function i4_log_10 ( i ) !*****************************************************************************80 ! !! i4_log_10() returns the integer part of the logarithm base 10 of an I4. ! ! Discussion: ! ! I4_LOG_10 ( I ) + 1 is the number of decimal digits in I. ! ! An I4 is an integer value. ! ! Example: ! ! I I4_LOG_10 ! ----- -------- ! 0 0 ! 1 0 ! 2 0 ! 9 0 ! 10 1 ! 11 1 ! 99 1 ! 100 2 ! 101 2 ! 999 2 ! 1000 3 ! 1001 3 ! 9999 3 ! 10000 4 ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 08 June 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer I, the number whose logarithm base 10 ! is desired. ! ! Output, integer I4_LOG_10, the integer part of the ! logarithm base 10 of the absolute value of X. ! implicit none integer i integer i_abs integer i4_log_10 integer ten_pow if ( i == 0 ) then i4_log_10 = 0 else i4_log_10 = 0 ten_pow = 10 i_abs = abs ( i ) do while ( ten_pow <= i_abs ) i4_log_10 = i4_log_10 + 1 ten_pow = ten_pow * 10 end do end if return end function r8_uniform_01 ( seed ) !*****************************************************************************80 ! !! R8_UNIFORM_01 returns a unit pseudorandom R8. ! ! Discussion: ! ! An R8 is a real ( kind = rk ) value. ! ! For now, the input quantity SEED is an integer variable. ! ! This routine implements the recursion ! ! seed = 16807 * seed mod ( 2^31 - 1 ) ! r8_uniform_01 = seed / ( 2^31 - 1 ) ! ! The integer arithmetic never requires more than 32 bits, ! including a sign bit. ! ! If the initial seed is 12345, then the first three computations are ! ! Input Output R8_UNIFORM_01 ! SEED SEED ! ! 12345 207482415 0.096616 ! 207482415 1790989824 0.833995 ! 1790989824 2035175616 0.947702 ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 05 July 2006 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Paul Bratley, Bennett Fox, Linus Schrage, ! A Guide to Simulation, ! Springer Verlag, pages 201-202, 1983. ! ! Pierre L'Ecuyer, ! Random Number Generation, ! in Handbook of Simulation, ! edited by Jerry Banks, ! Wiley Interscience, page 95, 1998. ! ! Bennett Fox, ! Algorithm 647: ! Implementation and Relative Efficiency of Quasirandom ! Sequence Generators, ! ACM Transactions on Mathematical Software, ! Volume 12, Number 4, pages 362-376, 1986. ! ! Peter Lewis, Allen Goodman, James Miller ! A Pseudo-Random Number Generator for the System/360, ! IBM Systems Journal, ! Volume 8, pages 136-143, 1969. ! ! Parameters: ! ! Input/output, integer SEED, the "seed" value, which should ! NOT be 0. On output, SEED has been updated. ! ! Output, real ( kind = rk ) R8_UNIFORM_01, a new pseudorandom variate, ! strictly between 0 and 1. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer, parameter :: i4_huge = 2147483647 integer k real ( kind = rk ) r8_uniform_01 integer seed if ( seed == 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8_UNIFORM_01 - Fatal error!' write ( *, '(a)' ) ' Input value of SEED = 0.' stop 1 end if k = seed / 127773 seed = 16807 * ( seed - k * 127773 ) - k * 2836 if ( seed < 0 ) then seed = seed + i4_huge end if r8_uniform_01 = real ( seed, kind = rk ) * 4.656612875D-10 return end subroutine r83_indicator ( m, n, a ) !*****************************************************************************80 ! !! R83_INDICATOR sets up an R83 indicator matrix. ! ! Discussion: ! ! The R83 storage format is used for a tridiagonal matrix. ! The superdiagonal is stored in entries (1,2:min(M+1,N)). ! The diagonal in entries (2,1:min(M,N)). ! The subdiagonal in (3,min(M-1,N)). ! R8GE A(I,J) = R83 A(I-J+2,J). ! ! The "indicator matrix" simply has a value like I*10+J at every ! entry of a dense matrix, or at every entry of a compressed storage ! matrix for which storage is allocated. ! ! Example: ! ! An R83 matrix of order 3x5 would be stored: ! ! * A12 A23 A34 * ! A11 A22 A33 * * ! A21 A32 * * * ! ! An R83 matrix of order 5x5 would be stored: ! ! * A12 A23 A34 A45 ! A11 A22 A33 A44 A55 ! A21 A32 A43 A54 * ! ! An R83 matrix of order 5x3 would be stored: ! ! * A12 A23 ! A11 A22 A33 ! A21 A32 A43 ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 05 September 2015 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer N, the order of the matrix. ! N must be at least 2. ! ! Output, real ( kind = rk ) A(3,N), the R83 indicator matrix. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer n real ( kind = rk ) a(3,n) integer fac integer i integer i4_log_10 integer j integer m fac = 10 ** ( i4_log_10 ( n ) + 1 ) a(1:3,1:n) = 0.0D+00 do j = 1, n do i = max ( 1, j - 1 ), min ( m, j + 1 ) a(i-j+2,j) = real ( fac * i + j, kind = rk ) end do end do return end subroutine r83_print ( m, n, a, title ) !*****************************************************************************80 ! !! R83_PRINT prints an R83 matrix. ! ! Discussion: ! ! The R83 storage format is used for a tridiagonal matrix. ! The superdiagonal is stored in entries (1,2:min(M+1,N)). ! The diagonal in entries (2,1:min(M,N)). ! The subdiagonal in (3,min(M-1,N)). ! R8GE A(I,J) = R83 A(I-J+2,J). ! ! Example: ! ! An R83 matrix of order 3x5 would be stored: ! ! * A12 A23 A34 * ! A11 A22 A33 * * ! A21 A32 * * * ! ! An R83 matrix of order 5x5 would be stored: ! ! * A12 A23 A34 A45 ! A11 A22 A33 A44 A55 ! A21 A32 A43 A54 * ! ! An R83 matrix of order 5x3 would be stored: ! ! * A12 A23 ! A11 A22 A33 ! A21 A32 A43 ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 31 August 2015 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, N, the order of the matrix. ! N must be positive. ! ! Input, real ( kind = rk ) A(3,N), the R83 matrix. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer n real ( kind = rk ) a(3,n) integer m character ( len = * ) title call r83_print_some ( m, n, a, 1, 1, m, n, title ) return end subroutine r83_print_some ( m, n, a, ilo, jlo, ihi, jhi, title ) !*****************************************************************************80 ! !! R83_PRINT_SOME prints some of an R83 matrix. ! ! Discussion: ! ! The R83 storage format is used for a tridiagonal matrix. ! The superdiagonal is stored in entries (1,2:min(M+1,N)). ! The diagonal in entries (2,1:min(M,N)). ! The subdiagonal in (3,min(M-1,N)). ! R8GE A(I,J) = R83 A(I-J+2,J). ! ! Example: ! ! An R83 matrix of order 3x5 would be stored: ! ! * A12 A23 A34 * ! A11 A22 A33 * * ! A21 A32 * * * ! ! An R83 matrix of order 5x5 would be stored: ! ! * A12 A23 A34 A45 ! A11 A22 A33 A44 A55 ! A21 A32 A43 A54 * ! ! An R83 matrix of order 5x3 would be stored: ! ! * A12 A23 ! A11 A22 A33 ! A21 A32 A43 ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 05 September 2015 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, N, the order of the matrix. ! N must be positive. ! ! Input, real ( kind = rk ) A(3,N), the R83 matrix. ! ! Input, integer ILO, JLO, IHI, JHI, the first row and ! column, and the last row and column, to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer, parameter :: incx = 5 integer n real ( kind = rk ) a(3,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 integer m character ( len = * ) title if ( 0 < len_trim ( title ) ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) end if ! ! Print the columns of the matrix, in strips of 5. ! do j2lo = jlo, jhi, 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 ( *, '(a,5a14)' ) ' Col: ', ( ctemp(j2), j2 = 1, inc ) write ( *, '(a)' ) ' Row' write ( *, '(a)' ) ' ---' ! ! Determine the range of the rows in this strip. ! i2lo = max ( ilo, 1 ) i2lo = max ( i2lo, j2lo - 1 ) i2hi = min ( ihi, m ) i2hi = min ( i2hi, j2hi + 1 ) do i = i2lo, i2hi ! ! Print out (up to) 5 entries in row I, that lie in the current strip. ! do j2 = 1, inc j = j2lo - 1 + j2 if ( i - j + 2 < 1 .or. 3 < i - j + 2 ) then ctemp(j2) = ' ' else write ( ctemp(j2), '(g14.6)' ) a(i-j+2,j) end if end do write ( *, '(i5,1x,5a14)' ) i, ( ctemp(j2), j2 = 1, inc ) end do end do return end subroutine r83v_cg ( n, a, b, c, ax, x ) !*****************************************************************************80 ! !! R83V_CG uses the conjugate gradient method on an R83V system. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Frank Beckman, ! The Solution of Linear Equations by the Conjugate Gradient Method, ! in Mathematical Methods for Digital Computers, ! edited by John Ralston, Herbert Wilf, ! Wiley, 1967, ! ISBN: 0471706892, ! LC: QA76.5.R3. ! ! Parameters: ! ! Input, integer N, the order of the matrix. ! N must be positive. ! ! Input, real ( kind = rk ) A(N-1)), B(N), C(N-1), ! the R83V matrix. ! ! Input, real ( kind = rk ) AX(N), the right hand side vector. ! ! Input/output, real ( kind = rk ) X(N). ! On input, an estimate for the solution, which may be 0. ! On output, the approximate solution vector. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer n real ( kind = rk ) a(n-1) real ( kind = rk ) alpha real ( kind = rk ) ap(n) real ( kind = rk ) ax(n) real ( kind = rk ) b(n) real ( kind = rk ) beta real ( kind = rk ) c(n-1) integer it real ( kind = rk ) p(n) real ( kind = rk ) pap real ( kind = rk ) pr real ( kind = rk ) r(n) real ( kind = rk ) rap real ( kind = rk ) x(n) ! ! Initialize ! AP = A * x, ! R = b - A * x, ! P = b - A * x. ! call r83v_mv ( n, n, a, b, c, x, ap ) r(1:n) = ax(1:n) - ap(1:n) p(1:n) = ax(1:n) - ap(1:n) ! ! Do the N steps of the conjugate gradient method. ! do it = 1, n ! ! Compute the matrix*vector product AP=A*P. ! call r83v_mv ( n, n, a, b, c, p, ap ) ! ! Compute the dot products ! PAP = P*AP, ! PR = P*R ! Set ! ALPHA = PR / PAP. ! pap = dot_product ( p(1:n), ap(1:n) ) pr = dot_product ( p(1:n), r(1:n) ) if ( pap == 0.0D+00 ) then return end if alpha = pr / pap ! ! Set ! X = X + ALPHA * P ! R = R - ALPHA * AP. ! x(1:n) = x(1:n) + alpha * p(1:n) r(1:n) = r(1:n) - alpha * ap(1:n) ! ! Compute the vector dot product ! RAP = R*AP ! Set ! BETA = - RAP / PAP. ! rap = dot_product ( r(1:n), ap(1:n) ) beta = - rap / pap ! ! Update the perturbation vector ! P = R + BETA * P. ! p(1:n) = r(1:n) + beta * p(1:n) end do return end subroutine r83v_copy ( m, n, a1, a2, a3, b1, b2, b3 ) !*****************************************************************************80 ! !! R83V_COPY copies an R83V matrix. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, N, the order of the linear system. ! ! Input, real ( kind = rk ) A1(min(M-1,N)), A2(min(M,N)), A3(min(M,N-1)), ! the R83V matrix. ! ! Output, real ( kind = rk ) B1(min(M-1,N)), B2(min(M,N)), B3(min(M,N-1)), ! the R83V matrix. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a1(min(m-1,n)) real ( kind = rk ) a2(min(m,n)) real ( kind = rk ) a3(min(m,n-1)) real ( kind = rk ) b1(min(m-1,n)) real ( kind = rk ) b2(min(m,n)) real ( kind = rk ) b3(min(m,n-1)) b1 = a1(1:min(m-1,n )) b2 = a2(1:min(m, n )) b3 = a3(1:min(m, n-1)) return end subroutine r83v_cr_fa ( n, a, b, c, a_cr ) !*****************************************************************************80 ! !! R83V_CR_FA decomposes an R83V matrix using cyclic reduction. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Once R83_CR_FA has decomposed a matrix A, then R83_CR_SL may be used to ! solve linear systems A * x = b. ! ! R83_CR_FA does not employ pivoting. Hence, the results can be more ! sensitive to ill-conditioning than standard Gauss elimination. In ! particular, R83_CR_FA will fail if any diagonal element of the matrix ! is zero. Other matrices may also cause R83_CR_FA to fail. ! ! R83_CR_FA can be guaranteed to work properly if the matrix is strictly ! diagonally dominant, that is, if the absolute value of the diagonal ! element is strictly greater than the sum of the absolute values of ! the offdiagonal elements, for each equation. ! ! The algorithm may be illustrated by the following figures: ! ! The initial matrix is given by: ! ! D1 U1 ! L1 D2 U2 ! L2 R83 U3 ! L3 D4 U4 ! L4 D5 U5 ! L5 D6 ! ! Rows and columns are permuted in an odd/even way to yield: ! ! D1 U1 ! R83 L2 U3 ! D5 L4 U5 ! L1 U2 D2 ! L3 U4 D4 ! L5 D6 ! ! A block LU decomposition is performed to yield: ! ! D1 |U1 ! R83 |L2 U3 ! D5| L4 U5 ! --------+-------- ! |D2'F3 ! |F1 D4'F4 ! | F2 D6' ! ! For large systems, this reduction is repeated on the lower right hand ! tridiagonal subsystem until a completely upper triangular system ! is obtained. The system has now been factored into the product of a ! lower triangular system and an upper triangular one, and the information ! defining this factorization may be used by R83_CR_SL to solve linear ! systems. ! ! Example: ! ! An R83 matrix of order 3x5 would be stored: ! ! * A12 A23 A34 * ! A11 A22 A33 * * ! A21 A32 * * * ! ! An R83 matrix of order 5x5 would be stored: ! ! * A12 A23 A34 A45 ! A11 A22 A33 A44 A55 ! A21 A32 A43 A54 * ! ! An R83 matrix of order 5x3 would be stored: ! ! * A12 A23 ! A11 A22 A33 ! A21 A32 A43 ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2015 ! ! Author: ! ! John Burkardt. ! ! Reference: ! ! Roger Hockney, ! A fast direct solution of Poisson's equation using Fourier Analysis, ! Journal of the ACM, ! Volume 12, Number 1, pages 95-113, January 1965. ! ! Parameters: ! ! Input, integer N, the order of the matrix. ! N must be positive. ! ! Input, real ( kind = rk ) A(min(M-1,N)), B(min(M,N)), C(min(M,N-1)), ! the R83V matrix. ! ! Output, real ( kind = rk ) A_CR(3,2*N+1), factorization information. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer n real ( kind = rk ) a(n-1) real ( kind = rk ) a_cr(3,2*n+1) real ( kind = rk ) b(n) real ( kind = rk ) c(n-1) integer iful integer ifulp integer ihaf integer il integer ilp integer inc integer incr integer ipnt integer ipntp if ( n <= 0 ) then write ( *, '(a)' ) '' write ( *, '(a)' ) 'R83V_CR_FA - Fatal error!' write ( *, '(a,i4)' ) ' Nonpositive N = ', n stop 1 end if a_cr(1:3,1:2*n+1) = 0.0D+00 if ( n == 1 ) then a_cr(2,2) = 1.0D+00 / b(1) return end if ! ! Set the workspace entries. ! a_cr(1,2:n) = c(1:n-1) a_cr(2,2:n+1) = b(1:n) a_cr(3,2:n) = a(1:n-1) il = n ipntp = 0 do while ( 1 < il ) ipnt = ipntp ipntp = ipntp + il if ( mod ( il, 2 ) == 1 ) then inc = il + 1 else inc = il end if incr = ( inc / 2 ) il = ( il / 2 ) ihaf = ipntp + incr + 1 ifulp = ipnt + inc + 2 do ilp = incr, 1, -1 ifulp = ifulp - 2 iful = ifulp - 1 ihaf = ihaf - 1 a_cr(2,iful+1) = 1.0D+00 / a_cr(2,iful+1) a_cr(3,iful+1) = a_cr(3,iful+1) * a_cr(2,iful+1) a_cr(1,ifulp+1) = a_cr(1,ifulp+1) * a_cr(2,ifulp+2) a_cr(2,ihaf+1) = a_cr(2,ifulp+1) - a_cr(1,iful+1) * a_cr(3,iful+1) & - a_cr(1,ifulp+1) * a_cr(3,ifulp+1) a_cr(3,ihaf+1) = -a_cr(3,ifulp+1) * a_cr(3,ifulp+2) a_cr(1,ihaf+1) = -a_cr(1,ifulp+1) * a_cr(1,ifulp+2) end do end do a_cr(2,ipntp+2) = 1.0D+00 / a_cr(2,ipntp+2) return end subroutine r83v_cr_sl ( n, a_cr, ax, x ) !*****************************************************************************80 ! !! R83V_CR_SL solves a real linear system factored by R83V_CR_FA. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! The matrix A must be tridiagonal. R83_CR_FA is called to compute the ! LU factors of A. It does so using a form of cyclic reduction. If ! the factors computed by R83_CR_FA are passed to R83_CR_SL, then one or many ! linear systems involving the matrix A may be solved. ! ! Note that R83_CR_FA does not perform pivoting, and so the solution ! produced by R83_CR_SL may be less accurate than a solution produced ! by a standard Gauss algorithm. However, such problems can be ! guaranteed not to occur if the matrix A is strictly diagonally ! dominant, that is, if the absolute value of the diagonal coefficient ! is greater than the sum of the absolute values of the two off diagonal ! coefficients, for each row of the matrix. ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt. ! ! Reference: ! ! Roger Hockney, ! A fast direct solution of Poisson's equation using Fourier Analysis, ! Journal of the ACM, ! Volume 12, Number 1, pages 95-113, January 1965. ! ! Parameters: ! ! Input, integer N, the order of the matrix. ! N must be positive. ! ! Input, real ( kind = rk ) A_CR(3,2*N+1), factorization information computed ! by R83V_CR_FA. ! ! Input, real ( kind = rk ) AX(N), the right hand side vector. ! ! Output, real ( kind = rk ) X(N), the solution of the linear system. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer n real ( kind = rk ) a_cr(3,2*n+1) real ( kind = rk ) ax(n) integer iful integer ifulm integer ihaf integer il integer ipnt integer ipntp integer ndiv real ( kind = rk ) rhs(2*n+1) real ( kind = rk ) x(n) if ( n <= 0 ) then write ( *, '(a)' ) '' write ( *, '(a)' ) 'R83V_CR_SL - Fatal error!' write ( *, '(a,i4)' ) ' Nonpositive N = ', n stop 1 end if if ( n == 1 ) then x(1) = a_cr(2,2) * ax(1) return end if ! ! Set up RHS. ! rhs(1) = 0.0D+00 rhs(2:n+1) = ax(1:n) rhs(n+2:2*n+1) = 0.0D+00 il = n ndiv = 1 ipntp = 0 do while ( 1 < il ) ipnt = ipntp ipntp = ipntp + il il = ( il / 2 ) ndiv = ndiv * 2 ihaf = ipntp do iful = ipnt + 2, ipntp, 2 ihaf = ihaf + 1 rhs(ihaf+1) = rhs(iful+1) - a_cr(3,iful) * rhs(iful) & - a_cr(1,iful+1) * rhs(iful+2) end do end do rhs(ihaf+1) = rhs(ihaf+1) * a_cr(2,ihaf+1) ipnt = ipntp do while ( 0 < ipnt ) ipntp = ipnt ndiv = ( ndiv / 2 ) il = ( n / ndiv ) ipnt = ipnt - il ihaf = ipntp do ifulm = ipnt + 1, ipntp, 2 iful = ifulm + 1 ihaf = ihaf + 1 rhs(iful+1) = rhs(ihaf+1) rhs(ifulm+1) = a_cr(2,ifulm+1) * ( rhs(ifulm+1) & - a_cr(3,ifulm) * rhs(ifulm) & - a_cr(1,ifulm+1) * rhs(iful+1) ) end do end do x(1:n) = rhs(2:n+1) return end subroutine r83v_cr_sls ( n, a_cr, nb, ax, x ) !*****************************************************************************80 ! !! R83V_CR_SLS solves several real linear systems factored by R83V_CR_FA. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! The matrix A must be tridiagonal. R83_CR_FA is called to compute the ! LU factors of A. It does so using a form of cyclic reduction. If ! the factors computed by R83_CR_FA are passed to R83_CR_SLS, then one or ! many linear systems involving the matrix A may be solved. ! ! Note that R83_CR_FA does not perform pivoting, and so the solution ! produced by R83_CR_SLS may be less accurate than a solution produced ! by a standard Gauss algorithm. However, such problems can be ! guaranteed not to occur if the matrix A is strictly diagonally ! dominant, that is, if the absolute value of the diagonal coefficient ! is greater than the sum of the absolute values of the two off diagonal ! coefficients, for each row of the matrix. ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt. ! ! Reference: ! ! Roger Hockney, ! A fast direct solution of Poisson's equation using Fourier Analysis, ! Journal of the ACM, ! Volume 12, Number 1, pages 95-113, January 1965. ! ! Parameters: ! ! Input, integer N, the order of the matrix. ! N must be positive. ! ! Input, real ( kind = rk ) A_CR(3,2*N+1), factorization information computed ! by R83V_CR_FA. ! ! Input, integer NB, the number of right hand sides. ! ! Input, real ( kind = rk ) AX(N,NB), the right hand side vectors. ! ! Output, real ( kind = rk ) X(N,NB), the solutions of the linear system. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer n integer nb real ( kind = rk ) a_cr(3,2*n+1) real ( kind = rk ) ax(n,nb) integer iful integer ifulm integer ihaf integer il integer ipnt integer ipntp integer ndiv real ( kind = rk ) rhs(2*n+1,nb) real ( kind = rk ) x(n,nb) if ( n <= 0 ) then write ( *, '(a)' ) '' write ( *, '(a)' ) 'R83V_CR_SLS - Fatal error!' write ( *, '(a,i4)' ) ' Nonpositive N = ', n stop 1 end if if ( n == 1 ) then x(1,1:nb) = a_cr(2,2) * ax(1,1:nb) return end if ! ! Set up RHS. ! rhs(1,1:nb) = 0.0D+00 rhs(2:n+1,1:nb) = ax(1:n,1:nb) rhs(n+2:2*n+1,1:nb) = 0.0D+00 il = n ndiv = 1 ipntp = 0 do while ( 1 < il ) ipnt = ipntp ipntp = ipntp + il il = ( il / 2 ) ndiv = ndiv * 2 ihaf = ipntp do iful = ipnt + 2, ipntp, 2 ihaf = ihaf + 1 rhs(ihaf+1,1:nb) = rhs(iful+1,1:nb) & - a_cr(3,iful) * rhs(iful,1:nb) & - a_cr(1,iful+1) * rhs(iful+2,1:nb) end do end do rhs(ihaf+1,1:nb) = rhs(ihaf+1,1:nb) * a_cr(2,ihaf+1) ipnt = ipntp do while ( 0 < ipnt ) ipntp = ipnt ndiv = ( ndiv / 2 ) il = ( n / ndiv ) ipnt = ipnt - il ihaf = ipntp do ifulm = ipnt + 1, ipntp, 2 iful = ifulm + 1 ihaf = ihaf + 1 rhs(iful+1,1:nb) = rhs(ihaf+1,1:nb) rhs(ifulm+1,1:nb) = a_cr(2,ifulm+1) * ( rhs(ifulm+1,1:nb) & - a_cr(3,ifulm) * rhs(ifulm,1:nb) & - a_cr(1,ifulm+1) * rhs(iful+1,1:nb) ) end do end do x(1:n,1:nb) = rhs(2:n+1,1:nb) return end subroutine r83v_dif2 ( m, n, a, b, c ) !*****************************************************************************80 ! !! R83V_DIF2 returns the DIF2 matrix in R83V format. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Properties: ! ! A is banded, with bandwidth 3. ! ! A is tridiagonal. ! ! Because A is tridiagonal, it has property A (bipartite). ! ! A is a special case of the TRIS or tridiagonal scalar matrix. ! ! A is integral, therefore det ( A ) is integral, and ! det ( A ) * inverse ( A ) is integral. ! ! A is Toeplitz: constant along diagonals. ! ! A is symmetric: A' = A. ! ! Because A is symmetric, it is normal. ! ! Because A is normal, it is diagonalizable. ! ! A is persymmetric: A(I,J) = A(N+1-J,N+1-I). ! ! A is positive definite. ! ! A is an M matrix. ! ! A is weakly diagonally dominant, but not strictly diagonally dominant. ! ! A has an LU factorization A = L * U, without pivoting. ! ! The matrix L is lower bidiagonal with subdiagonal elements: ! ! L(I+1,I) = -I/(I+1) ! ! The matrix U is upper bidiagonal, with diagonal elements ! ! U(I,I) = (I+1)/I ! ! and superdiagonal elements which are all -1. ! ! A has a Cholesky factorization A = L * L', with L lower bidiagonal. ! ! L(I,I) = sqrt ( (I+1) / I ) ! L(I,I-1) = -sqrt ( (I-1) / I ) ! ! The eigenvalues are ! ! LAMBDA(I) = 2 + 2 * COS(I*PI/(N+1)) ! = 4 SIN^2(I*PI/(2*N+2)) ! ! The corresponding eigenvector X(I) has entries ! ! X(I)(J) = sqrt(2/(N+1)) * sin ( I*J*PI/(N+1) ). ! ! Simple linear systems: ! ! x = (1,1,1,...,1,1), A*x=(1,0,0,...,0,1) ! ! x = (1,2,3,...,n-1,n), A*x=(0,0,0,...,0,n+1) ! ! det ( A ) = N + 1. ! ! The value of the determinant can be seen by induction, ! and expanding the determinant across the first row: ! ! det ( A(N) ) = 2 * det ( A(N-1) ) - (-1) * (-1) * det ( A(N-2) ) ! = 2 * N - (N-1) ! = N + 1 ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Robert Gregory, David Karney, ! A Collection of Matrices for Testing Computational Algorithms, ! Wiley, 1969, ! ISBN: 0882756494, ! LC: QA263.68 ! ! Morris Newman, John Todd, ! Example A8, ! The evaluation of matrix inversion programs, ! Journal of the Society for Industrial and Applied Mathematics, ! Volume 6, Number 4, pages 466-476, 1958. ! ! John Todd, ! Basic Numerical Mathematics, ! Volume 2: Numerical Algebra, ! Birkhauser, 1980, ! ISBN: 0817608117, ! LC: QA297.T58. ! ! Joan Westlake, ! A Handbook of Numerical Matrix Inversion and Solution of ! Linear Equations, ! John Wiley, 1968, ! ISBN13: 978-0471936756, ! LC: QA263.W47. ! ! Parameters: ! ! Input, integer M, N, the order of the matrix. ! ! Output, real ( kind = rk ) A(min(M-1,N)), B(min(M,N)), C(min(M,N-1)), ! the R83V matrix. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a(min(m-1,n)) real ( kind = rk ) b(min(m,n)) real ( kind = rk ) c(min(m,n-1)) a(1:min(m-1,n) ) = -1.0D+00 b(1:min(m, n) ) = +2.0D+00 c(1:min(m, n-1)) = -1.0D+00 return end subroutine r83v_fs ( n, a1, a2, a3, b, x ) !*****************************************************************************80 ! !! R83V_FS solves a linear system with R83V matrix. ! ! Discussion: ! ! This function is based on the LINPACK SGTSL routine. ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 19 February 2016 ! ! Author: ! ! John Burkardt, based on the LINPACK SGTSL function. ! ! Reference: ! ! Jack Dongarra, Cleve Moler, Jim Bunch and Pete Stewart, ! LINPACK User's Guide, ! SIAM, (Society for Industrial and Applied Mathematics), ! 3600 University City Science Center, ! Philadelphia, PA, 19104-2688. ! ISBN 0-89871-172-X ! ! Parameters: ! ! Input, integer N, the order of the tridiagonal matrix. ! ! Input, real ( kind = rk ) A1(N-1), A2(N), A3(N-1), the R83V matrix. ! ! Input, real ( kind = rk ) B(N), the right hand side. ! ! Output, real ( kind = rk ) X(N), the solution. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer n real ( kind = rk ) a1(n-1) real ( kind = rk ) a2(n) real ( kind = rk ) a3(n-1) real ( kind = rk ) b(n) real ( kind = rk ) c(n) real ( kind = rk ) d(n) real ( kind = rk ) e(n) integer k real ( kind = rk ) t real ( kind = rk ) x(n) ! ! Copy the input data. ! c(1) = 0.0D+00 c(2:n) = a1(1:n-1) d(1:n) = a2(1:n) e(1:n-1) = a3(1:n-1) e(n) = 0.0D+00 x(1:n) = b(1:n) ! ! Factor. ! c(1) = a2(1) if ( 2 <= n ) then d(1) = e(1) e(1) = 0.0D+00 e(n) = 0.0D+00 do k = 2, n ! ! Find the larger of the two rows and interchange if necessary. ! if ( abs ( c(k-1) ) <= abs ( c(k) ) ) then t = c(k) c(k) = c(k-1) c(k-1) = t t = d(k) d(k) = d(k-1) d(k-1) = t t = e(k) e(k) = e(k-1) e(k-1) = t t = x(k) x(k) = x(k-1) x(k-1) = t end if ! ! Zero elements. ! if ( c(k-1) == 0.0D+00 ) then write ( *, '(a)' ) '' write ( *, '(a)' ) 'R83V_FS - Fatal error!' write ( *, '(a,i4)' ) ' Zero pivot on step K = ', k stop 1 end if t = - c(k) / c(k-1) c(k) = d(k) + t * d(k-1) d(k) = e(k) + t * e(k-1) e(k) = 0.0D+00 x(k) = x(k) + t * x(k-1) end do end if if ( c(n) == 0.0D+00 ) then write ( *, '(a)' ) '' write ( *, '(a)' ) 'R83V_FS - Fatal error!' write ( *, '(a,i4)' ) ' Zero pivot on step K = ', n stop 1 end if ! ! Back solve. ! x(n) = x(n) / c(n) if ( 1 < n ) then x(n-1) = ( x(n-1) - d(n-1) * x(n) ) / c(n-1) do k = n - 2, 1, -1 x(k) = ( x(k) - d(k) * x(k+1) - e(k) * x(k+2) ) / c(k) end do end if return end subroutine r83v_gs_sl ( n, a, b, c, ax, x, it_max ) !*****************************************************************************80 ! !! R83V_GS_SL solves a R83V system using Gauss-Seidel iteration. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer N, the order of the matrix. ! N must be at least 2. ! ! Input, real ( kind = rk ) A(min(M-1,N)), B(min(M,N)), C(min(M,N-1)), ! the R83V matrix. ! ! Input, real ( kind = rk ) AX(N), the right hand side of the linear system. ! ! Input/output, real ( kind = rk ) X(N), on input, ! an approximate solution to the system. On output, an improved solution ! estimate. ! ! Input, integer IT_MAX, the maximum number of iterations. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer n real ( kind = rk ) a(n-1) real ( kind = rk ) ax(n) real ( kind = rk ) b(n) real ( kind = rk ) c(n-1) integer i integer it_max integer it_num real ( kind = rk ) x(n) real ( kind = rk ) x_new(n) ! ! No diagonal matrix entry can be zero. ! do i = 1, n if ( b(i) == 0.0D+00 ) then write ( *, '(a)' ) '' write ( *, '(a)' ) 'R83V_GS_SL - Fatal error!' write ( *, '(a,i4)' ) ' Zero diagonal entry, index = ', i stop 1 end if end do do it_num = 1, it_max x_new(1) = ( ax(1) - c(1) * x(2) ) / b(1) do i = 2, n - 1 x_new(i) = ( ax(i) - a(i-1) * x_new(i-1) - c(i) * x(i+1) ) / b(i) end do x_new(n) = ( ax(n) - a(n-1) * x_new(n-1) ) / b(n) x(1:n) = x_new(1:n) end do return end subroutine r83v_indicator ( m, n, a, b, c ) !*****************************************************************************80 ! !! R83V_INDICATOR sets up an R83V indicator matrix. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, N, the order of the matrix. ! ! Output, real ( kind = rk ) A(min(M-1,N)), B(min(M,N)), C(min(M,N-1)), ! the R83V matrix. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a(min(m-1,n)) real ( kind = rk ) b(min(m,n)) real ( kind = rk ) c(min(m,n-1)) integer fac integer i integer i4_log_10 fac = 10 ** ( i4_log_10 ( n ) + 1 ) do i = 1, min ( m - 1, n ) a(i) = real ( fac * ( i + 1 ) + i, kind = rk ) end do do i = 1, min ( m, n ) b(i) = real ( fac * i + i, kind = rk ) end do do i = 1, min ( m, n - 1 ) c(i) = real ( fac * i + i + 1, kind = rk ) end do return end subroutine r83v_jac_sl ( n, a, b, c, ax, x, it_max ) !*****************************************************************************80 ! !! R83V_JAC_SL solves a R83V system A*x=b using Jacobi iteration. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer N, the order of the matrix. ! N must be at least 2. ! ! Input, real ( kind = rk ) A(min(M-1,N)), B(min(M,N)), C(min(M,N-1)), the ! R83V matrix. ! ! Input, real ( kind = rk ) AX(N), the right hand side of the linear system. ! ! Input/output, real ( kind = rk ) X(N), an approximate solution to the ! system, which is updated on output. ! ! Input, integer IT_MAX, the maximum number of iterations. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer n real ( kind = rk ) a(n-1) real ( kind = rk ) ax(n) real ( kind = rk ) b(n) real ( kind = rk ) c(n-1) integer i integer it_max integer it_num real ( kind = rk ) x(n) real ( kind = rk ) x_new(n) ! ! No diagonal matrix entry can be zero. ! do i = 1, n if ( b(i) == 0.0D+00 ) then write ( *, '(a)' ) '' write ( *, '(a)' ) 'R83V_JAC_SL - Fatal error!' write ( *, '(a,i4)' ) ' Zero diagonal entry, index = ', i stop 1 end if end do ! ! Iterate IT_MAX times. ! do it_num = 1, it_max x_new(1:n) = ax(1:n) x_new(1:n-1) = x_new(1:n-1) - c(1:n-1) * x(2:n) x_new(2:n) = x_new(2:n) - a(1:n-1) * x(1:n-1) ! ! Divide by diagonal terms. ! x_new(1:n) = x_new(1:n) / b(1:n) ! ! Update. ! x(1:n) = x_new(1:n) end do return end subroutine r83v_mtv ( m, n, a, b, c, x, ax ) !*****************************************************************************80 ! !! R83V_MTV multiplies a vector by an R83V matrix. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, N, the order of the linear system. ! ! Input, real ( kind = rk ) A(min(M-1,N)), B(min(M,N)), C(min(M,N-1)), ! the R83V matrix. ! ! Input, real ( kind = rk ) X(M), the vector to be multiplied by A'. ! ! Output, real ( kind = rk ) AX(N), the product A' * x. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a(min(m-1,n)) integer ahi real ( kind = rk ) ax(n) real ( kind = rk ) b(min(m,n)) integer bhi real ( kind = rk ) c(min(m,n-1)) integer chi real ( kind = rk ) x(m) ! ! Find each nonzero A(I,J), multiply by X(I), add to B(J). ! ! A(K) = A(K+1,K) = A'(K,K+1) ! ahi = min ( m - 1, n ) bhi = min ( m, n ) chi = min ( m, n - 1 ) ax(1:n) = 0.0D+00 ax(1:ahi) = ax(1:ahi) + a(1:ahi) * x(2:ahi+1) ax(1:bhi) = ax(1:bhi) + b(1:bhi) * x(1:bhi) ax(2:chi+1) = ax(2:chi+1) + c(1:chi) * x(1:chi) return end subroutine r83v_mv ( m, n, a, b, c, x, ax ) !*****************************************************************************80 ! !! R83V_MV multiplies a R83V matrix times a vector. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, N, the order of the linear system. ! ! Input, real ( kind = rk ) A(min(M-1,N)), B(min(M,N)), C(min(M,N-1)), ! the R83V matrix. ! ! Input, real ( kind = rk ) X(N), the vector to be multiplied. ! ! Output, real ( kind = rk ) AX(M), the product A * x. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a(min(m-1,n)) integer ahi real ( kind = rk ) ax(m) real ( kind = rk ) b(min(m,n)) integer bhi real ( kind = rk ) c(min(m,n-1)) integer chi real ( kind = rk ) x(n) ahi = min ( m - 1, n ) bhi = min ( m, n ) chi = min ( m, n - 1 ) ax(1:m) = 0.0D+00 ax(2:ahi+1) = ax(2:ahi+1) + a(1:ahi) * x(1:ahi) ax(1:bhi) = ax(1:bhi) + b(1:bhi) * x(1:bhi) ax(1:chi) = ax(1:chi) + c(1:chi) * x(2:chi+1) return end subroutine r83v_print ( m, n, a, b, c, title ) !*****************************************************************************80 ! !! R83V_PRINT prints an R83V matrix. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, N, the order of the matrix. ! ! Input, real ( kind = rk ) A(min(M-1,N)), B(min(M,N)), C(min(M,N-1)), ! the R83V matrix. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a(min(m-1,n)) real ( kind = rk ) b(min(m,n)) real ( kind = rk ) c(min(m,n-1)) character ( len = * ) title call r83v_print_some ( m, n, a, b, c, 1, 1, m, n, title ) return end subroutine r83v_print_some ( m, n, a, b, c, ilo, jlo, ihi, jhi, title ) !*****************************************************************************80 ! !! R83V_PRINT_SOME prints some of a R83V matrix. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer N, the order of the matrix. ! N must be positive. ! ! Input, real ( kind = rk ) A(min(M-1,N)), B(min(M,N)), C(min(M+1,N)), ! the R83V matrix. ! ! Input, integer ILO, JLO, IHI, JHI, the first row and ! column, and the last row and column, to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a(min(m-1,n)) real ( kind = rk ) b(min(m,n)) real ( kind = rk ) c(min(m,n-1)) integer i integer i2hi integer i2lo integer ihi integer ilo integer inc integer incx integer j integer j2 integer j2hi integer j2lo integer jhi integer jlo character ( len = * ) title if ( 0 < len_trim ( title ) ) then write ( *, '(a)' ) '' write ( *, '(a)' ) trim( title ) end if incx = 5 ! ! Print the columns of the matrix, in strips of 5. ! do j2lo = jlo, jhi, incx j2hi = j2lo + incx - 1 j2hi = min ( j2hi, n ) j2hi = min ( j2hi, jhi ) inc = j2hi + 1 - j2lo write ( *, '(a)' ) '' write ( *, '(a)', advance = 'no' ) ' Col: ' do j = j2lo, j2hi write ( *, '(i7,7x)', advance = 'no' ) j end do write ( *, '(a)' ) '' write ( *, '(a)' ) ' Row' write ( *, '(a)' ) ' ---' ! ! Determine the range of the rows in this strip. ! i2lo = max ( ilo, 1 ) i2lo = max ( i2lo, j2lo - 1 ) i2hi = min ( ihi, m ) i2hi = min ( i2hi, j2hi + 1 ) do i = i2lo, i2hi ! ! Print out (up to) 5 entries in row I, that lie in the current strip. ! write ( *, '(i5,a)', advance = 'no' ) i, ':' do j2 = 1, inc j = j2lo - 1 + j2 if ( j < i - 1 ) then write ( *, '(a)', advance = 'no' ) ' ' else if ( j == i - 1 ) then write ( *, '(g14.6)', advance = 'no' ) a(i-1) else if ( j == i ) then write ( *, '(g14.6)', advance = 'no' ) b(i) else if ( j == i + 1 ) then write ( *, '(g14.6)', advance = 'no' ) c(i) else end if end do write ( *, '(a)' ) '' end do end do return end subroutine r83v_random ( m, n, seed, a, b, c ) !*****************************************************************************80 ! !! R83V_RANDOM randomizes an R83V matrix. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, N, the order of the matrix. ! ! Input, integer SEED, a seed for the random number generator. ! ! Output, real ( kind = rk ) A(min(M-1,N)), B(min(M,N)), C(min(M,N-1)), ! the R83V matrix. ! ! Output, integer SEED, an updated seed for the random ! number generator. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a(min(m-1,n)) real ( kind = rk ) b(min(m,n)) real ( kind = rk ) c(min(m,n-1)) integer seed call r8vec_uniform_01 ( min ( m - 1, n ), seed, a ) call r8vec_uniform_01 ( min ( m, n ), seed, b ) call r8vec_uniform_01 ( min ( m, n - 1 ), seed, c ) return end subroutine r83v_res ( m, n, a, b, c, x, ax, r ) !*****************************************************************************80 ! !! R83V_RES computes the residual R = b-A*x for R83V matrices. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, the number of rows of the matrix. ! M must be positive. ! ! Input, integer N, the number of columns of the matrix. ! N must be positive. ! ! Input, real ( kind = rk ) A(min(M-1,N)), B(min(M,N)), C(min(M,N-1)), ! the R83V matrix. ! ! Input, real ( kind = rk ) X(N), the vector to be multiplied by A. ! ! Input, real ( kind = rk ) AX(M), the desired result A * x. ! ! Output, real ( kind = rk ) R(M), the residual R = b - A * x. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a(min(m-1,n)) real ( kind = rk ) ax(m) real ( kind = rk ) b(min(m,n)) real ( kind = rk ) c(min(m,n-1)) real ( kind = rk ) r(m) real ( kind = rk ) x(n) call r83v_mv ( m, n, a, b, c, x, r ) r(1:m) = ax(1:m) - r(1:m) return end subroutine r83v_to_r8ge ( m, n, a_83v, b_83v, c_83v, a_ge ) !*****************************************************************************80 ! !! R83V_TO_R8GE copies an R83V matrix to a R8GE matrix. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer N, the order of the matrix. ! N must be at least 2. ! ! Input, real ( kind = rk ) A_83V(min(M-1,N)), B_83V(min(M,N)), ! C_83V(min(M,N-1)), the R83V matrix. ! ! Output, real ( kind = rk ) A_GE(M,N), the R8GE matrix. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a_83v(min(m-1,n)) real ( kind = rk ) a_ge(m,n) integer ahi real ( kind = rk ) b_83v(min(m,n)) integer bhi real ( kind = rk ) c_83v(min(m,n-1)) integer chi integer k a_ge(1:m,1:n) = 0.0D+00 ahi = min ( m - 1, n ) do k = 1, ahi a_ge(k+1,k) = a_83v(k) end do bhi = min ( m, n ) do k = 1, bhi a_ge(k,k) = b_83v(k) end do chi = min ( m, n - 1 ) do k = 1, chi a_ge(k,k+1) = c_83v(k) end do return end subroutine r83v_to_r8vec ( m, n, a1, a2, a3, a ) !*****************************************************************************80 ! !! R83V_TO_R8VEC copies an R83V matrix to an R8VEC. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 18 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, N, the order of the matrix. ! ! Input, real ( kind = rk ) A1(min(M-1,N)), A2(min(M,N)), A3(min(M,N-1)), ! the matrix. ! ! Output, real ( kind = rk ) R83V_TO_R8VEC(min(M-1,N)+min(M,N)+min(M,N-1)), ! the vector. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a(min(m-1,n)+min(m,n)+min(m,n-1)) real ( kind = rk ) a1(min(m-1,n)) real ( kind = rk ) a2(min(m,n)) real ( kind = rk ) a3(min(m,n-1)) integer j integer k k = 0 do j = 1, n if ( j <= m + 1 .and. 2 <= j ) then k = k + 1 a(k) = a3(j-1) end if if ( j <= m ) then k = k + 1 a(k) = a2(j) end if if ( j <= m - 1 ) then k = k + 1 a(k) = a1(j) end if end do return end subroutine r83v_transpose ( m, n, a1, a2, a3, b1, b2, b3 ) !*****************************************************************************80 ! !! R83V_TRANSPOSE makes a transposed copy of an R83V matrix. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, N, the order of the linear system. ! ! Input, real ( kind = rk ) A1(min(M-1,N)), A2(min(M,N)), A3(min(M,N-1)), ! the R83V matrix. ! ! Output, real ( kind = rk ) B1(min(N-1,M)), B2(min(N,M)), B3(min(N,M-1)), ! the R83V matrix. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a1(min(m-1,n)) real ( kind = rk ) a2(min(m,n)) real ( kind = rk ) a3(min(m,n-1)) real ( kind = rk ) b1(min(n-1,m)) real ( kind = rk ) b2(min(n,m)) real ( kind = rk ) b3(min(n,m-1)) b1 = a3(1:min(m, n-1)) b2 = a2(1:min(m, n )) b3 = a1(1:min(m-1,n )) return end subroutine r83v_zeros ( m, n, a, b, c ) !*****************************************************************************80 ! !! R83V_ZEROS zeros an R83V matrix. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, N, the order of the linear system. ! ! Output, real ( kind = rk ) A(min(M-1,N)), B(min(M,N)), C(min(M,N-1)), ! the R83V matrix. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a(min(m-1,n)) real ( kind = rk ) b(min(m,n)) real ( kind = rk ) c(min(m,n-1)) a(1:min(m-1,n)) = 0.0D+00 b(1:min(m,n)) = 0.0D+00 c(1:min(m,n-1)) = 0.0D+00 return end subroutine r8ge_indicator ( m, n, a ) !*****************************************************************************80 ! !! R8GE_INDICATOR sets up an R8GE indicator matrix. ! ! Discussion: ! ! The "indicator matrix" simply has a value like I*10+J at every ! entry of a dense matrix, or at every entry of a compressed storage ! matrix for which storage is allocated. ! ! The R8GE storage format is used for a general M by N matrix. A storage ! space is made for each entry. The two dimensional logical ! array can be thought of as a vector of M*N entries, starting with ! the M entries in the column 1, then the M entries in column 2 ! and so on. Considered as a vector, the entry A(I,J) is then stored ! in vector location I+(J-1)*M. ! ! R8GE storage is used by LINPACK and LAPACK. ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 January 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, the number of rows of the matrix. ! M must be positive. ! ! Input, integer N, the number of columns of the matrix. ! N must be positive. ! ! Output, real ( kind = rk ) A(M,N), the R8GE matrix. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a(m,n) integer fac integer i integer i4_log_10 integer j fac = 10 ** ( i4_log_10 ( n ) + 1 ) do i = 1, m do j = 1, n a(i,j) = real ( fac * i + j, kind = rk ) end do end do return end subroutine r8ge_mtv ( m, n, a, x, b ) !*****************************************************************************80 ! !! R8GE_MTV multiplies an R8VEC by an R8GE matrix. ! ! Discussion: ! ! The R8GE storage format is used for a general M by N matrix. A storage ! space is made for each entry. The two dimensional logical ! array can be thought of as a vector of M*N entries, starting with ! the M entries in the column 1, then the M entries in column 2 ! and so on. Considered as a vector, the entry A(I,J) is then stored ! in vector location I+(J-1)*M. ! ! R8GE storage is used by LINPACK and LAPACK. ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 January 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, the number of rows of the matrix. ! M must be positive. ! ! Input, integer N, the number of columns of the matrix. ! N must be positive. ! ! Input, real ( kind = rk ) A(M,N), the R8GE matrix. ! ! Input, real ( kind = rk ) X(M), the vector to be multiplied by A. ! ! Output, real ( kind = rk ) B(N), the product A' * x. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a(m,n) real ( kind = rk ) b(n) real ( kind = rk ) x(m) b(1:n) = matmul ( transpose ( a(1:m,1:n) ), x(1:m) ) return end subroutine r8ge_mv ( m, n, a, x, b ) !*****************************************************************************80 ! !! R8GE_MV multiplies an R8GE matrix by an R8VEC. ! ! Discussion: ! ! The R8GE storage format is used for a general M by N matrix. A storage ! space is made for each entry. The two dimensional logical ! array can be thought of as a vector of M*N entries, starting with ! the M entries in the column 1, then the M entries in column 2 ! and so on. Considered as a vector, the entry A(I,J) is then stored ! in vector location I+(J-1)*M. ! ! R8GE storage is used by LINPACK and LAPACK. ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 11 January 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, the number of rows of the matrix. ! M must be positive. ! ! Input, integer N, the number of columns of the matrix. ! N must be positive. ! ! Input, real ( kind = rk ) A(M,N), the R8GE matrix. ! ! Input, real ( kind = rk ) X(N), the vector to be multiplied by A. ! ! Output, real ( kind = rk ) B(M), the product A * x. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a(m,n) real ( kind = rk ) b(m) real ( kind = rk ) x(n) b(1:m) = matmul ( a(1:m,1:n), x(1:n) ) return end subroutine r8ge_print ( m, n, a, title ) !*****************************************************************************80 ! !! R8GE_PRINT prints an R8GE matrix. ! ! Discussion: ! ! The R8GE storage format is used for a general M by N matrix. A storage ! space is made for each entry. The two dimensional logical ! array can be thought of as a vector of M*N entries, starting with ! the M entries in the column 1, then the M entries in column 2 ! and so on. Considered as a vector, the entry A(I,J) is then stored ! in vector location I+(J-1)*M. ! ! R8GE storage is used by LINPACK and LAPACK. ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 07 May 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, the number of rows of the matrix. ! M must be positive. ! ! Input, integer N, the number of columns of the matrix. ! N must be positive. ! ! Input, real ( kind = rk ) A(M,N), the R8GE matrix. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a(m,n) character ( len = * ) title call r8ge_print_some ( m, n, a, 1, 1, m, n, title ) return end subroutine r8ge_print_some ( m, n, a, ilo, jlo, ihi, jhi, title ) !*****************************************************************************80 ! !! R8GE_PRINT_SOME prints some of an R8GE matrix. ! ! Discussion: ! ! The R8GE storage format is used for a general M by N matrix. A storage ! space is made for each entry. The two dimensional logical ! array can be thought of as a vector of M*N entries, starting with ! the M entries in the column 1, then the M entries in column 2 ! and so on. Considered as a vector, the entry A(I,J) is then stored ! in vector location I+(J-1)*M. ! ! R8GE storage is used by LINPACK and LAPACK. ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 21 March 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, the number of rows of the matrix. ! M must be positive. ! ! Input, integer N, the number of columns of the matrix. ! N must be positive. ! ! Input, real ( kind = rk ) A(M,N), the R8GE matrix. ! ! Input, integer ILO, JLO, IHI, JHI, the first row and ! column, and the last row and column to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer, parameter :: incx = 5 integer m integer n real ( kind = rk ) 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 character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) ! ! Print the columns of the matrix, in strips of 5. ! do j2lo = jlo, jhi, 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(j2), j2 = 1, inc ) write ( *, '(a)' ) ' Row' write ( *, '(a)' ) ' ---' ! ! Determine the range of the rows in this strip. ! i2lo = max ( ilo, 1 ) i2hi = min ( ihi, m ) do i = i2lo, i2hi ! ! Print out (up to) 5 entries in row I, that lie in the current strip. ! do j2 = 1, inc j = j2lo - 1 + j2 write ( ctemp(j2), '(g14.6)' ) a(i,j) end do write ( *, '(i5,1x,5a14)' ) i, ( ctemp(j2), j2 = 1, inc ) end do end do return end subroutine r8ge_to_r83v ( m, n, a, a1, a2, a3 ) !*****************************************************************************80 ! !! R8GE_TO_R83V copies (some of) an R8GE matrix to an R83V matrix. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 19 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, N, the order of the matrix. ! ! Input, real ( kind = rk ) A(M,N), the R8GE matrix. ! ! Output, real ( kind = rk ) A1(min(M-1,N)), A2(min(M,N)), A3(min(M,N-1)), ! the R83V matrix. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a(m,n) real ( kind = rk ) a1(min(m-1,n)) real ( kind = rk ) a2(min(m,n)) real ( kind = rk ) a3(min(m,n-1)) integer ahi integer bhi integer chi integer k ahi = min ( m - 1, n ) bhi = min ( m, n ) chi = min ( m, n - 1 ) do k = 1, ahi a1(k) = a(k+1,k) end do do k = 1, bhi a2(k) = a(k,k) end do do k = 1, chi a3(k) = a(k,k+1) end do return end subroutine r8vec_indicator1 ( n, a ) !*****************************************************************************80 ! !! R8VEC_INDICATOR1 sets an R8VEC to the indicator vector (1,2,3,...). ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 27 September 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer N, the number of elements of A. ! ! Output, real ( kind = rk ) A(N), the array. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer n real ( kind = rk ) a(n) integer i do i = 1, n a(i) = real ( i, kind = rk ) end do return end function r8vec_norm ( n, a ) !*****************************************************************************80 ! !! R8VEC_NORM returns the L2 norm of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The vector L2 norm is defined as: ! ! R8VEC_NORM = sqrt ( sum ( 1 <= I <= N ) A(I)^2 ). ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 21 August 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer N, the number of entries in A. ! ! Input, real ( kind = rk ) A(N), the vector whose L2 norm is desired. ! ! Output, real ( kind = rk ) R8VEC_NORM, the L2 norm of A. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer n real ( kind = rk ) a(n) real ( kind = rk ) r8vec_norm r8vec_norm = sqrt ( sum ( a(1:n)**2 ) ) return end function r8vec_norm_affine ( n, v0, v1 ) !*****************************************************************************80 ! !! R8VEC_NORM_AFFINE returns the affine norm of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The affine vector L2 norm is defined as: ! ! R8VEC_NORM_AFFINE(V0,V1) ! = sqrt ( sum ( 1 <= I <= N ) ( V1(I) - V0(I) )^2 ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 27 October 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer N, the order of the vectors. ! ! Input, real ( kind = rk ) V0(N), the base vector. ! ! Input, real ( kind = rk ) V1(N), the vector whose affine norm is desired. ! ! Output, real ( kind = rk ) R8VEC_NORM_AFFINE, the L2 norm of V1-V0. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer n real ( kind = rk ) r8vec_norm_affine real ( kind = rk ) v0(n) real ( kind = rk ) v1(n) r8vec_norm_affine = sqrt ( sum ( ( v0(1:n) - v1(1:n) )**2 ) ) return end subroutine r8vec_print ( n, a, title ) !*****************************************************************************80 ! !! R8VEC_PRINT prints an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 22 August 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer N, the number of components of the vector. ! ! Input, real ( kind = rk ) A(N), the vector to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer n real ( kind = rk ) a(n) integer i character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' do i = 1, n write ( *, '(2x,i8,a,1x,g16.8)' ) i, ':', a(i) end do return end subroutine r8vec_to_r83v ( m, n, a, a1, a2, a3 ) !*****************************************************************************80 ! !! R8VEC_TO_R83V copies an R8VEC to an R83V matrix. ! ! Discussion: ! ! The R83V storage format is used for a tridiagonal matrix. ! The subdiagonal is in A(min(M-1,N)). ! The diagonal is in B(min(M,N)). ! The superdiagonal is in C(min(M,N-1)). ! ! Example: ! ! An R83V matrix of order 3x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ! An R83 matrix of order 5x5 would be stored: ! ! B1 C1 ** ** ** ! A1 B2 C2 ** ** ! ** A2 B3 C3 ** ! ** ** A3 B4 C4 ! ** ** ** A4 B5 ! ! An R83 matrix of order 5x3 would be stored: ! ! B1 C1 ** ! A1 B2 C2 ! ** A2 B3 ! ** ** A3 ! ** ** ** ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 19 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer M, N, the order of the matrix. ! ! Input, real ( kind = rk ) A(min(N-1,M)+min(N,M)+min(N,M-1)), the vector. ! ! Output, real ( kind = rk ) A1(min(M-1,N)), A2(min(M,N)), A3(min(M,N-1)), ! the matrix. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer m integer n real ( kind = rk ) a(min(m-1,n)+min(m,n)+min(m,n-1)) real ( kind = rk ) a1(min(m-1,n)) real ( kind = rk ) a2(min(m,n)) real ( kind = rk ) a3(min(m,n-1)) integer ahi integer bhi integer chi integer j integer k ahi = min ( m - 1, n ) bhi = min ( m, n ) chi = min ( m, n - 1 ) k = 0 do j = 1, n if ( j <= m + 1 .and. 2 <= j ) then k = k + 1 a3(j-1) = a(k) end if if ( j <= m ) then k = k + 1 a2(j) = a(k) end if if ( j <= m - 1 ) then k = k + 1 a1(j) = a(k) end if end do return end subroutine r8vec_uniform_01 ( n, seed, r ) !*****************************************************************************80 ! !! R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 13 August 2014 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Paul Bratley, Bennett Fox, Linus Schrage, ! A Guide to Simulation, ! Springer Verlag, pages 201-202, 1983. ! ! Bennett Fox, ! Algorithm 647: ! Implementation and Relative Efficiency of Quasirandom ! Sequence Generators, ! ACM Transactions on Mathematical Software, ! Volume 12, Number 4, pages 362-376, 1986. ! ! Peter Lewis, Allen Goodman, James Miller ! A Pseudo-Random Number Generator for the System/360, ! IBM Systems Journal, ! Volume 8, pages 136-143, 1969. ! ! Parameters: ! ! Input, integer N, the number of entries in the vector. ! ! Input/output, integer SEED, the "seed" value, which ! should NOT be 0. On output, SEED has been updated. ! ! Output, real ( kind = rk ) R(N), the vector of pseudorandom values. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer n integer i integer, parameter :: i4_huge = 2147483647 integer k integer seed real ( kind = rk ) r(n) if ( seed == 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_UNIFORM_01 - Fatal error!' write ( *, '(a)' ) ' Input value of SEED = 0.' stop 1 end if do i = 1, n k = seed / 127773 seed = 16807 * ( seed - k * 127773 ) - k * 2836 if ( seed < 0 ) then seed = seed + i4_huge end if r(i) = real ( seed, kind = rk ) * 4.656612875D-10 end do return end subroutine r8vec2_print ( n, a1, a2, title ) !*****************************************************************************80 ! !! R8VEC2_PRINT prints an R8VEC2. ! ! Discussion: ! ! An R8VEC2 is a dataset consisting of N pairs of R8's, stored ! as two separate vectors A1 and A2. ! ! Licensing: ! ! This code is distributed under the MIT license. ! ! Modified: ! ! 13 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer N, the number of components of the vector. ! ! Input, real ( kind = rk ) A1(N), A2(N), the vectors to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer, parameter :: rk = kind ( 1.0D+00 ) integer n real ( kind = rk ) a1(n) real ( kind = rk ) a2(n) integer i character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' do i = 1, n write ( *, '(2x,i4,2x,g14.6,2x,g14.6)' ) i, a1(i), a2(i) end do return end