CGESDD(3) LAPACK driver routine (version 3.1) CGESDD(3)
NAME
CGESDD - 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 vec-
tors, 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 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**H 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**H are returned in the arrays
U and VT; = 'O': If M >= N, the first N columns of U are over-
written in 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 A;
= '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
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) 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 con-
tains the M-by-M unitary 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 contains 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 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) COMPLEX 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 >= 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 per-
formance, LWORK should generally be larger.
If LWORK = -1, a workspace query is assumed. The optimal size
for the WORK array is calculated and stored in WORK(1), and no
other work except argument checking is performed.
RWORK (workspace) REAL array, dimension (MAX(1,LRWORK))
If JOBZ = 'N', LRWORK >= 5*min(M,N). Otherwise, LRWORK >=
5*min(M,N)*min(M,N) + 7*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 illegal value.
> 0: The updating process of SBDSDC did not converge.
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 CGESDD(3)