# dgesdd (l) - Linux Manuals

## NAME

DGESDD - computes the singular value decomposition (SVD) of a real M-by-N matrix A, optionally computing the left and right singular vectors

## SYNOPSIS

SUBROUTINE DGESDD(
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( * )

DOUBLE PRECISION A( LDA, * ), S( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )

## PURPOSE

DGESDD 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

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 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**T, 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**T 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**T are returned in the arrays U and VT; = aqOaq: 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 A; = aqNaq: 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) DOUBLE PRECISION 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**T (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) DOUBLE PRECISION 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 orthogonal 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) DOUBLE PRECISION array, dimension (LDVT,N)
If JOBZ = aqAaq or JOBZ = aqOaq and M >= N, VT contains the N-by-N orthogonal matrix V**T; if JOBZ = aqSaq, VT contains the first min(M,N) rows of V**T (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) DOUBLE PRECISION 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 >= 3*min(M,N) + max(max(M,N),7*min(M,N)). If JOBZ = aqOaq, 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 = aqSaq or aqAaq 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 = -1 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 illegal value.
> 0: DBDSDC did not converge, updating process failed.

## FURTHER DETAILS

Based on contributions by

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