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version 1.6, 2010/08/13 21:03:45 version 1.10, 2012/08/22 09:48:14
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   *> \brief \b DGESDD
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DGESDD + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesdd.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesdd.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesdd.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
   *                          LWORK, IWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          JOBZ
   *       INTEGER            INFO, LDA, LDU, LDVT, LWORK, M, N
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IWORK( * )
   *       DOUBLE PRECISION   A( LDA, * ), S( * ), U( LDU, * ),
   *      $                   VT( LDVT, * ), WORK( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> 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 digits, but we know of none.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] JOBZ
   *> \verbatim
   *>          JOBZ is 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 A;
   *>          = 'N':  no columns of U or rows of V**T are computed.
   *> \endverbatim
   *>
   *> \param[in] M
   *> \verbatim
   *>          M is INTEGER
   *>          The number of rows of the input matrix A.  M >= 0.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The number of columns of the input matrix A.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is 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.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= max(1,M).
   *> \endverbatim
   *>
   *> \param[out] S
   *> \verbatim
   *>          S is DOUBLE PRECISION array, dimension (min(M,N))
   *>          The singular values of A, sorted so that S(i) >= S(i+1).
   *> \endverbatim
   *>
   *> \param[out] U
   *> \verbatim
   *>          U is 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 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 columnwise);
   *>          if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.
   *> \endverbatim
   *>
   *> \param[in] LDU
   *> \verbatim
   *>          LDU is INTEGER
   *>          The leading dimension of the array U.  LDU >= 1; if
   *>          JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.
   *> \endverbatim
   *>
   *> \param[out] VT
   *> \verbatim
   *>          VT is 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 rowwise);
   *>          if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.
   *> \endverbatim
   *>
   *> \param[in] LDVT
   *> \verbatim
   *>          LDVT is 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).
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
   *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK;
   *> \endverbatim
   *>
   *> \param[in] LWORK
   *> \verbatim
   *>          LWORK is 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) + 
   *>                     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) +
   *>                     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.
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension (8*min(M,N))
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0:  successful exit.
   *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
   *>          > 0:  DBDSDC did not converge, updating process failed.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleGEsing
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Ming Gu and Huan Ren, Computer Science Division, University of
   *>     California at Berkeley, USA
   *>
   *  =====================================================================
       SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,        SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
      $                   LWORK, IWORK, INFO )       $                   LWORK, IWORK, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.2.1)                                  --  *  -- LAPACK driver routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     March 2009  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          JOBZ        CHARACTER          JOBZ
Line 16 Line 232
      $                   VT( LDVT, * ), WORK( * )       $                   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 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 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 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 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 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) 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) +   
 *                     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) +  
 *                     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  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.6  
changed lines
  Added in v.1.10

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