.TH DGESDD 1 "November 2006" " LAPACK driver routine (version 3.1) " " LAPACK driver routine (version 3.1) " .SH NAME DGESDD - the singular value decomposition (SVD) of a real M-by-N matrix A, optionally computing the left and right singular vectors .SH SYNOPSIS .TP 19 SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, IWORK, INFO ) .TP 19 .ti +4 CHARACTER JOBZ .TP 19 .ti +4 INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N .TP 19 .ti +4 INTEGER IWORK( * ) .TP 19 .ti +4 DOUBLE PRECISION A( LDA, * ), S( * ), U( LDU, * ), VT( LDVT, * ), WORK( * ) .SH 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. .br The SVD is written .br A = U * SIGMA * transpose(V) .br 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. .br Note that the routine returns VT = V**T, not V. .br 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. .br .SH ARGUMENTS .TP 8 JOBZ (input) CHARACTER*1 Specifies options for computing all or part of the matrix U: .br = \(aqA\(aq: all M columns of U and all N rows of V**T are returned in the arrays U and VT; = \(aqS\(aq: 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; = \(aqO\(aq: 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; = \(aqN\(aq: no columns of U or rows of V**T are computed. .TP 8 M (input) INTEGER The number of rows of the input matrix A. M >= 0. .TP 8 N (input) INTEGER The number of columns of the input matrix A. N >= 0. .TP 8 A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, if JOBZ = \(aqO\(aq, 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. \(aqO\(aq, the contents of A are destroyed. .TP 8 LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,M). .TP 8 S (output) DOUBLE PRECISION array, dimension (min(M,N)) The singular values of A, sorted so that S(i) >= S(i+1). .TP 8 U (output) DOUBLE PRECISION array, dimension (LDU,UCOL) UCOL = M if JOBZ = \(aqA\(aq or JOBZ = \(aqO\(aq and M < N; UCOL = min(M,N) if JOBZ = \(aqS\(aq. If JOBZ = \(aqA\(aq or JOBZ = \(aqO\(aq and M < N, U contains the M-by-M orthogonal matrix U; if JOBZ = \(aqS\(aq, U contains the first min(M,N) columns of U (the left singular vectors, stored columnwise); if JOBZ = \(aqO\(aq and M >= N, or JOBZ = \(aqN\(aq, U is not referenced. .TP 8 LDU (input) INTEGER The leading dimension of the array U. LDU >= 1; if JOBZ = \(aqS\(aq or \(aqA\(aq or JOBZ = \(aqO\(aq and M < N, LDU >= M. .TP 8 VT (output) DOUBLE PRECISION array, dimension (LDVT,N) If JOBZ = \(aqA\(aq or JOBZ = \(aqO\(aq and M >= N, VT contains the N-by-N orthogonal matrix V**T; if JOBZ = \(aqS\(aq, VT contains the first min(M,N) rows of V**T (the right singular vectors, stored rowwise); if JOBZ = \(aqO\(aq and M < N, or JOBZ = \(aqN\(aq, VT is not referenced. .TP 8 LDVT (input) INTEGER The leading dimension of the array VT. LDVT >= 1; if JOBZ = \(aqA\(aq or JOBZ = \(aqO\(aq and M >= N, LDVT >= N; if JOBZ = \(aqS\(aq, LDVT >= min(M,N). .TP 8 WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK; .TP 8 LWORK (input) INTEGER The dimension of the array WORK. LWORK >= 1. If JOBZ = \(aqN\(aq, LWORK >= 3*min(M,N) + max(max(M,N),7*min(M,N)). If JOBZ = \(aqO\(aq, 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 = \(aqS\(aq or \(aqA\(aq 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. .TP 8 IWORK (workspace) INTEGER array, dimension (8*min(M,N)) .TP 8 INFO (output) INTEGER = 0: successful exit. .br < 0: if INFO = -i, the i-th argument had an illegal value. .br > 0: DBDSDC did not converge, updating process failed. .SH FURTHER DETAILS Based on contributions by .br Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA .br