colum

SGESDD(l)								 )								 SGESDD(l)

NAME

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

SYNOPSIS

       SUBROUTINE SGESDD( 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( * )

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

PURPOSE

       SGESDD 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

	    A = U * 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:
	       = 'A':  all M columns of U and all N rows of V**T 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**T are returned in the arrays U and VT; = 'O':  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 VT; = 'N':  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) REAL 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**T (the right singular vectors,  stored  rowwise)  other-
	       wise.  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) REAL 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 orthogonal matrix U; if JOBZ = 'S', U contains the first min(M,N) columns of U (the left singular vectors, stored colum-
	       nwise); 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) REAL array, dimension (LDVT,N)
	       If  JOBZ  =  'A'  or  JOBZ  =  'O'  and M >= N, VT contains the N-by-N orthogonal matrix V**T; if JOBZ = 'S', VT contains the first
	       min(M,N) rows of V**T (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) REAL array, dimension (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 >= 3*min(M,N) + max(max(M,N),6*min(M,N)).  If JOBZ = 'O', 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	=  'S'	or  'A'  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 < 0 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:  SBDSDC 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

LAPACK version 3.0						   15 June 2000 							 SGESDD(l)