# zgesdd (l) - Linux Man Pages

## NAME

ZGESDD - 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

## SYNOPSIS

SUBROUTINE ZGESDD(
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( * )

DOUBLE PRECISION RWORK( * ), S( * )

COMPLEX*16 A( LDA, * ), U( LDU, * ), VT( LDVT, * ), WORK( * )

## PURPOSE

ZGESDD 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
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 digits, but we know of none.

## ARGUMENTS

JOBZ (input) CHARACTER*1
Specifies options for computing all or part of the matrix U:
= aqAaq: all M columns of U and all N rows of V**H are returned in the arrays U and VT; = aqSaq: 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; = aqOaq: If M >= N, the first N columns of U are overwritten 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; = aqNaq: 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*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, if JOBZ = aqOaq, 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. aqOaq, 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) COMPLEX*16 array, dimension (LDU,UCOL)
UCOL = M if JOBZ = aqAaq or JOBZ = aqOaq and M < N; UCOL = min(M,N) if JOBZ = aqSaq. If JOBZ = aqAaq or JOBZ = aqOaq and M < N, U contains the M-by-M unitary matrix U; if JOBZ = aqSaq, U contains the first min(M,N) columns of U (the left singular vectors, stored columnwise); if JOBZ = aqOaq and M >= N, or JOBZ = aqNaq, U is not referenced.
LDU (input) INTEGER
The leading dimension of the array U. LDU >= 1; if JOBZ = aqSaq or aqAaq or JOBZ = aqOaq and M < N, LDU >= M.
VT (output) COMPLEX*16 array, dimension (LDVT,N)
If JOBZ = aqAaq or JOBZ = aqOaq and M >= N, VT contains the N-by-N unitary matrix V**H; if JOBZ = aqSaq, VT contains the first min(M,N) rows of V**H (the right singular vectors, stored rowwise); if JOBZ = aqOaq and M < N, or JOBZ = aqNaq, VT is not referenced.
LDVT (input) INTEGER
The leading dimension of the array VT. LDVT >= 1; if JOBZ = aqAaq or JOBZ = aqOaq and M >= N, LDVT >= N; if JOBZ = aqSaq, LDVT >= min(M,N).
WORK (workspace/output) COMPLEX*16 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 = aqNaq, LWORK >= 2*min(M,N)+max(M,N). if JOBZ = aqOaq, LWORK >= 2*min(M,N)*min(M,N)+2*min(M,N)+max(M,N). if JOBZ = aqSaq or aqAaq, LWORK >= min(M,N)*min(M,N)+2*min(M,N)+max(M,N). For good performance, 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) DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
If JOBZ = aqNaq, 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 DBDSDC did not converge.

## FURTHER DETAILS

Based on contributions by

Ming Gu and Huan Ren, Computer Science Division, University of
California at Berkeley, USA