dgesdd(3)

NAME

DGESDD - compute the singular value decomposition (SVD) of

a real M-by-N matrix A, optionally computing the left and right

singular vectors

SYNOPSIS

SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT,  LDVT, WORK, LWORK, IWORK, INFO )
    CHARACTER      JOBZ
    INTEGER        INFO, LDA, LDU, LDVT, LWORK, M, N
    INTEGER        IWORK( * )
    DOUBLE          PRECISION A( LDA, * ), S( * ), U( LDU,
* ), VT( LDVT, * ), WORK( * )

PURPOSE

DGESDD 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 di

vide-and-conquer algorithm.

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 VT = V**T, not V.

The divide and conquer algorithm makes very mild assump

tions 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 hexadeci

mal or decimal machines without guard digits, but we know of

none.

ARGUMENTS

JOBZ (input) CHARACTER*1

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) 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 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 de

stroyed.

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)

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 vec

tors, stored columnwise); if JOBZ = 'O' and M >= N, or JOBZ =

'N', U is not referenced.

LDU (input) INTEGER

The leading dimension of the array U. LDU >= 1;

if JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.

VT (output) DOUBLE PRECISION array, dimension (LD

VT,N)

If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT con

tains the N-by-N orthogonal matrix V**T; if JOBZ = 'S', VT con

tains the first min(M,N) rows of V**T (the right singular vec

tors, stored rowwise); if JOBZ = 'O' and M < N, or JOBZ = 'N', VT

is not referenced.

LDVT (input) INTEGER

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/output) DOUBLE PRECISION array, dimen

sion (LWORK)

On exit, if INFO = 0, WORK(1) returns the optimal

LWORK;

LWORK (input) INTEGER

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 perfor

mance, LWORK should generally be larger. If LWORK = -1 but other

input arguments are legal, WORK(1) returns the optimal LWORK.

IWORK (workspace) INTEGER array, dimension (8*min(M,N))

INFO (output) INTEGER

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

gal value.
> 0: DBDSDC did not converge, updating process

failed.

FURTHER DETAILS

Based on contributions by

Ming Gu and Huan Ren, Computer Science Division, Uni

versity of
California at Berkeley, USA

LAPACK version 3.0 15 June 2000

DGESDD(3)