NAME

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

SYNOPSIS

SUBROUTINE DGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, INFO )

CHARACTER JOBU, JOBVT

INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N

DOUBLE PRECISION A( LDA, ∗ ), S( ∗ ), U( LDU, ∗ ), VT( LDVT, ∗ ), WORK( ∗ )

#include <sunperf.h>

void dgesvd(char jobu, char jobvt, int m, int n, double ∗da, int lda, double ∗s, double ∗du, int ldu, double ∗dvt, int ldvt, int ∗info);

PURPOSE

DGESVD computes the singular value decomposition (SVD) of a real M-by-N matrix A, optionally computing the left and/or right singular vectors. The SVD is written
 
     A = 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 V∗∗T, not V.
 

ARGUMENTS

JOBU (input) CHARACTER∗1
Specifies options for computing all or part of the matrix U:
= ’A’:  all M columns of U are returned in array U:
= ’S’:  the first min(m,n) columns of U (the left singular vectors) are returned in the array U; = ’O’:  the first min(m,n) columns of U (the left singular vectors) are overwritten on the array A; = ’N’:  no columns of U (no left singular vectors) are computed.

JOBVT (input) CHARACTER∗1
Specifies options for computing all or part of the matrix V∗∗T:
= ’A’:  all N rows of V∗∗T are returned in the array VT;
= ’S’:  the first min(m,n) rows of V∗∗T (the right singular vectors) are returned in the array VT; = ’O’:  the first min(m,n) rows of V∗∗T (the right singular vectors) are overwritten on the array A; = ’N’:  no rows of V∗∗T (no right singular vectors) are computed.
 
JOBVT and JOBU cannot both be ’O’.

M (input) INTEGER
The number of rows of the input matrix A.  M >= 0.

N (input) INTEGER
The number of columns of the input matrix A.  N >= 0.

A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, if JOBU = ’O’,  A is overwritten with the first min(m,n) columns of U (the left singular vectors, stored columnwise); if JOBVT = ’O’, A is overwritten with the first min(m,n) rows of V∗∗T (the right singular vectors, stored rowwise); if JOBU .ne. ’O’ and JOBVT .ne. ’O’, the contents of A are destroyed.

LDA (input) INTEGER
The leading dimension of the array A.  LDA >= max(1,M).

S (output) DOUBLE PRECISION array, dimension (min(M,N))
The singular values of A, sorted so that S(i) >= S(i+1).

U (output) DOUBLE PRECISION array, dimension (LDU,UCOL)
(LDU,M) if JOBU = ’A’ or (LDU,min(M,N)) if JOBU = ’S’. If JOBU = ’A’, U contains the M-by-M orthogonal matrix U; if JOBU = ’S’, U contains the first min(m,n) columns of U (the left singular vectors, stored columnwise); if JOBU = ’N’ or ’O’, U is not referenced.

LDU (input) INTEGER
The leading dimension of the array U.  LDU >= 1; if JOBU = ’S’ or ’A’, LDU >= M.

VT (output) DOUBLE PRECISION array, dimension (LDVT,N)
If JOBVT = ’A’, VT contains the N-by-N orthogonal matrix V∗∗T; if JOBVT = ’S’, VT contains the first min(m,n) rows of V∗∗T (the right singular vectors, stored rowwise); if JOBVT = ’N’ or ’O’, VT is not referenced.

LDVT (input) INTEGER
The leading dimension of the array VT.  LDVT >= 1; if JOBVT = ’A’, LDVT >= N; if JOBVT = ’S’, LDVT >= min(M,N).

WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK; if INFO > 0, WORK(2:MIN(M,N)) contains the unconverged superdiagonal elements of an upper bidiagonal matrix B whose diagonal is in S (not necessarily sorted). B satisfies A = U ∗ B ∗ VT, so it has the same singular values as A, and singular vectors related by U and VT.

LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 1. LWORK >= MAX(3∗MIN(M,N)+MAX(M,N),5∗MIN(M,N)-4). For good performance, LWORK should generally be larger.

INFO (output) INTEGER
= 0:  successful exit.
< 0:  if INFO = -i, the i-th argument had an illegal value.
> 0:  if DBDSQR did not converge, INFO specifies how many superdiagonals of an intermediate bidiagonal form B did not converge to zero. See the description of WORK above for details.

SunOS 5.0  —  Last change: 10 Dec 1998