# include # include # include # include # include "r8sd.h" /******************************************************************************/ int i4_log_10 ( int i ) /******************************************************************************/ /* Purpose: I4_LOG_10 returns the integer part of the logarithm base 10 of an I4. 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 Discussion: I4_LOG_10 ( I ) + 1 is the number of decimal digits in I. Licensing: This code is distributed under the MIT license. Modified: 23 October 2007 Author: John Burkardt Parameters: Input, int I, the number whose logarithm base 10 is desired. Output, int I4_LOG_10, the integer part of the logarithm base 10 of the absolute value of X. */ { int i_abs; int ten_pow; int value; if ( i == 0 ) { value = 0; } else { value = 0; ten_pow = 10; i_abs = abs ( i ); while ( ten_pow <= i_abs ) { value = value + 1; ten_pow = ten_pow * 10; } } return value; } /******************************************************************************/ int i4_max ( int i1, int i2 ) /******************************************************************************/ /* Purpose: I4_MAX returns the maximum of two I4's. Licensing: This code is distributed under the MIT license. Modified: 29 August 2006 Author: John Burkardt Parameters: Input, int I1, I2, are two integers to be compared. Output, int I4_MAX, the larger of I1 and I2. */ { int value; if ( i2 < i1 ) { value = i1; } else { value = i2; } return value; } /******************************************************************************/ int i4_min ( int i1, int i2 ) /******************************************************************************/ /* Purpose: I4_MIN returns the smaller of two I4's. Licensing: This code is distributed under the MIT license. Modified: 29 August 2006 Author: John Burkardt Parameters: Input, int I1, I2, two integers to be compared. Output, int I4_MIN, the smaller of I1 and I2. */ { int value; if ( i1 < i2 ) { value = i1; } else { value = i2; } return value; } /******************************************************************************/ int i4_power ( int i, int j ) /******************************************************************************/ /* Purpose: I4_POWER returns the value of I^J. Licensing: This code is distributed under the MIT license. Modified: 23 October 2007 Author: John Burkardt Parameters: Input, int I, J, the base and the power. J should be nonnegative. Output, int I4_POWER, the value of I^J. */ { int k; int value; if ( j < 0 ) { if ( i == 1 ) { value = 1; } else if ( i == 0 ) { fprintf ( stderr, "\n" ); fprintf ( stderr, "I4_POWER - Fatal error!\n" ); fprintf ( stderr, " I^J requested, with I = 0 and J negative.\n" ); exit ( 1 ); } else { value = 0; } } else if ( j == 0 ) { if ( i == 0 ) { fprintf ( stderr, "\n" ); fprintf ( stderr, "I4_POWER - Fatal error!\n" ); fprintf ( stderr, " I^J requested, with I = 0 and J = 0.\n" ); exit ( 1 ); } else { value = 1; } } else if ( j == 1 ) { value = i; } else { value = 1; for ( k = 1; k <= j; k++ ) { value = value * i; } } return value; } /******************************************************************************/ double r8_max ( double x, double y ) /******************************************************************************/ /* Purpose: R8_MAX returns the maximum of two R8's. Licensing: This code is distributed under the MIT license. Modified: 07 May 2006 Author: John Burkardt Parameters: Input, double X, Y, the quantities to compare. Output, double R8_MAX, the maximum of X and Y. */ { double value; if ( y < x ) { value = x; } else { value = y; } return value; } /******************************************************************************/ double r8_min ( double x, double y ) /******************************************************************************/ /* Purpose: R8_MIN returns the minimum of two R8's. Discussion: The C math library provides the function fmin() which should be preferred. Licensing: This code is distributed under the MIT license. Modified: 07 May 2006 Author: John Burkardt Parameters: Input, double X, Y, the quantities to compare. Output, double R8_MIN, the minimum of X and Y. */ { double value; if ( y < x ) { value = y; } else { value = x; } return value; } /******************************************************************************/ double r8_uniform_01 ( int *seed ) /******************************************************************************/ /* Purpose: R8_UNIFORM_01 returns a unit pseudorandom R8. Discussion: 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: 11 August 2004 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. P A Lewis, A S Goodman, J M Miller, A Pseudo-Random Number Generator for the System/360, IBM Systems Journal, Volume 8, pages 136-143, 1969. Parameters: Input/output, int *SEED, the "seed" value. Normally, this value should not be 0. On output, SEED has been updated. Output, double R8_UNIFORM_01, a new pseudorandom variate, strictly between 0 and 1. */ { int k; double r; if ( *seed == 0 ) { fprintf ( stderr, "\n" ); fprintf ( stderr, "R8_UNIFORM_01 - Fatal error!\n" ); fprintf ( stderr, " Input value of SEED = 0\n" ); exit ( 1 ); } k = *seed / 127773; *seed = 16807 * ( *seed - k * 127773 ) - k * 2836; if ( *seed < 0 ) { *seed = *seed + 2147483647; } r = ( ( double ) ( *seed ) ) * 4.656612875E-10; return r; } /******************************************************************************/ void r8ge_print ( int m, int n, double a[], char *title ) /******************************************************************************/ /* Purpose: R8GE_PRINT prints an R8GE matrix. Discussion: The R8GE storage format is used for a "general" M by N matrix. A physical storage space is made for each logical entry. The two dimensional logical array is mapped to a vector, in which storage is by columns. Licensing: This code is distributed under the MIT license. Modified: 28 February 2012 Author: John Burkardt Parameters: Input, int M, the number of rows of the matrix. M must be positive. Input, int N, the number of columns of the matrix. N must be positive. Input, double A[M*N], the R8GE matrix. Input, char *TITLE, a title. */ { r8ge_print_some ( m, n, a, 1, 1, m, n, title ); return; } /******************************************************************************/ void r8ge_print_some ( int m, int n, double a[], int ilo, int jlo, int ihi, int jhi, char *title ) /******************************************************************************/ /* Purpose: R8GE_PRINT_SOME prints some of an R8GE matrix. Discussion: The R8GE storage format is used for a "general" M by N matrix. A physical storage space is made for each logical entry. The two dimensional logical array is mapped to a vector, in which storage is by columns. Licensing: This code is distributed under the MIT license. Modified: 28 February 2012 Author: John Burkardt Parameters: Input, int M, the number of rows of the matrix. M must be positive. Input, int N, the number of columns of the matrix. N must be positive. Input, double A[M*N], the R8GE matrix. Input, int ILO, JLO, IHI, JHI, designate the first row and column, and the last row and column to be printed. Input, char *TITLE, a title. */ { # define INCX 5 int i; int i2hi; int i2lo; int j; int j2hi; int j2lo; printf ( "\n" ); printf ( "%s\n", title ); /* Print the columns of the matrix, in strips of 5. */ for ( j2lo = jlo; j2lo <= jhi; j2lo = j2lo + INCX ) { j2hi = j2lo + INCX - 1; j2hi = i4_min ( j2hi, n ); j2hi = i4_min ( j2hi, jhi ); printf ( "\n" ); /* For each column J in the current range... Write the header. */ printf ( " Col: " ); for ( j = j2lo; j <= j2hi; j++ ) { printf ( "%7d ", j ); } printf ( "\n" ); printf ( " Row\n" ); printf ( " ---\n" ); /* Determine the range of the rows in this strip. */ i2lo = i4_max ( ilo, 1 ); i2hi = i4_min ( ihi, m ); for ( i = i2lo; i <= i2hi; i++ ) { /* Print out (up to) 5 entries in row I, that lie in the current strip. */ printf ( "%5d ", i ); for ( j = j2lo; j <= j2hi; j++ ) { printf ( "%12g ", a[i-1+(j-1)*m] ); } printf ( "\n" ); } } return; # undef INCX } /******************************************************************************/ double *r8sd_cg ( int n, int ndiag, int offset[], double a[], double b[], double x_init[] ) /******************************************************************************/ /* Purpose: R8SD_CG uses the conjugate gradient method on an R8SD linear system. Discussion: The R8SD storage format is used for symmetric matrices whose only nonzero entries occur along a few diagonals, but for which these diagonals are not all close enough to the main diagonal for band storage to be efficient. In that case, we assign the main diagonal the offset value 0, and each successive superdiagonal gets an offset value 1 higher, until the highest superdiagonal (the A(1,N) entry) is assigned the offset N-1. Assuming there are NDIAG nonzero diagonals (ignoring subdiagonals!), we then create an array B that has N rows and NDIAG columns, and simply "collapse" the matrix A to the left: For the conjugate gradient method to be applicable, the matrix A must be a positive definite symmetric matrix. The method is designed to reach the solution to the linear system A * x = b after N computational steps. However, roundoff may introduce unacceptably large errors for some problems. In such a case, calling the routine a second time, using the current solution estimate as the new starting guess, should result in improved results. Licensing: This code is distributed under the MIT license. Modified: 29 July 2015 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, int N, the order of the matrix. N must be positive. Input, int NDIAG, the number of diagonals that are stored. NDIAG must be at least 1 and no more than N. Input, int OFFSET[NDIAG], the offsets for the diagonal storage. Input, double A[N*NDIAG], the matrix. Input, double B[N], the right hand side vector. Input, double X_INIT[N], an estimate for the solution, which may be 0. Output, double R8SD_CG[N], the approximate solution vector. */ { double alpha; double *ap; double beta; int i; int it; double *p; double pap; double pr; double *r; double rap; double *x; x = r8vec_zeros_new ( n ); for ( i = 0; i < n; i++ ) { x[i] = x_init[i]; } /* Initialize AP = A * x, R = b - A * x, P = b - A * x. */ ap = r8sd_mv ( n, n, ndiag, offset, a, x ); r = r8vec_zeros_new ( n ); for ( i = 0; i < n; i++ ) { r[i] = b[i] - ap[i]; } p = r8vec_zeros_new ( n ); for ( i = 0; i < n; i++ ) { p[i] = b[i] - ap[i]; } /* Do the N steps of the conjugate gradient method. */ for ( it = 1; it < n; it++ ) { /* Compute the matrix*vector product AP = A*P. */ free ( ap ); ap = r8sd_mv ( n, n, ndiag, offset, a, p ); /* Compute the dot products PAP = P*AP, PR = P*R Set ALPHA = PR / PAP. */ pap = r8vec_dot_product ( n, p, ap ); if ( pap == 0.0 ) { free ( ap ); break; } pr = r8vec_dot_product ( n, p, r ); alpha = pr / pap; /* Set X = X + ALPHA * P R = R - ALPHA * AP. */ for ( i = 0; i < n; i++ ) { x[i] = x[i] + alpha * p[i]; } for ( i = 0; i < n; i++ ) { r[i] = r[i] - alpha * ap[i]; } /* Compute the vector dot product RAP = R*AP Set BETA = - RAP / PAP. */ rap = r8vec_dot_product ( n, r, ap ); beta = -rap / pap; /* Update the perturbation vector P = R + BETA * P. */ for ( i = 0; i < n; i++ ) { p[i] = r[i] + beta * p[i]; } } free ( p ); free ( r ); return x; } /******************************************************************************/ double *r8sd_dif2 ( int n, int ndiag, int offset[] ) /******************************************************************************/ /* Purpose: R8SD_DIF2 returns the DIF2 matrix in R8SD format. Example: N = 5 2 -1 . . . -1 2 -1 . . . -1 2 -1 . . . -1 2 -1 . . . -1 2 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: 05 June 2014 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, int M, N, the number of rows and columns. Input, int NDIAG, the number of diagonals available for storage. Input, int OFFSET[NDIAG], the indices of the diagonals. It is presumed that OFFSET[0] = 0 and OFFSET[1] = 1. Output, double R8SD_DIF2[N*NDIAG], the matrix. */ { double *a; int i; int jdiag; a = r8vec_zeros_new ( n * ndiag ); for ( i = 0; i < n; i++ ) { jdiag = 0; a[i+jdiag*n] = 2.0; } for ( i = 0; i < n - 1; i++ ) { jdiag = 1; a[i+jdiag*n] = -1.0; } return a; } /******************************************************************************/ double *r8sd_indicator ( int n, int ndiag, int offset[] ) /******************************************************************************/ /* Purpose: R8SD_INDICATOR sets up an R8SD indicator matrix. Discussion: The R8SD storage format is used for symmetric matrices whose only nonzero entries occur along a few diagonals, but for which these diagonals are not all close enough to the main diagonal for band storage to be efficient. In that case, we assign the main diagonal the offset value 0, and each successive superdiagonal gets an offset value 1 higher, until the highest superdiagonal (the A(1,N) entry) is assigned the offset N-1. Assuming there are NDIAG nonzero diagonals (ignoring subdiagonals!), we then create an array B that has N rows and NDIAG columns, and simply "collapse" the matrix A to the left. Licensing: This code is distributed under the MIT license. Modified: 16 February 2013 Author: John Burkardt Parameters: Input, int N, the order of the matrix. N must be positive. Input, int NDIAG, the number of diagonals that are stored. NDIAG must be at least 1 and no more than N. Input, int OFFSET[NDIAG], the offsets for the diagonal storage. Output, double R8SD_INDICATOR[N*NDIAG], the R8SD matrix. */ { double *a; int fac; int i; int j; int jdiag; a = r8vec_zeros_new ( n * ndiag ); fac = i4_power ( 10, i4_log_10 ( n ) + 1 ); for ( i = 0; i < n; i++ ) { for ( jdiag = 0; jdiag < ndiag; jdiag++ ) { j = i + offset[jdiag]; if ( 0 <= j && j < n ) { a[i+jdiag*n] = ( double ) ( fac * ( i + 1 ) + ( j + 1 ) ); } } } return a; } /******************************************************************************/ double *r8sd_mv ( int m, int n, int ndiag, int offset[], double a[], double x[] ) /******************************************************************************/ /* Purpose: R8SD_MV multiplies an R8SD matrix times a vector. Discussion: The R8SD storage format is used for symmetric matrices whose only nonzero entries occur along a few diagonals, but for which these diagonals are not all close enough to the main diagonal for band storage to be efficient. In that case, we assign the main diagonal the offset value 0, and each successive superdiagonal gets an offset value 1 higher, until the highest superdiagonal (the A(1,N) entry) is assigned the offset N-1. Assuming there are NDIAG nonzero diagonals (ignoring subdiagonals!), we then create an array B that has N rows and NDIAG columns, and simply "collapse" the matrix A to the left. Example: The "offset" value is printed above each column. Original matrix New Matrix 0 1 2 3 4 5 0 1 3 5 11 12 0 14 0 16 11 12 14 16 21 22 23 0 25 0 22 23 25 -- 0 32 33 34 0 36 33 34 36 -- 41 0 43 44 45 0 44 45 -- -- 0 52 0 54 55 56 55 56 -- -- 61 0 63 0 65 66 66 -- -- -- Licensing: This code is distributed under the MIT license. Modified: 16 February 2013 Author: John Burkardt Parameters: Input, int M, N, the number of rows and columns. Input, int NDIAG, the number of diagonals that are stored. NDIAG must be at least 1 and no more than N. Input, int OFFSET[NDIAG], the offsets for the diagonal storage. Input, double A[N*NDIAG], the matrix. Input, double X[N], the vector to be multiplied by A. Output, double R8SD_MV[M], the product A * x. */ { double *b; int i; int j; int jdiag; b = r8vec_zeros_new ( m ); for ( i = 0; i < n; i++ ) { for ( jdiag = 0; jdiag < ndiag; jdiag++ ) { j = i + offset[jdiag]; if ( 0 <= j && j < n ) { b[i] = b[i] + a[i+jdiag*n] * x[j]; if ( offset[jdiag] != 0 ) { b[j] = b[j] + a[i+jdiag*n] * x[i]; } } } } return b; } /******************************************************************************/ void r8sd_print ( int n, int ndiag, int offset[], double a[], char *title ) /******************************************************************************/ /* Purpose: R8SD_PRINT prints an R8SD matrix. Discussion: The R8SD storage format is used for symmetric matrices whose only nonzero entries occur along a few diagonals, but for which these diagonals are not all close enough to the main diagonal for band storage to be efficient. In that case, we assign the main diagonal the offset value 0, and each successive superdiagonal gets an offset value 1 higher, until the highest superdiagonal (the A(1,N) entry) is assigned the offset N-1. Assuming there are NDIAG nonzero diagonals (ignoring subdiagonals!), we then create an array B that has N rows and NDIAG columns, and simply "collapse" the matrix A to the left: Example: The "offset" value is printed above each column. Original matrix New Matrix 0 1 2 3 4 5 0 1 3 5 11 12 0 14 0 16 11 12 14 16 21 22 23 0 25 0 22 23 25 -- 0 32 33 34 0 36 33 34 36 -- 41 0 43 44 45 0 44 45 -- -- 0 52 0 54 55 56 55 56 -- -- 61 0 63 0 65 66 66 -- -- -- Licensing: This code is distributed under the MIT license. Modified: 16 February 2013 Author: John Burkardt Parameters: Input, int N, the number of columns of the matrix. N must be positive. Input, int NDIAG, the number of diagonals of the matrix that are stored in the array. NDIAG must be at least 1, and no more than N. Input, int OFFSET[NDIAG], the offsets for the diagonal storage. Input, double A[N*NDIAG], the R8SD matrix. Input, char *TITLE, a title. */ { r8sd_print_some ( n, ndiag, offset, a, 1, 1, n, n, title ); return; } /******************************************************************************/ void r8sd_print_some ( int n, int ndiag, int offset[], double a[], int ilo, int jlo, int ihi, int jhi, char *title ) /******************************************************************************/ /* Purpose: R8SD_PRINT_SOME prints some of an R8SD matrix. Discussion: The R8SD storage format is used for symmetric matrices whose only nonzero entries occur along a few diagonals, but for which these diagonals are not all close enough to the main diagonal for band storage to be efficient. In that case, we assign the main diagonal the offset value 0, and each successive superdiagonal gets an offset value 1 higher, until the highest superdiagonal (the A(1,N) entry) is assigned the offset N-1. Assuming there are NDIAG nonzero diagonals (ignoring subdiagonals!), we then create an array B that has N rows and NDIAG columns, and simply "collapse" the matrix A to the left: Example: The "offset" value is printed above each column. Original matrix New Matrix 0 1 2 3 4 5 0 1 3 5 11 12 0 14 0 16 11 12 14 16 21 22 23 0 25 0 22 23 25 -- 0 32 33 34 0 36 33 34 36 -- 41 0 43 44 45 0 44 45 -- -- 0 52 0 54 55 56 55 56 -- -- 61 0 63 0 65 66 66 -- -- -- Licensing: This code is distributed under the MIT license. Modified: 16 February 2013 Author: John Burkardt Parameters: Input, int N, the number of columns of the matrix. N must be positive. Input, int NDIAG, the number of diagonals of the matrix that are stored in the array. NDIAG must be at least 1, and no more than N. Input, int OFFSET[NDIAG], the offsets for the diagonal storage. Input, double A[N*NDIAG], the R8SD matrix. Input, int ILO, JLO, IHI, JHI, designate the first row and column, and the last row and column to be printed. Input, char *TITLE, a title. */ { # define INCX 5 double aij; int i; int i2hi; int i2lo; int j; int j2hi; int j2lo; int jdiag; printf ( "\n" ); printf ( "%s\n", title ); /* Print the columns of the matrix, in strips of 5. */ for ( j2lo = jlo; j2lo <= jhi; j2lo = j2lo + INCX ) { j2hi = j2lo + INCX - 1; j2hi = i4_min ( j2hi, n ); j2hi = i4_min ( j2hi, jhi ); printf ( " Col: " ); for ( j = j2lo; j <= j2hi; j++ ) { printf ( "%7d ", j ); } printf ( "\n" ); printf ( " Row\n" ); printf ( " ---\n" ); /* Determine the range of the rows in this strip. */ i2lo = i4_max ( ilo, 1 ); i2hi = i4_min ( ihi, n ); for ( i = i2lo; i <= i2hi; i++ ) { printf ( "%4d ", i ); /* Print out (up to) 5 entries in row I, that lie in the current strip. */ for ( j = j2lo; j <= j2hi; j++ ) { aij = 0.0; for ( jdiag = 0; jdiag < ndiag; jdiag++ ) { if ( j - i == offset[jdiag] ) { aij = a[i-1+jdiag*n]; } else if ( j - i == - offset[jdiag] ) { aij = a[j-1+jdiag*n]; } } printf ( "%12g ", aij ); } printf ( "\n" ); } } return; # undef INCX } /******************************************************************************/ double *r8sd_random ( int n, int ndiag, int offset[], int *seed ) /******************************************************************************/ /* Purpose: R8SD_RANDOM randomizes an R8SD matrix. Discussion: The R8SD storage format is used for symmetric matrices whose only nonzero entries occur along a few diagonals, but for which these diagonals are not all close enough to the main diagonal for band storage to be efficient. In that case, we assign the main diagonal the offset value 0, and each successive superdiagonal gets an offset value 1 higher, until the highest superdiagonal (the A(1,N) entry) is assigned the offset N-1. Assuming there are NDIAG nonzero diagonals (ignoring subdiagonals!), we then create an array B that has N rows and NDIAG columns, and simply "collapse" the matrix A to the left. Licensing: This code is distributed under the MIT license. Modified: 16 February 2013 Author: John Burkardt Parameters: Input, int N, the order of the matrix. N must be positive. Input, int NDIAG, the number of diagonals that are stored. NDIAG must be at least 1 and no more than N. Input, int OFFSET[NDIAG], the offsets for the diagonal storage. Input/output, int *SEED, a seed for the random number generator. Output, double R8SD_RANDOM[N*NDIAG], the R8SD matrix. */ { double *a; int i; int j; int jdiag; a = r8vec_zeros_new ( n * ndiag ); for ( i = 0; i < n; i++ ) { for ( jdiag = 0; jdiag < ndiag; jdiag++ ) { j = i + offset[jdiag]; if ( 0 <= j && j < n ) { a[i+jdiag*n] = r8_uniform_01 ( seed ); } } } return a; } /******************************************************************************/ double *r8sd_res ( int m, int n, int ndiag, int offset[], double a[], double x[], double b[] ) /******************************************************************************/ /* Purpose: R8SD_RES computes the residual R = B-A*X for R8SD matrices. Licensing: This code is distributed under the MIT license. Modified: 05 June 2014 Author: John Burkardt Parameters: Input, int M, the number of rows of the matrix. M must be positive. Input, int N, the number of columns of the matrix. N must be positive. Input, int NDIAG, the number of diagonals that are stored. NDIAG must be at least 1 and no more than N. Input, int OFFSET[NDIAG], the offsets for the diagonal storage. Input, double A[N*NDIAG], the matrix. Input, double X[N], the vector to be multiplied by A. Input, double B[M], the desired result A * x. Output, double R8SD_RES[M], the residual R = B - A * X. */ { int i; double *r; r = r8sd_mv ( m, n, ndiag, offset, a, x ); for ( i = 0; i < m; i++ ) { r[i] = b[i] - r[i]; } return r; } /******************************************************************************/ double *r8sd_to_r8ge ( int n, int ndiag, int offset[], double a[] ) /******************************************************************************/ /* Purpose: R8SD_TO_R8GE copies an R8SD matrix to an R8GE matrix. Discussion: The R8SD storage format is used for symmetric matrices whose only nonzero entries occur along a few diagonals, but for which these diagonals are not all close enough to the main diagonal for band storage to be efficient. In that case, we assign the main diagonal the offset value 0, and each successive superdiagonal gets an offset value 1 higher, until the highest superdiagonal (the A(1,N) entry) is assigned the offset N-1. Assuming there are NDIAG nonzero diagonals (ignoring subdiagonals!), we then create an array B that has N rows and NDIAG columns, and simply "collapse" the matrix A to the left: Example: The "offset" value is printed above each column. Original matrix New Matrix 0 1 2 3 4 5 0 1 3 5 11 12 0 14 0 16 11 12 14 16 21 22 23 0 25 0 22 23 25 -- 0 32 33 34 0 36 33 34 36 -- 41 0 43 44 45 0 44 45 -- -- 0 52 0 54 55 56 55 56 -- -- 61 0 63 0 65 66 66 -- -- -- Licensing: This code is distributed under the MIT license. Modified: 16 February 2013 Author: John Burkardt Parameters: Input, int N, the order of the matrix. N must be positive. Input, int NDIAG, the number of diagonals that are stored. NDIAG must be at least 1 and no more than N. Input, int OFFSET[NDIAG], the offsets for the diagonal storage. Input, double A[N*NDIAG], the R8SD matrix. Output, double R8SD_TO_R8GE[N*N], the R8GE matrix. */ { double *b; int i; int j; int jdiag; b = r8vec_zeros_new ( n * n ); for ( i = 0; i < n; i++ ) { for ( jdiag = 0; jdiag < ndiag; jdiag++ ) { j = i + offset[jdiag]; if ( 0 <= j && j < n ) { b[i+j*n] = a[i+jdiag*n]; if ( i != j ) { b[j+i*n] = a[i+jdiag*n]; } } } } return b; } /******************************************************************************/ double *r8sd_zeros ( int n, int ndiag, int offset[] ) /******************************************************************************/ /* Purpose: R8SD_ZEROS zeros an R8SD matrix. Discussion: The R8SD storage format is used for symmetric matrices whose only nonzero entries occur along a few diagonals, but for which these diagonals are not all close enough to the main diagonal for band storage to be efficient. In that case, we assign the main diagonal the offset value 0, and each successive superdiagonal gets an offset value 1 higher, until the highest superdiagonal (the A(1,N) entry) is assigned the offset N-1. Assuming there are NDIAG nonzero diagonals (ignoring subdiagonals!), we then create an array B that has N rows and NDIAG columns, and simply "collapse" the matrix A to the left. Licensing: This code is distributed under the MIT license. Modified: 16 February 2013 Author: John Burkardt Parameters: Input, int N, the order of the matrix. Input, int NDIAG, the number of diagonals that are stored. NDIAG must be at least 1 and no more than N. Input, int OFFSET[NDIAG], the offsets for the diagonal storage. Output, double R8SD_ZEROS[N*NDIAG], the R8SD matrix. */ { double *a; a = r8vec_zeros_new ( n * ndiag ); return a; } /******************************************************************************/ double r8vec_dot_product ( int n, double a1[], double a2[] ) /******************************************************************************/ /* Purpose: R8VEC_DOT_PRODUCT computes the dot product of a pair of R8VEC's. Licensing: This code is distributed under the MIT license. Modified: 26 July 2007 Author: John Burkardt Parameters: Input, int N, the number of entries in the vectors. Input, double A1[N], A2[N], the two vectors to be considered. Output, double R8VEC_DOT_PRODUCT, the dot product of the vectors. */ { int i; double value; value = 0.0; for ( i = 0; i < n; i++ ) { value = value + a1[i] * a2[i]; } return value; } /******************************************************************************/ double *r8vec_indicator1_new ( int n ) /******************************************************************************/ /* Purpose: R8VEC_INDICATOR1_NEW sets an R8VEC to the indicator1 vector {1,2,3...}. Licensing: This code is distributed under the MIT license. Modified: 26 August 2008 Author: John Burkardt Parameters: Input, int N, the number of elements of A. Output, double R8VEC_INDICATOR1_NEW[N], the array. */ { double *a; int i; a = r8vec_zeros_new ( n ); for ( i = 0; i <= n - 1; i++ ) { a[i] = ( double ) ( i + 1 ); } return a; } /******************************************************************************/ void r8vec_print ( int n, double a[], char *title ) /******************************************************************************/ /* Purpose: R8VEC_PRINT prints an R8VEC. Discussion: An R8VEC is a vector of R8's. Licensing: This code is distributed under the MIT license. Modified: 08 April 2009 Author: John Burkardt Parameters: Input, int N, the number of components of the vector. Input, double A[N], the vector to be printed. Input, char *TITLE, a title. */ { int i; printf ( "\n" ); printf ( "%s\n", title ); printf ( "\n" ); for ( i = 0; i < n; i++ ) { printf ( " %8d %14f\n", i, a[i] ); } return; } /******************************************************************************/ void r8vec_print_some ( int n, double a[], int max_print, char *title ) /******************************************************************************/ /* Purpose: R8VEC_PRINT_SOME prints "some" of an R8VEC. Discussion: The user specifies MAX_PRINT, the maximum number of lines to print. If N, the size of the vector, is no more than MAX_PRINT, then the entire vector is printed, one entry per line. Otherwise, if possible, the first MAX_PRINT-2 entries are printed, followed by a line of periods suggesting an omission, and the last entry. Licensing: This code is distributed under the MIT license. Modified: 27 February 2010 Author: John Burkardt Parameters: Input, int N, the number of entries of the vector. Input, double A[N], the vector to be printed. Input, int MAX_PRINT, the maximum number of lines to print. Input, char *TITLE, a title. */ { int i; if ( max_print <= 0 ) { return; } if ( n <= 0 ) { return; } fprintf ( stdout, "\n" ); fprintf ( stdout, "%s\n", title ); fprintf ( stdout, "\n" ); if ( n <= max_print ) { for ( i = 0; i < n; i++ ) { fprintf ( stdout, " %8d: %14g\n", i, a[i] ); } } else if ( 3 <= max_print ) { for ( i = 0; i < max_print - 2; i++ ) { fprintf ( stdout, " %8d: %14g\n", i, a[i] ); } fprintf ( stdout, " ........ ..............\n" ); i = n - 1; fprintf ( stdout, " %8d: %14g\n", i, a[i] ); } else { for ( i = 0; i < max_print - 1; i++ ) { fprintf ( stdout, " %8d: %14g\n", i, a[i] ); } i = max_print - 1; fprintf ( stdout, " %8d: %14g ...more entries...\n", i, a[i] ); } return; } /******************************************************************************/ double *r8vec_zeros_new ( int n ) /******************************************************************************/ /* Purpose: R8VEC_ZEROS_NEW creates and zeroes an R8VEC. Discussion: An R8VEC is a vector of R8's. Licensing: This code is distributed under the MIT license. Modified: 25 March 2009 Author: John Burkardt Parameters: Input, int N, the number of entries in the vector. Output, double R8VEC_ZEROS_NEW[N], a vector of zeroes. */ { double *a; int i; a = ( double * ) malloc ( n * sizeof ( double ) ); for ( i = 0; i < n; i++ ) { a[i] = 0.0; } return a; }