## Pseudo inverse of complex matrix

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### Pseudo inverse of complex matrix

Dear LAPACK users,

I need some help please. I need a subroutine (FORTRAN 77) to find the Pseudo-inverse of a complex rectangular matrix. I am wondering if anyone can guide me.

Your time and help are very appreciated.

Regards

Nawras
nawras

Posts: 4
Joined: Thu Sep 22, 2011 10:39 am

### Re: Pseudo inverse of complex matrix

You can have a look at our forum. There are a few discussions on how to compute the pseudoinverse using LAPACK.
See for example viewtopic.php?f=2&t=160
Julien.

Posts: 615
Joined: Wed Dec 08, 2004 7:07 pm

### Re: Pseudo inverse of complex matrix

But all discussions were made for real matrices and I am interested in the Pseudoinverse of complex matrix.
Thanks very much.
nawras

Posts: 4
Joined: Thu Sep 22, 2011 10:39 am

### Re: Pseudo inverse of complex matrix

Hi Nawras, Henc gave it a shot and wrote a quick example of computation of the pseudoinverse. First he computes the SVD of A. ( A = U S V^H .) Then he computes the pseudoinverse of A with A = V * S^(-1) * U^H. The codes are in Fortran and in C using double complex precision. See below. Henc is using DGESVD for the SVD, you can also use DGESDD. Henc is assuming that the matrix is full rank. (So all singular values of A are different from zero so the 1/S(i) is safe.) If some singular values are zero (or very small), (then A is not numerically nonsingular), and you should set 1/S(i) to zero in Henc's code. In the case of A full rank, another way (faster) is to go with a QR factorization instead of an SVD. Cheers, Julien.

{ Thanks Henc! }

Code: Select all
Program PseudoInverse

Implicit none

external ZLANGE
double precision ZLANGE

integer i, j, M, N, K, L, LWORK, INFO
parameter (M=15)
parameter (N=10)

parameter (K = MIN(M,N))
parameter (L = MAX(M,N))

parameter (LWORK = MAX(1,2*K+L))

double complex, dimension(M,N) :: A1, A2, SIGMA
double complex, dimension(N,M) :: PINV
double complex, dimension(M,K) :: U
double complex, dimension(K,N) :: VT
double complex, dimension(N,N) :: BUFF
double complex, dimension(LWORK) :: WORK
double precision, dimension(5*K) :: RWORK
double precision, dimension(K) :: S
integer, dimension(4) :: ISEED

double precision :: normA, normAPA, normPAP

data ISEED/0,0,0,1/

c  Fill A1 with random values and copy into A2
call ZLARNV( 1, ISEED, M*N, A1 )
do i=1,M
do j=1,N
A2(i,j) = A1(i,j)
end do
end do

c  Compute the SVD of A1
call ZGESVD( 'S', 'S', M, N, A1, M, S, U, M, VT, K, WORK, LWORK,
\$   RWORK, INFO)

c  Compute PINV = VT**T * SIGMA * U**T in two steps
do j = 1, K
call ZSCAL( M, dcmplx(1 / S( j )), U( 1, j ), 1 )
end do
call ZGEMM( 'C', 'C', N, M, K, dcmplx(1.0), VT, K, U, M,
\$   dcmplx(0.0), PINV, N)

c   check the result
normA = ZLANGE( 'F', M, N, A2, M, NULL() )
call ZGEMM( 'N', 'N', N, N, M, dcmplx(1.0), PINV, N, A2, M,
\$ dcmplx(0.0), BUFF, N )

call ZGEMM( 'N', 'N', M, N, N, dcmplx(-1.0), A2, M, BUFF, N,
\$ dcmplx(1.0), A2, M );
normAPA = ZLANGE( 'F', M, N, A2, M, NULL() )

call ZGEMM( 'N', 'N', N, M, N, dcmplx(-1.0), BUFF, N, PINV, N,
\$ dcmplx(1.0), PINV, N );
normPAP = ZLANGE( 'F', N, M, PINV, N, NULL() )

write(*,"(A, e10.4)") '|| A - A*P*A || = ', normAPA/normA
write(*,"(A, e10.4)") '|| P - P*A*P || = ', normPAP/normA
end

Code: Select all
#include <math.h>
#include <stdlib.h>
#include <stdio.h>
#include <string.h>
#include <unistd.h>
#include <sys/time.h>
#include <sys/resource.h>
#include <complex.h>

int cprintmatrix( char *matname, int m, int n, double complex *A);
int dprintmatrix( char *matname, int m, int n, double *A);

void zlarnv_( int* idist, int* iseed, int* n, double complex* x);
void zgesvd_( char* jobu, char* jobvt, int* m, int* n,
double complex* a, int* lda, double* s,
double complex* u, int* ldu,
double complex* vt, int* ldvt,
double complex* work, int* lwork,
double* rwork, int *info );
void zgemm_( char* transa, char* transb, int* m, int* n, int* k,
double complex* alpha, double complex* a, int* lda,
double complex* b, int* ldb, double complex* beta,
double complex* c, int* ldc );
double zlange_( char* norm, int* m, int* n, double complex* a,
int* lda, double* work );
void zscal_( int* n, double complex* alpha, double complex* x, int* incx );

void print_help(){
printf( "Usage: ./example_pinv   -h             : help (this text)\n" );
printf( "                        -n <width>   : nb of columns\n" );
printf( "                        -m <height>  : nb of rows\n" );
printf( "                        -print       : output in matlab format\n" );
}

int main(int argc, char **argv){

int i, j;
int IONE = 1;
int M, N, K, L, LDA, LDAxN;
int LDU, LDVT, LWORK, INFO;
int out2matlab;
char JOBU, JOBVT;
char NOTRANS, TRANS, CONJTRANS;
char NORM;
double complex zpone, znone, zzero, tempS;
double normA, normAPA, normPAP;

M     = 15;
N     = 10;
out2matlab = 0;

for ( i = 1; i < argc; i++ ) {
if( strcmp( argv[i], "-h" ) == 0 ) {print_help(); return EXIT_SUCCESS; };
if( strcmp( argv[i], "-m" ) == 0 ) { M = atoi(argv[i+1]); i++; }
if( strcmp( argv[i], "-n" ) == 0 ) { N = atoi(argv[i+1]); i++; }
if( strcmp( argv[i], "-print" ) == 0 ) { out2matlab = 1; }
}

K     = M < N ? M : N;
L     = M > N ? M : N;
LDA   = M;
LDAxN = LDA * N;
LDU   = M;
LDVT  = K;
JOBU  = 'S';
JOBVT = 'S';
LWORK = 1 > 2*K+L ? 1 : 2*K+L;

NOTRANS   = 'N';
TRANS     = 'T';
CONJTRANS = 'C';

NORM = 'F';

zpone = 1.0; znone = -1.0; zzero = 0.0;

int            *ISEED  = ( int            * )malloc(        4 * sizeof( int            ));
double complex *A1     = ( double complex * )malloc( LDA  * N * sizeof( double complex ));
double complex *A2     = ( double complex * )malloc( LDA  * N * sizeof( double complex ));
double complex *PINV   = ( double complex * )malloc( LDA  * N * sizeof( double complex ));
double complex *BUFF   = ( double complex * )malloc( N    * N * sizeof( double complex ));
double complex *U      = ( double complex * )malloc( LDU  * K * sizeof( double complex ));
double complex *VT     = ( double complex * )malloc( LDVT * N * sizeof( double complex ));
double complex *WORK   = ( double complex * )malloc(    LWORK * sizeof( double complex ));
double         *S      = ( double         * )malloc(        K * sizeof( double         ));
double         *RWORK  = ( double         * )malloc( 5    * K * sizeof( double         ));

ISEED[0] = 0; ISEED[1] = 0; ISEED[2] = 0; ISEED[3] = 1;
/* Check if unable to allocate memory */
if ((!A1)||(!A2)||(!PINV)){
printf("Out of Memory \n ");
exit(0);
}
if ((!U)||(!VT)||(!WORK)||(!S)||(!RWORK)){
printf("Out of Memory \n ");
exit(0);
}

/* Initialize A1 and A2 */
zlarnv_( &IONE, ISEED, &LDAxN, A1);
for (i = 0; i < M; i++)
for (j = 0; j < N; j++)
A2[LDA*j+i] = A1[LDA*j+i] ;

/* Compute the SVD of A1 */
zgesvd_( &JOBU, &JOBVT, &M, &N, A1, &LDA, S, U, &LDU, VT, &LDVT, WORK, &LWORK, RWORK, &INFO );

/* Compute the pseudo inverse */
for (i = 0; i < K; i++){
tempS = (double complex)(1 / S[i]);
zscal_( &M, &tempS, &(U[i*LDU]), &IONE );
}
zgemm_( &CONJTRANS, &CONJTRANS, &N, &M, &K, &zpone, VT, &LDVT, U, &M, &zzero, PINV, &N );

if (out2matlab){
printf("clear;\n");
cprintmatrix("A1",LDA,N,A1);
cprintmatrix("A2",LDA,N,A2);
cprintmatrix("US",LDU,K,U);
cprintmatrix("VT",LDVT,N,VT);
dprintmatrix("S",K,1,S);
cprintmatrix("P",N,M,PINV);
printf("PINV = VT'*US';\n");
printf("fprintf('|| A - A*pinv(A)*A || = %%1.4e\\n', norm(A2 - A2*PINV*A2,'fro'))\n");
printf("fprintf('|| pinv(A) - pinv(A)*A*pinv(A) || = %%1.4e\\n',norm(PINV-PINV*A2*PINV,'fro'))\n");
printf("fprintf('|| A - A*P*A || = %%1.4e\\n', norm(A2 - A2*P*A2,'fro'))\n");
printf("fprintf('|| P - P*A*P || = %%1.4e\\n',norm(P-P*A2*P,'fro'))\n");
}

/* check the result */
normA = zlange_( &NORM, &M, &N, A2, &M, NULL );
zgemm_( &NOTRANS, &NOTRANS, &N, &N, &M, &zpone, PINV, &N, A2, &LDA, &zzero, BUFF, &N );

/* || A - A * pinv(A) * A || */
zgemm_( &NOTRANS, &NOTRANS, &M, &N, &N, &znone, A2, &LDA, BUFF, &N, &zpone, A2, &LDA );
normAPA = zlange_( &NORM, &M, &N, A2, &M, NULL );

/* || pinv(A) - pinv(A) * A * pinv(A) || */
zgemm_( &NOTRANS, &NOTRANS, &N, &M, &N, &znone, BUFF, &N, PINV, &N, &zpone, PINV, &N );
normPAP = zlange_( &NORM, &N, &M, PINV, &N, NULL );

printf( "%% || A - A*pinv(A)*A || / || A ||             = %1.4e\n", normAPA / normA );
printf( "%% || pinv(A) - pinv(A)*A*pinv(A) || / || A || = %1.4e\n", normPAP / normA );

free( A1    );
free( A2    );
free( PINV  );
free( BUFF   );
free( U     );
free( VT    );
free( WORK  );
free( S     );
free( RWORK );

return 0;

}

int dprintmatrix( char *matname, int m, int n, double *A){
int i,j;

printf("%s = [\n", matname);
for( i = 0; i < m; i++){
for( j = 0; j < n; j++ )
printf("%1.16e ",A[i+j*m]);
printf("\n");
}
printf("]; \n");

return 0;
}

int cprintmatrix( char *matname, int m, int n, double complex *A){
int i,j;

printf("%s = [\n", matname);
for( i = 0; i < m; i++){
for( j = 0; j < n; j++ )
printf("%1.16e + %1.16ei ",creal(A[i+j*m]),cimag(A[i+j*m]));
printf("\n");
}
printf("]; \n");

return 0;
}
Julien Langou

Posts: 832
Joined: Thu Dec 09, 2004 12:32 pm
Location: Denver, CO, USA

### Re: Pseudo inverse of complex matrix

Thanks very much for Henc and also for Julien. I will try it soon!!
nawras

Posts: 4
Joined: Thu Sep 22, 2011 10:39 am

### Re: Pseudo inverse of complex matrix

What are the size limits for matrices that LAPACK can handle? Can the C code provided in this thread calculate the pseudoinverse of a 300k X 300k matrix?
MM2011

Posts: 2
Joined: Tue Dec 27, 2011 8:52 pm