CGESDD - compute the singular value decomposition (SVD) of
a complex M-by-N matrix A, optionally computing the left
and/or right singular vectors, by using divide-and-conquer
method
SYNOPSIS
SUBROUTINE CGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT,
LDVT, WORK, LWORK, RWORK, IWORK, INFO )
CHARACTER JOBZ
INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
INTEGER IWORK( * )
REAL RWORK( * ), S( * )
COMPLEX A( LDA, * ), U( LDU, * ), VT( LDVT, *
), WORK( * )
PURPOSE
CGESDD computes the singular value decomposition (SVD) of
a complex M-by-N matrix A, optionally computing the left
and/or right singular vectors, by using divide-and-conquer
method. The SVD is written
A = U * SIGMA * conjugate-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 unitary
matrix, and V is an N-by-N unitary 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**H, 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**H
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**H
are returned in the array VT; otherwise, all
columns of U are returned in the array U and the
first M rows of V**H are overwritten in the array
VT; = 'N': no columns of U or rows of V**H 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) COMPLEX 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**H (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) COMPLEX 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 uni
tary 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) COMPLEX array, dimension (LDVT,N)
If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT con
tains the N-by-N unitary matrix V**H; if JOBZ =
'S', VT contains the first min(M,N) rows of V**H
(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) COMPLEX 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 >= 2*min(M,N)+max(M,N). if JOBZ
= 'O', LWORK >=
2*min(M,N)*min(M,N)+2*min(M,N)+max(M,N). if JOBZ
= 'S' or 'A', LWORK >=
min(M,N)*min(M,N)+2*min(M,N)+max(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.
RWORK (workspace) REAL array, dimension (LRWORK)
If JOBZ = 'N', LRWORK >= 7*min(M,N). Otherwise,
LRWORK >= 5*min(M,N)*min(M,N) + 5*min(M,N)
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: The updating process of SBDSDC did not con
verge.
FURTHER DETAILS
Based on contributions by
Ming Gu and Huan Ren, Computer Science Division, Uni
versity of
California at Berkeley, USA
Man(1) output converted with
man2html