sgesvd(3)

NAME

SGESVD - 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 SGESVD( 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
    REAL            A(  LDA, * ), S( * ), U( LDU, * ), VT(
LDVT, * ), WORK( * )

PURPOSE

SGESVD 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-nega

tive, 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 vec

tors) 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 ar

ray 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 overwrit

ten 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) REAL 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 sin

gular 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) REAL array, dimension (min(M,N))

The singular values of A, sorted so that S(i) >=

S(i+1).

U (output) REAL 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) REAL 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) REAL 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 su

perdiagonal elements of an upper bidiagonal matrix B whose diago

nal is in S (not necessarily sorted). B satisfies A = U * B * VT,

so it has the same singular values as A, and singular vectors re

lated 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)). For good perfor

mance, LWORK should generally be larger.

If LWORK = -1, then a workspace query is assumed;

the routine only calculates the optimal size of the WORK array,

returns this value as the first entry of the WORK array, and no

error message related to LWORK is issued by XERBLA.

INFO (output) INTEGER

= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an ille

gal value.
> 0: if SBDSQR 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 de

tails.

LAPACK version 3.0 15 June 2000

SGESVD(3)