NAME

sgesdd - compute the singular value decomposition (SVD) of a real M-by-N matrix A, optionally computing the left and right singular vectors

SYNOPSIS

  SUBROUTINE SGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, 
 *      LWORK, IWORK, INFO)
  CHARACTER * 1 JOBZ
  INTEGER M, N, LDA, LDU, LDVT, LWORK, INFO
  INTEGER IWORK(*)
  REAL A(LDA,*), S(*), U(LDU,*), VT(LDVT,*), WORK(*)

  SUBROUTINE SGESDD_64( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, 
 *      LWORK, IWORK, INFO)
  CHARACTER * 1 JOBZ
  INTEGER*8 M, N, LDA, LDU, LDVT, LWORK, INFO
  INTEGER*8 IWORK(*)
  REAL A(LDA,*), S(*), U(LDU,*), VT(LDVT,*), WORK(*)

F95 INTERFACE

  SUBROUTINE GESDD( JOBZ, [M], [N], A, [LDA], S, U, [LDU], VT, [LDVT], 
 *       [WORK], [LWORK], [IWORK], [INFO])
  CHARACTER(LEN=1) :: JOBZ
  INTEGER :: M, N, LDA, LDU, LDVT, LWORK, INFO
  INTEGER, DIMENSION(:) :: IWORK
  REAL, DIMENSION(:) :: S, WORK
  REAL, DIMENSION(:,:) :: A, U, VT

  SUBROUTINE GESDD_64( JOBZ, [M], [N], A, [LDA], S, U, [LDU], VT, 
 *       [LDVT], [WORK], [LWORK], [IWORK], [INFO])
  CHARACTER(LEN=1) :: JOBZ
  INTEGER(8) :: M, N, LDA, LDU, LDVT, LWORK, INFO
  INTEGER(8), DIMENSION(:) :: IWORK
  REAL, DIMENSION(:) :: S, WORK
  REAL, DIMENSION(:,:) :: A, U, VT

C INTERFACE

#include <sunperf.h>

void sgesdd(char jobz, int m, int n, float *a, int lda, float *s, float *u, int ldu, float *vt, int ldvt, int *info);

void sgesdd_64(char jobz, long m, long n, float *a, long lda, float *s, float *u, long ldu, float *vt, long ldvt, long *info);

PURPOSE

sgesdd computes the singular value decomposition (SVD) of a real M-by-N matrix A, optionally computing the left and right singular vectors. If singular vectors are desired, it uses a divide-and-conquer algorithm.

The SVD is written

 = U * SIGMA * transpose(V)

where SIGMA is an M-by-N matrix which is zero except for its min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA are the singular values of A; they are real and non-negative, and are returned in descending order. The first min(m,n) columns of U and V are the left and right singular vectors of A.

Note that the routine returns VT = V**T, not V.

The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

ARGUMENTS

JOBZ (input)
Specifies options for computing all or part of the matrix U:

 = 'A':  all M columns of U and all N rows of V**T are
returned in the arrays U and VT;
 = 'S':  the first min(M,N) columns of U and the first
min(M,N) rows of V**T are returned in the arrays U
and VT;
 = 'O':  If M  > = N, the first N columns of U are overwritten
on the array A and all rows of V**T are returned in
the array VT;
otherwise, all columns of U are returned in the
array U and the first M rows of V**T are overwritten
in the array VT;
 = 'N':  no columns of U or rows of V**T are computed.

M (input)
The number of rows of the input matrix A. M > = 0.
N (input)
The number of columns of the input matrix A. N > = 0.
A (input/output)
On entry, the M-by-N matrix A. On exit, if JOBZ = 'O', A is overwritten with the first N columns of U (the left singular vectors, stored columnwise) if M > = N; A is overwritten with the first M rows of V**T (the right singular vectors, stored rowwise) otherwise. if JOBZ .ne. 'O', the contents of A are destroyed.
LDA (input)
The leading dimension of the array A. LDA > = max(1,M).
S (output)
The singular values of A, sorted so that S(i) > = S(i+1).
U (output)
UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N; UCOL = min(M,N) if JOBZ = 'S'. If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M orthogonal matrix U; if JOBZ = 'S', U contains the first min(M,N) columns of U (the left singular vectors, stored columnwise); if JOBZ = 'O' and M > = N, or JOBZ = 'N', U is not referenced.
LDU (input)
The leading dimension of the array U. LDU > = 1; if JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU > = M.
VT (output)
If JOBZ = 'A' or JOBZ = 'O' and M > = N, VT contains the N-by-N orthogonal matrix V**T; if JOBZ = 'S', VT contains the first min(M,N) rows of V**T (the right singular vectors, stored rowwise); if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.
LDVT (input)
The leading dimension of the array VT. LDVT > = 1; if JOBZ = 'A' or JOBZ = 'O' and M > = N, LDVT > = N; if JOBZ = 'S', LDVT > = min(M,N).
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK;
LWORK (input)
The dimension of the array WORK. LWORK > = 1. If JOBZ = 'N', LWORK > = 3*min(M,N) + max(max(M,N),6*min(M,N)). If JOBZ = 'O', LWORK > = 3*min(M,N)*min(M,N) + max(max(M,N),5*min(M,N)* min(M,N)+4*min(M,N)). If JOBZ = 'S' or 'A' LWORK > = 3*min(M,N)*min(M,N) + max(max(M,N),4*min(M,N)* min(M,N)+4*min(M,N)). For good performance, LWORK should generally be larger. If LWORK < 0 but other input arguments are legal, WORK(1) returns optimal LWORK.
IWORK (workspace)
dimension(8*MIN(M,N))

INFO (output)

 = 0:  successful exit.

 < 0:  if INFO  = -i, the i-th argument had an illegal value.

 > 0:  SBDSDC did not converge, updating process failed.

FURTHER DETAILS

Based on contributions by

   Ming Gu and Huan Ren, Computer Science Division, University of
   California at Berkeley, USA