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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: computes the singular value decomposition (SVD) of a complex M-by-N matrix A, optionally computing the left and/or

##### NAME
LAPACK-3 - 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 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 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) rows of V**H are returned in the arrays U and VT; = 'O': 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; = '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*16 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) 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 = '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 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*16 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 rowwise); if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced. 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*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 = '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 = -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 = 'N', LRWORK >= 5*min(M,N). Otherwise, LRWORK >= min(M,N)*max(5*min(M,N)+7,2*max(M,N)+2*min(M,N)+1) 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 LAPACK driver routine (version 3.March 2011 LAPACK-3(3)