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( * )
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 diago
nal 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
= 'N': no rows of V**T (no right singular vec
tors) 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 singu
lar 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 orthog
onal 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.
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 sin
gular 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)). For
good performance, 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 DBDSQR did not converge, INFO specifies
how many superdiagonals of an intermediate bidiag
onal form B did not converge to zero. See the
description of WORK above for details.
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