Last Update: Oct. 4, 2007
/***********************************************/
/* T.Kouya's GSL sample program collection */
/* Solving Linear System of Equations */
/* QR decomposition */
/* Written by Tomonori Kouya */
/* */
/* Version 0.01: 2007-10-02 */
/***********************************************/
#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_linalg.h>
/* Dimension of Matrix and Vectors */
#define DIM 5
int main(void)
{
int i, j, lotkin_signum, frank_signum;
gsl_matrix *lotkin_a, *frank_a;
gsl_vector *x, *lotkin_b, *frank_b, *lotkin_x, *frank_x;
gsl_vector *lotkin_tau, *frank_tau;
/* allocate a, x, b */
lotkin_a = gsl_matrix_alloc(DIM, DIM);
frank_a = gsl_matrix_alloc(DIM, DIM);
x = gsl_vector_alloc(DIM);
lotkin_b = gsl_vector_alloc(DIM);
frank_b = gsl_vector_alloc(DIM);
lotkin_x = gsl_vector_alloc(DIM);
frank_x = gsl_vector_alloc(DIM);
/* set x = [1 2 ... DIM] */
for(i = 0; i < DIM; i++)
gsl_vector_set(x, i, (double)i);
/* set Lotkin matrix */
/* a_ij = 1 (i = 1) or 1/(i+j-1) (i != 1) */
for(i = 0; i < DIM; i++)
gsl_matrix_set(lotkin_a, 0, i, 1.0);
for(i = 1; i < DIM; i++)
for(j = 0; j < DIM; j++)
gsl_matrix_set(lotkin_a, i, j, 1.0 / (double)(i + j + 1));
/* set Frank matrix */
/* a_ij = DIM - min(i,j) + 1 */
for(i = 0; i < DIM; i++)
for(j = 0; j < DIM; j++)
gsl_matrix_set(frank_a, i, j, (double)DIM - (double)GSL_MAX(i, j) );
/* Print matrix */
printf("Lotkin Matrix(DIM = %d)\n", DIM);
for(i = 0; i < DIM; i++)
{
printf("%3d: ", i);
for(j = 0; j < DIM; j++)
printf("%g ", gsl_matrix_get(lotkin_a, i, j));
printf("\n");
}
printf("\n");
printf("Frank Matrix(DIM = %d)\n", DIM);
for(i = 0; i < DIM; i++)
{
printf("%3d: ", i);
for(j = 0; j < DIM; j++)
printf("%g ", gsl_matrix_get(frank_a, i, j));
printf("\n");
}
printf("\n");
/* b = A * x */
gsl_blas_dgemv(CblasNoTrans, 1.0, lotkin_a, x, 0.0, lotkin_b);
gsl_blas_dgemv(CblasNoTrans, 1.0, frank_a, x, 0.0, frank_b);
/* QR decomposition and solve */
lotkin_tau = gsl_vector_alloc(DIM);
gsl_linalg_QR_decomp(lotkin_a, lotkin_tau);
gsl_linalg_QR_solve(lotkin_a, lotkin_tau, lotkin_b, lotkin_x);
gsl_vector_free(lotkin_tau);
frank_tau = gsl_vector_alloc(DIM);
gsl_linalg_QR_decomp(frank_a, frank_tau);
gsl_linalg_QR_solve(frank_a, frank_tau, frank_b, frank_x);
gsl_vector_free(frank_tau);
/* print */
printf("index Lotkin Frank True\n");
for(i = 0; i < DIM; i++)
printf("%3d %25.17e %25.17e %25.17e\n", i, gsl_vector_get(lotkin_x, i), gsl_vector_get(frank_x, i), gsl_vector_get(x, i));
/* free a, x, b */
gsl_matrix_free(lotkin_a);
gsl_matrix_free(frank_a);
gsl_vector_free(x);
gsl_vector_free(lotkin_b);
gsl_vector_free(frank_b);
gsl_vector_free(lotkin_x);
gsl_vector_free(frank_x);
return 0;
}
$ ./linear_system_qr Lotkin Matrix(DIM = 5) 0: 1 1 1 1 1 1: 0.5 0.333333 0.25 0.2 0.166667 2: 0.333333 0.25 0.2 0.166667 0.142857 3: 0.25 0.2 0.166667 0.142857 0.125 4: 0.2 0.166667 0.142857 0.125 0.111111 Frank Matrix(DIM = 5) 0: 5 4 3 2 1 1: 4 4 3 2 1 2: 3 3 3 2 1 3: 2 2 2 2 1 4: 1 1 1 1 1 index Lotkin Frank True 0 -1.11187674128441738e-12 7.18571722085572954e-16 0.00000000000000000e+00 1 1.00000000000953282e+00 1.00000000000000311e+00 1.00000000000000000e+00 2 1.99999999997410893e+00 1.99999999999999378e+00 2.00000000000000000e+00 3 3.00000000002813660e+00 3.00000000000000311e+00 3.00000000000000000e+00 4 3.99999999998933298e+00 3.99999999999999911e+00 4.00000000000000000e+00