SGESDD - compute the singular value decomposition (SVD) of
a real M-by-N matrix A, optionally computing the left and
right singular vectors
SYNOPSIS
SUBROUTINE SGESDD( 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( * )
REAL A( LDA, * ), S( * ), U( LDU, * ), VT(
LDVT, * ), WORK( * )
PURPOSE
SGESDD 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 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 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 hexadecimal or decimal machines with
out 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)
= '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) REAL 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 row
wise) 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) 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)
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
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) REAL array, dimension (LDVT,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 contains the first min(M,N) rows of V**T
(the right singular vectors, stored rowwise); if
JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not
referenced.
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) REAL array, dimension (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 performance, LWORK should generally be
larger. If LWORK < 0 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: SBDSDC 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
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