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DGESDD(1)	      LAPACK driver routine (version 3.2)	     DGESDD(1)

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
	    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 dig‐
       its, 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 over‐
	       written 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;
	       = '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) DOUBLE PRECISION 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)	 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) DOUBLE PRECISION 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 con‐
	       tains 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 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) DOUBLE PRECISION 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  row‐
	       wise); if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not refer‐
	       enced.

       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)   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 = 'N',
	       LWORK >= 3*min(M,N) + max(max(M,N),7*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 perfor‐
	       mance,  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

 LAPACK driver routine (version 3November 2008			     DGESDD(1)
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