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### Pseudo inverse of complex matrix Posted: Thu Sep 22, 2011 11:30 am
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

### Re: Pseudo inverse of complex matrix Posted: Fri Sep 23, 2011 11:46 am
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.

### Re: Pseudo inverse of complex matrix Posted: Fri Sep 23, 2011 6:15 pm
But all discussions were made for real matrices and I am interested in the Pseudoinverse of complex matrix.
Thanks very much.

### Re: Pseudo inverse of complex matrix Posted: Mon Sep 26, 2011 11:33 am
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 doc  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; ISEED = 0; ISEED = 0; ISEED = 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;}`

### Re: Pseudo inverse of complex matrix Posted: Mon Sep 26, 2011 12:21 pm
Thanks very much for Henc and also for Julien. I will try it soon!!

### Re: Pseudo inverse of complex matrix Posted: Wed Dec 28, 2011 3:35 am
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?