Developer Reference

## Intel® oneAPI Math Kernel Library LAPACK Examples

ID 766877
Date 3/31/2023
Public

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## LAPACKE_zgelsd Example Program in C for Column Major Data Layout

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/*
LAPACKE_zgelsd Example.
=======================

Program computes the minimum norm-solution to a complex linear least squares
problem using the singular value decomposition of A,
where A is the coefficient matrix:

(  4.55, -0.32) ( -4.36, -4.76) (  3.99, -6.84) (  8.03, -6.47)
(  8.87, -3.11) (  0.02,  8.43) (  5.43, -9.30) (  2.28,  8.94)
( -0.74,  1.16) (  3.80, -6.12) ( -7.24,  0.72) (  2.21,  9.52)

and B is the right-hand side matrix:

( -8.25,  7.98) (  2.91, -8.81)
( -5.04,  3.33) (  6.19,  0.19)
(  7.98, -4.38) ( -5.96,  7.18)

Description.
============

The routine computes the minimum-norm solution to a complex linear least
squares problem: minimize ||b - A*x|| using the singular value
decomposition (SVD) of A. A is an m-by-n matrix which may be rank-deficient.

Several right hand side vectors b and solution vectors x can be handled
in a single call; they are stored as the columns of the m-by-nrhs right
hand side matrix B and the n-by-nrhs solution matrix X.

The effective rank of A is determined by treating as zero those singular
values which are less than rcond times the largest singular value.

Example Program Results.
========================

LAPACKE_zgelsd (column-major, high-level) Example Program Results

Minimum norm solution
( -0.08,  0.09) (  0.04,  0.16)
( -0.17,  0.10) (  0.17, -0.47)
( -0.92, -0.01) (  0.71, -0.41)
( -0.47, -0.26) (  0.69,  0.02)

Effective rank =      3

Singular values
20.01  18.21   7.88
*/
#include <stdlib.h>
#include <stdio.h>
#include "mkl_lapacke.h"

/* Auxiliary routines prototypes */
extern void print_matrix( char* desc, MKL_INT m, MKL_INT n, MKL_Complex16* a, MKL_INT lda );
extern void print_rmatrix( char* desc, MKL_INT m, MKL_INT n, double* a, MKL_INT lda );

/* Parameters */
#define M 3
#define N 4
#define NRHS 2
#define LDA M
#define LDB N

/* Main program */
int main() {
/* Locals */
MKL_INT m = M, n = N, nrhs = NRHS, lda = LDA, ldb = LDB, info, rank;
/* Negative rcond means using default (machine precision) value */
double rcond = -1.0;
/* Local arrays */
double s[M];
MKL_Complex16 a[LDA*N] = {
{ 4.55, -0.32}, { 8.87, -3.11}, {-0.74,  1.16},
{-4.36, -4.76}, { 0.02,  8.43}, { 3.80, -6.12},
{ 3.99, -6.84}, { 5.43, -9.30}, {-7.24,  0.72},
{ 8.03, -6.47}, { 2.28,  8.94}, { 2.21,  9.52}
};
MKL_Complex16 b[LDB*NRHS] = {
{-8.25,  7.98}, {-5.04,  3.33}, { 7.98, -4.38}, { 0.00,  0.00},
{ 2.91, -8.81}, { 6.19,  0.19}, {-5.96,  7.18}, { 0.00,  0.00}
};
/* Executable statements */
printf( "LAPACKE_zgelsd (column-major, high-level) Example Program Results\n" );
/* Solve the equations A*X = B */
info = LAPACKE_zgelsd( LAPACK_COL_MAJOR, m, n, nrhs, a, lda, b, ldb,
s, rcond, &rank );
/* Check for convergence */
if( info > 0 ) {
printf( "The algorithm computing SVD failed to converge;\n" );
printf( "the least squares solution could not be computed.\n" );
exit( 1 );
}
/* Print minimum norm solution */
print_matrix( "Minimum norm solution", n, nrhs, b, ldb );
/* Print effective rank */
printf( "\n Effective rank = %6i\n", rank );
/* Print singular values */
print_rmatrix( "Singular values", 1, m, s, 1 );
exit( 0 );
} /* End of LAPACKE_zgelsd Example */

/* Auxiliary routine: printing a matrix */
void print_matrix( char* desc, MKL_INT m, MKL_INT n, MKL_Complex16* a, MKL_INT lda ) {
MKL_INT i, j;
printf( "\n %s\n", desc );
for( i = 0; i < m; i++ ) {
for( j = 0; j < n; j++ )
printf( " (%6.2f,%6.2f)", a[i+j*lda].real, a[i+j*lda].imag );
printf( "\n" );
}
}

/* Auxiliary routine: printing a real matrix */
void print_rmatrix( char* desc, MKL_INT m, MKL_INT n, double* a, MKL_INT lda ) {
MKL_INT i, j;
printf( "\n %s\n", desc );
for( i = 0; i < m; i++ ) {
for( j = 0; j < n; j++ ) printf( " %6.2f", a[i+j*lda] );
printf( "\n" );
}
}