Apply orthogonal matrix used in reduction to tridiagonal ! form to eigenvectors returned by DSTEIN. ! IF( WANTZ .AND. INFO.EQ.0 ) THEN INDWKN = INDE LLWRKN = LWORK - INDWKN + 1 CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA, & WORK( INDTAU ), Z, LDZ, WORK( INDWKN ), & LLWRKN, IINFO ) END IF END IF ! ! IF( INFO.EQ.0 ) THEN M = N GO TO 30 END IF INFO = 0 END IF ! ! Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN. ! Also call DSTEBZ and SSTEIN if SSTEGR fails. ! IF( WANTZ ) THEN ORDER = 'B' ELSE ORDER = 'E' END IF INDIFL = 1 INDIBL = INDIFL + N INDISP = INDIBL + N INDIWO = INDISP + N CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, & WORK( INDD ), WORK( INDE ), M, NSPLIT, W, & IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWK ), & IWORK( INDIWO ), INFO ) ! IF( WANTZ ) THEN CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W, & IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, & WORK( INDWK ), IWORK( INDIWO ), IWORK( INDIFL ), & INFO ) ! ! Apply orthogonal matrix used in reduction to tridiagonal ! form to eigenvectors returned by DSTEIN. ! INDWKN = INDE LLWRKN = LWORK - INDWKN + 1 CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z, & LDZ, WORK( INDWKN ), LLWRKN, IINFO ) END IF ! ! If matrix was scaled, then rescale eigenvalues appropriately. ! 30 CONTINUE IF( ISCALE.EQ.1 ) THEN IF( INFO.EQ.0 ) THEN IMAX = M ELSE IMAX = INFO - 1 END IF CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) END IF ! ! If eigenvalues are not in order, then sort them, along with ! eigenvectors. ! IF( WANTZ ) THEN DO 50 J = 1, M - 1 I = 0 TMP1 = W( J ) DO 40 JJ = J + 1, M IF( W( JJ ).LT.TMP1 ) THEN I = JJ TMP1 = W( JJ ) END IF 40 CONTINUE ! IF( I.NE.0 ) THEN ITMP1 = IWORK( INDIBL+I-1 ) W( I ) = W( J ) IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) W( J ) = TMP1 IWORK( INDIBL+J-1 ) = ITMP1 CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) END IF 50 CONTINUE END IF ! ! Set WORK(1) to optimal workspace size. ! WORK( 1 ) = LWKOPT IWORK( 1 ) = LIWMIN ! RETURN ! ! End of DSYEVR ! END SUBROUTINE DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, & ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, & IFAIL, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, RANGE, UPLO INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N DOUBLE PRECISION ABSTOL, VL, VU ! .. ! .. Array Arguments .. INTEGER IFAIL( * ), IWORK( * ) DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSYEVX computes selected eigenvalues and, optionally, eigenvectors ! of a real symmetric matrix A. Eigenvalues and eigenvectors can be ! selected by specifying either a range of values or a range of indices ! for the desired eigenvalues. ! ! Arguments ! ========= ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! RANGE (input) CHARACTER*1 ! = 'A': all eigenvalues will be found. ! = 'V': all eigenvalues in the half-open interval (VL,VU] ! will be found. ! = 'I': the IL-th through IU-th eigenvalues will be found. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the symmetric matrix A. If UPLO = 'U', the ! leading N-by-N upper triangular part of A contains the ! upper triangular part of the matrix A. If UPLO = 'L', ! the leading N-by-N lower triangular part of A contains ! the lower triangular part of the matrix A. ! On exit, the lower triangle (if UPLO='L') or the upper ! triangle (if UPLO='U') of A, including the diagonal, is ! destroyed. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! VL (input) DOUBLE PRECISION ! VU (input) DOUBLE PRECISION ! If RANGE='V', the lower and upper bounds of the interval to ! be searched for eigenvalues. VL < VU. ! Not referenced if RANGE = 'A' or 'I'. ! ! IL (input) INTEGER ! IU (input) INTEGER ! If RANGE='I', the indices (in ascending order) of the ! smallest and largest eigenvalues to be returned. ! 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. ! Not referenced if RANGE = 'A' or 'V'. ! ! ABSTOL (input) DOUBLE PRECISION ! The absolute error tolerance for the eigenvalues. ! An approximate eigenvalue is accepted as converged ! when it is determined to lie in an interval [a,b] ! of width less than or equal to ! ! ABSTOL + EPS * max( |a|,|b| ) , ! ! where EPS is the machine precision. If ABSTOL is less than ! or equal to zero, then EPS*|T| will be used in its place, ! where |T| is the 1-norm of the tridiagonal matrix obtained ! by reducing A to tridiagonal form. ! ! Eigenvalues will be computed most accurately when ABSTOL is ! set to twice the underflow threshold 2*DLAMCH('S'), not zero. ! If this routine returns with INFO>0, indicating that some ! eigenvectors did not converge, try setting ABSTOL to ! 2*DLAMCH('S'). ! ! See "Computing Small Singular Values of Bidiagonal Matrices ! with Guaranteed High Relative Accuracy," by Demmel and ! Kahan, LAPACK Working Note #3. ! ! M (output) INTEGER ! The total number of eigenvalues found. 0 <= M <= N. ! If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! On normal exit, the first M elements contain the selected ! eigenvalues in ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M)) ! If JOBZ = 'V', then if INFO = 0, the first M columns of Z ! contain the orthonormal eigenvectors of the matrix A ! corresponding to the selected eigenvalues, with the i-th ! column of Z holding the eigenvector associated with W(i). ! If an eigenvector fails to converge, then that column of Z ! contains the latest approximation to the eigenvector, and the ! index of the eigenvector is returned in IFAIL. ! If JOBZ = 'N', then Z is not referenced. ! Note: the user must ensure that at least max(1,M) columns are ! supplied in the array Z; if RANGE = 'V', the exact value of M ! is not known in advance and an upper bound must be used. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The length of the array WORK. LWORK >= max(1,8*N). ! For optimal efficiency, LWORK >= (NB+3)*N, ! where NB is the max of the blocksize for DSYTRD and DORMTR ! returned by ILAENV. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace) INTEGER array, dimension (5*N) ! ! IFAIL (output) INTEGER array, dimension (N) ! If JOBZ = 'V', then if INFO = 0, the first M elements of ! IFAIL are zero. If INFO > 0, then IFAIL contains the ! indices of the eigenvectors that failed to converge. ! If JOBZ = 'N', then IFAIL is not referenced. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, then i eigenvectors failed to converge. ! Their indices are stored in array IFAIL. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, VALEIG, WANTZ CHARACTER ORDER INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL, & INDISP, INDIWO, INDTAU, INDWKN, INDWRK, ISCALE, & ITMP1, J, JJ, LLWORK, LLWRKN, LOPT, LWKOPT, NB, & NSPLIT DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, & SIGMA, SMLNUM, TMP1, VLL, VUU ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANSY EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLACPY, DORGTR, DORMTR, DSCAL, DSTEBZ, & DSTEIN, DSTEQR, DSTERF, DSWAP, DSYTRD, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! LOWER = LSAME( UPLO, 'L' ) WANTZ = LSAME( JOBZ, 'V' ) ALLEIG = LSAME( RANGE, 'A' ) VALEIG = LSAME( RANGE, 'V' ) INDEIG = LSAME( RANGE, 'I' ) LQUERY = ( LWORK.EQ.-1 ) ! INFO = 0 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN INFO = -2 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( VALEIG ) THEN IF( N.GT.0 .AND. VU.LE.VL ) & INFO = -8 ELSE IF( INDEIG ) THEN IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN INFO = -10 END IF END IF END IF IF( INFO.EQ.0 ) THEN IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -15 ELSE IF( LWORK.LT.MAX( 1, 8*N ) .AND. .NOT.LQUERY ) THEN INFO = -17 END IF END IF ! IF( INFO.EQ.0 ) THEN NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 ) NB = MAX( NB, ILAENV( 1, 'DORMTR', UPLO, N, -1, -1, -1 ) ) LWKOPT = ( NB+3 )*N WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYEVX', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! M = 0 IF( N.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! IF( N.EQ.1 ) THEN WORK( 1 ) = 7 IF( ALLEIG .OR. INDEIG ) THEN M = 1 W( 1 ) = A( 1, 1 ) ELSE IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN M = 1 W( 1 ) = A( 1, 1 ) END IF END IF IF( WANTZ ) & Z( 1, 1 ) = ONE RETURN END IF ! ! Get machine constants. ! SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) ! ! Scale matrix to allowable range, if necessary. ! ISCALE = 0 ABSTLL = ABSTOL VLL = VL VUU = VU ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK ) IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / ANRM ELSE IF( ANRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / ANRM END IF IF( ISCALE.EQ.1 ) THEN IF( LOWER ) THEN DO 10 J = 1, N CALL DSCAL( N-J+1, SIGMA, A( J, J ), 1 ) 10 CONTINUE ELSE DO 20 J = 1, N CALL DSCAL( J, SIGMA, A( 1, J ), 1 ) 20 CONTINUE END IF IF( ABSTOL.GT.0 ) & ABSTLL = ABSTOL*SIGMA IF( VALEIG ) THEN VLL = VL*SIGMA VUU = VU*SIGMA END IF END IF ! ! Call DSYTRD to reduce symmetric matrix to tridiagonal form. ! INDTAU = 1 INDE = INDTAU + N INDD = INDE + N INDWRK = INDD + N LLWORK = LWORK - INDWRK + 1 CALL DSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ), & WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO ) LOPT = 3*N + WORK( INDWRK ) ! ! If all eigenvalues are desired and ABSTOL is less than or equal to ! zero, then call DSTERF or DORGTR and SSTEQR. If this fails for ! some eigenvalue, then try DSTEBZ. ! IF( ( ALLEIG .OR. ( INDEIG .AND. IL.EQ.1 .AND. IU.EQ.N ) ) .AND. & ( ABSTOL.LE.ZERO ) ) THEN CALL DCOPY( N, WORK( INDD ), 1, W, 1 ) INDEE = INDWRK + 2*N IF( .NOT.WANTZ ) THEN CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) CALL DSTERF( N, W, WORK( INDEE ), INFO ) ELSE CALL DLACPY( 'A', N, N, A, LDA, Z, LDZ ) CALL DORGTR( UPLO, N, Z, LDZ, WORK( INDTAU ), & WORK( INDWRK ), LLWORK, IINFO ) CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ, & WORK( INDWRK ), INFO ) IF( INFO.EQ.0 ) THEN DO 30 I = 1, N IFAIL( I ) = 0 30 CONTINUE END IF END IF IF( INFO.EQ.0 ) THEN M = N GO TO 40 END IF INFO = 0 END IF ! ! Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN. ! IF( WANTZ ) THEN ORDER = 'B' ELSE ORDER = 'E' END IF INDIBL = 1 INDISP = INDIBL + N INDIWO = INDISP + N CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, & WORK( INDD ), WORK( INDE ), M, NSPLIT, W, & IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ), & IWORK( INDIWO ), INFO ) ! IF( WANTZ ) THEN CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W, & IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, & WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO ) ! ! Apply orthogonal matrix used in reduction to tridiagonal ! form to eigenvectors returned by DSTEIN. ! INDWKN = INDE LLWRKN = LWORK - INDWKN + 1 CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z, & LDZ, WORK( INDWKN ), LLWRKN, IINFO ) END IF ! ! If matrix was scaled, then rescale eigenvalues appropriately. ! 40 CONTINUE IF( ISCALE.EQ.1 ) THEN IF( INFO.EQ.0 ) THEN IMAX = M ELSE IMAX = INFO - 1 END IF CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) END IF ! ! If eigenvalues are not in order, then sort them, along with ! eigenvectors. ! IF( WANTZ ) THEN DO 60 J = 1, M - 1 I = 0 TMP1 = W( J ) DO 50 JJ = J + 1, M IF( W( JJ ).LT.TMP1 ) THEN I = JJ TMP1 = W( JJ ) END IF 50 CONTINUE ! IF( I.NE.0 ) THEN ITMP1 = IWORK( INDIBL+I-1 ) W( I ) = W( J ) IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 ) W( J ) = TMP1 IWORK( INDIBL+J-1 ) = ITMP1 CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) IF( INFO.NE.0 ) THEN ITMP1 = IFAIL( I ) IFAIL( I ) = IFAIL( J ) IFAIL( J ) = ITMP1 END IF END IF 60 CONTINUE END IF ! ! Set WORK(1) to optimal workspace size. ! WORK( 1 ) = LWKOPT ! RETURN ! ! End of DSYEVX ! END SUBROUTINE DSYGS2( ITYPE, UPLO, N, A, LDA, B, LDB, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, ITYPE, LDA, LDB, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DSYGS2 reduces a real symmetric-definite generalized eigenproblem ! to standard form. ! ! If ITYPE = 1, the problem is A*x = lambda*B*x, ! and A is overwritten by inv(U')*A*inv(U) or inv(L)*A*inv(L') ! ! If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or ! B*A*x = lambda*x, and A is overwritten by U*A*U` or L'*A*L. ! ! B must have been previously factorized as U'*U or L*L' by DPOTRF. ! ! Arguments ! ========= ! ! ITYPE (input) INTEGER ! = 1: compute inv(U')*A*inv(U) or inv(L)*A*inv(L'); ! = 2 or 3: compute U*A*U' or L'*A*L. ! ! UPLO (input) CHARACTER ! Specifies whether the upper or lower triangular part of the ! symmetric matrix A is stored, and how B has been factorized. ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the symmetric matrix A. If UPLO = 'U', the leading ! n by n upper triangular part of A contains the upper ! triangular part of the matrix A, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading n by n lower triangular part of A contains the lower ! triangular part of the matrix A, and the strictly upper ! triangular part of A is not referenced. ! ! On exit, if INFO = 0, the transformed matrix, stored in the ! same format as A. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! B (input) DOUBLE PRECISION array, dimension (LDB,N) ! The triangular factor from the Cholesky factorization of B, ! as returned by DPOTRF. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, HALF PARAMETER ( ONE = 1.0D0, HALF = 0.5D0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER K DOUBLE PRECISION AKK, BKK, CT ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DSCAL, DSYR2, DTRMV, DTRSV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN INFO = -1 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYGS2', -INFO ) RETURN END IF ! IF( ITYPE.EQ.1 ) THEN IF( UPPER ) THEN ! ! Compute inv(U')*A*inv(U) ! DO 10 K = 1, N ! ! Update the upper triangle of A(k:n,k:n) ! AKK = A( K, K ) BKK = B( K, K ) AKK = AKK / BKK**2 A( K, K ) = AKK IF( K.LT.N ) THEN CALL DSCAL( N-K, ONE / BKK, A( K, K+1 ), LDA ) CT = -HALF*AKK CALL DAXPY( N-K, CT, B( K, K+1 ), LDB, A( K, K+1 ), & LDA ) CALL DSYR2( UPLO, N-K, -ONE, A( K, K+1 ), LDA, & B( K, K+1 ), LDB, A( K+1, K+1 ), LDA ) CALL DAXPY( N-K, CT, B( K, K+1 ), LDB, A( K, K+1 ), & LDA ) CALL DTRSV( UPLO, 'Transpose', 'Non-unit', N-K, & B( K+1, K+1 ), LDB, A( K, K+1 ), LDA ) END IF 10 CONTINUE ELSE ! ! Compute inv(L)*A*inv(L') ! DO 20 K = 1, N ! ! Update the lower triangle of A(k:n,k:n) ! AKK = A( K, K ) BKK = B( K, K ) AKK = AKK / BKK**2 A( K, K ) = AKK IF( K.LT.N ) THEN CALL DSCAL( N-K, ONE / BKK, A( K+1, K ), 1 ) CT = -HALF*AKK CALL DAXPY( N-K, CT, B( K+1, K ), 1, A( K+1, K ), 1 ) CALL DSYR2( UPLO, N-K, -ONE, A( K+1, K ), 1, & B( K+1, K ), 1, A( K+1, K+1 ), LDA ) CALL DAXPY( N-K, CT, B( K+1, K ), 1, A( K+1, K ), 1 ) CALL DTRSV( UPLO, 'No transpose', 'Non-unit', N-K, & B( K+1, K+1 ), LDB, A( K+1, K ), 1 ) END IF 20 CONTINUE END IF ELSE IF( UPPER ) THEN ! ! Compute U*A*U' ! DO 30 K = 1, N ! ! Update the upper triangle of A(1:k,1:k) ! AKK = A( K, K ) BKK = B( K, K ) CALL DTRMV( UPLO, 'No transpose', 'Non-unit', K-1, B, & LDB, A( 1, K ), 1 ) CT = HALF*AKK CALL DAXPY( K-1, CT, B( 1, K ), 1, A( 1, K ), 1 ) CALL DSYR2( UPLO, K-1, ONE, A( 1, K ), 1, B( 1, K ), 1, & A, LDA ) CALL DAXPY( K-1, CT, B( 1, K ), 1, A( 1, K ), 1 ) CALL DSCAL( K-1, BKK, A( 1, K ), 1 ) A( K, K ) = AKK*BKK**2 30 CONTINUE ELSE ! ! Compute L'*A*L ! DO 40 K = 1, N ! ! Update the lower triangle of A(1:k,1:k) ! AKK = A( K, K ) BKK = B( K, K ) CALL DTRMV( UPLO, 'Transpose', 'Non-unit', K-1, B, LDB, & A( K, 1 ), LDA ) CT = HALF*AKK CALL DAXPY( K-1, CT, B( K, 1 ), LDB, A( K, 1 ), LDA ) CALL DSYR2( UPLO, K-1, ONE, A( K, 1 ), LDA, B( K, 1 ), & LDB, A, LDA ) CALL DAXPY( K-1, CT, B( K, 1 ), LDB, A( K, 1 ), LDA ) CALL DSCAL( K-1, BKK, A( K, 1 ), LDA ) A( K, K ) = AKK*BKK**2 40 CONTINUE END IF END IF RETURN ! ! End of DSYGS2 ! END SUBROUTINE DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, ITYPE, LDA, LDB, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DSYGST reduces a real symmetric-definite generalized eigenproblem ! to standard form. ! ! If ITYPE = 1, the problem is A*x = lambda*B*x, ! and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T) ! ! If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or ! B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L. ! ! B must have been previously factorized as U**T*U or L*L**T by DPOTRF. ! ! Arguments ! ========= ! ! ITYPE (input) INTEGER ! = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T); ! = 2 or 3: compute U*A*U**T or L**T*A*L. ! ! UPLO (input) CHARACTER ! = 'U': Upper triangle of A is stored and B is factored as ! U**T*U; ! = 'L': Lower triangle of A is stored and B is factored as ! L*L**T. ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the symmetric matrix A. If UPLO = 'U', the leading ! N-by-N upper triangular part of A contains the upper ! triangular part of the matrix A, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading N-by-N lower triangular part of A contains the lower ! triangular part of the matrix A, and the strictly upper ! triangular part of A is not referenced. ! ! On exit, if INFO = 0, the transformed matrix, stored in the ! same format as A. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! B (input) DOUBLE PRECISION array, dimension (LDB,N) ! The triangular factor from the Cholesky factorization of B, ! as returned by DPOTRF. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, HALF PARAMETER ( ONE = 1.0D0, HALF = 0.5D0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER K, KB, NB ! .. ! .. External Subroutines .. EXTERNAL DSYGS2, DSYMM, DSYR2K, DTRMM, DTRSM, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN INFO = -1 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYGST', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Determine the block size for this environment. ! NB = ILAENV( 1, 'DSYGST', UPLO, N, -1, -1, -1 ) ! IF( NB.LE.1 .OR. NB.GE.N ) THEN ! ! Use unblocked code ! CALL DSYGS2( ITYPE, UPLO, N, A, LDA, B, LDB, INFO ) ELSE ! ! Use blocked code ! IF( ITYPE.EQ.1 ) THEN IF( UPPER ) THEN ! ! Compute inv(U')*A*inv(U) ! DO 10 K = 1, N, NB KB = MIN( N-K+1, NB ) ! ! Update the upper triangle of A(k:n,k:n) ! CALL DSYGS2( ITYPE, UPLO, KB, A( K, K ), LDA, & B( K, K ), LDB, INFO ) IF( K+KB.LE.N ) THEN CALL DTRSM( 'Left', UPLO, 'Transpose', 'Non-unit', & KB, N-K-KB+1, ONE, B( K, K ), LDB, & A( K, K+KB ), LDA ) CALL DSYMM( 'Left', UPLO, KB, N-K-KB+1, -HALF, & A( K, K ), LDA, B( K, K+KB ), LDB, ONE, & A( K, K+KB ), LDA ) CALL DSYR2K( UPLO, 'Transpose', N-K-KB+1, KB, -ONE, & A( K, K+KB ), LDA, B( K, K+KB ), LDB, & ONE, A( K+KB, K+KB ), LDA ) CALL DSYMM( 'Left', UPLO, KB, N-K-KB+1, -HALF, & A( K, K ), LDA, B( K, K+KB ), LDB, ONE, & A( K, K+KB ), LDA ) CALL DTRSM( 'Right', UPLO, 'No transpose', & 'Non-unit', KB, N-K-KB+1, ONE, & B( K+KB, K+KB ), LDB, A( K, K+KB ), & LDA ) END IF 10 CONTINUE ELSE ! ! Compute inv(L)*A*inv(L') ! DO 20 K = 1, N, NB KB = MIN( N-K+1, NB ) ! ! Update the lower triangle of A(k:n,k:n) ! CALL DSYGS2( ITYPE, UPLO, KB, A( K, K ), LDA, & B( K, K ), LDB, INFO ) IF( K+KB.LE.N ) THEN CALL DTRSM( 'Right', UPLO, 'Transpose', 'Non-unit', & N-K-KB+1, KB, ONE, B( K, K ), LDB, & A( K+KB, K ), LDA ) CALL DSYMM( 'Right', UPLO, N-K-KB+1, KB, -HALF, & A( K, K ), LDA, B( K+KB, K ), LDB, ONE, & A( K+KB, K ), LDA ) CALL DSYR2K( UPLO, 'No transpose', N-K-KB+1, KB, & -ONE, A( K+KB, K ), LDA, B( K+KB, K ), & LDB, ONE, A( K+KB, K+KB ), LDA ) CALL DSYMM( 'Right', UPLO, N-K-KB+1, KB, -HALF, & A( K, K ), LDA, B( K+KB, K ), LDB, ONE, & A( K+KB, K ), LDA ) CALL DTRSM( 'Left', UPLO, 'No transpose', & 'Non-unit', N-K-KB+1, KB, ONE, & B( K+KB, K+KB ), LDB, A( K+KB, K ), & LDA ) END IF 20 CONTINUE END IF ELSE IF( UPPER ) THEN ! ! Compute U*A*U' ! DO 30 K = 1, N, NB KB = MIN( N-K+1, NB ) ! ! Update the upper triangle of A(1:k+kb-1,1:k+kb-1) ! CALL DTRMM( 'Left', UPLO, 'No transpose', 'Non-unit', & K-1, KB, ONE, B, LDB, A( 1, K ), LDA ) CALL DSYMM( 'Right', UPLO, K-1, KB, HALF, A( K, K ), & LDA, B( 1, K ), LDB, ONE, A( 1, K ), LDA ) CALL DSYR2K( UPLO, 'No transpose', K-1, KB, ONE, & A( 1, K ), LDA, B( 1, K ), LDB, ONE, A, & LDA ) CALL DSYMM( 'Right', UPLO, K-1, KB, HALF, A( K, K ), & LDA, B( 1, K ), LDB, ONE, A( 1, K ), LDA ) CALL DTRMM( 'Right', UPLO, 'Transpose', 'Non-unit', & K-1, KB, ONE, B( K, K ), LDB, A( 1, K ), & LDA ) CALL DSYGS2( ITYPE, UPLO, KB, A( K, K ), LDA, & B( K, K ), LDB, INFO ) 30 CONTINUE ELSE ! ! Compute L'*A*L ! DO 40 K = 1, N, NB KB = MIN( N-K+1, NB ) ! ! Update the lower triangle of A(1:k+kb-1,1:k+kb-1) ! CALL DTRMM( 'Right', UPLO, 'No transpose', 'Non-unit', & KB, K-1, ONE, B, LDB, A( K, 1 ), LDA ) CALL DSYMM( 'Left', UPLO, KB, K-1, HALF, A( K, K ), & LDA, B( K, 1 ), LDB, ONE, A( K, 1 ), LDA ) CALL DSYR2K( UPLO, 'Transpose', K-1, KB, ONE, & A( K, 1 ), LDA, B( K, 1 ), LDB, ONE, A, & LDA ) CALL DSYMM( 'Left', UPLO, KB, K-1, HALF, A( K, K ), & LDA, B( K, 1 ), LDB, ONE, A( K, 1 ), LDA ) CALL DTRMM( 'Left', UPLO, 'Transpose', 'Non-unit', KB, & K-1, ONE, B( K, K ), LDB, A( K, 1 ), LDA ) CALL DSYGS2( ITYPE, UPLO, KB, A( K, K ), LDA, & B( K, K ), LDB, INFO ) 40 CONTINUE END IF END IF END IF RETURN ! ! End of DSYGST ! END SUBROUTINE DSYGV( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, & LWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, ITYPE, LDA, LDB, LWORK, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DSYGV computes all the eigenvalues, and optionally, the eigenvectors ! of a real generalized symmetric-definite eigenproblem, of the form ! A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. ! Here A and B are assumed to be symmetric and B is also ! positive definite. ! ! Arguments ! ========= ! ! ITYPE (input) INTEGER ! Specifies the problem type to be solved: ! = 1: A*x = (lambda)*B*x ! = 2: A*B*x = (lambda)*x ! = 3: B*A*x = (lambda)*x ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangles of A and B are stored; ! = 'L': Lower triangles of A and B are stored. ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the symmetric matrix A. If UPLO = 'U', the ! leading N-by-N upper triangular part of A contains the ! upper triangular part of the matrix A. If UPLO = 'L', ! the leading N-by-N lower triangular part of A contains ! the lower triangular part of the matrix A. ! ! On exit, if JOBZ = 'V', then if INFO = 0, A contains the ! matrix Z of eigenvectors. The eigenvectors are normalized ! as follows: ! if ITYPE = 1 or 2, Z**T*B*Z = I; ! if ITYPE = 3, Z**T*inv(B)*Z = I. ! If JOBZ = 'N', then on exit the upper triangle (if UPLO='U') ! or the lower triangle (if UPLO='L') of A, including the ! diagonal, is destroyed. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB, N) ! On entry, the symmetric positive definite matrix B. ! If UPLO = 'U', the leading N-by-N upper triangular part of B ! contains the upper triangular part of the matrix B. ! If UPLO = 'L', the leading N-by-N lower triangular part of B ! contains the lower triangular part of the matrix B. ! ! On exit, if INFO <= N, the part of B containing the matrix is ! overwritten by the triangular factor U or L from the Cholesky ! factorization B = U**T*U or B = L*L**T. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! W (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, the eigenvalues in ascending order. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The length of the array WORK. LWORK >= max(1,3*N-1). ! For optimal efficiency, LWORK >= (NB+2)*N, ! where NB is the blocksize for DSYTRD returned by ILAENV. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: DPOTRF or DSYEV returned an error code: ! <= N: if INFO = i, DSYEV failed to converge; ! i off-diagonal elements of an intermediate ! tridiagonal form did not converge to zero; ! > N: if INFO = N + i, for 1 <= i <= N, then the leading ! minor of order i of B is not positive definite. ! The factorization of B could not be completed and ! no eigenvalues or eigenvectors were computed. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, UPPER, WANTZ CHARACTER TRANS INTEGER LWKOPT, NB, NEIG ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DPOTRF, DSYEV, DSYGST, DTRMM, DTRSM, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) UPPER = LSAME( UPLO, 'U' ) LQUERY = ( LWORK.EQ.-1 ) ! INFO = 0 IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN INFO = -1 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LWORK.LT.MAX( 1, 3*N-1 ) .AND. .NOT.LQUERY ) THEN INFO = -11 END IF ! IF( INFO.EQ.0 ) THEN NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 ) LWKOPT = ( NB+2 )*N WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYGV ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Form a Cholesky factorization of B. ! CALL DPOTRF( UPLO, N, B, LDB, INFO ) IF( INFO.NE.0 ) THEN INFO = N + INFO RETURN END IF ! ! Transform problem to standard eigenvalue problem and solve. ! CALL DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO ) CALL DSYEV( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, INFO ) ! IF( WANTZ ) THEN ! ! Backtransform eigenvectors to the original problem. ! NEIG = N IF( INFO.GT.0 ) & NEIG = INFO - 1 IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN ! ! For A*x=(lambda)*B*x and A*B*x=(lambda)*x; ! backtransform eigenvectors: x = inv(L)'*y or inv(U)*y ! IF( UPPER ) THEN TRANS = 'N' ELSE TRANS = 'T' END IF ! CALL DTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG, ONE, & B, LDB, A, LDA ) ! ELSE IF( ITYPE.EQ.3 ) THEN ! ! For B*A*x=(lambda)*x; ! backtransform eigenvectors: x = L*y or U'*y ! IF( UPPER ) THEN TRANS = 'T' ELSE TRANS = 'N' END IF ! CALL DTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG, ONE, & B, LDB, A, LDA ) END IF END IF ! WORK( 1 ) = LWKOPT RETURN ! ! End of DSYGV ! END SUBROUTINE DSYGVD( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, & LWORK, IWORK, LIWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, ITYPE, LDA, LDB, LIWORK, LWORK, N ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DSYGVD computes all the eigenvalues, and optionally, the eigenvectors ! of a real generalized symmetric-definite eigenproblem, of the form ! A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and ! B are assumed to be symmetric and B is also positive definite. ! If eigenvectors are desired, it uses a divide and conquer algorithm. ! ! The divide and conquer algorithm makes very mild assumptions about ! floating point arithmetic. It will work on machines with a guard ! digit in add/subtract, or on those binary machines without guard ! digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or ! Cray-2. It could conceivably fail on hexadecimal or decimal machines ! without guard digits, but we know of none. ! ! Arguments ! ========= ! ! ITYPE (input) INTEGER ! Specifies the problem type to be solved: ! = 1: A*x = (lambda)*B*x ! = 2: A*B*x = (lambda)*x ! = 3: B*A*x = (lambda)*x ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangles of A and B are stored; ! = 'L': Lower triangles of A and B are stored. ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the symmetric matrix A. If UPLO = 'U', the ! leading N-by-N upper triangular part of A contains the ! upper triangular part of the matrix A. If UPLO = 'L', ! the leading N-by-N lower triangular part of A contains ! the lower triangular part of the matrix A. ! ! On exit, if JOBZ = 'V', then if INFO = 0, A contains the ! matrix Z of eigenvectors. The eigenvectors are normalized ! as follows: ! if ITYPE = 1 or 2, Z**T*B*Z = I; ! if ITYPE = 3, Z**T*inv(B)*Z = I. ! If JOBZ = 'N', then on exit the upper triangle (if UPLO='U') ! or the lower triangle (if UPLO='L') of A, including the ! diagonal, is destroyed. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB, N) ! On entry, the symmetric matrix B. If UPLO = 'U', the ! leading N-by-N upper triangular part of B contains the ! upper triangular part of the matrix B. If UPLO = 'L', ! the leading N-by-N lower triangular part of B contains ! the lower triangular part of the matrix B. ! ! On exit, if INFO <= N, the part of B containing the matrix is ! overwritten by the triangular factor U or L from the Cholesky ! factorization B = U**T*U or B = L*L**T. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! W (output) DOUBLE PRECISION array, dimension (N) ! If INFO = 0, the eigenvalues in ascending order. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! If N <= 1, LWORK >= 1. ! If JOBZ = 'N' and N > 1, LWORK >= 2*N+1. ! If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace/output) INTEGER array, dimension (LIWORK) ! On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. ! ! LIWORK (input) INTEGER ! The dimension of the array IWORK. ! If N <= 1, LIWORK >= 1. ! If JOBZ = 'N' and N > 1, LIWORK >= 1. ! If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N. ! ! If LIWORK = -1, then a workspace query is assumed; the ! routine only calculates the optimal size of the IWORK array, ! returns this value as the first entry of the IWORK array, and ! no error message related to LIWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: DPOTRF or DSYEVD returned an error code: ! <= N: if INFO = i, DSYEVD failed to converge; ! i off-diagonal elements of an intermediate ! tridiagonal form did not converge to zero; ! > N: if INFO = N + i, for 1 <= i <= N, then the leading ! minor of order i of B is not positive definite. ! The factorization of B could not be completed and ! no eigenvalues or eigenvectors were computed. ! ! Further Details ! =============== ! ! Based on contributions by ! Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, UPPER, WANTZ CHARACTER TRANS INTEGER LIOPT, LIWMIN, LOPT, LWMIN, NEIG ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DPOTRF, DSYEVD, DSYGST, DTRMM, DTRSM, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC DBLE, MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! WANTZ = LSAME( JOBZ, 'V' ) UPPER = LSAME( UPLO, 'U' ) LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) ! INFO = 0 IF( N.LE.1 ) THEN LIWMIN = 1 LWMIN = 1 LOPT = LWMIN LIOPT = LIWMIN ELSE IF( WANTZ ) THEN LIWMIN = 3 + 5*N LWMIN = 1 + 6*N + 2*N**2 ELSE LIWMIN = 1 LWMIN = 2*N + 1 END IF LOPT = LWMIN LIOPT = LIWMIN END IF IF( ITYPE.LT.0 .OR. ITYPE.GT.3 ) THEN INFO = -1 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -11 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -13 END IF ! IF( INFO.EQ.0 ) THEN WORK( 1 ) = LOPT IWORK( 1 ) = LIOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYGVD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Form a Cholesky factorization of B. ! CALL DPOTRF( UPLO, N, B, LDB, INFO ) IF( INFO.NE.0 ) THEN INFO = N + INFO RETURN END IF ! ! Transform problem to standard eigenvalue problem and solve. ! CALL DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO ) CALL DSYEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK, LIWORK, & INFO ) LOPT = MAX( DBLE( LOPT ), DBLE( WORK( 1 ) ) ) LIOPT = MAX( DBLE( LIOPT ), DBLE( IWORK( 1 ) ) ) ! IF( WANTZ ) THEN ! ! Backtransform eigenvectors to the original problem. ! NEIG = N IF( INFO.GT.0 ) & NEIG = INFO - 1 IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN ! ! For A*x=(lambda)*B*x and A*B*x=(lambda)*x; ! backtransform eigenvectors: x = inv(L)'*y or inv(U)*y ! IF( UPPER ) THEN TRANS = 'N' ELSE TRANS = 'T' END IF ! CALL DTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG, ONE, & B, LDB, A, LDA ) ! ELSE IF( ITYPE.EQ.3 ) THEN ! ! For B*A*x=(lambda)*x; ! backtransform eigenvectors: x = L*y or U'*y ! IF( UPPER ) THEN TRANS = 'T' ELSE TRANS = 'N' END IF ! CALL DTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, NEIG, ONE, & B, LDB, A, LDA ) END IF END IF ! WORK( 1 ) = LOPT IWORK( 1 ) = LIOPT ! RETURN ! ! End of DSYGVD ! END SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, & VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, & LWORK, IWORK, IFAIL, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBZ, RANGE, UPLO INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N DOUBLE PRECISION ABSTOL, VL, VU ! .. ! .. Array Arguments .. INTEGER IFAIL( * ), IWORK( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * ), & Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DSYGVX computes selected eigenvalues, and optionally, eigenvectors ! of a real generalized symmetric-definite eigenproblem, of the form ! A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A ! and B are assumed to be symmetric and B is also positive definite. ! Eigenvalues and eigenvectors can be selected by specifying either a ! range of values or a range of indices for the desired eigenvalues. ! ! Arguments ! ========= ! ! ITYPE (input) INTEGER ! Specifies the problem type to be solved: ! = 1: A*x = (lambda)*B*x ! = 2: A*B*x = (lambda)*x ! = 3: B*A*x = (lambda)*x ! ! JOBZ (input) CHARACTER*1 ! = 'N': Compute eigenvalues only; ! = 'V': Compute eigenvalues and eigenvectors. ! ! RANGE (input) CHARACTER*1 ! = 'A': all eigenvalues will be found. ! = 'V': all eigenvalues in the half-open interval (VL,VU] ! will be found. ! = 'I': the IL-th through IU-th eigenvalues will be found. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A and B are stored; ! = 'L': Lower triangle of A and B are stored. ! ! N (input) INTEGER ! The order of the matrix pencil (A,B). N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the symmetric matrix A. If UPLO = 'U', the ! leading N-by-N upper triangular part of A contains the ! upper triangular part of the matrix A. If UPLO = 'L', ! the leading N-by-N lower triangular part of A contains ! the lower triangular part of the matrix A. ! ! On exit, the lower triangle (if UPLO='L') or the upper ! triangle (if UPLO='U') of A, including the diagonal, is ! destroyed. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDA, N) ! On entry, the symmetric matrix B. If UPLO = 'U', the ! leading N-by-N upper triangular part of B contains the ! upper triangular part of the matrix B. If UPLO = 'L', ! the leading N-by-N lower triangular part of B contains ! the lower triangular part of the matrix B. ! ! On exit, if INFO <= N, the part of B containing the matrix is ! overwritten by the triangular factor U or L from the Cholesky ! factorization B = U**T*U or B = L*L**T. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! VL (input) DOUBLE PRECISION ! VU (input) DOUBLE PRECISION ! If RANGE='V', the lower and upper bounds of the interval to ! be searched for eigenvalues. VL < VU. ! Not referenced if RANGE = 'A' or 'I'. ! ! IL (input) INTEGER ! IU (input) INTEGER ! If RANGE='I', the indices (in ascending order) of the ! smallest and largest eigenvalues to be returned. ! 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. ! Not referenced if RANGE = 'A' or 'V'. ! ! ABSTOL (input) DOUBLE PRECISION ! The absolute error tolerance for the eigenvalues. ! An approximate eigenvalue is accepted as converged ! when it is determined to lie in an interval [a,b] ! of width less than or equal to ! ! ABSTOL + EPS * max( |a|,|b| ) , ! ! where EPS is the machine precision. If ABSTOL is less than ! or equal to zero, then EPS*|T| will be used in its place, ! where |T| is the 1-norm of the tridiagonal matrix obtained ! by reducing A to tridiagonal form. ! ! Eigenvalues will be computed most accurately when ABSTOL is ! set to twice the underflow threshold 2*DLAMCH('S'), not zero. ! If this routine returns with INFO>0, indicating that some ! eigenvectors did not converge, try setting ABSTOL to ! 2*DLAMCH('S'). ! ! M (output) INTEGER ! The total number of eigenvalues found. 0 <= M <= N. ! If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. ! ! W (output) DOUBLE PRECISION array, dimension (N) ! On normal exit, the first M elements contain the selected ! eigenvalues in ascending order. ! ! Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M)) ! If JOBZ = 'N', then Z is not referenced. ! If JOBZ = 'V', then if INFO = 0, the first M columns of Z ! contain the orthonormal eigenvectors of the matrix A ! corresponding to the selected eigenvalues, with the i-th ! column of Z holding the eigenvector associated with W(i). ! The eigenvectors are normalized as follows: ! if ITYPE = 1 or 2, Z**T*B*Z = I; ! if ITYPE = 3, Z**T*inv(B)*Z = I. ! ! If an eigenvector fails to converge, then that column of Z ! contains the latest approximation to the eigenvector, and the ! index of the eigenvector is returned in IFAIL. ! Note: the user must ensure that at least max(1,M) columns are ! supplied in the array Z; if RANGE = 'V', the exact value of M ! is not known in advance and an upper bound must be used. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1, and if ! JOBZ = 'V', LDZ >= max(1,N). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The length of the array WORK. LWORK >= max(1,8*N). ! For optimal efficiency, LWORK >= (NB+3)*N, ! where NB is the blocksize for DSYTRD returned by ILAENV. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace) INTEGER array, dimension (5*N) ! ! IFAIL (output) INTEGER array, dimension (N) ! If JOBZ = 'V', then if INFO = 0, the first M elements of ! IFAIL are zero. If INFO > 0, then IFAIL contains the ! indices of the eigenvectors that failed to converge. ! If JOBZ = 'N', then IFAIL is not referenced. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: DPOTRF or DSYEVX returned an error code: ! <= N: if INFO = i, DSYEVX failed to converge; ! i eigenvectors failed to converge. Their indices ! are stored in array IFAIL. ! > N: if INFO = N + i, for 1 <= i <= N, then the leading ! minor of order i of B is not positive definite. ! The factorization of B could not be completed and ! no eigenvalues or eigenvectors were computed. ! ! Further Details ! =============== ! ! Based on contributions by ! Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ CHARACTER TRANS INTEGER LOPT, LWKOPT, NB ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DPOTRF, DSYEVX, DSYGST, DTRMM, DTRSM, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! UPPER = LSAME( UPLO, 'U' ) WANTZ = LSAME( JOBZ, 'V' ) ALLEIG = LSAME( RANGE, 'A' ) VALEIG = LSAME( RANGE, 'V' ) INDEIG = LSAME( RANGE, 'I' ) LQUERY = ( LWORK.EQ.-1 ) ! INFO = 0 IF( ITYPE.LT.0 .OR. ITYPE.GT.3 ) THEN INFO = -1 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN INFO = -3 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( VALEIG .AND. N.GT.0 ) THEN IF( VU.LE.VL ) & INFO = -11 ELSE IF( INDEIG .AND. IL.LT.1 ) THEN INFO = -12 ELSE IF( INDEIG .AND. ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN INFO = -13 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -18 ELSE IF( LWORK.LT.MAX( 1, 8*N ) .AND. .NOT.LQUERY ) THEN INFO = -20 END IF ! IF( INFO.EQ.0 ) THEN NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 ) LWKOPT = ( NB+3 )*N WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYGVX', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! M = 0 IF( N.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! ! Form a Cholesky factorization of B. ! CALL DPOTRF( UPLO, N, B, LDB, INFO ) IF( INFO.NE.0 ) THEN INFO = N + INFO RETURN END IF ! ! Transform problem to standard eigenvalue problem and solve. ! CALL DSYGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO ) CALL DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, & M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO ) LOPT = WORK( 1 ) ! IF( WANTZ ) THEN ! ! Backtransform eigenvectors to the original problem. ! IF( INFO.GT.0 ) & M = INFO - 1 IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN ! ! For A*x=(lambda)*B*x and A*B*x=(lambda)*x; ! backtransform eigenvectors: x = inv(L)'*y or inv(U)*y ! IF( UPPER ) THEN TRANS = 'N' ELSE TRANS = 'T' END IF ! CALL DTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, M, ONE, B, & LDB, Z, LDZ ) ! ELSE IF( ITYPE.EQ.3 ) THEN ! ! For B*A*x=(lambda)*x; ! backtransform eigenvectors: x = L*y or U'*y ! IF( UPPER ) THEN TRANS = 'T' ELSE TRANS = 'N' END IF ! CALL DTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, M, ONE, B, & LDB, Z, LDZ ) END IF END IF ! ! Set WORK(1) to optimal workspace size. ! WORK( 1 ) = LWKOPT ! RETURN ! ! End of DSYGVX ! END SUBROUTINE DSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, & X, LDX, FERR, BERR, WORK, IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS ! .. ! .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), & BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DSYRFS improves the computed solution to a system of linear ! equations when the coefficient matrix is symmetric indefinite, and ! provides error bounds and backward error estimates for the solution. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The symmetric matrix A. If UPLO = 'U', the leading N-by-N ! upper triangular part of A contains the upper triangular part ! of the matrix A, and the strictly lower triangular part of A ! is not referenced. If UPLO = 'L', the leading N-by-N lower ! triangular part of A contains the lower triangular part of ! the matrix A, and the strictly upper triangular part of A is ! not referenced. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! AF (input) DOUBLE PRECISION array, dimension (LDAF,N) ! The factored form of the matrix A. AF contains the block ! diagonal matrix D and the multipliers used to obtain the ! factor U or L from the factorization A = U*D*U**T or ! A = L*D*L**T as computed by DSYTRF. ! ! LDAF (input) INTEGER ! The leading dimension of the array AF. LDAF >= max(1,N). ! ! IPIV (input) INTEGER array, dimension (N) ! Details of the interchanges and the block structure of D ! as determined by DSYTRF. ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) ! The right hand side matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! On entry, the solution matrix X, as computed by DSYTRS. ! On exit, the improved solution matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Internal Parameters ! =================== ! ! ITMAX is the maximum number of steps of iterative refinement. ! ! ===================================================================== ! ! .. Parameters .. INTEGER ITMAX PARAMETER ( ITMAX = 5 ) DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D+0 ) DOUBLE PRECISION THREE PARAMETER ( THREE = 3.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER COUNT, I, J, K, KASE, NZ DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DLACON, DSYMV, DSYTRS, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -10 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -12 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYRFS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN DO 10 J = 1, NRHS FERR( J ) = ZERO BERR( J ) = ZERO 10 CONTINUE RETURN END IF ! ! NZ = maximum number of nonzero elements in each row of A, plus 1 ! NZ = N + 1 EPS = DLAMCH( 'Epsilon' ) SAFMIN = DLAMCH( 'Safe minimum' ) SAFE1 = NZ*SAFMIN SAFE2 = SAFE1 / EPS ! ! Do for each right hand side ! DO 140 J = 1, NRHS ! COUNT = 1 LSTRES = THREE 20 CONTINUE ! ! Loop until stopping criterion is satisfied. ! ! Compute residual R = B - A * X ! CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, & WORK( N+1 ), 1 ) ! ! Compute componentwise relative backward error from formula ! ! max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) ! ! where abs(Z) is the componentwise absolute value of the matrix ! or vector Z. If the i-th component of the denominator is less ! than SAFE2, then SAFE1 is added to the i-th components of the ! numerator and denominator before dividing. ! DO 30 I = 1, N WORK( I ) = ABS( B( I, J ) ) 30 CONTINUE ! ! Compute abs(A)*abs(X) + abs(B). ! IF( UPPER ) THEN DO 50 K = 1, N S = ZERO XK = ABS( X( K, J ) ) DO 40 I = 1, K - 1 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 40 CONTINUE WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S 50 CONTINUE ELSE DO 70 K = 1, N S = ZERO XK = ABS( X( K, J ) ) WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK DO 60 I = K + 1, N WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 60 CONTINUE WORK( K ) = WORK( K ) + S 70 CONTINUE END IF S = ZERO DO 80 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) ELSE S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / & ( WORK( I )+SAFE1 ) ) END IF 80 CONTINUE BERR( J ) = S ! ! Test stopping criterion. Continue iterating if ! 1) The residual BERR(J) is larger than machine epsilon, and ! 2) BERR(J) decreased by at least a factor of 2 during the ! last iteration, and ! 3) At most ITMAX iterations tried. ! IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. & COUNT.LE.ITMAX ) THEN ! ! Update solution and try again. ! CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N, & INFO ) CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) LSTRES = BERR( J ) COUNT = COUNT + 1 GO TO 20 END IF ! ! Bound error from formula ! ! norm(X - XTRUE) / norm(X) .le. FERR = ! norm( abs(inv(A))* ! ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) ! ! where ! norm(Z) is the magnitude of the largest component of Z ! inv(A) is the inverse of A ! abs(Z) is the componentwise absolute value of the matrix or ! vector Z ! NZ is the maximum number of nonzeros in any row of A, plus 1 ! EPS is machine epsilon ! ! The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) ! is incremented by SAFE1 if the i-th component of ! abs(A)*abs(X) + abs(B) is less than SAFE2. ! ! Use DLACON to estimate the infinity-norm of the matrix ! inv(A) * diag(W), ! where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) ! DO 90 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) ELSE WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 END IF 90 CONTINUE ! KASE = 0 100 CONTINUE CALL DLACON( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), & KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN ! ! Multiply by diag(W)*inv(A'). ! CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N, & INFO ) DO 110 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 110 CONTINUE ELSE IF( KASE.EQ.2 ) THEN ! ! Multiply by inv(A)*diag(W). ! DO 120 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 120 CONTINUE CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N, & INFO ) END IF GO TO 100 END IF ! ! Normalize error. ! LSTRES = ZERO DO 130 I = 1, N LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 130 CONTINUE IF( LSTRES.NE.ZERO ) & FERR( J ) = FERR( J ) / LSTRES ! 140 CONTINUE ! RETURN ! ! End of DSYRFS ! END SUBROUTINE DSYSV( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, & LWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, LDB, LWORK, N, NRHS ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DSYSV computes the solution to a real system of linear equations ! A * X = B, ! where A is an N-by-N symmetric matrix and X and B are N-by-NRHS ! matrices. ! ! The diagonal pivoting method is used to factor A as ! A = U * D * U**T, if UPLO = 'U', or ! A = L * D * L**T, if UPLO = 'L', ! where U (or L) is a product of permutation and unit upper (lower) ! triangular matrices, and D is symmetric and block diagonal with ! 1-by-1 and 2-by-2 diagonal blocks. The factored form of A is then ! used to solve the system of equations A * X = B. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The number of linear equations, i.e., the order of the ! matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the symmetric matrix A. If UPLO = 'U', the leading ! N-by-N upper triangular part of A contains the upper ! triangular part of the matrix A, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading N-by-N lower triangular part of A contains the lower ! triangular part of the matrix A, and the strictly upper ! triangular part of A is not referenced. ! ! On exit, if INFO = 0, the block diagonal matrix D and the ! multipliers used to obtain the factor U or L from the ! factorization A = U*D*U**T or A = L*D*L**T as computed by ! DSYTRF. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! IPIV (output) INTEGER array, dimension (N) ! Details of the interchanges and the block structure of D, as ! determined by DSYTRF. If IPIV(k) > 0, then rows and columns ! k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1 ! diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, ! then rows and columns k-1 and -IPIV(k) were interchanged and ! D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and ! IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and ! -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 ! diagonal block. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the N-by-NRHS right hand side matrix B. ! On exit, if INFO = 0, the N-by-NRHS solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The length of WORK. LWORK >= 1, and for best performance ! LWORK >= N*NB, where NB is the optimal blocksize for ! DSYTRF. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, D(i,i) is exactly zero. The factorization ! has been completed, but the block diagonal matrix D is ! exactly singular, so the solution could not be computed. ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL LQUERY INTEGER LWKOPT, NB ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DSYTRF, DSYTRS, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 LQUERY = ( LWORK.EQ.-1 ) IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN INFO = -10 END IF ! IF( INFO.EQ.0 ) THEN NB = ILAENV( 1, 'DSYTRF', UPLO, N, -1, -1, -1 ) LWKOPT = N*NB WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYSV ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Compute the factorization A = U*D*U' or A = L*D*L'. ! CALL DSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO ) IF( INFO.EQ.0 ) THEN ! ! Solve the system A*X = B, overwriting B with X. ! CALL DSYTRS( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO ) ! END IF ! WORK( 1 ) = LWKOPT ! RETURN ! ! End of DSYSV ! END SUBROUTINE DSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, & LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK, & IWORK, INFO ) ! ! -- LAPACK driver routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER FACT, UPLO INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), & BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DSYSVX uses the diagonal pivoting factorization to compute the ! solution to a real system of linear equations A * X = B, ! where A is an N-by-N symmetric matrix and X and B are N-by-NRHS ! matrices. ! ! Error bounds on the solution and a condition estimate are also ! provided. ! ! Description ! =========== ! ! The following steps are performed: ! ! 1. If FACT = 'N', the diagonal pivoting method is used to factor A. ! The form of the factorization is ! A = U * D * U**T, if UPLO = 'U', or ! A = L * D * L**T, if UPLO = 'L', ! where U (or L) is a product of permutation and unit upper (lower) ! triangular matrices, and D is symmetric and block diagonal with ! 1-by-1 and 2-by-2 diagonal blocks. ! ! 2. If some D(i,i)=0, so that D is exactly singular, then the routine ! returns with INFO = i. Otherwise, the factored form of A is used ! to estimate the condition number of the matrix A. If the ! reciprocal of the condition number is less than machine precision, ! INFO = N+1 is returned as a warning, but the routine still goes on ! to solve for X and compute error bounds as described below. ! ! 3. The system of equations is solved for X using the factored form ! of A. ! ! 4. Iterative refinement is applied to improve the computed solution ! matrix and calculate error bounds and backward error estimates ! for it. ! ! Arguments ! ========= ! ! FACT (input) CHARACTER*1 ! Specifies whether or not the factored form of A has been ! supplied on entry. ! = 'F': On entry, AF and IPIV contain the factored form of ! A. AF and IPIV will not be modified. ! = 'N': The matrix A will be copied to AF and factored. ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The number of linear equations, i.e., the order of the ! matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The symmetric matrix A. If UPLO = 'U', the leading N-by-N ! upper triangular part of A contains the upper triangular part ! of the matrix A, and the strictly lower triangular part of A ! is not referenced. If UPLO = 'L', the leading N-by-N lower ! triangular part of A contains the lower triangular part of ! the matrix A, and the strictly upper triangular part of A is ! not referenced. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N) ! If FACT = 'F', then AF is an input argument and on entry ! contains the block diagonal matrix D and the multipliers used ! to obtain the factor U or L from the factorization ! A = U*D*U**T or A = L*D*L**T as computed by DSYTRF. ! ! If FACT = 'N', then AF is an output argument and on exit ! returns the block diagonal matrix D and the multipliers used ! to obtain the factor U or L from the factorization ! A = U*D*U**T or A = L*D*L**T. ! ! LDAF (input) INTEGER ! The leading dimension of the array AF. LDAF >= max(1,N). ! ! IPIV (input or output) INTEGER array, dimension (N) ! If FACT = 'F', then IPIV is an input argument and on entry ! contains details of the interchanges and the block structure ! of D, as determined by DSYTRF. ! If IPIV(k) > 0, then rows and columns k and IPIV(k) were ! interchanged and D(k,k) is a 1-by-1 diagonal block. ! If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and ! columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) ! is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = ! IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were ! interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. ! ! If FACT = 'N', then IPIV is an output argument and on exit ! contains details of the interchanges and the block structure ! of D, as determined by DSYTRF. ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) ! The N-by-NRHS right hand side matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) ! If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! RCOND (output) DOUBLE PRECISION ! The estimate of the reciprocal condition number of the matrix ! A. If RCOND is less than the machine precision (in ! particular, if RCOND = 0), the matrix is singular to working ! precision. This condition is indicated by a return code of ! INFO > 0. ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The length of WORK. LWORK >= 3*N, and for best performance ! LWORK >= N*NB, where NB is the optimal blocksize for ! DSYTRF. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, and i is ! <= N: D(i,i) is exactly zero. The factorization ! has been completed but the factor D is exactly ! singular, so the solution and error bounds could ! not be computed. RCOND = 0 is returned. ! = N+1: D is nonsingular, but RCOND is less than machine ! precision, meaning that the matrix is singular ! to working precision. Nevertheless, the ! solution and error bounds are computed because ! there are a number of situations where the ! computed solution can be more accurate than the ! value of RCOND would suggest. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, NOFACT INTEGER LWKOPT, NB DOUBLE PRECISION ANORM ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANSY EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY ! .. ! .. External Subroutines .. EXTERNAL DLACPY, DSYCON, DSYRFS, DSYTRF, DSYTRS, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 NOFACT = LSAME( FACT, 'N' ) LQUERY = ( LWORK.EQ.-1 ) IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN INFO = -1 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) & THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -11 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -13 ELSE IF( LWORK.LT.MAX( 1, 3*N ) .AND. .NOT.LQUERY ) THEN INFO = -18 END IF ! IF( INFO.EQ.0 ) THEN NB = ILAENV( 1, 'DSYTRF', UPLO, N, -1, -1, -1 ) LWKOPT = N*NB WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYSVX', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! IF( NOFACT ) THEN ! ! Compute the factorization A = U*D*U' or A = L*D*L'. ! CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF ) CALL DSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, LWORK, INFO ) ! ! Return if INFO is non-zero. ! IF( INFO.NE.0 ) THEN IF( INFO.GT.0 ) & RCOND = ZERO RETURN END IF END IF ! ! Compute the norm of the matrix A. ! ANORM = DLANSY( 'I', UPLO, N, A, LDA, WORK ) ! ! Compute the reciprocal of the condition number of A. ! CALL DSYCON( UPLO, N, AF, LDAF, IPIV, ANORM, RCOND, WORK, IWORK, & INFO ) ! ! Set INFO = N+1 if the matrix is singular to working precision. ! IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) & INFO = N + 1 ! ! Compute the solution vectors X. ! CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL DSYTRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO ) ! ! Use iterative refinement to improve the computed solutions and ! compute error bounds and backward error estimates for them. ! CALL DSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, & LDX, FERR, BERR, WORK, IWORK, INFO ) ! RETURN ! ! End of DSYSVX ! END SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ) ! .. ! ! Purpose ! ======= ! ! DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal ! form T by an orthogonal similarity transformation: Q' * A * Q = T. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the upper or lower triangular part of the ! symmetric matrix A is stored: ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the symmetric matrix A. If UPLO = 'U', the leading ! n-by-n upper triangular part of A contains the upper ! triangular part of the matrix A, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading n-by-n lower triangular part of A contains the lower ! triangular part of the matrix A, and the strictly upper ! triangular part of A is not referenced. ! On exit, if UPLO = 'U', the diagonal and first superdiagonal ! of A are overwritten by the corresponding elements of the ! tridiagonal matrix T, and the elements above the first ! superdiagonal, with the array TAU, represent the orthogonal ! matrix Q as a product of elementary reflectors; if UPLO ! = 'L', the diagonal and first subdiagonal of A are over- ! written by the corresponding elements of the tridiagonal ! matrix T, and the elements below the first subdiagonal, with ! the array TAU, represent the orthogonal matrix Q as a product ! of elementary reflectors. See Further Details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! D (output) DOUBLE PRECISION array, dimension (N) ! The diagonal elements of the tridiagonal matrix T: ! D(i) = A(i,i). ! ! E (output) DOUBLE PRECISION array, dimension (N-1) ! The off-diagonal elements of the tridiagonal matrix T: ! E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. ! ! TAU (output) DOUBLE PRECISION array, dimension (N-1) ! The scalar factors of the elementary reflectors (see Further ! Details). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! If UPLO = 'U', the matrix Q is represented as a product of elementary ! reflectors ! ! Q = H(n-1) . . . H(2) H(1). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in ! A(1:i-1,i+1), and tau in TAU(i). ! ! If UPLO = 'L', the matrix Q is represented as a product of elementary ! reflectors ! ! Q = H(1) H(2) . . . H(n-1). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), ! and tau in TAU(i). ! ! The contents of A on exit are illustrated by the following examples ! with n = 5: ! ! if UPLO = 'U': if UPLO = 'L': ! ! ( d e v2 v3 v4 ) ( d ) ! ( d e v3 v4 ) ( e d ) ! ( d e v4 ) ( v1 e d ) ! ( d e ) ( v1 v2 e d ) ! ( d ) ( v1 v2 v3 e d ) ! ! where d and e denote diagonal and off-diagonal elements of T, and vi ! denotes an element of the vector defining H(i). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO, HALF PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0, & HALF = 1.0D0 / 2.0D0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER I DOUBLE PRECISION ALPHA, TAUI ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DLARFG, DSYMV, DSYR2, XERBLA ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT EXTERNAL LSAME, DDOT ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYTD2', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.LE.0 ) & RETURN ! IF( UPPER ) THEN ! ! Reduce the upper triangle of A ! DO 10 I = N - 1, 1, -1 ! ! Generate elementary reflector H(i) = I - tau * v * v' ! to annihilate A(1:i-1,i+1) ! CALL DLARFG( I, A( I, I+1 ), A( 1, I+1 ), 1, TAUI ) E( I ) = A( I, I+1 ) ! IF( TAUI.NE.ZERO ) THEN ! ! Apply H(i) from both sides to A(1:i,1:i) ! A( I, I+1 ) = ONE ! ! Compute x := tau * A * v storing x in TAU(1:i) ! CALL DSYMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO, & TAU, 1 ) ! ! Compute w := x - 1/2 * tau * (x'*v) * v ! ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, A( 1, I+1 ), 1 ) CALL DAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 ) ! ! Apply the transformation as a rank-2 update: ! A := A - v * w' - w * v' ! CALL DSYR2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A, & LDA ) ! A( I, I+1 ) = E( I ) END IF D( I+1 ) = A( I+1, I+1 ) TAU( I ) = TAUI 10 CONTINUE D( 1 ) = A( 1, 1 ) ELSE ! ! Reduce the lower triangle of A ! DO 20 I = 1, N - 1 ! ! Generate elementary reflector H(i) = I - tau * v * v' ! to annihilate A(i+2:n,i) ! CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1, & TAUI ) E( I ) = A( I+1, I ) ! IF( TAUI.NE.ZERO ) THEN ! ! Apply H(i) from both sides to A(i+1:n,i+1:n) ! A( I+1, I ) = ONE ! ! Compute x := tau * A * v storing y in TAU(i:n-1) ! CALL DSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA, & A( I+1, I ), 1, ZERO, TAU( I ), 1 ) ! ! Compute w := x - 1/2 * tau * (x'*v) * v ! ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, A( I+1, I ), & 1 ) CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 ) ! ! Apply the transformation as a rank-2 update: ! A := A - v * w' - w * v' ! CALL DSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1, & A( I+1, I+1 ), LDA ) ! A( I+1, I ) = E( I ) END IF D( I ) = A( I, I ) TAU( I ) = TAUI 20 CONTINUE D( N ) = A( N, N ) END IF ! RETURN ! ! End of DSYTD2 ! END SUBROUTINE DSYTF2( UPLO, N, A, LDA, IPIV, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION A( LDA, * ) ! .. ! ! Purpose ! ======= ! ! DSYTF2 computes the factorization of a real symmetric matrix A using ! the Bunch-Kaufman diagonal pivoting method: ! ! A = U*D*U' or A = L*D*L' ! ! where U (or L) is a product of permutation and unit upper (lower) ! triangular matrices, U' is the transpose of U, and D is symmetric and ! block diagonal with 1-by-1 and 2-by-2 diagonal blocks. ! ! This is the unblocked version of the algorithm, calling Level 2 BLAS. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the upper or lower triangular part of the ! symmetric matrix A is stored: ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the symmetric matrix A. If UPLO = 'U', the leading ! n-by-n upper triangular part of A contains the upper ! triangular part of the matrix A, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading n-by-n lower triangular part of A contains the lower ! triangular part of the matrix A, and the strictly upper ! triangular part of A is not referenced. ! ! On exit, the block diagonal matrix D and the multipliers used ! to obtain the factor U or L (see below for further details). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! IPIV (output) INTEGER array, dimension (N) ! Details of the interchanges and the block structure of D. ! If IPIV(k) > 0, then rows and columns k and IPIV(k) were ! interchanged and D(k,k) is a 1-by-1 diagonal block. ! If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and ! columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) ! is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = ! IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were ! interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -k, the k-th argument had an illegal value ! > 0: if INFO = k, D(k,k) is exactly zero. The factorization ! has been completed, but the block diagonal matrix D is ! exactly singular, and division by zero will occur if it ! is used to solve a system of equations. ! ! Further Details ! =============== ! ! 1-96 - Based on modifications by J. Lewis, Boeing Computer Services ! Company ! ! If UPLO = 'U', then A = U*D*U', where ! U = P(n)*U(n)* ... *P(k)U(k)* ..., ! i.e., U is a product of terms P(k)*U(k), where k decreases from n to ! 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 ! and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as ! defined by IPIV(k), and U(k) is a unit upper triangular matrix, such ! that if the diagonal block D(k) is of order s (s = 1 or 2), then ! ! ( I v 0 ) k-s ! U(k) = ( 0 I 0 ) s ! ( 0 0 I ) n-k ! k-s s n-k ! ! If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). ! If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), ! and A(k,k), and v overwrites A(1:k-2,k-1:k). ! ! If UPLO = 'L', then A = L*D*L', where ! L = P(1)*L(1)* ... *P(k)*L(k)* ..., ! i.e., L is a product of terms P(k)*L(k), where k increases from 1 to ! n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 ! and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as ! defined by IPIV(k), and L(k) is a unit lower triangular matrix, such ! that if the diagonal block D(k) is of order s (s = 1 or 2), then ! ! ( I 0 0 ) k-1 ! L(k) = ( 0 I 0 ) s ! ( 0 v I ) n-k-s+1 ! k-1 s n-k-s+1 ! ! If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). ! If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), ! and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) DOUBLE PRECISION EIGHT, SEVTEN PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER I, IMAX, J, JMAX, K, KK, KP, KSTEP DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1, & ROWMAX, T, WK, WKM1, WKP1 ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX EXTERNAL LSAME, IDAMAX ! .. ! .. External Subroutines .. EXTERNAL DSCAL, DSWAP, DSYR, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYTF2', -INFO ) RETURN END IF ! ! Initialize ALPHA for use in choosing pivot block size. ! ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT ! IF( UPPER ) THEN ! ! Factorize A as U*D*U' using the upper triangle of A ! ! K is the main loop index, decreasing from N to 1 in steps of ! 1 or 2 ! K = N 10 CONTINUE ! ! If K < 1, exit from loop ! IF( K.LT.1 ) & GO TO 70 KSTEP = 1 ! ! Determine rows and columns to be interchanged and whether ! a 1-by-1 or 2-by-2 pivot block will be used ! ABSAKK = ABS( A( K, K ) ) ! ! IMAX is the row-index of the largest off-diagonal element in ! column K, and COLMAX is its absolute value ! IF( K.GT.1 ) THEN IMAX = IDAMAX( K-1, A( 1, K ), 1 ) COLMAX = ABS( A( IMAX, K ) ) ELSE COLMAX = ZERO END IF ! IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN ! ! Column K is zero: set INFO and continue ! IF( INFO.EQ.0 ) & INFO = K KP = K ELSE IF( ABSAKK.GE.ALPHA*COLMAX ) THEN ! ! no interchange, use 1-by-1 pivot block ! KP = K ELSE ! ! JMAX is the column-index of the largest off-diagonal ! element in row IMAX, and ROWMAX is its absolute value ! JMAX = IMAX + IDAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA ) ROWMAX = ABS( A( IMAX, JMAX ) ) IF( IMAX.GT.1 ) THEN JMAX = IDAMAX( IMAX-1, A( 1, IMAX ), 1 ) ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) ) END IF ! IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN ! ! no interchange, use 1-by-1 pivot block ! KP = K ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN ! ! interchange rows and columns K and IMAX, use 1-by-1 ! pivot block ! KP = IMAX ELSE ! ! interchange rows and columns K-1 and IMAX, use 2-by-2 ! pivot block ! KP = IMAX KSTEP = 2 END IF END IF ! KK = K - KSTEP + 1 IF( KP.NE.KK ) THEN ! ! Interchange rows and columns KK and KP in the leading ! submatrix A(1:k,1:k) ! CALL DSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 ) CALL DSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ), & LDA ) T = A( KK, KK ) A( KK, KK ) = A( KP, KP ) A( KP, KP ) = T IF( KSTEP.EQ.2 ) THEN T = A( K-1, K ) A( K-1, K ) = A( KP, K ) A( KP, K ) = T END IF END IF ! ! Update the leading submatrix ! IF( KSTEP.EQ.1 ) THEN ! ! 1-by-1 pivot block D(k): column k now holds ! ! W(k) = U(k)*D(k) ! ! where U(k) is the k-th column of U ! ! Perform a rank-1 update of A(1:k-1,1:k-1) as ! ! A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)' ! R1 = ONE / A( K, K ) CALL DSYR( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA ) ! ! Store U(k) in column k ! CALL DSCAL( K-1, R1, A( 1, K ), 1 ) ELSE ! ! 2-by-2 pivot block D(k): columns k and k-1 now hold ! ! ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k) ! ! where U(k) and U(k-1) are the k-th and (k-1)-th columns ! of U ! ! Perform a rank-2 update of A(1:k-2,1:k-2) as ! ! A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )' ! = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )' ! IF( K.GT.2 ) THEN ! D12 = A( K-1, K ) D22 = A( K-1, K-1 ) / D12 D11 = A( K, K ) / D12 T = ONE / ( D11*D22-ONE ) D12 = T / D12 ! DO 30 J = K - 2, 1, -1 WKM1 = D12*( D11*A( J, K-1 )-A( J, K ) ) WK = D12*( D22*A( J, K )-A( J, K-1 ) ) DO 20 I = J, 1, -1 A( I, J ) = A( I, J ) - A( I, K )*WK - & A( I, K-1 )*WKM1 20 CONTINUE A( J, K ) = WK A( J, K-1 ) = WKM1 30 CONTINUE ! END IF ! END IF END IF ! ! Store details of the interchanges in IPIV ! IF( KSTEP.EQ.1 ) THEN IPIV( K ) = KP ELSE IPIV( K ) = -KP IPIV( K-1 ) = -KP END IF ! ! Decrease K and return to the start of the main loop ! K = K - KSTEP GO TO 10 ! ELSE ! ! Factorize A as L*D*L' using the lower triangle of A ! ! K is the main loop index, increasing from 1 to N in steps of ! 1 or 2 ! K = 1 40 CONTINUE ! ! If K > N, exit from loop ! IF( K.GT.N ) & GO TO 70 KSTEP = 1 ! ! Determine rows and columns to be interchanged and whether ! a 1-by-1 or 2-by-2 pivot block will be used ! ABSAKK = ABS( A( K, K ) ) ! ! IMAX is the row-index of the largest off-diagonal element in ! column K, and COLMAX is its absolute value ! IF( K.LT.N ) THEN IMAX = K + IDAMAX( N-K, A( K+1, K ), 1 ) COLMAX = ABS( A( IMAX, K ) ) ELSE COLMAX = ZERO END IF ! IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN ! ! Column K is zero: set INFO and continue ! IF( INFO.EQ.0 ) & INFO = K KP = K ELSE IF( ABSAKK.GE.ALPHA*COLMAX ) THEN ! ! no interchange, use 1-by-1 pivot block ! KP = K ELSE ! ! JMAX is the column-index of the largest off-diagonal ! element in row IMAX, and ROWMAX is its absolute value ! JMAX = K - 1 + IDAMAX( IMAX-K, A( IMAX, K ), LDA ) ROWMAX = ABS( A( IMAX, JMAX ) ) IF( IMAX.LT.N ) THEN JMAX = IMAX + IDAMAX( N-IMAX, A( IMAX+1, IMAX ), 1 ) ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) ) END IF ! IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN ! ! no interchange, use 1-by-1 pivot block ! KP = K ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN ! ! interchange rows and columns K and IMAX, use 1-by-1 ! pivot block ! KP = IMAX ELSE ! ! interchange rows and columns K+1 and IMAX, use 2-by-2 ! pivot block ! KP = IMAX KSTEP = 2 END IF END IF ! KK = K + KSTEP - 1 IF( KP.NE.KK ) THEN ! ! Interchange rows and columns KK and KP in the trailing ! submatrix A(k:n,k:n) ! IF( KP.LT.N ) & CALL DSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 ) CALL DSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ), & LDA ) T = A( KK, KK ) A( KK, KK ) = A( KP, KP ) A( KP, KP ) = T IF( KSTEP.EQ.2 ) THEN T = A( K+1, K ) A( K+1, K ) = A( KP, K ) A( KP, K ) = T END IF END IF ! ! Update the trailing submatrix ! IF( KSTEP.EQ.1 ) THEN ! ! 1-by-1 pivot block D(k): column k now holds ! ! W(k) = L(k)*D(k) ! ! where L(k) is the k-th column of L ! IF( K.LT.N ) THEN ! ! Perform a rank-1 update of A(k+1:n,k+1:n) as ! ! A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)' ! D11 = ONE / A( K, K ) CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1, & A( K+1, K+1 ), LDA ) ! ! Store L(k) in column K ! CALL DSCAL( N-K, D11, A( K+1, K ), 1 ) END IF ELSE ! ! 2-by-2 pivot block D(k) ! IF( K.LT.N-1 ) THEN ! ! Perform a rank-2 update of A(k+2:n,k+2:n) as ! ! A := A - ( (A(k) A(k+1))*D(k)**(-1) ) * (A(k) A(k+1))' ! ! where L(k) and L(k+1) are the k-th and (k+1)-th ! columns of L ! D21 = A( K+1, K ) D11 = A( K+1, K+1 ) / D21 D22 = A( K, K ) / D21 T = ONE / ( D11*D22-ONE ) D21 = T / D21 ! DO 60 J = K + 2, N ! WK = D21*( D11*A( J, K )-A( J, K+1 ) ) WKP1 = D21*( D22*A( J, K+1 )-A( J, K ) ) ! DO 50 I = J, N A( I, J ) = A( I, J ) - A( I, K )*WK - & A( I, K+1 )*WKP1 50 CONTINUE ! A( J, K ) = WK A( J, K+1 ) = WKP1 ! 60 CONTINUE END IF END IF END IF ! ! Store details of the interchanges in IPIV ! IF( KSTEP.EQ.1 ) THEN IPIV( K ) = KP ELSE IPIV( K ) = -KP IPIV( K+1 ) = -KP END IF ! ! Increase K and return to the start of the main loop ! K = K + KSTEP GO TO 40 ! END IF ! 70 CONTINUE ! RETURN ! ! End of DSYTF2 ! END SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, LWORK, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ), & WORK( * ) ! .. ! ! Purpose ! ======= ! ! DSYTRD reduces a real symmetric matrix A to real symmetric ! tridiagonal form T by an orthogonal similarity transformation: ! Q**T * A * Q = T. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the symmetric matrix A. If UPLO = 'U', the leading ! N-by-N upper triangular part of A contains the upper ! triangular part of the matrix A, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading N-by-N lower triangular part of A contains the lower ! triangular part of the matrix A, and the strictly upper ! triangular part of A is not referenced. ! On exit, if UPLO = 'U', the diagonal and first superdiagonal ! of A are overwritten by the corresponding elements of the ! tridiagonal matrix T, and the elements above the first ! superdiagonal, with the array TAU, represent the orthogonal ! matrix Q as a product of elementary reflectors; if UPLO ! = 'L', the diagonal and first subdiagonal of A are over- ! written by the corresponding elements of the tridiagonal ! matrix T, and the elements below the first subdiagonal, with ! the array TAU, represent the orthogonal matrix Q as a product ! of elementary reflectors. See Further Details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! D (output) DOUBLE PRECISION array, dimension (N) ! The diagonal elements of the tridiagonal matrix T: ! D(i) = A(i,i). ! ! E (output) DOUBLE PRECISION array, dimension (N-1) ! The off-diagonal elements of the tridiagonal matrix T: ! E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. ! ! TAU (output) DOUBLE PRECISION array, dimension (N-1) ! The scalar factors of the elementary reflectors (see Further ! Details). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= 1. ! For optimum performance LWORK >= N*NB, where NB is the ! optimal blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! If UPLO = 'U', the matrix Q is represented as a product of elementary ! reflectors ! ! Q = H(n-1) . . . H(2) H(1). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in ! A(1:i-1,i+1), and tau in TAU(i). ! ! If UPLO = 'L', the matrix Q is represented as a product of elementary ! reflectors ! ! Q = H(1) H(2) . . . H(n-1). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), ! and tau in TAU(i). ! ! The contents of A on exit are illustrated by the following examples ! with n = 5: ! ! if UPLO = 'U': if UPLO = 'L': ! ! ( d e v2 v3 v4 ) ( d ) ! ( d e v3 v4 ) ( e d ) ! ( d e v4 ) ( v1 e d ) ! ( d e ) ( v1 v2 e d ) ! ( d ) ( v1 v2 v3 e d ) ! ! where d and e denote diagonal and off-diagonal elements of T, and vi ! denotes an element of the vector defining H(i). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, UPPER INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB, & NBMIN, NX ! .. ! .. External Subroutines .. EXTERNAL DLATRD, DSYR2K, DSYTD2, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) LQUERY = ( LWORK.EQ.-1 ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN INFO = -9 END IF ! IF( INFO.EQ.0 ) THEN ! ! Determine the block size. ! NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 ) LWKOPT = N*NB WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYTRD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! NX = N IWS = 1 IF( NB.GT.1 .AND. NB.LT.N ) THEN ! ! Determine when to cross over from blocked to unblocked code ! (last block is always handled by unblocked code). ! NX = MAX( NB, ILAENV( 3, 'DSYTRD', UPLO, N, -1, -1, -1 ) ) IF( NX.LT.N ) THEN ! ! Determine if workspace is large enough for blocked code. ! LDWORK = N IWS = LDWORK*NB IF( LWORK.LT.IWS ) THEN ! ! Not enough workspace to use optimal NB: determine the ! minimum value of NB, and reduce NB or force use of ! unblocked code by setting NX = N. ! NB = MAX( LWORK / LDWORK, 1 ) NBMIN = ILAENV( 2, 'DSYTRD', UPLO, N, -1, -1, -1 ) IF( NB.LT.NBMIN ) & NX = N END IF ELSE NX = N END IF ELSE NB = 1 END IF ! IF( UPPER ) THEN ! ! Reduce the upper triangle of A. ! Columns 1:kk are handled by the unblocked method. ! KK = N - ( ( N-NX+NB-1 ) / NB )*NB DO 20 I = N - NB + 1, KK + 1, -NB ! ! Reduce columns i:i+nb-1 to tridiagonal form and form the ! matrix W which is needed to update the unreduced part of ! the matrix ! CALL DLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK, & LDWORK ) ! ! Update the unreduced submatrix A(1:i-1,1:i-1), using an ! update of the form: A := A - V*W' - W*V' ! CALL DSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1, I ), & LDA, WORK, LDWORK, ONE, A, LDA ) ! ! Copy superdiagonal elements back into A, and diagonal ! elements into D ! DO 10 J = I, I + NB - 1 A( J-1, J ) = E( J-1 ) D( J ) = A( J, J ) 10 CONTINUE 20 CONTINUE ! ! Use unblocked code to reduce the last or only block ! CALL DSYTD2( UPLO, KK, A, LDA, D, E, TAU, IINFO ) ELSE ! ! Reduce the lower triangle of A ! DO 40 I = 1, N - NX, NB ! ! Reduce columns i:i+nb-1 to tridiagonal form and form the ! matrix W which is needed to update the unreduced part of ! the matrix ! CALL DLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ), & TAU( I ), WORK, LDWORK ) ! ! Update the unreduced submatrix A(i+ib:n,i+ib:n), using ! an update of the form: A := A - V*W' - W*V' ! CALL DSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE, & A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE, & A( I+NB, I+NB ), LDA ) ! ! Copy subdiagonal elements back into A, and diagonal ! elements into D ! DO 30 J = I, I + NB - 1 A( J+1, J ) = E( J ) D( J ) = A( J, J ) 30 CONTINUE 40 CONTINUE ! ! Use unblocked code to reduce the last or only block ! CALL DSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ), & TAU( I ), IINFO ) END IF ! WORK( 1 ) = LWKOPT RETURN ! ! End of DSYTRD ! END SUBROUTINE DSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, LWORK, N ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION A( LDA, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DSYTRF computes the factorization of a real symmetric matrix A using ! the Bunch-Kaufman diagonal pivoting method. The form of the ! factorization is ! ! A = U*D*U**T or A = L*D*L**T ! ! where U (or L) is a product of permutation and unit upper (lower) ! triangular matrices, and D is symmetric and block diagonal with ! 1-by-1 and 2-by-2 diagonal blocks. ! ! This is the blocked version of the algorithm, calling Level 3 BLAS. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the symmetric matrix A. If UPLO = 'U', the leading ! N-by-N upper triangular part of A contains the upper ! triangular part of the matrix A, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading N-by-N lower triangular part of A contains the lower ! triangular part of the matrix A, and the strictly upper ! triangular part of A is not referenced. ! ! On exit, the block diagonal matrix D and the multipliers used ! to obtain the factor U or L (see below for further details). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! IPIV (output) INTEGER array, dimension (N) ! Details of the interchanges and the block structure of D. ! If IPIV(k) > 0, then rows and columns k and IPIV(k) were ! interchanged and D(k,k) is a 1-by-1 diagonal block. ! If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and ! columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) ! is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = ! IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were ! interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The length of WORK. LWORK >=1. For best performance ! LWORK >= N*NB, where NB is the block size returned by ILAENV. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, D(i,i) is exactly zero. The factorization ! has been completed, but the block diagonal matrix D is ! exactly singular, and division by zero will occur if it ! is used to solve a system of equations. ! ! Further Details ! =============== ! ! If UPLO = 'U', then A = U*D*U', where ! U = P(n)*U(n)* ... *P(k)U(k)* ..., ! i.e., U is a product of terms P(k)*U(k), where k decreases from n to ! 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 ! and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as ! defined by IPIV(k), and U(k) is a unit upper triangular matrix, such ! that if the diagonal block D(k) is of order s (s = 1 or 2), then ! ! ( I v 0 ) k-s ! U(k) = ( 0 I 0 ) s ! ( 0 0 I ) n-k ! k-s s n-k ! ! If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). ! If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), ! and A(k,k), and v overwrites A(1:k-2,k-1:k). ! ! If UPLO = 'L', then A = L*D*L', where ! L = P(1)*L(1)* ... *P(k)*L(k)* ..., ! i.e., L is a product of terms P(k)*L(k), where k increases from 1 to ! n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 ! and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as ! defined by IPIV(k), and L(k) is a unit lower triangular matrix, such ! that if the diagonal block D(k) is of order s (s = 1 or 2), then ! ! ( I 0 0 ) k-1 ! L(k) = ( 0 I 0 ) s ! ( 0 v I ) n-k-s+1 ! k-1 s n-k-s+1 ! ! If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). ! If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), ! and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1). ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL LQUERY, UPPER INTEGER IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DLASYF, DSYTF2, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) LQUERY = ( LWORK.EQ.-1 ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN INFO = -7 END IF ! IF( INFO.EQ.0 ) THEN ! ! Determine the block size ! NB = ILAENV( 1, 'DSYTRF', UPLO, N, -1, -1, -1 ) LWKOPT = N*NB WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYTRF', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! NBMIN = 2 LDWORK = N IF( NB.GT.1 .AND. NB.LT.N ) THEN IWS = LDWORK*NB IF( LWORK.LT.IWS ) THEN NB = MAX( LWORK / LDWORK, 1 ) NBMIN = MAX( 2, ILAENV( 2, 'DSYTRF', UPLO, N, -1, -1, -1 ) ) END IF ELSE IWS = 1 END IF IF( NB.LT.NBMIN ) & NB = N ! IF( UPPER ) THEN ! ! Factorize A as U*D*U' using the upper triangle of A ! ! K is the main loop index, decreasing from N to 1 in steps of ! KB, where KB is the number of columns factorized by DLASYF; ! KB is either NB or NB-1, or K for the last block ! K = N 10 CONTINUE ! ! If K < 1, exit from loop ! IF( K.LT.1 ) & GO TO 40 ! IF( K.GT.NB ) THEN ! ! Factorize columns k-kb+1:k of A and use blocked code to ! update columns 1:k-kb ! CALL DLASYF( UPLO, K, NB, KB, A, LDA, IPIV, WORK, LDWORK, & IINFO ) ELSE ! ! Use unblocked code to factorize columns 1:k of A ! CALL DSYTF2( UPLO, K, A, LDA, IPIV, IINFO ) KB = K END IF ! ! Set INFO on the first occurrence of a zero pivot ! IF( INFO.EQ.0 .AND. IINFO.GT.0 ) & INFO = IINFO ! ! Decrease K and return to the start of the main loop ! K = K - KB GO TO 10 ! ELSE ! ! Factorize A as L*D*L' using the lower triangle of A ! ! K is the main loop index, increasing from 1 to N in steps of ! KB, where KB is the number of columns factorized by DLASYF; ! KB is either NB or NB-1, or N-K+1 for the last block ! K = 1 20 CONTINUE ! ! If K > N, exit from loop ! IF( K.GT.N ) & GO TO 40 ! IF( K.LE.N-NB ) THEN ! ! Factorize columns k:k+kb-1 of A and use blocked code to ! update columns k+kb:n ! CALL DLASYF( UPLO, N-K+1, NB, KB, A( K, K ), LDA, IPIV( K ), & WORK, LDWORK, IINFO ) ELSE ! ! Use unblocked code to factorize columns k:n of A ! CALL DSYTF2( UPLO, N-K+1, A( K, K ), LDA, IPIV( K ), IINFO ) KB = N - K + 1 END IF ! ! Set INFO on the first occurrence of a zero pivot ! IF( INFO.EQ.0 .AND. IINFO.GT.0 ) & INFO = IINFO + K - 1 ! ! Adjust IPIV ! DO 30 J = K, K + KB - 1 IF( IPIV( J ).GT.0 ) THEN IPIV( J ) = IPIV( J ) + K - 1 ELSE IPIV( J ) = IPIV( J ) - K + 1 END IF 30 CONTINUE ! ! Increase K and return to the start of the main loop ! K = K + KB GO TO 20 ! END IF ! 40 CONTINUE WORK( 1 ) = LWKOPT RETURN ! ! End of DSYTRF ! END SUBROUTINE DSYTRI( UPLO, N, A, LDA, IPIV, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION A( LDA, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DSYTRI computes the inverse of a real symmetric indefinite matrix ! A using the factorization A = U*D*U**T or A = L*D*L**T computed by ! DSYTRF. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the details of the factorization are stored ! as an upper or lower triangular matrix. ! = 'U': Upper triangular, form is A = U*D*U**T; ! = 'L': Lower triangular, form is A = L*D*L**T. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the block diagonal matrix D and the multipliers ! used to obtain the factor U or L as computed by DSYTRF. ! ! On exit, if INFO = 0, the (symmetric) inverse of the original ! matrix. If UPLO = 'U', the upper triangular part of the ! inverse is formed and the part of A below the diagonal is not ! referenced; if UPLO = 'L' the lower triangular part of the ! inverse is formed and the part of A above the diagonal is ! not referenced. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! IPIV (input) INTEGER array, dimension (N) ! Details of the interchanges and the block structure of D ! as determined by DSYTRF. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its ! inverse could not be computed. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER K, KP, KSTEP DOUBLE PRECISION AK, AKKP1, AKP1, D, T, TEMP ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT EXTERNAL LSAME, DDOT ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DSWAP, DSYMV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYTRI', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Check that the diagonal matrix D is nonsingular. ! IF( UPPER ) THEN ! ! Upper triangular storage: examine D from bottom to top ! DO 10 INFO = N, 1, -1 IF( IPIV( INFO ).GT.0 .AND. A( INFO, INFO ).EQ.ZERO ) & RETURN 10 CONTINUE ELSE ! ! Lower triangular storage: examine D from top to bottom. ! DO 20 INFO = 1, N IF( IPIV( INFO ).GT.0 .AND. A( INFO, INFO ).EQ.ZERO ) & RETURN 20 CONTINUE END IF INFO = 0 ! IF( UPPER ) THEN ! ! Compute inv(A) from the factorization A = U*D*U'. ! ! K is the main loop index, increasing from 1 to N in steps of ! 1 or 2, depending on the size of the diagonal blocks. ! K = 1 30 CONTINUE ! ! If K > N, exit from loop. ! IF( K.GT.N ) & GO TO 40 ! IF( IPIV( K ).GT.0 ) THEN ! ! 1 x 1 diagonal block ! ! Invert the diagonal block. ! A( K, K ) = ONE / A( K, K ) ! ! Compute column K of the inverse. ! IF( K.GT.1 ) THEN CALL DCOPY( K-1, A( 1, K ), 1, WORK, 1 ) CALL DSYMV( UPLO, K-1, -ONE, A, LDA, WORK, 1, ZERO, & A( 1, K ), 1 ) A( K, K ) = A( K, K ) - DDOT( K-1, WORK, 1, A( 1, K ), & 1 ) END IF KSTEP = 1 ELSE ! ! 2 x 2 diagonal block ! ! Invert the diagonal block. ! T = ABS( A( K, K+1 ) ) AK = A( K, K ) / T AKP1 = A( K+1, K+1 ) / T AKKP1 = A( K, K+1 ) / T D = T*( AK*AKP1-ONE ) A( K, K ) = AKP1 / D A( K+1, K+1 ) = AK / D A( K, K+1 ) = -AKKP1 / D ! ! Compute columns K and K+1 of the inverse. ! IF( K.GT.1 ) THEN CALL DCOPY( K-1, A( 1, K ), 1, WORK, 1 ) CALL DSYMV( UPLO, K-1, -ONE, A, LDA, WORK, 1, ZERO, & A( 1, K ), 1 ) A( K, K ) = A( K, K ) - DDOT( K-1, WORK, 1, A( 1, K ), & 1 ) A( K, K+1 ) = A( K, K+1 ) - & DDOT( K-1, A( 1, K ), 1, A( 1, K+1 ), 1 ) CALL DCOPY( K-1, A( 1, K+1 ), 1, WORK, 1 ) CALL DSYMV( UPLO, K-1, -ONE, A, LDA, WORK, 1, ZERO, & A( 1, K+1 ), 1 ) A( K+1, K+1 ) = A( K+1, K+1 ) - & DDOT( K-1, WORK, 1, A( 1, K+1 ), 1 ) END IF KSTEP = 2 END IF ! KP = ABS( IPIV( K ) ) IF( KP.NE.K ) THEN ! ! Interchange rows and columns K and KP in the leading ! submatrix A(1:k+1,1:k+1) ! CALL DSWAP( KP-1, A( 1, K ), 1, A( 1, KP ), 1 ) CALL DSWAP( K-KP-1, A( KP+1, K ), 1, A( KP, KP+1 ), LDA ) TEMP = A( K, K ) A( K, K ) = A( KP, KP ) A( KP, KP ) = TEMP IF( KSTEP.EQ.2 ) THEN TEMP = A( K, K+1 ) A( K, K+1 ) = A( KP, K+1 ) A( KP, K+1 ) = TEMP END IF END IF ! K = K + KSTEP GO TO 30 40 CONTINUE ! ELSE ! ! Compute inv(A) from the factorization A = L*D*L'. ! ! K is the main loop index, increasing from 1 to N in steps of ! 1 or 2, depending on the size of the diagonal blocks. ! K = N 50 CONTINUE ! ! If K < 1, exit from loop. ! IF( K.LT.1 ) & GO TO 60 ! IF( IPIV( K ).GT.0 ) THEN ! ! 1 x 1 diagonal block ! ! Invert the diagonal block. ! A( K, K ) = ONE / A( K, K ) ! ! Compute column K of the inverse. ! IF( K.LT.N ) THEN CALL DCOPY( N-K, A( K+1, K ), 1, WORK, 1 ) CALL DSYMV( UPLO, N-K, -ONE, A( K+1, K+1 ), LDA, WORK, 1, & ZERO, A( K+1, K ), 1 ) A( K, K ) = A( K, K ) - DDOT( N-K, WORK, 1, A( K+1, K ), & 1 ) END IF KSTEP = 1 ELSE ! ! 2 x 2 diagonal block ! ! Invert the diagonal block. ! T = ABS( A( K, K-1 ) ) AK = A( K-1, K-1 ) / T AKP1 = A( K, K ) / T AKKP1 = A( K, K-1 ) / T D = T*( AK*AKP1-ONE ) A( K-1, K-1 ) = AKP1 / D A( K, K ) = AK / D A( K, K-1 ) = -AKKP1 / D ! ! Compute columns K-1 and K of the inverse. ! IF( K.LT.N ) THEN CALL DCOPY( N-K, A( K+1, K ), 1, WORK, 1 ) CALL DSYMV( UPLO, N-K, -ONE, A( K+1, K+1 ), LDA, WORK, 1, & ZERO, A( K+1, K ), 1 ) A( K, K ) = A( K, K ) - DDOT( N-K, WORK, 1, A( K+1, K ), & 1 ) A( K, K-1 ) = A( K, K-1 ) - & DDOT( N-K, A( K+1, K ), 1, A( K+1, K-1 ), & 1 ) CALL DCOPY( N-K, A( K+1, K-1 ), 1, WORK, 1 ) CALL DSYMV( UPLO, N-K, -ONE, A( K+1, K+1 ), LDA, WORK, 1, & ZERO, A( K+1, K-1 ), 1 ) A( K-1, K-1 ) = A( K-1, K-1 ) - & DDOT( N-K, WORK, 1, A( K+1, K-1 ), 1 ) END IF KSTEP = 2 END IF ! KP = ABS( IPIV( K ) ) IF( KP.NE.K ) THEN ! ! Interchange rows and columns K and KP in the trailing ! submatrix A(k-1:n,k-1:n) ! IF( KP.LT.N ) & CALL DSWAP( N-KP, A( KP+1, K ), 1, A( KP+1, KP ), 1 ) CALL DSWAP( KP-K-1, A( K+1, K ), 1, A( KP, K+1 ), LDA ) TEMP = A( K, K ) A( K, K ) = A( KP, KP ) A( KP, KP ) = TEMP IF( KSTEP.EQ.2 ) THEN TEMP = A( K, K-1 ) A( K, K-1 ) = A( KP, K-1 ) A( KP, K-1 ) = TEMP END IF END IF ! K = K - KSTEP GO TO 50 60 CONTINUE END IF ! RETURN ! ! End of DSYTRI ! END SUBROUTINE DSYTRS( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, LDB, N, NRHS ! .. ! .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DSYTRS solves a system of linear equations A*X = B with a real ! symmetric matrix A using the factorization A = U*D*U**T or ! A = L*D*L**T computed by DSYTRF. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the details of the factorization are stored ! as an upper or lower triangular matrix. ! = 'U': Upper triangular, form is A = U*D*U**T; ! = 'L': Lower triangular, form is A = L*D*L**T. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The block diagonal matrix D and the multipliers used to ! obtain the factor U or L as computed by DSYTRF. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! IPIV (input) INTEGER array, dimension (N) ! Details of the interchanges and the block structure of D ! as determined by DSYTRF. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the right hand side matrix B. ! On exit, the solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER J, K, KP DOUBLE PRECISION AK, AKM1, AKM1K, BK, BKM1, DENOM ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DGEMV, DGER, DSCAL, DSWAP, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYTRS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) & RETURN ! IF( UPPER ) THEN ! ! Solve A*X = B, where A = U*D*U'. ! ! First solve U*D*X = B, overwriting B with X. ! ! K is the main loop index, decreasing from N to 1 in steps of ! 1 or 2, depending on the size of the diagonal blocks. ! K = N 10 CONTINUE ! ! If K < 1, exit from loop. ! IF( K.LT.1 ) & GO TO 30 ! IF( IPIV( K ).GT.0 ) THEN ! ! 1 x 1 diagonal block ! ! Interchange rows K and IPIV(K). ! KP = IPIV( K ) IF( KP.NE.K ) & CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB ) ! ! Multiply by inv(U(K)), where U(K) is the transformation ! stored in column K of A. ! CALL DGER( K-1, NRHS, -ONE, A( 1, K ), 1, B( K, 1 ), LDB, & B( 1, 1 ), LDB ) ! ! Multiply by the inverse of the diagonal block. ! CALL DSCAL( NRHS, ONE / A( K, K ), B( K, 1 ), LDB ) K = K - 1 ELSE ! ! 2 x 2 diagonal block ! ! Interchange rows K-1 and -IPIV(K). ! KP = -IPIV( K ) IF( KP.NE.K-1 ) & CALL DSWAP( NRHS, B( K-1, 1 ), LDB, B( KP, 1 ), LDB ) ! ! Multiply by inv(U(K)), where U(K) is the transformation ! stored in columns K-1 and K of A. ! CALL DGER( K-2, NRHS, -ONE, A( 1, K ), 1, B( K, 1 ), LDB, & B( 1, 1 ), LDB ) CALL DGER( K-2, NRHS, -ONE, A( 1, K-1 ), 1, B( K-1, 1 ), & LDB, B( 1, 1 ), LDB ) ! ! Multiply by the inverse of the diagonal block. ! AKM1K = A( K-1, K ) AKM1 = A( K-1, K-1 ) / AKM1K AK = A( K, K ) / AKM1K DENOM = AKM1*AK - ONE DO 20 J = 1, NRHS BKM1 = B( K-1, J ) / AKM1K BK = B( K, J ) / AKM1K B( K-1, J ) = ( AK*BKM1-BK ) / DENOM B( K, J ) = ( AKM1*BK-BKM1 ) / DENOM 20 CONTINUE K = K - 2 END IF ! GO TO 10 30 CONTINUE ! ! Next solve U'*X = B, overwriting B with X. ! ! K is the main loop index, increasing from 1 to N in steps of ! 1 or 2, depending on the size of the diagonal blocks. ! K = 1 40 CONTINUE ! ! If K > N, exit from loop. ! IF( K.GT.N ) & GO TO 50 ! IF( IPIV( K ).GT.0 ) THEN ! ! 1 x 1 diagonal block ! ! Multiply by inv(U'(K)), where U(K) is the transformation ! stored in column K of A. ! CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB, A( 1, K ), & 1, ONE, B( K, 1 ), LDB ) ! ! Interchange rows K and IPIV(K). ! KP = IPIV( K ) IF( KP.NE.K ) & CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB ) K = K + 1 ELSE ! ! 2 x 2 diagonal block ! ! Multiply by inv(U'(K+1)), where U(K+1) is the transformation ! stored in columns K and K+1 of A. ! CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB, A( 1, K ), & 1, ONE, B( K, 1 ), LDB ) CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB, & A( 1, K+1 ), 1, ONE, B( K+1, 1 ), LDB ) ! ! Interchange rows K and -IPIV(K). ! KP = -IPIV( K ) IF( KP.NE.K ) & CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB ) K = K + 2 END IF ! GO TO 40 50 CONTINUE ! ELSE ! ! Solve A*X = B, where A = L*D*L'. ! ! First solve L*D*X = B, overwriting B with X. ! ! K is the main loop index, increasing from 1 to N in steps of ! 1 or 2, depending on the size of the diagonal blocks. ! K = 1 60 CONTINUE ! ! If K > N, exit from loop. ! IF( K.GT.N ) & GO TO 80 ! IF( IPIV( K ).GT.0 ) THEN ! ! 1 x 1 diagonal block ! ! Interchange rows K and IPIV(K). ! KP = IPIV( K ) IF( KP.NE.K ) & CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB ) ! ! Multiply by inv(L(K)), where L(K) is the transformation ! stored in column K of A. ! IF( K.LT.N ) & CALL DGER( N-K, NRHS, -ONE, A( K+1, K ), 1, B( K, 1 ), & LDB, B( K+1, 1 ), LDB ) ! ! Multiply by the inverse of the diagonal block. ! CALL DSCAL( NRHS, ONE / A( K, K ), B( K, 1 ), LDB ) K = K + 1 ELSE ! ! 2 x 2 diagonal block ! ! Interchange rows K+1 and -IPIV(K). ! KP = -IPIV( K ) IF( KP.NE.K+1 ) & CALL DSWAP( NRHS, B( K+1, 1 ), LDB, B( KP, 1 ), LDB ) ! ! Multiply by inv(L(K)), where L(K) is the transformation ! stored in columns K and K+1 of A. ! IF( K.LT.N-1 ) THEN CALL DGER( N-K-1, NRHS, -ONE, A( K+2, K ), 1, B( K, 1 ), & LDB, B( K+2, 1 ), LDB ) CALL DGER( N-K-1, NRHS, -ONE, A( K+2, K+1 ), 1, & B( K+1, 1 ), LDB, B( K+2, 1 ), LDB ) END IF ! ! Multiply by the inverse of the diagonal block. ! AKM1K = A( K+1, K ) AKM1 = A( K, K ) / AKM1K AK = A( K+1, K+1 ) / AKM1K DENOM = AKM1*AK - ONE DO 70 J = 1, NRHS BKM1 = B( K, J ) / AKM1K BK = B( K+1, J ) / AKM1K B( K, J ) = ( AK*BKM1-BK ) / DENOM B( K+1, J ) = ( AKM1*BK-BKM1 ) / DENOM 70 CONTINUE K = K + 2 END IF ! GO TO 60 80 CONTINUE ! ! Next solve L'*X = B, overwriting B with X. ! ! K is the main loop index, decreasing from N to 1 in steps of ! 1 or 2, depending on the size of the diagonal blocks. ! K = N 90 CONTINUE ! ! If K < 1, exit from loop. ! IF( K.LT.1 ) & GO TO 100 ! IF( IPIV( K ).GT.0 ) THEN ! ! 1 x 1 diagonal block ! ! Multiply by inv(L'(K)), where L(K) is the transformation ! stored in column K of A. ! IF( K.LT.N ) & CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ), & LDB, A( K+1, K ), 1, ONE, B( K, 1 ), LDB ) ! ! Interchange rows K and IPIV(K). ! KP = IPIV( K ) IF( KP.NE.K ) & CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB ) K = K - 1 ELSE ! ! 2 x 2 diagonal block ! ! Multiply by inv(L'(K-1)), where L(K-1) is the transformation ! stored in columns K-1 and K of A. ! IF( K.LT.N ) THEN CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ), & LDB, A( K+1, K ), 1, ONE, B( K, 1 ), LDB ) CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ), & LDB, A( K+1, K-1 ), 1, ONE, B( K-1, 1 ), & LDB ) END IF ! ! Interchange rows K and -IPIV(K). ! KP = -IPIV( K ) IF( KP.NE.K ) & CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB ) K = K - 2 END IF ! GO TO 90 100 CONTINUE END IF ! RETURN ! ! End of DSYTRS ! END SUBROUTINE DTBCON( NORM, UPLO, DIAG, N, KD, AB, LDAB, RCOND, WORK, & IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER DIAG, NORM, UPLO INTEGER INFO, KD, LDAB, N DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION AB( LDAB, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DTBCON estimates the reciprocal of the condition number of a ! triangular band matrix A, in either the 1-norm or the infinity-norm. ! ! The norm of A is computed and an estimate is obtained for ! norm(inv(A)), then the reciprocal of the condition number is ! computed as ! RCOND = 1 / ( norm(A) * norm(inv(A)) ). ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies whether the 1-norm condition number or the ! infinity-norm condition number is required: ! = '1' or 'O': 1-norm; ! = 'I': Infinity-norm. ! ! UPLO (input) CHARACTER*1 ! = 'U': A is upper triangular; ! = 'L': A is lower triangular. ! ! DIAG (input) CHARACTER*1 ! = 'N': A is non-unit triangular; ! = 'U': A is unit triangular. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of superdiagonals or subdiagonals of the ! triangular band matrix A. KD >= 0. ! ! AB (input) DOUBLE PRECISION array, dimension (LDAB,N) ! The upper or lower triangular band matrix A, stored in the ! first kd+1 rows of the array. The j-th column of A is stored ! in the j-th column of the array AB as follows: ! if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). ! If DIAG = 'U', the diagonal elements of A are not referenced ! and are assumed to be 1. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KD+1. ! ! RCOND (output) DOUBLE PRECISION ! The reciprocal of the condition number of the matrix A, ! computed as RCOND = 1/(norm(A) * norm(inv(A))). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOUNIT, ONENRM, UPPER CHARACTER NORMIN INTEGER IX, KASE, KASE1 DOUBLE PRECISION AINVNM, ANORM, SCALE, SMLNUM, XNORM ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DLAMCH, DLANTB EXTERNAL LSAME, IDAMAX, DLAMCH, DLANTB ! .. ! .. External Subroutines .. EXTERNAL DLACON, DLATBS, DRSCL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' ) NOUNIT = LSAME( DIAG, 'N' ) ! IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN INFO = -1 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -2 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( KD.LT.0 ) THEN INFO = -5 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTBCON', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) THEN RCOND = ONE RETURN END IF ! RCOND = ZERO SMLNUM = DLAMCH( 'Safe minimum' )*DBLE( MAX( 1, N ) ) ! ! Compute the norm of the triangular matrix A. ! ANORM = DLANTB( NORM, UPLO, DIAG, N, KD, AB, LDAB, WORK ) ! ! Continue only if ANORM > 0. ! IF( ANORM.GT.ZERO ) THEN ! ! Estimate the norm of the inverse of A. ! AINVNM = ZERO NORMIN = 'N' IF( ONENRM ) THEN KASE1 = 1 ELSE KASE1 = 2 END IF KASE = 0 10 CONTINUE CALL DLACON( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.KASE1 ) THEN ! ! Multiply by inv(A). ! CALL DLATBS( UPLO, 'No transpose', DIAG, NORMIN, N, KD, & AB, LDAB, WORK, SCALE, WORK( 2*N+1 ), INFO ) ELSE ! ! Multiply by inv(A'). ! CALL DLATBS( UPLO, 'Transpose', DIAG, NORMIN, N, KD, AB, & LDAB, WORK, SCALE, WORK( 2*N+1 ), INFO ) END IF NORMIN = 'Y' ! ! Multiply by 1/SCALE if doing so will not cause overflow. ! IF( SCALE.NE.ONE ) THEN IX = IDAMAX( N, WORK, 1 ) XNORM = ABS( WORK( IX ) ) IF( SCALE.LT.XNORM*SMLNUM .OR. SCALE.EQ.ZERO ) & GO TO 20 CALL DRSCL( N, SCALE, WORK, 1 ) END IF GO TO 10 END IF ! ! Compute the estimate of the reciprocal condition number. ! IF( AINVNM.NE.ZERO ) & RCOND = ( ONE / ANORM ) / AINVNM END IF ! 20 CONTINUE RETURN ! ! End of DTBCON ! END SUBROUTINE DTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, & LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER DIAG, TRANS, UPLO INTEGER INFO, KD, LDAB, LDB, LDX, N, NRHS ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ), & FERR( * ), WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DTBRFS provides error bounds and backward error estimates for the ! solution to a system of linear equations with a triangular band ! coefficient matrix. ! ! The solution matrix X must be computed by DTBTRS or some other ! means before entering this routine. DTBRFS does not do iterative ! refinement because doing so cannot improve the backward error. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': A is upper triangular; ! = 'L': A is lower triangular. ! ! TRANS (input) CHARACTER*1 ! Specifies the form of the system of equations: ! = 'N': A * X = B (No transpose) ! = 'T': A**T * X = B (Transpose) ! = 'C': A**H * X = B (Conjugate transpose = Transpose) ! ! DIAG (input) CHARACTER*1 ! = 'N': A is non-unit triangular; ! = 'U': A is unit triangular. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of superdiagonals or subdiagonals of the ! triangular band matrix A. KD >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! AB (input) DOUBLE PRECISION array, dimension (LDAB,N) ! The upper or lower triangular band matrix A, stored in the ! first kd+1 rows of the array. The j-th column of A is stored ! in the j-th column of the array AB as follows: ! if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). ! If DIAG = 'U', the diagonal elements of A are not referenced ! and are assumed to be 1. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KD+1. ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) ! The right hand side matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) ! The solution matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOTRAN, NOUNIT, UPPER CHARACTER TRANST INTEGER I, J, K, KASE, NZ DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DLACON, DTBMV, DTBSV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) NOTRAN = LSAME( TRANS, 'N' ) NOUNIT = LSAME( DIAG, 'N' ) ! IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. & LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( KD.LT.0 ) THEN INFO = -5 ELSE IF( NRHS.LT.0 ) THEN INFO = -6 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -8 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -10 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -12 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTBRFS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN DO 10 J = 1, NRHS FERR( J ) = ZERO BERR( J ) = ZERO 10 CONTINUE RETURN END IF ! IF( NOTRAN ) THEN TRANST = 'T' ELSE TRANST = 'N' END IF ! ! NZ = maximum number of nonzero elements in each row of A, plus 1 ! NZ = KD + 2 EPS = DLAMCH( 'Epsilon' ) SAFMIN = DLAMCH( 'Safe minimum' ) SAFE1 = NZ*SAFMIN SAFE2 = SAFE1 / EPS ! ! Do for each right hand side ! DO 250 J = 1, NRHS ! ! Compute residual R = B - op(A) * X, ! where op(A) = A or A', depending on TRANS. ! CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 ) CALL DTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK( N+1 ), & 1 ) CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 ) ! ! Compute componentwise relative backward error from formula ! ! max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) ! ! where abs(Z) is the componentwise absolute value of the matrix ! or vector Z. If the i-th component of the denominator is less ! than SAFE2, then SAFE1 is added to the i-th components of the ! numerator and denominator before dividing. ! DO 20 I = 1, N WORK( I ) = ABS( B( I, J ) ) 20 CONTINUE ! IF( NOTRAN ) THEN ! ! Compute abs(A)*abs(X) + abs(B). ! IF( UPPER ) THEN IF( NOUNIT ) THEN DO 40 K = 1, N XK = ABS( X( K, J ) ) DO 30 I = MAX( 1, K-KD ), K WORK( I ) = WORK( I ) + & ABS( AB( KD+1+I-K, K ) )*XK 30 CONTINUE 40 CONTINUE ELSE DO 60 K = 1, N XK = ABS( X( K, J ) ) DO 50 I = MAX( 1, K-KD ), K - 1 WORK( I ) = WORK( I ) + & ABS( AB( KD+1+I-K, K ) )*XK 50 CONTINUE WORK( K ) = WORK( K ) + XK 60 CONTINUE END IF ELSE IF( NOUNIT ) THEN DO 80 K = 1, N XK = ABS( X( K, J ) ) DO 70 I = K, MIN( N, K+KD ) WORK( I ) = WORK( I ) + ABS( AB( 1+I-K, K ) )*XK 70 CONTINUE 80 CONTINUE ELSE DO 100 K = 1, N XK = ABS( X( K, J ) ) DO 90 I = K + 1, MIN( N, K+KD ) WORK( I ) = WORK( I ) + ABS( AB( 1+I-K, K ) )*XK 90 CONTINUE WORK( K ) = WORK( K ) + XK 100 CONTINUE END IF END IF ELSE ! ! Compute abs(A')*abs(X) + abs(B). ! IF( UPPER ) THEN IF( NOUNIT ) THEN DO 120 K = 1, N S = ZERO DO 110 I = MAX( 1, K-KD ), K S = S + ABS( AB( KD+1+I-K, K ) )* & ABS( X( I, J ) ) 110 CONTINUE WORK( K ) = WORK( K ) + S 120 CONTINUE ELSE DO 140 K = 1, N S = ABS( X( K, J ) ) DO 130 I = MAX( 1, K-KD ), K - 1 S = S + ABS( AB( KD+1+I-K, K ) )* & ABS( X( I, J ) ) 130 CONTINUE WORK( K ) = WORK( K ) + S 140 CONTINUE END IF ELSE IF( NOUNIT ) THEN DO 160 K = 1, N S = ZERO DO 150 I = K, MIN( N, K+KD ) S = S + ABS( AB( 1+I-K, K ) )*ABS( X( I, J ) ) 150 CONTINUE WORK( K ) = WORK( K ) + S 160 CONTINUE ELSE DO 180 K = 1, N S = ABS( X( K, J ) ) DO 170 I = K + 1, MIN( N, K+KD ) S = S + ABS( AB( 1+I-K, K ) )*ABS( X( I, J ) ) 170 CONTINUE WORK( K ) = WORK( K ) + S 180 CONTINUE END IF END IF END IF S = ZERO DO 190 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) ELSE S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / & ( WORK( I )+SAFE1 ) ) END IF 190 CONTINUE BERR( J ) = S ! ! Bound error from formula ! ! norm(X - XTRUE) / norm(X) .le. FERR = ! norm( abs(inv(op(A)))* ! ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) ! ! where ! norm(Z) is the magnitude of the largest component of Z ! inv(op(A)) is the inverse of op(A) ! abs(Z) is the componentwise absolute value of the matrix or ! vector Z ! NZ is the maximum number of nonzeros in any row of A, plus 1 ! EPS is machine epsilon ! ! The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) ! is incremented by SAFE1 if the i-th component of ! abs(op(A))*abs(X) + abs(B) is less than SAFE2. ! ! Use DLACON to estimate the infinity-norm of the matrix ! inv(op(A)) * diag(W), ! where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) ! DO 200 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) ELSE WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 END IF 200 CONTINUE ! KASE = 0 210 CONTINUE CALL DLACON( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), & KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN ! ! Multiply by diag(W)*inv(op(A)'). ! CALL DTBSV( UPLO, TRANST, DIAG, N, KD, AB, LDAB, & WORK( N+1 ), 1 ) DO 220 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 220 CONTINUE ELSE ! ! Multiply by inv(op(A))*diag(W). ! DO 230 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 230 CONTINUE CALL DTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, & WORK( N+1 ), 1 ) END IF GO TO 210 END IF ! ! Normalize error. ! LSTRES = ZERO DO 240 I = 1, N LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 240 CONTINUE IF( LSTRES.NE.ZERO ) & FERR( J ) = FERR( J ) / LSTRES ! 250 CONTINUE ! RETURN ! ! End of DTBRFS ! END SUBROUTINE DTBTRS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, & LDB, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER DIAG, TRANS, UPLO INTEGER INFO, KD, LDAB, LDB, N, NRHS ! .. ! .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DTBTRS solves a triangular system of the form ! ! A * X = B or A**T * X = B, ! ! where A is a triangular band matrix of order N, and B is an ! N-by NRHS matrix. A check is made to verify that A is nonsingular. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': A is upper triangular; ! = 'L': A is lower triangular. ! ! TRANS (input) CHARACTER*1 ! Specifies the form the system of equations: ! = 'N': A * X = B (No transpose) ! = 'T': A**T * X = B (Transpose) ! = 'C': A**H * X = B (Conjugate transpose = Transpose) ! ! DIAG (input) CHARACTER*1 ! = 'N': A is non-unit triangular; ! = 'U': A is unit triangular. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! KD (input) INTEGER ! The number of superdiagonals or subdiagonals of the ! triangular band matrix A. KD >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! AB (input) DOUBLE PRECISION array, dimension (LDAB,N) ! The upper or lower triangular band matrix A, stored in the ! first kd+1 rows of AB. The j-th column of A is stored ! in the j-th column of the array AB as follows: ! if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; ! if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). ! If DIAG = 'U', the diagonal elements of A are not referenced ! and are assumed to be 1. ! ! LDAB (input) INTEGER ! The leading dimension of the array AB. LDAB >= KD+1. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the right hand side matrix B. ! On exit, if INFO = 0, the solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the i-th diagonal element of A is zero, ! indicating that the matrix is singular and the ! solutions X have not been computed. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOUNIT, UPPER INTEGER J ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DTBSV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 NOUNIT = LSAME( DIAG, 'N' ) UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND. .NOT. & LSAME( TRANS, 'T' ) .AND. .NOT.LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( KD.LT.0 ) THEN INFO = -5 ELSE IF( NRHS.LT.0 ) THEN INFO = -6 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -8 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -10 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTBTRS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Check for singularity. ! IF( NOUNIT ) THEN IF( UPPER ) THEN DO 10 INFO = 1, N IF( AB( KD+1, INFO ).EQ.ZERO ) & RETURN 10 CONTINUE ELSE DO 20 INFO = 1, N IF( AB( 1, INFO ).EQ.ZERO ) & RETURN 20 CONTINUE END IF END IF INFO = 0 ! ! Solve A * X = B or A' * X = B. ! DO 30 J = 1, NRHS CALL DTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, B( 1, J ), 1 ) 30 CONTINUE ! RETURN ! ! End of DTBTRS ! END SUBROUTINE DTGEVC( SIDE, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, & LDVL, VR, LDVR, MM, M, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER HOWMNY, SIDE INTEGER INFO, LDA, LDB, LDVL, LDVR, M, MM, N ! .. ! .. Array Arguments .. LOGICAL SELECT( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), VL( LDVL, * ), & VR( LDVR, * ), WORK( * ) ! .. ! ! ! Purpose ! ======= ! ! DTGEVC computes some or all of the right and/or left generalized ! eigenvectors of a pair of real upper triangular matrices (A,B). ! ! The right generalized eigenvector x and the left generalized ! eigenvector y of (A,B) corresponding to a generalized eigenvalue ! w are defined by: ! ! (A - wB) * x = 0 and y**H * (A - wB) = 0 ! ! where y**H denotes the conjugate tranpose of y. ! ! If an eigenvalue w is determined by zero diagonal elements of both A ! and B, a unit vector is returned as the corresponding eigenvector. ! ! If all eigenvectors are requested, the routine may either return ! the matrices X and/or Y of right or left eigenvectors of (A,B), or ! the products Z*X and/or Q*Y, where Z and Q are input orthogonal ! matrices. If (A,B) was obtained from the generalized real-Schur ! factorization of an original pair of matrices ! (A0,B0) = (Q*A*Z**H,Q*B*Z**H), ! then Z*X and Q*Y are the matrices of right or left eigenvectors of ! A. ! ! A must be block upper triangular, with 1-by-1 and 2-by-2 diagonal ! blocks. Corresponding to each 2-by-2 diagonal block is a complex ! conjugate pair of eigenvalues and eigenvectors; only one ! eigenvector of the pair is computed, namely the one corresponding ! to the eigenvalue with positive imaginary part. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'R': compute right eigenvectors only; ! = 'L': compute left eigenvectors only; ! = 'B': compute both right and left eigenvectors. ! ! HOWMNY (input) CHARACTER*1 ! = 'A': compute all right and/or left eigenvectors; ! = 'B': compute all right and/or left eigenvectors, and ! backtransform them using the input matrices supplied ! in VR and/or VL; ! = 'S': compute selected right and/or left eigenvectors, ! specified by the logical array SELECT. ! ! SELECT (input) LOGICAL array, dimension (N) ! If HOWMNY='S', SELECT specifies the eigenvectors to be ! computed. ! If HOWMNY='A' or 'B', SELECT is not referenced. ! To select the real eigenvector corresponding to the real ! eigenvalue w(j), SELECT(j) must be set to .TRUE. To select ! the complex eigenvector corresponding to a complex conjugate ! pair w(j) and w(j+1), either SELECT(j) or SELECT(j+1) must ! be set to .TRUE.. ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The upper quasi-triangular matrix A. ! ! LDA (input) INTEGER ! The leading dimension of array A. LDA >= max(1, N). ! ! B (input) DOUBLE PRECISION array, dimension (LDB,N) ! The upper triangular matrix B. If A has a 2-by-2 diagonal ! block, then the corresponding 2-by-2 block of B must be ! diagonal with positive elements. ! ! LDB (input) INTEGER ! The leading dimension of array B. LDB >= max(1,N). ! ! VL (input/output) DOUBLE PRECISION array, dimension (LDVL,MM) ! On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must ! contain an N-by-N matrix Q (usually the orthogonal matrix Q ! of left Schur vectors returned by DHGEQZ). ! On exit, if SIDE = 'L' or 'B', VL contains: ! if HOWMNY = 'A', the matrix Y of left eigenvectors of (A,B); ! if HOWMNY = 'B', the matrix Q*Y; ! if HOWMNY = 'S', the left eigenvectors of (A,B) specified by ! SELECT, stored consecutively in the columns of ! VL, in the same order as their eigenvalues. ! If SIDE = 'R', VL is not referenced. ! ! A complex eigenvector corresponding to a complex eigenvalue ! is stored in two consecutive columns, the first holding the ! real part, and the second the imaginary part. ! ! LDVL (input) INTEGER ! The leading dimension of array VL. ! LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise. ! ! VR (input/output) DOUBLE PRECISION array, dimension (LDVR,MM) ! On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must ! contain an N-by-N matrix Q (usually the orthogonal matrix Z ! of right Schur vectors returned by DHGEQZ). ! On exit, if SIDE = 'R' or 'B', VR contains: ! if HOWMNY = 'A', the matrix X of right eigenvectors of (A,B); ! if HOWMNY = 'B', the matrix Z*X; ! if HOWMNY = 'S', the right eigenvectors of (A,B) specified by ! SELECT, stored consecutively in the columns of ! VR, in the same order as their eigenvalues. ! If SIDE = 'L', VR is not referenced. ! ! A complex eigenvector corresponding to a complex eigenvalue ! is stored in two consecutive columns, the first holding the ! real part and the second the imaginary part. ! ! LDVR (input) INTEGER ! The leading dimension of the array VR. ! LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise. ! ! MM (input) INTEGER ! The number of columns in the arrays VL and/or VR. MM >= M. ! ! M (output) INTEGER ! The number of columns in the arrays VL and/or VR actually ! used to store the eigenvectors. If HOWMNY = 'A' or 'B', M ! is set to N. Each selected real eigenvector occupies one ! column and each selected complex eigenvector occupies two ! columns. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (6*N) ! ! INFO (output) INTEGER ! = 0: successful exit. ! < 0: if INFO = -i, the i-th argument had an illegal value. ! > 0: the 2-by-2 block (INFO:INFO+1) does not have a complex ! eigenvalue. ! ! Further Details ! =============== ! ! Allocation of workspace: ! ---------- -- --------- ! ! WORK( j ) = 1-norm of j-th column of A, above the diagonal ! WORK( N+j ) = 1-norm of j-th column of B, above the diagonal ! WORK( 2*N+1:3*N ) = real part of eigenvector ! WORK( 3*N+1:4*N ) = imaginary part of eigenvector ! WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector ! WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector ! ! Rowwise vs. columnwise solution methods: ! ------- -- ---------- -------- ------- ! ! Finding a generalized eigenvector consists basically of solving the ! singular triangular system ! ! (A - w B) x = 0 (for right) or: (A - w B)**H y = 0 (for left) ! ! Consider finding the i-th right eigenvector (assume all eigenvalues ! are real). The equation to be solved is: ! n i ! 0 = sum C(j,k) v(k) = sum C(j,k) v(k) for j = i,. . .,1 ! k=j k=j ! ! where C = (A - w B) (The components v(i+1:n) are 0.) ! ! The "rowwise" method is: ! ! (1) v(i) := 1 ! for j = i-1,. . .,1: ! i ! (2) compute s = - sum C(j,k) v(k) and ! k=j+1 ! ! (3) v(j) := s / C(j,j) ! ! Step 2 is sometimes called the "dot product" step, since it is an ! inner product between the j-th row and the portion of the eigenvector ! that has been computed so far. ! ! The "columnwise" method consists basically in doing the sums ! for all the rows in parallel. As each v(j) is computed, the ! contribution of v(j) times the j-th column of C is added to the ! partial sums. Since FORTRAN arrays are stored columnwise, this has ! the advantage that at each step, the elements of C that are accessed ! are adjacent to one another, whereas with the rowwise method, the ! elements accessed at a step are spaced LDA (and LDB) words apart. ! ! When finding left eigenvectors, the matrix in question is the ! transpose of the one in storage, so the rowwise method then ! actually accesses columns of A and B at each step, and so is the ! preferred method. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, SAFETY PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, & SAFETY = 1.0D+2 ) ! .. ! .. Local Scalars .. LOGICAL COMPL, COMPR, IL2BY2, ILABAD, ILALL, ILBACK, & ILBBAD, ILCOMP, ILCPLX, LSA, LSB INTEGER I, IBEG, IEIG, IEND, IHWMNY, IINFO, IM, ISIDE, & J, JA, JC, JE, JR, JW, NA, NW DOUBLE PRECISION ACOEF, ACOEFA, ANORM, ASCALE, BCOEFA, BCOEFI, & BCOEFR, BIG, BIGNUM, BNORM, BSCALE, CIM2A, & CIM2B, CIMAGA, CIMAGB, CRE2A, CRE2B, CREALA, & CREALB, DMIN, SAFMIN, SALFAR, SBETA, SCALE, & SMALL, TEMP, TEMP2, TEMP2I, TEMP2R, ULP, XMAX, & XSCALE ! .. ! .. Local Arrays .. DOUBLE PRECISION BDIAG( 2 ), SUM( 2, 2 ), SUMA( 2, 2 ), & SUMB( 2, 2 ) ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DGEMV, DLACPY, DLAG2, DLALN2, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Executable Statements .. ! ! Decode and Test the input parameters ! IF( LSAME( HOWMNY, 'A' ) ) THEN IHWMNY = 1 ILALL = .TRUE. ILBACK = .FALSE. ELSE IF( LSAME( HOWMNY, 'S' ) ) THEN IHWMNY = 2 ILALL = .FALSE. ILBACK = .FALSE. ELSE IF( LSAME( HOWMNY, 'B' ) .OR. LSAME( HOWMNY, 'T' ) ) THEN IHWMNY = 3 ILALL = .TRUE. ILBACK = .TRUE. ELSE IHWMNY = -1 ILALL = .TRUE. END IF ! IF( LSAME( SIDE, 'R' ) ) THEN ISIDE = 1 COMPL = .FALSE. COMPR = .TRUE. ELSE IF( LSAME( SIDE, 'L' ) ) THEN ISIDE = 2 COMPL = .TRUE. COMPR = .FALSE. ELSE IF( LSAME( SIDE, 'B' ) ) THEN ISIDE = 3 COMPL = .TRUE. COMPR = .TRUE. ELSE ISIDE = -1 END IF ! INFO = 0 IF( ISIDE.LT.0 ) THEN INFO = -1 ELSE IF( IHWMNY.LT.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTGEVC', -INFO ) RETURN END IF ! ! Count the number of eigenvectors to be computed ! IF( .NOT.ILALL ) THEN IM = 0 ILCPLX = .FALSE. DO 10 J = 1, N IF( ILCPLX ) THEN ILCPLX = .FALSE. GO TO 10 END IF IF( J.LT.N ) THEN IF( A( J+1, J ).NE.ZERO ) & ILCPLX = .TRUE. END IF IF( ILCPLX ) THEN IF( SELECT( J ) .OR. SELECT( J+1 ) ) & IM = IM + 2 ELSE IF( SELECT( J ) ) & IM = IM + 1 END IF 10 CONTINUE ELSE IM = N END IF ! ! Check 2-by-2 diagonal blocks of A, B ! ILABAD = .FALSE. ILBBAD = .FALSE. DO 20 J = 1, N - 1 IF( A( J+1, J ).NE.ZERO ) THEN IF( B( J, J ).EQ.ZERO .OR. B( J+1, J+1 ).EQ.ZERO .OR. & B( J, J+1 ).NE.ZERO )ILBBAD = .TRUE. IF( J.LT.N-1 ) THEN IF( A( J+2, J+1 ).NE.ZERO ) & ILABAD = .TRUE. END IF END IF 20 CONTINUE ! IF( ILABAD ) THEN INFO = -5 ELSE IF( ILBBAD ) THEN INFO = -7 ELSE IF( COMPL .AND. LDVL.LT.N .OR. LDVL.LT.1 ) THEN INFO = -10 ELSE IF( COMPR .AND. LDVR.LT.N .OR. LDVR.LT.1 ) THEN INFO = -12 ELSE IF( MM.LT.IM ) THEN INFO = -13 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTGEVC', -INFO ) RETURN END IF ! ! Quick return if possible ! M = IM IF( N.EQ.0 ) & RETURN ! ! Machine Constants ! SAFMIN = DLAMCH( 'Safe minimum' ) BIG = ONE / SAFMIN CALL DLABAD( SAFMIN, BIG ) ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' ) SMALL = SAFMIN*N / ULP BIG = ONE / SMALL BIGNUM = ONE / ( SAFMIN*N ) ! ! Compute the 1-norm of each column of the strictly upper triangular ! part (i.e., excluding all elements belonging to the diagonal ! blocks) of A and B to check for possible overflow in the ! triangular solver. ! ANORM = ABS( A( 1, 1 ) ) IF( N.GT.1 ) & ANORM = ANORM + ABS( A( 2, 1 ) ) BNORM = ABS( B( 1, 1 ) ) WORK( 1 ) = ZERO WORK( N+1 ) = ZERO ! DO 50 J = 2, N TEMP = ZERO TEMP2 = ZERO IF( A( J, J-1 ).EQ.ZERO ) THEN IEND = J - 1 ELSE IEND = J - 2 END IF DO 30 I = 1, IEND TEMP = TEMP + ABS( A( I, J ) ) TEMP2 = TEMP2 + ABS( B( I, J ) ) 30 CONTINUE WORK( J ) = TEMP WORK( N+J ) = TEMP2 DO 40 I = IEND + 1, MIN( J+1, N ) TEMP = TEMP + ABS( A( I, J ) ) TEMP2 = TEMP2 + ABS( B( I, J ) ) 40 CONTINUE ANORM = MAX( ANORM, TEMP ) BNORM = MAX( BNORM, TEMP2 ) 50 CONTINUE ! ASCALE = ONE / MAX( ANORM, SAFMIN ) BSCALE = ONE / MAX( BNORM, SAFMIN ) ! ! Left eigenvectors ! IF( COMPL ) THEN IEIG = 0 ! ! Main loop over eigenvalues ! ILCPLX = .FALSE. DO 220 JE = 1, N ! ! Skip this iteration if (a) HOWMNY='S' and SELECT=.FALSE., or ! (b) this would be the second of a complex pair. ! Check for complex eigenvalue, so as to be sure of which ! entry(-ies) of SELECT to look at. ! IF( ILCPLX ) THEN ILCPLX = .FALSE. GO TO 220 END IF NW = 1 IF( JE.LT.N ) THEN IF( A( JE+1, JE ).NE.ZERO ) THEN ILCPLX = .TRUE. NW = 2 END IF END IF IF( ILALL ) THEN ILCOMP = .TRUE. ELSE IF( ILCPLX ) THEN ILCOMP = SELECT( JE ) .OR. SELECT( JE+1 ) ELSE ILCOMP = SELECT( JE ) END IF IF( .NOT.ILCOMP ) & GO TO 220 ! ! Decide if (a) singular pencil, (b) real eigenvalue, or ! (c) complex eigenvalue. ! IF( .NOT.ILCPLX ) THEN IF( ABS( A( JE, JE ) ).LE.SAFMIN .AND. & ABS( B( JE, JE ) ).LE.SAFMIN ) THEN ! ! Singular matrix pencil -- return unit eigenvector ! IEIG = IEIG + 1 DO 60 JR = 1, N VL( JR, IEIG ) = ZERO 60 CONTINUE VL( IEIG, IEIG ) = ONE GO TO 220 END IF END IF ! ! Clear vector ! DO 70 JR = 1, NW*N WORK( 2*N+JR ) = ZERO 70 CONTINUE ! T ! Compute coefficients in ( a A - b B ) y = 0 ! a is ACOEF ! b is BCOEFR + i*BCOEFI ! IF( .NOT.ILCPLX ) THEN ! ! Real eigenvalue ! TEMP = ONE / MAX( ABS( A( JE, JE ) )*ASCALE, & ABS( B( JE, JE ) )*BSCALE, SAFMIN ) SALFAR = ( TEMP*A( JE, JE ) )*ASCALE SBETA = ( TEMP*B( JE, JE ) )*BSCALE ACOEF = SBETA*ASCALE BCOEFR = SALFAR*BSCALE BCOEFI = ZERO ! ! Scale to avoid underflow ! SCALE = ONE LSA = ABS( SBETA ).GE.SAFMIN .AND. ABS( ACOEF ).LT.SMALL LSB = ABS( SALFAR ).GE.SAFMIN .AND. ABS( BCOEFR ).LT. & SMALL IF( LSA ) & SCALE = ( SMALL / ABS( SBETA ) )*MIN( ANORM, BIG ) IF( LSB ) & SCALE = MAX( SCALE, ( SMALL / ABS( SALFAR ) )* & MIN( BNORM, BIG ) ) IF( LSA .OR. LSB ) THEN SCALE = MIN( SCALE, ONE / & ( SAFMIN*MAX( ONE, ABS( ACOEF ), & ABS( BCOEFR ) ) ) ) IF( LSA ) THEN ACOEF = ASCALE*( SCALE*SBETA ) ELSE ACOEF = SCALE*ACOEF END IF IF( LSB ) THEN BCOEFR = BSCALE*( SCALE*SALFAR ) ELSE BCOEFR = SCALE*BCOEFR END IF END IF ACOEFA = ABS( ACOEF ) BCOEFA = ABS( BCOEFR ) ! ! First component is 1 ! WORK( 2*N+JE ) = ONE XMAX = ONE ELSE ! ! Complex eigenvalue ! CALL DLAG2( A( JE, JE ), LDA, B( JE, JE ), LDB, & SAFMIN*SAFETY, ACOEF, TEMP, BCOEFR, TEMP2, & BCOEFI ) BCOEFI = -BCOEFI IF( BCOEFI.EQ.ZERO ) THEN INFO = JE RETURN END IF ! ! Scale to avoid over/underflow ! ACOEFA = ABS( ACOEF ) BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI ) SCALE = ONE IF( ACOEFA*ULP.LT.SAFMIN .AND. ACOEFA.GE.SAFMIN ) & SCALE = ( SAFMIN / ULP ) / ACOEFA IF( BCOEFA*ULP.LT.SAFMIN .AND. BCOEFA.GE.SAFMIN ) & SCALE = MAX( SCALE, ( SAFMIN / ULP ) / BCOEFA ) IF( SAFMIN*ACOEFA.GT.ASCALE ) & SCALE = ASCALE / ( SAFMIN*ACOEFA ) IF( SAFMIN*BCOEFA.GT.BSCALE ) & SCALE = MIN( SCALE, BSCALE / ( SAFMIN*BCOEFA ) ) IF( SCALE.NE.ONE ) THEN ACOEF = SCALE*ACOEF ACOEFA = ABS( ACOEF ) BCOEFR = SCALE*BCOEFR BCOEFI = SCALE*BCOEFI BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI ) END IF ! ! Compute first two components of eigenvector ! TEMP = ACOEF*A( JE+1, JE ) TEMP2R = ACOEF*A( JE, JE ) - BCOEFR*B( JE, JE ) TEMP2I = -BCOEFI*B( JE, JE ) IF( ABS( TEMP ).GT.ABS( TEMP2R )+ABS( TEMP2I ) ) THEN WORK( 2*N+JE ) = ONE WORK( 3*N+JE ) = ZERO WORK( 2*N+JE+1 ) = -TEMP2R / TEMP WORK( 3*N+JE+1 ) = -TEMP2I / TEMP ELSE WORK( 2*N+JE+1 ) = ONE WORK( 3*N+JE+1 ) = ZERO TEMP = ACOEF*A( JE, JE+1 ) WORK( 2*N+JE ) = ( BCOEFR*B( JE+1, JE+1 )-ACOEF* & A( JE+1, JE+1 ) ) / TEMP WORK( 3*N+JE ) = BCOEFI*B( JE+1, JE+1 ) / TEMP END IF XMAX = MAX( ABS( WORK( 2*N+JE ) )+ABS( WORK( 3*N+JE ) ), & ABS( WORK( 2*N+JE+1 ) )+ABS( WORK( 3*N+JE+1 ) ) ) END IF ! DMIN = MAX( ULP*ACOEFA*ANORM, ULP*BCOEFA*BNORM, SAFMIN ) ! ! T ! Triangular solve of (a A - b B) y = 0 ! ! T ! (rowwise in (a A - b B) , or columnwise in (a A - b B) ) ! IL2BY2 = .FALSE. ! DO 160 J = JE + NW, N IF( IL2BY2 ) THEN IL2BY2 = .FALSE. GO TO 160 END IF ! NA = 1 BDIAG( 1 ) = B( J, J ) IF( J.LT.N ) THEN IF( A( J+1, J ).NE.ZERO ) THEN IL2BY2 = .TRUE. BDIAG( 2 ) = B( J+1, J+1 ) NA = 2 END IF END IF ! ! Check whether scaling is necessary for dot products ! XSCALE = ONE / MAX( ONE, XMAX ) TEMP = MAX( WORK( J ), WORK( N+J ), & ACOEFA*WORK( J )+BCOEFA*WORK( N+J ) ) IF( IL2BY2 ) & TEMP = MAX( TEMP, WORK( J+1 ), WORK( N+J+1 ), & ACOEFA*WORK( J+1 )+BCOEFA*WORK( N+J+1 ) ) IF( TEMP.GT.BIGNUM*XSCALE ) THEN DO 90 JW = 0, NW - 1 DO 80 JR = JE, J - 1 WORK( ( JW+2 )*N+JR ) = XSCALE* & WORK( ( JW+2 )*N+JR ) 80 CONTINUE 90 CONTINUE XMAX = XMAX*XSCALE END IF ! ! Compute dot products ! ! j-1 ! SUM = sum conjg( a*A(k,j) - b*B(k,j) )*x(k) ! k=je ! ! To reduce the op count, this is done as ! ! _ j-1 _ j-1 ! a*conjg( sum A(k,j)*x(k) ) - b*conjg( sum B(k,j)*x(k) ) ! k=je k=je ! ! which may cause underflow problems if A or B are close ! to underflow. (E.g., less than SMALL.) ! ! ! A series of compiler directives to defeat vectorization ! for the next loop ! !$PL$ CMCHAR=' ' !DIR$ NEXTSCALAR !$DIR SCALAR !DIR$ NEXT SCALAR !VD$L NOVECTOR !DEC$ NOVECTOR !VD$ NOVECTOR !VDIR NOVECTOR !VOCL LOOP,SCALAR !IBM PREFER SCALAR !$PL$ CMCHAR='*' ! DO 120 JW = 1, NW ! !$PL$ CMCHAR=' ' !DIR$ NEXTSCALAR !$DIR SCALAR !DIR$ NEXT SCALAR !VD$L NOVECTOR !DEC$ NOVECTOR !VD$ NOVECTOR !VDIR NOVECTOR !VOCL LOOP,SCALAR !IBM PREFER SCALAR !$PL$ CMCHAR='*' ! DO 110 JA = 1, NA SUMA( JA, JW ) = ZERO SUMB( JA, JW ) = ZERO ! DO 100 JR = JE, J - 1 SUMA( JA, JW ) = SUMA( JA, JW ) + & A( JR, J+JA-1 )* & WORK( ( JW+1 )*N+JR ) SUMB( JA, JW ) = SUMB( JA, JW ) + & B( JR, J+JA-1 )* & WORK( ( JW+1 )*N+JR ) 100 CONTINUE 110 CONTINUE 120 CONTINUE ! !$PL$ CMCHAR=' ' !DIR$ NEXTSCALAR !$DIR SCALAR !DIR$ NEXT SCALAR !VD$L NOVECTOR !DEC$ NOVECTOR !VD$ NOVECTOR !VDIR NOVECTOR !VOCL LOOP,SCALAR !IBM PREFER SCALAR !$PL$ CMCHAR='*' ! DO 130 JA = 1, NA IF( ILCPLX ) THEN SUM( JA, 1 ) = -ACOEF*SUMA( JA, 1 ) + & BCOEFR*SUMB( JA, 1 ) - & BCOEFI*SUMB( JA, 2 ) SUM( JA, 2 ) = -ACOEF*SUMA( JA, 2 ) + & BCOEFR*SUMB( JA, 2 ) + & BCOEFI*SUMB( JA, 1 ) ELSE SUM( JA, 1 ) = -ACOEF*SUMA( JA, 1 ) + & BCOEFR*SUMB( JA, 1 ) END IF 130 CONTINUE ! ! T ! Solve ( a A - b B ) y = SUM(,) ! with scaling and perturbation of the denominator ! CALL DLALN2( .TRUE., NA, NW, DMIN, ACOEF, A( J, J ), LDA, & BDIAG( 1 ), BDIAG( 2 ), SUM, 2, BCOEFR, & BCOEFI, WORK( 2*N+J ), N, SCALE, TEMP, & IINFO ) IF( SCALE.LT.ONE ) THEN DO 150 JW = 0, NW - 1 DO 140 JR = JE, J - 1 WORK( ( JW+2 )*N+JR ) = SCALE* & WORK( ( JW+2 )*N+JR ) 140 CONTINUE 150 CONTINUE XMAX = SCALE*XMAX END IF XMAX = MAX( XMAX, TEMP ) 160 CONTINUE ! ! Copy eigenvector to VL, back transforming if ! HOWMNY='B'. ! IEIG = IEIG + 1 IF( ILBACK ) THEN DO 170 JW = 0, NW - 1 CALL DGEMV( 'N', N, N+1-JE, ONE, VL( 1, JE ), LDVL, & WORK( ( JW+2 )*N+JE ), 1, ZERO, & WORK( ( JW+4 )*N+1 ), 1 ) 170 CONTINUE CALL DLACPY( ' ', N, NW, WORK( 4*N+1 ), N, VL( 1, JE ), & LDVL ) IBEG = 1 ELSE CALL DLACPY( ' ', N, NW, WORK( 2*N+1 ), N, VL( 1, IEIG ), & LDVL ) IBEG = JE END IF ! ! Scale eigenvector ! XMAX = ZERO IF( ILCPLX ) THEN DO 180 J = IBEG, N XMAX = MAX( XMAX, ABS( VL( J, IEIG ) )+ & ABS( VL( J, IEIG+1 ) ) ) 180 CONTINUE ELSE DO 190 J = IBEG, N XMAX = MAX( XMAX, ABS( VL( J, IEIG ) ) ) 190 CONTINUE END IF ! IF( XMAX.GT.SAFMIN ) THEN XSCALE = ONE / XMAX ! DO 210 JW = 0, NW - 1 DO 200 JR = IBEG, N VL( JR, IEIG+JW ) = XSCALE*VL( JR, IEIG+JW ) 200 CONTINUE 210 CONTINUE END IF IEIG = IEIG + NW - 1 ! 220 CONTINUE END IF ! ! Right eigenvectors ! IF( COMPR ) THEN IEIG = IM + 1 ! ! Main loop over eigenvalues ! ILCPLX = .FALSE. DO 500 JE = N, 1, -1 ! ! Skip this iteration if (a) HOWMNY='S' and SELECT=.FALSE., or ! (b) this would be the second of a complex pair. ! Check for complex eigenvalue, so as to be sure of which ! entry(-ies) of SELECT to look at -- if complex, SELECT(JE) ! or SELECT(JE-1). ! If this is a complex pair, the 2-by-2 diagonal block ! corresponding to the eigenvalue is in rows/columns JE-1:JE ! IF( ILCPLX ) THEN ILCPLX = .FALSE. GO TO 500 END IF NW = 1 IF( JE.GT.1 ) THEN IF( A( JE, JE-1 ).NE.ZERO ) THEN ILCPLX = .TRUE. NW = 2 END IF END IF IF( ILALL ) THEN ILCOMP = .TRUE. ELSE IF( ILCPLX ) THEN ILCOMP = SELECT( JE ) .OR. SELECT( JE-1 ) ELSE ILCOMP = SELECT( JE ) END IF IF( .NOT.ILCOMP ) & GO TO 500 ! ! Decide if (a) singular pencil, (b) real eigenvalue, or ! (c) complex eigenvalue. ! IF( .NOT.ILCPLX ) THEN IF( ABS( A( JE, JE ) ).LE.SAFMIN .AND. & ABS( B( JE, JE ) ).LE.SAFMIN ) THEN ! ! Singular matrix pencil -- unit eigenvector ! IEIG = IEIG - 1 DO 230 JR = 1, N VR( JR, IEIG ) = ZERO 230 CONTINUE VR( IEIG, IEIG ) = ONE GO TO 500 END IF END IF ! ! Clear vector ! DO 250 JW = 0, NW - 1 DO 240 JR = 1, N WORK( ( JW+2 )*N+JR ) = ZERO 240 CONTINUE 250 CONTINUE ! ! Compute coefficients in ( a A - b B ) x = 0 ! a is ACOEF ! b is BCOEFR + i*BCOEFI ! IF( .NOT.ILCPLX ) THEN ! ! Real eigenvalue ! TEMP = ONE / MAX( ABS( A( JE, JE ) )*ASCALE, & ABS( B( JE, JE ) )*BSCALE, SAFMIN ) SALFAR = ( TEMP*A( JE, JE ) )*ASCALE SBETA = ( TEMP*B( JE, JE ) )*BSCALE ACOEF = SBETA*ASCALE BCOEFR = SALFAR*BSCALE BCOEFI = ZERO ! ! Scale to avoid underflow ! SCALE = ONE LSA = ABS( SBETA ).GE.SAFMIN .AND. ABS( ACOEF ).LT.SMALL LSB = ABS( SALFAR ).GE.SAFMIN .AND. ABS( BCOEFR ).LT. & SMALL IF( LSA ) & SCALE = ( SMALL / ABS( SBETA ) )*MIN( ANORM, BIG ) IF( LSB ) & SCALE = MAX( SCALE, ( SMALL / ABS( SALFAR ) )* & MIN( BNORM, BIG ) ) IF( LSA .OR. LSB ) THEN SCALE = MIN( SCALE, ONE / & ( SAFMIN*MAX( ONE, ABS( ACOEF ), & ABS( BCOEFR ) ) ) ) IF( LSA ) THEN ACOEF = ASCALE*( SCALE*SBETA ) ELSE ACOEF = SCALE*ACOEF END IF IF( LSB ) THEN BCOEFR = BSCALE*( SCALE*SALFAR ) ELSE BCOEFR = SCALE*BCOEFR END IF END IF ACOEFA = ABS( ACOEF ) BCOEFA = ABS( BCOEFR ) ! ! First component is 1 ! WORK( 2*N+JE ) = ONE XMAX = ONE ! ! Compute contribution from column JE of A and B to sum ! (See "Further Details", above.) ! DO 260 JR = 1, JE - 1 WORK( 2*N+JR ) = BCOEFR*B( JR, JE ) - & ACOEF*A( JR, JE ) 260 CONTINUE ELSE ! ! Complex eigenvalue ! CALL DLAG2( A( JE-1, JE-1 ), LDA, B( JE-1, JE-1 ), LDB, & SAFMIN*SAFETY, ACOEF, TEMP, BCOEFR, TEMP2, & BCOEFI ) IF( BCOEFI.EQ.ZERO ) THEN INFO = JE - 1 RETURN END IF ! ! Scale to avoid over/underflow ! ACOEFA = ABS( ACOEF ) BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI ) SCALE = ONE IF( ACOEFA*ULP.LT.SAFMIN .AND. ACOEFA.GE.SAFMIN ) & SCALE = ( SAFMIN / ULP ) / ACOEFA IF( BCOEFA*ULP.LT.SAFMIN .AND. BCOEFA.GE.SAFMIN ) & SCALE = MAX( SCALE, ( SAFMIN / ULP ) / BCOEFA ) IF( SAFMIN*ACOEFA.GT.ASCALE ) & SCALE = ASCALE / ( SAFMIN*ACOEFA ) IF( SAFMIN*BCOEFA.GT.BSCALE ) & SCALE = MIN( SCALE, BSCALE / ( SAFMIN*BCOEFA ) ) IF( SCALE.NE.ONE ) THEN ACOEF = SCALE*ACOEF ACOEFA = ABS( ACOEF ) BCOEFR = SCALE*BCOEFR BCOEFI = SCALE*BCOEFI BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI ) END IF ! ! Compute first two components of eigenvector ! and contribution to sums ! TEMP = ACOEF*A( JE, JE-1 ) TEMP2R = ACOEF*A( JE, JE ) - BCOEFR*B( JE, JE ) TEMP2I = -BCOEFI*B( JE, JE ) IF( ABS( TEMP ).GE.ABS( TEMP2R )+ABS( TEMP2I ) ) THEN WORK( 2*N+JE ) = ONE WORK( 3*N+JE ) = ZERO WORK( 2*N+JE-1 ) = -TEMP2R / TEMP WORK( 3*N+JE-1 ) = -TEMP2I / TEMP ELSE WORK( 2*N+JE-1 ) = ONE WORK( 3*N+JE-1 ) = ZERO TEMP = ACOEF*A( JE-1, JE ) WORK( 2*N+JE ) = ( BCOEFR*B( JE-1, JE-1 )-ACOEF* & A( JE-1, JE-1 ) ) / TEMP WORK( 3*N+JE ) = BCOEFI*B( JE-1, JE-1 ) / TEMP END IF ! XMAX = MAX( ABS( WORK( 2*N+JE ) )+ABS( WORK( 3*N+JE ) ), & ABS( WORK( 2*N+JE-1 ) )+ABS( WORK( 3*N+JE-1 ) ) ) ! ! Compute contribution from columns JE and JE-1 ! of A and B to the sums. ! CREALA = ACOEF*WORK( 2*N+JE-1 ) CIMAGA = ACOEF*WORK( 3*N+JE-1 ) CREALB = BCOEFR*WORK( 2*N+JE-1 ) - & BCOEFI*WORK( 3*N+JE-1 ) CIMAGB = BCOEFI*WORK( 2*N+JE-1 ) + & BCOEFR*WORK( 3*N+JE-1 ) CRE2A = ACOEF*WORK( 2*N+JE ) CIM2A = ACOEF*WORK( 3*N+JE ) CRE2B = BCOEFR*WORK( 2*N+JE ) - BCOEFI*WORK( 3*N+JE ) CIM2B = BCOEFI*WORK( 2*N+JE ) + BCOEFR*WORK( 3*N+JE ) DO 270 JR = 1, JE - 2 WORK( 2*N+JR ) = -CREALA*A( JR, JE-1 ) + & CREALB*B( JR, JE-1 ) - & CRE2A*A( JR, JE ) + CRE2B*B( JR, JE ) WORK( 3*N+JR ) = -CIMAGA*A( JR, JE-1 ) + & CIMAGB*B( JR, JE-1 ) - & CIM2A*A( JR, JE ) + CIM2B*B( JR, JE ) 270 CONTINUE END IF ! DMIN = MAX( ULP*ACOEFA*ANORM, ULP*BCOEFA*BNORM, SAFMIN ) ! ! Columnwise triangular solve of (a A - b B) x = 0 ! IL2BY2 = .FALSE. DO 370 J = JE - NW, 1, -1 ! ! If a 2-by-2 block, is in position j-1:j, wait until ! next iteration to process it (when it will be j:j+1) ! IF( .NOT.IL2BY2 .AND. J.GT.1 ) THEN IF( A( J, J-1 ).NE.ZERO ) THEN IL2BY2 = .TRUE. GO TO 370 END IF END IF BDIAG( 1 ) = B( J, J ) IF( IL2BY2 ) THEN NA = 2 BDIAG( 2 ) = B( J+1, J+1 ) ELSE NA = 1 END IF ! ! Compute x(j) (and x(j+1), if 2-by-2 block) ! CALL DLALN2( .FALSE., NA, NW, DMIN, ACOEF, A( J, J ), & LDA, BDIAG( 1 ), BDIAG( 2 ), WORK( 2*N+J ), & N, BCOEFR, BCOEFI, SUM, 2, SCALE, TEMP, & IINFO ) IF( SCALE.LT.ONE ) THEN ! DO 290 JW = 0, NW - 1 DO 280 JR = 1, JE WORK( ( JW+2 )*N+JR ) = SCALE* & WORK( ( JW+2 )*N+JR ) 280 CONTINUE 290 CONTINUE END IF XMAX = MAX( SCALE*XMAX, TEMP ) ! DO 310 JW = 1, NW DO 300 JA = 1, NA WORK( ( JW+1 )*N+J+JA-1 ) = SUM( JA, JW ) 300 CONTINUE 310 CONTINUE ! ! w = w + x(j)*(a A(*,j) - b B(*,j) ) with scaling ! IF( J.GT.1 ) THEN ! ! Check whether scaling is necessary for sum. ! XSCALE = ONE / MAX( ONE, XMAX ) TEMP = ACOEFA*WORK( J ) + BCOEFA*WORK( N+J ) IF( IL2BY2 ) & TEMP = MAX( TEMP, ACOEFA*WORK( J+1 )+BCOEFA* & WORK( N+J+1 ) ) TEMP = MAX( TEMP, ACOEFA, BCOEFA ) IF( TEMP.GT.BIGNUM*XSCALE ) THEN ! DO 330 JW = 0, NW - 1 DO 320 JR = 1, JE WORK( ( JW+2 )*N+JR ) = XSCALE* & WORK( ( JW+2 )*N+JR ) 320 CONTINUE 330 CONTINUE XMAX = XMAX*XSCALE END IF ! ! Compute the contributions of the off-diagonals of ! column j (and j+1, if 2-by-2 block) of A and B to the ! sums. ! ! DO 360 JA = 1, NA IF( ILCPLX ) THEN CREALA = ACOEF*WORK( 2*N+J+JA-1 ) CIMAGA = ACOEF*WORK( 3*N+J+JA-1 ) CREALB = BCOEFR*WORK( 2*N+J+JA-1 ) - & BCOEFI*WORK( 3*N+J+JA-1 ) CIMAGB = BCOEFI*WORK( 2*N+J+JA-1 ) + & BCOEFR*WORK( 3*N+J+JA-1 ) DO 340 JR = 1, J - 1 WORK( 2*N+JR ) = WORK( 2*N+JR ) - & CREALA*A( JR, J+JA-1 ) + & CREALB*B( JR, J+JA-1 ) WORK( 3*N+JR ) = WORK( 3*N+JR ) - & CIMAGA*A( JR, J+JA-1 ) + & CIMAGB*B( JR, J+JA-1 ) 340 CONTINUE ELSE CREALA = ACOEF*WORK( 2*N+J+JA-1 ) CREALB = BCOEFR*WORK( 2*N+J+JA-1 ) DO 350 JR = 1, J - 1 WORK( 2*N+JR ) = WORK( 2*N+JR ) - & CREALA*A( JR, J+JA-1 ) + & CREALB*B( JR, J+JA-1 ) 350 CONTINUE END IF 360 CONTINUE END IF ! IL2BY2 = .FALSE. 370 CONTINUE ! ! Copy eigenvector to VR, back transforming if ! HOWMNY='B'. ! IEIG = IEIG - NW IF( ILBACK ) THEN ! DO 410 JW = 0, NW - 1 DO 380 JR = 1, N WORK( ( JW+4 )*N+JR ) = WORK( ( JW+2 )*N+1 )* & VR( JR, 1 ) 380 CONTINUE ! ! A series of compiler directives to defeat ! vectorization for the next loop ! ! DO 400 JC = 2, JE DO 390 JR = 1, N WORK( ( JW+4 )*N+JR ) = WORK( ( JW+4 )*N+JR ) + & WORK( ( JW+2 )*N+JC )*VR( JR, JC ) 390 CONTINUE 400 CONTINUE 410 CONTINUE ! DO 430 JW = 0, NW - 1 DO 420 JR = 1, N VR( JR, IEIG+JW ) = WORK( ( JW+4 )*N+JR ) 420 CONTINUE 430 CONTINUE ! IEND = N ELSE DO 450 JW = 0, NW - 1 DO 440 JR = 1, N VR( JR, IEIG+JW ) = WORK( ( JW+2 )*N+JR ) 440 CONTINUE 450 CONTINUE ! IEND = JE END IF ! ! Scale eigenvector ! XMAX = ZERO IF( ILCPLX ) THEN DO 460 J = 1, IEND XMAX = MAX( XMAX, ABS( VR( J, IEIG ) )+ & ABS( VR( J, IEIG+1 ) ) ) 460 CONTINUE ELSE DO 470 J = 1, IEND XMAX = MAX( XMAX, ABS( VR( J, IEIG ) ) ) 470 CONTINUE END IF ! IF( XMAX.GT.SAFMIN ) THEN XSCALE = ONE / XMAX DO 490 JW = 0, NW - 1 DO 480 JR = 1, IEND VR( JR, IEIG+JW ) = XSCALE*VR( JR, IEIG+JW ) 480 CONTINUE 490 CONTINUE END IF 500 CONTINUE END IF ! RETURN ! ! End of DTGEVC ! END SUBROUTINE DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, & LDZ, J1, N1, N2, WORK, LWORK, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. LOGICAL WANTQ, WANTZ INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, LWORK, N, N1, N2 ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), & WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DTGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22) ! of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair ! (A, B) by an orthogonal equivalence transformation. ! ! (A, B) must be in generalized real Schur canonical form (as returned ! by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2 ! diagonal blocks. B is upper triangular. ! ! Optionally, the matrices Q and Z of generalized Schur vectors are ! updated. ! ! Q(in) * A(in) * Z(in)' = Q(out) * A(out) * Z(out)' ! Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)' ! ! ! Arguments ! ========= ! ! WANTQ (input) LOGICAL ! .TRUE. : update the left transformation matrix Q; ! .FALSE.: do not update Q. ! ! WANTZ (input) LOGICAL ! .TRUE. : update the right transformation matrix Z; ! .FALSE.: do not update Z. ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! A (input/output) DOUBLE PRECISION arrays, dimensions (LDA,N) ! On entry, the matrix A in the pair (A, B). ! On exit, the updated matrix A. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION arrays, dimensions (LDB,N) ! On entry, the matrix B in the pair (A, B). ! On exit, the updated matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! Q (input/output) DOUBLE PRECISION array, dimension (LDZ,N) ! On entry, if WANTQ = .TRUE., the orthogonal matrix Q. ! On exit, the updated matrix Q. ! Not referenced if WANTQ = .FALSE.. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. LDQ >= 1. ! If WANTQ = .TRUE., LDQ >= N. ! ! Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N) ! On entry, if WANTZ =.TRUE., the orthogonal matrix Z. ! On exit, the updated matrix Z. ! Not referenced if WANTZ = .FALSE.. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1. ! If WANTZ = .TRUE., LDZ >= N. ! ! J1 (input) INTEGER ! The index to the first block (A11, B11). 1 <= J1 <= N. ! ! N1 (input) INTEGER ! The order of the first block (A11, B11). N1 = 0, 1 or 2. ! ! N2 (input) INTEGER ! The order of the second block (A22, B22). N2 = 0, 1 or 2. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (LWORK). ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! LWORK >= MAX( N*(N2+N1), (N2+N1)*(N2+N1)*2 ) ! ! INFO (output) INTEGER ! =0: Successful exit ! >0: If INFO = 1, the transformed matrix (A, B) would be ! too far from generalized Schur form; the blocks are ! not swapped and (A, B) and (Q, Z) are unchanged. ! The problem of swapping is too ill-conditioned. ! <0: If INFO = -16: LWORK is too small. Appropriate value ! for LWORK is returned in WORK(1). ! ! Further Details ! =============== ! ! Based on contributions by ! Bo Kagstrom and Peter Poromaa, Department of Computing Science, ! Umea University, S-901 87 Umea, Sweden. ! ! In the current code both weak and strong stability tests are ! performed. The user can omit the strong stability test by changing ! the internal logical parameter WANDS to .FALSE.. See ref. [2] for ! details. ! ! [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the ! Generalized Real Schur Form of a Regular Matrix Pair (A, B), in ! M.S. Moonen et al (eds), Linear Algebra for Large Scale and ! Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. ! ! [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified ! Eigenvalues of a Regular Matrix Pair (A, B) and Condition ! Estimation: Theory, Algorithms and Software, ! Report UMINF - 94.04, Department of Computing Science, Umea ! University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working ! Note 87. To appear in Numerical Algorithms, 1996. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) DOUBLE PRECISION TEN PARAMETER ( TEN = 1.0D+01 ) INTEGER LDST PARAMETER ( LDST = 4 ) LOGICAL WANDS PARAMETER ( WANDS = .TRUE. ) ! .. ! .. Local Scalars .. LOGICAL DTRONG, WEAK INTEGER I, IDUM, LINFO, M DOUBLE PRECISION BQRA21, BRQA21, DDUM, DNORM, DSCALE, DSUM, EPS, & F, G, SA, SB, SCALE, SMLNUM, SS, THRESH, WS ! .. ! .. Local Arrays .. INTEGER IWORK( LDST ) DOUBLE PRECISION AI( 2 ), AR( 2 ), BE( 2 ), IR( LDST, LDST ), & IRCOP( LDST, LDST ), LI( LDST, LDST ), & LICOP( LDST, LDST ), S( LDST, LDST ), & SCPY( LDST, LDST ), T( LDST, LDST ), & TAUL( LDST ), TAUR( LDST ), TCPY( LDST, LDST ) double precision zero_vec(1) ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEMM, DGEQR2, DGERQ2, DLACPY, DLAGV2, & DLARTG, DLASSQ, DORG2R, DORGR2, DORM2R, DORMR2, & DROT, DSCAL, DTGSY2 ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! INFO = 0 zero_vec(1) = 0.0D+00 ! ! Quick return if possible ! IF( N.LE.1 .OR. N1.LE.0 .OR. N2.LE.0 ) & RETURN IF( N1.GT.N .OR. ( J1+N1 ).GT.N ) & RETURN M = N1 + N2 IF( LWORK.LT.MAX( N*M, M*M*2 ) ) THEN INFO = -16 WORK( 1 ) = MAX( N*M, M*M*2 ) RETURN END IF ! WEAK = .FALSE. DTRONG = .FALSE. ! ! Make a local copy of selected block ! CALL DCOPY( LDST*LDST, zero_vec, 0, LI, 1 ) CALL DCOPY( LDST*LDST, zero_vec, 0, IR, 1 ) CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, S, LDST ) CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, T, LDST ) ! ! Compute threshold for testing acceptance of swapping. ! EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) / EPS DSCALE = ZERO DSUM = ONE CALL DLACPY( 'Full', M, M, S, LDST, WORK, M ) CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM ) CALL DLACPY( 'Full', M, M, T, LDST, WORK, M ) CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM ) DNORM = DSCALE*SQRT( DSUM ) THRESH = MAX( TEN*EPS*DNORM, SMLNUM ) ! IF( M.EQ.2 ) THEN ! ! CASE 1: Swap 1-by-1 and 1-by-1 blocks. ! ! Compute orthogonal QL and RQ that swap 1-by-1 and 1-by-1 blocks ! using Givens rotations and perform the swap tentatively. ! F = S( 2, 2 )*T( 1, 1 ) - T( 2, 2 )*S( 1, 1 ) G = S( 2, 2 )*T( 1, 2 ) - T( 2, 2 )*S( 1, 2 ) SB = ABS( T( 2, 2 ) ) SA = ABS( S( 2, 2 ) ) CALL DLARTG( F, G, IR( 1, 2 ), IR( 1, 1 ), DDUM ) IR( 2, 1 ) = -IR( 1, 2 ) IR( 2, 2 ) = IR( 1, 1 ) CALL DROT( 2, S( 1, 1 ), 1, S( 1, 2 ), 1, IR( 1, 1 ), & IR( 2, 1 ) ) CALL DROT( 2, T( 1, 1 ), 1, T( 1, 2 ), 1, IR( 1, 1 ), & IR( 2, 1 ) ) IF( SA.GE.SB ) THEN CALL DLARTG( S( 1, 1 ), S( 2, 1 ), LI( 1, 1 ), LI( 2, 1 ), & DDUM ) ELSE CALL DLARTG( T( 1, 1 ), T( 2, 1 ), LI( 1, 1 ), LI( 2, 1 ), & DDUM ) END IF CALL DROT( 2, S( 1, 1 ), LDST, S( 2, 1 ), LDST, LI( 1, 1 ), & LI( 2, 1 ) ) CALL DROT( 2, T( 1, 1 ), LDST, T( 2, 1 ), LDST, LI( 1, 1 ), & LI( 2, 1 ) ) LI( 2, 2 ) = LI( 1, 1 ) LI( 1, 2 ) = -LI( 2, 1 ) ! ! Weak stability test: ! |S21| + |T21| <= O(EPS * F-norm((S, T))) ! WS = ABS( S( 2, 1 ) ) + ABS( T( 2, 1 ) ) WEAK = WS.LE.THRESH IF( .NOT.WEAK ) & GO TO 70 ! IF( WANDS ) THEN ! ! Strong stability test: ! F-norm((A-QL'*S*QR, B-QL'*T*QR)) <= O(EPS*F-norm((A,B))) ! CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ), & M ) CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO, & WORK, M ) CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST, ONE, & WORK( M*M+1 ), M ) DSCALE = ZERO DSUM = ONE CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM ) ! CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ), & M ) CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO, & WORK, M ) CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST, ONE, & WORK( M*M+1 ), M ) CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM ) SS = DSCALE*SQRT( DSUM ) DTRONG = SS.LE.THRESH IF( .NOT.DTRONG ) & GO TO 70 END IF ! ! Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and ! (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)). ! CALL DROT( J1+1, A( 1, J1 ), 1, A( 1, J1+1 ), 1, IR( 1, 1 ), & IR( 2, 1 ) ) CALL DROT( J1+1, B( 1, J1 ), 1, B( 1, J1+1 ), 1, IR( 1, 1 ), & IR( 2, 1 ) ) CALL DROT( N-J1+1, A( J1, J1 ), LDA, A( J1+1, J1 ), LDA, & LI( 1, 1 ), LI( 2, 1 ) ) CALL DROT( N-J1+1, B( J1, J1 ), LDB, B( J1+1, J1 ), LDB, & LI( 1, 1 ), LI( 2, 1 ) ) ! ! Set N1-by-N2 (2,1) - blocks to ZERO. ! A( J1+1, J1 ) = ZERO B( J1+1, J1 ) = ZERO ! ! Accumulate transformations into Q and Z if requested. ! IF( WANTZ ) & CALL DROT( N, Z( 1, J1 ), 1, Z( 1, J1+1 ), 1, IR( 1, 1 ), & IR( 2, 1 ) ) IF( WANTQ ) & CALL DROT( N, Q( 1, J1 ), 1, Q( 1, J1+1 ), 1, LI( 1, 1 ), & LI( 2, 1 ) ) ! ! Exit with INFO = 0 if swap was successfully performed. ! RETURN ! ELSE ! ! CASE 2: Swap 1-by-1 and 2-by-2 blocks, or 2-by-2 ! and 2-by-2 blocks. ! ! Solve the generalized Sylvester equation ! S11 * R - L * S22 = SCALE * S12 ! T11 * R - L * T22 = SCALE * T12 ! for R and L. Solutions in LI and IR. ! CALL DLACPY( 'Full', N1, N2, T( 1, N1+1 ), LDST, LI, LDST ) CALL DLACPY( 'Full', N1, N2, S( 1, N1+1 ), LDST, & IR( N2+1, N1+1 ), LDST ) CALL DTGSY2( 'N', 0, N1, N2, S, LDST, S( N1+1, N1+1 ), LDST, & IR( N2+1, N1+1 ), LDST, T, LDST, T( N1+1, N1+1 ), & LDST, LI, LDST, SCALE, DSUM, DSCALE, IWORK, IDUM, & LINFO ) ! ! Compute orthogonal matrix QL: ! ! QL' * LI = [ TL ] ! [ 0 ] ! where ! LI = [ -L ] ! [ SCALE * identity(N2) ] ! DO I = 1, N2 CALL DSCAL( N1, -ONE, LI( 1, I ), 1 ) LI( N1+I, I ) = SCALE end do CALL DGEQR2( M, N2, LI, LDST, TAUL, WORK, LINFO ) IF( LINFO.NE.0 ) & GO TO 70 CALL DORG2R( M, M, N2, LI, LDST, TAUL, WORK, LINFO ) IF( LINFO.NE.0 ) & GO TO 70 ! ! Compute orthogonal matrix RQ: ! ! IR * RQ' = [ 0 TR], ! ! where IR = [ SCALE * identity(N1), R ] ! DO I = 1, N1 IR( N2+I, I ) = SCALE end do CALL DGERQ2( N1, M, IR( N2+1, 1 ), LDST, TAUR, WORK, LINFO ) IF( LINFO.NE.0 ) & GO TO 70 CALL DORGR2( M, M, N1, IR, LDST, TAUR, WORK, LINFO ) IF( LINFO.NE.0 ) & GO TO 70 ! ! Perform the swapping tentatively: ! CALL DGEMM( 'T', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO, & WORK, M ) CALL DGEMM( 'N', 'T', M, M, M, ONE, WORK, M, IR, LDST, ZERO, S, & LDST ) CALL DGEMM( 'T', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO, & WORK, M ) CALL DGEMM( 'N', 'T', M, M, M, ONE, WORK, M, IR, LDST, ZERO, T, & LDST ) CALL DLACPY( 'F', M, M, S, LDST, SCPY, LDST ) CALL DLACPY( 'F', M, M, T, LDST, TCPY, LDST ) CALL DLACPY( 'F', M, M, IR, LDST, IRCOP, LDST ) CALL DLACPY( 'F', M, M, LI, LDST, LICOP, LDST ) ! ! Triangularize the B-part by an RQ factorization. ! Apply transformation (from left) to A-part, giving S. ! CALL DGERQ2( M, M, T, LDST, TAUR, WORK, LINFO ) IF( LINFO.NE.0 ) & GO TO 70 CALL DORMR2( 'R', 'T', M, M, M, T, LDST, TAUR, S, LDST, WORK, & LINFO ) IF( LINFO.NE.0 ) & GO TO 70 CALL DORMR2( 'L', 'N', M, M, M, T, LDST, TAUR, IR, LDST, WORK, & LINFO ) IF( LINFO.NE.0 ) & GO TO 70 ! ! Compute F-norm(S21) in BRQA21. (T21 is 0.) ! DSCALE = ZERO DSUM = ONE DO I = 1, N2 CALL DLASSQ( N1, S( N2+1, I ), 1, DSCALE, DSUM ) end do BRQA21 = DSCALE*SQRT( DSUM ) ! ! Triangularize the B-part by a QR factorization. ! Apply transformation (from right) to A-part, giving S. ! CALL DGEQR2( M, M, TCPY, LDST, TAUL, WORK, LINFO ) IF( LINFO.NE.0 ) & GO TO 70 CALL DORM2R( 'L', 'T', M, M, M, TCPY, LDST, TAUL, SCPY, LDST, & WORK, INFO ) CALL DORM2R( 'R', 'N', M, M, M, TCPY, LDST, TAUL, LICOP, LDST, & WORK, INFO ) IF( LINFO.NE.0 ) & GO TO 70 ! ! Compute F-norm(S21) in BQRA21. (T21 is 0.) ! DSCALE = ZERO DSUM = ONE DO I = 1, N2 CALL DLASSQ( N1, SCPY( N2+1, I ), 1, DSCALE, DSUM ) end do BQRA21 = DSCALE*SQRT( DSUM ) ! ! Decide which method to use. ! Weak stability test: ! F-norm(S21) <= O(EPS * F-norm((S, T))) ! IF( BQRA21.LE.BRQA21 .AND. BQRA21.LE.THRESH ) THEN CALL DLACPY( 'F', M, M, SCPY, LDST, S, LDST ) CALL DLACPY( 'F', M, M, TCPY, LDST, T, LDST ) CALL DLACPY( 'F', M, M, IRCOP, LDST, IR, LDST ) CALL DLACPY( 'F', M, M, LICOP, LDST, LI, LDST ) ELSE IF( BRQA21.GE.THRESH ) THEN GO TO 70 END IF ! ! Set lower triangle of B-part to zero ! DO I = 2, M CALL DCOPY( M-I+1, zero_vec, 0, T( I, I-1 ), 1 ) end do ! IF( WANDS ) THEN ! ! Strong stability test: ! F-norm((A-QL*S*QR', B-QL*T*QR')) <= O(EPS*F-norm((A,B))) ! CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ), & M ) CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO, & WORK, M ) CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST, ONE, & WORK( M*M+1 ), M ) DSCALE = ZERO DSUM = ONE CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM ) ! CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ), & M ) CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO, & WORK, M ) CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST, ONE, & WORK( M*M+1 ), M ) CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM ) SS = DSCALE*SQRT( DSUM ) DTRONG = ( SS.LE.THRESH ) IF( .NOT.DTRONG ) & GO TO 70 ! END IF ! ! If the swap is accepted ("weakly" and "strongly"), apply the ! transformations and set N1-by-N2 (2,1)-block to zero. ! DO 60 I = 1, N2 CALL DCOPY( N1, zero_vec, 0, S( N2+1, I ), 1 ) 60 CONTINUE ! ! copy back M-by-M diagonal block starting at index J1 of (A, B) ! CALL DLACPY( 'F', M, M, S, LDST, A( J1, J1 ), LDA ) CALL DLACPY( 'F', M, M, T, LDST, B( J1, J1 ), LDB ) CALL DCOPY( LDST*LDST, zero_vec, 0, T, 1 ) ! ! Standardize existing 2-by-2 blocks. ! CALL DCOPY( M*M, zero_vec, 0, WORK, 1 ) WORK( 1 ) = ONE T( 1, 1 ) = ONE IDUM = LWORK - M*M - 2 IF( N2.GT.1 ) THEN CALL DLAGV2( A( J1, J1 ), LDA, B( J1, J1 ), LDB, AR, AI, BE, & WORK( 1 ), WORK( 2 ), T( 1, 1 ), T( 2, 1 ) ) WORK( M+1 ) = -WORK( 2 ) WORK( M+2 ) = WORK( 1 ) T( N2, N2 ) = T( 1, 1 ) T( 1, 2 ) = -T( 2, 1 ) END IF WORK( M*M ) = ONE T( M, M ) = ONE ! IF( N1.GT.1 ) THEN CALL DLAGV2( A( J1+N2, J1+N2 ), LDA, B( J1+N2, J1+N2 ), LDB, & TAUR, TAUL, WORK( M*M+1 ), WORK( N2*M+N2+1 ), & WORK( N2*M+N2+2 ), T( N2+1, N2+1 ), & T( M, M-1 ) ) WORK( M*M ) = WORK( N2*M+N2+1 ) WORK( M*M-1 ) = -WORK( N2*M+N2+2 ) T( M, M ) = T( N2+1, N2+1 ) T( M-1, M ) = -T( M, M-1 ) END IF CALL DGEMM( 'T', 'N', N2, N1, N2, ONE, WORK, M, A( J1, J1+N2 ), & LDA, ZERO, WORK( M*M+1 ), N2 ) CALL DLACPY( 'Full', N2, N1, WORK( M*M+1 ), N2, A( J1, J1+N2 ), & LDA ) CALL DGEMM( 'T', 'N', N2, N1, N2, ONE, WORK, M, B( J1, J1+N2 ), & LDB, ZERO, WORK( M*M+1 ), N2 ) CALL DLACPY( 'Full', N2, N1, WORK( M*M+1 ), N2, B( J1, J1+N2 ), & LDB ) CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, WORK, M, ZERO, & WORK( M*M+1 ), M ) CALL DLACPY( 'Full', M, M, WORK( M*M+1 ), M, LI, LDST ) CALL DGEMM( 'N', 'N', N2, N1, N1, ONE, A( J1, J1+N2 ), LDA, & T( N2+1, N2+1 ), LDST, ZERO, WORK, N2 ) CALL DLACPY( 'Full', N2, N1, WORK, N2, A( J1, J1+N2 ), LDA ) CALL DGEMM( 'N', 'N', N2, N1, N1, ONE, B( J1, J1+N2 ), LDA, & T( N2+1, N2+1 ), LDST, ZERO, WORK, N2 ) CALL DLACPY( 'Full', N2, N1, WORK, N2, B( J1, J1+N2 ), LDB ) CALL DGEMM( 'T', 'N', M, M, M, ONE, IR, LDST, T, LDST, ZERO, & WORK, M ) CALL DLACPY( 'Full', M, M, WORK, M, IR, LDST ) ! ! Accumulate transformations into Q and Z if requested. ! IF( WANTQ ) THEN CALL DGEMM( 'N', 'N', N, M, M, ONE, Q( 1, J1 ), LDQ, LI, & LDST, ZERO, WORK, N ) CALL DLACPY( 'Full', N, M, WORK, N, Q( 1, J1 ), LDQ ) ! END IF ! IF( WANTZ ) THEN CALL DGEMM( 'N', 'N', N, M, M, ONE, Z( 1, J1 ), LDZ, IR, & LDST, ZERO, WORK, N ) CALL DLACPY( 'Full', N, M, WORK, N, Z( 1, J1 ), LDZ ) ! END IF ! ! Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and ! (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)). ! I = J1 + M IF( I.LE.N ) THEN CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST, & A( J1, I ), LDA, ZERO, WORK, M ) CALL DLACPY( 'Full', M, N-I+1, WORK, M, A( J1, I ), LDA ) CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST, & B( J1, I ), LDA, ZERO, WORK, M ) CALL DLACPY( 'Full', M, N-I+1, WORK, M, B( J1, I ), LDA ) END IF I = J1 - 1 IF( I.GT.0 ) THEN CALL DGEMM( 'N', 'N', I, M, M, ONE, A( 1, J1 ), LDA, IR, & LDST, ZERO, WORK, I ) CALL DLACPY( 'Full', I, M, WORK, I, A( 1, J1 ), LDA ) CALL DGEMM( 'N', 'N', I, M, M, ONE, B( 1, J1 ), LDB, IR, & LDST, ZERO, WORK, I ) CALL DLACPY( 'Full', I, M, WORK, I, B( 1, J1 ), LDB ) END IF ! ! Exit with INFO = 0 if swap was successfully performed. ! RETURN ! END IF ! ! Exit with INFO = 1 if swap was rejected. ! 70 CONTINUE ! INFO = 1 RETURN ! ! End of DTGEX2 ! END SUBROUTINE DTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, & LDZ, IFST, ILST, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. LOGICAL WANTQ, WANTZ INTEGER IFST, ILST, INFO, LDA, LDB, LDQ, LDZ, LWORK, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), & WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DTGEXC reorders the generalized real Schur decomposition of a real ! matrix pair (A,B) using an orthogonal equivalence transformation ! ! (A, B) = Q * (A, B) * Z', ! ! so that the diagonal block of (A, B) with row index IFST is moved ! to row ILST. ! ! (A, B) must be in generalized real Schur canonical form (as returned ! by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2 ! diagonal blocks. B is upper triangular. ! ! Optionally, the matrices Q and Z of generalized Schur vectors are ! updated. ! ! Q(in) * A(in) * Z(in)' = Q(out) * A(out) * Z(out)' ! Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)' ! ! ! Arguments ! ========= ! ! WANTQ (input) LOGICAL ! .TRUE. : update the left transformation matrix Q; ! .FALSE.: do not update Q. ! ! WANTZ (input) LOGICAL ! .TRUE. : update the right transformation matrix Z; ! .FALSE.: do not update Z. ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the matrix A in generalized real Schur canonical ! form. ! On exit, the updated matrix A, again in generalized ! real Schur canonical form. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,N) ! On entry, the matrix B in generalized real Schur canonical ! form (A,B). ! On exit, the updated matrix B, again in generalized ! real Schur canonical form (A,B). ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! Q (input/output) DOUBLE PRECISION array, dimension (LDZ,N) ! On entry, if WANTQ = .TRUE., the orthogonal matrix Q. ! On exit, the updated matrix Q. ! If WANTQ = .FALSE., Q is not referenced. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. LDQ >= 1. ! If WANTQ = .TRUE., LDQ >= N. ! ! Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N) ! On entry, if WANTZ = .TRUE., the orthogonal matrix Z. ! On exit, the updated matrix Z. ! If WANTZ = .FALSE., Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1. ! If WANTZ = .TRUE., LDZ >= N. ! ! IFST (input/output) INTEGER ! ILST (input/output) INTEGER ! Specify the reordering of the diagonal blocks of (A, B). ! The block with row index IFST is moved to row ILST, by a ! sequence of swapping between adjacent blocks. ! On exit, if IFST pointed on entry to the second row of ! a 2-by-2 block, it is changed to point to the first row; ! ILST always points to the first row of the block in its ! final position (which may differ from its input value by ! +1 or -1). 1 <= IFST, ILST <= N. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= 4*N + 16. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! =0: successful exit. ! <0: if INFO = -i, the i-th argument had an illegal value. ! =1: The transformed matrix pair (A, B) would be too far ! from generalized Schur form; the problem is ill- ! conditioned. (A, B) may have been partially reordered, ! and ILST points to the first row of the current ! position of the block being moved. ! ! Further Details ! =============== ! ! Based on contributions by ! Bo Kagstrom and Peter Poromaa, Department of Computing Science, ! Umea University, S-901 87 Umea, Sweden. ! ! [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the ! Generalized Real Schur Form of a Regular Matrix Pair (A, B), in ! M.S. Moonen et al (eds), Linear Algebra for Large Scale and ! Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER HERE, LWMIN, NBF, NBL, NBNEXT ! .. ! .. External Subroutines .. EXTERNAL DTGEX2, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Decode and test input arguments. ! INFO = 0 LWMIN = MAX( 1, 4*N+16 ) LQUERY = ( LWORK.EQ.-1 ) IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. ( LDQ.LT.MAX( 1, N ) ) ) THEN INFO = -9 ELSE IF( LDZ.LT.1 .OR. WANTZ .AND. ( LDZ.LT.MAX( 1, N ) ) ) THEN INFO = -11 ELSE IF( IFST.LT.1 .OR. IFST.GT.N ) THEN INFO = -12 ELSE IF( ILST.LT.1 .OR. ILST.GT.N ) THEN INFO = -13 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -15 END IF ! IF( INFO.EQ.0 ) THEN WORK( 1 ) = LWMIN END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTGEXC', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.LE.1 ) & RETURN ! ! Determine the first row of the specified block and find out ! if it is 1-by-1 or 2-by-2. ! IF( IFST.GT.1 ) THEN IF( A( IFST, IFST-1 ).NE.ZERO ) & IFST = IFST - 1 END IF NBF = 1 IF( IFST.LT.N ) THEN IF( A( IFST+1, IFST ).NE.ZERO ) & NBF = 2 END IF ! ! Determine the first row of the final block ! and find out if it is 1-by-1 or 2-by-2. ! IF( ILST.GT.1 ) THEN IF( A( ILST, ILST-1 ).NE.ZERO ) & ILST = ILST - 1 END IF NBL = 1 IF( ILST.LT.N ) THEN IF( A( ILST+1, ILST ).NE.ZERO ) & NBL = 2 END IF IF( IFST.EQ.ILST ) & RETURN ! IF( IFST.LT.ILST ) THEN ! ! Update ILST. ! IF( NBF.EQ.2 .AND. NBL.EQ.1 ) & ILST = ILST - 1 IF( NBF.EQ.1 .AND. NBL.EQ.2 ) & ILST = ILST + 1 ! HERE = IFST ! 10 CONTINUE ! ! Swap with next one below. ! IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN ! ! Current block either 1-by-1 or 2-by-2. ! NBNEXT = 1 IF( HERE+NBF+1.LE.N ) THEN IF( A( HERE+NBF+1, HERE+NBF ).NE.ZERO ) & NBNEXT = 2 END IF CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, & LDZ, HERE, NBF, NBNEXT, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE + NBNEXT ! ! Test if 2-by-2 block breaks into two 1-by-1 blocks. ! IF( NBF.EQ.2 ) THEN IF( A( HERE+1, HERE ).EQ.ZERO ) & NBF = 3 END IF ! ELSE ! ! Current block consists of two 1-by-1 blocks, each of which ! must be swapped individually. ! NBNEXT = 1 IF( HERE+3.LE.N ) THEN IF( A( HERE+3, HERE+2 ).NE.ZERO ) & NBNEXT = 2 END IF CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, & LDZ, HERE+1, 1, NBNEXT, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF IF( NBNEXT.EQ.1 ) THEN ! ! Swap two 1-by-1 blocks. ! CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, & LDZ, HERE, 1, 1, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE + 1 ! ELSE ! ! Recompute NBNEXT in case of 2-by-2 split. ! IF( A( HERE+2, HERE+1 ).EQ.ZERO ) & NBNEXT = 1 IF( NBNEXT.EQ.2 ) THEN ! ! 2-by-2 block did not split. ! CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, & Z, LDZ, HERE, 1, NBNEXT, WORK, LWORK, & INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE + 2 ELSE ! ! 2-by-2 block did split. ! CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, & Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE + 1 CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, & Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE + 1 END IF ! END IF END IF IF( HERE.LT.ILST ) & GO TO 10 ELSE HERE = IFST ! 20 CONTINUE ! ! Swap with next one below. ! IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN ! ! Current block either 1-by-1 or 2-by-2. ! NBNEXT = 1 IF( HERE.GE.3 ) THEN IF( A( HERE-1, HERE-2 ).NE.ZERO ) & NBNEXT = 2 END IF CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, & LDZ, HERE-NBNEXT, NBNEXT, NBF, WORK, LWORK, & INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE - NBNEXT ! ! Test if 2-by-2 block breaks into two 1-by-1 blocks. ! IF( NBF.EQ.2 ) THEN IF( A( HERE+1, HERE ).EQ.ZERO ) & NBF = 3 END IF ! ELSE ! ! Current block consists of two 1-by-1 blocks, each of which ! must be swapped individually. ! NBNEXT = 1 IF( HERE.GE.3 ) THEN IF( A( HERE-1, HERE-2 ).NE.ZERO ) & NBNEXT = 2 END IF CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, & LDZ, HERE-NBNEXT, NBNEXT, 1, WORK, LWORK, & INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF IF( NBNEXT.EQ.1 ) THEN ! ! Swap two 1-by-1 blocks. ! CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, & LDZ, HERE, NBNEXT, 1, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE - 1 ELSE ! ! Recompute NBNEXT in case of 2-by-2 split. ! IF( A( HERE, HERE-1 ).EQ.ZERO ) & NBNEXT = 1 IF( NBNEXT.EQ.2 ) THEN ! ! 2-by-2 block did not split. ! CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, & Z, LDZ, HERE-1, 2, 1, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE - 2 ELSE ! ! 2-by-2 block did split. ! CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, & Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE - 1 CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, & Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE - 1 END IF END IF END IF IF( HERE.GT.ILST ) & GO TO 20 END IF ILST = HERE WORK( 1 ) = LWMIN RETURN ! ! End of DTGEXC ! END SUBROUTINE DTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, & ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL, & PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. LOGICAL WANTQ, WANTZ INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK, & M, N DOUBLE PRECISION PL, PR ! .. ! .. Array Arguments .. LOGICAL SELECT( * ) INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), & B( LDB, * ), BETA( * ), DIF( * ), Q( LDQ, * ), & WORK( * ), Z( LDZ, * ) ! .. ! ! Purpose ! ======= ! ! DTGSEN reorders the generalized real Schur decomposition of a real ! matrix pair (A, B) (in terms of an orthonormal equivalence trans- ! formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues ! appears in the leading diagonal blocks of the upper quasi-triangular ! matrix A and the upper triangular B. The leading columns of Q and ! Z form orthonormal bases of the corresponding left and right eigen- ! spaces (deflating subspaces). (A, B) must be in generalized real ! Schur canonical form (as returned by DGGES), i.e. A is block upper ! triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper ! triangular. ! ! DTGSEN also computes the generalized eigenvalues ! ! w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j) ! ! of the reordered matrix pair (A, B). ! ! Optionally, DTGSEN computes the estimates of reciprocal condition ! numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), ! (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) ! between the matrix pairs (A11, B11) and (A22,B22) that correspond to ! the selected cluster and the eigenvalues outside the cluster, resp., ! and norms of "projections" onto left and right eigenspaces w.r.t. ! the selected cluster in the (1,1)-block. ! ! Arguments ! ========= ! ! IJOB (input) INTEGER ! Specifies whether condition numbers are required for the ! cluster of eigenvalues (PL and PR) or the deflating subspaces ! (Difu and Difl): ! =0: Only reorder w.r.t. SELECT. No extras. ! =1: Reciprocal of norms of "projections" onto left and right ! eigenspaces w.r.t. the selected cluster (PL and PR). ! =2: Upper bounds on Difu and Difl. F-norm-based estimate ! (DIF(1:2)). ! =3: Estimate of Difu and Difl. 1-norm-based estimate ! (DIF(1:2)). ! About 5 times as expensive as IJOB = 2. ! =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic ! version to get it all. ! =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above) ! ! WANTQ (input) LOGICAL ! .TRUE. : update the left transformation matrix Q; ! .FALSE.: do not update Q. ! ! WANTZ (input) LOGICAL ! .TRUE. : update the right transformation matrix Z; ! .FALSE.: do not update Z. ! ! SELECT (input) LOGICAL array, dimension (N) ! SELECT specifies the eigenvalues in the selected cluster. ! To select a real eigenvalue w(j), SELECT(j) must be set to ! .TRUE.. To select a complex conjugate pair of eigenvalues ! w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, ! either SELECT(j) or SELECT(j+1) or both must be set to ! .TRUE.; a complex conjugate pair of eigenvalues must be ! either both included in the cluster or both excluded. ! ! N (input) INTEGER ! The order of the matrices A and B. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension(LDA,N) ! On entry, the upper quasi-triangular matrix A, with (A, B) in ! generalized real Schur canonical form. ! On exit, A is overwritten by the reordered matrix A. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension(LDB,N) ! On entry, the upper triangular matrix B, with (A, B) in ! generalized real Schur canonical form. ! On exit, B is overwritten by the reordered matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! ALPHAR (output) DOUBLE PRECISION array, dimension (N) ! ALPHAI (output) DOUBLE PRECISION array, dimension (N) ! BETA (output) DOUBLE PRECISION array, dimension (N) ! On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will ! be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i ! and BETA(j),j=1,...,N are the diagonals of the complex Schur ! form (S,T) that would result if the 2-by-2 diagonal blocks of ! the real generalized Schur form of (A,B) were further reduced ! to triangular form using complex unitary transformations. ! If ALPHAI(j) is zero, then the j-th eigenvalue is real; if ! positive, then the j-th and (j+1)-st eigenvalues are a ! complex conjugate pair, with ALPHAI(j+1) negative. ! ! Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) ! On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. ! On exit, Q has been postmultiplied by the left orthogonal ! transformation matrix which reorder (A, B); The leading M ! columns of Q form orthonormal bases for the specified pair of ! left eigenspaces (deflating subspaces). ! If WANTQ = .FALSE., Q is not referenced. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. LDQ >= 1; ! and if WANTQ = .TRUE., LDQ >= N. ! ! Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N) ! On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. ! On exit, Z has been postmultiplied by the left orthogonal ! transformation matrix which reorder (A, B); The leading M ! columns of Z form orthonormal bases for the specified pair of ! left eigenspaces (deflating subspaces). ! If WANTZ = .FALSE., Z is not referenced. ! ! LDZ (input) INTEGER ! The leading dimension of the array Z. LDZ >= 1; ! If WANTZ = .TRUE., LDZ >= N. ! ! M (output) INTEGER ! The dimension of the specified pair of left and right eigen- ! spaces (deflating subspaces). 0 <= M <= N. ! ! PL, PR (output) DOUBLE PRECISION ! If IJOB = 1, 4 or 5, PL, PR are lower bounds on the ! reciprocal of the norm of "projections" onto left and right ! eigenspaces with respect to the selected cluster. ! 0 < PL, PR <= 1. ! If M = 0 or M = N, PL = PR = 1. ! If IJOB = 0, 2 or 3, PL and PR are not referenced. ! ! DIF (output) DOUBLE PRECISION array, dimension (2). ! If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. ! If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on ! Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based ! estimates of Difu and Difl. ! If M = 0 or N, DIF(1:2) = F-norm([A, B]). ! If IJOB = 0 or 1, DIF is not referenced. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! IF IJOB = 0, WORK is not referenced. Otherwise, ! on exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= 4*N+16. ! If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)). ! If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)). ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace/output) INTEGER array, dimension (LIWORK) ! IF IJOB = 0, IWORK is not referenced. Otherwise, ! on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. ! ! LIWORK (input) INTEGER ! The dimension of the array IWORK. LIWORK >= 1. ! If IJOB = 1, 2 or 4, LIWORK >= N+6. ! If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6). ! ! If LIWORK = -1, then a workspace query is assumed; the ! routine only calculates the optimal size of the IWORK array, ! returns this value as the first entry of the IWORK array, and ! no error message related to LIWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! =0: Successful exit. ! <0: If INFO = -i, the i-th argument had an illegal value. ! =1: Reordering of (A, B) failed because the transformed ! matrix pair (A, B) would be too far from generalized ! Schur form; the problem is very ill-conditioned. ! (A, B) may have been partially reordered. ! If requested, 0 is returned in DIF(*), PL and PR. ! ! Further Details ! =============== ! ! DTGSEN first collects the selected eigenvalues by computing ! orthogonal U and W that move them to the top left corner of (A, B). ! In other words, the selected eigenvalues are the eigenvalues of ! (A11, B11) in: ! ! U'*(A, B)*W = (A11 A12) (B11 B12) n1 ! ( 0 A22),( 0 B22) n2 ! n1 n2 n1 n2 ! ! where N = n1+n2 and U' means the transpose of U. The first n1 columns ! of U and W span the specified pair of left and right eigenspaces ! (deflating subspaces) of (A, B). ! ! If (A, B) has been obtained from the generalized real Schur ! decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the ! reordered generalized real Schur form of (C, D) is given by ! ! (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)', ! ! and the first n1 columns of Q*U and Z*W span the corresponding ! deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). ! ! Note that if the selected eigenvalue is sufficiently ill-conditioned, ! then its value may differ significantly from its value before ! reordering. ! ! The reciprocal condition numbers of the left and right eigenspaces ! spanned by the first n1 columns of U and W (or Q*U and Z*W) may ! be returned in DIF(1:2), corresponding to Difu and Difl, resp. ! ! The Difu and Difl are defined as: ! ! Difu[(A11, B11), (A22, B22)] = sigma-min( Zu ) ! and ! Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)], ! ! where sigma-min(Zu) is the smallest singular value of the ! (2*n1*n2)-by-(2*n1*n2) matrix ! ! Zu = [ kron(In2, A11) -kron(A22', In1) ] ! [ kron(In2, B11) -kron(B22', In1) ]. ! ! Here, Inx is the identity matrix of size nx and A22' is the ! transpose of A22. kron(X, Y) is the Kronecker product between ! the matrices X and Y. ! ! When DIF(2) is small, small changes in (A, B) can cause large changes ! in the deflating subspace. An approximate (asymptotic) bound on the ! maximum angular error in the computed deflating subspaces is ! ! EPS * norm((A, B)) / DIF(2), ! ! where EPS is the machine precision. ! ! The reciprocal norm of the projectors on the left and right ! eigenspaces associated with (A11, B11) may be returned in PL and PR. ! They are computed as follows. First we compute L and R so that ! P*(A, B)*Q is block diagonal, where ! ! P = ( I -L ) n1 Q = ( I R ) n1 ! ( 0 I ) n2 and ( 0 I ) n2 ! n1 n2 n1 n2 ! ! and (L, R) is the solution to the generalized Sylvester equation ! ! A11*R - L*A22 = -A12 ! B11*R - L*B22 = -B12 ! ! Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). ! An approximate (asymptotic) bound on the average absolute error of ! the selected eigenvalues is ! ! EPS * norm((A, B)) / PL. ! ! There are also global error bounds which valid for perturbations up ! to a certain restriction: A lower bound (x) on the smallest ! F-norm(E,F) for which an eigenvalue of (A11, B11) may move and ! coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), ! (i.e. (A + E, B + F), is ! ! x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)). ! ! An approximate bound on x can be computed from DIF(1:2), PL and PR. ! ! If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed ! (L', R') and unperturbed (L, R) left and right deflating subspaces ! associated with the selected cluster in the (1,1)-blocks can be ! bounded as ! ! max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2)) ! max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2)) ! ! See LAPACK User's Guide section 4.11 or the following references ! for more information. ! ! Note that if the default method for computing the Frobenius-norm- ! based estimate DIF is not wanted (see DLATDF), then the parameter ! IDIFJB (see below) should be changed from 3 to 4 (routine DLATDF ! (IJOB = 2 will be used)). See DTGSYL for more details. ! ! Based on contributions by ! Bo Kagstrom and Peter Poromaa, Department of Computing Science, ! Umea University, S-901 87 Umea, Sweden. ! ! References ! ========== ! ! [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the ! Generalized Real Schur Form of a Regular Matrix Pair (A, B), in ! M.S. Moonen et al (eds), Linear Algebra for Large Scale and ! Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. ! ! [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified ! Eigenvalues of a Regular Matrix Pair (A, B) and Condition ! Estimation: Theory, Algorithms and Software, ! Report UMINF - 94.04, Department of Computing Science, Umea ! University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working ! Note 87. To appear in Numerical Algorithms, 1996. ! ! [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software ! for Solving the Generalized Sylvester Equation and Estimating the ! Separation between Regular Matrix Pairs, Report UMINF - 93.23, ! Department of Computing Science, Umea University, S-901 87 Umea, ! Sweden, December 1993, Revised April 1994, Also as LAPACK Working ! Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, ! 1996. ! ! ===================================================================== ! ! .. Parameters .. INTEGER IDIFJB PARAMETER ( IDIFJB = 3 ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, PAIR, SWAP, WANTD, WANTD1, WANTD2, & WANTP INTEGER I, IERR, IJB, K, KASE, KK, KS, LIWMIN, LWMIN, & MN2, N1, N2 DOUBLE PRECISION DSCALE, DSUM, EPS, RDSCAL, SMLNUM ! .. ! .. External Subroutines .. EXTERNAL DLACON, DLACPY, DLAG2, DLASSQ, DTGEXC, DTGSYL, & XERBLA ! .. ! .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, SIGN, SQRT ! .. ! .. Executable Statements .. ! ! Decode and test the input parameters ! INFO = 0 LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) ! IF( IJOB.LT.0 .OR. IJOB.GT.5 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN INFO = -14 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -16 END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTGSEN', -INFO ) RETURN END IF ! ! Get machine constants ! EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) / EPS IERR = 0 ! WANTP = IJOB.EQ.1 .OR. IJOB.GE.4 WANTD1 = IJOB.EQ.2 .OR. IJOB.EQ.4 WANTD2 = IJOB.EQ.3 .OR. IJOB.EQ.5 WANTD = WANTD1 .OR. WANTD2 ! ! Set M to the dimension of the specified pair of deflating ! subspaces. ! M = 0 PAIR = .FALSE. DO 10 K = 1, N IF( PAIR ) THEN PAIR = .FALSE. ELSE IF( K.LT.N ) THEN IF( A( K+1, K ).EQ.ZERO ) THEN IF( SELECT( K ) ) & M = M + 1 ELSE PAIR = .TRUE. IF( SELECT( K ) .OR. SELECT( K+1 ) ) & M = M + 2 END IF ELSE IF( SELECT( N ) ) & M = M + 1 END IF END IF 10 CONTINUE ! IF( IJOB.EQ.1 .OR. IJOB.EQ.2 .OR. IJOB.EQ.4 ) THEN LWMIN = MAX( 1, 4*N+16, 2*M*( N-M ) ) LIWMIN = MAX( 1, N+6 ) ELSE IF( IJOB.EQ.3 .OR. IJOB.EQ.5 ) THEN LWMIN = MAX( 1, 4*N+16, 4*M*( N-M ) ) LIWMIN = MAX( 1, 2*M*( N-M ), N+6 ) ELSE LWMIN = MAX( 1, 4*N+16 ) LIWMIN = 1 END IF ! WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN ! IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -22 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -24 END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTGSEN', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible. ! IF( M.EQ.N .OR. M.EQ.0 ) THEN IF( WANTP ) THEN PL = ONE PR = ONE END IF IF( WANTD ) THEN DSCALE = ZERO DSUM = ONE DO 20 I = 1, N CALL DLASSQ( N, A( 1, I ), 1, DSCALE, DSUM ) CALL DLASSQ( N, B( 1, I ), 1, DSCALE, DSUM ) 20 CONTINUE DIF( 1 ) = DSCALE*SQRT( DSUM ) DIF( 2 ) = DIF( 1 ) END IF GO TO 60 END IF ! ! Collect the selected blocks at the top-left corner of (A, B). ! KS = 0 PAIR = .FALSE. DO 30 K = 1, N IF( PAIR ) THEN PAIR = .FALSE. ELSE ! SWAP = SELECT( K ) IF( K.LT.N ) THEN IF( A( K+1, K ).NE.ZERO ) THEN PAIR = .TRUE. SWAP = SWAP .OR. SELECT( K+1 ) END IF END IF ! IF( SWAP ) THEN KS = KS + 1 ! ! Swap the K-th block to position KS. ! Perform the reordering of diagonal blocks in (A, B) ! by orthogonal transformation matrices and update ! Q and Z accordingly (if requested): ! KK = K IF( K.NE.KS ) & CALL DTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, & Z, LDZ, KK, KS, WORK, LWORK, IERR ) ! IF( IERR.GT.0 ) THEN ! ! Swap is rejected: exit. ! INFO = 1 IF( WANTP ) THEN PL = ZERO PR = ZERO END IF IF( WANTD ) THEN DIF( 1 ) = ZERO DIF( 2 ) = ZERO END IF GO TO 60 END IF ! IF( PAIR ) & KS = KS + 1 END IF END IF 30 CONTINUE IF( WANTP ) THEN ! ! Solve generalized Sylvester equation for R and L ! and compute PL and PR. ! N1 = M N2 = N - M I = N1 + 1 IJB = 0 CALL DLACPY( 'Full', N1, N2, A( 1, I ), LDA, WORK, N1 ) CALL DLACPY( 'Full', N1, N2, B( 1, I ), LDB, WORK( N1*N2+1 ), & N1 ) CALL DTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, WORK, & N1, B, LDB, B( I, I ), LDB, WORK( N1*N2+1 ), N1, & DSCALE, DIF( 1 ), WORK( N1*N2*2+1 ), & LWORK-2*N1*N2, IWORK, IERR ) ! ! Estimate the reciprocal of norms of "projections" onto left ! and right eigenspaces. ! RDSCAL = ZERO DSUM = ONE CALL DLASSQ( N1*N2, WORK, 1, RDSCAL, DSUM ) PL = RDSCAL*SQRT( DSUM ) IF( PL.EQ.ZERO ) THEN PL = ONE ELSE PL = DSCALE / ( SQRT( DSCALE*DSCALE / PL+PL )*SQRT( PL ) ) END IF RDSCAL = ZERO DSUM = ONE CALL DLASSQ( N1*N2, WORK( N1*N2+1 ), 1, RDSCAL, DSUM ) PR = RDSCAL*SQRT( DSUM ) IF( PR.EQ.ZERO ) THEN PR = ONE ELSE PR = DSCALE / ( SQRT( DSCALE*DSCALE / PR+PR )*SQRT( PR ) ) END IF END IF ! IF( WANTD ) THEN ! ! Compute estimates of Difu and Difl. ! IF( WANTD1 ) THEN N1 = M N2 = N - M I = N1 + 1 IJB = IDIFJB ! ! Frobenius norm-based Difu-estimate. ! CALL DTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, WORK, & N1, B, LDB, B( I, I ), LDB, WORK( N1*N2+1 ), & N1, DSCALE, DIF( 1 ), WORK( 2*N1*N2+1 ), & LWORK-2*N1*N2, IWORK, IERR ) ! ! Frobenius norm-based Difl-estimate. ! CALL DTGSYL( 'N', IJB, N2, N1, A( I, I ), LDA, A, LDA, WORK, & N2, B( I, I ), LDB, B, LDB, WORK( N1*N2+1 ), & N2, DSCALE, DIF( 2 ), WORK( 2*N1*N2+1 ), & LWORK-2*N1*N2, IWORK, IERR ) ELSE ! ! ! Compute 1-norm-based estimates of Difu and Difl using ! reversed communication with DLACON. In each step a ! generalized Sylvester equation or a transposed variant ! is solved. ! KASE = 0 N1 = M N2 = N - M I = N1 + 1 IJB = 0 MN2 = 2*N1*N2 ! ! 1-norm-based estimate of Difu. ! 40 CONTINUE CALL DLACON( MN2, WORK( MN2+1 ), WORK, IWORK, DIF( 1 ), & KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN ! ! Solve generalized Sylvester equation. ! CALL DTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, & WORK, N1, B, LDB, B( I, I ), LDB, & WORK( N1*N2+1 ), N1, DSCALE, DIF( 1 ), & WORK( 2*N1*N2+1 ), LWORK-2*N1*N2, IWORK, & IERR ) ELSE ! ! Solve the transposed variant. ! CALL DTGSYL( 'T', IJB, N1, N2, A, LDA, A( I, I ), LDA, & WORK, N1, B, LDB, B( I, I ), LDB, & WORK( N1*N2+1 ), N1, DSCALE, DIF( 1 ), & WORK( 2*N1*N2+1 ), LWORK-2*N1*N2, IWORK, & IERR ) END IF GO TO 40 END IF DIF( 1 ) = DSCALE / DIF( 1 ) ! ! 1-norm-based estimate of Difl. ! 50 CONTINUE CALL DLACON( MN2, WORK( MN2+1 ), WORK, IWORK, DIF( 2 ), & KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN ! ! Solve generalized Sylvester equation. ! CALL DTGSYL( 'N', IJB, N2, N1, A( I, I ), LDA, A, LDA, & WORK, N2, B( I, I ), LDB, B, LDB, & WORK( N1*N2+1 ), N2, DSCALE, DIF( 2 ), & WORK( 2*N1*N2+1 ), LWORK-2*N1*N2, IWORK, & IERR ) ELSE ! ! Solve the transposed variant. ! CALL DTGSYL( 'T', IJB, N2, N1, A( I, I ), LDA, A, LDA, & WORK, N2, B( I, I ), LDB, B, LDB, & WORK( N1*N2+1 ), N2, DSCALE, DIF( 2 ), & WORK( 2*N1*N2+1 ), LWORK-2*N1*N2, IWORK, & IERR ) END IF GO TO 50 END IF DIF( 2 ) = DSCALE / DIF( 2 ) ! END IF END IF ! 60 CONTINUE ! ! Compute generalized eigenvalues of reordered pair (A, B) and ! normalize the generalized Schur form. ! PAIR = .FALSE. DO 80 K = 1, N IF( PAIR ) THEN PAIR = .FALSE. ELSE ! IF( K.LT.N ) THEN IF( A( K+1, K ).NE.ZERO ) THEN PAIR = .TRUE. END IF END IF ! IF( PAIR ) THEN ! ! Compute the eigenvalue(s) at position K. ! WORK( 1 ) = A( K, K ) WORK( 2 ) = A( K+1, K ) WORK( 3 ) = A( K, K+1 ) WORK( 4 ) = A( K+1, K+1 ) WORK( 5 ) = B( K, K ) WORK( 6 ) = B( K+1, K ) WORK( 7 ) = B( K, K+1 ) WORK( 8 ) = B( K+1, K+1 ) CALL DLAG2( WORK, 2, WORK( 5 ), 2, SMLNUM*EPS, BETA( K ), & BETA( K+1 ), ALPHAR( K ), ALPHAR( K+1 ), & ALPHAI( K ) ) ALPHAI( K+1 ) = -ALPHAI( K ) ! ELSE ! IF( SIGN( ONE, B( K, K ) ).LT.ZERO ) THEN ! ! If B(K,K) is negative, make it positive ! DO 70 I = 1, N A( K, I ) = -A( K, I ) B( K, I ) = -B( K, I ) Q( I, K ) = -Q( I, K ) 70 CONTINUE END IF ! ALPHAR( K ) = A( K, K ) ALPHAI( K ) = ZERO BETA( K ) = B( K, K ) ! END IF END IF 80 CONTINUE ! WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN ! RETURN ! ! End of DTGSEN ! END SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, & LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, & Q, LDQ, WORK, NCYCLE, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER JOBQ, JOBU, JOBV INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, & NCYCLE, P DOUBLE PRECISION TOLA, TOLB ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ), & BETA( * ), Q( LDQ, * ), U( LDU, * ), & V( LDV, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DTGSJA computes the generalized singular value decomposition (GSVD) ! of two real upper triangular (or trapezoidal) matrices A and B. ! ! On entry, it is assumed that matrices A and B have the following ! forms, which may be obtained by the preprocessing subroutine DGGSVP ! from a general M-by-N matrix A and P-by-N matrix B: ! ! N-K-L K L ! A = K ( 0 A12 A13 ) if M-K-L >= 0; ! L ( 0 0 A23 ) ! M-K-L ( 0 0 0 ) ! ! N-K-L K L ! A = K ( 0 A12 A13 ) if M-K-L < 0; ! M-K ( 0 0 A23 ) ! ! N-K-L K L ! B = L ( 0 0 B13 ) ! P-L ( 0 0 0 ) ! ! where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular ! upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, ! otherwise A23 is (M-K)-by-L upper trapezoidal. ! ! On exit, ! ! U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ), ! ! where U, V and Q are orthogonal matrices, Z' denotes the transpose ! of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are ! ``diagonal'' matrices, which are of the following structures: ! ! If M-K-L >= 0, ! ! K L ! D1 = K ( I 0 ) ! L ( 0 C ) ! M-K-L ( 0 0 ) ! ! K L ! D2 = L ( 0 S ) ! P-L ( 0 0 ) ! ! N-K-L K L ! ( 0 R ) = K ( 0 R11 R12 ) K ! L ( 0 0 R22 ) L ! ! where ! ! C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), ! S = diag( BETA(K+1), ... , BETA(K+L) ), ! C**2 + S**2 = I. ! ! R is stored in A(1:K+L,N-K-L+1:N) on exit. ! ! If M-K-L < 0, ! ! K M-K K+L-M ! D1 = K ( I 0 0 ) ! M-K ( 0 C 0 ) ! ! K M-K K+L-M ! D2 = M-K ( 0 S 0 ) ! K+L-M ( 0 0 I ) ! P-L ( 0 0 0 ) ! ! N-K-L K M-K K+L-M ! ( 0 R ) = K ( 0 R11 R12 R13 ) ! M-K ( 0 0 R22 R23 ) ! K+L-M ( 0 0 0 R33 ) ! ! where ! C = diag( ALPHA(K+1), ... , ALPHA(M) ), ! S = diag( BETA(K+1), ... , BETA(M) ), ! C**2 + S**2 = I. ! ! R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored ! ( 0 R22 R23 ) ! in B(M-K+1:L,N+M-K-L+1:N) on exit. ! ! The computation of the orthogonal transformation matrices U, V or Q ! is optional. These matrices may either be formed explicitly, or they ! may be postmultiplied into input matrices U1, V1, or Q1. ! ! Arguments ! ========= ! ! JOBU (input) CHARACTER*1 ! = 'U': U must contain an orthogonal matrix U1 on entry, and ! the product U1*U is returned; ! = 'I': U is initialized to the unit matrix, and the ! orthogonal matrix U is returned; ! = 'N': U is not computed. ! ! JOBV (input) CHARACTER*1 ! = 'V': V must contain an orthogonal matrix V1 on entry, and ! the product V1*V is returned; ! = 'I': V is initialized to the unit matrix, and the ! orthogonal matrix V is returned; ! = 'N': V is not computed. ! ! JOBQ (input) CHARACTER*1 ! = 'Q': Q must contain an orthogonal matrix Q1 on entry, and ! the product Q1*Q is returned; ! = 'I': Q is initialized to the unit matrix, and the ! orthogonal matrix Q is returned; ! = 'N': Q is not computed. ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! P (input) INTEGER ! The number of rows of the matrix B. P >= 0. ! ! N (input) INTEGER ! The number of columns of the matrices A and B. N >= 0. ! ! K (input) INTEGER ! L (input) INTEGER ! K and L specify the subblocks in the input matrices A and B: ! A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N) ! of A and B, whose GSVD is going to be computed by DTGSJA. ! See Further details. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the M-by-N matrix A. ! On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular ! matrix R or part of R. See Purpose for details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,N) ! On entry, the P-by-N matrix B. ! On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains ! a part of R. See Purpose for details. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,P). ! ! TOLA (input) DOUBLE PRECISION ! TOLB (input) DOUBLE PRECISION ! TOLA and TOLB are the convergence criteria for the Jacobi- ! Kogbetliantz iteration procedure. Generally, they are the ! same as used in the preprocessing step, say ! TOLA = max(M,N)*norm(A)*MAZHEPS, ! TOLB = max(P,N)*norm(B)*MAZHEPS. ! ! ALPHA (output) DOUBLE PRECISION array, dimension (N) ! BETA (output) DOUBLE PRECISION array, dimension (N) ! On exit, ALPHA and BETA contain the generalized singular ! value pairs of A and B; ! ALPHA(1:K) = 1, ! BETA(1:K) = 0, ! and if M-K-L >= 0, ! ALPHA(K+1:K+L) = diag(C), ! BETA(K+1:K+L) = diag(S), ! or if M-K-L < 0, ! ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0 ! BETA(K+1:M) = S, BETA(M+1:K+L) = 1. ! Furthermore, if K+L < N, ! ALPHA(K+L+1:N) = 0 and ! BETA(K+L+1:N) = 0. ! ! U (input/output) DOUBLE PRECISION array, dimension (LDU,M) ! On entry, if JOBU = 'U', U must contain a matrix U1 (usually ! the orthogonal matrix returned by DGGSVP). ! On exit, ! if JOBU = 'I', U contains the orthogonal matrix U; ! if JOBU = 'U', U contains the product U1*U. ! If JOBU = 'N', U is not referenced. ! ! LDU (input) INTEGER ! The leading dimension of the array U. LDU >= max(1,M) if ! JOBU = 'U'; LDU >= 1 otherwise. ! ! V (input/output) DOUBLE PRECISION array, dimension (LDV,P) ! On entry, if JOBV = 'V', V must contain a matrix V1 (usually ! the orthogonal matrix returned by DGGSVP). ! On exit, ! if JOBV = 'I', V contains the orthogonal matrix V; ! if JOBV = 'V', V contains the product V1*V. ! If JOBV = 'N', V is not referenced. ! ! LDV (input) INTEGER ! The leading dimension of the array V. LDV >= max(1,P) if ! JOBV = 'V'; LDV >= 1 otherwise. ! ! Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) ! On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually ! the orthogonal matrix returned by DGGSVP). ! On exit, ! if JOBQ = 'I', Q contains the orthogonal matrix Q; ! if JOBQ = 'Q', Q contains the product Q1*Q. ! If JOBQ = 'N', Q is not referenced. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. LDQ >= max(1,N) if ! JOBQ = 'Q'; LDQ >= 1 otherwise. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (2*N) ! ! NCYCLE (output) INTEGER ! The number of cycles required for convergence. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! = 1: the procedure does not converge after MAXIT cycles. ! ! Internal Parameters ! =================== ! ! MAXIT INTEGER ! MAXIT specifies the total loops that the iterative procedure ! may take. If after MAXIT cycles, the routine fails to ! converge, we return INFO = 1. ! ! Further Details ! =============== ! ! DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce ! min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L ! matrix B13 to the form: ! ! U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1, ! ! where U1, V1 and Q1 are orthogonal matrix, and Z' is the transpose ! of Z. C1 and S1 are diagonal matrices satisfying ! ! C1**2 + S1**2 = I, ! ! and R1 is an L-by-L nonsingular upper triangular matrix. ! ! ===================================================================== ! ! .. Parameters .. INTEGER MAXIT PARAMETER ( MAXIT = 40 ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. ! LOGICAL INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV INTEGER I, J, KCYCLE DOUBLE PRECISION A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, ERROR, & GAMMA, RWK, SNQ, SNU, SNV, SSMIN ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLAGS2, DLAPLL, DLARTG, DLASET, DROT, & DSCAL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN ! .. ! .. Executable Statements .. ! ! Decode and test the input parameters ! INITU = LSAME( JOBU, 'I' ) WANTU = INITU .OR. LSAME( JOBU, 'U' ) ! INITV = LSAME( JOBV, 'I' ) WANTV = INITV .OR. LSAME( JOBV, 'V' ) ! INITQ = LSAME( JOBQ, 'I' ) WANTQ = INITQ .OR. LSAME( JOBQ, 'Q' ) ! INFO = 0 IF( .NOT.( INITU .OR. WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( INITV .OR. WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( INITQ .OR. WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN INFO = -3 ELSE IF( M.LT.0 ) THEN INFO = -4 ELSE IF( P.LT.0 ) THEN INFO = -5 ELSE IF( N.LT.0 ) THEN INFO = -6 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -10 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN INFO = -12 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN INFO = -18 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN INFO = -20 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN INFO = -22 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTGSJA', -INFO ) RETURN END IF ! ! Initialize U, V and Q, if necessary ! IF( INITU ) & CALL DLASET( 'Full', M, M, ZERO, ONE, U, LDU ) IF( INITV ) & CALL DLASET( 'Full', P, P, ZERO, ONE, V, LDV ) IF( INITQ ) & CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ ) ! ! Loop until convergence ! UPPER = .FALSE. DO 40 KCYCLE = 1, MAXIT ! UPPER = .NOT.UPPER ! DO 20 I = 1, L - 1 DO 10 J = I + 1, L ! A1 = ZERO A2 = ZERO A3 = ZERO IF( K+I.LE.M ) & A1 = A( K+I, N-L+I ) IF( K+J.LE.M ) & A3 = A( K+J, N-L+J ) ! B1 = B( I, N-L+I ) B3 = B( J, N-L+J ) ! IF( UPPER ) THEN IF( K+I.LE.M ) & A2 = A( K+I, N-L+J ) B2 = B( I, N-L+J ) ELSE IF( K+J.LE.M ) & A2 = A( K+J, N-L+I ) B2 = B( J, N-L+I ) END IF ! CALL DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, & CSV, SNV, CSQ, SNQ ) ! ! Update (K+I)-th and (K+J)-th rows of matrix A: U'*A ! IF( K+J.LE.M ) & CALL DROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ), & LDA, CSU, SNU ) ! ! Update I-th and J-th rows of matrix B: V'*B ! CALL DROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB, & CSV, SNV ) ! ! Update (N-L+I)-th and (N-L+J)-th columns of matrices ! A and B: A*Q and B*Q ! CALL DROT( MIN( K+L, M ), A( 1, N-L+J ), 1, & A( 1, N-L+I ), 1, CSQ, SNQ ) ! CALL DROT( L, B( 1, N-L+J ), 1, B( 1, N-L+I ), 1, CSQ, & SNQ ) ! IF( UPPER ) THEN IF( K+I.LE.M ) & A( K+I, N-L+J ) = ZERO B( I, N-L+J ) = ZERO ELSE IF( K+J.LE.M ) & A( K+J, N-L+I ) = ZERO B( J, N-L+I ) = ZERO END IF ! ! Update orthogonal matrices U, V, Q, if desired. ! IF( WANTU .AND. K+J.LE.M ) & CALL DROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU, & SNU ) ! IF( WANTV ) & CALL DROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV, SNV ) ! IF( WANTQ ) & CALL DROT( N, Q( 1, N-L+J ), 1, Q( 1, N-L+I ), 1, CSQ, & SNQ ) ! 10 CONTINUE 20 CONTINUE ! IF( .NOT.UPPER ) THEN ! ! The matrices A13 and B13 were lower triangular at the start ! of the cycle, and are now upper triangular. ! ! Convergence test: test the parallelism of the corresponding ! rows of A and B. ! ERROR = ZERO DO 30 I = 1, MIN( L, M-K ) CALL DCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 ) CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ), 1 ) CALL DLAPLL( L-I+1, WORK, 1, WORK( L+1 ), 1, SSMIN ) ERROR = MAX( ERROR, SSMIN ) 30 CONTINUE ! IF( ABS( ERROR ).LE.MIN( TOLA, TOLB ) ) & GO TO 50 END IF ! ! End of cycle loop ! 40 CONTINUE ! ! The algorithm has not converged after MAXIT cycles. ! INFO = 1 GO TO 100 ! 50 CONTINUE ! ! If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged. ! Compute the generalized singular value pairs (ALPHA, BETA), and ! set the triangular matrix R to array A. ! DO 60 I = 1, K ALPHA( I ) = ONE BETA( I ) = ZERO 60 CONTINUE ! DO 70 I = 1, MIN( L, M-K ) ! A1 = A( K+I, N-L+I ) B1 = B( I, N-L+I ) ! IF( A1.NE.ZERO ) THEN GAMMA = B1 / A1 ! ! change sign if necessary ! IF( GAMMA.LT.ZERO ) THEN CALL DSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB ) IF( WANTV ) & CALL DSCAL( P, -ONE, V( 1, I ), 1 ) END IF ! CALL DLARTG( ABS( GAMMA ), ONE, BETA( K+I ), ALPHA( K+I ), & RWK ) ! IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN CALL DSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I, N-L+I ), & LDA ) ELSE CALL DSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ), & LDB ) CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ), & LDA ) END IF ! ELSE ! ALPHA( K+I ) = ZERO BETA( K+I ) = ONE CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ), & LDA ) ! END IF ! 70 CONTINUE ! ! Post-assignment ! DO 80 I = M + 1, K + L ALPHA( I ) = ZERO BETA( I ) = ONE 80 CONTINUE ! IF( K+L.LT.N ) THEN DO 90 I = K + L + 1, N ALPHA( I ) = ZERO BETA( I ) = ZERO 90 CONTINUE END IF ! 100 CONTINUE NCYCLE = KCYCLE RETURN ! ! End of DTGSJA ! END SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, & LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, & IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER HOWMNY, JOB INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N ! .. ! .. Array Arguments .. LOGICAL SELECT( * ) INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DIF( * ), S( * ), & VL( LDVL, * ), VR( LDVR, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DTGSNA estimates reciprocal condition numbers for specified ! eigenvalues and/or eigenvectors of a matrix pair (A, B) in ! generalized real Schur canonical form (or of any matrix pair ! (Q*A*Z', Q*B*Z') with orthogonal matrices Q and Z, where ! Z' denotes the transpose of Z. ! ! (A, B) must be in generalized real Schur form (as returned by DGGES), ! i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal ! blocks. B is upper triangular. ! ! ! Arguments ! ========= ! ! JOB (input) CHARACTER*1 ! Specifies whether condition numbers are required for ! eigenvalues (S) or eigenvectors (DIF): ! = 'E': for eigenvalues only (S); ! = 'V': for eigenvectors only (DIF); ! = 'B': for both eigenvalues and eigenvectors (S and DIF). ! ! HOWMNY (input) CHARACTER*1 ! = 'A': compute condition numbers for all eigenpairs; ! = 'S': compute condition numbers for selected eigenpairs ! specified by the array SELECT. ! ! SELECT (input) LOGICAL array, dimension (N) ! If HOWMNY = 'S', SELECT specifies the eigenpairs for which ! condition numbers are required. To select condition numbers ! for the eigenpair corresponding to a real eigenvalue w(j), ! SELECT(j) must be set to .TRUE.. To select condition numbers ! corresponding to a complex conjugate pair of eigenvalues w(j) ! and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be ! set to .TRUE.. ! If HOWMNY = 'A', SELECT is not referenced. ! ! N (input) INTEGER ! The order of the square matrix pair (A, B). N >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The upper quasi-triangular matrix A in the pair (A,B). ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! B (input) DOUBLE PRECISION array, dimension (LDB,N) ! The upper triangular matrix B in the pair (A,B). ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! VL (input) DOUBLE PRECISION array, dimension (LDVL,M) ! If JOB = 'E' or 'B', VL must contain left eigenvectors of ! (A, B), corresponding to the eigenpairs specified by HOWMNY ! and SELECT. The eigenvectors must be stored in consecutive ! columns of VL, as returned by DTGEVC. ! If JOB = 'V', VL is not referenced. ! ! LDVL (input) INTEGER ! The leading dimension of the array VL. LDVL >= 1. ! If JOB = 'E' or 'B', LDVL >= N. ! ! VR (input) DOUBLE PRECISION array, dimension (LDVR,M) ! If JOB = 'E' or 'B', VR must contain right eigenvectors of ! (A, B), corresponding to the eigenpairs specified by HOWMNY ! and SELECT. The eigenvectors must be stored in consecutive ! columns ov VR, as returned by DTGEVC. ! If JOB = 'V', VR is not referenced. ! ! LDVR (input) INTEGER ! The leading dimension of the array VR. LDVR >= 1. ! If JOB = 'E' or 'B', LDVR >= N. ! ! S (output) DOUBLE PRECISION array, dimension (MM) ! If JOB = 'E' or 'B', the reciprocal condition numbers of the ! selected eigenvalues, stored in consecutive elements of the ! array. For a complex conjugate pair of eigenvalues two ! consecutive elements of S are set to the same value. Thus ! S(j), DIF(j), and the j-th columns of VL and VR all ! correspond to the same eigenpair (but not in general the ! j-th eigenpair, unless all eigenpairs are selected). ! If JOB = 'V', S is not referenced. ! ! DIF (output) DOUBLE PRECISION array, dimension (MM) ! If JOB = 'V' or 'B', the estimated reciprocal condition ! numbers of the selected eigenvectors, stored in consecutive ! elements of the array. For a complex eigenvector two ! consecutive elements of DIF are set to the same value. If ! the eigenvalues cannot be reordered to compute DIF(j), DIF(j) ! is set to 0; this can only occur when the true value would be ! very small anyway. ! If JOB = 'E', DIF is not referenced. ! ! MM (input) INTEGER ! The number of elements in the arrays S and DIF. MM >= M. ! ! M (output) INTEGER ! The number of elements of the arrays S and DIF used to store ! the specified condition numbers; for each selected real ! eigenvalue one element is used, and for each selected complex ! conjugate pair of eigenvalues, two elements are used. ! If HOWMNY = 'A', M is set to N. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! If JOB = 'E', WORK is not referenced. Otherwise, ! on exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= N. ! If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace) INTEGER array, dimension (N + 6) ! If JOB = 'E', IWORK is not referenced. ! ! INFO (output) INTEGER ! =0: Successful exit ! <0: If INFO = -i, the i-th argument had an illegal value ! ! ! Further Details ! =============== ! ! The reciprocal of the condition number of a generalized eigenvalue ! w = (a, b) is defined as ! ! S(w) = (|u'Av|**2 + |u'Bv|**2)**(1/2) / (norm(u)*norm(v)) ! ! where u and v are the left and right eigenvectors of (A, B) ! corresponding to w; |z| denotes the absolute value of the complex ! number, and norm(u) denotes the 2-norm of the vector u. ! The pair (a, b) corresponds to an eigenvalue w = a/b (= u'Av/u'Bv) ! of the matrix pair (A, B). If both a and b equal zero, then (A B) is ! singular and S(I) = -1 is returned. ! ! An approximate error bound on the chordal distance between the i-th ! computed generalized eigenvalue w and the corresponding exact ! eigenvalue lambda is ! ! chord(w, lambda) <= EPS * norm(A, B) / S(I) ! ! where EPS is the machine precision. ! ! The reciprocal of the condition number DIF(i) of right eigenvector u ! and left eigenvector v corresponding to the generalized eigenvalue w ! is defined as follows: ! ! a) If the i-th eigenvalue w = (a,b) is real ! ! Suppose U and V are orthogonal transformations such that ! ! U'*(A, B)*V = (S, T) = ( a * ) ( b * ) 1 ! ( 0 S22 ),( 0 T22 ) n-1 ! 1 n-1 1 n-1 ! ! Then the reciprocal condition number DIF(i) is ! ! Difl((a, b), (S22, T22)) = sigma-min( Zl ), ! ! where sigma-min(Zl) denotes the smallest singular value of the ! 2(n-1)-by-2(n-1) matrix ! ! Zl = [ kron(a, In-1) -kron(1, S22) ] ! [ kron(b, In-1) -kron(1, T22) ] . ! ! Here In-1 is the identity matrix of size n-1. kron(X, Y) is the ! Kronecker product between the matrices X and Y. ! ! Note that if the default method for computing DIF(i) is wanted ! (see DLATDF), then the parameter DIFDRI (see below) should be ! changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). ! See DTGSYL for more details. ! ! b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair, ! ! Suppose U and V are orthogonal transformations such that ! ! U'*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2 ! ( 0 S22 ),( 0 T22) n-2 ! 2 n-2 2 n-2 ! ! and (S11, T11) corresponds to the complex conjugate eigenvalue ! pair (w, conjg(w)). There exist unitary matrices U1 and V1 such ! that ! ! U1'*S11*V1 = ( s11 s12 ) and U1'*T11*V1 = ( t11 t12 ) ! ( 0 s22 ) ( 0 t22 ) ! ! where the generalized eigenvalues w = s11/t11 and ! conjg(w) = s22/t22. ! ! Then the reciprocal condition number DIF(i) is bounded by ! ! min( d1, max( 1, |real(s11)/real(s22)| )*d2 ) ! ! where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where ! Z1 is the complex 2-by-2 matrix ! ! Z1 = [ s11 -s22 ] ! [ t11 -t22 ], ! ! This is done by computing (using real arithmetic) the ! roots of the characteristical polynomial det(Z1' * Z1 - lambda I), ! where Z1' denotes the conjugate transpose of Z1 and det(X) denotes ! the determinant of X. ! ! and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an ! upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2) ! ! Z2 = [ kron(S11', In-2) -kron(I2, S22) ] ! [ kron(T11', In-2) -kron(I2, T22) ] ! ! Note that if the default method for computing DIF is wanted (see ! DLATDF), then the parameter DIFDRI (see below) should be changed ! from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL ! for more details. ! ! For each eigenvalue/vector specified by SELECT, DIF stores a ! Frobenius norm-based estimate of Difl. ! ! An approximate error bound for the i-th computed eigenvector VL(i) or ! VR(i) is given by ! ! EPS * norm(A, B) / DIF(i). ! ! See ref. [2-3] for more details and further references. ! ! Based on contributions by ! Bo Kagstrom and Peter Poromaa, Department of Computing Science, ! Umea University, S-901 87 Umea, Sweden. ! ! References ! ========== ! ! [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the ! Generalized Real Schur Form of a Regular Matrix Pair (A, B), in ! M.S. Moonen et al (eds), Linear Algebra for Large Scale and ! Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. ! ! [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified ! Eigenvalues of a Regular Matrix Pair (A, B) and Condition ! Estimation: Theory, Algorithms and Software, ! Report UMINF - 94.04, Department of Computing Science, Umea ! University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working ! Note 87. To appear in Numerical Algorithms, 1996. ! ! [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software ! for Solving the Generalized Sylvester Equation and Estimating the ! Separation between Regular Matrix Pairs, Report UMINF - 93.23, ! Department of Computing Science, Umea University, S-901 87 Umea, ! Sweden, December 1993, Revised April 1994, Also as LAPACK Working ! Note 75. To appear in ACM Trans. on Math. Software, Vol 22, ! No 1, 1996. ! ! ===================================================================== ! ! .. Parameters .. INTEGER DIFDRI PARAMETER ( DIFDRI = 3 ) DOUBLE PRECISION ZERO, ONE, TWO, FOUR PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0, & FOUR = 4.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, PAIR, SOMCON, WANTBH, WANTDF, WANTS INTEGER I, IERR, IFST, ILST, IZ, K, KS, LWMIN, N1, N2 DOUBLE PRECISION ALPHAI, ALPHAR, ALPRQT, BETA, C1, C2, COND, & EPS, LNRM, RNRM, ROOT1, ROOT2, SCALE, SMLNUM, & TMPII, TMPIR, TMPRI, TMPRR, UHAV, UHAVI, UHBV, & UHBVI ! .. ! .. Local Arrays .. DOUBLE PRECISION DUMMY( 1 ), DUMMY1( 1 ) ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT, DLAMCH, DLAPY2, DNRM2 EXTERNAL LSAME, DDOT, DLAMCH, DLAPY2, DNRM2 ! .. ! .. External Subroutines .. EXTERNAL DGEMV, DLACPY, DLAG2, DTGEXC, DTGSYL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT ! .. ! .. Executable Statements .. ! ! Decode and test the input parameters ! WANTBH = LSAME( JOB, 'B' ) WANTS = LSAME( JOB, 'E' ) .OR. WANTBH WANTDF = LSAME( JOB, 'V' ) .OR. WANTBH ! SOMCON = LSAME( HOWMNY, 'S' ) ! INFO = 0 LQUERY = ( LWORK.EQ.-1 ) ! IF( LSAME( JOB, 'V' ) .OR. LSAME( JOB, 'B' ) ) THEN LWMIN = MAX( 1, 2*N*( N+2 )+16 ) ELSE LWMIN = 1 END IF ! IF( .NOT.WANTS .AND. .NOT.WANTDF ) THEN INFO = -1 ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( WANTS .AND. LDVL.LT.N ) THEN INFO = -10 ELSE IF( WANTS .AND. LDVR.LT.N ) THEN INFO = -12 ELSE ! ! Set M to the number of eigenpairs for which condition numbers ! are required, and test MM. ! IF( SOMCON ) THEN M = 0 PAIR = .FALSE. DO 10 K = 1, N IF( PAIR ) THEN PAIR = .FALSE. ELSE IF( K.LT.N ) THEN IF( A( K+1, K ).EQ.ZERO ) THEN IF( SELECT( K ) ) & M = M + 1 ELSE PAIR = .TRUE. IF( SELECT( K ) .OR. SELECT( K+1 ) ) & M = M + 2 END IF ELSE IF( SELECT( N ) ) & M = M + 1 END IF END IF 10 CONTINUE ELSE M = N END IF ! IF( MM.LT.M ) THEN INFO = -15 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -18 ! ELSE IF( WANTDF .AND. LWORK.LT.2*N*( N+2 )+16 ) THEN ! INFO = -18 END IF END IF ! IF( INFO.EQ.0 ) THEN WORK( 1 ) = LWMIN END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTGSNA', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Get machine constants ! EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) / EPS KS = 0 PAIR = .FALSE. ! DO 20 K = 1, N ! ! Determine whether A(k,k) begins a 1-by-1 or 2-by-2 block. ! IF( PAIR ) THEN PAIR = .FALSE. GO TO 20 ELSE IF( K.LT.N ) & PAIR = A( K+1, K ).NE.ZERO END IF ! ! Determine whether condition numbers are required for the k-th ! eigenpair. ! IF( SOMCON ) THEN IF( PAIR ) THEN IF( .NOT.SELECT( K ) .AND. .NOT.SELECT( K+1 ) ) & GO TO 20 ELSE IF( .NOT.SELECT( K ) ) & GO TO 20 END IF END IF ! KS = KS + 1 ! IF( WANTS ) THEN ! ! Compute the reciprocal condition number of the k-th ! eigenvalue. ! IF( PAIR ) THEN ! ! Complex eigenvalue pair. ! RNRM = DLAPY2( DNRM2( N, VR( 1, KS ), 1 ), & DNRM2( N, VR( 1, KS+1 ), 1 ) ) LNRM = DLAPY2( DNRM2( N, VL( 1, KS ), 1 ), & DNRM2( N, VL( 1, KS+1 ), 1 ) ) CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS ), 1, ZERO, & WORK, 1 ) TMPRR = DDOT( N, WORK, 1, VL( 1, KS ), 1 ) TMPRI = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 ) CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS+1 ), 1, & ZERO, WORK, 1 ) TMPII = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 ) TMPIR = DDOT( N, WORK, 1, VL( 1, KS ), 1 ) UHAV = TMPRR + TMPII UHAVI = TMPIR - TMPRI CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS ), 1, ZERO, & WORK, 1 ) TMPRR = DDOT( N, WORK, 1, VL( 1, KS ), 1 ) TMPRI = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 ) CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS+1 ), 1, & ZERO, WORK, 1 ) TMPII = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 ) TMPIR = DDOT( N, WORK, 1, VL( 1, KS ), 1 ) UHBV = TMPRR + TMPII UHBVI = TMPIR - TMPRI UHAV = DLAPY2( UHAV, UHAVI ) UHBV = DLAPY2( UHBV, UHBVI ) COND = DLAPY2( UHAV, UHBV ) S( KS ) = COND / ( RNRM*LNRM ) S( KS+1 ) = S( KS ) ! ELSE ! ! Real eigenvalue. ! RNRM = DNRM2( N, VR( 1, KS ), 1 ) LNRM = DNRM2( N, VL( 1, KS ), 1 ) CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS ), 1, ZERO, & WORK, 1 ) UHAV = DDOT( N, WORK, 1, VL( 1, KS ), 1 ) CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS ), 1, ZERO, & WORK, 1 ) UHBV = DDOT( N, WORK, 1, VL( 1, KS ), 1 ) COND = DLAPY2( UHAV, UHBV ) IF( COND.EQ.ZERO ) THEN S( KS ) = -ONE ELSE S( KS ) = COND / ( RNRM*LNRM ) END IF END IF END IF ! IF( WANTDF ) THEN IF( N.EQ.1 ) THEN DIF( KS ) = DLAPY2( A( 1, 1 ), B( 1, 1 ) ) GO TO 20 END IF ! ! Estimate the reciprocal condition number of the k-th ! eigenvectors. IF( PAIR ) THEN ! ! Copy the 2-by 2 pencil beginning at (A(k,k), B(k, k)). ! Compute the eigenvalue(s) at position K. ! WORK( 1 ) = A( K, K ) WORK( 2 ) = A( K+1, K ) WORK( 3 ) = A( K, K+1 ) WORK( 4 ) = A( K+1, K+1 ) WORK( 5 ) = B( K, K ) WORK( 6 ) = B( K+1, K ) WORK( 7 ) = B( K, K+1 ) WORK( 8 ) = B( K+1, K+1 ) CALL DLAG2( WORK, 2, WORK( 5 ), 2, SMLNUM*EPS, BETA, & DUMMY1( 1 ), ALPHAR, DUMMY( 1 ), ALPHAI ) ALPRQT = ONE C1 = TWO*( ALPHAR*ALPHAR+ALPHAI*ALPHAI+BETA*BETA ) C2 = FOUR*BETA*BETA*ALPHAI*ALPHAI ROOT1 = C1 + SQRT( C1*C1-4.0D0*C2 ) ROOT2 = C2 / ROOT1 ROOT1 = ROOT1 / TWO COND = MIN( SQRT( ROOT1 ), SQRT( ROOT2 ) ) END IF ! ! Copy the matrix (A, B) to the array WORK and swap the ! diagonal block beginning at A(k,k) to the (1,1) position. ! CALL DLACPY( 'Full', N, N, A, LDA, WORK, N ) CALL DLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N ) IFST = K ILST = 1 ! CALL DTGEXC( .FALSE., .FALSE., N, WORK, N, WORK( N*N+1 ), N, & DUMMY, 1, DUMMY1, 1, IFST, ILST, & WORK( N*N*2+1 ), LWORK-2*N*N, IERR ) ! IF( IERR.GT.0 ) THEN ! ! Ill-conditioned problem - swap rejected. ! DIF( KS ) = ZERO ELSE ! ! Reordering successful, solve generalized Sylvester ! equation for R and L, ! A22 * R - L * A11 = A12 ! B22 * R - L * B11 = B12, ! and compute estimate of Difl((A11,B11), (A22, B22)). ! N1 = 1 IF( WORK( 2 ).NE.ZERO ) & N1 = 2 N2 = N - N1 IF( N2.EQ.0 ) THEN DIF( KS ) = COND ELSE I = N*N + 1 IZ = 2*N*N + 1 CALL DTGSYL( 'N', DIFDRI, N2, N1, WORK( N*N1+N1+1 ), & N, WORK, N, WORK( N1+1 ), N, & WORK( N*N1+N1+I ), N, WORK( I ), N, & WORK( N1+I ), N, SCALE, DIF( KS ), & WORK( IZ+1 ), LWORK-2*N*N, IWORK, IERR ) ! IF( PAIR ) & DIF( KS ) = MIN( MAX( ONE, ALPRQT )*DIF( KS ), & COND ) END IF END IF IF( PAIR ) & DIF( KS+1 ) = DIF( KS ) END IF IF( PAIR ) & KS = KS + 1 ! 20 CONTINUE WORK( 1 ) = LWMIN RETURN ! ! End of DTGSNA ! END SUBROUTINE DTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, & LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, & IWORK, PQ, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER TRANS INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N, & PQ DOUBLE PRECISION RDSCAL, RDSUM, SCALE ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ), & D( LDD, * ), E( LDE, * ), F( LDF, * ) ! .. ! ! Purpose ! ======= ! ! DTGSY2 solves the generalized Sylvester equation: ! ! A * R - L * B = scale * C (1) ! D * R - L * E = scale * F, ! ! using Level 1 and 2 BLAS. where R and L are unknown M-by-N matrices, ! (A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M, ! N-by-N and M-by-N, respectively, with real entries. (A, D) and (B, E) ! must be in generalized Schur canonical form, i.e. A, B are upper ! quasi triangular and D, E are upper triangular. The solution (R, L) ! overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor ! chosen to avoid overflow. ! ! In matrix notation solving equation (1) corresponds to solve ! Z*x = scale*b, where Z is defined as ! ! Z = [ kron(In, A) -kron(B', Im) ] (2) ! [ kron(In, D) -kron(E', Im) ], ! ! Ik is the identity matrix of size k and X' is the transpose of X. ! kron(X, Y) is the Kronecker product between the matrices X and Y. ! In the process of solving (1), we solve a number of such systems ! where Dim(In), Dim(In) = 1 or 2. ! ! If TRANS = 'T', solve the transposed system Z'*y = scale*b for y, ! which is equivalent to solve for R and L in ! ! A' * R + D' * L = scale * C (3) ! R * B' + L * E' = scale * -F ! ! This case is used to compute an estimate of Dif[(A, D), (B, E)] = ! sigma_min(Z) using reverse communicaton with DLACON. ! ! DTGSY2 also (IJOB >= 1) contributes to the computation in STGSYL ! of an upper bound on the separation between to matrix pairs. Then ! the input (A, D), (B, E) are sub-pencils of the matrix pair in ! DTGSYL. See STGSYL for details. ! ! Arguments ! ========= ! ! TRANS (input) CHARACTER ! = 'N', solve the generalized Sylvester equation (1). ! = 'T': solve the 'transposed' system (3). ! ! IJOB (input) INTEGER ! Specifies what kind of functionality to be performed. ! = 0: solve (1) only. ! = 1: A contribution from this subsystem to a Frobenius ! norm-based estimate of the separation between two matrix ! pairs is computed. (look ahead strategy is used). ! = 2: A contribution from this subsystem to a Frobenius ! norm-based estimate of the separation between two matrix ! pairs is computed. (DGECON on sub-systems is used.) ! Not referenced if TRANS = 'T'. ! ! M (input) INTEGER ! On entry, M specifies the order of A and D, and the row ! dimension of C, F, R and L. ! ! N (input) INTEGER ! On entry, N specifies the order of B and E, and the column ! dimension of C, F, R and L. ! ! A (input) DOUBLE PRECISION array, dimension (LDA, M) ! On entry, A contains an upper quasi triangular matrix. ! ! LDA (input) INTEGER ! The leading dimension of the matrix A. LDA >= max(1, M). ! ! B (input) DOUBLE PRECISION array, dimension (LDB, N) ! On entry, B contains an upper quasi triangular matrix. ! ! LDB (input) INTEGER ! The leading dimension of the matrix B. LDB >= max(1, N). ! ! C (input/ output) DOUBLE PRECISION array, dimension (LDC, N) ! On entry, C contains the right-hand-side of the first matrix ! equation in (1). ! On exit, if IJOB = 0, C has been overwritten by the ! solution R. ! ! LDC (input) INTEGER ! The leading dimension of the matrix C. LDC >= max(1, M). ! ! D (input) DOUBLE PRECISION array, dimension (LDD, M) ! On entry, D contains an upper triangular matrix. ! ! LDD (input) INTEGER ! The leading dimension of the matrix D. LDD >= max(1, M). ! ! E (input) DOUBLE PRECISION array, dimension (LDE, N) ! On entry, E contains an upper triangular matrix. ! ! LDE (input) INTEGER ! The leading dimension of the matrix E. LDE >= max(1, N). ! ! F (input/ output) DOUBLE PRECISION array, dimension (LDF, N) ! On entry, F contains the right-hand-side of the second matrix ! equation in (1). ! On exit, if IJOB = 0, F has been overwritten by the ! solution L. ! ! LDF (input) INTEGER ! The leading dimension of the matrix F. LDF >= max(1, M). ! ! SCALE (output) DOUBLE PRECISION ! On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions ! R and L (C and F on entry) will hold the solutions to a ! slightly perturbed system but the input matrices A, B, D and ! E have not been changed. If SCALE = 0, R and L will hold the ! solutions to the homogeneous system with C = F = 0. Normally, ! SCALE = 1. ! ! RDSUM (input/output) DOUBLE PRECISION ! On entry, the sum of squares of computed contributions to ! the Dif-estimate under computation by DTGSYL, where the ! scaling factor RDSCAL (see below) has been factored out. ! On exit, the corresponding sum of squares updated with the ! contributions from the current sub-system. ! If TRANS = 'T' RDSUM is not touched. ! NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL. ! ! RDSCAL (input/output) DOUBLE PRECISION ! On entry, scaling factor used to prevent overflow in RDSUM. ! On exit, RDSCAL is updated w.r.t. the current contributions ! in RDSUM. ! If TRANS = 'T', RDSCAL is not touched. ! NOTE: RDSCAL only makes sense when DTGSY2 is called by ! DTGSYL. ! ! IWORK (workspace) INTEGER array, dimension (M+N+2) ! ! PQ (output) INTEGER ! On exit, the number of subsystems (of size 2-by-2, 4-by-4 and ! 8-by-8) solved by this routine. ! ! INFO (output) INTEGER ! On exit, if INFO is set to ! =0: Successful exit ! <0: If INFO = -i, the i-th argument had an illegal value. ! >0: The matrix pairs (A, D) and (B, E) have common or very ! close eigenvalues. ! ! Further Details ! =============== ! ! Based on contributions by ! Bo Kagstrom and Peter Poromaa, Department of Computing Science, ! Umea University, S-901 87 Umea, Sweden. ! ! ===================================================================== ! ! .. Parameters .. INTEGER LDZ PARAMETER ( LDZ = 8 ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOTRAN INTEGER I, IE, IERR, II, IS, ISP1, J, JE, JJ, JS, JSP1, & K, MB, NB, P, Q, ZDIM DOUBLE PRECISION ALPHA, SCALOC ! .. ! .. Local Arrays .. INTEGER IPIV( LDZ ), JPIV( LDZ ) DOUBLE PRECISION RHS( LDZ ), Z( LDZ, LDZ ) double precision zero_vec(1) ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DGEMM, DGEMV, DGER, DGESC2, & DGETC2, DLATDF, DSCAL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! zero_vec(1) = 0.0D+00 ! ! Decode and test input parameters ! INFO = 0 IERR = 0 NOTRAN = LSAME( TRANS, 'N' ) IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN INFO = -1 ELSE IF( ( IJOB.LT.0 ) .OR. ( IJOB.GT.2 ) ) THEN INFO = -2 ELSE IF( M.LE.0 ) THEN INFO = -3 ELSE IF( N.LE.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -10 ELSE IF( LDD.LT.MAX( 1, M ) ) THEN INFO = -12 ELSE IF( LDE.LT.MAX( 1, N ) ) THEN INFO = -14 ELSE IF( LDF.LT.MAX( 1, M ) ) THEN INFO = -16 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTGSY2', -INFO ) RETURN END IF ! ! Determine block structure of A ! PQ = 0 P = 0 I = 1 10 CONTINUE IF( I.GT.M ) & GO TO 20 P = P + 1 IWORK( P ) = I IF( I.EQ.M ) & GO TO 20 IF( A( I+1, I ).NE.ZERO ) THEN I = I + 2 ELSE I = I + 1 END IF GO TO 10 20 CONTINUE IWORK( P+1 ) = M + 1 ! ! Determine block structure of B ! Q = P + 1 J = 1 30 CONTINUE IF( J.GT.N ) & GO TO 40 Q = Q + 1 IWORK( Q ) = J IF( J.EQ.N ) & GO TO 40 IF( B( J+1, J ).NE.ZERO ) THEN J = J + 2 ELSE J = J + 1 END IF GO TO 30 40 CONTINUE IWORK( Q+1 ) = N + 1 PQ = P*( Q-P-1 ) ! IF( NOTRAN ) THEN ! ! Solve (I, J) - subsystem ! A(I, I) * R(I, J) - L(I, J) * B(J, J) = C(I, J) ! D(I, I) * R(I, J) - L(I, J) * E(J, J) = F(I, J) ! for I = P, P - 1, ..., 1; J = 1, 2, ..., Q ! SCALE = ONE SCALOC = ONE DO 120 J = P + 2, Q JS = IWORK( J ) JSP1 = JS + 1 JE = IWORK( J+1 ) - 1 NB = JE - JS + 1 DO 110 I = P, 1, -1 ! IS = IWORK( I ) ISP1 = IS + 1 IE = IWORK( I+1 ) - 1 MB = IE - IS + 1 ZDIM = MB*NB*2 ! IF( ( MB.EQ.1 ) .AND. ( NB.EQ.1 ) ) THEN ! ! Build a 2-by-2 system Z * x = RHS ! Z( 1, 1 ) = A( IS, IS ) Z( 2, 1 ) = D( IS, IS ) Z( 1, 2 ) = -B( JS, JS ) Z( 2, 2 ) = -E( JS, JS ) ! ! Set up right hand side(s) ! RHS( 1 ) = C( IS, JS ) RHS( 2 ) = F( IS, JS ) ! ! Solve Z * x = RHS ! CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) & INFO = IERR ! IF( IJOB.EQ.0 ) THEN CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, & SCALOC ) IF( SCALOC.NE.ONE ) THEN DO K = 1, N CALL DSCAL( M, SCALOC, C( 1, K ), 1 ) CALL DSCAL( M, SCALOC, F( 1, K ), 1 ) end do SCALE = SCALE*SCALOC END IF ELSE CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM, & RDSCAL, IPIV, JPIV ) END IF ! ! Unpack solution vector(s) ! C( IS, JS ) = RHS( 1 ) F( IS, JS ) = RHS( 2 ) ! ! Substitute R(I, J) and L(I, J) into remaining ! equation. ! IF( I.GT.1 ) THEN ALPHA = -RHS( 1 ) CALL DAXPY( IS-1, ALPHA, A( 1, IS ), 1, C( 1, JS ), & 1 ) CALL DAXPY( IS-1, ALPHA, D( 1, IS ), 1, F( 1, JS ), & 1 ) END IF IF( J.LT.Q ) THEN CALL DAXPY( N-JE, RHS( 2 ), B( JS, JE+1 ), LDB, & C( IS, JE+1 ), LDC ) CALL DAXPY( N-JE, RHS( 2 ), E( JS, JE+1 ), LDE, & F( IS, JE+1 ), LDF ) END IF ! ELSE IF( ( MB.EQ.1 ) .AND. ( NB.EQ.2 ) ) THEN ! ! Build a 4-by-4 system Z * x = RHS ! Z( 1, 1 ) = A( IS, IS ) Z( 2, 1 ) = ZERO Z( 3, 1 ) = D( IS, IS ) Z( 4, 1 ) = ZERO ! Z( 1, 2 ) = ZERO Z( 2, 2 ) = A( IS, IS ) Z( 3, 2 ) = ZERO Z( 4, 2 ) = D( IS, IS ) ! Z( 1, 3 ) = -B( JS, JS ) Z( 2, 3 ) = -B( JS, JSP1 ) Z( 3, 3 ) = -E( JS, JS ) Z( 4, 3 ) = -E( JS, JSP1 ) ! Z( 1, 4 ) = -B( JSP1, JS ) Z( 2, 4 ) = -B( JSP1, JSP1 ) Z( 3, 4 ) = ZERO Z( 4, 4 ) = -E( JSP1, JSP1 ) ! ! Set up right hand side(s) ! RHS( 1 ) = C( IS, JS ) RHS( 2 ) = C( IS, JSP1 ) RHS( 3 ) = F( IS, JS ) RHS( 4 ) = F( IS, JSP1 ) ! ! Solve Z * x = RHS ! CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) & INFO = IERR ! IF( IJOB.EQ.0 ) THEN CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, & SCALOC ) IF( SCALOC.NE.ONE ) THEN DO 60 K = 1, N CALL DSCAL( M, SCALOC, C( 1, K ), 1 ) CALL DSCAL( M, SCALOC, F( 1, K ), 1 ) 60 CONTINUE SCALE = SCALE*SCALOC END IF ELSE CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM, & RDSCAL, IPIV, JPIV ) END IF ! ! Unpack solution vector(s) ! C( IS, JS ) = RHS( 1 ) C( IS, JSP1 ) = RHS( 2 ) F( IS, JS ) = RHS( 3 ) F( IS, JSP1 ) = RHS( 4 ) ! ! Substitute R(I, J) and L(I, J) into remaining ! equation. ! IF( I.GT.1 ) THEN CALL DGER( IS-1, NB, -ONE, A( 1, IS ), 1, RHS( 1 ), & 1, C( 1, JS ), LDC ) CALL DGER( IS-1, NB, -ONE, D( 1, IS ), 1, RHS( 1 ), & 1, F( 1, JS ), LDF ) END IF IF( J.LT.Q ) THEN CALL DAXPY( N-JE, RHS( 3 ), B( JS, JE+1 ), LDB, & C( IS, JE+1 ), LDC ) CALL DAXPY( N-JE, RHS( 3 ), E( JS, JE+1 ), LDE, & F( IS, JE+1 ), LDF ) CALL DAXPY( N-JE, RHS( 4 ), B( JSP1, JE+1 ), LDB, & C( IS, JE+1 ), LDC ) CALL DAXPY( N-JE, RHS( 4 ), E( JSP1, JE+1 ), LDE, & F( IS, JE+1 ), LDF ) END IF ! ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.1 ) ) THEN ! ! Build a 4-by-4 system Z * x = RHS ! Z( 1, 1 ) = A( IS, IS ) Z( 2, 1 ) = A( ISP1, IS ) Z( 3, 1 ) = D( IS, IS ) Z( 4, 1 ) = ZERO ! Z( 1, 2 ) = A( IS, ISP1 ) Z( 2, 2 ) = A( ISP1, ISP1 ) Z( 3, 2 ) = D( IS, ISP1 ) Z( 4, 2 ) = D( ISP1, ISP1 ) ! Z( 1, 3 ) = -B( JS, JS ) Z( 2, 3 ) = ZERO Z( 3, 3 ) = -E( JS, JS ) Z( 4, 3 ) = ZERO ! Z( 1, 4 ) = ZERO Z( 2, 4 ) = -B( JS, JS ) Z( 3, 4 ) = ZERO Z( 4, 4 ) = -E( JS, JS ) ! ! Set up right hand side(s) ! RHS( 1 ) = C( IS, JS ) RHS( 2 ) = C( ISP1, JS ) RHS( 3 ) = F( IS, JS ) RHS( 4 ) = F( ISP1, JS ) ! ! Solve Z * x = RHS ! CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) & INFO = IERR IF( IJOB.EQ.0 ) THEN CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, & SCALOC ) IF( SCALOC.NE.ONE ) THEN DO 70 K = 1, N CALL DSCAL( M, SCALOC, C( 1, K ), 1 ) CALL DSCAL( M, SCALOC, F( 1, K ), 1 ) 70 CONTINUE SCALE = SCALE*SCALOC END IF ELSE CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM, & RDSCAL, IPIV, JPIV ) END IF ! ! Unpack solution vector(s) ! C( IS, JS ) = RHS( 1 ) C( ISP1, JS ) = RHS( 2 ) F( IS, JS ) = RHS( 3 ) F( ISP1, JS ) = RHS( 4 ) ! ! Substitute R(I, J) and L(I, J) into remaining ! equation. ! IF( I.GT.1 ) THEN CALL DGEMV( 'N', IS-1, MB, -ONE, A( 1, IS ), LDA, & RHS( 1 ), 1, ONE, C( 1, JS ), 1 ) CALL DGEMV( 'N', IS-1, MB, -ONE, D( 1, IS ), LDD, & RHS( 1 ), 1, ONE, F( 1, JS ), 1 ) END IF IF( J.LT.Q ) THEN CALL DGER( MB, N-JE, ONE, RHS( 3 ), 1, & B( JS, JE+1 ), LDB, C( IS, JE+1 ), LDC ) CALL DGER( MB, N-JE, ONE, RHS( 3 ), 1, & E( JS, JE+1 ), LDB, F( IS, JE+1 ), LDC ) END IF ! ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.2 ) ) THEN ! ! Build an 8-by-8 system Z * x = RHS ! CALL DCOPY( LDZ*LDZ, zero_vec, 0, Z, 1 ) ! Z( 1, 1 ) = A( IS, IS ) Z( 2, 1 ) = A( ISP1, IS ) Z( 5, 1 ) = D( IS, IS ) ! Z( 1, 2 ) = A( IS, ISP1 ) Z( 2, 2 ) = A( ISP1, ISP1 ) Z( 5, 2 ) = D( IS, ISP1 ) Z( 6, 2 ) = D( ISP1, ISP1 ) ! Z( 3, 3 ) = A( IS, IS ) Z( 4, 3 ) = A( ISP1, IS ) Z( 7, 3 ) = D( IS, IS ) ! Z( 3, 4 ) = A( IS, ISP1 ) Z( 4, 4 ) = A( ISP1, ISP1 ) Z( 7, 4 ) = D( IS, ISP1 ) Z( 8, 4 ) = D( ISP1, ISP1 ) ! Z( 1, 5 ) = -B( JS, JS ) Z( 3, 5 ) = -B( JS, JSP1 ) Z( 5, 5 ) = -E( JS, JS ) Z( 7, 5 ) = -E( JS, JSP1 ) ! Z( 2, 6 ) = -B( JS, JS ) Z( 4, 6 ) = -B( JS, JSP1 ) Z( 6, 6 ) = -E( JS, JS ) Z( 8, 6 ) = -E( JS, JSP1 ) ! Z( 1, 7 ) = -B( JSP1, JS ) Z( 3, 7 ) = -B( JSP1, JSP1 ) Z( 7, 7 ) = -E( JSP1, JSP1 ) ! Z( 2, 8 ) = -B( JSP1, JS ) Z( 4, 8 ) = -B( JSP1, JSP1 ) Z( 8, 8 ) = -E( JSP1, JSP1 ) ! ! Set up right hand side(s) ! K = 1 II = MB*NB + 1 DO 80 JJ = 0, NB - 1 CALL DCOPY( MB, C( IS, JS+JJ ), 1, RHS( K ), 1 ) CALL DCOPY( MB, F( IS, JS+JJ ), 1, RHS( II ), 1 ) K = K + MB II = II + MB 80 CONTINUE ! ! Solve Z * x = RHS ! CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) & INFO = IERR IF( IJOB.EQ.0 ) THEN CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, & SCALOC ) IF( SCALOC.NE.ONE ) THEN DO K = 1, N CALL DSCAL( M, SCALOC, C( 1, K ), 1 ) CALL DSCAL( M, SCALOC, F( 1, K ), 1 ) end do SCALE = SCALE*SCALOC END IF ELSE CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM, & RDSCAL, IPIV, JPIV ) END IF ! ! Unpack solution vector(s) ! K = 1 II = MB*NB + 1 DO 100 JJ = 0, NB - 1 CALL DCOPY( MB, RHS( K ), 1, C( IS, JS+JJ ), 1 ) CALL DCOPY( MB, RHS( II ), 1, F( IS, JS+JJ ), 1 ) K = K + MB II = II + MB 100 CONTINUE ! ! Substitute R(I, J) and L(I, J) into remaining ! equation. ! IF( I.GT.1 ) THEN CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE, & A( 1, IS ), LDA, RHS( 1 ), MB, ONE, & C( 1, JS ), LDC ) CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE, & D( 1, IS ), LDD, RHS( 1 ), MB, ONE, & F( 1, JS ), LDF ) END IF IF( J.LT.Q ) THEN K = MB*NB + 1 CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE, RHS( K ), & MB, B( JS, JE+1 ), LDB, ONE, & C( IS, JE+1 ), LDC ) CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE, RHS( K ), & MB, E( JS, JE+1 ), LDE, ONE, & F( IS, JE+1 ), LDF ) END IF ! END IF ! 110 CONTINUE 120 CONTINUE ELSE ! ! Solve (I, J) - subsystem ! A(I, I)' * R(I, J) + D(I, I)' * L(J, J) = C(I, J) ! R(I, I) * B(J, J) + L(I, J) * E(J, J) = -F(I, J) ! for I = 1, 2, ..., P, J = Q, Q - 1, ..., 1 ! SCALE = ONE SCALOC = ONE DO 200 I = 1, P ! IS = IWORK( I ) ISP1 = IS + 1 IE = IWORK( I+1 ) - 1 MB = IE - IS + 1 DO 190 J = Q, P + 2, -1 ! JS = IWORK( J ) JSP1 = JS + 1 JE = IWORK( J+1 ) - 1 NB = JE - JS + 1 ZDIM = MB*NB*2 IF( ( MB.EQ.1 ) .AND. ( NB.EQ.1 ) ) THEN ! ! Build a 2-by-2 system Z' * x = RHS ! Z( 1, 1 ) = A( IS, IS ) Z( 2, 1 ) = -B( JS, JS ) Z( 1, 2 ) = D( IS, IS ) Z( 2, 2 ) = -E( JS, JS ) ! ! Set up right hand side(s) ! RHS( 1 ) = C( IS, JS ) RHS( 2 ) = F( IS, JS ) ! ! Solve Z' * x = RHS ! CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) & INFO = IERR ! CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, SCALOC ) IF( SCALOC.NE.ONE ) THEN DO 130 K = 1, N CALL DSCAL( M, SCALOC, C( 1, K ), 1 ) CALL DSCAL( M, SCALOC, F( 1, K ), 1 ) 130 CONTINUE SCALE = SCALE*SCALOC END IF ! ! Unpack solution vector(s) ! C( IS, JS ) = RHS( 1 ) F( IS, JS ) = RHS( 2 ) ! ! Substitute R(I, J) and L(I, J) into remaining ! equation. ! IF( J.GT.P+2 ) THEN ALPHA = RHS( 1 ) CALL DAXPY( JS-1, ALPHA, B( 1, JS ), 1, F( IS, 1 ), & LDF ) ALPHA = RHS( 2 ) CALL DAXPY( JS-1, ALPHA, E( 1, JS ), 1, F( IS, 1 ), & LDF ) END IF IF( I.LT.P ) THEN ALPHA = -RHS( 1 ) CALL DAXPY( M-IE, ALPHA, A( IS, IE+1 ), LDA, & C( IE+1, JS ), 1 ) ALPHA = -RHS( 2 ) CALL DAXPY( M-IE, ALPHA, D( IS, IE+1 ), LDD, & C( IE+1, JS ), 1 ) END IF ! ELSE IF( ( MB.EQ.1 ) .AND. ( NB.EQ.2 ) ) THEN ! ! Build a 4-by-4 system Z' * x = RHS ! Z( 1, 1 ) = A( IS, IS ) Z( 2, 1 ) = ZERO Z( 3, 1 ) = -B( JS, JS ) Z( 4, 1 ) = -B( JSP1, JS ) ! Z( 1, 2 ) = ZERO Z( 2, 2 ) = A( IS, IS ) Z( 3, 2 ) = -B( JS, JSP1 ) Z( 4, 2 ) = -B( JSP1, JSP1 ) ! Z( 1, 3 ) = D( IS, IS ) Z( 2, 3 ) = ZERO Z( 3, 3 ) = -E( JS, JS ) Z( 4, 3 ) = ZERO ! Z( 1, 4 ) = ZERO Z( 2, 4 ) = D( IS, IS ) Z( 3, 4 ) = -E( JS, JSP1 ) Z( 4, 4 ) = -E( JSP1, JSP1 ) ! ! Set up right hand side(s) ! RHS( 1 ) = C( IS, JS ) RHS( 2 ) = C( IS, JSP1 ) RHS( 3 ) = F( IS, JS ) RHS( 4 ) = F( IS, JSP1 ) ! ! Solve Z' * x = RHS ! CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) & INFO = IERR CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, SCALOC ) IF( SCALOC.NE.ONE ) THEN DO 140 K = 1, N CALL DSCAL( M, SCALOC, C( 1, K ), 1 ) CALL DSCAL( M, SCALOC, F( 1, K ), 1 ) 140 CONTINUE SCALE = SCALE*SCALOC END IF ! ! Unpack solution vector(s) ! C( IS, JS ) = RHS( 1 ) C( IS, JSP1 ) = RHS( 2 ) F( IS, JS ) = RHS( 3 ) F( IS, JSP1 ) = RHS( 4 ) ! ! Substitute R(I, J) and L(I, J) into remaining ! equation. ! IF( J.GT.P+2 ) THEN CALL DAXPY( JS-1, RHS( 1 ), B( 1, JS ), 1, & F( IS, 1 ), LDF ) CALL DAXPY( JS-1, RHS( 2 ), B( 1, JSP1 ), 1, & F( IS, 1 ), LDF ) CALL DAXPY( JS-1, RHS( 3 ), E( 1, JS ), 1, & F( IS, 1 ), LDF ) CALL DAXPY( JS-1, RHS( 4 ), E( 1, JSP1 ), 1, & F( IS, 1 ), LDF ) END IF IF( I.LT.P ) THEN CALL DGER( M-IE, NB, -ONE, A( IS, IE+1 ), LDA, & RHS( 1 ), 1, C( IE+1, JS ), LDC ) CALL DGER( M-IE, NB, -ONE, D( IS, IE+1 ), LDD, & RHS( 3 ), 1, C( IE+1, JS ), LDC ) END IF ! ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.1 ) ) THEN ! ! Build a 4-by-4 system Z' * x = RHS ! Z( 1, 1 ) = A( IS, IS ) Z( 2, 1 ) = A( IS, ISP1 ) Z( 3, 1 ) = -B( JS, JS ) Z( 4, 1 ) = ZERO ! Z( 1, 2 ) = A( ISP1, IS ) Z( 2, 2 ) = A( ISP1, ISP1 ) Z( 3, 2 ) = ZERO Z( 4, 2 ) = -B( JS, JS ) ! Z( 1, 3 ) = D( IS, IS ) Z( 2, 3 ) = D( IS, ISP1 ) Z( 3, 3 ) = -E( JS, JS ) Z( 4, 3 ) = ZERO ! Z( 1, 4 ) = ZERO Z( 2, 4 ) = D( ISP1, ISP1 ) Z( 3, 4 ) = ZERO Z( 4, 4 ) = -E( JS, JS ) ! ! Set up right hand side(s) ! RHS( 1 ) = C( IS, JS ) RHS( 2 ) = C( ISP1, JS ) RHS( 3 ) = F( IS, JS ) RHS( 4 ) = F( ISP1, JS ) ! ! Solve Z' * x = RHS ! CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) & INFO = IERR ! CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, SCALOC ) IF( SCALOC.NE.ONE ) THEN DO K = 1, N CALL DSCAL( M, SCALOC, C( 1, K ), 1 ) CALL DSCAL( M, SCALOC, F( 1, K ), 1 ) end do SCALE = SCALE*SCALOC END IF ! ! Unpack solution vector(s) ! C( IS, JS ) = RHS( 1 ) C( ISP1, JS ) = RHS( 2 ) F( IS, JS ) = RHS( 3 ) F( ISP1, JS ) = RHS( 4 ) ! ! Substitute R(I, J) and L(I, J) into remaining ! equation. ! IF( J.GT.P+2 ) THEN CALL DGER( MB, JS-1, ONE, RHS( 1 ), 1, B( 1, JS ), & 1, F( IS, 1 ), LDF ) CALL DGER( MB, JS-1, ONE, RHS( 3 ), 1, E( 1, JS ), & 1, F( IS, 1 ), LDF ) END IF IF( I.LT.P ) THEN CALL DGEMV( 'T', MB, M-IE, -ONE, A( IS, IE+1 ), & LDA, RHS( 1 ), 1, ONE, C( IE+1, JS ), & 1 ) CALL DGEMV( 'T', MB, M-IE, -ONE, D( IS, IE+1 ), & LDD, RHS( 3 ), 1, ONE, C( IE+1, JS ), & 1 ) END IF ! ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.2 ) ) THEN ! ! Build an 8-by-8 system Z' * x = RHS ! CALL DCOPY( LDZ*LDZ, zero_vec, 0, Z, 1 ) ! Z( 1, 1 ) = A( IS, IS ) Z( 2, 1 ) = A( IS, ISP1 ) Z( 5, 1 ) = -B( JS, JS ) Z( 7, 1 ) = -B( JSP1, JS ) ! Z( 1, 2 ) = A( ISP1, IS ) Z( 2, 2 ) = A( ISP1, ISP1 ) Z( 6, 2 ) = -B( JS, JS ) Z( 8, 2 ) = -B( JSP1, JS ) ! Z( 3, 3 ) = A( IS, IS ) Z( 4, 3 ) = A( IS, ISP1 ) Z( 5, 3 ) = -B( JS, JSP1 ) Z( 7, 3 ) = -B( JSP1, JSP1 ) ! Z( 3, 4 ) = A( ISP1, IS ) Z( 4, 4 ) = A( ISP1, ISP1 ) Z( 6, 4 ) = -B( JS, JSP1 ) Z( 8, 4 ) = -B( JSP1, JSP1 ) ! Z( 1, 5 ) = D( IS, IS ) Z( 2, 5 ) = D( IS, ISP1 ) Z( 5, 5 ) = -E( JS, JS ) ! Z( 2, 6 ) = D( ISP1, ISP1 ) Z( 6, 6 ) = -E( JS, JS ) ! Z( 3, 7 ) = D( IS, IS ) Z( 4, 7 ) = D( IS, ISP1 ) Z( 5, 7 ) = -E( JS, JSP1 ) Z( 7, 7 ) = -E( JSP1, JSP1 ) ! Z( 4, 8 ) = D( ISP1, ISP1 ) Z( 6, 8 ) = -E( JS, JSP1 ) Z( 8, 8 ) = -E( JSP1, JSP1 ) ! ! Set up right hand side(s) ! K = 1 II = MB*NB + 1 DO 160 JJ = 0, NB - 1 CALL DCOPY( MB, C( IS, JS+JJ ), 1, RHS( K ), 1 ) CALL DCOPY( MB, F( IS, JS+JJ ), 1, RHS( II ), 1 ) K = K + MB II = II + MB 160 CONTINUE ! ! ! Solve Z' * x = RHS ! CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) & INFO = IERR ! CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, SCALOC ) IF( SCALOC.NE.ONE ) THEN DO 170 K = 1, N CALL DSCAL( M, SCALOC, C( 1, K ), 1 ) CALL DSCAL( M, SCALOC, F( 1, K ), 1 ) 170 CONTINUE SCALE = SCALE*SCALOC END IF ! ! Unpack solution vector(s) ! K = 1 II = MB*NB + 1 DO 180 JJ = 0, NB - 1 CALL DCOPY( MB, RHS( K ), 1, C( IS, JS+JJ ), 1 ) CALL DCOPY( MB, RHS( II ), 1, F( IS, JS+JJ ), 1 ) K = K + MB II = II + MB 180 CONTINUE ! ! Substitute R(I, J) and L(I, J) into remaining ! equation. ! IF( J.GT.P+2 ) THEN CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE, & C( IS, JS ), LDC, B( 1, JS ), LDB, ONE, & F( IS, 1 ), LDF ) CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE, & F( IS, JS ), LDF, E( 1, JS ), LDE, ONE, & F( IS, 1 ), LDF ) END IF IF( I.LT.P ) THEN CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE, & A( IS, IE+1 ), LDA, C( IS, JS ), LDC, & ONE, C( IE+1, JS ), LDC ) CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE, & D( IS, IE+1 ), LDD, F( IS, JS ), LDF, & ONE, C( IE+1, JS ), LDC ) END IF ! END IF ! 190 CONTINUE 200 CONTINUE ! END IF RETURN ! ! End of DTGSY2 ! END SUBROUTINE DTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, & LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, & IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER TRANS INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, & LWORK, M, N DOUBLE PRECISION DIF, SCALE ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ), & D( LDD, * ), E( LDE, * ), F( LDF, * ), & WORK( * ) ! .. ! ! Purpose ! ======= ! ! DTGSYL solves the generalized Sylvester equation: ! ! A * R - L * B = scale * C (1) ! D * R - L * E = scale * F ! ! where R and L are unknown m-by-n matrices, (A, D), (B, E) and ! (C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n, ! respectively, with real entries. (A, D) and (B, E) must be in ! generalized (real) Schur canonical form, i.e. A, B are upper quasi ! triangular and D, E are upper triangular. ! ! The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output ! scaling factor chosen to avoid overflow. ! ! In matrix notation (1) is equivalent to solve Zx = scale b, where ! Z is defined as ! ! Z = [ kron(In, A) -kron(B', Im) ] (2) ! [ kron(In, D) -kron(E', Im) ]. ! ! Here Ik is the identity matrix of size k and X' is the transpose of ! X. kron(X, Y) is the Kronecker product between the matrices X and Y. ! ! If TRANS = 'T', DTGSYL solves the transposed system Z'*y = scale*b, ! which is equivalent to solve for R and L in ! ! A' * R + D' * L = scale * C (3) ! R * B' + L * E' = scale * (-F) ! ! This case (TRANS = 'T') is used to compute an one-norm-based estimate ! of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D) ! and (B,E), using DLACON. ! ! If IJOB >= 1, DTGSYL computes a Frobenius norm-based estimate ! of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the ! reciprocal of the smallest singular value of Z. See [1-2] for more ! information. ! ! This is a level 3 BLAS algorithm. ! ! Arguments ! ========= ! ! TRANS (input) CHARACTER*1 ! = 'N', solve the generalized Sylvester equation (1). ! = 'T', solve the 'transposed' system (3). ! ! IJOB (input) INTEGER ! Specifies what kind of functionality to be performed. ! =0: solve (1) only. ! =1: The functionality of 0 and 3. ! =2: The functionality of 0 and 4. ! =3: Only an estimate of Dif[(A,D), (B,E)] is computed. ! (look ahead strategy IJOB = 1 is used). ! =4: Only an estimate of Dif[(A,D), (B,E)] is computed. ! ( DGECON on sub-systems is used ). ! Not referenced if TRANS = 'T'. ! ! M (input) INTEGER ! The order of the matrices A and D, and the row dimension of ! the matrices C, F, R and L. ! ! N (input) INTEGER ! The order of the matrices B and E, and the column dimension ! of the matrices C, F, R and L. ! ! A (input) DOUBLE PRECISION array, dimension (LDA, M) ! The upper quasi triangular matrix A. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1, M). ! ! B (input) DOUBLE PRECISION array, dimension (LDB, N) ! The upper quasi triangular matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1, N). ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC, N) ! On entry, C contains the right-hand-side of the first matrix ! equation in (1) or (3). ! On exit, if IJOB = 0, 1 or 2, C has been overwritten by ! the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R, ! the solution achieved during the computation of the ! Dif-estimate. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1, M). ! ! D (input) DOUBLE PRECISION array, dimension (LDD, M) ! The upper triangular matrix D. ! ! LDD (input) INTEGER ! The leading dimension of the array D. LDD >= max(1, M). ! ! E (input) DOUBLE PRECISION array, dimension (LDE, N) ! The upper triangular matrix E. ! ! LDE (input) INTEGER ! The leading dimension of the array E. LDE >= max(1, N). ! ! F (input/output) DOUBLE PRECISION array, dimension (LDF, N) ! On entry, F contains the right-hand-side of the second matrix ! equation in (1) or (3). ! On exit, if IJOB = 0, 1 or 2, F has been overwritten by ! the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L, ! the solution achieved during the computation of the ! Dif-estimate. ! ! LDF (input) INTEGER ! The leading dimension of the array F. LDF >= max(1, M). ! ! DIF (output) DOUBLE PRECISION ! On exit DIF is the reciprocal of a lower bound of the ! reciprocal of the Dif-function, i.e. DIF is an upper bound of ! Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2). ! IF IJOB = 0 or TRANS = 'T', DIF is not touched. ! ! SCALE (output) DOUBLE PRECISION ! On exit SCALE is the scaling factor in (1) or (3). ! If 0 < SCALE < 1, C and F hold the solutions R and L, resp., ! to a slightly perturbed system but the input matrices A, B, D ! and E have not been changed. If SCALE = 0, C and F hold the ! solutions R and L, respectively, to the homogeneous system ! with C = F = 0. Normally, SCALE = 1. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! If IJOB = 0, WORK is not referenced. Otherwise, ! on exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK > = 1. ! If IJOB = 1 or 2 and TRANS = 'N', LWORK >= 2*M*N. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace) INTEGER array, dimension (M+N+6) ! ! INFO (output) INTEGER ! =0: successful exit ! <0: If INFO = -i, the i-th argument had an illegal value. ! >0: (A, D) and (B, E) have common or close eigenvalues. ! ! Further Details ! =============== ! ! Based on contributions by ! Bo Kagstrom and Peter Poromaa, Department of Computing Science, ! Umea University, S-901 87 Umea, Sweden. ! ! [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software ! for Solving the Generalized Sylvester Equation and Estimating the ! Separation between Regular Matrix Pairs, Report UMINF - 93.23, ! Department of Computing Science, Umea University, S-901 87 Umea, ! Sweden, December 1993, Revised April 1994, Also as LAPACK Working ! Note 75. To appear in ACM Trans. on Math. Software, Vol 22, ! No 1, 1996. ! ! [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester ! Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal. ! Appl., 15(4):1045-1060, 1994 ! ! [3] B. Kagstrom and L. Westin, Generalized Schur Methods with ! Condition Estimators for Solving the Generalized Sylvester ! Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, ! July 1989, pp 745-751. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, NOTRAN INTEGER I, IE, IFUNC, IROUND, IS, ISOLVE, J, JE, JS, K, & LINFO, LWMIN, MB, NB, P, PPQQ, PQ, Q DOUBLE PRECISION DSCALE, DSUM, SCALE2, SCALOC double precision zero_vec(1) ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DGEMM, DLACPY, DSCAL, DTGSY2, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC DBLE, MAX, SQRT ! .. ! .. Executable Statements .. ! zero_vec(1) = 0.0D+00 ! ! Decode and test input parameters ! INFO = 0 NOTRAN = LSAME( TRANS, 'N' ) LQUERY = ( LWORK.EQ.-1 ) ! IF( ( IJOB.EQ.1 .OR. IJOB.EQ.2 ) .AND. NOTRAN ) THEN LWMIN = MAX( 1, 2*M*N ) ELSE LWMIN = 1 END IF ! IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN INFO = -1 ELSE IF( ( IJOB.LT.0 ) .OR. ( IJOB.GT.4 ) ) THEN INFO = -2 ELSE IF( M.LE.0 ) THEN INFO = -3 ELSE IF( N.LE.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -6 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -10 ELSE IF( LDD.LT.MAX( 1, M ) ) THEN INFO = -12 ELSE IF( LDE.LT.MAX( 1, N ) ) THEN INFO = -14 ELSE IF( LDF.LT.MAX( 1, M ) ) THEN INFO = -16 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -20 END IF ! IF( INFO.EQ.0 ) THEN WORK( 1 ) = LWMIN END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTGSYL', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Determine optimal block sizes MB and NB ! MB = ILAENV( 2, 'DTGSYL', TRANS, M, N, -1, -1 ) NB = ILAENV( 5, 'DTGSYL', TRANS, M, N, -1, -1 ) ! ISOLVE = 1 IFUNC = 0 IF( IJOB.GE.3 .AND. NOTRAN ) THEN IFUNC = IJOB - 2 DO J = 1, N CALL DCOPY( M, zero_vec, 0, C( 1, J ), 1 ) CALL DCOPY( M, zero_vec, 0, F( 1, J ), 1 ) end do ELSE IF( IJOB.GE.1 .AND. NOTRAN ) THEN ISOLVE = 2 END IF ! IF( ( MB.LE.1 .AND. NB.LE.1 ) .OR. ( MB.GE.M .AND. NB.GE.N ) ) & THEN ! DO 30 IROUND = 1, ISOLVE ! ! Use unblocked Level 2 solver ! DSCALE = ZERO DSUM = ONE PQ = 0 CALL DTGSY2( TRANS, IFUNC, M, N, A, LDA, B, LDB, C, LDC, D, & LDD, E, LDE, F, LDF, SCALE, DSUM, DSCALE, & IWORK, PQ, INFO ) IF( DSCALE.NE.ZERO ) THEN IF( IJOB.EQ.1 .OR. IJOB.EQ.3 ) THEN DIF = SQRT( DBLE( 2*M*N ) ) / ( DSCALE*SQRT( DSUM ) ) ELSE DIF = SQRT( DBLE( PQ ) ) / ( DSCALE*SQRT( DSUM ) ) END IF END IF ! IF( ISOLVE.EQ.2 .AND. IROUND.EQ.1 ) THEN IFUNC = IJOB SCALE2 = SCALE CALL DLACPY( 'F', M, N, C, LDC, WORK, M ) CALL DLACPY( 'F', M, N, F, LDF, WORK( M*N+1 ), M ) DO J = 1, N CALL DCOPY( M, zero_vec, 0, C( 1, J ), 1 ) CALL DCOPY( M, zero_vec, 0, F( 1, J ), 1 ) end do ELSE IF( ISOLVE.EQ.2 .AND. IROUND.EQ.2 ) THEN CALL DLACPY( 'F', M, N, WORK, M, C, LDC ) CALL DLACPY( 'F', M, N, WORK( M*N+1 ), M, F, LDF ) SCALE = SCALE2 END IF 30 CONTINUE ! RETURN END IF ! ! Determine block structure of A ! P = 0 I = 1 40 CONTINUE IF( I.GT.M ) & GO TO 50 P = P + 1 IWORK( P ) = I I = I + MB IF( I.GE.M ) & GO TO 50 IF( A( I, I-1 ).NE.ZERO ) & I = I + 1 GO TO 40 50 CONTINUE ! IWORK( P+1 ) = M + 1 IF( IWORK( P ).EQ.IWORK( P+1 ) ) & P = P - 1 ! ! Determine block structure of B ! Q = P + 1 J = 1 60 CONTINUE IF( J.GT.N ) & GO TO 70 Q = Q + 1 IWORK( Q ) = J J = J + NB IF( J.GE.N ) & GO TO 70 IF( B( J, J-1 ).NE.ZERO ) & J = J + 1 GO TO 60 70 CONTINUE ! IWORK( Q+1 ) = N + 1 IF( IWORK( Q ).EQ.IWORK( Q+1 ) ) & Q = Q - 1 ! IF( NOTRAN ) THEN ! DO 150 IROUND = 1, ISOLVE ! ! Solve (I, J)-subsystem ! A(I, I) * R(I, J) - L(I, J) * B(J, J) = C(I, J) ! D(I, I) * R(I, J) - L(I, J) * E(J, J) = F(I, J) ! for I = P, P - 1,..., 1; J = 1, 2,..., Q ! DSCALE = ZERO DSUM = ONE PQ = 0 SCALE = ONE DO 130 J = P + 2, Q JS = IWORK( J ) JE = IWORK( J+1 ) - 1 NB = JE - JS + 1 DO 120 I = P, 1, -1 IS = IWORK( I ) IE = IWORK( I+1 ) - 1 MB = IE - IS + 1 PPQQ = 0 CALL DTGSY2( TRANS, IFUNC, MB, NB, A( IS, IS ), LDA, & B( JS, JS ), LDB, C( IS, JS ), LDC, & D( IS, IS ), LDD, E( JS, JS ), LDE, & F( IS, JS ), LDF, SCALOC, DSUM, DSCALE, & IWORK( Q+2 ), PPQQ, LINFO ) IF( LINFO.GT.0 ) & INFO = LINFO ! PQ = PQ + PPQQ IF( SCALOC.NE.ONE ) THEN DO 80 K = 1, JS - 1 CALL DSCAL( M, SCALOC, C( 1, K ), 1 ) CALL DSCAL( M, SCALOC, F( 1, K ), 1 ) 80 CONTINUE DO 90 K = JS, JE CALL DSCAL( IS-1, SCALOC, C( 1, K ), 1 ) CALL DSCAL( IS-1, SCALOC, F( 1, K ), 1 ) 90 CONTINUE DO 100 K = JS, JE CALL DSCAL( M-IE, SCALOC, C( IE+1, K ), 1 ) CALL DSCAL( M-IE, SCALOC, F( IE+1, K ), 1 ) 100 CONTINUE DO 110 K = JE + 1, N CALL DSCAL( M, SCALOC, C( 1, K ), 1 ) CALL DSCAL( M, SCALOC, F( 1, K ), 1 ) 110 CONTINUE SCALE = SCALE*SCALOC END IF ! ! Substitute R(I, J) and L(I, J) into remaining ! equation. ! IF( I.GT.1 ) THEN CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE, & A( 1, IS ), LDA, C( IS, JS ), LDC, ONE, & C( 1, JS ), LDC ) CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE, & D( 1, IS ), LDD, C( IS, JS ), LDC, ONE, & F( 1, JS ), LDF ) END IF IF( J.LT.Q ) THEN CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE, & F( IS, JS ), LDF, B( JS, JE+1 ), LDB, & ONE, C( IS, JE+1 ), LDC ) CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE, & F( IS, JS ), LDF, E( JS, JE+1 ), LDE, & ONE, F( IS, JE+1 ), LDF ) END IF 120 CONTINUE 130 CONTINUE IF( DSCALE.NE.ZERO ) THEN IF( IJOB.EQ.1 .OR. IJOB.EQ.3 ) THEN DIF = SQRT( DBLE( 2*M*N ) ) / ( DSCALE*SQRT( DSUM ) ) ELSE DIF = SQRT( DBLE( PQ ) ) / ( DSCALE*SQRT( DSUM ) ) END IF END IF IF( ISOLVE.EQ.2 .AND. IROUND.EQ.1 ) THEN IFUNC = IJOB SCALE2 = SCALE CALL DLACPY( 'F', M, N, C, LDC, WORK, M ) CALL DLACPY( 'F', M, N, F, LDF, WORK( M*N+1 ), M ) DO J = 1, N CALL DCOPY( M, zero_vec, 0, C( 1, J ), 1 ) CALL DCOPY( M, zero_vec, 0, F( 1, J ), 1 ) end do ELSE IF( ISOLVE.EQ.2 .AND. IROUND.EQ.2 ) THEN CALL DLACPY( 'F', M, N, WORK, M, C, LDC ) CALL DLACPY( 'F', M, N, WORK( M*N+1 ), M, F, LDF ) SCALE = SCALE2 END IF 150 CONTINUE ! ELSE ! ! Solve transposed (I, J)-subsystem ! A(I, I)' * R(I, J) + D(I, I)' * L(I, J) = C(I, J) ! R(I, J) * B(J, J)' + L(I, J) * E(J, J)' = -F(I, J) ! for I = 1,2,..., P; J = Q, Q-1,..., 1 ! SCALE = ONE DO 210 I = 1, P IS = IWORK( I ) IE = IWORK( I+1 ) - 1 MB = IE - IS + 1 DO 200 J = Q, P + 2, -1 JS = IWORK( J ) JE = IWORK( J+1 ) - 1 NB = JE - JS + 1 CALL DTGSY2( TRANS, IFUNC, MB, NB, A( IS, IS ), LDA, & B( JS, JS ), LDB, C( IS, JS ), LDC, & D( IS, IS ), LDD, E( JS, JS ), LDE, & F( IS, JS ), LDF, SCALOC, DSUM, DSCALE, & IWORK( Q+2 ), PPQQ, LINFO ) IF( LINFO.GT.0 ) & INFO = LINFO IF( SCALOC.NE.ONE ) THEN DO 160 K = 1, JS - 1 CALL DSCAL( M, SCALOC, C( 1, K ), 1 ) CALL DSCAL( M, SCALOC, F( 1, K ), 1 ) 160 CONTINUE DO 170 K = JS, JE CALL DSCAL( IS-1, SCALOC, C( 1, K ), 1 ) CALL DSCAL( IS-1, SCALOC, F( 1, K ), 1 ) 170 CONTINUE DO 180 K = JS, JE CALL DSCAL( M-IE, SCALOC, C( IE+1, K ), 1 ) CALL DSCAL( M-IE, SCALOC, F( IE+1, K ), 1 ) 180 CONTINUE DO 190 K = JE + 1, N CALL DSCAL( M, SCALOC, C( 1, K ), 1 ) CALL DSCAL( M, SCALOC, F( 1, K ), 1 ) 190 CONTINUE SCALE = SCALE*SCALOC END IF ! ! Substitute R(I, J) and L(I, J) into remaining equation. ! IF( J.GT.P+2 ) THEN CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE, C( IS, JS ), & LDC, B( 1, JS ), LDB, ONE, F( IS, 1 ), & LDF ) CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE, F( IS, JS ), & LDF, E( 1, JS ), LDE, ONE, F( IS, 1 ), & LDF ) END IF IF( I.LT.P ) THEN CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE, & A( IS, IE+1 ), LDA, C( IS, JS ), LDC, ONE, & C( IE+1, JS ), LDC ) CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE, & D( IS, IE+1 ), LDD, F( IS, JS ), LDF, ONE, & C( IE+1, JS ), LDC ) END IF 200 CONTINUE 210 CONTINUE ! END IF ! WORK( 1 ) = LWMIN ! RETURN ! ! End of DTGSYL ! END SUBROUTINE DTPCON( NORM, UPLO, DIAG, N, AP, RCOND, WORK, IWORK, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER DIAG, NORM, UPLO INTEGER INFO, N DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION AP( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DTPCON estimates the reciprocal of the condition number of a packed ! triangular matrix A, in either the 1-norm or the infinity-norm. ! ! The norm of A is computed and an estimate is obtained for ! norm(inv(A)), then the reciprocal of the condition number is ! computed as ! RCOND = 1 / ( norm(A) * norm(inv(A)) ). ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies whether the 1-norm condition number or the ! infinity-norm condition number is required: ! = '1' or 'O': 1-norm; ! = 'I': Infinity-norm. ! ! UPLO (input) CHARACTER*1 ! = 'U': A is upper triangular; ! = 'L': A is lower triangular. ! ! DIAG (input) CHARACTER*1 ! = 'N': A is non-unit triangular; ! = 'U': A is unit triangular. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The upper or lower triangular matrix A, packed columnwise in ! a linear array. The j-th column of A is stored in the array ! AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. ! If DIAG = 'U', the diagonal elements of A are not referenced ! and are assumed to be 1. ! ! RCOND (output) DOUBLE PRECISION ! The reciprocal of the condition number of the matrix A, ! computed as RCOND = 1/(norm(A) * norm(inv(A))). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOUNIT, ONENRM, UPPER CHARACTER NORMIN INTEGER IX, KASE, KASE1 DOUBLE PRECISION AINVNM, ANORM, SCALE, SMLNUM, XNORM ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DLAMCH, DLANTP EXTERNAL LSAME, IDAMAX, DLAMCH, DLANTP ! .. ! .. External Subroutines .. EXTERNAL DLACON, DLATPS, DRSCL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' ) NOUNIT = LSAME( DIAG, 'N' ) ! IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN INFO = -1 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -2 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTPCON', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) THEN RCOND = ONE RETURN END IF ! RCOND = ZERO SMLNUM = DLAMCH( 'Safe minimum' )*DBLE( MAX( 1, N ) ) ! ! Compute the norm of the triangular matrix A. ! ANORM = DLANTP( NORM, UPLO, DIAG, N, AP, WORK ) ! ! Continue only if ANORM > 0. ! IF( ANORM.GT.ZERO ) THEN ! ! Estimate the norm of the inverse of A. ! AINVNM = ZERO NORMIN = 'N' IF( ONENRM ) THEN KASE1 = 1 ELSE KASE1 = 2 END IF KASE = 0 10 CONTINUE CALL DLACON( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.KASE1 ) THEN ! ! Multiply by inv(A). ! CALL DLATPS( UPLO, 'No transpose', DIAG, NORMIN, N, AP, & WORK, SCALE, WORK( 2*N+1 ), INFO ) ELSE ! ! Multiply by inv(A'). ! CALL DLATPS( UPLO, 'Transpose', DIAG, NORMIN, N, AP, & WORK, SCALE, WORK( 2*N+1 ), INFO ) END IF NORMIN = 'Y' ! ! Multiply by 1/SCALE if doing so will not cause overflow. ! IF( SCALE.NE.ONE ) THEN IX = IDAMAX( N, WORK, 1 ) XNORM = ABS( WORK( IX ) ) IF( SCALE.LT.XNORM*SMLNUM .OR. SCALE.EQ.ZERO ) & GO TO 20 CALL DRSCL( N, SCALE, WORK, 1 ) END IF GO TO 10 END IF ! ! Compute the estimate of the reciprocal condition number. ! IF( AINVNM.NE.ZERO ) & RCOND = ( ONE / ANORM ) / AINVNM END IF ! 20 CONTINUE RETURN ! ! End of DTPCON ! END SUBROUTINE DTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, & FERR, BERR, WORK, IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER DIAG, TRANS, UPLO INTEGER INFO, LDB, LDX, N, NRHS ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ), & WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DTPRFS provides error bounds and backward error estimates for the ! solution to a system of linear equations with a triangular packed ! coefficient matrix. ! ! The solution matrix X must be computed by DTPTRS or some other ! means before entering this routine. DTPRFS does not do iterative ! refinement because doing so cannot improve the backward error. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': A is upper triangular; ! = 'L': A is lower triangular. ! ! TRANS (input) CHARACTER*1 ! Specifies the form of the system of equations: ! = 'N': A * X = B (No transpose) ! = 'T': A**T * X = B (Transpose) ! = 'C': A**H * X = B (Conjugate transpose = Transpose) ! ! DIAG (input) CHARACTER*1 ! = 'N': A is non-unit triangular; ! = 'U': A is unit triangular. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The upper or lower triangular matrix A, packed columnwise in ! a linear array. The j-th column of A is stored in the array ! AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. ! If DIAG = 'U', the diagonal elements of A are not referenced ! and are assumed to be 1. ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) ! The right hand side matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) ! The solution matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOTRAN, NOUNIT, UPPER CHARACTER TRANST INTEGER I, J, K, KASE, KC, NZ DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DLACON, DTPMV, DTPSV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) NOTRAN = LSAME( TRANS, 'N' ) NOUNIT = LSAME( DIAG, 'N' ) ! IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. & LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( NRHS.LT.0 ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -10 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTPRFS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN DO 10 J = 1, NRHS FERR( J ) = ZERO BERR( J ) = ZERO 10 CONTINUE RETURN END IF ! IF( NOTRAN ) THEN TRANST = 'T' ELSE TRANST = 'N' END IF ! ! NZ = maximum number of nonzero elements in each row of A, plus 1 ! NZ = N + 1 EPS = DLAMCH( 'Epsilon' ) SAFMIN = DLAMCH( 'Safe minimum' ) SAFE1 = NZ*SAFMIN SAFE2 = SAFE1 / EPS ! ! Do for each right hand side ! DO 250 J = 1, NRHS ! ! Compute residual R = B - op(A) * X, ! where op(A) = A or A', depending on TRANS. ! CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 ) CALL DTPMV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 ) CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 ) ! ! Compute componentwise relative backward error from formula ! ! max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) ! ! where abs(Z) is the componentwise absolute value of the matrix ! or vector Z. If the i-th component of the denominator is less ! than SAFE2, then SAFE1 is added to the i-th components of the ! numerator and denominator before dividing. ! DO 20 I = 1, N WORK( I ) = ABS( B( I, J ) ) 20 CONTINUE ! IF( NOTRAN ) THEN ! ! Compute abs(A)*abs(X) + abs(B). ! IF( UPPER ) THEN KC = 1 IF( NOUNIT ) THEN DO 40 K = 1, N XK = ABS( X( K, J ) ) DO 30 I = 1, K WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK 30 CONTINUE KC = KC + K 40 CONTINUE ELSE DO 60 K = 1, N XK = ABS( X( K, J ) ) DO 50 I = 1, K - 1 WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK 50 CONTINUE WORK( K ) = WORK( K ) + XK KC = KC + K 60 CONTINUE END IF ELSE KC = 1 IF( NOUNIT ) THEN DO 80 K = 1, N XK = ABS( X( K, J ) ) DO 70 I = K, N WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK 70 CONTINUE KC = KC + N - K + 1 80 CONTINUE ELSE DO 100 K = 1, N XK = ABS( X( K, J ) ) DO 90 I = K + 1, N WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK 90 CONTINUE WORK( K ) = WORK( K ) + XK KC = KC + N - K + 1 100 CONTINUE END IF END IF ELSE ! ! Compute abs(A')*abs(X) + abs(B). ! IF( UPPER ) THEN KC = 1 IF( NOUNIT ) THEN DO 120 K = 1, N S = ZERO DO 110 I = 1, K S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) ) 110 CONTINUE WORK( K ) = WORK( K ) + S KC = KC + K 120 CONTINUE ELSE DO 140 K = 1, N S = ABS( X( K, J ) ) DO 130 I = 1, K - 1 S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) ) 130 CONTINUE WORK( K ) = WORK( K ) + S KC = KC + K 140 CONTINUE END IF ELSE KC = 1 IF( NOUNIT ) THEN DO 160 K = 1, N S = ZERO DO 150 I = K, N S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) ) 150 CONTINUE WORK( K ) = WORK( K ) + S KC = KC + N - K + 1 160 CONTINUE ELSE DO 180 K = 1, N S = ABS( X( K, J ) ) DO 170 I = K + 1, N S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) ) 170 CONTINUE WORK( K ) = WORK( K ) + S KC = KC + N - K + 1 180 CONTINUE END IF END IF END IF S = ZERO DO 190 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) ELSE S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / & ( WORK( I )+SAFE1 ) ) END IF 190 CONTINUE BERR( J ) = S ! ! Bound error from formula ! ! norm(X - XTRUE) / norm(X) .le. FERR = ! norm( abs(inv(op(A)))* ! ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) ! ! where ! norm(Z) is the magnitude of the largest component of Z ! inv(op(A)) is the inverse of op(A) ! abs(Z) is the componentwise absolute value of the matrix or ! vector Z ! NZ is the maximum number of nonzeros in any row of A, plus 1 ! EPS is machine epsilon ! ! The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) ! is incremented by SAFE1 if the i-th component of ! abs(op(A))*abs(X) + abs(B) is less than SAFE2. ! ! Use DLACON to estimate the infinity-norm of the matrix ! inv(op(A)) * diag(W), ! where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) ! DO 200 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) ELSE WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 END IF 200 CONTINUE ! KASE = 0 210 CONTINUE CALL DLACON( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), & KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN ! ! Multiply by diag(W)*inv(op(A)'). ! CALL DTPSV( UPLO, TRANST, DIAG, N, AP, WORK( N+1 ), 1 ) DO 220 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 220 CONTINUE ELSE ! ! Multiply by inv(op(A))*diag(W). ! DO 230 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 230 CONTINUE CALL DTPSV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 ) END IF GO TO 210 END IF ! ! Normalize error. ! LSTRES = ZERO DO 240 I = 1, N LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 240 CONTINUE IF( LSTRES.NE.ZERO ) & FERR( J ) = FERR( J ) / LSTRES ! 250 CONTINUE ! RETURN ! ! End of DTPRFS ! END SUBROUTINE DTPTRI( UPLO, DIAG, N, AP, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER DIAG, UPLO INTEGER INFO, N ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ) ! .. ! ! Purpose ! ======= ! ! DTPTRI computes the inverse of a real upper or lower triangular ! matrix A stored in packed format. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': A is upper triangular; ! = 'L': A is lower triangular. ! ! DIAG (input) CHARACTER*1 ! = 'N': A is non-unit triangular; ! = 'U': A is unit triangular. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! On entry, the upper or lower triangular matrix A, stored ! columnwise in a linear array. The j-th column of A is stored ! in the array AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*((2*n-j)/2) = A(i,j) for j<=i<=n. ! See below for further details. ! On exit, the (triangular) inverse of the original matrix, in ! the same packed storage format. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, A(i,i) is exactly zero. The triangular ! matrix is singular and its inverse can not be computed. ! ! Further Details ! =============== ! ! A triangular matrix A can be transferred to packed storage using one ! of the following program segments: ! ! UPLO = 'U': UPLO = 'L': ! ! JC = 1 JC = 1 ! DO 2 J = 1, N DO 2 J = 1, N ! DO 1 I = 1, J DO 1 I = J, N ! AP(JC+I-1) = A(I,J) AP(JC+I-J) = A(I,J) ! 1 CONTINUE 1 CONTINUE ! JC = JC + J JC = JC + N - J + 1 ! 2 CONTINUE 2 CONTINUE ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOUNIT, UPPER INTEGER J, JC, JCLAST, JJ DOUBLE PRECISION AJJ ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DSCAL, DTPMV, XERBLA ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) NOUNIT = LSAME( DIAG, 'N' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTPTRI', -INFO ) RETURN END IF ! ! Check for singularity if non-unit. ! IF( NOUNIT ) THEN IF( UPPER ) THEN JJ = 0 DO 10 INFO = 1, N JJ = JJ + INFO IF( AP( JJ ).EQ.ZERO ) & RETURN 10 CONTINUE ELSE JJ = 1 DO 20 INFO = 1, N IF( AP( JJ ).EQ.ZERO ) & RETURN JJ = JJ + N - INFO + 1 20 CONTINUE END IF INFO = 0 END IF ! IF( UPPER ) THEN ! ! Compute inverse of upper triangular matrix. ! JC = 1 DO 30 J = 1, N IF( NOUNIT ) THEN AP( JC+J-1 ) = ONE / AP( JC+J-1 ) AJJ = -AP( JC+J-1 ) ELSE AJJ = -ONE END IF ! ! Compute elements 1:j-1 of j-th column. ! CALL DTPMV( 'Upper', 'No transpose', DIAG, J-1, AP, & AP( JC ), 1 ) CALL DSCAL( J-1, AJJ, AP( JC ), 1 ) JC = JC + J 30 CONTINUE ! ELSE ! ! Compute inverse of lower triangular matrix. ! JC = N*( N+1 ) / 2 DO 40 J = N, 1, -1 IF( NOUNIT ) THEN AP( JC ) = ONE / AP( JC ) AJJ = -AP( JC ) ELSE AJJ = -ONE END IF IF( J.LT.N ) THEN ! ! Compute elements j+1:n of j-th column. ! CALL DTPMV( 'Lower', 'No transpose', DIAG, N-J, & AP( JCLAST ), AP( JC+1 ), 1 ) CALL DSCAL( N-J, AJJ, AP( JC+1 ), 1 ) END IF JCLAST = JC JC = JC - N + J - 2 40 CONTINUE END IF ! RETURN ! ! End of DTPTRI ! END SUBROUTINE DTPTRS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER DIAG, TRANS, UPLO INTEGER INFO, LDB, N, NRHS ! .. ! .. Array Arguments .. DOUBLE PRECISION AP( * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DTPTRS solves a triangular system of the form ! ! A * X = B or A**T * X = B, ! ! where A is a triangular matrix of order N stored in packed format, ! and B is an N-by-NRHS matrix. A check is made to verify that A is ! nonsingular. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': A is upper triangular; ! = 'L': A is lower triangular. ! ! TRANS (input) CHARACTER*1 ! Specifies the form of the system of equations: ! = 'N': A * X = B (No transpose) ! = 'T': A**T * X = B (Transpose) ! = 'C': A**H * X = B (Conjugate transpose = Transpose) ! ! DIAG (input) CHARACTER*1 ! = 'N': A is non-unit triangular; ! = 'U': A is unit triangular. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) ! The upper or lower triangular matrix A, packed columnwise in ! a linear array. The j-th column of A is stored in the array ! AP as follows: ! if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; ! if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the right hand side matrix B. ! On exit, if INFO = 0, the solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the i-th diagonal element of A is zero, ! indicating that the matrix is singular and the ! solutions X have not been computed. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOUNIT, UPPER INTEGER J, JC ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DTPSV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) NOUNIT = LSAME( DIAG, 'N' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND. .NOT. & LSAME( TRANS, 'T' ) .AND. .NOT.LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( NRHS.LT.0 ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTPTRS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Check for singularity. ! IF( NOUNIT ) THEN IF( UPPER ) THEN JC = 1 DO 10 INFO = 1, N IF( AP( JC+INFO-1 ).EQ.ZERO ) & RETURN JC = JC + INFO 10 CONTINUE ELSE JC = 1 DO 20 INFO = 1, N IF( AP( JC ).EQ.ZERO ) & RETURN JC = JC + N - INFO + 1 20 CONTINUE END IF END IF INFO = 0 ! ! Solve A * x = b or A' * x = b. ! DO 30 J = 1, NRHS CALL DTPSV( UPLO, TRANS, DIAG, N, AP, B( 1, J ), 1 ) 30 CONTINUE ! RETURN ! ! End of DTPTRS ! END SUBROUTINE DTRCON( NORM, UPLO, DIAG, N, A, LDA, RCOND, WORK, & IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER DIAG, NORM, UPLO INTEGER INFO, LDA, N DOUBLE PRECISION RCOND ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DTRCON estimates the reciprocal of the condition number of a ! triangular matrix A, in either the 1-norm or the infinity-norm. ! ! The norm of A is computed and an estimate is obtained for ! norm(inv(A)), then the reciprocal of the condition number is ! computed as ! RCOND = 1 / ( norm(A) * norm(inv(A)) ). ! ! Arguments ! ========= ! ! NORM (input) CHARACTER*1 ! Specifies whether the 1-norm condition number or the ! infinity-norm condition number is required: ! = '1' or 'O': 1-norm; ! = 'I': Infinity-norm. ! ! UPLO (input) CHARACTER*1 ! = 'U': A is upper triangular; ! = 'L': A is lower triangular. ! ! DIAG (input) CHARACTER*1 ! = 'N': A is non-unit triangular; ! = 'U': A is unit triangular. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The triangular matrix A. If UPLO = 'U', the leading N-by-N ! upper triangular part of the array A contains the upper ! triangular matrix, and the strictly lower triangular part of ! A is not referenced. If UPLO = 'L', the leading N-by-N lower ! triangular part of the array A contains the lower triangular ! matrix, and the strictly upper triangular part of A is not ! referenced. If DIAG = 'U', the diagonal elements of A are ! also not referenced and are assumed to be 1. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! RCOND (output) DOUBLE PRECISION ! The reciprocal of the condition number of the matrix A, ! computed as RCOND = 1/(norm(A) * norm(inv(A))). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOUNIT, ONENRM, UPPER CHARACTER NORMIN INTEGER IX, KASE, KASE1 DOUBLE PRECISION AINVNM, ANORM, SCALE, SMLNUM, XNORM ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DLAMCH, DLANTR EXTERNAL LSAME, IDAMAX, DLAMCH, DLANTR ! .. ! .. External Subroutines .. EXTERNAL DLACON, DLATRS, DRSCL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) ONENRM = NORM.EQ.'1' .OR. LSAME( NORM, 'O' ) NOUNIT = LSAME( DIAG, 'N' ) ! IF( .NOT.ONENRM .AND. .NOT.LSAME( NORM, 'I' ) ) THEN INFO = -1 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -2 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTRCON', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) THEN RCOND = ONE RETURN END IF ! RCOND = ZERO SMLNUM = DLAMCH( 'Safe minimum' )*DBLE( MAX( 1, N ) ) ! ! Compute the norm of the triangular matrix A. ! ANORM = DLANTR( NORM, UPLO, DIAG, N, N, A, LDA, WORK ) ! ! Continue only if ANORM > 0. ! IF( ANORM.GT.ZERO ) THEN ! ! Estimate the norm of the inverse of A. ! AINVNM = ZERO NORMIN = 'N' IF( ONENRM ) THEN KASE1 = 1 ELSE KASE1 = 2 END IF KASE = 0 10 CONTINUE CALL DLACON( N, WORK( N+1 ), WORK, IWORK, AINVNM, KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.KASE1 ) THEN ! ! Multiply by inv(A). ! CALL DLATRS( UPLO, 'No transpose', DIAG, NORMIN, N, A, & LDA, WORK, SCALE, WORK( 2*N+1 ), INFO ) ELSE ! ! Multiply by inv(A'). ! CALL DLATRS( UPLO, 'Transpose', DIAG, NORMIN, N, A, LDA, & WORK, SCALE, WORK( 2*N+1 ), INFO ) END IF NORMIN = 'Y' ! ! Multiply by 1/SCALE if doing so will not cause overflow. ! IF( SCALE.NE.ONE ) THEN IX = IDAMAX( N, WORK, 1 ) XNORM = ABS( WORK( IX ) ) IF( SCALE.LT.XNORM*SMLNUM .OR. SCALE.EQ.ZERO ) & GO TO 20 CALL DRSCL( N, SCALE, WORK, 1 ) END IF GO TO 10 END IF ! ! Compute the estimate of the reciprocal condition number. ! IF( AINVNM.NE.ZERO ) & RCOND = ( ONE / ANORM ) / AINVNM END IF ! 20 CONTINUE RETURN ! ! End of DTRCON ! END SUBROUTINE DTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, & LDVR, MM, M, WORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER HOWMNY, SIDE INTEGER INFO, LDT, LDVL, LDVR, M, MM, N ! .. ! .. Array Arguments .. LOGICAL SELECT( * ) DOUBLE PRECISION T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), & WORK( * ) ! .. ! ! Purpose ! ======= ! ! DTREVC computes some or all of the right and/or left eigenvectors of ! a real upper quasi-triangular matrix T. ! ! The right eigenvector x and the left eigenvector y of T corresponding ! to an eigenvalue w are defined by: ! ! T*x = w*x, y'*T = w*y' ! ! where y' denotes the conjugate transpose of the vector y. ! ! If all eigenvectors are requested, the routine may either return the ! matrices X and/or Y of right or left eigenvectors of T, or the ! products Q*X and/or Q*Y, where Q is an input orthogonal ! matrix. If T was obtained from the real-Schur factorization of an ! original matrix A = Q*T*Q', then Q*X and Q*Y are the matrices of ! right or left eigenvectors of A. ! ! T must be in Schur canonical form (as returned by DHSEQR), that is, ! block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each ! 2-by-2 diagonal block has its diagonal elements equal and its ! off-diagonal elements of opposite sign. Corresponding to each 2-by-2 ! diagonal block is a complex conjugate pair of eigenvalues and ! eigenvectors; only one eigenvector of the pair is computed, namely ! the one corresponding to the eigenvalue with positive imaginary part. ! ! Arguments ! ========= ! ! SIDE (input) CHARACTER*1 ! = 'R': compute right eigenvectors only; ! = 'L': compute left eigenvectors only; ! = 'B': compute both right and left eigenvectors. ! ! HOWMNY (input) CHARACTER*1 ! = 'A': compute all right and/or left eigenvectors; ! = 'B': compute all right and/or left eigenvectors, ! and backtransform them using the input matrices ! supplied in VR and/or VL; ! = 'S': compute selected right and/or left eigenvectors, ! specified by the logical array SELECT. ! ! SELECT (input/output) LOGICAL array, dimension (N) ! If HOWMNY = 'S', SELECT specifies the eigenvectors to be ! computed. ! If HOWMNY = 'A' or 'B', SELECT is not referenced. ! To select the real eigenvector corresponding to a real ! eigenvalue w(j), SELECT(j) must be set to .TRUE.. To select ! the complex eigenvector corresponding to a complex conjugate ! pair w(j) and w(j+1), either SELECT(j) or SELECT(j+1) must be ! set to .TRUE.; then on exit SELECT(j) is .TRUE. and ! SELECT(j+1) is .FALSE.. ! ! N (input) INTEGER ! The order of the matrix T. N >= 0. ! ! T (input) DOUBLE PRECISION array, dimension (LDT,N) ! The upper quasi-triangular matrix T in Schur canonical form. ! ! LDT (input) INTEGER ! The leading dimension of the array T. LDT >= max(1,N). ! ! VL (input/output) DOUBLE PRECISION array, dimension (LDVL,MM) ! On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must ! contain an N-by-N matrix Q (usually the orthogonal matrix Q ! of Schur vectors returned by DHSEQR). ! On exit, if SIDE = 'L' or 'B', VL contains: ! if HOWMNY = 'A', the matrix Y of left eigenvectors of T; ! VL has the same quasi-lower triangular form ! as T'. If T(i,i) is a real eigenvalue, then ! the i-th column VL(i) of VL is its ! corresponding eigenvector. If T(i:i+1,i:i+1) ! is a 2-by-2 block whose eigenvalues are ! complex-conjugate eigenvalues of T, then ! VL(i)+sqrt(-1)*VL(i+1) is the complex ! eigenvector corresponding to the eigenvalue ! with positive real part. ! if HOWMNY = 'B', the matrix Q*Y; ! if HOWMNY = 'S', the left eigenvectors of T specified by ! SELECT, stored consecutively in the columns ! of VL, in the same order as their ! eigenvalues. ! A complex eigenvector corresponding to a complex eigenvalue ! is stored in two consecutive columns, the first holding the ! real part, and the second the imaginary part. ! If SIDE = 'R', VL is not referenced. ! ! LDVL (input) INTEGER ! The leading dimension of the array VL. LDVL >= max(1,N) if ! SIDE = 'L' or 'B'; LDVL >= 1 otherwise. ! ! VR (input/output) DOUBLE PRECISION array, dimension (LDVR,MM) ! On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must ! contain an N-by-N matrix Q (usually the orthogonal matrix Q ! of Schur vectors returned by DHSEQR). ! On exit, if SIDE = 'R' or 'B', VR contains: ! if HOWMNY = 'A', the matrix X of right eigenvectors of T; ! VR has the same quasi-upper triangular form ! as T. If T(i,i) is a real eigenvalue, then ! the i-th column VR(i) of VR is its ! corresponding eigenvector. If T(i:i+1,i:i+1) ! is a 2-by-2 block whose eigenvalues are ! complex-conjugate eigenvalues of T, then ! VR(i)+sqrt(-1)*VR(i+1) is the complex ! eigenvector corresponding to the eigenvalue ! with positive real part. ! if HOWMNY = 'B', the matrix Q*X; ! if HOWMNY = 'S', the right eigenvectors of T specified by ! SELECT, stored consecutively in the columns ! of VR, in the same order as their ! eigenvalues. ! A complex eigenvector corresponding to a complex eigenvalue ! is stored in two consecutive columns, the first holding the ! real part and the second the imaginary part. ! If SIDE = 'L', VR is not referenced. ! ! LDVR (input) INTEGER ! The leading dimension of the array VR. LDVR >= max(1,N) if ! SIDE = 'R' or 'B'; LDVR >= 1 otherwise. ! ! MM (input) INTEGER ! The number of columns in the arrays VL and/or VR. MM >= M. ! ! M (output) INTEGER ! The number of columns in the arrays VL and/or VR actually ! used to store the eigenvectors. ! If HOWMNY = 'A' or 'B', M is set to N. ! Each selected real eigenvector occupies one column and each ! selected complex eigenvector occupies two columns. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! The algorithm used in this program is basically backward (forward) ! substitution, with scaling to make the the code robust against ! possible overflow. ! ! Each eigenvector is normalized so that the element of largest ! magnitude has magnitude 1; here the magnitude of a complex number ! (x,y) is taken to be |x| + |y|. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL ALLV, BOTHV, LEFTV, OVER, PAIR, RIGHTV, SOMEV INTEGER I, IERR, II, IP, IS, J, J1, J2, JNXT, K, KI, N2 DOUBLE PRECISION BETA, BIGNUM, EMAX, OVFL, REC, REMAX, SCALE, & SMIN, SMLNUM, ULP, UNFL, VCRIT, VMAX, WI, WR, & XNORM ! .. ! .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DDOT, DLAMCH EXTERNAL LSAME, IDAMAX, DDOT, DLAMCH ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DGEMV, DLALN2, DSCAL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Local Arrays .. DOUBLE PRECISION X( 2, 2 ) ! .. ! .. Executable Statements .. ! ! Decode and test the input parameters ! BOTHV = LSAME( SIDE, 'B' ) RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV ! ALLV = LSAME( HOWMNY, 'A' ) OVER = LSAME( HOWMNY, 'B' ) SOMEV = LSAME( HOWMNY, 'S' ) ! INFO = 0 IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN INFO = -1 ELSE IF( .NOT.ALLV .AND. .NOT.OVER .AND. .NOT.SOMEV ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDT.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN INFO = -8 ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN INFO = -10 ELSE ! ! Set M to the number of columns required to store the selected ! eigenvectors, standardize the array SELECT if necessary, and ! test MM. ! IF( SOMEV ) THEN M = 0 PAIR = .FALSE. DO 10 J = 1, N IF( PAIR ) THEN PAIR = .FALSE. SELECT( J ) = .FALSE. ELSE IF( J.LT.N ) THEN IF( T( J+1, J ).EQ.ZERO ) THEN IF( SELECT( J ) ) & M = M + 1 ELSE PAIR = .TRUE. IF( SELECT( J ) .OR. SELECT( J+1 ) ) THEN SELECT( J ) = .TRUE. M = M + 2 END IF END IF ELSE IF( SELECT( N ) ) & M = M + 1 END IF END IF 10 CONTINUE ELSE M = N END IF ! IF( MM.LT.M ) THEN INFO = -11 END IF END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTREVC', -INFO ) RETURN END IF ! ! Quick return if possible. ! IF( N.EQ.0 ) & RETURN ! ! Set the constants to control overflow. ! UNFL = DLAMCH( 'Safe minimum' ) OVFL = ONE / UNFL CALL DLABAD( UNFL, OVFL ) ULP = DLAMCH( 'Precision' ) SMLNUM = UNFL*( N / ULP ) BIGNUM = ( ONE-ULP ) / SMLNUM ! ! Compute 1-norm of each column of strictly upper triangular ! part of T to control overflow in triangular solver. ! WORK( 1 ) = ZERO DO 30 J = 2, N WORK( J ) = ZERO DO 20 I = 1, J - 1 WORK( J ) = WORK( J ) + ABS( T( I, J ) ) 20 CONTINUE 30 CONTINUE ! ! Index IP is used to specify the real or complex eigenvalue: ! IP = 0, real eigenvalue, ! 1, first of conjugate complex pair: (wr,wi) ! -1, second of conjugate complex pair: (wr,wi) ! N2 = 2*N ! IF( RIGHTV ) THEN ! ! Compute right eigenvectors. ! IP = 0 IS = M DO 140 KI = N, 1, -1 ! IF( IP.EQ.1 ) & GO TO 130 IF( KI.EQ.1 ) & GO TO 40 IF( T( KI, KI-1 ).EQ.ZERO ) & GO TO 40 IP = -1 ! 40 CONTINUE IF( SOMEV ) THEN IF( IP.EQ.0 ) THEN IF( .NOT.SELECT( KI ) ) & GO TO 130 ELSE IF( .NOT.SELECT( KI-1 ) ) & GO TO 130 END IF END IF ! ! Compute the KI-th eigenvalue (WR,WI). ! WR = T( KI, KI ) WI = ZERO IF( IP.NE.0 ) & WI = SQRT( ABS( T( KI, KI-1 ) ) )* & SQRT( ABS( T( KI-1, KI ) ) ) SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM ) ! IF( IP.EQ.0 ) THEN ! ! Real right eigenvector ! WORK( KI+N ) = ONE ! ! Form right-hand side ! DO 50 K = 1, KI - 1 WORK( K+N ) = -T( K, KI ) 50 CONTINUE ! ! Solve the upper quasi-triangular system: ! (T(1:KI-1,1:KI-1) - WR)*X = SCALE*WORK. ! JNXT = KI - 1 DO 60 J = KI - 1, 1, -1 IF( J.GT.JNXT ) & GO TO 60 J1 = J J2 = J JNXT = J - 1 IF( J.GT.1 ) THEN IF( T( J, J-1 ).NE.ZERO ) THEN J1 = J - 1 JNXT = J - 2 END IF END IF ! IF( J1.EQ.J2 ) THEN ! ! 1-by-1 diagonal block ! CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J, J ), & LDT, ONE, ONE, WORK( J+N ), N, WR, & ZERO, X, 2, SCALE, XNORM, IERR ) ! ! Scale X(1,1) to avoid overflow when updating ! the right-hand side. ! IF( XNORM.GT.ONE ) THEN IF( WORK( J ).GT.BIGNUM / XNORM ) THEN X( 1, 1 ) = X( 1, 1 ) / XNORM SCALE = SCALE / XNORM END IF END IF ! ! Scale if necessary ! IF( SCALE.NE.ONE ) & CALL DSCAL( KI, SCALE, WORK( 1+N ), 1 ) WORK( J+N ) = X( 1, 1 ) ! ! Update right-hand side ! CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1, & WORK( 1+N ), 1 ) ! ELSE ! ! 2-by-2 diagonal block ! CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, & T( J-1, J-1 ), LDT, ONE, ONE, & WORK( J-1+N ), N, WR, ZERO, X, 2, & SCALE, XNORM, IERR ) ! ! Scale X(1,1) and X(2,1) to avoid overflow when ! updating the right-hand side. ! IF( XNORM.GT.ONE ) THEN BETA = MAX( WORK( J-1 ), WORK( J ) ) IF( BETA.GT.BIGNUM / XNORM ) THEN X( 1, 1 ) = X( 1, 1 ) / XNORM X( 2, 1 ) = X( 2, 1 ) / XNORM SCALE = SCALE / XNORM END IF END IF ! ! Scale if necessary ! IF( SCALE.NE.ONE ) & CALL DSCAL( KI, SCALE, WORK( 1+N ), 1 ) WORK( J-1+N ) = X( 1, 1 ) WORK( J+N ) = X( 2, 1 ) ! ! Update right-hand side ! CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1, & WORK( 1+N ), 1 ) CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1, & WORK( 1+N ), 1 ) END IF 60 CONTINUE ! ! Copy the vector x or Q*x to VR and normalize. ! IF( .NOT.OVER ) THEN CALL DCOPY( KI, WORK( 1+N ), 1, VR( 1, IS ), 1 ) ! II = IDAMAX( KI, VR( 1, IS ), 1 ) REMAX = ONE / ABS( VR( II, IS ) ) CALL DSCAL( KI, REMAX, VR( 1, IS ), 1 ) ! DO 70 K = KI + 1, N VR( K, IS ) = ZERO 70 CONTINUE ELSE IF( KI.GT.1 ) & CALL DGEMV( 'N', N, KI-1, ONE, VR, LDVR, & WORK( 1+N ), 1, WORK( KI+N ), & VR( 1, KI ), 1 ) ! II = IDAMAX( N, VR( 1, KI ), 1 ) REMAX = ONE / ABS( VR( II, KI ) ) CALL DSCAL( N, REMAX, VR( 1, KI ), 1 ) END IF ! ELSE ! ! Complex right eigenvector. ! ! Initial solve ! [ (T(KI-1,KI-1) T(KI-1,KI) ) - (WR + I* WI)]*X = 0. ! [ (T(KI,KI-1) T(KI,KI) ) ] ! IF( ABS( T( KI-1, KI ) ).GE.ABS( T( KI, KI-1 ) ) ) THEN WORK( KI-1+N ) = ONE WORK( KI+N2 ) = WI / T( KI-1, KI ) ELSE WORK( KI-1+N ) = -WI / T( KI, KI-1 ) WORK( KI+N2 ) = ONE END IF WORK( KI+N ) = ZERO WORK( KI-1+N2 ) = ZERO ! ! Form right-hand side ! DO 80 K = 1, KI - 2 WORK( K+N ) = -WORK( KI-1+N )*T( K, KI-1 ) WORK( K+N2 ) = -WORK( KI+N2 )*T( K, KI ) 80 CONTINUE ! ! Solve upper quasi-triangular system: ! (T(1:KI-2,1:KI-2) - (WR+i*WI))*X = SCALE*(WORK+i*WORK2) ! JNXT = KI - 2 DO 90 J = KI - 2, 1, -1 IF( J.GT.JNXT ) & GO TO 90 J1 = J J2 = J JNXT = J - 1 IF( J.GT.1 ) THEN IF( T( J, J-1 ).NE.ZERO ) THEN J1 = J - 1 JNXT = J - 2 END IF END IF ! IF( J1.EQ.J2 ) THEN ! ! 1-by-1 diagonal block ! CALL DLALN2( .FALSE., 1, 2, SMIN, ONE, T( J, J ), & LDT, ONE, ONE, WORK( J+N ), N, WR, WI, & X, 2, SCALE, XNORM, IERR ) ! ! Scale X(1,1) and X(1,2) to avoid overflow when ! updating the right-hand side. ! IF( XNORM.GT.ONE ) THEN IF( WORK( J ).GT.BIGNUM / XNORM ) THEN X( 1, 1 ) = X( 1, 1 ) / XNORM X( 1, 2 ) = X( 1, 2 ) / XNORM SCALE = SCALE / XNORM END IF END IF ! ! Scale if necessary ! IF( SCALE.NE.ONE ) THEN CALL DSCAL( KI, SCALE, WORK( 1+N ), 1 ) CALL DSCAL( KI, SCALE, WORK( 1+N2 ), 1 ) END IF WORK( J+N ) = X( 1, 1 ) WORK( J+N2 ) = X( 1, 2 ) ! ! Update the right-hand side ! CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1, & WORK( 1+N ), 1 ) CALL DAXPY( J-1, -X( 1, 2 ), T( 1, J ), 1, & WORK( 1+N2 ), 1 ) ! ELSE ! ! 2-by-2 diagonal block ! CALL DLALN2( .FALSE., 2, 2, SMIN, ONE, & T( J-1, J-1 ), LDT, ONE, ONE, & WORK( J-1+N ), N, WR, WI, X, 2, SCALE, & XNORM, IERR ) ! ! Scale X to avoid overflow when updating ! the right-hand side. ! IF( XNORM.GT.ONE ) THEN BETA = MAX( WORK( J-1 ), WORK( J ) ) IF( BETA.GT.BIGNUM / XNORM ) THEN REC = ONE / XNORM X( 1, 1 ) = X( 1, 1 )*REC X( 1, 2 ) = X( 1, 2 )*REC X( 2, 1 ) = X( 2, 1 )*REC X( 2, 2 ) = X( 2, 2 )*REC SCALE = SCALE*REC END IF END IF ! ! Scale if necessary ! IF( SCALE.NE.ONE ) THEN CALL DSCAL( KI, SCALE, WORK( 1+N ), 1 ) CALL DSCAL( KI, SCALE, WORK( 1+N2 ), 1 ) END IF WORK( J-1+N ) = X( 1, 1 ) WORK( J+N ) = X( 2, 1 ) WORK( J-1+N2 ) = X( 1, 2 ) WORK( J+N2 ) = X( 2, 2 ) ! ! Update the right-hand side ! CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1, & WORK( 1+N ), 1 ) CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1, & WORK( 1+N ), 1 ) CALL DAXPY( J-2, -X( 1, 2 ), T( 1, J-1 ), 1, & WORK( 1+N2 ), 1 ) CALL DAXPY( J-2, -X( 2, 2 ), T( 1, J ), 1, & WORK( 1+N2 ), 1 ) END IF 90 CONTINUE ! ! Copy the vector x or Q*x to VR and normalize. ! IF( .NOT.OVER ) THEN CALL DCOPY( KI, WORK( 1+N ), 1, VR( 1, IS-1 ), 1 ) CALL DCOPY( KI, WORK( 1+N2 ), 1, VR( 1, IS ), 1 ) ! EMAX = ZERO DO 100 K = 1, KI EMAX = MAX( EMAX, ABS( VR( K, IS-1 ) )+ & ABS( VR( K, IS ) ) ) 100 CONTINUE ! REMAX = ONE / EMAX CALL DSCAL( KI, REMAX, VR( 1, IS-1 ), 1 ) CALL DSCAL( KI, REMAX, VR( 1, IS ), 1 ) ! DO 110 K = KI + 1, N VR( K, IS-1 ) = ZERO VR( K, IS ) = ZERO 110 CONTINUE ! ELSE ! IF( KI.GT.2 ) THEN CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR, & WORK( 1+N ), 1, WORK( KI-1+N ), & VR( 1, KI-1 ), 1 ) CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR, & WORK( 1+N2 ), 1, WORK( KI+N2 ), & VR( 1, KI ), 1 ) ELSE CALL DSCAL( N, WORK( KI-1+N ), VR( 1, KI-1 ), 1 ) CALL DSCAL( N, WORK( KI+N2 ), VR( 1, KI ), 1 ) END IF ! EMAX = ZERO DO 120 K = 1, N EMAX = MAX( EMAX, ABS( VR( K, KI-1 ) )+ & ABS( VR( K, KI ) ) ) 120 CONTINUE REMAX = ONE / EMAX CALL DSCAL( N, REMAX, VR( 1, KI-1 ), 1 ) CALL DSCAL( N, REMAX, VR( 1, KI ), 1 ) END IF END IF ! IS = IS - 1 IF( IP.NE.0 ) & IS = IS - 1 130 CONTINUE IF( IP.EQ.1 ) & IP = 0 IF( IP.EQ.-1 ) & IP = 1 140 CONTINUE END IF ! IF( LEFTV ) THEN ! ! Compute left eigenvectors. ! IP = 0 IS = 1 DO 260 KI = 1, N ! IF( IP.EQ.-1 ) & GO TO 250 IF( KI.EQ.N ) & GO TO 150 IF( T( KI+1, KI ).EQ.ZERO ) & GO TO 150 IP = 1 ! 150 CONTINUE IF( SOMEV ) THEN IF( .NOT.SELECT( KI ) ) & GO TO 250 END IF ! ! Compute the KI-th eigenvalue (WR,WI). ! WR = T( KI, KI ) WI = ZERO IF( IP.NE.0 ) & WI = SQRT( ABS( T( KI, KI+1 ) ) )* & SQRT( ABS( T( KI+1, KI ) ) ) SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM ) ! IF( IP.EQ.0 ) THEN ! ! Real left eigenvector. ! WORK( KI+N ) = ONE ! ! Form right-hand side ! DO 160 K = KI + 1, N WORK( K+N ) = -T( KI, K ) 160 CONTINUE ! ! Solve the quasi-triangular system: ! (T(KI+1:N,KI+1:N) - WR)'*X = SCALE*WORK ! VMAX = ONE VCRIT = BIGNUM ! JNXT = KI + 1 DO 170 J = KI + 1, N IF( J.LT.JNXT ) & GO TO 170 J1 = J J2 = J JNXT = J + 1 IF( J.LT.N ) THEN IF( T( J+1, J ).NE.ZERO ) THEN J2 = J + 1 JNXT = J + 2 END IF END IF ! IF( J1.EQ.J2 ) THEN ! ! 1-by-1 diagonal block ! ! Scale if necessary to avoid overflow when forming ! the right-hand side. ! IF( WORK( J ).GT.VCRIT ) THEN REC = ONE / VMAX CALL DSCAL( N-KI+1, REC, WORK( KI+N ), 1 ) VMAX = ONE VCRIT = BIGNUM END IF ! WORK( J+N ) = WORK( J+N ) - & DDOT( J-KI-1, T( KI+1, J ), 1, & WORK( KI+1+N ), 1 ) ! ! Solve (T(J,J)-WR)'*X = WORK ! CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J, J ), & LDT, ONE, ONE, WORK( J+N ), N, WR, & ZERO, X, 2, SCALE, XNORM, IERR ) ! ! Scale if necessary ! IF( SCALE.NE.ONE ) & CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 ) WORK( J+N ) = X( 1, 1 ) VMAX = MAX( ABS( WORK( J+N ) ), VMAX ) VCRIT = BIGNUM / VMAX ! ELSE ! ! 2-by-2 diagonal block ! ! Scale if necessary to avoid overflow when forming ! the right-hand side. ! BETA = MAX( WORK( J ), WORK( J+1 ) ) IF( BETA.GT.VCRIT ) THEN REC = ONE / VMAX CALL DSCAL( N-KI+1, REC, WORK( KI+N ), 1 ) VMAX = ONE VCRIT = BIGNUM END IF ! WORK( J+N ) = WORK( J+N ) - & DDOT( J-KI-1, T( KI+1, J ), 1, & WORK( KI+1+N ), 1 ) ! WORK( J+1+N ) = WORK( J+1+N ) - & DDOT( J-KI-1, T( KI+1, J+1 ), 1, & WORK( KI+1+N ), 1 ) ! ! Solve ! [T(J,J)-WR T(J,J+1) ]'* X = SCALE*( WORK1 ) ! [T(J+1,J) T(J+1,J+1)-WR] ( WORK2 ) ! CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, T( J, J ), & LDT, ONE, ONE, WORK( J+N ), N, WR, & ZERO, X, 2, SCALE, XNORM, IERR ) ! ! Scale if necessary ! IF( SCALE.NE.ONE ) & CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 ) WORK( J+N ) = X( 1, 1 ) WORK( J+1+N ) = X( 2, 1 ) ! VMAX = MAX( ABS( WORK( J+N ) ), & ABS( WORK( J+1+N ) ), VMAX ) VCRIT = BIGNUM / VMAX ! END IF 170 CONTINUE ! ! Copy the vector x or Q*x to VL and normalize. ! IF( .NOT.OVER ) THEN CALL DCOPY( N-KI+1, WORK( KI+N ), 1, VL( KI, IS ), 1 ) ! II = IDAMAX( N-KI+1, VL( KI, IS ), 1 ) + KI - 1 REMAX = ONE / ABS( VL( II, IS ) ) CALL DSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 ) ! DO 180 K = 1, KI - 1 VL( K, IS ) = ZERO 180 CONTINUE ! ELSE ! IF( KI.LT.N ) & CALL DGEMV( 'N', N, N-KI, ONE, VL( 1, KI+1 ), LDVL, & WORK( KI+1+N ), 1, WORK( KI+N ), & VL( 1, KI ), 1 ) ! II = IDAMAX( N, VL( 1, KI ), 1 ) REMAX = ONE / ABS( VL( II, KI ) ) CALL DSCAL( N, REMAX, VL( 1, KI ), 1 ) ! END IF ! ELSE ! ! Complex left eigenvector. ! ! Initial solve: ! ((T(KI,KI) T(KI,KI+1) )' - (WR - I* WI))*X = 0. ! ((T(KI+1,KI) T(KI+1,KI+1)) ) ! IF( ABS( T( KI, KI+1 ) ).GE.ABS( T( KI+1, KI ) ) ) THEN WORK( KI+N ) = WI / T( KI, KI+1 ) WORK( KI+1+N2 ) = ONE ELSE WORK( KI+N ) = ONE WORK( KI+1+N2 ) = -WI / T( KI+1, KI ) END IF WORK( KI+1+N ) = ZERO WORK( KI+N2 ) = ZERO ! ! Form right-hand side ! DO 190 K = KI + 2, N WORK( K+N ) = -WORK( KI+N )*T( KI, K ) WORK( K+N2 ) = -WORK( KI+1+N2 )*T( KI+1, K ) 190 CONTINUE ! ! Solve complex quasi-triangular system: ! ( T(KI+2,N:KI+2,N) - (WR-i*WI) )*X = WORK1+i*WORK2 ! VMAX = ONE VCRIT = BIGNUM ! JNXT = KI + 2 DO 200 J = KI + 2, N IF( J.LT.JNXT ) & GO TO 200 J1 = J J2 = J JNXT = J + 1 IF( J.LT.N ) THEN IF( T( J+1, J ).NE.ZERO ) THEN J2 = J + 1 JNXT = J + 2 END IF END IF ! IF( J1.EQ.J2 ) THEN ! ! 1-by-1 diagonal block ! ! Scale if necessary to avoid overflow when ! forming the right-hand side elements. ! IF( WORK( J ).GT.VCRIT ) THEN REC = ONE / VMAX CALL DSCAL( N-KI+1, REC, WORK( KI+N ), 1 ) CALL DSCAL( N-KI+1, REC, WORK( KI+N2 ), 1 ) VMAX = ONE VCRIT = BIGNUM END IF ! WORK( J+N ) = WORK( J+N ) - & DDOT( J-KI-2, T( KI+2, J ), 1, & WORK( KI+2+N ), 1 ) WORK( J+N2 ) = WORK( J+N2 ) - & DDOT( J-KI-2, T( KI+2, J ), 1, & WORK( KI+2+N2 ), 1 ) ! ! Solve (T(J,J)-(WR-i*WI))*(X11+i*X12)= WK+I*WK2 ! CALL DLALN2( .FALSE., 1, 2, SMIN, ONE, T( J, J ), & LDT, ONE, ONE, WORK( J+N ), N, WR, & -WI, X, 2, SCALE, XNORM, IERR ) ! ! Scale if necessary ! IF( SCALE.NE.ONE ) THEN CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 ) CALL DSCAL( N-KI+1, SCALE, WORK( KI+N2 ), 1 ) END IF WORK( J+N ) = X( 1, 1 ) WORK( J+N2 ) = X( 1, 2 ) VMAX = MAX( ABS( WORK( J+N ) ), & ABS( WORK( J+N2 ) ), VMAX ) VCRIT = BIGNUM / VMAX ! ELSE ! ! 2-by-2 diagonal block ! ! Scale if necessary to avoid overflow when forming ! the right-hand side elements. ! BETA = MAX( WORK( J ), WORK( J+1 ) ) IF( BETA.GT.VCRIT ) THEN REC = ONE / VMAX CALL DSCAL( N-KI+1, REC, WORK( KI+N ), 1 ) CALL DSCAL( N-KI+1, REC, WORK( KI+N2 ), 1 ) VMAX = ONE VCRIT = BIGNUM END IF ! WORK( J+N ) = WORK( J+N ) - & DDOT( J-KI-2, T( KI+2, J ), 1, & WORK( KI+2+N ), 1 ) ! WORK( J+N2 ) = WORK( J+N2 ) - & DDOT( J-KI-2, T( KI+2, J ), 1, & WORK( KI+2+N2 ), 1 ) ! WORK( J+1+N ) = WORK( J+1+N ) - & DDOT( J-KI-2, T( KI+2, J+1 ), 1, & WORK( KI+2+N ), 1 ) ! WORK( J+1+N2 ) = WORK( J+1+N2 ) - & DDOT( J-KI-2, T( KI+2, J+1 ), 1, & WORK( KI+2+N2 ), 1 ) ! ! Solve 2-by-2 complex linear equation ! ([T(j,j) T(j,j+1) ]'-(wr-i*wi)*I)*X = SCALE*B ! ([T(j+1,j) T(j+1,j+1)] ) ! CALL DLALN2( .TRUE., 2, 2, SMIN, ONE, T( J, J ), & LDT, ONE, ONE, WORK( J+N ), N, WR, & -WI, X, 2, SCALE, XNORM, IERR ) ! ! Scale if necessary ! IF( SCALE.NE.ONE ) THEN CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 ) CALL DSCAL( N-KI+1, SCALE, WORK( KI+N2 ), 1 ) END IF WORK( J+N ) = X( 1, 1 ) WORK( J+N2 ) = X( 1, 2 ) WORK( J+1+N ) = X( 2, 1 ) WORK( J+1+N2 ) = X( 2, 2 ) VMAX = MAX( ABS( X( 1, 1 ) ), ABS( X( 1, 2 ) ), & ABS( X( 2, 1 ) ), ABS( X( 2, 2 ) ), VMAX ) VCRIT = BIGNUM / VMAX ! END IF 200 CONTINUE ! ! Copy the vector x or Q*x to VL and normalize. ! 210 CONTINUE IF( .NOT.OVER ) THEN CALL DCOPY( N-KI+1, WORK( KI+N ), 1, VL( KI, IS ), 1 ) CALL DCOPY( N-KI+1, WORK( KI+N2 ), 1, VL( KI, IS+1 ), & 1 ) ! EMAX = ZERO DO 220 K = KI, N EMAX = MAX( EMAX, ABS( VL( K, IS ) )+ & ABS( VL( K, IS+1 ) ) ) 220 CONTINUE REMAX = ONE / EMAX CALL DSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 ) CALL DSCAL( N-KI+1, REMAX, VL( KI, IS+1 ), 1 ) ! DO 230 K = 1, KI - 1 VL( K, IS ) = ZERO VL( K, IS+1 ) = ZERO 230 CONTINUE ELSE IF( KI.LT.N-1 ) THEN CALL DGEMV( 'N', N, N-KI-1, ONE, VL( 1, KI+2 ), & LDVL, WORK( KI+2+N ), 1, WORK( KI+N ), & VL( 1, KI ), 1 ) CALL DGEMV( 'N', N, N-KI-1, ONE, VL( 1, KI+2 ), & LDVL, WORK( KI+2+N2 ), 1, & WORK( KI+1+N2 ), VL( 1, KI+1 ), 1 ) ELSE CALL DSCAL( N, WORK( KI+N ), VL( 1, KI ), 1 ) CALL DSCAL( N, WORK( KI+1+N2 ), VL( 1, KI+1 ), 1 ) END IF ! EMAX = ZERO DO 240 K = 1, N EMAX = MAX( EMAX, ABS( VL( K, KI ) )+ & ABS( VL( K, KI+1 ) ) ) 240 CONTINUE REMAX = ONE / EMAX CALL DSCAL( N, REMAX, VL( 1, KI ), 1 ) CALL DSCAL( N, REMAX, VL( 1, KI+1 ), 1 ) ! END IF ! END IF ! IS = IS + 1 IF( IP.NE.0 ) & IS = IS + 1 250 CONTINUE IF( IP.EQ.-1 ) & IP = 0 IF( IP.EQ.1 ) & IP = -1 ! 260 CONTINUE ! END IF ! RETURN ! ! End of DTREVC ! END SUBROUTINE DTREXC( COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, WORK, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER COMPQ INTEGER IFST, ILST, INFO, LDQ, LDT, N ! .. ! .. Array Arguments .. DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DTREXC reorders the real Schur factorization of a real matrix ! A = Q*T*Q**T, so that the diagonal block of T with row index IFST is ! moved to row ILST. ! ! The real Schur form T is reordered by an orthogonal similarity ! transformation Z**T*T*Z, and optionally the matrix Q of Schur vectors ! is updated by postmultiplying it with Z. ! ! T must be in Schur canonical form (as returned by DHSEQR), that is, ! block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each ! 2-by-2 diagonal block has its diagonal elements equal and its ! off-diagonal elements of opposite sign. ! ! Arguments ! ========= ! ! COMPQ (input) CHARACTER*1 ! = 'V': update the matrix Q of Schur vectors; ! = 'N': do not update Q. ! ! N (input) INTEGER ! The order of the matrix T. N >= 0. ! ! T (input/output) DOUBLE PRECISION array, dimension (LDT,N) ! On entry, the upper quasi-triangular matrix T, in Schur ! Schur canonical form. ! On exit, the reordered upper quasi-triangular matrix, again ! in Schur canonical form. ! ! LDT (input) INTEGER ! The leading dimension of the array T. LDT >= max(1,N). ! ! Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) ! On entry, if COMPQ = 'V', the matrix Q of Schur vectors. ! On exit, if COMPQ = 'V', Q has been postmultiplied by the ! orthogonal transformation matrix Z which reorders T. ! If COMPQ = 'N', Q is not referenced. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. LDQ >= max(1,N). ! ! IFST (input/output) INTEGER ! ILST (input/output) INTEGER ! Specify the reordering of the diagonal blocks of T. ! The block with row index IFST is moved to row ILST, by a ! sequence of transpositions between adjacent blocks. ! On exit, if IFST pointed on entry to the second row of a ! 2-by-2 block, it is changed to point to the first row; ILST ! always points to the first row of the block in its final ! position (which may differ from its input value by +1 or -1). ! 1 <= IFST <= N; 1 <= ILST <= N. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! = 1: two adjacent blocks were too close to swap (the problem ! is very ill-conditioned); T may have been partially ! reordered, and ILST points to the first row of the ! current position of the block being moved. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL WANTQ INTEGER HERE, NBF, NBL, NBNEXT ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DLAEXC, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Decode and test the input arguments. ! INFO = 0 WANTQ = LSAME( COMPQ, 'V' ) IF( .NOT.WANTQ .AND. .NOT.LSAME( COMPQ, 'N' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDT.LT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.MAX( 1, N ) ) ) THEN INFO = -6 ELSE IF( IFST.LT.1 .OR. IFST.GT.N ) THEN INFO = -7 ELSE IF( ILST.LT.1 .OR. ILST.GT.N ) THEN INFO = -8 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTREXC', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.LE.1 ) & RETURN ! ! Determine the first row of specified block ! and find out it is 1 by 1 or 2 by 2. ! IF( IFST.GT.1 ) THEN IF( T( IFST, IFST-1 ).NE.ZERO ) & IFST = IFST - 1 END IF NBF = 1 IF( IFST.LT.N ) THEN IF( T( IFST+1, IFST ).NE.ZERO ) & NBF = 2 END IF ! ! Determine the first row of the final block ! and find out it is 1 by 1 or 2 by 2. ! IF( ILST.GT.1 ) THEN IF( T( ILST, ILST-1 ).NE.ZERO ) & ILST = ILST - 1 END IF NBL = 1 IF( ILST.LT.N ) THEN IF( T( ILST+1, ILST ).NE.ZERO ) & NBL = 2 END IF ! IF( IFST.EQ.ILST ) & RETURN ! IF( IFST.LT.ILST ) THEN ! ! Update ILST ! IF( NBF.EQ.2 .AND. NBL.EQ.1 ) & ILST = ILST - 1 IF( NBF.EQ.1 .AND. NBL.EQ.2 ) & ILST = ILST + 1 ! HERE = IFST ! 10 CONTINUE ! ! Swap block with next one below ! IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN ! ! Current block either 1 by 1 or 2 by 2 ! NBNEXT = 1 IF( HERE+NBF+1.LE.N ) THEN IF( T( HERE+NBF+1, HERE+NBF ).NE.ZERO ) & NBNEXT = 2 END IF CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, NBF, NBNEXT, & WORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE + NBNEXT ! ! Test if 2 by 2 block breaks into two 1 by 1 blocks ! IF( NBF.EQ.2 ) THEN IF( T( HERE+1, HERE ).EQ.ZERO ) & NBF = 3 END IF ! ELSE ! ! Current block consists of two 1 by 1 blocks each of which ! must be swapped individually ! NBNEXT = 1 IF( HERE+3.LE.N ) THEN IF( T( HERE+3, HERE+2 ).NE.ZERO ) & NBNEXT = 2 END IF CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE+1, 1, NBNEXT, & WORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF IF( NBNEXT.EQ.1 ) THEN ! ! Swap two 1 by 1 blocks, no problems possible ! CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, 1, NBNEXT, & WORK, INFO ) HERE = HERE + 1 ELSE ! ! Recompute NBNEXT in case 2 by 2 split ! IF( T( HERE+2, HERE+1 ).EQ.ZERO ) & NBNEXT = 1 IF( NBNEXT.EQ.2 ) THEN ! ! 2 by 2 Block did not split ! CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, 1, & NBNEXT, WORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE + 2 ELSE ! ! 2 by 2 Block did split ! CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, 1, 1, & WORK, INFO ) CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE+1, 1, 1, & WORK, INFO ) HERE = HERE + 2 END IF END IF END IF IF( HERE.LT.ILST ) & GO TO 10 ! ELSE ! HERE = IFST 20 CONTINUE ! ! Swap block with next one above ! IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN ! ! Current block either 1 by 1 or 2 by 2 ! NBNEXT = 1 IF( HERE.GE.3 ) THEN IF( T( HERE-1, HERE-2 ).NE.ZERO ) & NBNEXT = 2 END IF CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE-NBNEXT, NBNEXT, & NBF, WORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE - NBNEXT ! ! Test if 2 by 2 block breaks into two 1 by 1 blocks ! IF( NBF.EQ.2 ) THEN IF( T( HERE+1, HERE ).EQ.ZERO ) & NBF = 3 END IF ! ELSE ! ! Current block consists of two 1 by 1 blocks each of which ! must be swapped individually ! NBNEXT = 1 IF( HERE.GE.3 ) THEN IF( T( HERE-1, HERE-2 ).NE.ZERO ) & NBNEXT = 2 END IF CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE-NBNEXT, NBNEXT, & 1, WORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF IF( NBNEXT.EQ.1 ) THEN ! ! Swap two 1 by 1 blocks, no problems possible ! CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, NBNEXT, 1, & WORK, INFO ) HERE = HERE - 1 ELSE ! ! Recompute NBNEXT in case 2 by 2 split ! IF( T( HERE, HERE-1 ).EQ.ZERO ) & NBNEXT = 1 IF( NBNEXT.EQ.2 ) THEN ! ! 2 by 2 Block did not split ! CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE-1, 2, 1, & WORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE - 2 ELSE ! ! 2 by 2 Block did split ! CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE, 1, 1, & WORK, INFO ) CALL DLAEXC( WANTQ, N, T, LDT, Q, LDQ, HERE-1, 1, 1, & WORK, INFO ) HERE = HERE - 2 END IF END IF END IF IF( HERE.GT.ILST ) & GO TO 20 END IF ILST = HERE ! RETURN ! ! End of DTREXC ! END SUBROUTINE DTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X, & LDX, FERR, BERR, WORK, IWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER DIAG, TRANS, UPLO INTEGER INFO, LDA, LDB, LDX, N, NRHS ! .. ! .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ), & WORK( * ), X( LDX, * ) ! .. ! ! Purpose ! ======= ! ! DTRRFS provides error bounds and backward error estimates for the ! solution to a system of linear equations with a triangular ! coefficient matrix. ! ! The solution matrix X must be computed by DTRTRS or some other ! means before entering this routine. DTRRFS does not do iterative ! refinement because doing so cannot improve the backward error. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': A is upper triangular; ! = 'L': A is lower triangular. ! ! TRANS (input) CHARACTER*1 ! Specifies the form of the system of equations: ! = 'N': A * X = B (No transpose) ! = 'T': A**T * X = B (Transpose) ! = 'C': A**H * X = B (Conjugate transpose = Transpose) ! ! DIAG (input) CHARACTER*1 ! = 'N': A is non-unit triangular; ! = 'U': A is unit triangular. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrices B and X. NRHS >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The triangular matrix A. If UPLO = 'U', the leading N-by-N ! upper triangular part of the array A contains the upper ! triangular matrix, and the strictly lower triangular part of ! A is not referenced. If UPLO = 'L', the leading N-by-N lower ! triangular part of the array A contains the lower triangular ! matrix, and the strictly upper triangular part of A is not ! referenced. If DIAG = 'U', the diagonal elements of A are ! also not referenced and are assumed to be 1. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) ! The right hand side matrix B. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) ! The solution matrix X. ! ! LDX (input) INTEGER ! The leading dimension of the array X. LDX >= max(1,N). ! ! FERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The estimated forward error bound for each solution vector ! X(j) (the j-th column of the solution matrix X). ! If XTRUE is the true solution corresponding to X(j), FERR(j) ! is an estimated upper bound for the magnitude of the largest ! element in (X(j) - XTRUE) divided by the magnitude of the ! largest element in X(j). The estimate is as reliable as ! the estimate for RCOND, and is almost always a slight ! overestimate of the true error. ! ! BERR (output) DOUBLE PRECISION array, dimension (NRHS) ! The componentwise relative backward error of each solution ! vector X(j) (i.e., the smallest relative change in ! any element of A or B that makes X(j) an exact solution). ! ! WORK (workspace) DOUBLE PRECISION array, dimension (3*N) ! ! IWORK (workspace) INTEGER array, dimension (N) ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOTRAN, NOUNIT, UPPER CHARACTER TRANST INTEGER I, J, K, KASE, NZ DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DLACON, DTRMV, DTRSV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH EXTERNAL LSAME, DLAMCH ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) NOTRAN = LSAME( TRANS, 'N' ) NOUNIT = LSAME( DIAG, 'N' ) ! IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. & LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( NRHS.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -11 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTRRFS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN DO 10 J = 1, NRHS FERR( J ) = ZERO BERR( J ) = ZERO 10 CONTINUE RETURN END IF ! IF( NOTRAN ) THEN TRANST = 'T' ELSE TRANST = 'N' END IF ! ! NZ = maximum number of nonzero elements in each row of A, plus 1 ! NZ = N + 1 EPS = DLAMCH( 'Epsilon' ) SAFMIN = DLAMCH( 'Safe minimum' ) SAFE1 = NZ*SAFMIN SAFE2 = SAFE1 / EPS ! ! Do for each right hand side ! DO 250 J = 1, NRHS ! ! Compute residual R = B - op(A) * X, ! where op(A) = A or A', depending on TRANS. ! CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 ) CALL DTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ), 1 ) CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 ) ! ! Compute componentwise relative backward error from formula ! ! max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) ) ! ! where abs(Z) is the componentwise absolute value of the matrix ! or vector Z. If the i-th component of the denominator is less ! than SAFE2, then SAFE1 is added to the i-th components of the ! numerator and denominator before dividing. ! DO 20 I = 1, N WORK( I ) = ABS( B( I, J ) ) 20 CONTINUE ! IF( NOTRAN ) THEN ! ! Compute abs(A)*abs(X) + abs(B). ! IF( UPPER ) THEN IF( NOUNIT ) THEN DO 40 K = 1, N XK = ABS( X( K, J ) ) DO 30 I = 1, K WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 30 CONTINUE 40 CONTINUE ELSE DO 60 K = 1, N XK = ABS( X( K, J ) ) DO 50 I = 1, K - 1 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 50 CONTINUE WORK( K ) = WORK( K ) + XK 60 CONTINUE END IF ELSE IF( NOUNIT ) THEN DO 80 K = 1, N XK = ABS( X( K, J ) ) DO 70 I = K, N WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 70 CONTINUE 80 CONTINUE ELSE DO 100 K = 1, N XK = ABS( X( K, J ) ) DO 90 I = K + 1, N WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK 90 CONTINUE WORK( K ) = WORK( K ) + XK 100 CONTINUE END IF END IF ELSE ! ! Compute abs(A')*abs(X) + abs(B). ! IF( UPPER ) THEN IF( NOUNIT ) THEN DO 120 K = 1, N S = ZERO DO 110 I = 1, K S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 110 CONTINUE WORK( K ) = WORK( K ) + S 120 CONTINUE ELSE DO 140 K = 1, N S = ABS( X( K, J ) ) DO 130 I = 1, K - 1 S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 130 CONTINUE WORK( K ) = WORK( K ) + S 140 CONTINUE END IF ELSE IF( NOUNIT ) THEN DO 160 K = 1, N S = ZERO DO 150 I = K, N S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 150 CONTINUE WORK( K ) = WORK( K ) + S 160 CONTINUE ELSE DO 180 K = 1, N S = ABS( X( K, J ) ) DO 170 I = K + 1, N S = S + ABS( A( I, K ) )*ABS( X( I, J ) ) 170 CONTINUE WORK( K ) = WORK( K ) + S 180 CONTINUE END IF END IF END IF S = ZERO DO 190 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) ELSE S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / & ( WORK( I )+SAFE1 ) ) END IF 190 CONTINUE BERR( J ) = S ! ! Bound error from formula ! ! norm(X - XTRUE) / norm(X) .le. FERR = ! norm( abs(inv(op(A)))* ! ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) ! ! where ! norm(Z) is the magnitude of the largest component of Z ! inv(op(A)) is the inverse of op(A) ! abs(Z) is the componentwise absolute value of the matrix or ! vector Z ! NZ is the maximum number of nonzeros in any row of A, plus 1 ! EPS is machine epsilon ! ! The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B)) ! is incremented by SAFE1 if the i-th component of ! abs(op(A))*abs(X) + abs(B) is less than SAFE2. ! ! Use DLACON to estimate the infinity-norm of the matrix ! inv(op(A)) * diag(W), ! where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) ! DO 200 I = 1, N IF( WORK( I ).GT.SAFE2 ) THEN WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) ELSE WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 END IF 200 CONTINUE ! KASE = 0 210 CONTINUE CALL DLACON( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), & KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN ! ! Multiply by diag(W)*inv(op(A)'). ! CALL DTRSV( UPLO, TRANST, DIAG, N, A, LDA, WORK( N+1 ), & 1 ) DO 220 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 220 CONTINUE ELSE ! ! Multiply by inv(op(A))*diag(W). ! DO 230 I = 1, N WORK( N+I ) = WORK( I )*WORK( N+I ) 230 CONTINUE CALL DTRSV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ), & 1 ) END IF GO TO 210 END IF ! ! Normalize error. ! LSTRES = ZERO DO 240 I = 1, N LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 240 CONTINUE IF( LSTRES.NE.ZERO ) & FERR( J ) = FERR( J ) / LSTRES ! 250 CONTINUE ! RETURN ! ! End of DTRRFS ! END SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, & M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER COMPQ, JOB INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N DOUBLE PRECISION S, SEP ! .. ! .. Array Arguments .. LOGICAL SELECT( * ) INTEGER IWORK( * ) DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ), & WR( * ) ! .. ! ! Purpose ! ======= ! ! DTRSEN reorders the real Schur factorization of a real matrix ! A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in ! the leading diagonal blocks of the upper quasi-triangular matrix T, ! and the leading columns of Q form an orthonormal basis of the ! corresponding right invariant subspace. ! ! Optionally the routine computes the reciprocal condition numbers of ! the cluster of eigenvalues and/or the invariant subspace. ! ! T must be in Schur canonical form (as returned by DHSEQR), that is, ! block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each ! 2-by-2 diagonal block has its diagonal elemnts equal and its ! off-diagonal elements of opposite sign. ! ! Arguments ! ========= ! ! JOB (input) CHARACTER*1 ! Specifies whether condition numbers are required for the ! cluster of eigenvalues (S) or the invariant subspace (SEP): ! = 'N': none; ! = 'E': for eigenvalues only (S); ! = 'V': for invariant subspace only (SEP); ! = 'B': for both eigenvalues and invariant subspace (S and ! SEP). ! ! COMPQ (input) CHARACTER*1 ! = 'V': update the matrix Q of Schur vectors; ! = 'N': do not update Q. ! ! SELECT (input) LOGICAL array, dimension (N) ! SELECT specifies the eigenvalues in the selected cluster. To ! select a real eigenvalue w(j), SELECT(j) must be set to ! .TRUE.. To select a complex conjugate pair of eigenvalues ! w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, ! either SELECT(j) or SELECT(j+1) or both must be set to ! .TRUE.; a complex conjugate pair of eigenvalues must be ! either both included in the cluster or both excluded. ! ! N (input) INTEGER ! The order of the matrix T. N >= 0. ! ! T (input/output) DOUBLE PRECISION array, dimension (LDT,N) ! On entry, the upper quasi-triangular matrix T, in Schur ! canonical form. ! On exit, T is overwritten by the reordered matrix T, again in ! Schur canonical form, with the selected eigenvalues in the ! leading diagonal blocks. ! ! LDT (input) INTEGER ! The leading dimension of the array T. LDT >= max(1,N). ! ! Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) ! On entry, if COMPQ = 'V', the matrix Q of Schur vectors. ! On exit, if COMPQ = 'V', Q has been postmultiplied by the ! orthogonal transformation matrix which reorders T; the ! leading M columns of Q form an orthonormal basis for the ! specified invariant subspace. ! If COMPQ = 'N', Q is not referenced. ! ! LDQ (input) INTEGER ! The leading dimension of the array Q. ! LDQ >= 1; and if COMPQ = 'V', LDQ >= N. ! ! WR (output) DOUBLE PRECISION array, dimension (N) ! WI (output) DOUBLE PRECISION array, dimension (N) ! The real and imaginary parts, respectively, of the reordered ! eigenvalues of T. The eigenvalues are stored in the same ! order as on the diagonal of T, with WR(i) = T(i,i) and, if ! T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and ! WI(i+1) = -WI(i). Note that if a complex eigenvalue is ! sufficiently ill-conditioned, then its value may differ ! significantly from its value before reordering. ! ! M (output) INTEGER ! The dimension of the specified invariant subspace. ! 0 < = M <= N. ! ! S (output) DOUBLE PRECISION ! If JOB = 'E' or 'B', S is a lower bound on the reciprocal ! condition number for the selected cluster of eigenvalues. ! S cannot underestimate the true reciprocal condition number ! by more than a factor of sqrt(N). If M = 0 or N, S = 1. ! If JOB = 'N' or 'V', S is not referenced. ! ! SEP (output) DOUBLE PRECISION ! If JOB = 'V' or 'B', SEP is the estimated reciprocal ! condition number of the specified invariant subspace. If ! M = 0 or N, SEP = norm(T). ! If JOB = 'N' or 'E', SEP is not referenced. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. ! If JOB = 'N', LWORK >= max(1,N); ! if JOB = 'E', LWORK >= M*(N-M); ! if JOB = 'V' or 'B', LWORK >= 2*M*(N-M). ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! IWORK (workspace) INTEGER array, dimension (LIWORK) ! IF JOB = 'N' or 'E', IWORK is not referenced. ! ! LIWORK (input) INTEGER ! The dimension of the array IWORK. ! If JOB = 'N' or 'E', LIWORK >= 1; ! if JOB = 'V' or 'B', LIWORK >= M*(N-M). ! ! If LIWORK = -1, then a workspace query is assumed; the ! routine only calculates the optimal size of the IWORK array, ! returns this value as the first entry of the IWORK array, and ! no error message related to LIWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! = 1: reordering of T failed because some eigenvalues are too ! close to separate (the problem is very ill-conditioned); ! T may have been partially reordered, and WR and WI ! contain the eigenvalues in the same order as in T; S and ! SEP (if requested) are set to zero. ! ! Further Details ! =============== ! ! DTRSEN first collects the selected eigenvalues by computing an ! orthogonal transformation Z to move them to the top left corner of T. ! In other words, the selected eigenvalues are the eigenvalues of T11 ! in: ! ! Z'*T*Z = ( T11 T12 ) n1 ! ( 0 T22 ) n2 ! n1 n2 ! ! where N = n1+n2 and Z' means the transpose of Z. The first n1 columns ! of Z span the specified invariant subspace of T. ! ! If T has been obtained from the real Schur factorization of a matrix ! A = Q*T*Q', then the reordered real Schur factorization of A is given ! by A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span ! the corresponding invariant subspace of A. ! ! The reciprocal condition number of the average of the eigenvalues of ! T11 may be returned in S. S lies between 0 (very badly conditioned) ! and 1 (very well conditioned). It is computed as follows. First we ! compute R so that ! ! P = ( I R ) n1 ! ( 0 0 ) n2 ! n1 n2 ! ! is the projector on the invariant subspace associated with T11. ! R is the solution of the Sylvester equation: ! ! T11*R - R*T22 = T12. ! ! Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote ! the two-norm of M. Then S is computed as the lower bound ! ! (1 + F-norm(R)**2)**(-1/2) ! ! on the reciprocal of 2-norm(P), the true reciprocal condition number. ! S cannot underestimate 1 / 2-norm(P) by more than a factor of ! sqrt(N). ! ! An approximate error bound for the computed average of the ! eigenvalues of T11 is ! ! EPS * norm(T) / S ! ! where EPS is the machine precision. ! ! The reciprocal condition number of the right invariant subspace ! spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP. ! SEP is defined as the separation of T11 and T22: ! ! sep( T11, T22 ) = sigma-min( C ) ! ! where sigma-min(C) is the smallest singular value of the ! n1*n2-by-n1*n2 matrix ! ! C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) ) ! ! I(m) is an m by m identity matrix, and kprod denotes the Kronecker ! product. We estimate sigma-min(C) by the reciprocal of an estimate of ! the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C) ! cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2). ! ! When SEP is small, small changes in T can cause large changes in ! the invariant subspace. An approximate bound on the maximum angular ! error in the computed right invariant subspace is ! ! EPS * norm(T) / SEP ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS, & WANTSP INTEGER IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2, & NN DOUBLE PRECISION EST, RNORM, SCALE ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLANGE EXTERNAL LSAME, DLANGE ! .. ! .. External Subroutines .. EXTERNAL DLACON, DLACPY, DTREXC, DTRSYL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! ! Decode and test the input parameters ! WANTBH = LSAME( JOB, 'B' ) WANTS = LSAME( JOB, 'E' ) .OR. WANTBH WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH WANTQ = LSAME( COMPQ, 'V' ) ! INFO = 0 LQUERY = ( LWORK.EQ.-1 ) IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP ) & THEN INFO = -1 ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDT.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN INFO = -8 ELSE ! ! Set M to the dimension of the specified invariant subspace, ! and test LWORK and LIWORK. ! M = 0 PAIR = .FALSE. DO 10 K = 1, N IF( PAIR ) THEN PAIR = .FALSE. ELSE IF( K.LT.N ) THEN IF( T( K+1, K ).EQ.ZERO ) THEN IF( SELECT( K ) ) & M = M + 1 ELSE PAIR = .TRUE. IF( SELECT( K ) .OR. SELECT( K+1 ) ) & M = M + 2 END IF ELSE IF( SELECT( N ) ) & M = M + 1 END IF END IF 10 CONTINUE ! N1 = M N2 = N - M NN = N1*N2 ! IF( WANTSP ) THEN LWMIN = MAX( 1, 2*NN ) LIWMIN = MAX( 1, NN ) ELSE IF( LSAME( JOB, 'N' ) ) THEN LWMIN = MAX( 1, N ) LIWMIN = 1 ELSE IF( LSAME( JOB, 'E' ) ) THEN LWMIN = MAX( 1, NN ) LIWMIN = 1 END IF ! IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -15 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -17 END IF END IF ! IF( INFO.EQ.0 ) THEN WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTRSEN', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible. ! IF( M.EQ.N .OR. M.EQ.0 ) THEN IF( WANTS ) & S = ONE IF( WANTSP ) & SEP = DLANGE( '1', N, N, T, LDT, WORK ) GO TO 40 END IF ! ! Collect the selected blocks at the top-left corner of T. ! KS = 0 PAIR = .FALSE. DO 20 K = 1, N IF( PAIR ) THEN PAIR = .FALSE. ELSE SWAP = SELECT( K ) IF( K.LT.N ) THEN IF( T( K+1, K ).NE.ZERO ) THEN PAIR = .TRUE. SWAP = SWAP .OR. SELECT( K+1 ) END IF END IF IF( SWAP ) THEN KS = KS + 1 ! ! Swap the K-th block to position KS. ! IERR = 0 KK = K IF( K.NE.KS ) & CALL DTREXC( COMPQ, N, T, LDT, Q, LDQ, KK, KS, WORK, & IERR ) IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN ! ! Blocks too close to swap: exit. ! INFO = 1 IF( WANTS ) & S = ZERO IF( WANTSP ) & SEP = ZERO GO TO 40 END IF IF( PAIR ) & KS = KS + 1 END IF END IF 20 CONTINUE ! IF( WANTS ) THEN ! ! Solve Sylvester equation for R: ! ! T11*R - R*T22 = scale*T12 ! CALL DLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 ) CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ), & LDT, WORK, N1, SCALE, IERR ) ! ! Estimate the reciprocal of the condition number of the cluster ! of eigenvalues. ! RNORM = DLANGE( 'F', N1, N2, WORK, N1, WORK ) IF( RNORM.EQ.ZERO ) THEN S = ONE ELSE S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )* & SQRT( RNORM ) ) END IF END IF ! IF( WANTSP ) THEN ! ! Estimate sep(T11,T22). ! EST = ZERO KASE = 0 30 CONTINUE CALL DLACON( NN, WORK( NN+1 ), WORK, IWORK, EST, KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN ! ! Solve T11*R - R*T22 = scale*X. ! CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT, & T( N1+1, N1+1 ), LDT, WORK, N1, SCALE, & IERR ) ELSE ! ! Solve T11'*R - R*T22' = scale*X. ! CALL DTRSYL( 'T', 'T', -1, N1, N2, T, LDT, & T( N1+1, N1+1 ), LDT, WORK, N1, SCALE, & IERR ) END IF GO TO 30 END IF ! SEP = SCALE / EST END IF ! 40 CONTINUE ! ! Store the output eigenvalues in WR and WI. ! DO 50 K = 1, N WR( K ) = T( K, K ) WI( K ) = ZERO 50 CONTINUE DO 60 K = 1, N - 1 IF( T( K+1, K ).NE.ZERO ) THEN WI( K ) = SQRT( ABS( T( K, K+1 ) ) )* & SQRT( ABS( T( K+1, K ) ) ) WI( K+1 ) = -WI( K ) END IF 60 CONTINUE ! WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN ! RETURN ! ! End of DTRSEN ! END SUBROUTINE DTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, & LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER HOWMNY, JOB INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N ! .. ! .. Array Arguments .. LOGICAL SELECT( * ) INTEGER IWORK( * ) DOUBLE PRECISION S( * ), SEP( * ), T( LDT, * ), VL( LDVL, * ), & VR( LDVR, * ), WORK( LDWORK, * ) ! .. ! ! Purpose ! ======= ! ! DTRSNA estimates reciprocal condition numbers for specified ! eigenvalues and/or right eigenvectors of a real upper ! quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q ! orthogonal). ! ! T must be in Schur canonical form (as returned by DHSEQR), that is, ! block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each ! 2-by-2 diagonal block has its diagonal elements equal and its ! off-diagonal elements of opposite sign. ! ! Arguments ! ========= ! ! JOB (input) CHARACTER*1 ! Specifies whether condition numbers are required for ! eigenvalues (S) or eigenvectors (SEP): ! = 'E': for eigenvalues only (S); ! = 'V': for eigenvectors only (SEP); ! = 'B': for both eigenvalues and eigenvectors (S and SEP). ! ! HOWMNY (input) CHARACTER*1 ! = 'A': compute condition numbers for all eigenpairs; ! = 'S': compute condition numbers for selected eigenpairs ! specified by the array SELECT. ! ! SELECT (input) LOGICAL array, dimension (N) ! If HOWMNY = 'S', SELECT specifies the eigenpairs for which ! condition numbers are required. To select condition numbers ! for the eigenpair corresponding to a real eigenvalue w(j), ! SELECT(j) must be set to .TRUE.. To select condition numbers ! corresponding to a complex conjugate pair of eigenvalues w(j) ! and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be ! set to .TRUE.. ! If HOWMNY = 'A', SELECT is not referenced. ! ! N (input) INTEGER ! The order of the matrix T. N >= 0. ! ! T (input) DOUBLE PRECISION array, dimension (LDT,N) ! The upper quasi-triangular matrix T, in Schur canonical form. ! ! LDT (input) INTEGER ! The leading dimension of the array T. LDT >= max(1,N). ! ! VL (input) DOUBLE PRECISION array, dimension (LDVL,M) ! If JOB = 'E' or 'B', VL must contain left eigenvectors of T ! (or of any Q*T*Q**T with Q orthogonal), corresponding to the ! eigenpairs specified by HOWMNY and SELECT. The eigenvectors ! must be stored in consecutive columns of VL, as returned by ! DHSEIN or DTREVC. ! If JOB = 'V', VL is not referenced. ! ! LDVL (input) INTEGER ! The leading dimension of the array VL. ! LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N. ! ! VR (input) DOUBLE PRECISION array, dimension (LDVR,M) ! If JOB = 'E' or 'B', VR must contain right eigenvectors of T ! (or of any Q*T*Q**T with Q orthogonal), corresponding to the ! eigenpairs specified by HOWMNY and SELECT. The eigenvectors ! must be stored in consecutive columns of VR, as returned by ! DHSEIN or DTREVC. ! If JOB = 'V', VR is not referenced. ! ! LDVR (input) INTEGER ! The leading dimension of the array VR. ! LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N. ! ! S (output) DOUBLE PRECISION array, dimension (MM) ! If JOB = 'E' or 'B', the reciprocal condition numbers of the ! selected eigenvalues, stored in consecutive elements of the ! array. For a complex conjugate pair of eigenvalues two ! consecutive elements of S are set to the same value. Thus ! S(j), SEP(j), and the j-th columns of VL and VR all ! correspond to the same eigenpair (but not in general the ! j-th eigenpair, unless all eigenpairs are selected). ! If JOB = 'V', S is not referenced. ! ! SEP (output) DOUBLE PRECISION array, dimension (MM) ! If JOB = 'V' or 'B', the estimated reciprocal condition ! numbers of the selected eigenvectors, stored in consecutive ! elements of the array. For a complex eigenvector two ! consecutive elements of SEP are set to the same value. If ! the eigenvalues cannot be reordered to compute SEP(j), SEP(j) ! is set to 0; this can only occur when the true value would be ! very small anyway. ! If JOB = 'E', SEP is not referenced. ! ! MM (input) INTEGER ! The number of elements in the arrays S (if JOB = 'E' or 'B') ! and/or SEP (if JOB = 'V' or 'B'). MM >= M. ! ! M (output) INTEGER ! The number of elements of the arrays S and/or SEP actually ! used to store the estimated condition numbers. ! If HOWMNY = 'A', M is set to N. ! ! WORK (workspace) DOUBLE PRECISION array, dimension (LDWORK,N+1) ! If JOB = 'E', WORK is not referenced. ! ! LDWORK (input) INTEGER ! The leading dimension of the array WORK. ! LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. ! ! IWORK (workspace) INTEGER array, dimension (N) ! If JOB = 'E', IWORK is not referenced. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! The reciprocal of the condition number of an eigenvalue lambda is ! defined as ! ! S(lambda) = |v'*u| / (norm(u)*norm(v)) ! ! where u and v are the right and left eigenvectors of T corresponding ! to lambda; v' denotes the conjugate-transpose of v, and norm(u) ! denotes the Euclidean norm. These reciprocal condition numbers always ! lie between zero (very badly conditioned) and one (very well ! conditioned). If n = 1, S(lambda) is defined to be 1. ! ! An approximate error bound for a computed eigenvalue W(i) is given by ! ! EPS * norm(T) / S(i) ! ! where EPS is the machine precision. ! ! The reciprocal of the condition number of the right eigenvector u ! corresponding to lambda is defined as follows. Suppose ! ! T = ( lambda c ) ! ( 0 T22 ) ! ! Then the reciprocal condition number is ! ! SEP( lambda, T22 ) = sigma-min( T22 - lambda*I ) ! ! where sigma-min denotes the smallest singular value. We approximate ! the smallest singular value by the reciprocal of an estimate of the ! one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is ! defined to be abs(T(1,1)). ! ! An approximate error bound for a computed right eigenvector VR(i) ! is given by ! ! EPS * norm(T) / SEP(i) ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, TWO PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL PAIR, SOMCON, WANTBH, WANTS, WANTSP INTEGER I, IERR, IFST, ILST, J, K, KASE, KS, N2, NN DOUBLE PRECISION BIGNUM, COND, CS, DELTA, DUMM, EPS, EST, LNRM, & MU, PROD, PROD1, PROD2, RNRM, SCALE, SMLNUM, SN ! .. ! .. Local Arrays .. DOUBLE PRECISION DUMMY( 1 ) ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT, DLAMCH, DLAPY2, DNRM2 EXTERNAL LSAME, DDOT, DLAMCH, DLAPY2, DNRM2 ! .. ! .. External Subroutines .. EXTERNAL DLACON, DLACPY, DLAQTR, DTREXC, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT ! .. ! .. Executable Statements .. ! ! Decode and test the input parameters ! WANTBH = LSAME( JOB, 'B' ) WANTS = LSAME( JOB, 'E' ) .OR. WANTBH WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH ! SOMCON = LSAME( HOWMNY, 'S' ) ! INFO = 0 IF( .NOT.WANTS .AND. .NOT.WANTSP ) THEN INFO = -1 ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDT.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDVL.LT.1 .OR. ( WANTS .AND. LDVL.LT.N ) ) THEN INFO = -8 ELSE IF( LDVR.LT.1 .OR. ( WANTS .AND. LDVR.LT.N ) ) THEN INFO = -10 ELSE ! ! Set M to the number of eigenpairs for which condition numbers ! are required, and test MM. ! IF( SOMCON ) THEN M = 0 PAIR = .FALSE. DO 10 K = 1, N IF( PAIR ) THEN PAIR = .FALSE. ELSE IF( K.LT.N ) THEN IF( T( K+1, K ).EQ.ZERO ) THEN IF( SELECT( K ) ) & M = M + 1 ELSE PAIR = .TRUE. IF( SELECT( K ) .OR. SELECT( K+1 ) ) & M = M + 2 END IF ELSE IF( SELECT( N ) ) & M = M + 1 END IF END IF 10 CONTINUE ELSE M = N END IF ! IF( MM.LT.M ) THEN INFO = -13 ELSE IF( LDWORK.LT.1 .OR. ( WANTSP .AND. LDWORK.LT.N ) ) THEN INFO = -16 END IF END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTRSNA', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! IF( N.EQ.1 ) THEN IF( SOMCON ) THEN IF( .NOT.SELECT( 1 ) ) & RETURN END IF IF( WANTS ) & S( 1 ) = ONE IF( WANTSP ) & SEP( 1 ) = ABS( T( 1, 1 ) ) RETURN END IF ! ! Get machine constants ! EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) / EPS BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) ! KS = 0 PAIR = .FALSE. DO 60 K = 1, N ! ! Determine whether T(k,k) begins a 1-by-1 or 2-by-2 block. ! IF( PAIR ) THEN PAIR = .FALSE. GO TO 60 ELSE IF( K.LT.N ) & PAIR = T( K+1, K ).NE.ZERO END IF ! ! Determine whether condition numbers are required for the k-th ! eigenpair. ! IF( SOMCON ) THEN IF( PAIR ) THEN IF( .NOT.SELECT( K ) .AND. .NOT.SELECT( K+1 ) ) & GO TO 60 ELSE IF( .NOT.SELECT( K ) ) & GO TO 60 END IF END IF ! KS = KS + 1 ! IF( WANTS ) THEN ! ! Compute the reciprocal condition number of the k-th ! eigenvalue. ! IF( .NOT.PAIR ) THEN ! ! Real eigenvalue. ! PROD = DDOT( N, VR( 1, KS ), 1, VL( 1, KS ), 1 ) RNRM = DNRM2( N, VR( 1, KS ), 1 ) LNRM = DNRM2( N, VL( 1, KS ), 1 ) S( KS ) = ABS( PROD ) / ( RNRM*LNRM ) ELSE ! ! Complex eigenvalue. ! PROD1 = DDOT( N, VR( 1, KS ), 1, VL( 1, KS ), 1 ) PROD1 = PROD1 + DDOT( N, VR( 1, KS+1 ), 1, VL( 1, KS+1 ), & 1 ) PROD2 = DDOT( N, VL( 1, KS ), 1, VR( 1, KS+1 ), 1 ) PROD2 = PROD2 - DDOT( N, VL( 1, KS+1 ), 1, VR( 1, KS ), & 1 ) RNRM = DLAPY2( DNRM2( N, VR( 1, KS ), 1 ), & DNRM2( N, VR( 1, KS+1 ), 1 ) ) LNRM = DLAPY2( DNRM2( N, VL( 1, KS ), 1 ), & DNRM2( N, VL( 1, KS+1 ), 1 ) ) COND = DLAPY2( PROD1, PROD2 ) / ( RNRM*LNRM ) S( KS ) = COND S( KS+1 ) = COND END IF END IF ! IF( WANTSP ) THEN ! ! Estimate the reciprocal condition number of the k-th ! eigenvector. ! ! Copy the matrix T to the array WORK and swap the diagonal ! block beginning at T(k,k) to the (1,1) position. ! CALL DLACPY( 'Full', N, N, T, LDT, WORK, LDWORK ) IFST = K ILST = 1 CALL DTREXC( 'No Q', N, WORK, LDWORK, DUMMY, 1, IFST, ILST, & WORK( 1, N+1 ), IERR ) ! IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN ! ! Could not swap because blocks not well separated ! SCALE = ONE EST = BIGNUM ELSE ! ! Reordering successful ! IF( WORK( 2, 1 ).EQ.ZERO ) THEN ! ! Form C = T22 - lambda*I in WORK(2:N,2:N). ! DO 20 I = 2, N WORK( I, I ) = WORK( I, I ) - WORK( 1, 1 ) 20 CONTINUE N2 = 1 NN = N - 1 ELSE ! ! Triangularize the 2 by 2 block by unitary ! transformation U = [ cs i*ss ] ! [ i*ss cs ]. ! such that the (1,1) position of WORK is complex ! eigenvalue lambda with positive imaginary part. (2,2) ! position of WORK is the complex eigenvalue lambda ! with negative imaginary part. ! MU = SQRT( ABS( WORK( 1, 2 ) ) )* & SQRT( ABS( WORK( 2, 1 ) ) ) DELTA = DLAPY2( MU, WORK( 2, 1 ) ) CS = MU / DELTA SN = -WORK( 2, 1 ) / DELTA ! ! Form ! ! C' = WORK(2:N,2:N) + i*[rwork(1) ..... rwork(n-1) ] ! [ mu ] ! [ .. ] ! [ .. ] ! [ mu ] ! where C' is conjugate transpose of complex matrix C, ! and RWORK is stored starting in the N+1-st column of ! WORK. ! DO 30 J = 3, N WORK( 2, J ) = CS*WORK( 2, J ) WORK( J, J ) = WORK( J, J ) - WORK( 1, 1 ) 30 CONTINUE WORK( 2, 2 ) = ZERO ! WORK( 1, N+1 ) = TWO*MU DO 40 I = 2, N - 1 WORK( I, N+1 ) = SN*WORK( 1, I+1 ) 40 CONTINUE N2 = 2 NN = 2*( N-1 ) END IF ! ! Estimate norm(inv(C')) ! EST = ZERO KASE = 0 50 CONTINUE CALL DLACON( NN, WORK( 1, N+2 ), WORK( 1, N+4 ), IWORK, & EST, KASE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN IF( N2.EQ.1 ) THEN ! ! Real eigenvalue: solve C'*x = scale*c. ! CALL DLAQTR( .TRUE., .TRUE., N-1, WORK( 2, 2 ), & LDWORK, DUMMY, DUMM, SCALE, & WORK( 1, N+4 ), WORK( 1, N+6 ), & IERR ) ELSE ! ! Complex eigenvalue: solve ! C'*(p+iq) = scale*(c+id) in real arithmetic. ! CALL DLAQTR( .TRUE., .FALSE., N-1, WORK( 2, 2 ), & LDWORK, WORK( 1, N+1 ), MU, SCALE, & WORK( 1, N+4 ), WORK( 1, N+6 ), & IERR ) END IF ELSE IF( N2.EQ.1 ) THEN ! ! Real eigenvalue: solve C*x = scale*c. ! CALL DLAQTR( .FALSE., .TRUE., N-1, WORK( 2, 2 ), & LDWORK, DUMMY, DUMM, SCALE, & WORK( 1, N+4 ), WORK( 1, N+6 ), & IERR ) ELSE ! ! Complex eigenvalue: solve ! C*(p+iq) = scale*(c+id) in real arithmetic. ! CALL DLAQTR( .FALSE., .FALSE., N-1, & WORK( 2, 2 ), LDWORK, & WORK( 1, N+1 ), MU, SCALE, & WORK( 1, N+4 ), WORK( 1, N+6 ), & IERR ) ! END IF END IF ! GO TO 50 END IF END IF ! SEP( KS ) = SCALE / MAX( EST, SMLNUM ) IF( PAIR ) & SEP( KS+1 ) = SEP( KS ) END IF ! IF( PAIR ) & KS = KS + 1 ! 60 CONTINUE RETURN ! ! End of DTRSNA ! END SUBROUTINE DTRSYL( TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, & LDC, SCALE, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER TRANA, TRANB INTEGER INFO, ISGN, LDA, LDB, LDC, M, N DOUBLE PRECISION SCALE ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ) ! .. ! ! Purpose ! ======= ! ! DTRSYL solves the real Sylvester matrix equation: ! ! op(A)*X + X*op(B) = scale*C or ! op(A)*X - X*op(B) = scale*C, ! ! where op(A) = A or A**T, and A and B are both upper quasi- ! triangular. A is M-by-M and B is N-by-N; the right hand side C and ! the solution X are M-by-N; and scale is an output scale factor, set ! <= 1 to avoid overflow in X. ! ! A and B must be in Schur canonical form (as returned by DHSEQR), that ! is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; ! each 2-by-2 diagonal block has its diagonal elements equal and its ! off-diagonal elements of opposite sign. ! ! Arguments ! ========= ! ! TRANA (input) CHARACTER*1 ! Specifies the option op(A): ! = 'N': op(A) = A (No transpose) ! = 'T': op(A) = A**T (Transpose) ! = 'C': op(A) = A**H (Conjugate transpose = Transpose) ! ! TRANB (input) CHARACTER*1 ! Specifies the option op(B): ! = 'N': op(B) = B (No transpose) ! = 'T': op(B) = B**T (Transpose) ! = 'C': op(B) = B**H (Conjugate transpose = Transpose) ! ! ISGN (input) INTEGER ! Specifies the sign in the equation: ! = +1: solve op(A)*X + X*op(B) = scale*C ! = -1: solve op(A)*X - X*op(B) = scale*C ! ! M (input) INTEGER ! The order of the matrix A, and the number of rows in the ! matrices X and C. M >= 0. ! ! N (input) INTEGER ! The order of the matrix B, and the number of columns in the ! matrices X and C. N >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,M) ! The upper quasi-triangular matrix A, in Schur canonical form. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! B (input) DOUBLE PRECISION array, dimension (LDB,N) ! The upper quasi-triangular matrix B, in Schur canonical form. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! C (input/output) DOUBLE PRECISION array, dimension (LDC,N) ! On entry, the M-by-N right hand side matrix C. ! On exit, C is overwritten by the solution matrix X. ! ! LDC (input) INTEGER ! The leading dimension of the array C. LDC >= max(1,M) ! ! SCALE (output) DOUBLE PRECISION ! The scale factor, scale, set <= 1 to avoid overflow in X. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! = 1: A and B have common or very close eigenvalues; perturbed ! values were used to solve the equation (but the matrices ! A and B are unchanged). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOTRNA, NOTRNB INTEGER IERR, J, K, K1, K2, KNEXT, L, L1, L2, LNEXT DOUBLE PRECISION A11, BIGNUM, DA11, DB, EPS, SCALOC, SGN, SMIN, & SMLNUM, SUML, SUMR, XNORM ! .. ! .. Local Arrays .. DOUBLE PRECISION DUM( 1 ), VEC( 2, 2 ), X( 2, 2 ) ! .. ! .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT, DLAMCH, DLANGE EXTERNAL LSAME, DDOT, DLAMCH, DLANGE ! .. ! .. External Subroutines .. EXTERNAL DLALN2, DLASY2, DSCAL, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX, MIN ! .. ! .. Executable Statements .. ! ! Decode and Test input parameters ! NOTRNA = LSAME( TRANA, 'N' ) NOTRNB = LSAME( TRANB, 'N' ) ! INFO = 0 IF( .NOT.NOTRNA .AND. .NOT.LSAME( TRANA, 'T' ) .AND. .NOT. & LSAME( TRANA, 'C' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRNB .AND. .NOT.LSAME( TRANB, 'T' ) .AND. .NOT. & LSAME( TRANB, 'C' ) ) THEN INFO = -2 ELSE IF( ISGN.NE.1 .AND. ISGN.NE.-1 ) THEN INFO = -3 ELSE IF( M.LT.0 ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -7 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( LDC.LT.MAX( 1, M ) ) THEN INFO = -11 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTRSYL', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 .OR. N.EQ.0 ) & RETURN ! ! Set constants to control overflow ! EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) SMLNUM = SMLNUM*DBLE( M*N ) / EPS BIGNUM = ONE / SMLNUM ! SMIN = MAX( SMLNUM, EPS*DLANGE( 'M', M, M, A, LDA, DUM ), & EPS*DLANGE( 'M', N, N, B, LDB, DUM ) ) ! SCALE = ONE SGN = ISGN ! IF( NOTRNA .AND. NOTRNB ) THEN ! ! Solve A*X + ISGN*X*B = scale*C. ! ! The (K,L)th block of X is determined starting from ! bottom-left corner column by column by ! ! A(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L) ! ! Where ! M L-1 ! R(K,L) = SUM [A(K,I)*X(I,L)] + ISGN*SUM [X(K,J)*B(J,L)]. ! I=K+1 J=1 ! ! Start column loop (index = L) ! L1 (L2) : column index of the first (first) row of X(K,L). ! LNEXT = 1 DO 60 L = 1, N IF( L.LT.LNEXT ) & GO TO 60 IF( L.EQ.N ) THEN L1 = L L2 = L ELSE IF( B( L+1, L ).NE.ZERO ) THEN L1 = L L2 = L + 1 LNEXT = L + 2 ELSE L1 = L L2 = L LNEXT = L + 1 END IF END IF ! ! Start row loop (index = K) ! K1 (K2): row index of the first (last) row of X(K,L). ! KNEXT = M DO 50 K = M, 1, -1 IF( K.GT.KNEXT ) & GO TO 50 IF( K.EQ.1 ) THEN K1 = K K2 = K ELSE IF( A( K, K-1 ).NE.ZERO ) THEN K1 = K - 1 K2 = K KNEXT = K - 2 ELSE K1 = K K2 = K KNEXT = K - 1 END IF END IF ! IF( L1.EQ.L2 .AND. K1.EQ.K2 ) THEN SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA, & C( MIN( K1+1, M ), L1 ), 1 ) SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 ) VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR ) SCALOC = ONE ! A11 = A( K1, K1 ) + SGN*B( L1, L1 ) DA11 = ABS( A11 ) IF( DA11.LE.SMIN ) THEN A11 = SMIN DA11 = SMIN INFO = 1 END IF DB = ABS( VEC( 1, 1 ) ) IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN IF( DB.GT.BIGNUM*DA11 ) & SCALOC = ONE / DB END IF X( 1, 1 ) = ( VEC( 1, 1 )*SCALOC ) / A11 ! IF( SCALOC.NE.ONE ) THEN DO 10 J = 1, N CALL DSCAL( M, SCALOC, C( 1, J ), 1 ) 10 CONTINUE SCALE = SCALE*SCALOC END IF C( K1, L1 ) = X( 1, 1 ) ! ELSE IF( L1.EQ.L2 .AND. K1.NE.K2 ) THEN ! SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA, & C( MIN( K2+1, M ), L1 ), 1 ) SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 ) VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR ) ! SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA, & C( MIN( K2+1, M ), L1 ), 1 ) SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L1 ), 1 ) VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR ) ! CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, A( K1, K1 ), & LDA, ONE, ONE, VEC, 2, -SGN*B( L1, L1 ), & ZERO, X, 2, SCALOC, XNORM, IERR ) IF( IERR.NE.0 ) & INFO = 1 ! IF( SCALOC.NE.ONE ) THEN DO 20 J = 1, N CALL DSCAL( M, SCALOC, C( 1, J ), 1 ) 20 CONTINUE SCALE = SCALE*SCALOC END IF C( K1, L1 ) = X( 1, 1 ) C( K2, L1 ) = X( 2, 1 ) ! ELSE IF( L1.NE.L2 .AND. K1.EQ.K2 ) THEN ! SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA, & C( MIN( K1+1, M ), L1 ), 1 ) SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 ) VEC( 1, 1 ) = SGN*( C( K1, L1 )-( SUML+SGN*SUMR ) ) ! SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA, & C( MIN( K1+1, M ), L2 ), 1 ) SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L2 ), 1 ) VEC( 2, 1 ) = SGN*( C( K1, L2 )-( SUML+SGN*SUMR ) ) ! CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, B( L1, L1 ), & LDB, ONE, ONE, VEC, 2, -SGN*A( K1, K1 ), & ZERO, X, 2, SCALOC, XNORM, IERR ) IF( IERR.NE.0 ) & INFO = 1 ! IF( SCALOC.NE.ONE ) THEN DO 30 J = 1, N CALL DSCAL( M, SCALOC, C( 1, J ), 1 ) 30 CONTINUE SCALE = SCALE*SCALOC END IF C( K1, L1 ) = X( 1, 1 ) C( K1, L2 ) = X( 2, 1 ) ! ELSE IF( L1.NE.L2 .AND. K1.NE.K2 ) THEN ! SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA, & C( MIN( K2+1, M ), L1 ), 1 ) SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 ) VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR ) ! SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA, & C( MIN( K2+1, M ), L2 ), 1 ) SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L2 ), 1 ) VEC( 1, 2 ) = C( K1, L2 ) - ( SUML+SGN*SUMR ) ! SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA, & C( MIN( K2+1, M ), L1 ), 1 ) SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L1 ), 1 ) VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR ) ! SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA, & C( MIN( K2+1, M ), L2 ), 1 ) SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L2 ), 1 ) VEC( 2, 2 ) = C( K2, L2 ) - ( SUML+SGN*SUMR ) ! CALL DLASY2( .FALSE., .FALSE., ISGN, 2, 2, & A( K1, K1 ), LDA, B( L1, L1 ), LDB, VEC, & 2, SCALOC, X, 2, XNORM, IERR ) IF( IERR.NE.0 ) & INFO = 1 ! IF( SCALOC.NE.ONE ) THEN DO 40 J = 1, N CALL DSCAL( M, SCALOC, C( 1, J ), 1 ) 40 CONTINUE SCALE = SCALE*SCALOC END IF C( K1, L1 ) = X( 1, 1 ) C( K1, L2 ) = X( 1, 2 ) C( K2, L1 ) = X( 2, 1 ) C( K2, L2 ) = X( 2, 2 ) END IF ! 50 CONTINUE ! 60 CONTINUE ! ELSE IF( .NOT.NOTRNA .AND. NOTRNB ) THEN ! ! Solve A' *X + ISGN*X*B = scale*C. ! ! The (K,L)th block of X is determined starting from ! upper-left corner column by column by ! ! A(K,K)'*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L) ! ! Where ! K-1 L-1 ! R(K,L) = SUM [A(I,K)'*X(I,L)] +ISGN*SUM [X(K,J)*B(J,L)] ! I=1 J=1 ! ! Start column loop (index = L) ! L1 (L2): column index of the first (last) row of X(K,L) ! LNEXT = 1 DO 120 L = 1, N IF( L.LT.LNEXT ) & GO TO 120 IF( L.EQ.N ) THEN L1 = L L2 = L ELSE IF( B( L+1, L ).NE.ZERO ) THEN L1 = L L2 = L + 1 LNEXT = L + 2 ELSE L1 = L L2 = L LNEXT = L + 1 END IF END IF ! ! Start row loop (index = K) ! K1 (K2): row index of the first (last) row of X(K,L) ! KNEXT = 1 DO 110 K = 1, M IF( K.LT.KNEXT ) & GO TO 110 IF( K.EQ.M ) THEN K1 = K K2 = K ELSE IF( A( K+1, K ).NE.ZERO ) THEN K1 = K K2 = K + 1 KNEXT = K + 2 ELSE K1 = K K2 = K KNEXT = K + 1 END IF END IF ! IF( L1.EQ.L2 .AND. K1.EQ.K2 ) THEN SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 ) SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 ) VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR ) SCALOC = ONE ! A11 = A( K1, K1 ) + SGN*B( L1, L1 ) DA11 = ABS( A11 ) IF( DA11.LE.SMIN ) THEN A11 = SMIN DA11 = SMIN INFO = 1 END IF DB = ABS( VEC( 1, 1 ) ) IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN IF( DB.GT.BIGNUM*DA11 ) & SCALOC = ONE / DB END IF X( 1, 1 ) = ( VEC( 1, 1 )*SCALOC ) / A11 ! IF( SCALOC.NE.ONE ) THEN DO 70 J = 1, N CALL DSCAL( M, SCALOC, C( 1, J ), 1 ) 70 CONTINUE SCALE = SCALE*SCALOC END IF C( K1, L1 ) = X( 1, 1 ) ! ELSE IF( L1.EQ.L2 .AND. K1.NE.K2 ) THEN ! SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 ) SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 ) VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR ) ! SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L1 ), 1 ) SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L1 ), 1 ) VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR ) ! CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, A( K1, K1 ), & LDA, ONE, ONE, VEC, 2, -SGN*B( L1, L1 ), & ZERO, X, 2, SCALOC, XNORM, IERR ) IF( IERR.NE.0 ) & INFO = 1 ! IF( SCALOC.NE.ONE ) THEN DO 80 J = 1, N CALL DSCAL( M, SCALOC, C( 1, J ), 1 ) 80 CONTINUE SCALE = SCALE*SCALOC END IF C( K1, L1 ) = X( 1, 1 ) C( K2, L1 ) = X( 2, 1 ) ! ELSE IF( L1.NE.L2 .AND. K1.EQ.K2 ) THEN ! SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 ) SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 ) VEC( 1, 1 ) = SGN*( C( K1, L1 )-( SUML+SGN*SUMR ) ) ! SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L2 ), 1 ) SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L2 ), 1 ) VEC( 2, 1 ) = SGN*( C( K1, L2 )-( SUML+SGN*SUMR ) ) ! CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, B( L1, L1 ), & LDB, ONE, ONE, VEC, 2, -SGN*A( K1, K1 ), & ZERO, X, 2, SCALOC, XNORM, IERR ) IF( IERR.NE.0 ) & INFO = 1 ! IF( SCALOC.NE.ONE ) THEN DO 90 J = 1, N CALL DSCAL( M, SCALOC, C( 1, J ), 1 ) 90 CONTINUE SCALE = SCALE*SCALOC END IF C( K1, L1 ) = X( 1, 1 ) C( K1, L2 ) = X( 2, 1 ) ! ELSE IF( L1.NE.L2 .AND. K1.NE.K2 ) THEN ! SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 ) SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 ) VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR ) ! SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L2 ), 1 ) SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L2 ), 1 ) VEC( 1, 2 ) = C( K1, L2 ) - ( SUML+SGN*SUMR ) ! SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L1 ), 1 ) SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L1 ), 1 ) VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR ) ! SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L2 ), 1 ) SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L2 ), 1 ) VEC( 2, 2 ) = C( K2, L2 ) - ( SUML+SGN*SUMR ) ! CALL DLASY2( .TRUE., .FALSE., ISGN, 2, 2, A( K1, K1 ), & LDA, B( L1, L1 ), LDB, VEC, 2, SCALOC, X, & 2, XNORM, IERR ) IF( IERR.NE.0 ) & INFO = 1 ! IF( SCALOC.NE.ONE ) THEN DO 100 J = 1, N CALL DSCAL( M, SCALOC, C( 1, J ), 1 ) 100 CONTINUE SCALE = SCALE*SCALOC END IF C( K1, L1 ) = X( 1, 1 ) C( K1, L2 ) = X( 1, 2 ) C( K2, L1 ) = X( 2, 1 ) C( K2, L2 ) = X( 2, 2 ) END IF ! 110 CONTINUE 120 CONTINUE ! ELSE IF( .NOT.NOTRNA .AND. .NOT.NOTRNB ) THEN ! ! Solve A'*X + ISGN*X*B' = scale*C. ! ! The (K,L)th block of X is determined starting from ! top-right corner column by column by ! ! A(K,K)'*X(K,L) + ISGN*X(K,L)*B(L,L)' = C(K,L) - R(K,L) ! ! Where ! K-1 N ! R(K,L) = SUM [A(I,K)'*X(I,L)] + ISGN*SUM [X(K,J)*B(L,J)']. ! I=1 J=L+1 ! ! Start column loop (index = L) ! L1 (L2): column index of the first (last) row of X(K,L) ! LNEXT = N DO 180 L = N, 1, -1 IF( L.GT.LNEXT ) & GO TO 180 IF( L.EQ.1 ) THEN L1 = L L2 = L ELSE IF( B( L, L-1 ).NE.ZERO ) THEN L1 = L - 1 L2 = L LNEXT = L - 2 ELSE L1 = L L2 = L LNEXT = L - 1 END IF END IF ! ! Start row loop (index = K) ! K1 (K2): row index of the first (last) row of X(K,L) ! KNEXT = 1 DO 170 K = 1, M IF( K.LT.KNEXT ) & GO TO 170 IF( K.EQ.M ) THEN K1 = K K2 = K ELSE IF( A( K+1, K ).NE.ZERO ) THEN K1 = K K2 = K + 1 KNEXT = K + 2 ELSE K1 = K K2 = K KNEXT = K + 1 END IF END IF ! IF( L1.EQ.L2 .AND. K1.EQ.K2 ) THEN SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 ) SUMR = DDOT( N-L1, C( K1, MIN( L1+1, N ) ), LDC, & B( L1, MIN( L1+1, N ) ), LDB ) VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR ) SCALOC = ONE ! A11 = A( K1, K1 ) + SGN*B( L1, L1 ) DA11 = ABS( A11 ) IF( DA11.LE.SMIN ) THEN A11 = SMIN DA11 = SMIN INFO = 1 END IF DB = ABS( VEC( 1, 1 ) ) IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN IF( DB.GT.BIGNUM*DA11 ) & SCALOC = ONE / DB END IF X( 1, 1 ) = ( VEC( 1, 1 )*SCALOC ) / A11 ! IF( SCALOC.NE.ONE ) THEN DO 130 J = 1, N CALL DSCAL( M, SCALOC, C( 1, J ), 1 ) 130 CONTINUE SCALE = SCALE*SCALOC END IF C( K1, L1 ) = X( 1, 1 ) ! ELSE IF( L1.EQ.L2 .AND. K1.NE.K2 ) THEN ! SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 ) SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC, & B( L1, MIN( L2+1, N ) ), LDB ) VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR ) ! SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L1 ), 1 ) SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC, & B( L1, MIN( L2+1, N ) ), LDB ) VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR ) ! CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, A( K1, K1 ), & LDA, ONE, ONE, VEC, 2, -SGN*B( L1, L1 ), & ZERO, X, 2, SCALOC, XNORM, IERR ) IF( IERR.NE.0 ) & INFO = 1 ! IF( SCALOC.NE.ONE ) THEN DO 140 J = 1, N CALL DSCAL( M, SCALOC, C( 1, J ), 1 ) 140 CONTINUE SCALE = SCALE*SCALOC END IF C( K1, L1 ) = X( 1, 1 ) C( K2, L1 ) = X( 2, 1 ) ! ELSE IF( L1.NE.L2 .AND. K1.EQ.K2 ) THEN ! SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 ) SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC, & B( L1, MIN( L2+1, N ) ), LDB ) VEC( 1, 1 ) = SGN*( C( K1, L1 )-( SUML+SGN*SUMR ) ) ! SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L2 ), 1 ) SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC, & B( L2, MIN( L2+1, N ) ), LDB ) VEC( 2, 1 ) = SGN*( C( K1, L2 )-( SUML+SGN*SUMR ) ) ! CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, B( L1, L1 ), & LDB, ONE, ONE, VEC, 2, -SGN*A( K1, K1 ), & ZERO, X, 2, SCALOC, XNORM, IERR ) IF( IERR.NE.0 ) & INFO = 1 ! IF( SCALOC.NE.ONE ) THEN DO 150 J = 1, N CALL DSCAL( M, SCALOC, C( 1, J ), 1 ) 150 CONTINUE SCALE = SCALE*SCALOC END IF C( K1, L1 ) = X( 1, 1 ) C( K1, L2 ) = X( 2, 1 ) ! ELSE IF( L1.NE.L2 .AND. K1.NE.K2 ) THEN ! SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 ) SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC, & B( L1, MIN( L2+1, N ) ), LDB ) VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR ) ! SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L2 ), 1 ) SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC, & B( L2, MIN( L2+1, N ) ), LDB ) VEC( 1, 2 ) = C( K1, L2 ) - ( SUML+SGN*SUMR ) ! SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L1 ), 1 ) SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC, & B( L1, MIN( L2+1, N ) ), LDB ) VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR ) ! SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L2 ), 1 ) SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC, & B( L2, MIN( L2+1, N ) ), LDB ) VEC( 2, 2 ) = C( K2, L2 ) - ( SUML+SGN*SUMR ) ! CALL DLASY2( .TRUE., .TRUE., ISGN, 2, 2, A( K1, K1 ), & LDA, B( L1, L1 ), LDB, VEC, 2, SCALOC, X, & 2, XNORM, IERR ) IF( IERR.NE.0 ) & INFO = 1 ! IF( SCALOC.NE.ONE ) THEN DO 160 J = 1, N CALL DSCAL( M, SCALOC, C( 1, J ), 1 ) 160 CONTINUE SCALE = SCALE*SCALOC END IF C( K1, L1 ) = X( 1, 1 ) C( K1, L2 ) = X( 1, 2 ) C( K2, L1 ) = X( 2, 1 ) C( K2, L2 ) = X( 2, 2 ) END IF ! 170 CONTINUE 180 CONTINUE ! ELSE IF( NOTRNA .AND. .NOT.NOTRNB ) THEN ! ! Solve A*X + ISGN*X*B' = scale*C. ! ! The (K,L)th block of X is determined starting from ! bottom-right corner column by column by ! ! A(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L)' = C(K,L) - R(K,L) ! ! Where ! M N ! R(K,L) = SUM [A(K,I)*X(I,L)] + ISGN*SUM [X(K,J)*B(L,J)']. ! I=K+1 J=L+1 ! ! Start column loop (index = L) ! L1 (L2): column index of the first (last) row of X(K,L) ! LNEXT = N DO 240 L = N, 1, -1 IF( L.GT.LNEXT ) & GO TO 240 IF( L.EQ.1 ) THEN L1 = L L2 = L ELSE IF( B( L, L-1 ).NE.ZERO ) THEN L1 = L - 1 L2 = L LNEXT = L - 2 ELSE L1 = L L2 = L LNEXT = L - 1 END IF END IF ! ! Start row loop (index = K) ! K1 (K2): row index of the first (last) row of X(K,L) ! KNEXT = M DO 230 K = M, 1, -1 IF( K.GT.KNEXT ) & GO TO 230 IF( K.EQ.1 ) THEN K1 = K K2 = K ELSE IF( A( K, K-1 ).NE.ZERO ) THEN K1 = K - 1 K2 = K KNEXT = K - 2 ELSE K1 = K K2 = K KNEXT = K - 1 END IF END IF ! IF( L1.EQ.L2 .AND. K1.EQ.K2 ) THEN SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA, & C( MIN( K1+1, M ), L1 ), 1 ) SUMR = DDOT( N-L1, C( K1, MIN( L1+1, N ) ), LDC, & B( L1, MIN( L1+1, N ) ), LDB ) VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR ) SCALOC = ONE ! A11 = A( K1, K1 ) + SGN*B( L1, L1 ) DA11 = ABS( A11 ) IF( DA11.LE.SMIN ) THEN A11 = SMIN DA11 = SMIN INFO = 1 END IF DB = ABS( VEC( 1, 1 ) ) IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN IF( DB.GT.BIGNUM*DA11 ) & SCALOC = ONE / DB END IF X( 1, 1 ) = ( VEC( 1, 1 )*SCALOC ) / A11 ! IF( SCALOC.NE.ONE ) THEN DO 190 J = 1, N CALL DSCAL( M, SCALOC, C( 1, J ), 1 ) 190 CONTINUE SCALE = SCALE*SCALOC END IF C( K1, L1 ) = X( 1, 1 ) ! ELSE IF( L1.EQ.L2 .AND. K1.NE.K2 ) THEN ! SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA, & C( MIN( K2+1, M ), L1 ), 1 ) SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC, & B( L1, MIN( L2+1, N ) ), LDB ) VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR ) ! SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA, & C( MIN( K2+1, M ), L1 ), 1 ) SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC, & B( L1, MIN( L2+1, N ) ), LDB ) VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR ) ! CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, A( K1, K1 ), & LDA, ONE, ONE, VEC, 2, -SGN*B( L1, L1 ), & ZERO, X, 2, SCALOC, XNORM, IERR ) IF( IERR.NE.0 ) & INFO = 1 ! IF( SCALOC.NE.ONE ) THEN DO 200 J = 1, N CALL DSCAL( M, SCALOC, C( 1, J ), 1 ) 200 CONTINUE SCALE = SCALE*SCALOC END IF C( K1, L1 ) = X( 1, 1 ) C( K2, L1 ) = X( 2, 1 ) ! ELSE IF( L1.NE.L2 .AND. K1.EQ.K2 ) THEN ! SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA, & C( MIN( K1+1, M ), L1 ), 1 ) SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC, & B( L1, MIN( L2+1, N ) ), LDB ) VEC( 1, 1 ) = SGN*( C( K1, L1 )-( SUML+SGN*SUMR ) ) ! SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA, & C( MIN( K1+1, M ), L2 ), 1 ) SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC, & B( L2, MIN( L2+1, N ) ), LDB ) VEC( 2, 1 ) = SGN*( C( K1, L2 )-( SUML+SGN*SUMR ) ) ! CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, B( L1, L1 ), & LDB, ONE, ONE, VEC, 2, -SGN*A( K1, K1 ), & ZERO, X, 2, SCALOC, XNORM, IERR ) IF( IERR.NE.0 ) & INFO = 1 ! IF( SCALOC.NE.ONE ) THEN DO 210 J = 1, N CALL DSCAL( M, SCALOC, C( 1, J ), 1 ) 210 CONTINUE SCALE = SCALE*SCALOC END IF C( K1, L1 ) = X( 1, 1 ) C( K1, L2 ) = X( 2, 1 ) ! ELSE IF( L1.NE.L2 .AND. K1.NE.K2 ) THEN ! SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA, & C( MIN( K2+1, M ), L1 ), 1 ) SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC, & B( L1, MIN( L2+1, N ) ), LDB ) VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR ) ! SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA, & C( MIN( K2+1, M ), L2 ), 1 ) SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC, & B( L2, MIN( L2+1, N ) ), LDB ) VEC( 1, 2 ) = C( K1, L2 ) - ( SUML+SGN*SUMR ) ! SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA, & C( MIN( K2+1, M ), L1 ), 1 ) SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC, & B( L1, MIN( L2+1, N ) ), LDB ) VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR ) ! SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA, & C( MIN( K2+1, M ), L2 ), 1 ) SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC, & B( L2, MIN( L2+1, N ) ), LDB ) VEC( 2, 2 ) = C( K2, L2 ) - ( SUML+SGN*SUMR ) ! CALL DLASY2( .FALSE., .TRUE., ISGN, 2, 2, A( K1, K1 ), & LDA, B( L1, L1 ), LDB, VEC, 2, SCALOC, X, & 2, XNORM, IERR ) IF( IERR.NE.0 ) & INFO = 1 ! IF( SCALOC.NE.ONE ) THEN DO 220 J = 1, N CALL DSCAL( M, SCALOC, C( 1, J ), 1 ) 220 CONTINUE SCALE = SCALE*SCALOC END IF C( K1, L1 ) = X( 1, 1 ) C( K1, L2 ) = X( 1, 2 ) C( K2, L1 ) = X( 2, 1 ) C( K2, L2 ) = X( 2, 2 ) END IF ! 230 CONTINUE 240 CONTINUE ! END IF ! RETURN ! ! End of DTRSYL ! END SUBROUTINE DTRTI2( UPLO, DIAG, N, A, LDA, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! February 29, 1992 ! ! .. Scalar Arguments .. CHARACTER DIAG, UPLO INTEGER INFO, LDA, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ) ! .. ! ! Purpose ! ======= ! ! DTRTI2 computes the inverse of a real upper or lower triangular ! matrix. ! ! This is the Level 2 BLAS version of the algorithm. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the matrix A is upper or lower triangular. ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! DIAG (input) CHARACTER*1 ! Specifies whether or not the matrix A is unit triangular. ! = 'N': Non-unit triangular ! = 'U': Unit triangular ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the triangular matrix A. If UPLO = 'U', the ! leading n by n upper triangular part of the array A contains ! the upper triangular matrix, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading n by n lower triangular part of the array A contains ! the lower triangular matrix, and the strictly upper ! triangular part of A is not referenced. If DIAG = 'U', the ! diagonal elements of A are also not referenced and are ! assumed to be 1. ! ! On exit, the (triangular) inverse of the original matrix, in ! the same storage format. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -k, the k-th argument had an illegal value ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOUNIT, UPPER INTEGER J DOUBLE PRECISION AJJ ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DSCAL, DTRMV, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) NOUNIT = LSAME( DIAG, 'N' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTRTI2', -INFO ) RETURN END IF ! IF( UPPER ) THEN ! ! Compute inverse of upper triangular matrix. ! DO 10 J = 1, N IF( NOUNIT ) THEN A( J, J ) = ONE / A( J, J ) AJJ = -A( J, J ) ELSE AJJ = -ONE END IF ! ! Compute elements 1:j-1 of j-th column. ! CALL DTRMV( 'Upper', 'No transpose', DIAG, J-1, A, LDA, & A( 1, J ), 1 ) CALL DSCAL( J-1, AJJ, A( 1, J ), 1 ) 10 CONTINUE ELSE ! ! Compute inverse of lower triangular matrix. ! DO 20 J = N, 1, -1 IF( NOUNIT ) THEN A( J, J ) = ONE / A( J, J ) AJJ = -A( J, J ) ELSE AJJ = -ONE END IF IF( J.LT.N ) THEN ! ! Compute elements j+1:n of j-th column. ! CALL DTRMV( 'Lower', 'No transpose', DIAG, N-J, & A( J+1, J+1 ), LDA, A( J+1, J ), 1 ) CALL DSCAL( N-J, AJJ, A( J+1, J ), 1 ) END IF 20 CONTINUE END IF ! RETURN ! ! End of DTRTI2 ! END SUBROUTINE DTRTRI( UPLO, DIAG, N, A, LDA, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER DIAG, UPLO INTEGER INFO, LDA, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ) ! .. ! ! Purpose ! ======= ! ! DTRTRI computes the inverse of a real upper or lower triangular ! matrix A. ! ! This is the Level 3 BLAS version of the algorithm. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': A is upper triangular; ! = 'L': A is lower triangular. ! ! DIAG (input) CHARACTER*1 ! = 'N': A is non-unit triangular; ! = 'U': A is unit triangular. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the triangular matrix A. If UPLO = 'U', the ! leading N-by-N upper triangular part of the array A contains ! the upper triangular matrix, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading N-by-N lower triangular part of the array A contains ! the lower triangular matrix, and the strictly upper ! triangular part of A is not referenced. If DIAG = 'U', the ! diagonal elements of A are also not referenced and are ! assumed to be 1. ! On exit, the (triangular) inverse of the original matrix, in ! the same storage format. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, A(i,i) is exactly zero. The triangular ! matrix is singular and its inverse can not be computed. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOUNIT, UPPER INTEGER J, JB, NB, NN ! .. ! .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV ! .. ! .. External Subroutines .. EXTERNAL DTRMM, DTRSM, DTRTI2, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) NOUNIT = LSAME( DIAG, 'N' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTRTRI', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Check for singularity if non-unit. ! IF( NOUNIT ) THEN DO 10 INFO = 1, N IF( A( INFO, INFO ).EQ.ZERO ) & RETURN 10 CONTINUE INFO = 0 END IF ! ! Determine the block size for this environment. ! NB = ILAENV( 1, 'DTRTRI', UPLO // DIAG, N, -1, -1, -1 ) IF( NB.LE.1 .OR. NB.GE.N ) THEN ! ! Use unblocked code ! CALL DTRTI2( UPLO, DIAG, N, A, LDA, INFO ) ELSE ! ! Use blocked code ! IF( UPPER ) THEN ! ! Compute inverse of upper triangular matrix ! DO 20 J = 1, N, NB JB = MIN( NB, N-J+1 ) ! ! Compute rows 1:j-1 of current block column ! CALL DTRMM( 'Left', 'Upper', 'No transpose', DIAG, J-1, & JB, ONE, A, LDA, A( 1, J ), LDA ) CALL DTRSM( 'Right', 'Upper', 'No transpose', DIAG, J-1, & JB, -ONE, A( J, J ), LDA, A( 1, J ), LDA ) ! ! Compute inverse of current diagonal block ! CALL DTRTI2( 'Upper', DIAG, JB, A( J, J ), LDA, INFO ) 20 CONTINUE ELSE ! ! Compute inverse of lower triangular matrix ! NN = ( ( N-1 ) / NB )*NB + 1 DO 30 J = NN, 1, -NB JB = MIN( NB, N-J+1 ) IF( J+JB.LE.N ) THEN ! ! Compute rows j+jb:n of current block column ! CALL DTRMM( 'Left', 'Lower', 'No transpose', DIAG, & N-J-JB+1, JB, ONE, A( J+JB, J+JB ), LDA, & A( J+JB, J ), LDA ) CALL DTRSM( 'Right', 'Lower', 'No transpose', DIAG, & N-J-JB+1, JB, -ONE, A( J, J ), LDA, & A( J+JB, J ), LDA ) END IF ! ! Compute inverse of current diagonal block ! CALL DTRTI2( 'Lower', DIAG, JB, A( J, J ), LDA, INFO ) 30 CONTINUE END IF END IF ! RETURN ! ! End of DTRTRI ! END SUBROUTINE DTRTRS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, & INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! March 31, 1993 ! ! .. Scalar Arguments .. CHARACTER DIAG, TRANS, UPLO INTEGER INFO, LDA, LDB, N, NRHS ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DTRTRS solves a triangular system of the form ! ! A * X = B or A**T * X = B, ! ! where A is a triangular matrix of order N, and B is an N-by-NRHS ! matrix. A check is made to verify that A is nonsingular. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': A is upper triangular; ! = 'L': A is lower triangular. ! ! TRANS (input) CHARACTER*1 ! Specifies the form of the system of equations: ! = 'N': A * X = B (No transpose) ! = 'T': A**T * X = B (Transpose) ! = 'C': A**H * X = B (Conjugate transpose = Transpose) ! ! DIAG (input) CHARACTER*1 ! = 'N': A is non-unit triangular; ! = 'U': A is unit triangular. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! NRHS (input) INTEGER ! The number of right hand sides, i.e., the number of columns ! of the matrix B. NRHS >= 0. ! ! A (input) DOUBLE PRECISION array, dimension (LDA,N) ! The triangular matrix A. If UPLO = 'U', the leading N-by-N ! upper triangular part of the array A contains the upper ! triangular matrix, and the strictly lower triangular part of ! A is not referenced. If UPLO = 'L', the leading N-by-N lower ! triangular part of the array A contains the lower triangular ! matrix, and the strictly upper triangular part of A is not ! referenced. If DIAG = 'U', the diagonal elements of A are ! also not referenced and are assumed to be 1. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) ! On entry, the right hand side matrix B. ! On exit, if INFO = 0, the solution matrix X. ! ! LDB (input) INTEGER ! The leading dimension of the array B. LDB >= max(1,N). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! > 0: if INFO = i, the i-th diagonal element of A is zero, ! indicating that the matrix is singular and the solutions ! X have not been computed. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL NOUNIT ! .. ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. ! .. External Subroutines .. EXTERNAL DTRSM, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 NOUNIT = LSAME( DIAG, 'N' ) IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( .NOT.LSAME( TRANS, 'N' ) .AND. .NOT. & LSAME( TRANS, 'T' ) .AND. .NOT.LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( NRHS.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -9 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTRTRS', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.EQ.0 ) & RETURN ! ! Check for singularity. ! IF( NOUNIT ) THEN DO 10 INFO = 1, N IF( A( INFO, INFO ).EQ.ZERO ) & RETURN 10 CONTINUE END IF INFO = 0 ! ! Solve A * x = b or A' * x = b. ! CALL DTRSM( 'Left', UPLO, TRANS, DIAG, N, NRHS, ONE, A, LDA, B, & LDB ) ! RETURN ! ! End of DTRTRS ! END SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ) ! .. ! ! Purpose ! ======= ! ! This routine is deprecated and has been replaced by routine DTZRZF. ! ! DTZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A ! to upper triangular form by means of orthogonal transformations. ! ! The upper trapezoidal matrix A is factored as ! ! A = ( R 0 ) * Z, ! ! where Z is an N-by-N orthogonal matrix and R is an M-by-M upper ! triangular matrix. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= M. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the leading M-by-N upper trapezoidal part of the ! array A must contain the matrix to be factorized. ! On exit, the leading M-by-M upper triangular part of A ! contains the upper triangular matrix R, and elements M+1 to ! N of the first M rows of A, with the array TAU, represent the ! orthogonal matrix Z as a product of M elementary reflectors. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! TAU (output) DOUBLE PRECISION array, dimension (M) ! The scalar factors of the elementary reflectors. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! The factorization is obtained by Householder's method. The kth ! transformation matrix, Z( k ), which is used to introduce zeros into ! the ( m - k + 1 )th row of A, is given in the form ! ! Z( k ) = ( I 0 ), ! ( 0 T( k ) ) ! ! where ! ! T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ), ! ( 0 ) ! ( z( k ) ) ! ! tau is a scalar and z( k ) is an ( n - m ) element vector. ! tau and z( k ) are chosen to annihilate the elements of the kth row ! of X. ! ! The scalar tau is returned in the kth element of TAU and the vector ! u( k ) in the kth row of A, such that the elements of z( k ) are ! in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in ! the upper triangular part of A. ! ! Z is given by ! ! Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, K, M1 ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DGEMV, DGER, DLARFG, XERBLA ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.M ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTZRQF', -INFO ) RETURN END IF ! ! Perform the factorization. ! IF( M.EQ.0 ) & RETURN IF( M.EQ.N ) THEN DO 10 I = 1, N TAU( I ) = ZERO 10 CONTINUE ELSE M1 = MIN( M+1, N ) DO 20 K = M, 1, -1 ! ! Use a Householder reflection to zero the kth row of A. ! First set up the reflection. ! CALL DLARFG( N-M+1, A( K, K ), A( K, M1 ), LDA, TAU( K ) ) ! IF( ( TAU( K ).NE.ZERO ) .AND. ( K.GT.1 ) ) THEN ! ! We now perform the operation A := A*P( k ). ! ! Use the first ( k - 1 ) elements of TAU to store a( k ), ! where a( k ) consists of the first ( k - 1 ) elements of ! the kth column of A. Also let B denote the first ! ( k - 1 ) rows of the last ( n - m ) columns of A. ! CALL DCOPY( K-1, A( 1, K ), 1, TAU, 1 ) ! ! Form w = a( k ) + B*z( k ) in TAU. ! CALL DGEMV( 'No transpose', K-1, N-M, ONE, A( 1, M1 ), & LDA, A( K, M1 ), LDA, ONE, TAU, 1 ) ! ! Now form a( k ) := a( k ) - tau*w ! and B := B - tau*w*z( k )'. ! CALL DAXPY( K-1, -TAU( K ), TAU, 1, A( 1, K ), 1 ) CALL DGER( K-1, N-M, -TAU( K ), TAU, 1, A( K, M1 ), LDA, & A( 1, M1 ), LDA ) END IF 20 CONTINUE END IF ! RETURN ! ! End of DTZRQF ! END SUBROUTINE DTZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. INTEGER INFO, LDA, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) ! .. ! ! Purpose ! ======= ! ! DTZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A ! to upper triangular form by means of orthogonal transformations. ! ! The upper trapezoidal matrix A is factored as ! ! A = ( R 0 ) * Z, ! ! where Z is an N-by-N orthogonal matrix and R is an M-by-M upper ! triangular matrix. ! ! Arguments ! ========= ! ! M (input) INTEGER ! The number of rows of the matrix A. M >= 0. ! ! N (input) INTEGER ! The number of columns of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the leading M-by-N upper trapezoidal part of the ! array A must contain the matrix to be factorized. ! On exit, the leading M-by-M upper triangular part of A ! contains the upper triangular matrix R, and elements M+1 to ! N of the first M rows of A, with the array TAU, represent the ! orthogonal matrix Z as a product of M elementary reflectors. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,M). ! ! TAU (output) DOUBLE PRECISION array, dimension (M) ! The scalar factors of the elementary reflectors. ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) INTEGER ! The dimension of the array WORK. LWORK >= max(1,M). ! For optimum performance LWORK >= M*NB, where NB is ! the optimal blocksize. ! ! If LWORK = -1, then a workspace query is assumed; the routine ! only calculates the optimal size of the WORK array, returns ! this value as the first entry of the WORK array, and no error ! message related to LWORK is issued by XERBLA. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! Based on contributions by ! A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA ! ! The factorization is obtained by Householder's method. The kth ! transformation matrix, Z( k ), which is used to introduce zeros into ! the ( m - k + 1 )th row of A, is given in the form ! ! Z( k ) = ( I 0 ), ! ( 0 T( k ) ) ! ! where ! ! T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ), ! ( 0 ) ! ( z( k ) ) ! ! tau is a scalar and z( k ) is an ( n - m ) element vector. ! tau and z( k ) are chosen to annihilate the elements of the kth row ! of X. ! ! The scalar tau is returned in the kth element of TAU and the vector ! u( k ) in the kth row of A, such that the elements of z( k ) are ! in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in ! the upper triangular part of A. ! ! Z is given by ! ! Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, IB, IWS, KI, KK, LDWORK, LWKOPT, M1, MU, NB, & NBMIN, NX ! .. ! .. External Subroutines .. EXTERNAL DLARZB, DLARZT, DLATRZ, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Executable Statements .. ! ! Test the input arguments ! INFO = 0 LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.M ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 ELSE IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN INFO = -7 END IF ! IF( INFO.EQ.0 ) THEN ! ! Determine the block size. ! NB = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 ) LWKOPT = M*NB WORK( 1 ) = LWKOPT END IF ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTZRZF', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! IF( M.EQ.0 ) THEN WORK( 1 ) = 1 RETURN ELSE IF( M.EQ.N ) THEN DO 10 I = 1, N TAU( I ) = ZERO 10 CONTINUE WORK( 1 ) = 1 RETURN END IF ! NBMIN = 2 NX = 1 IWS = M IF( NB.GT.1 .AND. NB.LT.M ) THEN ! ! Determine when to cross over from blocked to unblocked code. ! NX = MAX( 0, ILAENV( 3, 'DGERQF', ' ', M, N, -1, -1 ) ) IF( NX.LT.M ) THEN ! ! Determine if workspace is large enough for blocked code. ! LDWORK = M IWS = LDWORK*NB IF( LWORK.LT.IWS ) THEN ! ! Not enough workspace to use optimal NB: reduce NB and ! determine the minimum value of NB. ! NB = LWORK / LDWORK NBMIN = MAX( 2, ILAENV( 2, 'DGERQF', ' ', M, N, -1, & -1 ) ) END IF END IF END IF ! IF( NB.GE.NBMIN .AND. NB.LT.M .AND. NX.LT.M ) THEN ! ! Use blocked code initially. ! The last kk rows are handled by the block method. ! M1 = MIN( M+1, N ) KI = ( ( M-NX-1 ) / NB )*NB KK = MIN( M, KI+NB ) ! DO 20 I = M - KK + KI + 1, M - KK + 1, -NB IB = MIN( M-I+1, NB ) ! ! Compute the TZ factorization of the current block ! A(i:i+ib-1,i:n) ! CALL DLATRZ( IB, N-I+1, N-M, A( I, I ), LDA, TAU( I ), & WORK ) IF( I.GT.1 ) THEN ! ! Form the triangular factor of the block reflector ! H = H(i+ib-1) . . . H(i+1) H(i) ! CALL DLARZT( 'Backward', 'Rowwise', N-M, IB, A( I, M1 ), & LDA, TAU( I ), WORK, LDWORK ) ! ! Apply H to A(1:i-1,i:n) from the right ! CALL DLARZB( 'Right', 'No transpose', 'Backward', & 'Rowwise', I-1, N-I+1, IB, N-M, A( I, M1 ), & LDA, WORK, LDWORK, A( 1, I ), LDA, & WORK( IB+1 ), LDWORK ) END IF 20 CONTINUE MU = I + NB - 1 ELSE MU = M END IF ! ! Use unblocked code to factor the last or only block ! IF( MU.GT.0 ) & CALL DLATRZ( MU, N, N-M, A, LDA, TAU, WORK ) ! WORK( 1 ) = LWKOPT ! RETURN ! ! End of DTZRZF ! END DOUBLE PRECISION FUNCTION DZSUM1( N, CX, INCX ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! October 31, 1992 ! ! .. Scalar Arguments .. INTEGER INCX, N ! .. ! .. Array Arguments .. COMPLEX*16 CX( * ) ! .. ! ! Purpose ! ======= ! ! DZSUM1 takes the sum of the absolute values of a complex ! vector and returns a double precision result. ! ! Based on DZASUM from the Level 1 BLAS. ! The change is to use the 'genuine' absolute value. ! ! Contributed by Nick Higham for use with ZLACON. ! ! Arguments ! ========= ! ! N (input) INTEGER ! The number of elements in the vector CX. ! ! CX (input) COMPLEX*16 array, dimension (N) ! The vector whose elements will be summed. ! ! INCX (input) INTEGER ! The spacing between successive values of CX. INCX > 0. ! ! ===================================================================== ! ! .. Local Scalars .. INTEGER I, NINCX DOUBLE PRECISION STEMP ! .. ! .. Intrinsic Functions .. INTRINSIC ABS ! .. ! .. Executable Statements .. ! DZSUM1 = 0.0D0 STEMP = 0.0D0 IF( N.LE.0 ) & RETURN IF( INCX.EQ.1 ) & GO TO 20 ! ! CODE FOR INCREMENT NOT EQUAL TO 1 ! NINCX = N*INCX DO 10 I = 1, NINCX, INCX ! ! NEXT LINE MODIFIED. ! STEMP = STEMP + ABS( CX( I ) ) 10 CONTINUE DZSUM1 = STEMP RETURN ! ! CODE FOR INCREMENT EQUAL TO 1 ! 20 CONTINUE DO 30 I = 1, N ! ! NEXT LINE MODIFIED. ! STEMP = STEMP + ABS( CX( I ) ) 30 CONTINUE DZSUM1 = STEMP RETURN ! ! End of DZSUM1 ! END LOGICAL FUNCTION LSAME( CA, CB ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER CA, CB ! .. ! ! Purpose ! ======= ! ! LSAME returns .TRUE. if CA is the same letter as CB regardless of ! case. ! ! Arguments ! ========= ! ! CA (input) CHARACTER*1 ! CB (input) CHARACTER*1 ! CA and CB specify the single characters to be compared. ! ! ===================================================================== ! ! .. Intrinsic Functions .. INTRINSIC ICHAR ! .. ! .. Local Scalars .. INTEGER INTA, INTB, ZCODE ! .. ! .. Executable Statements .. ! ! Test if the characters are equal ! LSAME = CA.EQ.CB IF( LSAME ) & RETURN ! ! Now test for equivalence if both characters are alphabetic. ! ZCODE = ICHAR( 'Z' ) ! ! Use 'Z' rather than 'A' so that ASCII can be detected on Prime ! machines, on which ICHAR returns a value with bit 8 set. ! ICHAR('A') on Prime machines returns 193 which is the same as ! ICHAR('A') on an EBCDIC machine. ! INTA = ICHAR( CA ) INTB = ICHAR( CB ) ! IF( ZCODE.EQ.90 .OR. ZCODE.EQ.122 ) THEN ! ! ASCII is assumed - ZCODE is the ASCII code of either lower or ! upper case 'Z'. ! IF( INTA.GE.97 .AND. INTA.LE.122 ) INTA = INTA - 32 IF( INTB.GE.97 .AND. INTB.LE.122 ) INTB = INTB - 32 ! ELSE IF( ZCODE.EQ.233 .OR. ZCODE.EQ.169 ) THEN ! ! EBCDIC is assumed - ZCODE is the EBCDIC code of either lower or ! upper case 'Z'. ! IF( INTA.GE.129 .AND. INTA.LE.137 .OR. & INTA.GE.145 .AND. INTA.LE.153 .OR. & INTA.GE.162 .AND. INTA.LE.169 ) INTA = INTA + 64 IF( INTB.GE.129 .AND. INTB.LE.137 .OR. & INTB.GE.145 .AND. INTB.LE.153 .OR. & INTB.GE.162 .AND. INTB.LE.169 ) INTB = INTB + 64 ! ELSE IF( ZCODE.EQ.218 .OR. ZCODE.EQ.250 ) THEN ! ! ASCII is assumed, on Prime machines - ZCODE is the ASCII code ! plus 128 of either lower or upper case 'Z'. ! IF( INTA.GE.225 .AND. INTA.LE.250 ) INTA = INTA - 32 IF( INTB.GE.225 .AND. INTB.LE.250 ) INTB = INTB - 32 END IF LSAME = INTA.EQ.INTB ! ! RETURN ! ! End of LSAME ! END double precision function dasum(n,dx,incx) ! ! takes the sum of the absolute values. ! jack dongarra, linpack, 3/11/78. ! modified 3/93 to return if incx .le. 0. ! modified 12/3/93, array(1) declarations changed to array(*) ! double precision dx(*),dtemp integer i,incx,m,mp1,n,nincx ! dasum = 0.0d0 dtemp = 0.0d0 if( n.le.0 .or. incx.le.0 )return if(incx.eq.1)go to 20 ! ! code for increment not equal to 1 ! nincx = n*incx do 10 i = 1,nincx,incx dtemp = dtemp + dabs(dx(i)) 10 continue dasum = dtemp return ! ! code for increment equal to 1 ! ! ! clean-up loop ! 20 m = mod(n,6) if( m .eq. 0 ) go to 40 do 30 i = 1,m dtemp = dtemp + dabs(dx(i)) 30 continue if( n .lt. 6 ) go to 60 40 mp1 = m + 1 do 50 i = mp1,n,6 dtemp = dtemp + dabs(dx(i)) + dabs(dx(i + 1)) + dabs(dx(i + 2)) & + dabs(dx(i + 3)) + dabs(dx(i + 4)) + dabs(dx(i + 5)) 50 continue 60 dasum = dtemp return end subroutine daxpy(n,da,dx,incx,dy,incy) ! ! constant times a vector plus a vector. ! uses unrolled loops for increments equal to one. ! jack dongarra, linpack, 3/11/78. ! modified 12/3/93, array(1) declarations changed to array(*) ! double precision dx(*),dy(*),da integer i,incx,incy,ix,iy,m,mp1,n ! if(n.le.0)return if (da .eq. 0.0d0) return if(incx.eq.1.and.incy.eq.1)go to 20 ! ! code for unequal increments or equal increments ! not equal to 1 ! ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n dy(iy) = dy(iy) + da*dx(ix) ix = ix + incx iy = iy + incy 10 continue return ! ! code for both increments equal to 1 ! ! ! clean-up loop ! 20 m = mod(n,4) if( m .eq. 0 ) go to 40 do 30 i = 1,m dy(i) = dy(i) + da*dx(i) 30 continue if( n .lt. 4 ) return 40 mp1 = m + 1 do 50 i = mp1,n,4 dy(i) = dy(i) + da*dx(i) dy(i + 1) = dy(i + 1) + da*dx(i + 1) dy(i + 2) = dy(i + 2) + da*dx(i + 2) dy(i + 3) = dy(i + 3) + da*dx(i + 3) 50 continue return end double precision function dcabs1(z) double complex z,zz double precision t(2) equivalence (zz,t(1)) zz = z dcabs1 = dabs(t(1)) + dabs(t(2)) return end subroutine dcopy(n,dx,incx,dy,incy) ! ! copies a vector, x, to a vector, y. ! uses unrolled loops for increments equal to one. ! jack dongarra, linpack, 3/11/78. ! modified 12/3/93, array(1) declarations changed to array(*) ! double precision dx(*),dy(*) integer i,incx,incy,ix,iy,m,mp1,n ! if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 ! ! code for unequal increments or equal increments ! not equal to 1 ! ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n dy(iy) = dx(ix) ix = ix + incx iy = iy + incy 10 continue return ! ! code for both increments equal to 1 ! ! ! clean-up loop ! 20 m = mod(n,7) if( m .eq. 0 ) go to 40 do 30 i = 1,m dy(i) = dx(i) 30 continue if( n .lt. 7 ) return 40 mp1 = m + 1 do 50 i = mp1,n,7 dy(i) = dx(i) dy(i + 1) = dx(i + 1) dy(i + 2) = dx(i + 2) dy(i + 3) = dx(i + 3) dy(i + 4) = dx(i + 4) dy(i + 5) = dx(i + 5) dy(i + 6) = dx(i + 6) 50 continue return end double precision function ddot(n,dx,incx,dy,incy) ! ! forms the dot product of two vectors. ! uses unrolled loops for increments equal to one. ! jack dongarra, linpack, 3/11/78. ! modified 12/3/93, array(1) declarations changed to array(*) ! double precision dx(*),dy(*),dtemp integer i,incx,incy,ix,iy,m,mp1,n ! ddot = 0.0d0 dtemp = 0.0d0 if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 ! ! code for unequal increments or equal increments ! not equal to 1 ! ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n dtemp = dtemp + dx(ix)*dy(iy) ix = ix + incx iy = iy + incy 10 continue ddot = dtemp return ! ! code for both increments equal to 1 ! ! ! clean-up loop ! 20 m = mod(n,5) if( m .eq. 0 ) go to 40 do 30 i = 1,m dtemp = dtemp + dx(i)*dy(i) 30 continue if( n .lt. 5 ) go to 60 40 mp1 = m + 1 do 50 i = mp1,n,5 dtemp = dtemp + dx(i)*dy(i) + dx(i + 1)*dy(i + 1) + & dx(i + 2)*dy(i + 2) + dx(i + 3)*dy(i + 3) + dx(i + 4)*dy(i + 4) 50 continue 60 ddot = dtemp return end SUBROUTINE DGBMV ( TRANS, M, N, KL, KU, ALPHA, A, LDA, X, INCX, & BETA, Y, INCY ) ! .. Scalar Arguments .. DOUBLE PRECISION ALPHA, BETA INTEGER INCX, INCY, KL, KU, LDA, M, N CHARACTER*1 TRANS ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), X( * ), Y( * ) ! .. ! ! Purpose ! ======= ! ! DGBMV performs one of the matrix-vector operations ! ! y := alpha*A*x + beta*y, or y := alpha*A'*x + beta*y, ! ! where alpha and beta are scalars, x and y are vectors and A is an ! m by n band matrix, with kl sub-diagonals and ku super-diagonals. ! ! Parameters ! ========== ! ! TRANS - CHARACTER*1. ! On entry, TRANS specifies the operation to be performed as ! follows: ! ! TRANS = 'N' or 'n' y := alpha*A*x + beta*y. ! ! TRANS = 'T' or 't' y := alpha*A'*x + beta*y. ! ! TRANS = 'C' or 'c' y := alpha*A'*x + beta*y. ! ! Unchanged on exit. ! ! M - INTEGER. ! On entry, M specifies the number of rows of the matrix A. ! M must be at least zero. ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the number of columns of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! KL - INTEGER. ! On entry, KL specifies the number of sub-diagonals of the ! matrix A. KL must satisfy 0 .le. KL. ! Unchanged on exit. ! ! KU - INTEGER. ! On entry, KU specifies the number of super-diagonals of the ! matrix A. KU must satisfy 0 .le. KU. ! Unchanged on exit. ! ! ALPHA - DOUBLE PRECISION. ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, n ). ! Before entry, the leading ( kl + ku + 1 ) by n part of the ! array A must contain the matrix of coefficients, supplied ! column by column, with the leading diagonal of the matrix in ! row ( ku + 1 ) of the array, the first super-diagonal ! starting at position 2 in row ku, the first sub-diagonal ! starting at position 1 in row ( ku + 2 ), and so on. ! Elements in the array A that do not correspond to elements ! in the band matrix (such as the top left ku by ku triangle) ! are not referenced. ! The following program segment will transfer a band matrix ! from conventional full matrix storage to band storage: ! ! DO 20, J = 1, N ! K = KU + 1 - J ! DO 10, I = MAX( 1, J - KU ), MIN( M, J + KL ) ! A( K + I, J ) = matrix( I, J ) ! 10 CONTINUE ! 20 CONTINUE ! ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. LDA must be at least ! ( kl + ku + 1 ). ! Unchanged on exit. ! ! X - DOUBLE PRECISION array of DIMENSION at least ! ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n' ! and at least ! ( 1 + ( m - 1 )*abs( INCX ) ) otherwise. ! Before entry, the incremented array X must contain the ! vector x. ! Unchanged on exit. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! BETA - DOUBLE PRECISION. ! On entry, BETA specifies the scalar beta. When BETA is ! supplied as zero then Y need not be set on input. ! Unchanged on exit. ! ! Y - DOUBLE PRECISION array of DIMENSION at least ! ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n' ! and at least ! ( 1 + ( n - 1 )*abs( INCY ) ) otherwise. ! Before entry, the incremented array Y must contain the ! vector y. On exit, Y is overwritten by the updated vector y. ! ! INCY - INTEGER. ! On entry, INCY specifies the increment for the elements of ! Y. INCY must not be zero. ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. Local Scalars .. DOUBLE PRECISION TEMP INTEGER I, INFO, IX, IY, J, JX, JY, K, KUP1, KX, KY, & LENX, LENY ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( TRANS, 'N' ).AND. & .NOT.LSAME( TRANS, 'T' ).AND. & .NOT.LSAME( TRANS, 'C' ) )THEN INFO = 1 ELSE IF( M.LT.0 )THEN INFO = 2 ELSE IF( N.LT.0 )THEN INFO = 3 ELSE IF( KL.LT.0 )THEN INFO = 4 ELSE IF( KU.LT.0 )THEN INFO = 5 ELSE IF( LDA.LT.( KL + KU + 1 ) )THEN INFO = 8 ELSE IF( INCX.EQ.0 )THEN INFO = 10 ELSE IF( INCY.EQ.0 )THEN INFO = 13 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DGBMV ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR. & ( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) ) & RETURN ! ! Set LENX and LENY, the lengths of the vectors x and y, and set ! up the start points in X and Y. ! IF( LSAME( TRANS, 'N' ) )THEN LENX = N LENY = M ELSE LENX = M LENY = N END IF IF( INCX.GT.0 )THEN KX = 1 ELSE KX = 1 - ( LENX - 1 )*INCX END IF IF( INCY.GT.0 )THEN KY = 1 ELSE KY = 1 - ( LENY - 1 )*INCY END IF ! ! Start the operations. In this version the elements of A are ! accessed sequentially with one pass through the band part of A. ! ! First form y := beta*y. ! IF( BETA.NE.ONE )THEN IF( INCY.EQ.1 )THEN IF( BETA.EQ.ZERO )THEN DO 10, I = 1, LENY Y( I ) = ZERO 10 CONTINUE ELSE DO 20, I = 1, LENY Y( I ) = BETA*Y( I ) 20 CONTINUE END IF ELSE IY = KY IF( BETA.EQ.ZERO )THEN DO 30, I = 1, LENY Y( IY ) = ZERO IY = IY + INCY 30 CONTINUE ELSE DO 40, I = 1, LENY Y( IY ) = BETA*Y( IY ) IY = IY + INCY 40 CONTINUE END IF END IF END IF IF( ALPHA.EQ.ZERO ) & RETURN KUP1 = KU + 1 IF( LSAME( TRANS, 'N' ) )THEN ! ! Form y := alpha*A*x + y. ! JX = KX IF( INCY.EQ.1 )THEN DO 60, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = ALPHA*X( JX ) K = KUP1 - J DO 50, I = MAX( 1, J - KU ), MIN( M, J + KL ) Y( I ) = Y( I ) + TEMP*A( K + I, J ) 50 CONTINUE END IF JX = JX + INCX 60 CONTINUE ELSE DO 80, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = ALPHA*X( JX ) IY = KY K = KUP1 - J DO 70, I = MAX( 1, J - KU ), MIN( M, J + KL ) Y( IY ) = Y( IY ) + TEMP*A( K + I, J ) IY = IY + INCY 70 CONTINUE END IF JX = JX + INCX IF( J.GT.KU ) & KY = KY + INCY 80 CONTINUE END IF ELSE ! ! Form y := alpha*A'*x + y. ! JY = KY IF( INCX.EQ.1 )THEN DO 100, J = 1, N TEMP = ZERO K = KUP1 - J DO 90, I = MAX( 1, J - KU ), MIN( M, J + KL ) TEMP = TEMP + A( K + I, J )*X( I ) 90 CONTINUE Y( JY ) = Y( JY ) + ALPHA*TEMP JY = JY + INCY 100 CONTINUE ELSE DO 120, J = 1, N TEMP = ZERO IX = KX K = KUP1 - J DO 110, I = MAX( 1, J - KU ), MIN( M, J + KL ) TEMP = TEMP + A( K + I, J )*X( IX ) IX = IX + INCX 110 CONTINUE Y( JY ) = Y( JY ) + ALPHA*TEMP JY = JY + INCY IF( J.GT.KU ) & KX = KX + INCX 120 CONTINUE END IF END IF ! RETURN ! ! End of DGBMV . ! END SUBROUTINE DGEMM ( TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, & BETA, C, LDC ) ! .. Scalar Arguments .. CHARACTER*1 TRANSA, TRANSB INTEGER M, N, K, LDA, LDB, LDC DOUBLE PRECISION ALPHA, BETA ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ) ! .. ! ! Purpose ! ======= ! ! DGEMM performs one of the matrix-matrix operations ! ! C := alpha*op( A )*op( B ) + beta*C, ! ! where op( X ) is one of ! ! op( X ) = X or op( X ) = X', ! ! alpha and beta are scalars, and A, B and C are matrices, with op( A ) ! an m by k matrix, op( B ) a k by n matrix and C an m by n matrix. ! ! Parameters ! ========== ! ! TRANSA - CHARACTER*1. ! On entry, TRANSA specifies the form of op( A ) to be used in ! the matrix multiplication as follows: ! ! TRANSA = 'N' or 'n', op( A ) = A. ! ! TRANSA = 'T' or 't', op( A ) = A'. ! ! TRANSA = 'C' or 'c', op( A ) = A'. ! ! Unchanged on exit. ! ! TRANSB - CHARACTER*1. ! On entry, TRANSB specifies the form of op( B ) to be used in ! the matrix multiplication as follows: ! ! TRANSB = 'N' or 'n', op( B ) = B. ! ! TRANSB = 'T' or 't', op( B ) = B'. ! ! TRANSB = 'C' or 'c', op( B ) = B'. ! ! Unchanged on exit. ! ! M - INTEGER. ! On entry, M specifies the number of rows of the matrix ! op( A ) and of the matrix C. M must be at least zero. ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the number of columns of the matrix ! op( B ) and the number of columns of the matrix C. N must be ! at least zero. ! Unchanged on exit. ! ! K - INTEGER. ! On entry, K specifies the number of columns of the matrix ! op( A ) and the number of rows of the matrix op( B ). K must ! be at least zero. ! Unchanged on exit. ! ! ALPHA - DOUBLE PRECISION. ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, ka ), where ka is ! k when TRANSA = 'N' or 'n', and is m otherwise. ! Before entry with TRANSA = 'N' or 'n', the leading m by k ! part of the array A must contain the matrix A, otherwise ! the leading k by m part of the array A must contain the ! matrix A. ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. When TRANSA = 'N' or 'n' then ! LDA must be at least max( 1, m ), otherwise LDA must be at ! least max( 1, k ). ! Unchanged on exit. ! ! B - DOUBLE PRECISION array of DIMENSION ( LDB, kb ), where kb is ! n when TRANSB = 'N' or 'n', and is k otherwise. ! Before entry with TRANSB = 'N' or 'n', the leading k by n ! part of the array B must contain the matrix B, otherwise ! the leading n by k part of the array B must contain the ! matrix B. ! Unchanged on exit. ! ! LDB - INTEGER. ! On entry, LDB specifies the first dimension of B as declared ! in the calling (sub) program. When TRANSB = 'N' or 'n' then ! LDB must be at least max( 1, k ), otherwise LDB must be at ! least max( 1, n ). ! Unchanged on exit. ! ! BETA - DOUBLE PRECISION. ! On entry, BETA specifies the scalar beta. When BETA is ! supplied as zero then C need not be set on input. ! Unchanged on exit. ! ! C - DOUBLE PRECISION array of DIMENSION ( LDC, n ). ! Before entry, the leading m by n part of the array C must ! contain the matrix C, except when beta is zero, in which ! case C need not be set on entry. ! On exit, the array C is overwritten by the m by n matrix ! ( alpha*op( A )*op( B ) + beta*C ). ! ! LDC - INTEGER. ! On entry, LDC specifies the first dimension of C as declared ! in the calling (sub) program. LDC must be at least ! max( 1, m ). ! Unchanged on exit. ! ! ! Level 3 Blas routine. ! ! -- Written on 8-February-1989. ! Jack Dongarra, Argonne National Laboratory. ! Iain Duff, AERE Harwell. ! Jeremy Du Croz, Numerical Algorithms Group Ltd. ! Sven Hammarling, Numerical Algorithms Group Ltd. ! ! ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX ! .. Local Scalars .. LOGICAL NOTA, NOTB INTEGER I, INFO, J, L, NCOLA, NROWA, NROWB DOUBLE PRECISION TEMP ! .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Executable Statements .. ! ! Set NOTA and NOTB as true if A and B respectively are not ! transposed and set NROWA, NCOLA and NROWB as the number of rows ! and columns of A and the number of rows of B respectively. ! NOTA = LSAME( TRANSA, 'N' ) NOTB = LSAME( TRANSB, 'N' ) IF( NOTA )THEN NROWA = M NCOLA = K ELSE NROWA = K NCOLA = M END IF IF( NOTB )THEN NROWB = K ELSE NROWB = N END IF ! ! Test the input parameters. ! INFO = 0 IF( ( .NOT.NOTA ).AND. & ( .NOT.LSAME( TRANSA, 'C' ) ).AND. & ( .NOT.LSAME( TRANSA, 'T' ) ) )THEN INFO = 1 ELSE IF( ( .NOT.NOTB ).AND. & ( .NOT.LSAME( TRANSB, 'C' ) ).AND. & ( .NOT.LSAME( TRANSB, 'T' ) ) )THEN INFO = 2 ELSE IF( M .LT.0 )THEN INFO = 3 ELSE IF( N .LT.0 )THEN INFO = 4 ELSE IF( K .LT.0 )THEN INFO = 5 ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN INFO = 8 ELSE IF( LDB.LT.MAX( 1, NROWB ) )THEN INFO = 10 ELSE IF( LDC.LT.MAX( 1, M ) )THEN INFO = 13 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DGEMM ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR. & ( ( ( ALPHA.EQ.ZERO ).OR.( K.EQ.0 ) ).AND.( BETA.EQ.ONE ) ) ) & RETURN ! ! And if alpha.eq.zero. ! IF( ALPHA.EQ.ZERO )THEN IF( BETA.EQ.ZERO )THEN DO 20, J = 1, N DO 10, I = 1, M C( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ELSE DO 40, J = 1, N DO 30, I = 1, M C( I, J ) = BETA*C( I, J ) 30 CONTINUE 40 CONTINUE END IF RETURN END IF ! ! Start the operations. ! IF( NOTB )THEN IF( NOTA )THEN ! ! Form C := alpha*A*B + beta*C. ! DO 90, J = 1, N IF( BETA.EQ.ZERO )THEN DO 50, I = 1, M C( I, J ) = ZERO 50 CONTINUE ELSE IF( BETA.NE.ONE )THEN DO 60, I = 1, M C( I, J ) = BETA*C( I, J ) 60 CONTINUE END IF DO 80, L = 1, K IF( B( L, J ).NE.ZERO )THEN TEMP = ALPHA*B( L, J ) DO 70, I = 1, M C( I, J ) = C( I, J ) + TEMP*A( I, L ) 70 CONTINUE END IF 80 CONTINUE 90 CONTINUE ELSE ! ! Form C := alpha*A'*B + beta*C ! DO 120, J = 1, N DO 110, I = 1, M TEMP = ZERO DO 100, L = 1, K TEMP = TEMP + A( L, I )*B( L, J ) 100 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = ALPHA*TEMP ELSE C( I, J ) = ALPHA*TEMP + BETA*C( I, J ) END IF 110 CONTINUE 120 CONTINUE END IF ELSE IF( NOTA )THEN ! ! Form C := alpha*A*B' + beta*C ! DO 170, J = 1, N IF( BETA.EQ.ZERO )THEN DO 130, I = 1, M C( I, J ) = ZERO 130 CONTINUE ELSE IF( BETA.NE.ONE )THEN DO 140, I = 1, M C( I, J ) = BETA*C( I, J ) 140 CONTINUE END IF DO 160, L = 1, K IF( B( J, L ).NE.ZERO )THEN TEMP = ALPHA*B( J, L ) DO 150, I = 1, M C( I, J ) = C( I, J ) + TEMP*A( I, L ) 150 CONTINUE END IF 160 CONTINUE 170 CONTINUE ELSE ! ! Form C := alpha*A'*B' + beta*C ! DO 200, J = 1, N DO 190, I = 1, M TEMP = ZERO DO 180, L = 1, K TEMP = TEMP + A( L, I )*B( J, L ) 180 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = ALPHA*TEMP ELSE C( I, J ) = ALPHA*TEMP + BETA*C( I, J ) END IF 190 CONTINUE 200 CONTINUE END IF END IF ! RETURN ! ! End of DGEMM . ! END SUBROUTINE DGEMV ( TRANS, M, N, ALPHA, A, LDA, X, INCX, & BETA, Y, INCY ) ! .. Scalar Arguments .. DOUBLE PRECISION ALPHA, BETA INTEGER INCX, INCY, LDA, M, N CHARACTER*1 TRANS ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), X( * ), Y( * ) ! .. ! ! Purpose ! ======= ! ! DGEMV performs one of the matrix-vector operations ! ! y := alpha*A*x + beta*y, or y := alpha*A'*x + beta*y, ! ! where alpha and beta are scalars, x and y are vectors and A is an ! m by n matrix. ! ! Parameters ! ========== ! ! TRANS - CHARACTER*1. ! On entry, TRANS specifies the operation to be performed as ! follows: ! ! TRANS = 'N' or 'n' y := alpha*A*x + beta*y. ! ! TRANS = 'T' or 't' y := alpha*A'*x + beta*y. ! ! TRANS = 'C' or 'c' y := alpha*A'*x + beta*y. ! ! Unchanged on exit. ! ! M - INTEGER. ! On entry, M specifies the number of rows of the matrix A. ! M must be at least zero. ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the number of columns of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! ALPHA - DOUBLE PRECISION. ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, n ). ! Before entry, the leading m by n part of the array A must ! contain the matrix of coefficients. ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. LDA must be at least ! max( 1, m ). ! Unchanged on exit. ! ! X - DOUBLE PRECISION array of DIMENSION at least ! ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n' ! and at least ! ( 1 + ( m - 1 )*abs( INCX ) ) otherwise. ! Before entry, the incremented array X must contain the ! vector x. ! Unchanged on exit. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! BETA - DOUBLE PRECISION. ! On entry, BETA specifies the scalar beta. When BETA is ! supplied as zero then Y need not be set on input. ! Unchanged on exit. ! ! Y - DOUBLE PRECISION array of DIMENSION at least ! ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n' ! and at least ! ( 1 + ( n - 1 )*abs( INCY ) ) otherwise. ! Before entry with BETA non-zero, the incremented array Y ! must contain the vector y. On exit, Y is overwritten by the ! updated vector y. ! ! INCY - INTEGER. ! On entry, INCY specifies the increment for the elements of ! Y. INCY must not be zero. ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. Local Scalars .. DOUBLE PRECISION TEMP INTEGER I, INFO, IX, IY, J, JX, JY, KX, KY, LENX, LENY ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( TRANS, 'N' ).AND. & .NOT.LSAME( TRANS, 'T' ).AND. & .NOT.LSAME( TRANS, 'C' ) )THEN INFO = 1 ELSE IF( M.LT.0 )THEN INFO = 2 ELSE IF( N.LT.0 )THEN INFO = 3 ELSE IF( LDA.LT.MAX( 1, M ) )THEN INFO = 6 ELSE IF( INCX.EQ.0 )THEN INFO = 8 ELSE IF( INCY.EQ.0 )THEN INFO = 11 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DGEMV ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR. & ( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) ) & RETURN ! ! Set LENX and LENY, the lengths of the vectors x and y, and set ! up the start points in X and Y. ! IF( LSAME( TRANS, 'N' ) )THEN LENX = N LENY = M ELSE LENX = M LENY = N END IF IF( INCX.GT.0 )THEN KX = 1 ELSE KX = 1 - ( LENX - 1 )*INCX END IF IF( INCY.GT.0 )THEN KY = 1 ELSE KY = 1 - ( LENY - 1 )*INCY END IF ! ! Start the operations. In this version the elements of A are ! accessed sequentially with one pass through A. ! ! First form y := beta*y. ! IF( BETA.NE.ONE )THEN IF( INCY.EQ.1 )THEN IF( BETA.EQ.ZERO )THEN DO 10, I = 1, LENY Y( I ) = ZERO 10 CONTINUE ELSE DO 20, I = 1, LENY Y( I ) = BETA*Y( I ) 20 CONTINUE END IF ELSE IY = KY IF( BETA.EQ.ZERO )THEN DO 30, I = 1, LENY Y( IY ) = ZERO IY = IY + INCY 30 CONTINUE ELSE DO 40, I = 1, LENY Y( IY ) = BETA*Y( IY ) IY = IY + INCY 40 CONTINUE END IF END IF END IF IF( ALPHA.EQ.ZERO ) & RETURN IF( LSAME( TRANS, 'N' ) )THEN ! ! Form y := alpha*A*x + y. ! JX = KX IF( INCY.EQ.1 )THEN DO 60, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = ALPHA*X( JX ) DO 50, I = 1, M Y( I ) = Y( I ) + TEMP*A( I, J ) 50 CONTINUE END IF JX = JX + INCX 60 CONTINUE ELSE DO 80, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = ALPHA*X( JX ) IY = KY DO 70, I = 1, M Y( IY ) = Y( IY ) + TEMP*A( I, J ) IY = IY + INCY 70 CONTINUE END IF JX = JX + INCX 80 CONTINUE END IF ELSE ! ! Form y := alpha*A'*x + y. ! JY = KY IF( INCX.EQ.1 )THEN DO 100, J = 1, N TEMP = ZERO DO 90, I = 1, M TEMP = TEMP + A( I, J )*X( I ) 90 CONTINUE Y( JY ) = Y( JY ) + ALPHA*TEMP JY = JY + INCY 100 CONTINUE ELSE DO 120, J = 1, N TEMP = ZERO IX = KX DO 110, I = 1, M TEMP = TEMP + A( I, J )*X( IX ) IX = IX + INCX 110 CONTINUE Y( JY ) = Y( JY ) + ALPHA*TEMP JY = JY + INCY 120 CONTINUE END IF END IF ! RETURN ! ! End of DGEMV . ! END SUBROUTINE DGER ( M, N, ALPHA, X, INCX, Y, INCY, A, LDA ) ! .. Scalar Arguments .. DOUBLE PRECISION ALPHA INTEGER INCX, INCY, LDA, M, N ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), X( * ), Y( * ) ! .. ! ! Purpose ! ======= ! ! DGER performs the rank 1 operation ! ! A := alpha*x*y' + A, ! ! where alpha is a scalar, x is an m element vector, y is an n element ! vector and A is an m by n matrix. ! ! Parameters ! ========== ! ! M - INTEGER. ! On entry, M specifies the number of rows of the matrix A. ! M must be at least zero. ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the number of columns of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! ALPHA - DOUBLE PRECISION. ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! X - DOUBLE PRECISION array of dimension at least ! ( 1 + ( m - 1 )*abs( INCX ) ). ! Before entry, the incremented array X must contain the m ! element vector x. ! Unchanged on exit. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! Y - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCY ) ). ! Before entry, the incremented array Y must contain the n ! element vector y. ! Unchanged on exit. ! ! INCY - INTEGER. ! On entry, INCY specifies the increment for the elements of ! Y. INCY must not be zero. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, n ). ! Before entry, the leading m by n part of the array A must ! contain the matrix of coefficients. On exit, A is ! overwritten by the updated matrix. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. LDA must be at least ! max( 1, m ). ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. Local Scalars .. DOUBLE PRECISION TEMP INTEGER I, INFO, IX, J, JY, KX ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( M.LT.0 )THEN INFO = 1 ELSE IF( N.LT.0 )THEN INFO = 2 ELSE IF( INCX.EQ.0 )THEN INFO = 5 ELSE IF( INCY.EQ.0 )THEN INFO = 7 ELSE IF( LDA.LT.MAX( 1, M ) )THEN INFO = 9 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DGER ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR.( ALPHA.EQ.ZERO ) ) & RETURN ! ! Start the operations. In this version the elements of A are ! accessed sequentially with one pass through A. ! IF( INCY.GT.0 )THEN JY = 1 ELSE JY = 1 - ( N - 1 )*INCY END IF IF( INCX.EQ.1 )THEN DO 20, J = 1, N IF( Y( JY ).NE.ZERO )THEN TEMP = ALPHA*Y( JY ) DO 10, I = 1, M A( I, J ) = A( I, J ) + X( I )*TEMP 10 CONTINUE END IF JY = JY + INCY 20 CONTINUE ELSE IF( INCX.GT.0 )THEN KX = 1 ELSE KX = 1 - ( M - 1 )*INCX END IF DO 40, J = 1, N IF( Y( JY ).NE.ZERO )THEN TEMP = ALPHA*Y( JY ) IX = KX DO 30, I = 1, M A( I, J ) = A( I, J ) + X( IX )*TEMP IX = IX + INCX 30 CONTINUE END IF JY = JY + INCY 40 CONTINUE END IF ! RETURN ! ! End of DGER . ! END DOUBLE PRECISION FUNCTION DNRM2 ( N, X, INCX ) ! .. Scalar Arguments .. INTEGER INCX, N ! .. Array Arguments .. DOUBLE PRECISION X( * ) ! .. ! ! DNRM2 returns the euclidean norm of a vector via the function ! name, so that ! ! DNRM2 := sqrt( x'*x ) ! ! ! ! -- This version written on 25-October-1982. ! Modified on 14-October-1993 to inline the call to DLASSQ. ! Sven Hammarling, Nag Ltd. ! ! ! .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. Local Scalars .. INTEGER IX DOUBLE PRECISION ABSXI, NORM, SCALE, SSQ ! .. Intrinsic Functions .. INTRINSIC ABS, SQRT ! .. ! .. Executable Statements .. IF( N.LT.1 .OR. INCX.LT.1 )THEN NORM = ZERO ELSE IF( N.EQ.1 )THEN NORM = ABS( X( 1 ) ) ELSE SCALE = ZERO SSQ = ONE ! The following loop is equivalent to this call to the LAPACK ! auxiliary routine: ! CALL DLASSQ( N, X, INCX, SCALE, SSQ ) ! DO 10, IX = 1, 1 + ( N - 1 )*INCX, INCX IF( X( IX ).NE.ZERO )THEN ABSXI = ABS( X( IX ) ) IF( SCALE.LT.ABSXI )THEN SSQ = ONE + SSQ*( SCALE/ABSXI )**2 SCALE = ABSXI ELSE SSQ = SSQ + ( ABSXI/SCALE )**2 END IF END IF 10 CONTINUE NORM = SCALE * SQRT( SSQ ) END IF ! DNRM2 = NORM RETURN ! ! End of DNRM2. ! END subroutine drot (n,dx,incx,dy,incy,c,s) ! ! applies a plane rotation. ! jack dongarra, linpack, 3/11/78. ! modified 12/3/93, array(1) declarations changed to array(*) ! double precision dx(*),dy(*),dtemp,c,s integer i,incx,incy,ix,iy,n ! if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 ! ! code for unequal increments or equal increments not equal ! to 1 ! ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n dtemp = c*dx(ix) + s*dy(iy) dy(iy) = c*dy(iy) - s*dx(ix) dx(ix) = dtemp ix = ix + incx iy = iy + incy 10 continue return ! ! code for both increments equal to 1 ! 20 do 30 i = 1,n dtemp = c*dx(i) + s*dy(i) dy(i) = c*dy(i) - s*dx(i) dx(i) = dtemp 30 continue return end subroutine drotg(da,db,c,s) ! ! construct givens plane rotation. ! jack dongarra, linpack, 3/11/78. ! double precision da,db,c,s,roe,scale,r,z ! roe = db if( dabs(da) .gt. dabs(db) ) roe = da scale = dabs(da) + dabs(db) if( scale .ne. 0.0d0 ) go to 10 c = 1.0d0 s = 0.0d0 r = 0.0d0 z = 0.0d0 go to 20 10 r = scale*dsqrt((da/scale)**2 + (db/scale)**2) r = dsign(1.0d0,roe)*r c = da/r s = db/r z = 1.0d0 if( dabs(da) .gt. dabs(db) ) z = s if( dabs(db) .ge. dabs(da) .and. c .ne. 0.0d0 ) z = 1.0d0/c 20 da = r db = z return end SUBROUTINE DSBMV ( UPLO, N, K, ALPHA, A, LDA, X, INCX, & BETA, Y, INCY ) ! .. Scalar Arguments .. DOUBLE PRECISION ALPHA, BETA INTEGER INCX, INCY, K, LDA, N CHARACTER*1 UPLO ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), X( * ), Y( * ) ! .. ! ! Purpose ! ======= ! ! DSBMV performs the matrix-vector operation ! ! y := alpha*A*x + beta*y, ! ! where alpha and beta are scalars, x and y are n element vectors and ! A is an n by n symmetric band matrix, with k super-diagonals. ! ! Parameters ! ========== ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the upper or lower ! triangular part of the band matrix A is being supplied as ! follows: ! ! UPLO = 'U' or 'u' The upper triangular part of A is ! being supplied. ! ! UPLO = 'L' or 'l' The lower triangular part of A is ! being supplied. ! ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the order of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! K - INTEGER. ! On entry, K specifies the number of super-diagonals of the ! matrix A. K must satisfy 0 .le. K. ! Unchanged on exit. ! ! ALPHA - DOUBLE PRECISION. ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, n ). ! Before entry with UPLO = 'U' or 'u', the leading ( k + 1 ) ! by n part of the array A must contain the upper triangular ! band part of the symmetric matrix, supplied column by ! column, with the leading diagonal of the matrix in row ! ( k + 1 ) of the array, the first super-diagonal starting at ! position 2 in row k, and so on. The top left k by k triangle ! of the array A is not referenced. ! The following program segment will transfer the upper ! triangular part of a symmetric band matrix from conventional ! full matrix storage to band storage: ! ! DO 20, J = 1, N ! M = K + 1 - J ! DO 10, I = MAX( 1, J - K ), J ! A( M + I, J ) = matrix( I, J ) ! 10 CONTINUE ! 20 CONTINUE ! ! Before entry with UPLO = 'L' or 'l', the leading ( k + 1 ) ! by n part of the array A must contain the lower triangular ! band part of the symmetric matrix, supplied column by ! column, with the leading diagonal of the matrix in row 1 of ! the array, the first sub-diagonal starting at position 1 in ! row 2, and so on. The bottom right k by k triangle of the ! array A is not referenced. ! The following program segment will transfer the lower ! triangular part of a symmetric band matrix from conventional ! full matrix storage to band storage: ! ! DO 20, J = 1, N ! M = 1 - J ! DO 10, I = J, MIN( N, J + K ) ! A( M + I, J ) = matrix( I, J ) ! 10 CONTINUE ! 20 CONTINUE ! ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. LDA must be at least ! ( k + 1 ). ! Unchanged on exit. ! ! X - DOUBLE PRECISION array of DIMENSION at least ! ( 1 + ( n - 1 )*abs( INCX ) ). ! Before entry, the incremented array X must contain the ! vector x. ! Unchanged on exit. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! BETA - DOUBLE PRECISION. ! On entry, BETA specifies the scalar beta. ! Unchanged on exit. ! ! Y - DOUBLE PRECISION array of DIMENSION at least ! ( 1 + ( n - 1 )*abs( INCY ) ). ! Before entry, the incremented array Y must contain the ! vector y. On exit, Y is overwritten by the updated vector y. ! ! INCY - INTEGER. ! On entry, INCY specifies the increment for the elements of ! Y. INCY must not be zero. ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. Local Scalars .. DOUBLE PRECISION TEMP1, TEMP2 INTEGER I, INFO, IX, IY, J, JX, JY, KPLUS1, KX, KY, L ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( UPLO, 'U' ).AND. & .NOT.LSAME( UPLO, 'L' ) )THEN INFO = 1 ELSE IF( N.LT.0 )THEN INFO = 2 ELSE IF( K.LT.0 )THEN INFO = 3 ELSE IF( LDA.LT.( K + 1 ) )THEN INFO = 6 ELSE IF( INCX.EQ.0 )THEN INFO = 8 ELSE IF( INCY.EQ.0 )THEN INFO = 11 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DSBMV ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( N.EQ.0 ).OR.( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) ) & RETURN ! ! Set up the start points in X and Y. ! IF( INCX.GT.0 )THEN KX = 1 ELSE KX = 1 - ( N - 1 )*INCX END IF IF( INCY.GT.0 )THEN KY = 1 ELSE KY = 1 - ( N - 1 )*INCY END IF ! ! Start the operations. In this version the elements of the array A ! are accessed sequentially with one pass through A. ! ! First form y := beta*y. ! IF( BETA.NE.ONE )THEN IF( INCY.EQ.1 )THEN IF( BETA.EQ.ZERO )THEN DO 10, I = 1, N Y( I ) = ZERO 10 CONTINUE ELSE DO 20, I = 1, N Y( I ) = BETA*Y( I ) 20 CONTINUE END IF ELSE IY = KY IF( BETA.EQ.ZERO )THEN DO 30, I = 1, N Y( IY ) = ZERO IY = IY + INCY 30 CONTINUE ELSE DO 40, I = 1, N Y( IY ) = BETA*Y( IY ) IY = IY + INCY 40 CONTINUE END IF END IF END IF IF( ALPHA.EQ.ZERO ) & RETURN IF( LSAME( UPLO, 'U' ) )THEN ! ! Form y when upper triangle of A is stored. ! KPLUS1 = K + 1 IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN DO 60, J = 1, N TEMP1 = ALPHA*X( J ) TEMP2 = ZERO L = KPLUS1 - J DO 50, I = MAX( 1, J - K ), J - 1 Y( I ) = Y( I ) + TEMP1*A( L + I, J ) TEMP2 = TEMP2 + A( L + I, J )*X( I ) 50 CONTINUE Y( J ) = Y( J ) + TEMP1*A( KPLUS1, J ) + ALPHA*TEMP2 60 CONTINUE ELSE JX = KX JY = KY DO 80, J = 1, N TEMP1 = ALPHA*X( JX ) TEMP2 = ZERO IX = KX IY = KY L = KPLUS1 - J DO 70, I = MAX( 1, J - K ), J - 1 Y( IY ) = Y( IY ) + TEMP1*A( L + I, J ) TEMP2 = TEMP2 + A( L + I, J )*X( IX ) IX = IX + INCX IY = IY + INCY 70 CONTINUE Y( JY ) = Y( JY ) + TEMP1*A( KPLUS1, J ) + ALPHA*TEMP2 JX = JX + INCX JY = JY + INCY IF( J.GT.K )THEN KX = KX + INCX KY = KY + INCY END IF 80 CONTINUE END IF ELSE ! ! Form y when lower triangle of A is stored. ! IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN DO 100, J = 1, N TEMP1 = ALPHA*X( J ) TEMP2 = ZERO Y( J ) = Y( J ) + TEMP1*A( 1, J ) L = 1 - J DO 90, I = J + 1, MIN( N, J + K ) Y( I ) = Y( I ) + TEMP1*A( L + I, J ) TEMP2 = TEMP2 + A( L + I, J )*X( I ) 90 CONTINUE Y( J ) = Y( J ) + ALPHA*TEMP2 100 CONTINUE ELSE JX = KX JY = KY DO 120, J = 1, N TEMP1 = ALPHA*X( JX ) TEMP2 = ZERO Y( JY ) = Y( JY ) + TEMP1*A( 1, J ) L = 1 - J IX = JX IY = JY DO 110, I = J + 1, MIN( N, J + K ) IX = IX + INCX IY = IY + INCY Y( IY ) = Y( IY ) + TEMP1*A( L + I, J ) TEMP2 = TEMP2 + A( L + I, J )*X( IX ) 110 CONTINUE Y( JY ) = Y( JY ) + ALPHA*TEMP2 JX = JX + INCX JY = JY + INCY 120 CONTINUE END IF END IF ! RETURN ! ! End of DSBMV . ! END subroutine dscal(n,da,dx,incx) ! ! scales a vector by a constant. ! uses unrolled loops for increment equal to one. ! jack dongarra, linpack, 3/11/78. ! modified 3/93 to return if incx .le. 0. ! modified 12/3/93, array(1) declarations changed to array(*) ! double precision da,dx(*) integer i,incx,m,mp1,n,nincx ! if( n.le.0 .or. incx.le.0 )return if(incx.eq.1)go to 20 ! ! code for increment not equal to 1 ! nincx = n*incx do 10 i = 1,nincx,incx dx(i) = da*dx(i) 10 continue return ! ! code for increment equal to 1 ! ! ! clean-up loop ! 20 m = mod(n,5) if( m .eq. 0 ) go to 40 do 30 i = 1,m dx(i) = da*dx(i) 30 continue if( n .lt. 5 ) return 40 mp1 = m + 1 do 50 i = mp1,n,5 dx(i) = da*dx(i) dx(i + 1) = da*dx(i + 1) dx(i + 2) = da*dx(i + 2) dx(i + 3) = da*dx(i + 3) dx(i + 4) = da*dx(i + 4) 50 continue return end SUBROUTINE DSPMV ( UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY ) ! .. Scalar Arguments .. DOUBLE PRECISION ALPHA, BETA INTEGER INCX, INCY, N CHARACTER*1 UPLO ! .. Array Arguments .. DOUBLE PRECISION AP( * ), X( * ), Y( * ) ! .. ! ! Purpose ! ======= ! ! DSPMV performs the matrix-vector operation ! ! y := alpha*A*x + beta*y, ! ! where alpha and beta are scalars, x and y are n element vectors and ! A is an n by n symmetric matrix, supplied in packed form. ! ! Parameters ! ========== ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the upper or lower ! triangular part of the matrix A is supplied in the packed ! array AP as follows: ! ! UPLO = 'U' or 'u' The upper triangular part of A is ! supplied in AP. ! ! UPLO = 'L' or 'l' The lower triangular part of A is ! supplied in AP. ! ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the order of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! ALPHA - DOUBLE PRECISION. ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! AP - DOUBLE PRECISION array of DIMENSION at least ! ( ( n*( n + 1 ) )/2 ). ! Before entry with UPLO = 'U' or 'u', the array AP must ! contain the upper triangular part of the symmetric matrix ! packed sequentially, column by column, so that AP( 1 ) ! contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 ) ! and a( 2, 2 ) respectively, and so on. ! Before entry with UPLO = 'L' or 'l', the array AP must ! contain the lower triangular part of the symmetric matrix ! packed sequentially, column by column, so that AP( 1 ) ! contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 ) ! and a( 3, 1 ) respectively, and so on. ! Unchanged on exit. ! ! X - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCX ) ). ! Before entry, the incremented array X must contain the n ! element vector x. ! Unchanged on exit. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! BETA - DOUBLE PRECISION. ! On entry, BETA specifies the scalar beta. When BETA is ! supplied as zero then Y need not be set on input. ! Unchanged on exit. ! ! Y - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCY ) ). ! Before entry, the incremented array Y must contain the n ! element vector y. On exit, Y is overwritten by the updated ! vector y. ! ! INCY - INTEGER. ! On entry, INCY specifies the increment for the elements of ! Y. INCY must not be zero. ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. Local Scalars .. DOUBLE PRECISION TEMP1, TEMP2 INTEGER I, INFO, IX, IY, J, JX, JY, K, KK, KX, KY ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( UPLO, 'U' ).AND. & .NOT.LSAME( UPLO, 'L' ) )THEN INFO = 1 ELSE IF( N.LT.0 )THEN INFO = 2 ELSE IF( INCX.EQ.0 )THEN INFO = 6 ELSE IF( INCY.EQ.0 )THEN INFO = 9 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DSPMV ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( N.EQ.0 ).OR.( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) ) & RETURN ! ! Set up the start points in X and Y. ! IF( INCX.GT.0 )THEN KX = 1 ELSE KX = 1 - ( N - 1 )*INCX END IF IF( INCY.GT.0 )THEN KY = 1 ELSE KY = 1 - ( N - 1 )*INCY END IF ! ! Start the operations. In this version the elements of the array AP ! are accessed sequentially with one pass through AP. ! ! First form y := beta*y. ! IF( BETA.NE.ONE )THEN IF( INCY.EQ.1 )THEN IF( BETA.EQ.ZERO )THEN DO 10, I = 1, N Y( I ) = ZERO 10 CONTINUE ELSE DO 20, I = 1, N Y( I ) = BETA*Y( I ) 20 CONTINUE END IF ELSE IY = KY IF( BETA.EQ.ZERO )THEN DO 30, I = 1, N Y( IY ) = ZERO IY = IY + INCY 30 CONTINUE ELSE DO 40, I = 1, N Y( IY ) = BETA*Y( IY ) IY = IY + INCY 40 CONTINUE END IF END IF END IF IF( ALPHA.EQ.ZERO ) & RETURN KK = 1 IF( LSAME( UPLO, 'U' ) )THEN ! ! Form y when AP contains the upper triangle. ! IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN DO 60, J = 1, N TEMP1 = ALPHA*X( J ) TEMP2 = ZERO K = KK DO 50, I = 1, J - 1 Y( I ) = Y( I ) + TEMP1*AP( K ) TEMP2 = TEMP2 + AP( K )*X( I ) K = K + 1 50 CONTINUE Y( J ) = Y( J ) + TEMP1*AP( KK + J - 1 ) + ALPHA*TEMP2 KK = KK + J 60 CONTINUE ELSE JX = KX JY = KY DO 80, J = 1, N TEMP1 = ALPHA*X( JX ) TEMP2 = ZERO IX = KX IY = KY DO 70, K = KK, KK + J - 2 Y( IY ) = Y( IY ) + TEMP1*AP( K ) TEMP2 = TEMP2 + AP( K )*X( IX ) IX = IX + INCX IY = IY + INCY 70 CONTINUE Y( JY ) = Y( JY ) + TEMP1*AP( KK + J - 1 ) + ALPHA*TEMP2 JX = JX + INCX JY = JY + INCY KK = KK + J 80 CONTINUE END IF ELSE ! ! Form y when AP contains the lower triangle. ! IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN DO 100, J = 1, N TEMP1 = ALPHA*X( J ) TEMP2 = ZERO Y( J ) = Y( J ) + TEMP1*AP( KK ) K = KK + 1 DO 90, I = J + 1, N Y( I ) = Y( I ) + TEMP1*AP( K ) TEMP2 = TEMP2 + AP( K )*X( I ) K = K + 1 90 CONTINUE Y( J ) = Y( J ) + ALPHA*TEMP2 KK = KK + ( N - J + 1 ) 100 CONTINUE ELSE JX = KX JY = KY DO 120, J = 1, N TEMP1 = ALPHA*X( JX ) TEMP2 = ZERO Y( JY ) = Y( JY ) + TEMP1*AP( KK ) IX = JX IY = JY DO 110, K = KK + 1, KK + N - J IX = IX + INCX IY = IY + INCY Y( IY ) = Y( IY ) + TEMP1*AP( K ) TEMP2 = TEMP2 + AP( K )*X( IX ) 110 CONTINUE Y( JY ) = Y( JY ) + ALPHA*TEMP2 JX = JX + INCX JY = JY + INCY KK = KK + ( N - J + 1 ) 120 CONTINUE END IF END IF ! RETURN ! ! End of DSPMV . ! END SUBROUTINE DSPR ( UPLO, N, ALPHA, X, INCX, AP ) ! .. Scalar Arguments .. DOUBLE PRECISION ALPHA INTEGER INCX, N CHARACTER*1 UPLO ! .. Array Arguments .. DOUBLE PRECISION AP( * ), X( * ) ! .. ! ! Purpose ! ======= ! ! DSPR performs the symmetric rank 1 operation ! ! A := alpha*x*x' + A, ! ! where alpha is a real scalar, x is an n element vector and A is an ! n by n symmetric matrix, supplied in packed form. ! ! Parameters ! ========== ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the upper or lower ! triangular part of the matrix A is supplied in the packed ! array AP as follows: ! ! UPLO = 'U' or 'u' The upper triangular part of A is ! supplied in AP. ! ! UPLO = 'L' or 'l' The lower triangular part of A is ! supplied in AP. ! ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the order of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! ALPHA - DOUBLE PRECISION. ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! X - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCX ) ). ! Before entry, the incremented array X must contain the n ! element vector x. ! Unchanged on exit. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! AP - DOUBLE PRECISION array of DIMENSION at least ! ( ( n*( n + 1 ) )/2 ). ! Before entry with UPLO = 'U' or 'u', the array AP must ! contain the upper triangular part of the symmetric matrix ! packed sequentially, column by column, so that AP( 1 ) ! contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 ) ! and a( 2, 2 ) respectively, and so on. On exit, the array ! AP is overwritten by the upper triangular part of the ! updated matrix. ! Before entry with UPLO = 'L' or 'l', the array AP must ! contain the lower triangular part of the symmetric matrix ! packed sequentially, column by column, so that AP( 1 ) ! contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 ) ! and a( 3, 1 ) respectively, and so on. On exit, the array ! AP is overwritten by the lower triangular part of the ! updated matrix. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. Local Scalars .. DOUBLE PRECISION TEMP INTEGER I, INFO, IX, J, JX, K, KK, KX ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( UPLO, 'U' ).AND. & .NOT.LSAME( UPLO, 'L' ) )THEN INFO = 1 ELSE IF( N.LT.0 )THEN INFO = 2 ELSE IF( INCX.EQ.0 )THEN INFO = 5 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DSPR ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( N.EQ.0 ).OR.( ALPHA.EQ.ZERO ) ) & RETURN ! ! Set the start point in X if the increment is not unity. ! IF( INCX.LE.0 )THEN KX = 1 - ( N - 1 )*INCX ELSE IF( INCX.NE.1 )THEN KX = 1 END IF ! ! Start the operations. In this version the elements of the array AP ! are accessed sequentially with one pass through AP. ! KK = 1 IF( LSAME( UPLO, 'U' ) )THEN ! ! Form A when upper triangle is stored in AP. ! IF( INCX.EQ.1 )THEN DO 20, J = 1, N IF( X( J ).NE.ZERO )THEN TEMP = ALPHA*X( J ) K = KK DO 10, I = 1, J AP( K ) = AP( K ) + X( I )*TEMP K = K + 1 10 CONTINUE END IF KK = KK + J 20 CONTINUE ELSE JX = KX DO 40, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = ALPHA*X( JX ) IX = KX DO 30, K = KK, KK + J - 1 AP( K ) = AP( K ) + X( IX )*TEMP IX = IX + INCX 30 CONTINUE END IF JX = JX + INCX KK = KK + J 40 CONTINUE END IF ELSE ! ! Form A when lower triangle is stored in AP. ! IF( INCX.EQ.1 )THEN DO 60, J = 1, N IF( X( J ).NE.ZERO )THEN TEMP = ALPHA*X( J ) K = KK DO 50, I = J, N AP( K ) = AP( K ) + X( I )*TEMP K = K + 1 50 CONTINUE END IF KK = KK + N - J + 1 60 CONTINUE ELSE JX = KX DO 80, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = ALPHA*X( JX ) IX = JX DO 70, K = KK, KK + N - J AP( K ) = AP( K ) + X( IX )*TEMP IX = IX + INCX 70 CONTINUE END IF JX = JX + INCX KK = KK + N - J + 1 80 CONTINUE END IF END IF ! RETURN ! ! End of DSPR . ! END SUBROUTINE DSPR2 ( UPLO, N, ALPHA, X, INCX, Y, INCY, AP ) ! .. Scalar Arguments .. DOUBLE PRECISION ALPHA INTEGER INCX, INCY, N CHARACTER*1 UPLO ! .. Array Arguments .. DOUBLE PRECISION AP( * ), X( * ), Y( * ) ! .. ! ! Purpose ! ======= ! ! DSPR2 performs the symmetric rank 2 operation ! ! A := alpha*x*y' + alpha*y*x' + A, ! ! where alpha is a scalar, x and y are n element vectors and A is an ! n by n symmetric matrix, supplied in packed form. ! ! Parameters ! ========== ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the upper or lower ! triangular part of the matrix A is supplied in the packed ! array AP as follows: ! ! UPLO = 'U' or 'u' The upper triangular part of A is ! supplied in AP. ! ! UPLO = 'L' or 'l' The lower triangular part of A is ! supplied in AP. ! ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the order of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! ALPHA - DOUBLE PRECISION. ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! X - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCX ) ). ! Before entry, the incremented array X must contain the n ! element vector x. ! Unchanged on exit. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! Y - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCY ) ). ! Before entry, the incremented array Y must contain the n ! element vector y. ! Unchanged on exit. ! ! INCY - INTEGER. ! On entry, INCY specifies the increment for the elements of ! Y. INCY must not be zero. ! Unchanged on exit. ! ! AP - DOUBLE PRECISION array of DIMENSION at least ! ( ( n*( n + 1 ) )/2 ). ! Before entry with UPLO = 'U' or 'u', the array AP must ! contain the upper triangular part of the symmetric matrix ! packed sequentially, column by column, so that AP( 1 ) ! contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 ) ! and a( 2, 2 ) respectively, and so on. On exit, the array ! AP is overwritten by the upper triangular part of the ! updated matrix. ! Before entry with UPLO = 'L' or 'l', the array AP must ! contain the lower triangular part of the symmetric matrix ! packed sequentially, column by column, so that AP( 1 ) ! contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 ) ! and a( 3, 1 ) respectively, and so on. On exit, the array ! AP is overwritten by the lower triangular part of the ! updated matrix. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. Local Scalars .. DOUBLE PRECISION TEMP1, TEMP2 INTEGER I, INFO, IX, IY, J, JX, JY, K, KK, KX, KY ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( UPLO, 'U' ).AND. & .NOT.LSAME( UPLO, 'L' ) )THEN INFO = 1 ELSE IF( N.LT.0 )THEN INFO = 2 ELSE IF( INCX.EQ.0 )THEN INFO = 5 ELSE IF( INCY.EQ.0 )THEN INFO = 7 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DSPR2 ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( N.EQ.0 ).OR.( ALPHA.EQ.ZERO ) ) & RETURN ! ! Set up the start points in X and Y if the increments are not both ! unity. ! IF( ( INCX.NE.1 ).OR.( INCY.NE.1 ) )THEN IF( INCX.GT.0 )THEN KX = 1 ELSE KX = 1 - ( N - 1 )*INCX END IF IF( INCY.GT.0 )THEN KY = 1 ELSE KY = 1 - ( N - 1 )*INCY END IF JX = KX JY = KY END IF ! ! Start the operations. In this version the elements of the array AP ! are accessed sequentially with one pass through AP. ! KK = 1 IF( LSAME( UPLO, 'U' ) )THEN ! ! Form A when upper triangle is stored in AP. ! IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN DO 20, J = 1, N IF( ( X( J ).NE.ZERO ).OR.( Y( J ).NE.ZERO ) )THEN TEMP1 = ALPHA*Y( J ) TEMP2 = ALPHA*X( J ) K = KK DO 10, I = 1, J AP( K ) = AP( K ) + X( I )*TEMP1 + Y( I )*TEMP2 K = K + 1 10 CONTINUE END IF KK = KK + J 20 CONTINUE ELSE DO 40, J = 1, N IF( ( X( JX ).NE.ZERO ).OR.( Y( JY ).NE.ZERO ) )THEN TEMP1 = ALPHA*Y( JY ) TEMP2 = ALPHA*X( JX ) IX = KX IY = KY DO 30, K = KK, KK + J - 1 AP( K ) = AP( K ) + X( IX )*TEMP1 + Y( IY )*TEMP2 IX = IX + INCX IY = IY + INCY 30 CONTINUE END IF JX = JX + INCX JY = JY + INCY KK = KK + J 40 CONTINUE END IF ELSE ! ! Form A when lower triangle is stored in AP. ! IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN DO 60, J = 1, N IF( ( X( J ).NE.ZERO ).OR.( Y( J ).NE.ZERO ) )THEN TEMP1 = ALPHA*Y( J ) TEMP2 = ALPHA*X( J ) K = KK DO 50, I = J, N AP( K ) = AP( K ) + X( I )*TEMP1 + Y( I )*TEMP2 K = K + 1 50 CONTINUE END IF KK = KK + N - J + 1 60 CONTINUE ELSE DO 80, J = 1, N IF( ( X( JX ).NE.ZERO ).OR.( Y( JY ).NE.ZERO ) )THEN TEMP1 = ALPHA*Y( JY ) TEMP2 = ALPHA*X( JX ) IX = JX IY = JY DO 70, K = KK, KK + N - J AP( K ) = AP( K ) + X( IX )*TEMP1 + Y( IY )*TEMP2 IX = IX + INCX IY = IY + INCY 70 CONTINUE END IF JX = JX + INCX JY = JY + INCY KK = KK + N - J + 1 80 CONTINUE END IF END IF ! RETURN ! ! End of DSPR2 . ! END subroutine dswap (n,dx,incx,dy,incy) ! ! interchanges two vectors. ! uses unrolled loops for increments equal one. ! jack dongarra, linpack, 3/11/78. ! modified 12/3/93, array(1) declarations changed to array(*) ! double precision dx(*),dy(*),dtemp integer i,incx,incy,ix,iy,m,mp1,n ! if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 ! ! code for unequal increments or equal increments not equal ! to 1 ! ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n dtemp = dx(ix) dx(ix) = dy(iy) dy(iy) = dtemp ix = ix + incx iy = iy + incy 10 continue return ! ! code for both increments equal to 1 ! ! ! clean-up loop ! 20 m = mod(n,3) if( m .eq. 0 ) go to 40 do 30 i = 1,m dtemp = dx(i) dx(i) = dy(i) dy(i) = dtemp 30 continue if( n .lt. 3 ) return 40 mp1 = m + 1 do 50 i = mp1,n,3 dtemp = dx(i) dx(i) = dy(i) dy(i) = dtemp dtemp = dx(i + 1) dx(i + 1) = dy(i + 1) dy(i + 1) = dtemp dtemp = dx(i + 2) dx(i + 2) = dy(i + 2) dy(i + 2) = dtemp 50 continue return end SUBROUTINE DSYMM ( SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, & BETA, C, LDC ) ! .. Scalar Arguments .. CHARACTER*1 SIDE, UPLO INTEGER M, N, LDA, LDB, LDC DOUBLE PRECISION ALPHA, BETA ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ) ! .. ! ! Purpose ! ======= ! ! DSYMM performs one of the matrix-matrix operations ! ! C := alpha*A*B + beta*C, ! ! or ! ! C := alpha*B*A + beta*C, ! ! where alpha and beta are scalars, A is a symmetric matrix and B and ! C are m by n matrices. ! ! Parameters ! ========== ! ! SIDE - CHARACTER*1. ! On entry, SIDE specifies whether the symmetric matrix A ! appears on the left or right in the operation as follows: ! ! SIDE = 'L' or 'l' C := alpha*A*B + beta*C, ! ! SIDE = 'R' or 'r' C := alpha*B*A + beta*C, ! ! Unchanged on exit. ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the upper or lower ! triangular part of the symmetric matrix A is to be ! referenced as follows: ! ! UPLO = 'U' or 'u' Only the upper triangular part of the ! symmetric matrix is to be referenced. ! ! UPLO = 'L' or 'l' Only the lower triangular part of the ! symmetric matrix is to be referenced. ! ! Unchanged on exit. ! ! M - INTEGER. ! On entry, M specifies the number of rows of the matrix C. ! M must be at least zero. ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the number of columns of the matrix C. ! N must be at least zero. ! Unchanged on exit. ! ! ALPHA - DOUBLE PRECISION. ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, ka ), where ka is ! m when SIDE = 'L' or 'l' and is n otherwise. ! Before entry with SIDE = 'L' or 'l', the m by m part of ! the array A must contain the symmetric matrix, such that ! when UPLO = 'U' or 'u', the leading m by m upper triangular ! part of the array A must contain the upper triangular part ! of the symmetric matrix and the strictly lower triangular ! part of A is not referenced, and when UPLO = 'L' or 'l', ! the leading m by m lower triangular part of the array A ! must contain the lower triangular part of the symmetric ! matrix and the strictly upper triangular part of A is not ! referenced. ! Before entry with SIDE = 'R' or 'r', the n by n part of ! the array A must contain the symmetric matrix, such that ! when UPLO = 'U' or 'u', the leading n by n upper triangular ! part of the array A must contain the upper triangular part ! of the symmetric matrix and the strictly lower triangular ! part of A is not referenced, and when UPLO = 'L' or 'l', ! the leading n by n lower triangular part of the array A ! must contain the lower triangular part of the symmetric ! matrix and the strictly upper triangular part of A is not ! referenced. ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. When SIDE = 'L' or 'l' then ! LDA must be at least max( 1, m ), otherwise LDA must be at ! least max( 1, n ). ! Unchanged on exit. ! ! B - DOUBLE PRECISION array of DIMENSION ( LDB, n ). ! Before entry, the leading m by n part of the array B must ! contain the matrix B. ! Unchanged on exit. ! ! LDB - INTEGER. ! On entry, LDB specifies the first dimension of B as declared ! in the calling (sub) program. LDB must be at least ! max( 1, m ). ! Unchanged on exit. ! ! BETA - DOUBLE PRECISION. ! On entry, BETA specifies the scalar beta. When BETA is ! supplied as zero then C need not be set on input. ! Unchanged on exit. ! ! C - DOUBLE PRECISION array of DIMENSION ( LDC, n ). ! Before entry, the leading m by n part of the array C must ! contain the matrix C, except when beta is zero, in which ! case C need not be set on entry. ! On exit, the array C is overwritten by the m by n updated ! matrix. ! ! LDC - INTEGER. ! On entry, LDC specifies the first dimension of C as declared ! in the calling (sub) program. LDC must be at least ! max( 1, m ). ! Unchanged on exit. ! ! ! Level 3 Blas routine. ! ! -- Written on 8-February-1989. ! Jack Dongarra, Argonne National Laboratory. ! Iain Duff, AERE Harwell. ! Jeremy Du Croz, Numerical Algorithms Group Ltd. ! Sven Hammarling, Numerical Algorithms Group Ltd. ! ! ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX ! .. Local Scalars .. LOGICAL UPPER INTEGER I, INFO, J, K, NROWA DOUBLE PRECISION TEMP1, TEMP2 ! .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Executable Statements .. ! ! Set NROWA as the number of rows of A. ! IF( LSAME( SIDE, 'L' ) )THEN NROWA = M ELSE NROWA = N END IF UPPER = LSAME( UPLO, 'U' ) ! ! Test the input parameters. ! INFO = 0 IF( ( .NOT.LSAME( SIDE, 'L' ) ).AND. & ( .NOT.LSAME( SIDE, 'R' ) ) )THEN INFO = 1 ELSE IF( ( .NOT.UPPER ).AND. & ( .NOT.LSAME( UPLO, 'L' ) ) )THEN INFO = 2 ELSE IF( M .LT.0 )THEN INFO = 3 ELSE IF( N .LT.0 )THEN INFO = 4 ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN INFO = 7 ELSE IF( LDB.LT.MAX( 1, M ) )THEN INFO = 9 ELSE IF( LDC.LT.MAX( 1, M ) )THEN INFO = 12 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DSYMM ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR. & ( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) ) & RETURN ! ! And when alpha.eq.zero. ! IF( ALPHA.EQ.ZERO )THEN IF( BETA.EQ.ZERO )THEN DO 20, J = 1, N DO 10, I = 1, M C( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ELSE DO 40, J = 1, N DO 30, I = 1, M C( I, J ) = BETA*C( I, J ) 30 CONTINUE 40 CONTINUE END IF RETURN END IF ! ! Start the operations. ! IF( LSAME( SIDE, 'L' ) )THEN ! ! Form C := alpha*A*B + beta*C. ! IF( UPPER )THEN DO 70, J = 1, N DO 60, I = 1, M TEMP1 = ALPHA*B( I, J ) TEMP2 = ZERO DO 50, K = 1, I - 1 C( K, J ) = C( K, J ) + TEMP1 *A( K, I ) TEMP2 = TEMP2 + B( K, J )*A( K, I ) 50 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = TEMP1*A( I, I ) + ALPHA*TEMP2 ELSE C( I, J ) = BETA *C( I, J ) + & TEMP1*A( I, I ) + ALPHA*TEMP2 END IF 60 CONTINUE 70 CONTINUE ELSE DO 100, J = 1, N DO 90, I = M, 1, -1 TEMP1 = ALPHA*B( I, J ) TEMP2 = ZERO DO 80, K = I + 1, M C( K, J ) = C( K, J ) + TEMP1 *A( K, I ) TEMP2 = TEMP2 + B( K, J )*A( K, I ) 80 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = TEMP1*A( I, I ) + ALPHA*TEMP2 ELSE C( I, J ) = BETA *C( I, J ) + & TEMP1*A( I, I ) + ALPHA*TEMP2 END IF 90 CONTINUE 100 CONTINUE END IF ELSE ! ! Form C := alpha*B*A + beta*C. ! DO 170, J = 1, N TEMP1 = ALPHA*A( J, J ) IF( BETA.EQ.ZERO )THEN DO 110, I = 1, M C( I, J ) = TEMP1*B( I, J ) 110 CONTINUE ELSE DO 120, I = 1, M C( I, J ) = BETA*C( I, J ) + TEMP1*B( I, J ) 120 CONTINUE END IF DO 140, K = 1, J - 1 IF( UPPER )THEN TEMP1 = ALPHA*A( K, J ) ELSE TEMP1 = ALPHA*A( J, K ) END IF DO 130, I = 1, M C( I, J ) = C( I, J ) + TEMP1*B( I, K ) 130 CONTINUE 140 CONTINUE DO 160, K = J + 1, N IF( UPPER )THEN TEMP1 = ALPHA*A( J, K ) ELSE TEMP1 = ALPHA*A( K, J ) END IF DO 150, I = 1, M C( I, J ) = C( I, J ) + TEMP1*B( I, K ) 150 CONTINUE 160 CONTINUE 170 CONTINUE END IF ! RETURN ! ! End of DSYMM . ! END SUBROUTINE DSYMV ( UPLO, N, ALPHA, A, LDA, X, INCX, & BETA, Y, INCY ) ! .. Scalar Arguments .. DOUBLE PRECISION ALPHA, BETA INTEGER INCX, INCY, LDA, N CHARACTER*1 UPLO ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), X( * ), Y( * ) ! .. ! ! Purpose ! ======= ! ! DSYMV performs the matrix-vector operation ! ! y := alpha*A*x + beta*y, ! ! where alpha and beta are scalars, x and y are n element vectors and ! A is an n by n symmetric matrix. ! ! Parameters ! ========== ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the upper or lower ! triangular part of the array A is to be referenced as ! follows: ! ! UPLO = 'U' or 'u' Only the upper triangular part of A ! is to be referenced. ! ! UPLO = 'L' or 'l' Only the lower triangular part of A ! is to be referenced. ! ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the order of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! ALPHA - DOUBLE PRECISION. ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, n ). ! Before entry with UPLO = 'U' or 'u', the leading n by n ! upper triangular part of the array A must contain the upper ! triangular part of the symmetric matrix and the strictly ! lower triangular part of A is not referenced. ! Before entry with UPLO = 'L' or 'l', the leading n by n ! lower triangular part of the array A must contain the lower ! triangular part of the symmetric matrix and the strictly ! upper triangular part of A is not referenced. ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. LDA must be at least ! max( 1, n ). ! Unchanged on exit. ! ! X - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCX ) ). ! Before entry, the incremented array X must contain the n ! element vector x. ! Unchanged on exit. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! BETA - DOUBLE PRECISION. ! On entry, BETA specifies the scalar beta. When BETA is ! supplied as zero then Y need not be set on input. ! Unchanged on exit. ! ! Y - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCY ) ). ! Before entry, the incremented array Y must contain the n ! element vector y. On exit, Y is overwritten by the updated ! vector y. ! ! INCY - INTEGER. ! On entry, INCY specifies the increment for the elements of ! Y. INCY must not be zero. ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. Local Scalars .. DOUBLE PRECISION TEMP1, TEMP2 INTEGER I, INFO, IX, IY, J, JX, JY, KX, KY ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( UPLO, 'U' ).AND. & .NOT.LSAME( UPLO, 'L' ) )THEN INFO = 1 ELSE IF( N.LT.0 )THEN INFO = 2 ELSE IF( LDA.LT.MAX( 1, N ) )THEN INFO = 5 ELSE IF( INCX.EQ.0 )THEN INFO = 7 ELSE IF( INCY.EQ.0 )THEN INFO = 10 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DSYMV ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( N.EQ.0 ).OR.( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) ) & RETURN ! ! Set up the start points in X and Y. ! IF( INCX.GT.0 )THEN KX = 1 ELSE KX = 1 - ( N - 1 )*INCX END IF IF( INCY.GT.0 )THEN KY = 1 ELSE KY = 1 - ( N - 1 )*INCY END IF ! ! Start the operations. In this version the elements of A are ! accessed sequentially with one pass through the triangular part ! of A. ! ! First form y := beta*y. ! IF( BETA.NE.ONE )THEN IF( INCY.EQ.1 )THEN IF( BETA.EQ.ZERO )THEN DO 10, I = 1, N Y( I ) = ZERO 10 CONTINUE ELSE DO 20, I = 1, N Y( I ) = BETA*Y( I ) 20 CONTINUE END IF ELSE IY = KY IF( BETA.EQ.ZERO )THEN DO 30, I = 1, N Y( IY ) = ZERO IY = IY + INCY 30 CONTINUE ELSE DO 40, I = 1, N Y( IY ) = BETA*Y( IY ) IY = IY + INCY 40 CONTINUE END IF END IF END IF IF( ALPHA.EQ.ZERO ) & RETURN IF( LSAME( UPLO, 'U' ) )THEN ! ! Form y when A is stored in upper triangle. ! IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN DO 60, J = 1, N TEMP1 = ALPHA*X( J ) TEMP2 = ZERO DO 50, I = 1, J - 1 Y( I ) = Y( I ) + TEMP1*A( I, J ) TEMP2 = TEMP2 + A( I, J )*X( I ) 50 CONTINUE Y( J ) = Y( J ) + TEMP1*A( J, J ) + ALPHA*TEMP2 60 CONTINUE ELSE JX = KX JY = KY DO 80, J = 1, N TEMP1 = ALPHA*X( JX ) TEMP2 = ZERO IX = KX IY = KY DO 70, I = 1, J - 1 Y( IY ) = Y( IY ) + TEMP1*A( I, J ) TEMP2 = TEMP2 + A( I, J )*X( IX ) IX = IX + INCX IY = IY + INCY 70 CONTINUE Y( JY ) = Y( JY ) + TEMP1*A( J, J ) + ALPHA*TEMP2 JX = JX + INCX JY = JY + INCY 80 CONTINUE END IF ELSE ! ! Form y when A is stored in lower triangle. ! IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN DO 100, J = 1, N TEMP1 = ALPHA*X( J ) TEMP2 = ZERO Y( J ) = Y( J ) + TEMP1*A( J, J ) DO 90, I = J + 1, N Y( I ) = Y( I ) + TEMP1*A( I, J ) TEMP2 = TEMP2 + A( I, J )*X( I ) 90 CONTINUE Y( J ) = Y( J ) + ALPHA*TEMP2 100 CONTINUE ELSE JX = KX JY = KY DO 120, J = 1, N TEMP1 = ALPHA*X( JX ) TEMP2 = ZERO Y( JY ) = Y( JY ) + TEMP1*A( J, J ) IX = JX IY = JY DO 110, I = J + 1, N IX = IX + INCX IY = IY + INCY Y( IY ) = Y( IY ) + TEMP1*A( I, J ) TEMP2 = TEMP2 + A( I, J )*X( IX ) 110 CONTINUE Y( JY ) = Y( JY ) + ALPHA*TEMP2 JX = JX + INCX JY = JY + INCY 120 CONTINUE END IF END IF ! RETURN ! ! End of DSYMV . ! END SUBROUTINE DSYR ( UPLO, N, ALPHA, X, INCX, A, LDA ) ! .. Scalar Arguments .. DOUBLE PRECISION ALPHA INTEGER INCX, LDA, N CHARACTER*1 UPLO ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), X( * ) ! .. ! ! Purpose ! ======= ! ! DSYR performs the symmetric rank 1 operation ! ! A := alpha*x*x' + A, ! ! where alpha is a real scalar, x is an n element vector and A is an ! n by n symmetric matrix. ! ! Parameters ! ========== ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the upper or lower ! triangular part of the array A is to be referenced as ! follows: ! ! UPLO = 'U' or 'u' Only the upper triangular part of A ! is to be referenced. ! ! UPLO = 'L' or 'l' Only the lower triangular part of A ! is to be referenced. ! ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the order of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! ALPHA - DOUBLE PRECISION. ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! X - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCX ) ). ! Before entry, the incremented array X must contain the n ! element vector x. ! Unchanged on exit. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, n ). ! Before entry with UPLO = 'U' or 'u', the leading n by n ! upper triangular part of the array A must contain the upper ! triangular part of the symmetric matrix and the strictly ! lower triangular part of A is not referenced. On exit, the ! upper triangular part of the array A is overwritten by the ! upper triangular part of the updated matrix. ! Before entry with UPLO = 'L' or 'l', the leading n by n ! lower triangular part of the array A must contain the lower ! triangular part of the symmetric matrix and the strictly ! upper triangular part of A is not referenced. On exit, the ! lower triangular part of the array A is overwritten by the ! lower triangular part of the updated matrix. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. LDA must be at least ! max( 1, n ). ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. Local Scalars .. DOUBLE PRECISION TEMP INTEGER I, INFO, IX, J, JX, KX ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( UPLO, 'U' ).AND. & .NOT.LSAME( UPLO, 'L' ) )THEN INFO = 1 ELSE IF( N.LT.0 )THEN INFO = 2 ELSE IF( INCX.EQ.0 )THEN INFO = 5 ELSE IF( LDA.LT.MAX( 1, N ) )THEN INFO = 7 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DSYR ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( N.EQ.0 ).OR.( ALPHA.EQ.ZERO ) ) & RETURN ! ! Set the start point in X if the increment is not unity. ! IF( INCX.LE.0 )THEN KX = 1 - ( N - 1 )*INCX ELSE IF( INCX.NE.1 )THEN KX = 1 END IF ! ! Start the operations. In this version the elements of A are ! accessed sequentially with one pass through the triangular part ! of A. ! IF( LSAME( UPLO, 'U' ) )THEN ! ! Form A when A is stored in upper triangle. ! IF( INCX.EQ.1 )THEN DO 20, J = 1, N IF( X( J ).NE.ZERO )THEN TEMP = ALPHA*X( J ) DO 10, I = 1, J A( I, J ) = A( I, J ) + X( I )*TEMP 10 CONTINUE END IF 20 CONTINUE ELSE JX = KX DO 40, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = ALPHA*X( JX ) IX = KX DO 30, I = 1, J A( I, J ) = A( I, J ) + X( IX )*TEMP IX = IX + INCX 30 CONTINUE END IF JX = JX + INCX 40 CONTINUE END IF ELSE ! ! Form A when A is stored in lower triangle. ! IF( INCX.EQ.1 )THEN DO 60, J = 1, N IF( X( J ).NE.ZERO )THEN TEMP = ALPHA*X( J ) DO 50, I = J, N A( I, J ) = A( I, J ) + X( I )*TEMP 50 CONTINUE END IF 60 CONTINUE ELSE JX = KX DO 80, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = ALPHA*X( JX ) IX = JX DO 70, I = J, N A( I, J ) = A( I, J ) + X( IX )*TEMP IX = IX + INCX 70 CONTINUE END IF JX = JX + INCX 80 CONTINUE END IF END IF ! RETURN ! ! End of DSYR . ! END SUBROUTINE DSYR2 ( UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA ) ! .. Scalar Arguments .. DOUBLE PRECISION ALPHA INTEGER INCX, INCY, LDA, N CHARACTER*1 UPLO ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), X( * ), Y( * ) ! .. ! ! Purpose ! ======= ! ! DSYR2 performs the symmetric rank 2 operation ! ! A := alpha*x*y' + alpha*y*x' + A, ! ! where alpha is a scalar, x and y are n element vectors and A is an n ! by n symmetric matrix. ! ! Parameters ! ========== ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the upper or lower ! triangular part of the array A is to be referenced as ! follows: ! ! UPLO = 'U' or 'u' Only the upper triangular part of A ! is to be referenced. ! ! UPLO = 'L' or 'l' Only the lower triangular part of A ! is to be referenced. ! ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the order of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! ALPHA - DOUBLE PRECISION. ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! X - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCX ) ). ! Before entry, the incremented array X must contain the n ! element vector x. ! Unchanged on exit. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! Y - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCY ) ). ! Before entry, the incremented array Y must contain the n ! element vector y. ! Unchanged on exit. ! ! INCY - INTEGER. ! On entry, INCY specifies the increment for the elements of ! Y. INCY must not be zero. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, n ). ! Before entry with UPLO = 'U' or 'u', the leading n by n ! upper triangular part of the array A must contain the upper ! triangular part of the symmetric matrix and the strictly ! lower triangular part of A is not referenced. On exit, the ! upper triangular part of the array A is overwritten by the ! upper triangular part of the updated matrix. ! Before entry with UPLO = 'L' or 'l', the leading n by n ! lower triangular part of the array A must contain the lower ! triangular part of the symmetric matrix and the strictly ! upper triangular part of A is not referenced. On exit, the ! lower triangular part of the array A is overwritten by the ! lower triangular part of the updated matrix. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. LDA must be at least ! max( 1, n ). ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. Local Scalars .. DOUBLE PRECISION TEMP1, TEMP2 INTEGER I, INFO, IX, IY, J, JX, JY, KX, KY ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( UPLO, 'U' ).AND. & .NOT.LSAME( UPLO, 'L' ) )THEN INFO = 1 ELSE IF( N.LT.0 )THEN INFO = 2 ELSE IF( INCX.EQ.0 )THEN INFO = 5 ELSE IF( INCY.EQ.0 )THEN INFO = 7 ELSE IF( LDA.LT.MAX( 1, N ) )THEN INFO = 9 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DSYR2 ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( N.EQ.0 ).OR.( ALPHA.EQ.ZERO ) ) & RETURN ! ! Set up the start points in X and Y if the increments are not both ! unity. ! IF( ( INCX.NE.1 ).OR.( INCY.NE.1 ) )THEN IF( INCX.GT.0 )THEN KX = 1 ELSE KX = 1 - ( N - 1 )*INCX END IF IF( INCY.GT.0 )THEN KY = 1 ELSE KY = 1 - ( N - 1 )*INCY END IF JX = KX JY = KY END IF ! ! Start the operations. In this version the elements of A are ! accessed sequentially with one pass through the triangular part ! of A. ! IF( LSAME( UPLO, 'U' ) )THEN ! ! Form A when A is stored in the upper triangle. ! IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN DO 20, J = 1, N IF( ( X( J ).NE.ZERO ).OR.( Y( J ).NE.ZERO ) )THEN TEMP1 = ALPHA*Y( J ) TEMP2 = ALPHA*X( J ) DO 10, I = 1, J A( I, J ) = A( I, J ) + X( I )*TEMP1 + Y( I )*TEMP2 10 CONTINUE END IF 20 CONTINUE ELSE DO 40, J = 1, N IF( ( X( JX ).NE.ZERO ).OR.( Y( JY ).NE.ZERO ) )THEN TEMP1 = ALPHA*Y( JY ) TEMP2 = ALPHA*X( JX ) IX = KX IY = KY DO 30, I = 1, J A( I, J ) = A( I, J ) + X( IX )*TEMP1 & + Y( IY )*TEMP2 IX = IX + INCX IY = IY + INCY 30 CONTINUE END IF JX = JX + INCX JY = JY + INCY 40 CONTINUE END IF ELSE ! ! Form A when A is stored in the lower triangle. ! IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN DO 60, J = 1, N IF( ( X( J ).NE.ZERO ).OR.( Y( J ).NE.ZERO ) )THEN TEMP1 = ALPHA*Y( J ) TEMP2 = ALPHA*X( J ) DO 50, I = J, N A( I, J ) = A( I, J ) + X( I )*TEMP1 + Y( I )*TEMP2 50 CONTINUE END IF 60 CONTINUE ELSE DO 80, J = 1, N IF( ( X( JX ).NE.ZERO ).OR.( Y( JY ).NE.ZERO ) )THEN TEMP1 = ALPHA*Y( JY ) TEMP2 = ALPHA*X( JX ) IX = JX IY = JY DO 70, I = J, N A( I, J ) = A( I, J ) + X( IX )*TEMP1 & + Y( IY )*TEMP2 IX = IX + INCX IY = IY + INCY 70 CONTINUE END IF JX = JX + INCX JY = JY + INCY 80 CONTINUE END IF END IF ! RETURN ! ! End of DSYR2 . ! END SUBROUTINE DSYR2K( UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, & BETA, C, LDC ) ! .. Scalar Arguments .. CHARACTER*1 UPLO, TRANS INTEGER N, K, LDA, LDB, LDC DOUBLE PRECISION ALPHA, BETA ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ) ! .. ! ! Purpose ! ======= ! ! DSYR2K performs one of the symmetric rank 2k operations ! ! C := alpha*A*B' + alpha*B*A' + beta*C, ! ! or ! ! C := alpha*A'*B + alpha*B'*A + beta*C, ! ! where alpha and beta are scalars, C is an n by n symmetric matrix ! and A and B are n by k matrices in the first case and k by n ! matrices in the second case. ! ! Parameters ! ========== ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the upper or lower ! triangular part of the array C is to be referenced as ! follows: ! ! UPLO = 'U' or 'u' Only the upper triangular part of C ! is to be referenced. ! ! UPLO = 'L' or 'l' Only the lower triangular part of C ! is to be referenced. ! ! Unchanged on exit. ! ! TRANS - CHARACTER*1. ! On entry, TRANS specifies the operation to be performed as ! follows: ! ! TRANS = 'N' or 'n' C := alpha*A*B' + alpha*B*A' + ! beta*C. ! ! TRANS = 'T' or 't' C := alpha*A'*B + alpha*B'*A + ! beta*C. ! ! TRANS = 'C' or 'c' C := alpha*A'*B + alpha*B'*A + ! beta*C. ! ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the order of the matrix C. N must be ! at least zero. ! Unchanged on exit. ! ! K - INTEGER. ! On entry with TRANS = 'N' or 'n', K specifies the number ! of columns of the matrices A and B, and on entry with ! TRANS = 'T' or 't' or 'C' or 'c', K specifies the number ! of rows of the matrices A and B. K must be at least zero. ! Unchanged on exit. ! ! ALPHA - DOUBLE PRECISION. ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, ka ), where ka is ! k when TRANS = 'N' or 'n', and is n otherwise. ! Before entry with TRANS = 'N' or 'n', the leading n by k ! part of the array A must contain the matrix A, otherwise ! the leading k by n part of the array A must contain the ! matrix A. ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. When TRANS = 'N' or 'n' ! then LDA must be at least max( 1, n ), otherwise LDA must ! be at least max( 1, k ). ! Unchanged on exit. ! ! B - DOUBLE PRECISION array of DIMENSION ( LDB, kb ), where kb is ! k when TRANS = 'N' or 'n', and is n otherwise. ! Before entry with TRANS = 'N' or 'n', the leading n by k ! part of the array B must contain the matrix B, otherwise ! the leading k by n part of the array B must contain the ! matrix B. ! Unchanged on exit. ! ! LDB - INTEGER. ! On entry, LDB specifies the first dimension of B as declared ! in the calling (sub) program. When TRANS = 'N' or 'n' ! then LDB must be at least max( 1, n ), otherwise LDB must ! be at least max( 1, k ). ! Unchanged on exit. ! ! BETA - DOUBLE PRECISION. ! On entry, BETA specifies the scalar beta. ! Unchanged on exit. ! ! C - DOUBLE PRECISION array of DIMENSION ( LDC, n ). ! Before entry with UPLO = 'U' or 'u', the leading n by n ! upper triangular part of the array C must contain the upper ! triangular part of the symmetric matrix and the strictly ! lower triangular part of C is not referenced. On exit, the ! upper triangular part of the array C is overwritten by the ! upper triangular part of the updated matrix. ! Before entry with UPLO = 'L' or 'l', the leading n by n ! lower triangular part of the array C must contain the lower ! triangular part of the symmetric matrix and the strictly ! upper triangular part of C is not referenced. On exit, the ! lower triangular part of the array C is overwritten by the ! lower triangular part of the updated matrix. ! ! LDC - INTEGER. ! On entry, LDC specifies the first dimension of C as declared ! in the calling (sub) program. LDC must be at least ! max( 1, n ). ! Unchanged on exit. ! ! ! Level 3 Blas routine. ! ! ! -- Written on 8-February-1989. ! Jack Dongarra, Argonne National Laboratory. ! Iain Duff, AERE Harwell. ! Jeremy Du Croz, Numerical Algorithms Group Ltd. ! Sven Hammarling, Numerical Algorithms Group Ltd. ! ! ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX ! .. Local Scalars .. LOGICAL UPPER INTEGER I, INFO, J, L, NROWA DOUBLE PRECISION TEMP1, TEMP2 ! .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! IF( LSAME( TRANS, 'N' ) )THEN NROWA = N ELSE NROWA = K END IF UPPER = LSAME( UPLO, 'U' ) ! INFO = 0 IF( ( .NOT.UPPER ).AND. & ( .NOT.LSAME( UPLO , 'L' ) ) )THEN INFO = 1 ELSE IF( ( .NOT.LSAME( TRANS, 'N' ) ).AND. & ( .NOT.LSAME( TRANS, 'T' ) ).AND. & ( .NOT.LSAME( TRANS, 'C' ) ) )THEN INFO = 2 ELSE IF( N .LT.0 )THEN INFO = 3 ELSE IF( K .LT.0 )THEN INFO = 4 ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN INFO = 7 ELSE IF( LDB.LT.MAX( 1, NROWA ) )THEN INFO = 9 ELSE IF( LDC.LT.MAX( 1, N ) )THEN INFO = 12 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DSYR2K', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( N.EQ.0 ).OR. & ( ( ( ALPHA.EQ.ZERO ).OR.( K.EQ.0 ) ).AND.( BETA.EQ.ONE ) ) ) & RETURN ! ! And when alpha.eq.zero. ! IF( ALPHA.EQ.ZERO )THEN IF( UPPER )THEN IF( BETA.EQ.ZERO )THEN DO 20, J = 1, N DO 10, I = 1, J C( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ELSE DO 40, J = 1, N DO 30, I = 1, J C( I, J ) = BETA*C( I, J ) 30 CONTINUE 40 CONTINUE END IF ELSE IF( BETA.EQ.ZERO )THEN DO 60, J = 1, N DO 50, I = J, N C( I, J ) = ZERO 50 CONTINUE 60 CONTINUE ELSE DO 80, J = 1, N DO 70, I = J, N C( I, J ) = BETA*C( I, J ) 70 CONTINUE 80 CONTINUE END IF END IF RETURN END IF ! ! Start the operations. ! IF( LSAME( TRANS, 'N' ) )THEN ! ! Form C := alpha*A*B' + alpha*B*A' + C. ! IF( UPPER )THEN DO 130, J = 1, N IF( BETA.EQ.ZERO )THEN DO 90, I = 1, J C( I, J ) = ZERO 90 CONTINUE ELSE IF( BETA.NE.ONE )THEN DO 100, I = 1, J C( I, J ) = BETA*C( I, J ) 100 CONTINUE END IF DO 120, L = 1, K IF( ( A( J, L ).NE.ZERO ).OR. & ( B( J, L ).NE.ZERO ) )THEN TEMP1 = ALPHA*B( J, L ) TEMP2 = ALPHA*A( J, L ) DO 110, I = 1, J C( I, J ) = C( I, J ) + & A( I, L )*TEMP1 + B( I, L )*TEMP2 110 CONTINUE END IF 120 CONTINUE 130 CONTINUE ELSE DO 180, J = 1, N IF( BETA.EQ.ZERO )THEN DO 140, I = J, N C( I, J ) = ZERO 140 CONTINUE ELSE IF( BETA.NE.ONE )THEN DO 150, I = J, N C( I, J ) = BETA*C( I, J ) 150 CONTINUE END IF DO 170, L = 1, K IF( ( A( J, L ).NE.ZERO ).OR. & ( B( J, L ).NE.ZERO ) )THEN TEMP1 = ALPHA*B( J, L ) TEMP2 = ALPHA*A( J, L ) DO 160, I = J, N C( I, J ) = C( I, J ) + & A( I, L )*TEMP1 + B( I, L )*TEMP2 160 CONTINUE END IF 170 CONTINUE 180 CONTINUE END IF ELSE ! ! Form C := alpha*A'*B + alpha*B'*A + C. ! IF( UPPER )THEN DO 210, J = 1, N DO 200, I = 1, J TEMP1 = ZERO TEMP2 = ZERO DO 190, L = 1, K TEMP1 = TEMP1 + A( L, I )*B( L, J ) TEMP2 = TEMP2 + B( L, I )*A( L, J ) 190 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = ALPHA*TEMP1 + ALPHA*TEMP2 ELSE C( I, J ) = BETA *C( I, J ) + & ALPHA*TEMP1 + ALPHA*TEMP2 END IF 200 CONTINUE 210 CONTINUE ELSE DO 240, J = 1, N DO 230, I = J, N TEMP1 = ZERO TEMP2 = ZERO DO 220, L = 1, K TEMP1 = TEMP1 + A( L, I )*B( L, J ) TEMP2 = TEMP2 + B( L, I )*A( L, J ) 220 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = ALPHA*TEMP1 + ALPHA*TEMP2 ELSE C( I, J ) = BETA *C( I, J ) + & ALPHA*TEMP1 + ALPHA*TEMP2 END IF 230 CONTINUE 240 CONTINUE END IF END IF ! RETURN ! ! End of DSYR2K. ! END SUBROUTINE DSYRK ( UPLO, TRANS, N, K, ALPHA, A, LDA, & BETA, C, LDC ) ! .. Scalar Arguments .. CHARACTER*1 UPLO, TRANS INTEGER N, K, LDA, LDC DOUBLE PRECISION ALPHA, BETA ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), C( LDC, * ) ! .. ! ! Purpose ! ======= ! ! DSYRK performs one of the symmetric rank k operations ! ! C := alpha*A*A' + beta*C, ! ! or ! ! C := alpha*A'*A + beta*C, ! ! where alpha and beta are scalars, C is an n by n symmetric matrix ! and A is an n by k matrix in the first case and a k by n matrix ! in the second case. ! ! Parameters ! ========== ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the upper or lower ! triangular part of the array C is to be referenced as ! follows: ! ! UPLO = 'U' or 'u' Only the upper triangular part of C ! is to be referenced. ! ! UPLO = 'L' or 'l' Only the lower triangular part of C ! is to be referenced. ! ! Unchanged on exit. ! ! TRANS - CHARACTER*1. ! On entry, TRANS specifies the operation to be performed as ! follows: ! ! TRANS = 'N' or 'n' C := alpha*A*A' + beta*C. ! ! TRANS = 'T' or 't' C := alpha*A'*A + beta*C. ! ! TRANS = 'C' or 'c' C := alpha*A'*A + beta*C. ! ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the order of the matrix C. N must be ! at least zero. ! Unchanged on exit. ! ! K - INTEGER. ! On entry with TRANS = 'N' or 'n', K specifies the number ! of columns of the matrix A, and on entry with ! TRANS = 'T' or 't' or 'C' or 'c', K specifies the number ! of rows of the matrix A. K must be at least zero. ! Unchanged on exit. ! ! ALPHA - DOUBLE PRECISION. ! On entry, ALPHA specifies the scalar alpha. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, ka ), where ka is ! k when TRANS = 'N' or 'n', and is n otherwise. ! Before entry with TRANS = 'N' or 'n', the leading n by k ! part of the array A must contain the matrix A, otherwise ! the leading k by n part of the array A must contain the ! matrix A. ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. When TRANS = 'N' or 'n' ! then LDA must be at least max( 1, n ), otherwise LDA must ! be at least max( 1, k ). ! Unchanged on exit. ! ! BETA - DOUBLE PRECISION. ! On entry, BETA specifies the scalar beta. ! Unchanged on exit. ! ! C - DOUBLE PRECISION array of DIMENSION ( LDC, n ). ! Before entry with UPLO = 'U' or 'u', the leading n by n ! upper triangular part of the array C must contain the upper ! triangular part of the symmetric matrix and the strictly ! lower triangular part of C is not referenced. On exit, the ! upper triangular part of the array C is overwritten by the ! upper triangular part of the updated matrix. ! Before entry with UPLO = 'L' or 'l', the leading n by n ! lower triangular part of the array C must contain the lower ! triangular part of the symmetric matrix and the strictly ! upper triangular part of C is not referenced. On exit, the ! lower triangular part of the array C is overwritten by the ! lower triangular part of the updated matrix. ! ! LDC - INTEGER. ! On entry, LDC specifies the first dimension of C as declared ! in the calling (sub) program. LDC must be at least ! max( 1, n ). ! Unchanged on exit. ! ! ! Level 3 Blas routine. ! ! -- Written on 8-February-1989. ! Jack Dongarra, Argonne National Laboratory. ! Iain Duff, AERE Harwell. ! Jeremy Du Croz, Numerical Algorithms Group Ltd. ! Sven Hammarling, Numerical Algorithms Group Ltd. ! ! ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX ! .. Local Scalars .. LOGICAL UPPER INTEGER I, INFO, J, L, NROWA DOUBLE PRECISION TEMP ! .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! IF( LSAME( TRANS, 'N' ) )THEN NROWA = N ELSE NROWA = K END IF UPPER = LSAME( UPLO, 'U' ) ! INFO = 0 IF( ( .NOT.UPPER ).AND. & ( .NOT.LSAME( UPLO , 'L' ) ) )THEN INFO = 1 ELSE IF( ( .NOT.LSAME( TRANS, 'N' ) ).AND. & ( .NOT.LSAME( TRANS, 'T' ) ).AND. & ( .NOT.LSAME( TRANS, 'C' ) ) )THEN INFO = 2 ELSE IF( N .LT.0 )THEN INFO = 3 ELSE IF( K .LT.0 )THEN INFO = 4 ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN INFO = 7 ELSE IF( LDC.LT.MAX( 1, N ) )THEN INFO = 10 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DSYRK ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( ( N.EQ.0 ).OR. & ( ( ( ALPHA.EQ.ZERO ).OR.( K.EQ.0 ) ).AND.( BETA.EQ.ONE ) ) ) & RETURN ! ! And when alpha.eq.zero. ! IF( ALPHA.EQ.ZERO )THEN IF( UPPER )THEN IF( BETA.EQ.ZERO )THEN DO 20, J = 1, N DO 10, I = 1, J C( I, J ) = ZERO 10 CONTINUE 20 CONTINUE ELSE DO 40, J = 1, N DO 30, I = 1, J C( I, J ) = BETA*C( I, J ) 30 CONTINUE 40 CONTINUE END IF ELSE IF( BETA.EQ.ZERO )THEN DO 60, J = 1, N DO 50, I = J, N C( I, J ) = ZERO 50 CONTINUE 60 CONTINUE ELSE DO 80, J = 1, N DO 70, I = J, N C( I, J ) = BETA*C( I, J ) 70 CONTINUE 80 CONTINUE END IF END IF RETURN END IF ! ! Start the operations. ! IF( LSAME( TRANS, 'N' ) )THEN ! ! Form C := alpha*A*A' + beta*C. ! IF( UPPER )THEN DO 130, J = 1, N IF( BETA.EQ.ZERO )THEN DO 90, I = 1, J C( I, J ) = ZERO 90 CONTINUE ELSE IF( BETA.NE.ONE )THEN DO 100, I = 1, J C( I, J ) = BETA*C( I, J ) 100 CONTINUE END IF DO 120, L = 1, K IF( A( J, L ).NE.ZERO )THEN TEMP = ALPHA*A( J, L ) DO 110, I = 1, J C( I, J ) = C( I, J ) + TEMP*A( I, L ) 110 CONTINUE END IF 120 CONTINUE 130 CONTINUE ELSE DO 180, J = 1, N IF( BETA.EQ.ZERO )THEN DO 140, I = J, N C( I, J ) = ZERO 140 CONTINUE ELSE IF( BETA.NE.ONE )THEN DO 150, I = J, N C( I, J ) = BETA*C( I, J ) 150 CONTINUE END IF DO 170, L = 1, K IF( A( J, L ).NE.ZERO )THEN TEMP = ALPHA*A( J, L ) DO 160, I = J, N C( I, J ) = C( I, J ) + TEMP*A( I, L ) 160 CONTINUE END IF 170 CONTINUE 180 CONTINUE END IF ELSE ! ! Form C := alpha*A'*A + beta*C. ! IF( UPPER )THEN DO 210, J = 1, N DO 200, I = 1, J TEMP = ZERO DO 190, L = 1, K TEMP = TEMP + A( L, I )*A( L, J ) 190 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = ALPHA*TEMP ELSE C( I, J ) = ALPHA*TEMP + BETA*C( I, J ) END IF 200 CONTINUE 210 CONTINUE ELSE DO 240, J = 1, N DO 230, I = J, N TEMP = ZERO DO 220, L = 1, K TEMP = TEMP + A( L, I )*A( L, J ) 220 CONTINUE IF( BETA.EQ.ZERO )THEN C( I, J ) = ALPHA*TEMP ELSE C( I, J ) = ALPHA*TEMP + BETA*C( I, J ) END IF 230 CONTINUE 240 CONTINUE END IF END IF ! RETURN ! ! End of DSYRK . ! END SUBROUTINE DTBMV ( UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX ) ! .. Scalar Arguments .. INTEGER INCX, K, LDA, N CHARACTER*1 DIAG, TRANS, UPLO ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), X( * ) ! .. ! ! Purpose ! ======= ! ! DTBMV performs one of the matrix-vector operations ! ! x := A*x, or x := A'*x, ! ! where x is an n element vector and A is an n by n unit, or non-unit, ! upper or lower triangular band matrix, with ( k + 1 ) diagonals. ! ! Parameters ! ========== ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the matrix is an upper or ! lower triangular matrix as follows: ! ! UPLO = 'U' or 'u' A is an upper triangular matrix. ! ! UPLO = 'L' or 'l' A is a lower triangular matrix. ! ! Unchanged on exit. ! ! TRANS - CHARACTER*1. ! On entry, TRANS specifies the operation to be performed as ! follows: ! ! TRANS = 'N' or 'n' x := A*x. ! ! TRANS = 'T' or 't' x := A'*x. ! ! TRANS = 'C' or 'c' x := A'*x. ! ! Unchanged on exit. ! ! DIAG - CHARACTER*1. ! On entry, DIAG specifies whether or not A is unit ! triangular as follows: ! ! DIAG = 'U' or 'u' A is assumed to be unit triangular. ! ! DIAG = 'N' or 'n' A is not assumed to be unit ! triangular. ! ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the order of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! K - INTEGER. ! On entry with UPLO = 'U' or 'u', K specifies the number of ! super-diagonals of the matrix A. ! On entry with UPLO = 'L' or 'l', K specifies the number of ! sub-diagonals of the matrix A. ! K must satisfy 0 .le. K. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, n ). ! Before entry with UPLO = 'U' or 'u', the leading ( k + 1 ) ! by n part of the array A must contain the upper triangular ! band part of the matrix of coefficients, supplied column by ! column, with the leading diagonal of the matrix in row ! ( k + 1 ) of the array, the first super-diagonal starting at ! position 2 in row k, and so on. The top left k by k triangle ! of the array A is not referenced. ! The following program segment will transfer an upper ! triangular band matrix from conventional full matrix storage ! to band storage: ! ! DO 20, J = 1, N ! M = K + 1 - J ! DO 10, I = MAX( 1, J - K ), J ! A( M + I, J ) = matrix( I, J ) ! 10 CONTINUE ! 20 CONTINUE ! ! Before entry with UPLO = 'L' or 'l', the leading ( k + 1 ) ! by n part of the array A must contain the lower triangular ! band part of the matrix of coefficients, supplied column by ! column, with the leading diagonal of the matrix in row 1 of ! the array, the first sub-diagonal starting at position 1 in ! row 2, and so on. The bottom right k by k triangle of the ! array A is not referenced. ! The following program segment will transfer a lower ! triangular band matrix from conventional full matrix storage ! to band storage: ! ! DO 20, J = 1, N ! M = 1 - J ! DO 10, I = J, MIN( N, J + K ) ! A( M + I, J ) = matrix( I, J ) ! 10 CONTINUE ! 20 CONTINUE ! ! Note that when DIAG = 'U' or 'u' the elements of the array A ! corresponding to the diagonal elements of the matrix are not ! referenced, but are assumed to be unity. ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. LDA must be at least ! ( k + 1 ). ! Unchanged on exit. ! ! X - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCX ) ). ! Before entry, the incremented array X must contain the n ! element vector x. On exit, X is overwritten with the ! tranformed vector x. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. Local Scalars .. DOUBLE PRECISION TEMP INTEGER I, INFO, IX, J, JX, KPLUS1, KX, L LOGICAL NOUNIT ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( UPLO , 'U' ).AND. & .NOT.LSAME( UPLO , 'L' ) )THEN INFO = 1 ELSE IF( .NOT.LSAME( TRANS, 'N' ).AND. & .NOT.LSAME( TRANS, 'T' ).AND. & .NOT.LSAME( TRANS, 'C' ) )THEN INFO = 2 ELSE IF( .NOT.LSAME( DIAG , 'U' ).AND. & .NOT.LSAME( DIAG , 'N' ) )THEN INFO = 3 ELSE IF( N.LT.0 )THEN INFO = 4 ELSE IF( K.LT.0 )THEN INFO = 5 ELSE IF( LDA.LT.( K + 1 ) )THEN INFO = 7 ELSE IF( INCX.EQ.0 )THEN INFO = 9 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DTBMV ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( N.EQ.0 ) & RETURN ! NOUNIT = LSAME( DIAG, 'N' ) ! ! Set up the start point in X if the increment is not unity. This ! will be ( N - 1 )*INCX too small for descending loops. ! IF( INCX.LE.0 )THEN KX = 1 - ( N - 1 )*INCX ELSE IF( INCX.NE.1 )THEN KX = 1 END IF ! ! Start the operations. In this version the elements of A are ! accessed sequentially with one pass through A. ! IF( LSAME( TRANS, 'N' ) )THEN ! ! Form x := A*x. ! IF( LSAME( UPLO, 'U' ) )THEN KPLUS1 = K + 1 IF( INCX.EQ.1 )THEN DO 20, J = 1, N IF( X( J ).NE.ZERO )THEN TEMP = X( J ) L = KPLUS1 - J DO 10, I = MAX( 1, J - K ), J - 1 X( I ) = X( I ) + TEMP*A( L + I, J ) 10 CONTINUE IF( NOUNIT ) & X( J ) = X( J )*A( KPLUS1, J ) END IF 20 CONTINUE ELSE JX = KX DO 40, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = X( JX ) IX = KX L = KPLUS1 - J DO 30, I = MAX( 1, J - K ), J - 1 X( IX ) = X( IX ) + TEMP*A( L + I, J ) IX = IX + INCX 30 CONTINUE IF( NOUNIT ) & X( JX ) = X( JX )*A( KPLUS1, J ) END IF JX = JX + INCX IF( J.GT.K ) & KX = KX + INCX 40 CONTINUE END IF ELSE IF( INCX.EQ.1 )THEN DO 60, J = N, 1, -1 IF( X( J ).NE.ZERO )THEN TEMP = X( J ) L = 1 - J DO 50, I = MIN( N, J + K ), J + 1, -1 X( I ) = X( I ) + TEMP*A( L + I, J ) 50 CONTINUE IF( NOUNIT ) & X( J ) = X( J )*A( 1, J ) END IF 60 CONTINUE ELSE KX = KX + ( N - 1 )*INCX JX = KX DO 80, J = N, 1, -1 IF( X( JX ).NE.ZERO )THEN TEMP = X( JX ) IX = KX L = 1 - J DO 70, I = MIN( N, J + K ), J + 1, -1 X( IX ) = X( IX ) + TEMP*A( L + I, J ) IX = IX - INCX 70 CONTINUE IF( NOUNIT ) & X( JX ) = X( JX )*A( 1, J ) END IF JX = JX - INCX IF( ( N - J ).GE.K ) & KX = KX - INCX 80 CONTINUE END IF END IF ELSE ! ! Form x := A'*x. ! IF( LSAME( UPLO, 'U' ) )THEN KPLUS1 = K + 1 IF( INCX.EQ.1 )THEN DO 100, J = N, 1, -1 TEMP = X( J ) L = KPLUS1 - J IF( NOUNIT ) & TEMP = TEMP*A( KPLUS1, J ) DO 90, I = J - 1, MAX( 1, J - K ), -1 TEMP = TEMP + A( L + I, J )*X( I ) 90 CONTINUE X( J ) = TEMP 100 CONTINUE ELSE KX = KX + ( N - 1 )*INCX JX = KX DO 120, J = N, 1, -1 TEMP = X( JX ) KX = KX - INCX IX = KX L = KPLUS1 - J IF( NOUNIT ) & TEMP = TEMP*A( KPLUS1, J ) DO 110, I = J - 1, MAX( 1, J - K ), -1 TEMP = TEMP + A( L + I, J )*X( IX ) IX = IX - INCX 110 CONTINUE X( JX ) = TEMP JX = JX - INCX 120 CONTINUE END IF ELSE IF( INCX.EQ.1 )THEN DO 140, J = 1, N TEMP = X( J ) L = 1 - J IF( NOUNIT ) & TEMP = TEMP*A( 1, J ) DO 130, I = J + 1, MIN( N, J + K ) TEMP = TEMP + A( L + I, J )*X( I ) 130 CONTINUE X( J ) = TEMP 140 CONTINUE ELSE JX = KX DO 160, J = 1, N TEMP = X( JX ) KX = KX + INCX IX = KX L = 1 - J IF( NOUNIT ) & TEMP = TEMP*A( 1, J ) DO 150, I = J + 1, MIN( N, J + K ) TEMP = TEMP + A( L + I, J )*X( IX ) IX = IX + INCX 150 CONTINUE X( JX ) = TEMP JX = JX + INCX 160 CONTINUE END IF END IF END IF ! RETURN ! ! End of DTBMV . ! END SUBROUTINE DTBSV ( UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX ) ! .. Scalar Arguments .. INTEGER INCX, K, LDA, N CHARACTER*1 DIAG, TRANS, UPLO ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), X( * ) ! .. ! ! Purpose ! ======= ! ! DTBSV solves one of the systems of equations ! ! A*x = b, or A'*x = b, ! ! where b and x are n element vectors and A is an n by n unit, or ! non-unit, upper or lower triangular band matrix, with ( k + 1 ) ! diagonals. ! ! No test for singularity or near-singularity is included in this ! routine. Such tests must be performed before calling this routine. ! ! Parameters ! ========== ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the matrix is an upper or ! lower triangular matrix as follows: ! ! UPLO = 'U' or 'u' A is an upper triangular matrix. ! ! UPLO = 'L' or 'l' A is a lower triangular matrix. ! ! Unchanged on exit. ! ! TRANS - CHARACTER*1. ! On entry, TRANS specifies the equations to be solved as ! follows: ! ! TRANS = 'N' or 'n' A*x = b. ! ! TRANS = 'T' or 't' A'*x = b. ! ! TRANS = 'C' or 'c' A'*x = b. ! ! Unchanged on exit. ! ! DIAG - CHARACTER*1. ! On entry, DIAG specifies whether or not A is unit ! triangular as follows: ! ! DIAG = 'U' or 'u' A is assumed to be unit triangular. ! ! DIAG = 'N' or 'n' A is not assumed to be unit ! triangular. ! ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the order of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! K - INTEGER. ! On entry with UPLO = 'U' or 'u', K specifies the number of ! super-diagonals of the matrix A. ! On entry with UPLO = 'L' or 'l', K specifies the number of ! sub-diagonals of the matrix A. ! K must satisfy 0 .le. K. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, n ). ! Before entry with UPLO = 'U' or 'u', the leading ( k + 1 ) ! by n part of the array A must contain the upper triangular ! band part of the matrix of coefficients, supplied column by ! column, with the leading diagonal of the matrix in row ! ( k + 1 ) of the array, the first super-diagonal starting at ! position 2 in row k, and so on. The top left k by k triangle ! of the array A is not referenced. ! The following program segment will transfer an upper ! triangular band matrix from conventional full matrix storage ! to band storage: ! ! DO 20, J = 1, N ! M = K + 1 - J ! DO 10, I = MAX( 1, J - K ), J ! A( M + I, J ) = matrix( I, J ) ! 10 CONTINUE ! 20 CONTINUE ! ! Before entry with UPLO = 'L' or 'l', the leading ( k + 1 ) ! by n part of the array A must contain the lower triangular ! band part of the matrix of coefficients, supplied column by ! column, with the leading diagonal of the matrix in row 1 of ! the array, the first sub-diagonal starting at position 1 in ! row 2, and so on. The bottom right k by k triangle of the ! array A is not referenced. ! The following program segment will transfer a lower ! triangular band matrix from conventional full matrix storage ! to band storage: ! ! DO 20, J = 1, N ! M = 1 - J ! DO 10, I = J, MIN( N, J + K ) ! A( M + I, J ) = matrix( I, J ) ! 10 CONTINUE ! 20 CONTINUE ! ! Note that when DIAG = 'U' or 'u' the elements of the array A ! corresponding to the diagonal elements of the matrix are not ! referenced, but are assumed to be unity. ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. LDA must be at least ! ( k + 1 ). ! Unchanged on exit. ! ! X - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCX ) ). ! Before entry, the incremented array X must contain the n ! element right-hand side vector b. On exit, X is overwritten ! with the solution vector x. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. Local Scalars .. DOUBLE PRECISION TEMP INTEGER I, INFO, IX, J, JX, KPLUS1, KX, L LOGICAL NOUNIT ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( UPLO , 'U' ).AND. & .NOT.LSAME( UPLO , 'L' ) )THEN INFO = 1 ELSE IF( .NOT.LSAME( TRANS, 'N' ).AND. & .NOT.LSAME( TRANS, 'T' ).AND. & .NOT.LSAME( TRANS, 'C' ) )THEN INFO = 2 ELSE IF( .NOT.LSAME( DIAG , 'U' ).AND. & .NOT.LSAME( DIAG , 'N' ) )THEN INFO = 3 ELSE IF( N.LT.0 )THEN INFO = 4 ELSE IF( K.LT.0 )THEN INFO = 5 ELSE IF( LDA.LT.( K + 1 ) )THEN INFO = 7 ELSE IF( INCX.EQ.0 )THEN INFO = 9 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DTBSV ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( N.EQ.0 ) & RETURN ! NOUNIT = LSAME( DIAG, 'N' ) ! ! Set up the start point in X if the increment is not unity. This ! will be ( N - 1 )*INCX too small for descending loops. ! IF( INCX.LE.0 )THEN KX = 1 - ( N - 1 )*INCX ELSE IF( INCX.NE.1 )THEN KX = 1 END IF ! ! Start the operations. In this version the elements of A are ! accessed by sequentially with one pass through A. ! IF( LSAME( TRANS, 'N' ) )THEN ! ! Form x := inv( A )*x. ! IF( LSAME( UPLO, 'U' ) )THEN KPLUS1 = K + 1 IF( INCX.EQ.1 )THEN DO 20, J = N, 1, -1 IF( X( J ).NE.ZERO )THEN L = KPLUS1 - J IF( NOUNIT ) & X( J ) = X( J )/A( KPLUS1, J ) TEMP = X( J ) DO 10, I = J - 1, MAX( 1, J - K ), -1 X( I ) = X( I ) - TEMP*A( L + I, J ) 10 CONTINUE END IF 20 CONTINUE ELSE KX = KX + ( N - 1 )*INCX JX = KX DO 40, J = N, 1, -1 KX = KX - INCX IF( X( JX ).NE.ZERO )THEN IX = KX L = KPLUS1 - J IF( NOUNIT ) & X( JX ) = X( JX )/A( KPLUS1, J ) TEMP = X( JX ) DO 30, I = J - 1, MAX( 1, J - K ), -1 X( IX ) = X( IX ) - TEMP*A( L + I, J ) IX = IX - INCX 30 CONTINUE END IF JX = JX - INCX 40 CONTINUE END IF ELSE IF( INCX.EQ.1 )THEN DO 60, J = 1, N IF( X( J ).NE.ZERO )THEN L = 1 - J IF( NOUNIT ) & X( J ) = X( J )/A( 1, J ) TEMP = X( J ) DO 50, I = J + 1, MIN( N, J + K ) X( I ) = X( I ) - TEMP*A( L + I, J ) 50 CONTINUE END IF 60 CONTINUE ELSE JX = KX DO 80, J = 1, N KX = KX + INCX IF( X( JX ).NE.ZERO )THEN IX = KX L = 1 - J IF( NOUNIT ) & X( JX ) = X( JX )/A( 1, J ) TEMP = X( JX ) DO 70, I = J + 1, MIN( N, J + K ) X( IX ) = X( IX ) - TEMP*A( L + I, J ) IX = IX + INCX 70 CONTINUE END IF JX = JX + INCX 80 CONTINUE END IF END IF ELSE ! ! Form x := inv( A')*x. ! IF( LSAME( UPLO, 'U' ) )THEN KPLUS1 = K + 1 IF( INCX.EQ.1 )THEN DO 100, J = 1, N TEMP = X( J ) L = KPLUS1 - J DO 90, I = MAX( 1, J - K ), J - 1 TEMP = TEMP - A( L + I, J )*X( I ) 90 CONTINUE IF( NOUNIT ) & TEMP = TEMP/A( KPLUS1, J ) X( J ) = TEMP 100 CONTINUE ELSE JX = KX DO 120, J = 1, N TEMP = X( JX ) IX = KX L = KPLUS1 - J DO 110, I = MAX( 1, J - K ), J - 1 TEMP = TEMP - A( L + I, J )*X( IX ) IX = IX + INCX 110 CONTINUE IF( NOUNIT ) & TEMP = TEMP/A( KPLUS1, J ) X( JX ) = TEMP JX = JX + INCX IF( J.GT.K ) & KX = KX + INCX 120 CONTINUE END IF ELSE IF( INCX.EQ.1 )THEN DO 140, J = N, 1, -1 TEMP = X( J ) L = 1 - J DO 130, I = MIN( N, J + K ), J + 1, -1 TEMP = TEMP - A( L + I, J )*X( I ) 130 CONTINUE IF( NOUNIT ) & TEMP = TEMP/A( 1, J ) X( J ) = TEMP 140 CONTINUE ELSE KX = KX + ( N - 1 )*INCX JX = KX DO 160, J = N, 1, -1 TEMP = X( JX ) IX = KX L = 1 - J DO 150, I = MIN( N, J + K ), J + 1, -1 TEMP = TEMP - A( L + I, J )*X( IX ) IX = IX - INCX 150 CONTINUE IF( NOUNIT ) & TEMP = TEMP/A( 1, J ) X( JX ) = TEMP JX = JX - INCX IF( ( N - J ).GE.K ) & KX = KX - INCX 160 CONTINUE END IF END IF END IF ! RETURN ! ! End of DTBSV . ! END SUBROUTINE DTPMV ( UPLO, TRANS, DIAG, N, AP, X, INCX ) ! .. Scalar Arguments .. INTEGER INCX, N CHARACTER*1 DIAG, TRANS, UPLO ! .. Array Arguments .. DOUBLE PRECISION AP( * ), X( * ) ! .. ! ! Purpose ! ======= ! ! DTPMV performs one of the matrix-vector operations ! ! x := A*x, or x := A'*x, ! ! where x is an n element vector and A is an n by n unit, or non-unit, ! upper or lower triangular matrix, supplied in packed form. ! ! Parameters ! ========== ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the matrix is an upper or ! lower triangular matrix as follows: ! ! UPLO = 'U' or 'u' A is an upper triangular matrix. ! ! UPLO = 'L' or 'l' A is a lower triangular matrix. ! ! Unchanged on exit. ! ! TRANS - CHARACTER*1. ! On entry, TRANS specifies the operation to be performed as ! follows: ! ! TRANS = 'N' or 'n' x := A*x. ! ! TRANS = 'T' or 't' x := A'*x. ! ! TRANS = 'C' or 'c' x := A'*x. ! ! Unchanged on exit. ! ! DIAG - CHARACTER*1. ! On entry, DIAG specifies whether or not A is unit ! triangular as follows: ! ! DIAG = 'U' or 'u' A is assumed to be unit triangular. ! ! DIAG = 'N' or 'n' A is not assumed to be unit ! triangular. ! ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the order of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! AP - DOUBLE PRECISION array of DIMENSION at least ! ( ( n*( n + 1 ) )/2 ). ! Before entry with UPLO = 'U' or 'u', the array AP must ! contain the upper triangular matrix packed sequentially, ! column by column, so that AP( 1 ) contains a( 1, 1 ), ! AP( 2 ) and AP( 3 ) contain a( 1, 2 ) and a( 2, 2 ) ! respectively, and so on. ! Before entry with UPLO = 'L' or 'l', the array AP must ! contain the lower triangular matrix packed sequentially, ! column by column, so that AP( 1 ) contains a( 1, 1 ), ! AP( 2 ) and AP( 3 ) contain a( 2, 1 ) and a( 3, 1 ) ! respectively, and so on. ! Note that when DIAG = 'U' or 'u', the diagonal elements of ! A are not referenced, but are assumed to be unity. ! Unchanged on exit. ! ! X - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCX ) ). ! Before entry, the incremented array X must contain the n ! element vector x. On exit, X is overwritten with the ! tranformed vector x. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. Local Scalars .. DOUBLE PRECISION TEMP INTEGER I, INFO, IX, J, JX, K, KK, KX LOGICAL NOUNIT ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( UPLO , 'U' ).AND. & .NOT.LSAME( UPLO , 'L' ) )THEN INFO = 1 ELSE IF( .NOT.LSAME( TRANS, 'N' ).AND. & .NOT.LSAME( TRANS, 'T' ).AND. & .NOT.LSAME( TRANS, 'C' ) )THEN INFO = 2 ELSE IF( .NOT.LSAME( DIAG , 'U' ).AND. & .NOT.LSAME( DIAG , 'N' ) )THEN INFO = 3 ELSE IF( N.LT.0 )THEN INFO = 4 ELSE IF( INCX.EQ.0 )THEN INFO = 7 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DTPMV ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( N.EQ.0 ) & RETURN ! NOUNIT = LSAME( DIAG, 'N' ) ! ! Set up the start point in X if the increment is not unity. This ! will be ( N - 1 )*INCX too small for descending loops. ! IF( INCX.LE.0 )THEN KX = 1 - ( N - 1 )*INCX ELSE IF( INCX.NE.1 )THEN KX = 1 END IF ! ! Start the operations. In this version the elements of AP are ! accessed sequentially with one pass through AP. ! IF( LSAME( TRANS, 'N' ) )THEN ! ! Form x:= A*x. ! IF( LSAME( UPLO, 'U' ) )THEN KK =1 IF( INCX.EQ.1 )THEN DO 20, J = 1, N IF( X( J ).NE.ZERO )THEN TEMP = X( J ) K = KK DO 10, I = 1, J - 1 X( I ) = X( I ) + TEMP*AP( K ) K = K + 1 10 CONTINUE IF( NOUNIT ) & X( J ) = X( J )*AP( KK + J - 1 ) END IF KK = KK + J 20 CONTINUE ELSE JX = KX DO 40, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = X( JX ) IX = KX DO 30, K = KK, KK + J - 2 X( IX ) = X( IX ) + TEMP*AP( K ) IX = IX + INCX 30 CONTINUE IF( NOUNIT ) & X( JX ) = X( JX )*AP( KK + J - 1 ) END IF JX = JX + INCX KK = KK + J 40 CONTINUE END IF ELSE KK = ( N*( N + 1 ) )/2 IF( INCX.EQ.1 )THEN DO 60, J = N, 1, -1 IF( X( J ).NE.ZERO )THEN TEMP = X( J ) K = KK DO 50, I = N, J + 1, -1 X( I ) = X( I ) + TEMP*AP( K ) K = K - 1 50 CONTINUE IF( NOUNIT ) & X( J ) = X( J )*AP( KK - N + J ) END IF KK = KK - ( N - J + 1 ) 60 CONTINUE ELSE KX = KX + ( N - 1 )*INCX JX = KX DO 80, J = N, 1, -1 IF( X( JX ).NE.ZERO )THEN TEMP = X( JX ) IX = KX DO 70, K = KK, KK - ( N - ( J + 1 ) ), -1 X( IX ) = X( IX ) + TEMP*AP( K ) IX = IX - INCX 70 CONTINUE IF( NOUNIT ) & X( JX ) = X( JX )*AP( KK - N + J ) END IF JX = JX - INCX KK = KK - ( N - J + 1 ) 80 CONTINUE END IF END IF ELSE ! ! Form x := A'*x. ! IF( LSAME( UPLO, 'U' ) )THEN KK = ( N*( N + 1 ) )/2 IF( INCX.EQ.1 )THEN DO 100, J = N, 1, -1 TEMP = X( J ) IF( NOUNIT ) & TEMP = TEMP*AP( KK ) K = KK - 1 DO 90, I = J - 1, 1, -1 TEMP = TEMP + AP( K )*X( I ) K = K - 1 90 CONTINUE X( J ) = TEMP KK = KK - J 100 CONTINUE ELSE JX = KX + ( N - 1 )*INCX DO 120, J = N, 1, -1 TEMP = X( JX ) IX = JX IF( NOUNIT ) & TEMP = TEMP*AP( KK ) DO 110, K = KK - 1, KK - J + 1, -1 IX = IX - INCX TEMP = TEMP + AP( K )*X( IX ) 110 CONTINUE X( JX ) = TEMP JX = JX - INCX KK = KK - J 120 CONTINUE END IF ELSE KK = 1 IF( INCX.EQ.1 )THEN DO 140, J = 1, N TEMP = X( J ) IF( NOUNIT ) & TEMP = TEMP*AP( KK ) K = KK + 1 DO 130, I = J + 1, N TEMP = TEMP + AP( K )*X( I ) K = K + 1 130 CONTINUE X( J ) = TEMP KK = KK + ( N - J + 1 ) 140 CONTINUE ELSE JX = KX DO 160, J = 1, N TEMP = X( JX ) IX = JX IF( NOUNIT ) & TEMP = TEMP*AP( KK ) DO 150, K = KK + 1, KK + N - J IX = IX + INCX TEMP = TEMP + AP( K )*X( IX ) 150 CONTINUE X( JX ) = TEMP JX = JX + INCX KK = KK + ( N - J + 1 ) 160 CONTINUE END IF END IF END IF ! RETURN ! ! End of DTPMV . ! END SUBROUTINE DTPSV ( UPLO, TRANS, DIAG, N, AP, X, INCX ) ! .. Scalar Arguments .. INTEGER INCX, N CHARACTER*1 DIAG, TRANS, UPLO ! .. Array Arguments .. DOUBLE PRECISION AP( * ), X( * ) ! .. ! ! Purpose ! ======= ! ! DTPSV solves one of the systems of equations ! ! A*x = b, or A'*x = b, ! ! where b and x are n element vectors and A is an n by n unit, or ! non-unit, upper or lower triangular matrix, supplied in packed form. ! ! No test for singularity or near-singularity is included in this ! routine. Such tests must be performed before calling this routine. ! ! Parameters ! ========== ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the matrix is an upper or ! lower triangular matrix as follows: ! ! UPLO = 'U' or 'u' A is an upper triangular matrix. ! ! UPLO = 'L' or 'l' A is a lower triangular matrix. ! ! Unchanged on exit. ! ! TRANS - CHARACTER*1. ! On entry, TRANS specifies the equations to be solved as ! follows: ! ! TRANS = 'N' or 'n' A*x = b. ! ! TRANS = 'T' or 't' A'*x = b. ! ! TRANS = 'C' or 'c' A'*x = b. ! ! Unchanged on exit. ! ! DIAG - CHARACTER*1. ! On entry, DIAG specifies whether or not A is unit ! triangular as follows: ! ! DIAG = 'U' or 'u' A is assumed to be unit triangular. ! ! DIAG = 'N' or 'n' A is not assumed to be unit ! triangular. ! ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the order of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! AP - DOUBLE PRECISION array of DIMENSION at least ! ( ( n*( n + 1 ) )/2 ). ! Before entry with UPLO = 'U' or 'u', the array AP must ! contain the upper triangular matrix packed sequentially, ! column by column, so that AP( 1 ) contains a( 1, 1 ), ! AP( 2 ) and AP( 3 ) contain a( 1, 2 ) and a( 2, 2 ) ! respectively, and so on. ! Before entry with UPLO = 'L' or 'l', the array AP must ! contain the lower triangular matrix packed sequentially, ! column by column, so that AP( 1 ) contains a( 1, 1 ), ! AP( 2 ) and AP( 3 ) contain a( 2, 1 ) and a( 3, 1 ) ! respectively, and so on. ! Note that when DIAG = 'U' or 'u', the diagonal elements of ! A are not referenced, but are assumed to be unity. ! Unchanged on exit. ! ! X - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCX ) ). ! Before entry, the incremented array X must contain the n ! element right-hand side vector b. On exit, X is overwritten ! with the solution vector x. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. Local Scalars .. DOUBLE PRECISION TEMP INTEGER I, INFO, IX, J, JX, K, KK, KX LOGICAL NOUNIT ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( UPLO , 'U' ).AND. & .NOT.LSAME( UPLO , 'L' ) )THEN INFO = 1 ELSE IF( .NOT.LSAME( TRANS, 'N' ).AND. & .NOT.LSAME( TRANS, 'T' ).AND. & .NOT.LSAME( TRANS, 'C' ) )THEN INFO = 2 ELSE IF( .NOT.LSAME( DIAG , 'U' ).AND. & .NOT.LSAME( DIAG , 'N' ) )THEN INFO = 3 ELSE IF( N.LT.0 )THEN INFO = 4 ELSE IF( INCX.EQ.0 )THEN INFO = 7 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DTPSV ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( N.EQ.0 ) & RETURN ! NOUNIT = LSAME( DIAG, 'N' ) ! ! Set up the start point in X if the increment is not unity. This ! will be ( N - 1 )*INCX too small for descending loops. ! IF( INCX.LE.0 )THEN KX = 1 - ( N - 1 )*INCX ELSE IF( INCX.NE.1 )THEN KX = 1 END IF ! ! Start the operations. In this version the elements of AP are ! accessed sequentially with one pass through AP. ! IF( LSAME( TRANS, 'N' ) )THEN ! ! Form x := inv( A )*x. ! IF( LSAME( UPLO, 'U' ) )THEN KK = ( N*( N + 1 ) )/2 IF( INCX.EQ.1 )THEN DO 20, J = N, 1, -1 IF( X( J ).NE.ZERO )THEN IF( NOUNIT ) & X( J ) = X( J )/AP( KK ) TEMP = X( J ) K = KK - 1 DO 10, I = J - 1, 1, -1 X( I ) = X( I ) - TEMP*AP( K ) K = K - 1 10 CONTINUE END IF KK = KK - J 20 CONTINUE ELSE JX = KX + ( N - 1 )*INCX DO 40, J = N, 1, -1 IF( X( JX ).NE.ZERO )THEN IF( NOUNIT ) & X( JX ) = X( JX )/AP( KK ) TEMP = X( JX ) IX = JX DO 30, K = KK - 1, KK - J + 1, -1 IX = IX - INCX X( IX ) = X( IX ) - TEMP*AP( K ) 30 CONTINUE END IF JX = JX - INCX KK = KK - J 40 CONTINUE END IF ELSE KK = 1 IF( INCX.EQ.1 )THEN DO 60, J = 1, N IF( X( J ).NE.ZERO )THEN IF( NOUNIT ) & X( J ) = X( J )/AP( KK ) TEMP = X( J ) K = KK + 1 DO 50, I = J + 1, N X( I ) = X( I ) - TEMP*AP( K ) K = K + 1 50 CONTINUE END IF KK = KK + ( N - J + 1 ) 60 CONTINUE ELSE JX = KX DO 80, J = 1, N IF( X( JX ).NE.ZERO )THEN IF( NOUNIT ) & X( JX ) = X( JX )/AP( KK ) TEMP = X( JX ) IX = JX DO 70, K = KK + 1, KK + N - J IX = IX + INCX X( IX ) = X( IX ) - TEMP*AP( K ) 70 CONTINUE END IF JX = JX + INCX KK = KK + ( N - J + 1 ) 80 CONTINUE END IF END IF ELSE ! ! Form x := inv( A' )*x. ! IF( LSAME( UPLO, 'U' ) )THEN KK = 1 IF( INCX.EQ.1 )THEN DO 100, J = 1, N TEMP = X( J ) K = KK DO 90, I = 1, J - 1 TEMP = TEMP - AP( K )*X( I ) K = K + 1 90 CONTINUE IF( NOUNIT ) & TEMP = TEMP/AP( KK + J - 1 ) X( J ) = TEMP KK = KK + J 100 CONTINUE ELSE JX = KX DO 120, J = 1, N TEMP = X( JX ) IX = KX DO 110, K = KK, KK + J - 2 TEMP = TEMP - AP( K )*X( IX ) IX = IX + INCX 110 CONTINUE IF( NOUNIT ) & TEMP = TEMP/AP( KK + J - 1 ) X( JX ) = TEMP JX = JX + INCX KK = KK + J 120 CONTINUE END IF ELSE KK = ( N*( N + 1 ) )/2 IF( INCX.EQ.1 )THEN DO 140, J = N, 1, -1 TEMP = X( J ) K = KK DO 130, I = N, J + 1, -1 TEMP = TEMP - AP( K )*X( I ) K = K - 1 130 CONTINUE IF( NOUNIT ) & TEMP = TEMP/AP( KK - N + J ) X( J ) = TEMP KK = KK - ( N - J + 1 ) 140 CONTINUE ELSE KX = KX + ( N - 1 )*INCX JX = KX DO 160, J = N, 1, -1 TEMP = X( JX ) IX = KX DO 150, K = KK, KK - ( N - ( J + 1 ) ), -1 TEMP = TEMP - AP( K )*X( IX ) IX = IX - INCX 150 CONTINUE IF( NOUNIT ) & TEMP = TEMP/AP( KK - N + J ) X( JX ) = TEMP JX = JX - INCX KK = KK - (N - J + 1 ) 160 CONTINUE END IF END IF END IF ! RETURN ! ! End of DTPSV . ! END SUBROUTINE DTRMM ( SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, & B, LDB ) ! .. Scalar Arguments .. CHARACTER*1 SIDE, UPLO, TRANSA, DIAG INTEGER M, N, LDA, LDB DOUBLE PRECISION ALPHA ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DTRMM performs one of the matrix-matrix operations ! ! B := alpha*op( A )*B, or B := alpha*B*op( A ), ! ! where alpha is a scalar, B is an m by n matrix, A is a unit, or ! non-unit, upper or lower triangular matrix and op( A ) is one of ! ! op( A ) = A or op( A ) = A'. ! ! Parameters ! ========== ! ! SIDE - CHARACTER*1. ! On entry, SIDE specifies whether op( A ) multiplies B from ! the left or right as follows: ! ! SIDE = 'L' or 'l' B := alpha*op( A )*B. ! ! SIDE = 'R' or 'r' B := alpha*B*op( A ). ! ! Unchanged on exit. ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the matrix A is an upper or ! lower triangular matrix as follows: ! ! UPLO = 'U' or 'u' A is an upper triangular matrix. ! ! UPLO = 'L' or 'l' A is a lower triangular matrix. ! ! Unchanged on exit. ! ! TRANSA - CHARACTER*1. ! On entry, TRANSA specifies the form of op( A ) to be used in ! the matrix multiplication as follows: ! ! TRANSA = 'N' or 'n' op( A ) = A. ! ! TRANSA = 'T' or 't' op( A ) = A'. ! ! TRANSA = 'C' or 'c' op( A ) = A'. ! ! Unchanged on exit. ! ! DIAG - CHARACTER*1. ! On entry, DIAG specifies whether or not A is unit triangular ! as follows: ! ! DIAG = 'U' or 'u' A is assumed to be unit triangular. ! ! DIAG = 'N' or 'n' A is not assumed to be unit ! triangular. ! ! Unchanged on exit. ! ! M - INTEGER. ! On entry, M specifies the number of rows of B. M must be at ! least zero. ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the number of columns of B. N must be ! at least zero. ! Unchanged on exit. ! ! ALPHA - DOUBLE PRECISION. ! On entry, ALPHA specifies the scalar alpha. When alpha is ! zero then A is not referenced and B need not be set before ! entry. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, k ), where k is m ! when SIDE = 'L' or 'l' and is n when SIDE = 'R' or 'r'. ! Before entry with UPLO = 'U' or 'u', the leading k by k ! upper triangular part of the array A must contain the upper ! triangular matrix and the strictly lower triangular part of ! A is not referenced. ! Before entry with UPLO = 'L' or 'l', the leading k by k ! lower triangular part of the array A must contain the lower ! triangular matrix and the strictly upper triangular part of ! A is not referenced. ! Note that when DIAG = 'U' or 'u', the diagonal elements of ! A are not referenced either, but are assumed to be unity. ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. When SIDE = 'L' or 'l' then ! LDA must be at least max( 1, m ), when SIDE = 'R' or 'r' ! then LDA must be at least max( 1, n ). ! Unchanged on exit. ! ! B - DOUBLE PRECISION array of DIMENSION ( LDB, n ). ! Before entry, the leading m by n part of the array B must ! contain the matrix B, and on exit is overwritten by the ! transformed matrix. ! ! LDB - INTEGER. ! On entry, LDB specifies the first dimension of B as declared ! in the calling (sub) program. LDB must be at least ! max( 1, m ). ! Unchanged on exit. ! ! ! Level 3 Blas routine. ! ! -- Written on 8-February-1989. ! Jack Dongarra, Argonne National Laboratory. ! Iain Duff, AERE Harwell. ! Jeremy Du Croz, Numerical Algorithms Group Ltd. ! Sven Hammarling, Numerical Algorithms Group Ltd. ! ! ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX ! .. Local Scalars .. LOGICAL LSIDE, NOUNIT, UPPER INTEGER I, INFO, J, K, NROWA DOUBLE PRECISION TEMP ! .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! LSIDE = LSAME( SIDE , 'L' ) IF( LSIDE )THEN NROWA = M ELSE NROWA = N END IF NOUNIT = LSAME( DIAG , 'N' ) UPPER = LSAME( UPLO , 'U' ) ! INFO = 0 IF( ( .NOT.LSIDE ).AND. & ( .NOT.LSAME( SIDE , 'R' ) ) )THEN INFO = 1 ELSE IF( ( .NOT.UPPER ).AND. & ( .NOT.LSAME( UPLO , 'L' ) ) )THEN INFO = 2 ELSE IF( ( .NOT.LSAME( TRANSA, 'N' ) ).AND. & ( .NOT.LSAME( TRANSA, 'T' ) ).AND. & ( .NOT.LSAME( TRANSA, 'C' ) ) )THEN INFO = 3 ELSE IF( ( .NOT.LSAME( DIAG , 'U' ) ).AND. & ( .NOT.LSAME( DIAG , 'N' ) ) )THEN INFO = 4 ELSE IF( M .LT.0 )THEN INFO = 5 ELSE IF( N .LT.0 )THEN INFO = 6 ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN INFO = 9 ELSE IF( LDB.LT.MAX( 1, M ) )THEN INFO = 11 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DTRMM ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( N.EQ.0 ) & RETURN ! ! And when alpha.eq.zero. ! IF( ALPHA.EQ.ZERO )THEN DO 20, J = 1, N DO 10, I = 1, M B( I, J ) = ZERO 10 CONTINUE 20 CONTINUE RETURN END IF ! ! Start the operations. ! IF( LSIDE )THEN IF( LSAME( TRANSA, 'N' ) )THEN ! ! Form B := alpha*A*B. ! IF( UPPER )THEN DO 50, J = 1, N DO 40, K = 1, M IF( B( K, J ).NE.ZERO )THEN TEMP = ALPHA*B( K, J ) DO 30, I = 1, K - 1 B( I, J ) = B( I, J ) + TEMP*A( I, K ) 30 CONTINUE IF( NOUNIT ) & TEMP = TEMP*A( K, K ) B( K, J ) = TEMP END IF 40 CONTINUE 50 CONTINUE ELSE DO 80, J = 1, N DO 70 K = M, 1, -1 IF( B( K, J ).NE.ZERO )THEN TEMP = ALPHA*B( K, J ) B( K, J ) = TEMP IF( NOUNIT ) & B( K, J ) = B( K, J )*A( K, K ) DO 60, I = K + 1, M B( I, J ) = B( I, J ) + TEMP*A( I, K ) 60 CONTINUE END IF 70 CONTINUE 80 CONTINUE END IF ELSE ! ! Form B := alpha*A'*B. ! IF( UPPER )THEN DO 110, J = 1, N DO 100, I = M, 1, -1 TEMP = B( I, J ) IF( NOUNIT ) & TEMP = TEMP*A( I, I ) DO 90, K = 1, I - 1 TEMP = TEMP + A( K, I )*B( K, J ) 90 CONTINUE B( I, J ) = ALPHA*TEMP 100 CONTINUE 110 CONTINUE ELSE DO 140, J = 1, N DO 130, I = 1, M TEMP = B( I, J ) IF( NOUNIT ) & TEMP = TEMP*A( I, I ) DO 120, K = I + 1, M TEMP = TEMP + A( K, I )*B( K, J ) 120 CONTINUE B( I, J ) = ALPHA*TEMP 130 CONTINUE 140 CONTINUE END IF END IF ELSE IF( LSAME( TRANSA, 'N' ) )THEN ! ! Form B := alpha*B*A. ! IF( UPPER )THEN DO 180, J = N, 1, -1 TEMP = ALPHA IF( NOUNIT ) & TEMP = TEMP*A( J, J ) DO 150, I = 1, M B( I, J ) = TEMP*B( I, J ) 150 CONTINUE DO 170, K = 1, J - 1 IF( A( K, J ).NE.ZERO )THEN TEMP = ALPHA*A( K, J ) DO 160, I = 1, M B( I, J ) = B( I, J ) + TEMP*B( I, K ) 160 CONTINUE END IF 170 CONTINUE 180 CONTINUE ELSE DO 220, J = 1, N TEMP = ALPHA IF( NOUNIT ) & TEMP = TEMP*A( J, J ) DO 190, I = 1, M B( I, J ) = TEMP*B( I, J ) 190 CONTINUE DO 210, K = J + 1, N IF( A( K, J ).NE.ZERO )THEN TEMP = ALPHA*A( K, J ) DO 200, I = 1, M B( I, J ) = B( I, J ) + TEMP*B( I, K ) 200 CONTINUE END IF 210 CONTINUE 220 CONTINUE END IF ELSE ! ! Form B := alpha*B*A'. ! IF( UPPER )THEN DO 260, K = 1, N DO 240, J = 1, K - 1 IF( A( J, K ).NE.ZERO )THEN TEMP = ALPHA*A( J, K ) DO 230, I = 1, M B( I, J ) = B( I, J ) + TEMP*B( I, K ) 230 CONTINUE END IF 240 CONTINUE TEMP = ALPHA IF( NOUNIT ) & TEMP = TEMP*A( K, K ) IF( TEMP.NE.ONE )THEN DO 250, I = 1, M B( I, K ) = TEMP*B( I, K ) 250 CONTINUE END IF 260 CONTINUE ELSE DO 300, K = N, 1, -1 DO 280, J = K + 1, N IF( A( J, K ).NE.ZERO )THEN TEMP = ALPHA*A( J, K ) DO 270, I = 1, M B( I, J ) = B( I, J ) + TEMP*B( I, K ) 270 CONTINUE END IF 280 CONTINUE TEMP = ALPHA IF( NOUNIT ) & TEMP = TEMP*A( K, K ) IF( TEMP.NE.ONE )THEN DO 290, I = 1, M B( I, K ) = TEMP*B( I, K ) 290 CONTINUE END IF 300 CONTINUE END IF END IF END IF ! RETURN ! ! End of DTRMM . ! END SUBROUTINE DTRMV ( UPLO, TRANS, DIAG, N, A, LDA, X, INCX ) ! .. Scalar Arguments .. INTEGER INCX, LDA, N CHARACTER*1 DIAG, TRANS, UPLO ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), X( * ) ! .. ! ! Purpose ! ======= ! ! DTRMV performs one of the matrix-vector operations ! ! x := A*x, or x := A'*x, ! ! where x is an n element vector and A is an n by n unit, or non-unit, ! upper or lower triangular matrix. ! ! Parameters ! ========== ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the matrix is an upper or ! lower triangular matrix as follows: ! ! UPLO = 'U' or 'u' A is an upper triangular matrix. ! ! UPLO = 'L' or 'l' A is a lower triangular matrix. ! ! Unchanged on exit. ! ! TRANS - CHARACTER*1. ! On entry, TRANS specifies the operation to be performed as ! follows: ! ! TRANS = 'N' or 'n' x := A*x. ! ! TRANS = 'T' or 't' x := A'*x. ! ! TRANS = 'C' or 'c' x := A'*x. ! ! Unchanged on exit. ! ! DIAG - CHARACTER*1. ! On entry, DIAG specifies whether or not A is unit ! triangular as follows: ! ! DIAG = 'U' or 'u' A is assumed to be unit triangular. ! ! DIAG = 'N' or 'n' A is not assumed to be unit ! triangular. ! ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the order of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, n ). ! Before entry with UPLO = 'U' or 'u', the leading n by n ! upper triangular part of the array A must contain the upper ! triangular matrix and the strictly lower triangular part of ! A is not referenced. ! Before entry with UPLO = 'L' or 'l', the leading n by n ! lower triangular part of the array A must contain the lower ! triangular matrix and the strictly upper triangular part of ! A is not referenced. ! Note that when DIAG = 'U' or 'u', the diagonal elements of ! A are not referenced either, but are assumed to be unity. ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. LDA must be at least ! max( 1, n ). ! Unchanged on exit. ! ! X - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCX ) ). ! Before entry, the incremented array X must contain the n ! element vector x. On exit, X is overwritten with the ! tranformed vector x. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. Local Scalars .. DOUBLE PRECISION TEMP INTEGER I, INFO, IX, J, JX, KX LOGICAL NOUNIT ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( UPLO , 'U' ).AND. & .NOT.LSAME( UPLO , 'L' ) )THEN INFO = 1 ELSE IF( .NOT.LSAME( TRANS, 'N' ).AND. & .NOT.LSAME( TRANS, 'T' ).AND. & .NOT.LSAME( TRANS, 'C' ) )THEN INFO = 2 ELSE IF( .NOT.LSAME( DIAG , 'U' ).AND. & .NOT.LSAME( DIAG , 'N' ) )THEN INFO = 3 ELSE IF( N.LT.0 )THEN INFO = 4 ELSE IF( LDA.LT.MAX( 1, N ) )THEN INFO = 6 ELSE IF( INCX.EQ.0 )THEN INFO = 8 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DTRMV ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( N.EQ.0 ) & RETURN ! NOUNIT = LSAME( DIAG, 'N' ) ! ! Set up the start point in X if the increment is not unity. This ! will be ( N - 1 )*INCX too small for descending loops. ! IF( INCX.LE.0 )THEN KX = 1 - ( N - 1 )*INCX ELSE IF( INCX.NE.1 )THEN KX = 1 END IF ! ! Start the operations. In this version the elements of A are ! accessed sequentially with one pass through A. ! IF( LSAME( TRANS, 'N' ) )THEN ! ! Form x := A*x. ! IF( LSAME( UPLO, 'U' ) )THEN IF( INCX.EQ.1 )THEN DO 20, J = 1, N IF( X( J ).NE.ZERO )THEN TEMP = X( J ) DO 10, I = 1, J - 1 X( I ) = X( I ) + TEMP*A( I, J ) 10 CONTINUE IF( NOUNIT ) & X( J ) = X( J )*A( J, J ) END IF 20 CONTINUE ELSE JX = KX DO 40, J = 1, N IF( X( JX ).NE.ZERO )THEN TEMP = X( JX ) IX = KX DO 30, I = 1, J - 1 X( IX ) = X( IX ) + TEMP*A( I, J ) IX = IX + INCX 30 CONTINUE IF( NOUNIT ) & X( JX ) = X( JX )*A( J, J ) END IF JX = JX + INCX 40 CONTINUE END IF ELSE IF( INCX.EQ.1 )THEN DO 60, J = N, 1, -1 IF( X( J ).NE.ZERO )THEN TEMP = X( J ) DO 50, I = N, J + 1, -1 X( I ) = X( I ) + TEMP*A( I, J ) 50 CONTINUE IF( NOUNIT ) & X( J ) = X( J )*A( J, J ) END IF 60 CONTINUE ELSE KX = KX + ( N - 1 )*INCX JX = KX DO 80, J = N, 1, -1 IF( X( JX ).NE.ZERO )THEN TEMP = X( JX ) IX = KX DO 70, I = N, J + 1, -1 X( IX ) = X( IX ) + TEMP*A( I, J ) IX = IX - INCX 70 CONTINUE IF( NOUNIT ) & X( JX ) = X( JX )*A( J, J ) END IF JX = JX - INCX 80 CONTINUE END IF END IF ELSE ! ! Form x := A'*x. ! IF( LSAME( UPLO, 'U' ) )THEN IF( INCX.EQ.1 )THEN DO 100, J = N, 1, -1 TEMP = X( J ) IF( NOUNIT ) & TEMP = TEMP*A( J, J ) DO 90, I = J - 1, 1, -1 TEMP = TEMP + A( I, J )*X( I ) 90 CONTINUE X( J ) = TEMP 100 CONTINUE ELSE JX = KX + ( N - 1 )*INCX DO 120, J = N, 1, -1 TEMP = X( JX ) IX = JX IF( NOUNIT ) & TEMP = TEMP*A( J, J ) DO 110, I = J - 1, 1, -1 IX = IX - INCX TEMP = TEMP + A( I, J )*X( IX ) 110 CONTINUE X( JX ) = TEMP JX = JX - INCX 120 CONTINUE END IF ELSE IF( INCX.EQ.1 )THEN DO 140, J = 1, N TEMP = X( J ) IF( NOUNIT ) & TEMP = TEMP*A( J, J ) DO 130, I = J + 1, N TEMP = TEMP + A( I, J )*X( I ) 130 CONTINUE X( J ) = TEMP 140 CONTINUE ELSE JX = KX DO 160, J = 1, N TEMP = X( JX ) IX = JX IF( NOUNIT ) & TEMP = TEMP*A( J, J ) DO 150, I = J + 1, N IX = IX + INCX TEMP = TEMP + A( I, J )*X( IX ) 150 CONTINUE X( JX ) = TEMP JX = JX + INCX 160 CONTINUE END IF END IF END IF ! RETURN ! ! End of DTRMV . ! END SUBROUTINE DTRSM ( SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, & B, LDB ) ! .. Scalar Arguments .. CHARACTER*1 SIDE, UPLO, TRANSA, DIAG INTEGER M, N, LDA, LDB DOUBLE PRECISION ALPHA ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ) ! .. ! ! Purpose ! ======= ! ! DTRSM solves one of the matrix equations ! ! op( A )*X = alpha*B, or X*op( A ) = alpha*B, ! ! where alpha is a scalar, X and B are m by n matrices, A is a unit, or ! non-unit, upper or lower triangular matrix and op( A ) is one of ! ! op( A ) = A or op( A ) = A'. ! ! The matrix X is overwritten on B. ! ! Parameters ! ========== ! ! SIDE - CHARACTER*1. ! On entry, SIDE specifies whether op( A ) appears on the left ! or right of X as follows: ! ! SIDE = 'L' or 'l' op( A )*X = alpha*B. ! ! SIDE = 'R' or 'r' X*op( A ) = alpha*B. ! ! Unchanged on exit. ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the matrix A is an upper or ! lower triangular matrix as follows: ! ! UPLO = 'U' or 'u' A is an upper triangular matrix. ! ! UPLO = 'L' or 'l' A is a lower triangular matrix. ! ! Unchanged on exit. ! ! TRANSA - CHARACTER*1. ! On entry, TRANSA specifies the form of op( A ) to be used in ! the matrix multiplication as follows: ! ! TRANSA = 'N' or 'n' op( A ) = A. ! ! TRANSA = 'T' or 't' op( A ) = A'. ! ! TRANSA = 'C' or 'c' op( A ) = A'. ! ! Unchanged on exit. ! ! DIAG - CHARACTER*1. ! On entry, DIAG specifies whether or not A is unit triangular ! as follows: ! ! DIAG = 'U' or 'u' A is assumed to be unit triangular. ! ! DIAG = 'N' or 'n' A is not assumed to be unit ! triangular. ! ! Unchanged on exit. ! ! M - INTEGER. ! On entry, M specifies the number of rows of B. M must be at ! least zero. ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the number of columns of B. N must be ! at least zero. ! Unchanged on exit. ! ! ALPHA - DOUBLE PRECISION. ! On entry, ALPHA specifies the scalar alpha. When alpha is ! zero then A is not referenced and B need not be set before ! entry. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, k ), where k is m ! when SIDE = 'L' or 'l' and is n when SIDE = 'R' or 'r'. ! Before entry with UPLO = 'U' or 'u', the leading k by k ! upper triangular part of the array A must contain the upper ! triangular matrix and the strictly lower triangular part of ! A is not referenced. ! Before entry with UPLO = 'L' or 'l', the leading k by k ! lower triangular part of the array A must contain the lower ! triangular matrix and the strictly upper triangular part of ! A is not referenced. ! Note that when DIAG = 'U' or 'u', the diagonal elements of ! A are not referenced either, but are assumed to be unity. ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. When SIDE = 'L' or 'l' then ! LDA must be at least max( 1, m ), when SIDE = 'R' or 'r' ! then LDA must be at least max( 1, n ). ! Unchanged on exit. ! ! B - DOUBLE PRECISION array of DIMENSION ( LDB, n ). ! Before entry, the leading m by n part of the array B must ! contain the right-hand side matrix B, and on exit is ! overwritten by the solution matrix X. ! ! LDB - INTEGER. ! On entry, LDB specifies the first dimension of B as declared ! in the calling (sub) program. LDB must be at least ! max( 1, m ). ! Unchanged on exit. ! ! ! Level 3 Blas routine. ! ! ! -- Written on 8-February-1989. ! Jack Dongarra, Argonne National Laboratory. ! Iain Duff, AERE Harwell. ! Jeremy Du Croz, Numerical Algorithms Group Ltd. ! Sven Hammarling, Numerical Algorithms Group Ltd. ! ! ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX ! .. Local Scalars .. LOGICAL LSIDE, NOUNIT, UPPER INTEGER I, INFO, J, K, NROWA DOUBLE PRECISION TEMP ! .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! LSIDE = LSAME( SIDE , 'L' ) IF( LSIDE )THEN NROWA = M ELSE NROWA = N END IF NOUNIT = LSAME( DIAG , 'N' ) UPPER = LSAME( UPLO , 'U' ) ! INFO = 0 IF( ( .NOT.LSIDE ).AND. & ( .NOT.LSAME( SIDE , 'R' ) ) )THEN INFO = 1 ELSE IF( ( .NOT.UPPER ).AND. & ( .NOT.LSAME( UPLO , 'L' ) ) )THEN INFO = 2 ELSE IF( ( .NOT.LSAME( TRANSA, 'N' ) ).AND. & ( .NOT.LSAME( TRANSA, 'T' ) ).AND. & ( .NOT.LSAME( TRANSA, 'C' ) ) )THEN INFO = 3 ELSE IF( ( .NOT.LSAME( DIAG , 'U' ) ).AND. & ( .NOT.LSAME( DIAG , 'N' ) ) )THEN INFO = 4 ELSE IF( M .LT.0 )THEN INFO = 5 ELSE IF( N .LT.0 )THEN INFO = 6 ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN INFO = 9 ELSE IF( LDB.LT.MAX( 1, M ) )THEN INFO = 11 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DTRSM ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( N.EQ.0 ) & RETURN ! ! And when alpha.eq.zero. ! IF( ALPHA.EQ.ZERO )THEN DO 20, J = 1, N DO 10, I = 1, M B( I, J ) = ZERO 10 CONTINUE 20 CONTINUE RETURN END IF ! ! Start the operations. ! IF( LSIDE )THEN IF( LSAME( TRANSA, 'N' ) )THEN ! ! Form B := alpha*inv( A )*B. ! IF( UPPER )THEN DO 60, J = 1, N IF( ALPHA.NE.ONE )THEN DO 30, I = 1, M B( I, J ) = ALPHA*B( I, J ) 30 CONTINUE END IF DO 50, K = M, 1, -1 IF( B( K, J ).NE.ZERO )THEN IF( NOUNIT ) & B( K, J ) = B( K, J )/A( K, K ) DO 40, I = 1, K - 1 B( I, J ) = B( I, J ) - B( K, J )*A( I, K ) 40 CONTINUE END IF 50 CONTINUE 60 CONTINUE ELSE DO 100, J = 1, N IF( ALPHA.NE.ONE )THEN DO 70, I = 1, M B( I, J ) = ALPHA*B( I, J ) 70 CONTINUE END IF DO 90 K = 1, M IF( B( K, J ).NE.ZERO )THEN IF( NOUNIT ) & B( K, J ) = B( K, J )/A( K, K ) DO 80, I = K + 1, M B( I, J ) = B( I, J ) - B( K, J )*A( I, K ) 80 CONTINUE END IF 90 CONTINUE 100 CONTINUE END IF ELSE ! ! Form B := alpha*inv( A' )*B. ! IF( UPPER )THEN DO 130, J = 1, N DO 120, I = 1, M TEMP = ALPHA*B( I, J ) DO 110, K = 1, I - 1 TEMP = TEMP - A( K, I )*B( K, J ) 110 CONTINUE IF( NOUNIT ) & TEMP = TEMP/A( I, I ) B( I, J ) = TEMP 120 CONTINUE 130 CONTINUE ELSE DO 160, J = 1, N DO 150, I = M, 1, -1 TEMP = ALPHA*B( I, J ) DO 140, K = I + 1, M TEMP = TEMP - A( K, I )*B( K, J ) 140 CONTINUE IF( NOUNIT ) & TEMP = TEMP/A( I, I ) B( I, J ) = TEMP 150 CONTINUE 160 CONTINUE END IF END IF ELSE IF( LSAME( TRANSA, 'N' ) )THEN ! ! Form B := alpha*B*inv( A ). ! IF( UPPER )THEN DO 210, J = 1, N IF( ALPHA.NE.ONE )THEN DO 170, I = 1, M B( I, J ) = ALPHA*B( I, J ) 170 CONTINUE END IF DO 190, K = 1, J - 1 IF( A( K, J ).NE.ZERO )THEN DO 180, I = 1, M B( I, J ) = B( I, J ) - A( K, J )*B( I, K ) 180 CONTINUE END IF 190 CONTINUE IF( NOUNIT )THEN TEMP = ONE/A( J, J ) DO 200, I = 1, M B( I, J ) = TEMP*B( I, J ) 200 CONTINUE END IF 210 CONTINUE ELSE DO 260, J = N, 1, -1 IF( ALPHA.NE.ONE )THEN DO 220, I = 1, M B( I, J ) = ALPHA*B( I, J ) 220 CONTINUE END IF DO 240, K = J + 1, N IF( A( K, J ).NE.ZERO )THEN DO 230, I = 1, M B( I, J ) = B( I, J ) - A( K, J )*B( I, K ) 230 CONTINUE END IF 240 CONTINUE IF( NOUNIT )THEN TEMP = ONE/A( J, J ) DO 250, I = 1, M B( I, J ) = TEMP*B( I, J ) 250 CONTINUE END IF 260 CONTINUE END IF ELSE ! ! Form B := alpha*B*inv( A' ). ! IF( UPPER )THEN DO 310, K = N, 1, -1 IF( NOUNIT )THEN TEMP = ONE/A( K, K ) DO 270, I = 1, M B( I, K ) = TEMP*B( I, K ) 270 CONTINUE END IF DO 290, J = 1, K - 1 IF( A( J, K ).NE.ZERO )THEN TEMP = A( J, K ) DO 280, I = 1, M B( I, J ) = B( I, J ) - TEMP*B( I, K ) 280 CONTINUE END IF 290 CONTINUE IF( ALPHA.NE.ONE )THEN DO 300, I = 1, M B( I, K ) = ALPHA*B( I, K ) 300 CONTINUE END IF 310 CONTINUE ELSE DO 360, K = 1, N IF( NOUNIT )THEN TEMP = ONE/A( K, K ) DO 320, I = 1, M B( I, K ) = TEMP*B( I, K ) 320 CONTINUE END IF DO 340, J = K + 1, N IF( A( J, K ).NE.ZERO )THEN TEMP = A( J, K ) DO 330, I = 1, M B( I, J ) = B( I, J ) - TEMP*B( I, K ) 330 CONTINUE END IF 340 CONTINUE IF( ALPHA.NE.ONE )THEN DO 350, I = 1, M B( I, K ) = ALPHA*B( I, K ) 350 CONTINUE END IF 360 CONTINUE END IF END IF END IF ! RETURN ! ! End of DTRSM . ! END SUBROUTINE DTRSV ( UPLO, TRANS, DIAG, N, A, LDA, X, INCX ) ! .. Scalar Arguments .. INTEGER INCX, LDA, N CHARACTER*1 DIAG, TRANS, UPLO ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), X( * ) ! .. ! ! Purpose ! ======= ! ! DTRSV solves one of the systems of equations ! ! A*x = b, or A'*x = b, ! ! where b and x are n element vectors and A is an n by n unit, or ! non-unit, upper or lower triangular matrix. ! ! No test for singularity or near-singularity is included in this ! routine. Such tests must be performed before calling this routine. ! ! Parameters ! ========== ! ! UPLO - CHARACTER*1. ! On entry, UPLO specifies whether the matrix is an upper or ! lower triangular matrix as follows: ! ! UPLO = 'U' or 'u' A is an upper triangular matrix. ! ! UPLO = 'L' or 'l' A is a lower triangular matrix. ! ! Unchanged on exit. ! ! TRANS - CHARACTER*1. ! On entry, TRANS specifies the equations to be solved as ! follows: ! ! TRANS = 'N' or 'n' A*x = b. ! ! TRANS = 'T' or 't' A'*x = b. ! ! TRANS = 'C' or 'c' A'*x = b. ! ! Unchanged on exit. ! ! DIAG - CHARACTER*1. ! On entry, DIAG specifies whether or not A is unit ! triangular as follows: ! ! DIAG = 'U' or 'u' A is assumed to be unit triangular. ! ! DIAG = 'N' or 'n' A is not assumed to be unit ! triangular. ! ! Unchanged on exit. ! ! N - INTEGER. ! On entry, N specifies the order of the matrix A. ! N must be at least zero. ! Unchanged on exit. ! ! A - DOUBLE PRECISION array of DIMENSION ( LDA, n ). ! Before entry with UPLO = 'U' or 'u', the leading n by n ! upper triangular part of the array A must contain the upper ! triangular matrix and the strictly lower triangular part of ! A is not referenced. ! Before entry with UPLO = 'L' or 'l', the leading n by n ! lower triangular part of the array A must contain the lower ! triangular matrix and the strictly upper triangular part of ! A is not referenced. ! Note that when DIAG = 'U' or 'u', the diagonal elements of ! A are not referenced either, but are assumed to be unity. ! Unchanged on exit. ! ! LDA - INTEGER. ! On entry, LDA specifies the first dimension of A as declared ! in the calling (sub) program. LDA must be at least ! max( 1, n ). ! Unchanged on exit. ! ! X - DOUBLE PRECISION array of dimension at least ! ( 1 + ( n - 1 )*abs( INCX ) ). ! Before entry, the incremented array X must contain the n ! element right-hand side vector b. On exit, X is overwritten ! with the solution vector x. ! ! INCX - INTEGER. ! On entry, INCX specifies the increment for the elements of ! X. INCX must not be zero. ! Unchanged on exit. ! ! ! Level 2 Blas routine. ! ! -- Written on 22-October-1986. ! Jack Dongarra, Argonne National Lab. ! Jeremy Du Croz, Nag Central Office. ! Sven Hammarling, Nag Central Office. ! Richard Hanson, Sandia National Labs. ! ! ! .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) ! .. Local Scalars .. DOUBLE PRECISION TEMP INTEGER I, INFO, IX, J, JX, KX LOGICAL NOUNIT ! .. External Functions .. LOGICAL LSAME EXTERNAL LSAME ! .. External Subroutines .. EXTERNAL XERBLA ! .. Intrinsic Functions .. INTRINSIC MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 IF ( .NOT.LSAME( UPLO , 'U' ).AND. & .NOT.LSAME( UPLO , 'L' ) )THEN INFO = 1 ELSE IF( .NOT.LSAME( TRANS, 'N' ).AND. & .NOT.LSAME( TRANS, 'T' ).AND. & .NOT.LSAME( TRANS, 'C' ) )THEN INFO = 2 ELSE IF( .NOT.LSAME( DIAG , 'U' ).AND. & .NOT.LSAME( DIAG , 'N' ) )THEN INFO = 3 ELSE IF( N.LT.0 )THEN INFO = 4 ELSE IF( LDA.LT.MAX( 1, N ) )THEN INFO = 6 ELSE IF( INCX.EQ.0 )THEN INFO = 8 END IF IF( INFO.NE.0 )THEN CALL XERBLA( 'DTRSV ', INFO ) RETURN END IF ! ! Quick return if possible. ! IF( N.EQ.0 ) & RETURN ! NOUNIT = LSAME( DIAG, 'N' ) ! ! Set up the start point in X if the increment is not unity. This ! will be ( N - 1 )*INCX too small for descending loops. ! IF( INCX.LE.0 )THEN KX = 1 - ( N - 1 )*INCX ELSE IF( INCX.NE.1 )THEN KX = 1 END IF ! ! Start the operations. In this version the elements of A are ! accessed sequentially with one pass through A. ! IF( LSAME( TRANS, 'N' ) )THEN ! ! Form x := inv( A )*x. ! IF( LSAME( UPLO, 'U' ) )THEN IF( INCX.EQ.1 )THEN DO 20, J = N, 1, -1 IF( X( J ).NE.ZERO )THEN IF( NOUNIT ) & X( J ) = X( J )/A( J, J ) TEMP = X( J ) DO 10, I = J - 1, 1, -1 X( I ) = X( I ) - TEMP*A( I, J ) 10 CONTINUE END IF 20 CONTINUE ELSE JX = KX + ( N - 1 )*INCX DO 40, J = N, 1, -1 IF( X( JX ).NE.ZERO )THEN IF( NOUNIT ) & X( JX ) = X( JX )/A( J, J ) TEMP = X( JX ) IX = JX DO 30, I = J - 1, 1, -1 IX = IX - INCX X( IX ) = X( IX ) - TEMP*A( I, J ) 30 CONTINUE END IF JX = JX - INCX 40 CONTINUE END IF ELSE IF( INCX.EQ.1 )THEN DO 60, J = 1, N IF( X( J ).NE.ZERO )THEN IF( NOUNIT ) & X( J ) = X( J )/A( J, J ) TEMP = X( J ) DO 50, I = J + 1, N X( I ) = X( I ) - TEMP*A( I, J ) 50 CONTINUE END IF 60 CONTINUE ELSE JX = KX DO 80, J = 1, N IF( X( JX ).NE.ZERO )THEN IF( NOUNIT ) & X( JX ) = X( JX )/A( J, J ) TEMP = X( JX ) IX = JX DO 70, I = J + 1, N IX = IX + INCX X( IX ) = X( IX ) - TEMP*A( I, J ) 70 CONTINUE END IF JX = JX + INCX 80 CONTINUE END IF END IF ELSE ! ! Form x := inv( A' )*x. ! IF( LSAME( UPLO, 'U' ) )THEN IF( INCX.EQ.1 )THEN DO 100, J = 1, N TEMP = X( J ) DO 90, I = 1, J - 1 TEMP = TEMP - A( I, J )*X( I ) 90 CONTINUE IF( NOUNIT ) & TEMP = TEMP/A( J, J ) X( J ) = TEMP 100 CONTINUE ELSE JX = KX DO 120, J = 1, N TEMP = X( JX ) IX = KX DO 110, I = 1, J - 1 TEMP = TEMP - A( I, J )*X( IX ) IX = IX + INCX 110 CONTINUE IF( NOUNIT ) & TEMP = TEMP/A( J, J ) X( JX ) = TEMP JX = JX + INCX 120 CONTINUE END IF ELSE IF( INCX.EQ.1 )THEN DO 140, J = N, 1, -1 TEMP = X( J ) DO 130, I = N, J + 1, -1 TEMP = TEMP - A( I, J )*X( I ) 130 CONTINUE IF( NOUNIT ) & TEMP = TEMP/A( J, J ) X( J ) = TEMP 140 CONTINUE ELSE KX = KX + ( N - 1 )*INCX JX = KX DO 160, J = N, 1, -1 TEMP = X( JX ) IX = KX DO 150, I = N, J + 1, -1 TEMP = TEMP - A( I, J )*X( IX ) IX = IX - INCX 150 CONTINUE IF( NOUNIT ) & TEMP = TEMP/A( J, J ) X( JX ) = TEMP JX = JX - INCX 160 CONTINUE END IF END IF END IF ! RETURN ! ! End of DTRSV . ! END subroutine dvec_print ( n, a, title ) ! !******************************************************************************* ! !! DVEC_PRINT prints a double precision vector. ! ! ! Modified: ! ! 22 August 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer N, the number of components of the vector. ! ! Input, real ( kind = 8 ) A(N), the vector to be printed. ! ! Input, character ( len = * ) TITLE, an optional title. ! implicit none ! integer n ! real ( kind = 8 ) a(n) integer i character ( len = * ) title ! if ( 0 < len_trim ( title ) ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) end if write ( *, '(a)' ) ' ' do i = 1, n write ( *, '(i6,g16.8)' ) i, a(i) end do return end double precision function dzasum(n,zx,incx) ! ! takes the sum of the absolute values. ! jack dongarra, 3/11/78. ! modified 3/93 to return if incx .le. 0. ! modified 12/3/93, array(1) declarations changed to array(*) ! double complex zx(*) double precision stemp,dcabs1 integer i,incx,ix,n ! dzasum = 0.0d0 stemp = 0.0d0 if( n.le.0 .or. incx.le.0 )return if(incx.eq.1)go to 20 ! ! code for increment not equal to 1 ! ix = 1 do 10 i = 1,n stemp = stemp + dcabs1(zx(ix)) ix = ix + incx 10 continue dzasum = stemp return ! ! code for increment equal to 1 ! 20 do 30 i = 1,n stemp = stemp + dcabs1(zx(i)) 30 continue dzasum = stemp return end DOUBLE PRECISION FUNCTION DZNRM2( N, X, INCX ) ! .. Scalar Arguments .. INTEGER INCX, N ! .. Array Arguments .. COMPLEX*16 X( * ) ! .. ! ! DZNRM2 returns the euclidean norm of a vector via the function ! name, so that ! ! DZNRM2 := sqrt( conjg( x' )*x ) ! ! ! ! -- This version written on 25-October-1982. ! Modified on 14-October-1993 to inline the call to ZLASSQ. ! Sven Hammarling, Nag Ltd. ! ! ! .. Parameters .. DOUBLE PRECISION ONE , ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. Local Scalars .. INTEGER IX DOUBLE PRECISION NORM, SCALE, SSQ, TEMP ! .. Intrinsic Functions .. INTRINSIC ABS, DIMAG, DBLE, SQRT ! .. ! .. Executable Statements .. IF( N.LT.1 .OR. INCX.LT.1 )THEN NORM = ZERO ELSE SCALE = ZERO SSQ = ONE ! The following loop is equivalent to this call to the LAPACK ! auxiliary routine: ! CALL ZLASSQ( N, X, INCX, SCALE, SSQ ) ! DO 10, IX = 1, 1 + ( N - 1 )*INCX, INCX IF( DBLE( X( IX ) ).NE.ZERO )THEN TEMP = ABS( DBLE( X( IX ) ) ) IF( SCALE.LT.TEMP )THEN SSQ = ONE + SSQ*( SCALE/TEMP )**2 SCALE = TEMP ELSE SSQ = SSQ + ( TEMP/SCALE )**2 END IF END IF IF( DIMAG( X( IX ) ).NE.ZERO )THEN TEMP = ABS( DIMAG( X( IX ) ) ) IF( SCALE.LT.TEMP )THEN SSQ = ONE + SSQ*( SCALE/TEMP )**2 SCALE = TEMP ELSE SSQ = SSQ + ( TEMP/SCALE )**2 END IF END IF 10 CONTINUE NORM = SCALE * SQRT( SSQ ) END IF ! DZNRM2 = NORM RETURN ! ! End of DZNRM2. ! END SUBROUTINE XERBLA( SRNAME, INFO ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! September 30, 1994 ! ! .. Scalar Arguments .. CHARACTER*6 SRNAME INTEGER INFO ! .. ! ! Purpose ! ======= ! ! XERBLA is an error handler for the LAPACK routines. ! It is called by an LAPACK routine if an input parameter has an ! invalid value. A message is printed and execution stops. ! ! Installers may consider modifying the STOP statement in order to ! call system-specific exception-handling facilities. ! ! Arguments ! ========= ! ! SRNAME (input) CHARACTER*6 ! The name of the routine which called XERBLA. ! ! INFO (input) INTEGER ! The position of the invalid parameter in the parameter list ! of the calling routine. ! ! ===================================================================== ! ! .. Executable Statements .. ! WRITE( *, FMT = 9999 )SRNAME, INFO ! STOP ! 9999 FORMAT( ' ** On entry to ', A6, ' parameter number ', I2, ' had ', & 'an illegal value' ) ! ! End of XERBLA ! END integer function idamax(n,dx,incx) ! ! finds the index of element having max. absolute value. ! jack dongarra, linpack, 3/11/78. ! modified 3/93 to return if incx .le. 0. ! modified 12/3/93, array(1) declarations changed to array(*) ! double precision dx(*),dmax integer i,incx,ix,n ! idamax = 0 if( n.lt.1 .or. incx.le.0 ) return idamax = 1 if(n.eq.1)return if(incx.eq.1)go to 20 ! ! code for increment not equal to 1 ! ix = 1 dmax = dabs(dx(1)) ix = ix + incx do 10 i = 2,n if(dabs(dx(ix)).le.dmax) go to 5 idamax = i dmax = dabs(dx(ix)) 5 ix = ix + incx 10 continue return ! ! code for increment equal to 1 ! 20 dmax = dabs(dx(1)) do 30 i = 2,n if(dabs(dx(i)).le.dmax) go to 30 idamax = i dmax = dabs(dx(i)) 30 continue return end INTEGER FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3, & N4 ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1999 ! ! .. Scalar Arguments .. CHARACTER*( * ) NAME, OPTS INTEGER ISPEC, N1, N2, N3, N4 ! .. ! ! Purpose ! ======= ! ! ILAENV is called from the LAPACK routines to choose problem-dependent ! parameters for the local environment. See ISPEC for a description of ! the parameters. ! ! This version provides a set of parameters which should give good, ! but not optimal, performance on many of the currently available ! computers. Users are encouraged to modify this subroutine to set ! the tuning parameters for their particular machine using the option ! and problem size information in the arguments. ! ! This routine will not function correctly if it is converted to all ! lower case. Converting it to all upper case is allowed. ! ! Arguments ! ========= ! ! ISPEC (input) INTEGER ! Specifies the parameter to be returned as the value of ! ILAENV. ! = 1: the optimal blocksize; if this value is 1, an unblocked ! algorithm will give the best performance. ! = 2: the minimum block size for which the block routine ! should be used; if the usable block size is less than ! this value, an unblocked routine should be used. ! = 3: the crossover point (in a block routine, for N less ! than this value, an unblocked routine should be used) ! = 4: the number of shifts, used in the nonsymmetric ! eigenvalue routines ! = 5: the minimum column dimension for blocking to be used; ! rectangular blocks must have dimension at least k by m, ! where k is given by ILAENV(2,...) and m by ILAENV(5,...) ! = 6: the crossover point for the SVD (when reducing an m by n ! matrix to bidiagonal form, if max(m,n)/min(m,n) exceeds ! this value, a QR factorization is used first to reduce ! the matrix to a triangular form.) ! = 7: the number of processors ! = 8: the crossover point for the multishift QR and QZ methods ! for nonsymmetric eigenvalue problems. ! = 9: maximum size of the subproblems at the bottom of the ! computation tree in the divide-and-conquer algorithm ! (used by xGELSD and xGESDD) ! =10: ieee NaN arithmetic can be trusted not to trap ! =11: infinity arithmetic can be trusted not to trap ! ! NAME (input) CHARACTER*(*) ! The name of the calling subroutine, in either upper case or ! lower case. ! ! OPTS (input) CHARACTER*(*) ! The character options to the subroutine NAME, concatenated ! into a single character string. For example, UPLO = 'U', ! TRANS = 'T', and DIAG = 'N' for a triangular routine would ! be specified as OPTS = 'UTN'. ! ! N1 (input) INTEGER ! N2 (input) INTEGER ! N3 (input) INTEGER ! N4 (input) INTEGER ! Problem dimensions for the subroutine NAME; these may not all ! be required. ! ! (ILAENV) (output) INTEGER ! >= 0: the value of the parameter specified by ISPEC ! < 0: if ILAENV = -k, the k-th argument had an illegal value. ! ! Further Details ! =============== ! ! The following conventions have been used when calling ILAENV from the ! LAPACK routines: ! 1) OPTS is a concatenation of all of the character options to ! subroutine NAME, in the same order that they appear in the ! argument list for NAME, even if they are not used in determining ! the value of the parameter specified by ISPEC. ! 2) The problem dimensions N1, N2, N3, N4 are specified in the order ! that they appear in the argument list for NAME. N1 is used ! first, N2 second, and so on, and unused problem dimensions are ! passed a value of -1. ! 3) The parameter value returned by ILAENV is checked for validity in ! the calling subroutine. For example, ILAENV is used to retrieve ! the optimal blocksize for STRTRI as follows: ! ! NB = ILAENV( 1, 'STRTRI', UPLO // DIAG, N, -1, -1, -1 ) ! IF( NB.LE.1 ) NB = MAX( 1, N ) ! ! ===================================================================== ! ! .. Local Scalars .. LOGICAL CNAME, SNAME CHARACTER*1 C1 CHARACTER*2 C2, C4 CHARACTER*3 C3 CHARACTER*6 SUBNAM INTEGER I, IC, IZ, NB, NBMIN, NX ! .. ! .. Intrinsic Functions .. INTRINSIC CHAR, ICHAR, INT, MIN, REAL ! .. ! .. External Functions .. INTEGER IEEECK EXTERNAL IEEECK ! .. ! .. Executable Statements .. ! GO TO ( 100, 100, 100, 400, 500, 600, 700, 800, 900, 1000, & 1100 ) ISPEC ! ! Invalid value for ISPEC ! ILAENV = -1 RETURN ! 100 CONTINUE ! ! Convert NAME to upper case if the first character is lower case. ! ILAENV = 1 SUBNAM = NAME IC = ICHAR( SUBNAM( 1:1 ) ) IZ = ICHAR( 'Z' ) IF( IZ.EQ.90 .OR. IZ.EQ.122 ) THEN ! ! ASCII character set ! IF( IC.GE.97 .AND. IC.LE.122 ) THEN SUBNAM( 1:1 ) = CHAR( IC-32 ) DO 10 I = 2, 6 IC = ICHAR( SUBNAM( I:I ) ) IF( IC.GE.97 .AND. IC.LE.122 ) & SUBNAM( I:I ) = CHAR( IC-32 ) 10 CONTINUE END IF ! ELSE IF( IZ.EQ.233 .OR. IZ.EQ.169 ) THEN ! ! EBCDIC character set ! IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR. & ( IC.GE.145 .AND. IC.LE.153 ) .OR. & ( IC.GE.162 .AND. IC.LE.169 ) ) THEN SUBNAM( 1:1 ) = CHAR( IC+64 ) DO 20 I = 2, 6 IC = ICHAR( SUBNAM( I:I ) ) IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR. & ( IC.GE.145 .AND. IC.LE.153 ) .OR. & ( IC.GE.162 .AND. IC.LE.169 ) ) & SUBNAM( I:I ) = CHAR( IC+64 ) 20 CONTINUE END IF ! ELSE IF( IZ.EQ.218 .OR. IZ.EQ.250 ) THEN ! ! Prime machines: ASCII+128 ! IF( IC.GE.225 .AND. IC.LE.250 ) THEN SUBNAM( 1:1 ) = CHAR( IC-32 ) DO 30 I = 2, 6 IC = ICHAR( SUBNAM( I:I ) ) IF( IC.GE.225 .AND. IC.LE.250 ) & SUBNAM( I:I ) = CHAR( IC-32 ) 30 CONTINUE END IF END IF ! C1 = SUBNAM( 1:1 ) SNAME = C1.EQ.'S' .OR. C1.EQ.'D' CNAME = C1.EQ.'C' .OR. C1.EQ.'Z' IF( .NOT.( CNAME .OR. SNAME ) ) & RETURN C2 = SUBNAM( 2:3 ) C3 = SUBNAM( 4:6 ) C4 = C3( 2:3 ) ! GO TO ( 110, 200, 300 ) ISPEC ! 110 CONTINUE ! ! ISPEC = 1: block size ! ! In these examples, separate code is provided for setting NB for ! real and complex. We assume that NB will take the same value in ! single or double precision. ! NB = 1 ! IF( C2.EQ.'GE' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF ELSE IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. & C3.EQ.'QLF' ) THEN IF( SNAME ) THEN NB = 32 ELSE NB = 32 END IF ELSE IF( C3.EQ.'HRD' ) THEN IF( SNAME ) THEN NB = 32 ELSE NB = 32 END IF ELSE IF( C3.EQ.'BRD' ) THEN IF( SNAME ) THEN NB = 32 ELSE NB = 32 END IF ELSE IF( C3.EQ.'TRI' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF END IF ELSE IF( C2.EQ.'PO' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF END IF ELSE IF( C2.EQ.'SY' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN NB = 32 ELSE IF( SNAME .AND. C3.EQ.'GST' ) THEN NB = 64 END IF ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN IF( C3.EQ.'TRF' ) THEN NB = 64 ELSE IF( C3.EQ.'TRD' ) THEN NB = 32 ELSE IF( C3.EQ.'GST' ) THEN NB = 64 END IF ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NB = 32 END IF ELSE IF( C3( 1:1 ).EQ.'M' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NB = 32 END IF END IF ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NB = 32 END IF ELSE IF( C3( 1:1 ).EQ.'M' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NB = 32 END IF END IF ELSE IF( C2.EQ.'GB' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN IF( N4.LE.64 ) THEN NB = 1 ELSE NB = 32 END IF ELSE IF( N4.LE.64 ) THEN NB = 1 ELSE NB = 32 END IF END IF END IF ELSE IF( C2.EQ.'PB' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN IF( N2.LE.64 ) THEN NB = 1 ELSE NB = 32 END IF ELSE IF( N2.LE.64 ) THEN NB = 1 ELSE NB = 32 END IF END IF END IF ELSE IF( C2.EQ.'TR' ) THEN IF( C3.EQ.'TRI' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF END IF ELSE IF( C2.EQ.'LA' ) THEN IF( C3.EQ.'UUM' ) THEN IF( SNAME ) THEN NB = 64 ELSE NB = 64 END IF END IF ELSE IF( SNAME .AND. C2.EQ.'ST' ) THEN IF( C3.EQ.'EBZ' ) THEN NB = 1 END IF END IF ILAENV = NB RETURN ! 200 CONTINUE ! ! ISPEC = 2: minimum block size ! NBMIN = 2 IF( C2.EQ.'GE' ) THEN IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. & C3.EQ.'QLF' ) THEN IF( SNAME ) THEN NBMIN = 2 ELSE NBMIN = 2 END IF ELSE IF( C3.EQ.'HRD' ) THEN IF( SNAME ) THEN NBMIN = 2 ELSE NBMIN = 2 END IF ELSE IF( C3.EQ.'BRD' ) THEN IF( SNAME ) THEN NBMIN = 2 ELSE NBMIN = 2 END IF ELSE IF( C3.EQ.'TRI' ) THEN IF( SNAME ) THEN NBMIN = 2 ELSE NBMIN = 2 END IF END IF ELSE IF( C2.EQ.'SY' ) THEN IF( C3.EQ.'TRF' ) THEN IF( SNAME ) THEN NBMIN = 8 ELSE NBMIN = 8 END IF ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN NBMIN = 2 END IF ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN IF( C3.EQ.'TRD' ) THEN NBMIN = 2 END IF ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NBMIN = 2 END IF ELSE IF( C3( 1:1 ).EQ.'M' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NBMIN = 2 END IF END IF ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NBMIN = 2 END IF ELSE IF( C3( 1:1 ).EQ.'M' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NBMIN = 2 END IF END IF END IF ILAENV = NBMIN RETURN ! 300 CONTINUE ! ! ISPEC = 3: crossover point ! NX = 0 IF( C2.EQ.'GE' ) THEN IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. & C3.EQ.'QLF' ) THEN IF( SNAME ) THEN NX = 128 ELSE NX = 128 END IF ELSE IF( C3.EQ.'HRD' ) THEN IF( SNAME ) THEN NX = 128 ELSE NX = 128 END IF ELSE IF( C3.EQ.'BRD' ) THEN IF( SNAME ) THEN NX = 128 ELSE NX = 128 END IF END IF ELSE IF( C2.EQ.'SY' ) THEN IF( SNAME .AND. C3.EQ.'TRD' ) THEN NX = 32 END IF ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN IF( C3.EQ.'TRD' ) THEN NX = 32 END IF ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NX = 128 END IF END IF ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN IF( C3( 1:1 ).EQ.'G' ) THEN IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. & C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. & C4.EQ.'BR' ) THEN NX = 128 END IF END IF END IF ILAENV = NX RETURN ! 400 CONTINUE ! ! ISPEC = 4: number of shifts (used by xHSEQR) ! ILAENV = 6 RETURN ! 500 CONTINUE ! ! ISPEC = 5: minimum column dimension (not used) ! ILAENV = 2 RETURN ! 600 CONTINUE ! ! ISPEC = 6: crossover point for SVD (used by xGELSS and xGESVD) ! ILAENV = INT( REAL( MIN( N1, N2 ) )*1.6E0 ) RETURN ! 700 CONTINUE ! ! ISPEC = 7: number of processors (not used) ! ILAENV = 1 RETURN ! 800 CONTINUE ! ! ISPEC = 8: crossover point for multishift (used by xHSEQR) ! ILAENV = 50 RETURN ! 900 CONTINUE ! ! ISPEC = 9: maximum size of the subproblems at the bottom of the ! computation tree in the divide-and-conquer algorithm ! (used by xGELSD and xGESDD) ! ILAENV = 25 RETURN ! 1000 CONTINUE ! ! ISPEC = 10: ieee NaN arithmetic can be trusted not to trap ! ! ILAENV = 0 ILAENV = 1 IF( ILAENV.EQ.1 ) THEN ILAENV = IEEECK( 0, 0.0, 1.0 ) END IF RETURN ! 1100 CONTINUE ! ! ISPEC = 11: infinity arithmetic can be trusted not to trap ! ! ILAENV = 0 ILAENV = 1 IF( ILAENV.EQ.1 ) THEN ILAENV = IEEECK( 1, 0.0, 1.0 ) END IF RETURN ! ! End of ILAENV ! END INTEGER FUNCTION IEEECK( ISPEC, ZERO, ONE ) ! ! -- LAPACK auxiliary routine (version 3.0) -- ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., ! Courant Institute, Argonne National Lab, and Rice University ! June 30, 1998 ! ! .. Scalar Arguments .. INTEGER ISPEC REAL ONE, ZERO ! .. ! ! Purpose ! ======= ! ! IEEECK is called from the ILAENV to verify that Infinity and ! possibly NaN arithmetic is safe (i.e. will not trap). ! ! Arguments ! ========= ! ! ISPEC (input) INTEGER ! Specifies whether to test just for inifinity arithmetic ! or whether to test for infinity and NaN arithmetic. ! = 0: Verify infinity arithmetic only. ! = 1: Verify infinity and NaN arithmetic. ! ! ZERO (input) REAL ! Must contain the value 0.0 ! This is passed to prevent the compiler from optimizing ! away this code. ! ! ONE (input) REAL ! Must contain the value 1.0 ! This is passed to prevent the compiler from optimizing ! away this code. ! ! RETURN VALUE: INTEGER ! = 0: Arithmetic failed to produce the correct answers ! = 1: Arithmetic produced the correct answers ! ! .. Local Scalars .. REAL NAN1, NAN2, NAN3, NAN4, NAN5, NAN6, NEGINF, & NEGZRO, NEWZRO, POSINF ! .. ! .. Executable Statements .. IEEECK = 1 ! POSINF = ONE / ZERO IF( POSINF.LE.ONE ) THEN IEEECK = 0 RETURN END IF ! NEGINF = -ONE / ZERO IF( NEGINF.GE.ZERO ) THEN IEEECK = 0 RETURN END IF ! NEGZRO = ONE / ( NEGINF+ONE ) IF( NEGZRO.NE.ZERO ) THEN IEEECK = 0 RETURN END IF ! NEGINF = ONE / NEGZRO IF( NEGINF.GE.ZERO ) THEN IEEECK = 0 RETURN END IF ! NEWZRO = NEGZRO + ZERO IF( NEWZRO.NE.ZERO ) THEN IEEECK = 0 RETURN END IF ! POSINF = ONE / NEWZRO IF( POSINF.LE.ONE ) THEN IEEECK = 0 RETURN END IF ! NEGINF = NEGINF*POSINF IF( NEGINF.GE.ZERO ) THEN IEEECK = 0 RETURN END IF ! POSINF = POSINF*POSINF IF( POSINF.LE.ONE ) THEN IEEECK = 0 RETURN END IF ! ! ! ! ! Return if we were only asked to check infinity arithmetic ! IF( ISPEC.EQ.0 ) & RETURN ! NAN1 = POSINF + NEGINF ! NAN2 = POSINF / NEGINF ! NAN3 = POSINF / POSINF ! NAN4 = POSINF*ZERO ! NAN5 = NEGINF*NEGZRO ! NAN6 = NAN5*0.0 ! IF( NAN1.EQ.NAN1 ) THEN IEEECK = 0 RETURN END IF ! IF( NAN2.EQ.NAN2 ) THEN IEEECK = 0 RETURN END IF ! IF( NAN3.EQ.NAN3 ) THEN IEEECK = 0 RETURN END IF ! IF( NAN4.EQ.NAN4 ) THEN IEEECK = 0 RETURN END IF ! IF( NAN5.EQ.NAN5 ) THEN IEEECK = 0 RETURN END IF ! IF( NAN6.EQ.NAN6 ) THEN IEEECK = 0 RETURN END IF ! RETURN END