DGESDD(3) LAPACK driver routine (version 3.1) DGESDD(3)
NAME
DGESDD - 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 divide-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-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 dig-
its, 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 over-
written 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 A;
= '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
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)
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 con-
tains 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) 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 (LDVT,N)
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 row-
wise); if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not refer-
enced.
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, dimension
(MAX(1,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),7*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 illegal value.
> 0: DBDSDC 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
LAPACK driver routine (version 3.N1o)vember 2006 DGESDD(3)