OpenBLAS/lapack-netlib/SRC/dgesdd.f

*> \brief \b DGESDD
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> [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 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.
*>
*>          Let mx = max(M,N) and mn = min(M,N).
*>          If JOBZ = 'N', LWORK >= 3*mn + max( mx, 7*mn ).
*>          If JOBZ = 'O', LWORK >= 3*mn + max( mx, 5*mn*mn + 4*mn ).
*>          If JOBZ = 'S', LWORK >= 4*mn*mn + 7*mn.
*>          If JOBZ = 'A', LWORK >= 4*mn*mn + 6*mn + mx.
*>          These are not tight minimums in all cases; see comments inside code.
*>          For good performance, LWORK should generally be larger;
*>          a query is recommended.
*> \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 June 2016
*
*> \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, LWORK, IWORK, INFO )
      implicit none
*
*  -- LAPACK driver routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2016
*
*     .. 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( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, WNTQA, WNTQAS, WNTQN, WNTQO, WNTQS
      INTEGER            BDSPAC, BLK, CHUNK, I, IE, IERR, IL,
     $                   IR, ISCL, ITAU, ITAUP, ITAUQ, IU, IVT, LDWKVT,
     $                   LDWRKL, LDWRKR, LDWRKU, MAXWRK, MINMN, MINWRK,
     $                   MNTHR, NWORK, WRKBL
      INTEGER            LWORK_DGEBRD_MN, LWORK_DGEBRD_MM,
     $                   LWORK_DGEBRD_NN, LWORK_DGELQF_MN,
     $                   LWORK_DGEQRF_MN,
     $                   LWORK_DORGBR_P_MM, LWORK_DORGBR_Q_NN,
     $                   LWORK_DORGLQ_MN, LWORK_DORGLQ_NN,
     $                   LWORK_DORGQR_MM, LWORK_DORGQR_MN,
     $                   LWORK_DORMBR_PRT_MM, LWORK_DORMBR_QLN_MM,
     $                   LWORK_DORMBR_PRT_MN, LWORK_DORMBR_QLN_MN,
     $                   LWORK_DORMBR_PRT_NN, LWORK_DORMBR_QLN_NN
      DOUBLE PRECISION   ANRM, BIGNUM, EPS, SMLNUM
*     ..
*     .. Local Arrays ..
      INTEGER            IDUM( 1 )
      DOUBLE PRECISION   DUM( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DBDSDC, DGEBRD, DGELQF, DGEMM, DGEQRF, DLACPY,
     $                   DLASCL, DLASET, DORGBR, DORGLQ, DORGQR, DORMBR,
     $                   XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           DLAMCH, DLANGE, LSAME
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          INT, MAX, MIN, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      INFO   = 0
      MINMN  = MIN( M, N )
      WNTQA  = LSAME( JOBZ, 'A' )
      WNTQS  = LSAME( JOBZ, 'S' )
      WNTQAS = WNTQA .OR. WNTQS
      WNTQO  = LSAME( JOBZ, 'O' )
      WNTQN  = LSAME( JOBZ, 'N' )
      LQUERY = ( LWORK.EQ.-1 )
*
      IF( .NOT.( WNTQA .OR. WNTQS .OR. WNTQO .OR. WNTQN ) ) THEN
         INFO = -1
      ELSE IF( M.LT.0 ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      ELSE IF( LDU.LT.1 .OR. ( WNTQAS .AND. LDU.LT.M ) .OR.
     $         ( WNTQO .AND. M.LT.N .AND. LDU.LT.M ) ) THEN
         INFO = -8
      ELSE IF( LDVT.LT.1 .OR. ( WNTQA .AND. LDVT.LT.N ) .OR.
     $         ( WNTQS .AND. LDVT.LT.MINMN ) .OR.
     $         ( WNTQO .AND. M.GE.N .AND. LDVT.LT.N ) ) THEN
         INFO = -10
      END IF
*
*     Compute workspace
*       Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace allocated at that point in the code,
*       as well as the preferred amount for good performance.
*       NB refers to the optimal block size for the immediately
*       following subroutine, as returned by ILAENV.
*
      IF( INFO.EQ.0 ) THEN
         MINWRK = 1
         MAXWRK = 1
         BDSPAC = 0
         MNTHR  = INT( MINMN*11.0D0 / 6.0D0 )
         IF( M.GE.N .AND. MINMN.GT.0 ) THEN
*
*           Compute space needed for DBDSDC
*
            IF( WNTQN ) THEN
*              dbdsdc needs only 4*N (or 6*N for uplo=L for LAPACK <= 3.6)
*              keep 7*N for backwards compatibility.
               BDSPAC = 7*N
            ELSE
               BDSPAC = 3*N*N + 4*N
            END IF
*
*           Compute space preferred for each routine
            CALL DGEBRD( M, N, DUM(1), M, DUM(1), DUM(1), DUM(1),
     $                   DUM(1), DUM(1), -1, IERR )
            LWORK_DGEBRD_MN = INT( DUM(1) )
*
            CALL DGEBRD( N, N, DUM(1), N, DUM(1), DUM(1), DUM(1),
     $                   DUM(1), DUM(1), -1, IERR )
            LWORK_DGEBRD_NN = INT( DUM(1) )
*
            CALL DGEQRF( M, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
            LWORK_DGEQRF_MN = INT( DUM(1) )
*
            CALL DORGBR( 'Q', N, N, N, DUM(1), N, DUM(1), DUM(1), -1,
     $                   IERR )
            LWORK_DORGBR_Q_NN = INT( DUM(1) )
*
            CALL DORGQR( M, M, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
            LWORK_DORGQR_MM = INT( DUM(1) )
*
            CALL DORGQR( M, N, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
            LWORK_DORGQR_MN = INT( DUM(1) )
*
            CALL DORMBR( 'P', 'R', 'T', N, N, N, DUM(1), N,
     $                   DUM(1), DUM(1), N, DUM(1), -1, IERR )
            LWORK_DORMBR_PRT_NN = INT( DUM(1) )
*
            CALL DORMBR( 'Q', 'L', 'N', N, N, N, DUM(1), N,
     $                   DUM(1), DUM(1), N, DUM(1), -1, IERR )
            LWORK_DORMBR_QLN_NN = INT( DUM(1) )
*
            CALL DORMBR( 'Q', 'L', 'N', M, N, N, DUM(1), M,
     $                   DUM(1), DUM(1), M, DUM(1), -1, IERR )
            LWORK_DORMBR_QLN_MN = INT( DUM(1) )
*
            CALL DORMBR( 'Q', 'L', 'N', M, M, N, DUM(1), M,
     $                   DUM(1), DUM(1), M, DUM(1), -1, IERR )
            LWORK_DORMBR_QLN_MM = INT( DUM(1) )
*
            IF( M.GE.MNTHR ) THEN
               IF( WNTQN ) THEN
*
*                 Path 1 (M >> N, JOBZ='N')
*
                  WRKBL = N + LWORK_DGEQRF_MN
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
                  MAXWRK = MAX( WRKBL, BDSPAC + N )
                  MINWRK = BDSPAC + N
               ELSE IF( WNTQO ) THEN
*
*                 Path 2 (M >> N, JOBZ='O')
*
                  WRKBL = N + LWORK_DGEQRF_MN
                  WRKBL = MAX( WRKBL,   N + LWORK_DORGQR_MN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
                  WRKBL = MAX( WRKBL, 3*N + BDSPAC )
                  MAXWRK = WRKBL + 2*N*N
                  MINWRK = BDSPAC + 2*N*N + 3*N
               ELSE IF( WNTQS ) THEN
*
*                 Path 3 (M >> N, JOBZ='S')
*
                  WRKBL = N + LWORK_DGEQRF_MN
                  WRKBL = MAX( WRKBL,   N + LWORK_DORGQR_MN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
                  WRKBL = MAX( WRKBL, 3*N + BDSPAC )
                  MAXWRK = WRKBL + N*N
                  MINWRK = BDSPAC + N*N + 3*N
               ELSE IF( WNTQA ) THEN
*
*                 Path 4 (M >> N, JOBZ='A')
*
                  WRKBL = N + LWORK_DGEQRF_MN
                  WRKBL = MAX( WRKBL,   N + LWORK_DORGQR_MM )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
                  WRKBL = MAX( WRKBL, 3*N + BDSPAC )
                  MAXWRK = WRKBL + N*N
                  MINWRK = N*N + MAX( 3*N + BDSPAC, N + M )
               END IF
            ELSE
*
*              Path 5 (M >= N, but not much larger)
*
               WRKBL = 3*N + LWORK_DGEBRD_MN
               IF( WNTQN ) THEN
*                 Path 5n (M >= N, jobz='N')
                  MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
                  MINWRK = 3*N + MAX( M, BDSPAC )
               ELSE IF( WNTQO ) THEN
*                 Path 5o (M >= N, jobz='O')
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MN )
                  WRKBL = MAX( WRKBL, 3*N + BDSPAC )
                  MAXWRK = WRKBL + M*N
                  MINWRK = 3*N + MAX( M, N*N + BDSPAC )
               ELSE IF( WNTQS ) THEN
*                 Path 5s (M >= N, jobz='S')
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MN )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
                  MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
                  MINWRK = 3*N + MAX( M, BDSPAC )
               ELSE IF( WNTQA ) THEN
*                 Path 5a (M >= N, jobz='A')
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MM )
                  WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
                  MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
                  MINWRK = 3*N + MAX( M, BDSPAC )
               END IF
            END IF
         ELSE IF( MINMN.GT.0 ) THEN
*
*           Compute space needed for DBDSDC
*
            IF( WNTQN ) THEN
*              dbdsdc needs only 4*N (or 6*N for uplo=L for LAPACK <= 3.6)
*              keep 7*N for backwards compatibility.
               BDSPAC = 7*M
            ELSE
               BDSPAC = 3*M*M + 4*M
            END IF
*
*           Compute space preferred for each routine
            CALL DGEBRD( M, N, DUM(1), M, DUM(1), DUM(1), DUM(1),
     $                   DUM(1), DUM(1), -1, IERR )
            LWORK_DGEBRD_MN = INT( DUM(1) )
*
            CALL DGEBRD( M, M, A, M, S, DUM(1), DUM(1),
     $                   DUM(1), DUM(1), -1, IERR )
            LWORK_DGEBRD_MM = INT( DUM(1) )
*
            CALL DGELQF( M, N, A, M, DUM(1), DUM(1), -1, IERR )
            LWORK_DGELQF_MN = INT( DUM(1) )
*
            CALL DORGLQ( N, N, M, DUM(1), N, DUM(1), DUM(1), -1, IERR )
            LWORK_DORGLQ_NN = INT( DUM(1) )
*
            CALL DORGLQ( M, N, M, A, M, DUM(1), DUM(1), -1, IERR )
            LWORK_DORGLQ_MN = INT( DUM(1) )
*
            CALL DORGBR( 'P', M, M, M, A, N, DUM(1), DUM(1), -1, IERR )
            LWORK_DORGBR_P_MM = INT( DUM(1) )
*
            CALL DORMBR( 'P', 'R', 'T', M, M, M, DUM(1), M,
     $                   DUM(1), DUM(1), M, DUM(1), -1, IERR )
            LWORK_DORMBR_PRT_MM = INT( DUM(1) )
*
            CALL DORMBR( 'P', 'R', 'T', M, N, M, DUM(1), M,
     $                   DUM(1), DUM(1), M, DUM(1), -1, IERR )
            LWORK_DORMBR_PRT_MN = INT( DUM(1) )
*
            CALL DORMBR( 'P', 'R', 'T', N, N, M, DUM(1), N,
     $                   DUM(1), DUM(1), N, DUM(1), -1, IERR )
            LWORK_DORMBR_PRT_NN = INT( DUM(1) )
*
            CALL DORMBR( 'Q', 'L', 'N', M, M, M, DUM(1), M,
     $                   DUM(1), DUM(1), M, DUM(1), -1, IERR )
            LWORK_DORMBR_QLN_MM = INT( DUM(1) )
*
            IF( N.GE.MNTHR ) THEN
               IF( WNTQN ) THEN
*
*                 Path 1t (N >> M, JOBZ='N')
*
                  WRKBL = M + LWORK_DGELQF_MN
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
                  MAXWRK = MAX( WRKBL, BDSPAC + M )
                  MINWRK = BDSPAC + M
               ELSE IF( WNTQO ) THEN
*
*                 Path 2t (N >> M, JOBZ='O')
*
                  WRKBL = M + LWORK_DGELQF_MN
                  WRKBL = MAX( WRKBL,   M + LWORK_DORGLQ_MN )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
                  WRKBL = MAX( WRKBL, 3*M + BDSPAC )
                  MAXWRK = WRKBL + 2*M*M
                  MINWRK = BDSPAC + 2*M*M + 3*M
               ELSE IF( WNTQS ) THEN
*
*                 Path 3t (N >> M, JOBZ='S')
*
                  WRKBL = M + LWORK_DGELQF_MN
                  WRKBL = MAX( WRKBL,   M + LWORK_DORGLQ_MN )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
                  WRKBL = MAX( WRKBL, 3*M + BDSPAC )
                  MAXWRK = WRKBL + M*M
                  MINWRK = BDSPAC + M*M + 3*M
               ELSE IF( WNTQA ) THEN
*
*                 Path 4t (N >> M, JOBZ='A')
*
                  WRKBL = M + LWORK_DGELQF_MN
                  WRKBL = MAX( WRKBL,   M + LWORK_DORGLQ_NN )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
                  WRKBL = MAX( WRKBL, 3*M + BDSPAC )
                  MAXWRK = WRKBL + M*M
                  MINWRK = M*M + MAX( 3*M + BDSPAC, M + N )
               END IF
            ELSE
*
*              Path 5t (N > M, but not much larger)
*
               WRKBL = 3*M + LWORK_DGEBRD_MN
               IF( WNTQN ) THEN
*                 Path 5tn (N > M, jobz='N')
                  MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
                  MINWRK = 3*M + MAX( N, BDSPAC )
               ELSE IF( WNTQO ) THEN
*                 Path 5to (N > M, jobz='O')
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MN )
                  WRKBL = MAX( WRKBL, 3*M + BDSPAC )
                  MAXWRK = WRKBL + M*N
                  MINWRK = 3*M + MAX( N, M*M + BDSPAC )
               ELSE IF( WNTQS ) THEN
*                 Path 5ts (N > M, jobz='S')
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MN )
                  MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
                  MINWRK = 3*M + MAX( N, BDSPAC )
               ELSE IF( WNTQA ) THEN
*                 Path 5ta (N > M, jobz='A')
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
                  WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_NN )
                  MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
                  MINWRK = 3*M + MAX( N, BDSPAC )
               END IF
            END IF
         END IF

         MAXWRK = MAX( MAXWRK, MINWRK )
         WORK( 1 ) = MAXWRK
*
         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
            INFO = -12
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DGESDD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         RETURN
      END IF
*
*     Get machine constants
*
      EPS = DLAMCH( 'P' )
      SMLNUM = SQRT( DLAMCH( 'S' ) ) / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = DLANGE( 'M', M, N, A, LDA, DUM )
      ISCL = 0
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
         ISCL = 1
         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, IERR )
      ELSE IF( ANRM.GT.BIGNUM ) THEN
         ISCL = 1
         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, IERR )
      END IF
*
      IF( M.GE.N ) THEN
*
*        A has at least as many rows as columns. If A has sufficiently
*        more rows than columns, first reduce using the QR
*        decomposition (if sufficient workspace available)
*
         IF( M.GE.MNTHR ) THEN
*
            IF( WNTQN ) THEN
*
*              Path 1 (M >> N, JOBZ='N')
*              No singular vectors to be computed
*
               ITAU = 1
               NWORK = ITAU + N
*
*              Compute A=Q*R
*              Workspace: need   N [tau] + N    [work]
*              Workspace: prefer N [tau] + N*NB [work]
*
               CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Zero out below R
*
               CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA )
               IE = 1
               ITAUQ = IE + N
               ITAUP = ITAUQ + N
               NWORK = ITAUP + N
*
*              Bidiagonalize R in A
*              Workspace: need   3*N [e, tauq, taup] + N      [work]
*              Workspace: prefer 3*N [e, tauq, taup] + 2*N*NB [work]
*
               CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
     $                      WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
     $                      IERR )
               NWORK = IE + N
*
*              Perform bidiagonal SVD, computing singular values only
*              Workspace: need   N [e] + BDSPAC
*
               CALL DBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM, 1,
     $                      DUM, IDUM, WORK( NWORK ), IWORK, INFO )
*
            ELSE IF( WNTQO ) THEN
*
*              Path 2 (M >> N, JOBZ = 'O')
*              N left singular vectors to be overwritten on A and
*              N right singular vectors to be computed in VT
*
               IR = 1
*
*              WORK(IR) is LDWRKR by N
*
               IF( LWORK .GE. LDA*N + N*N + 3*N + BDSPAC ) THEN
                  LDWRKR = LDA
               ELSE
                  LDWRKR = ( LWORK - N*N - 3*N - BDSPAC ) / N
               END IF
               ITAU = IR + LDWRKR*N
               NWORK = ITAU + N
*
*              Compute A=Q*R
*              Workspace: need   N*N [R] + N [tau] + N    [work]
*              Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
               CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Copy R to WORK(IR), zeroing out below it
*
               CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
               CALL DLASET( 'L', N - 1, N - 1, ZERO, ZERO, WORK(IR+1),
     $                      LDWRKR )
*
*              Generate Q in A
*              Workspace: need   N*N [R] + N [tau] + N    [work]
*              Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
               CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
               IE = ITAU
               ITAUQ = IE + N
               ITAUP = ITAUQ + N
               NWORK = ITAUP + N
*
*              Bidiagonalize R in WORK(IR)
*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N      [work]
*              Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
               CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
     $                      WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              WORK(IU) is N by N
*
               IU = NWORK
               NWORK = IU + N*N
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in WORK(IU) and computing right
*              singular vectors of bidiagonal matrix in VT
*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N*N [U] + BDSPAC
*
               CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ), N,
     $                      VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
     $                      INFO )
*
*              Overwrite WORK(IU) by left singular vectors of R
*              and VT by right singular vectors of R
*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N*N [U] + N    [work]
*              Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR,
     $                      WORK( ITAUQ ), WORK( IU ), N, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ), LDWRKR,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Multiply Q in A by left singular vectors of R in
*              WORK(IU), storing result in WORK(IR) and copying to A
*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N*N [U]
*              Workspace: prefer M*N [R] + 3*N [e, tauq, taup] + N*N [U]
*
               DO 10 I = 1, M, LDWRKR
                  CHUNK = MIN( M - I + 1, LDWRKR )
                  CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ),
     $                        LDA, WORK( IU ), N, ZERO, WORK( IR ),
     $                        LDWRKR )
                  CALL DLACPY( 'F', CHUNK, N, WORK( IR ), LDWRKR,
     $                         A( I, 1 ), LDA )
   10          CONTINUE
*
            ELSE IF( WNTQS ) THEN
*
*              Path 3 (M >> N, JOBZ='S')
*              N left singular vectors to be computed in U and
*              N right singular vectors to be computed in VT
*
               IR = 1
*
*              WORK(IR) is N by N
*
               LDWRKR = N
               ITAU = IR + LDWRKR*N
               NWORK = ITAU + N
*
*              Compute A=Q*R
*              Workspace: need   N*N [R] + N [tau] + N    [work]
*              Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
               CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Copy R to WORK(IR), zeroing out below it
*
               CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
               CALL DLASET( 'L', N - 1, N - 1, ZERO, ZERO, WORK(IR+1),
     $                      LDWRKR )
*
*              Generate Q in A
*              Workspace: need   N*N [R] + N [tau] + N    [work]
*              Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
               CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
               IE = ITAU
               ITAUQ = IE + N
               ITAUP = ITAUQ + N
               NWORK = ITAUP + N
*
*              Bidiagonalize R in WORK(IR)
*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N      [work]
*              Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
               CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
     $                      WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagoal matrix in U and computing right singular
*              vectors of bidiagonal matrix in VT
*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + BDSPAC
*
               CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
     $                      LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
     $                      INFO )
*
*              Overwrite U by left singular vectors of R and VT
*              by right singular vectors of R
*              Workspace: need   N*N [R] + 3*N [e, tauq, taup] + N    [work]
*              Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + N*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
               CALL DORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ), LDWRKR,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Multiply Q in A by left singular vectors of R in
*              WORK(IR), storing result in U
*              Workspace: need   N*N [R]
*
               CALL DLACPY( 'F', N, N, U, LDU, WORK( IR ), LDWRKR )
               CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA, WORK( IR ),
     $                     LDWRKR, ZERO, U, LDU )
*
            ELSE IF( WNTQA ) THEN
*
*              Path 4 (M >> N, JOBZ='A')
*              M left singular vectors to be computed in U and
*              N right singular vectors to be computed in VT
*
               IU = 1
*
*              WORK(IU) is N by N
*
               LDWRKU = N
               ITAU = IU + LDWRKU*N
               NWORK = ITAU + N
*
*              Compute A=Q*R, copying result to U
*              Workspace: need   N*N [U] + N [tau] + N    [work]
*              Workspace: prefer N*N [U] + N [tau] + N*NB [work]
*
               CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
*              Generate Q in U
*              Workspace: need   N*N [U] + N [tau] + M    [work]
*              Workspace: prefer N*N [U] + N [tau] + M*NB [work]
               CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
*              Produce R in A, zeroing out other entries
*
               CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA )
               IE = ITAU
               ITAUQ = IE + N
               ITAUP = ITAUQ + N
               NWORK = ITAUP + N
*
*              Bidiagonalize R in A
*              Workspace: need   N*N [U] + 3*N [e, tauq, taup] + N      [work]
*              Workspace: prefer N*N [U] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
               CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
     $                      WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
     $                      IERR )
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in WORK(IU) and computing right
*              singular vectors of bidiagonal matrix in VT
*              Workspace: need   N*N [U] + 3*N [e, tauq, taup] + BDSPAC
*
               CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ), N,
     $                      VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
     $                      INFO )
*
*              Overwrite WORK(IU) by left singular vectors of R and VT
*              by right singular vectors of R
*              Workspace: need   N*N [U] + 3*N [e, tauq, taup] + N    [work]
*              Workspace: prefer N*N [U] + 3*N [e, tauq, taup] + N*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', N, N, N, A, LDA,
     $                      WORK( ITAUQ ), WORK( IU ), LDWRKU,
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Multiply Q in U by left singular vectors of R in
*              WORK(IU), storing result in A
*              Workspace: need   N*N [U]
*
               CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU, WORK( IU ),
     $                     LDWRKU, ZERO, A, LDA )
*
*              Copy left singular vectors of A from A to U
*
               CALL DLACPY( 'F', M, N, A, LDA, U, LDU )
*
            END IF
*
         ELSE
*
*           M .LT. MNTHR
*
*           Path 5 (M >= N, but not much larger)
*           Reduce to bidiagonal form without QR decomposition
*
            IE = 1
            ITAUQ = IE + N
            ITAUP = ITAUQ + N
            NWORK = ITAUP + N
*
*           Bidiagonalize A
*           Workspace: need   3*N [e, tauq, taup] + M        [work]
*           Workspace: prefer 3*N [e, tauq, taup] + (M+N)*NB [work]
*
            CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
     $                   WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
     $                   IERR )
            IF( WNTQN ) THEN
*
*              Path 5n (M >= N, JOBZ='N')
*              Perform bidiagonal SVD, only computing singular values
*              Workspace: need   3*N [e, tauq, taup] + BDSPAC
*
               CALL DBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM, 1,
     $                      DUM, IDUM, WORK( NWORK ), IWORK, INFO )
            ELSE IF( WNTQO ) THEN
*              Path 5o (M >= N, JOBZ='O')
               IU = NWORK
               IF( LWORK .GE. M*N + 3*N + BDSPAC ) THEN
*
*                 WORK( IU ) is M by N
*
                  LDWRKU = M
                  NWORK = IU + LDWRKU*N
                  CALL DLASET( 'F', M, N, ZERO, ZERO, WORK( IU ),
     $                         LDWRKU )
*                 IR is unused; silence compile warnings
                  IR = -1
               ELSE
*
*                 WORK( IU ) is N by N
*
                  LDWRKU = N
                  NWORK = IU + LDWRKU*N
*
*                 WORK(IR) is LDWRKR by N
*
                  IR = NWORK
                  LDWRKR = ( LWORK - N*N - 3*N ) / N
               END IF
               NWORK = IU + LDWRKU*N
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in WORK(IU) and computing right
*              singular vectors of bidiagonal matrix in VT
*              Workspace: need   3*N [e, tauq, taup] + N*N [U] + BDSPAC
*
               CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ),
     $                      LDWRKU, VT, LDVT, DUM, IDUM, WORK( NWORK ),
     $                      IWORK, INFO )
*
*              Overwrite VT by right singular vectors of A
*              Workspace: need   3*N [e, tauq, taup] + N*N [U] + N    [work]
*              Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
               CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
               IF( LWORK .GE. M*N + 3*N + BDSPAC ) THEN
*
*                 Path 5o-fast
*                 Overwrite WORK(IU) by left singular vectors of A
*                 Workspace: need   3*N [e, tauq, taup] + M*N [U] + N    [work]
*                 Workspace: prefer 3*N [e, tauq, taup] + M*N [U] + N*NB [work]
*
                  CALL DORMBR( 'Q', 'L', 'N', M, N, N, A, LDA,
     $                         WORK( ITAUQ ), WORK( IU ), LDWRKU,
     $                         WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
*                 Copy left singular vectors of A from WORK(IU) to A
*
                  CALL DLACPY( 'F', M, N, WORK( IU ), LDWRKU, A, LDA )
               ELSE
*
*                 Path 5o-slow
*                 Generate Q in A
*                 Workspace: need   3*N [e, tauq, taup] + N*N [U] + N    [work]
*                 Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
                  CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
     $                         WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
*                 Multiply Q in A by left singular vectors of
*                 bidiagonal matrix in WORK(IU), storing result in
*                 WORK(IR) and copying to A
*                 Workspace: need   3*N [e, tauq, taup] + N*N [U] + NB*N [R]
*                 Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + M*N  [R]
*
                  DO 20 I = 1, M, LDWRKR
                     CHUNK = MIN( M - I + 1, LDWRKR )
                     CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ),
     $                           LDA, WORK( IU ), LDWRKU, ZERO,
     $                           WORK( IR ), LDWRKR )
                     CALL DLACPY( 'F', CHUNK, N, WORK( IR ), LDWRKR,
     $                            A( I, 1 ), LDA )
   20             CONTINUE
               END IF
*
            ELSE IF( WNTQS ) THEN
*
*              Path 5s (M >= N, JOBZ='S')
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U and computing right singular
*              vectors of bidiagonal matrix in VT
*              Workspace: need   3*N [e, tauq, taup] + BDSPAC
*
               CALL DLASET( 'F', M, N, ZERO, ZERO, U, LDU )
               CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
     $                      LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
     $                      INFO )
*
*              Overwrite U by left singular vectors of A and VT
*              by right singular vectors of A
*              Workspace: need   3*N [e, tauq, taup] + N    [work]
*              Workspace: prefer 3*N [e, tauq, taup] + N*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', M, N, N, A, LDA,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
            ELSE IF( WNTQA ) THEN
*
*              Path 5a (M >= N, JOBZ='A')
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U and computing right singular
*              vectors of bidiagonal matrix in VT
*              Workspace: need   3*N [e, tauq, taup] + BDSPAC
*
               CALL DLASET( 'F', M, M, ZERO, ZERO, U, LDU )
               CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
     $                      LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
     $                      INFO )
*
*              Set the right corner of U to identity matrix
*
               IF( M.GT.N ) THEN
                  CALL DLASET( 'F', M - N, M - N, ZERO, ONE, U(N+1,N+1),
     $                         LDU )
               END IF
*
*              Overwrite U by left singular vectors of A and VT
*              by right singular vectors of A
*              Workspace: need   3*N [e, tauq, taup] + M    [work]
*              Workspace: prefer 3*N [e, tauq, taup] + M*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', N, N, M, A, LDA,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
            END IF
*
         END IF
*
      ELSE
*
*        A has more columns than rows. If A has sufficiently more
*        columns than rows, first reduce using the LQ decomposition (if
*        sufficient workspace available)
*
         IF( N.GE.MNTHR ) THEN
*
            IF( WNTQN ) THEN
*
*              Path 1t (N >> M, JOBZ='N')
*              No singular vectors to be computed
*
               ITAU = 1
               NWORK = ITAU + M
*
*              Compute A=L*Q
*              Workspace: need   M [tau] + M [work]
*              Workspace: prefer M [tau] + M*NB [work]
*
               CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Zero out above L
*
               CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ), LDA )
               IE = 1
               ITAUQ = IE + M
               ITAUP = ITAUQ + M
               NWORK = ITAUP + M
*
*              Bidiagonalize L in A
*              Workspace: need   3*M [e, tauq, taup] + M      [work]
*              Workspace: prefer 3*M [e, tauq, taup] + 2*M*NB [work]
*
               CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
     $                      WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
     $                      IERR )
               NWORK = IE + M
*
*              Perform bidiagonal SVD, computing singular values only
*              Workspace: need   M [e] + BDSPAC
*
               CALL DBDSDC( 'U', 'N', M, S, WORK( IE ), DUM, 1, DUM, 1,
     $                      DUM, IDUM, WORK( NWORK ), IWORK, INFO )
*
            ELSE IF( WNTQO ) THEN
*
*              Path 2t (N >> M, JOBZ='O')
*              M right singular vectors to be overwritten on A and
*              M left singular vectors to be computed in U
*
               IVT = 1
*
*              WORK(IVT) is M by M
*              WORK(IL)  is M by M; it is later resized to M by chunk for gemm
*
               IL = IVT + M*M
               IF( LWORK .GE. M*N + M*M + 3*M + BDSPAC ) THEN
                  LDWRKL = M
                  CHUNK = N
               ELSE
                  LDWRKL = M
                  CHUNK = ( LWORK - M*M ) / M
               END IF
               ITAU = IL + LDWRKL*M
               NWORK = ITAU + M
*
*              Compute A=L*Q
*              Workspace: need   M*M [VT] + M*M [L] + M [tau] + M    [work]
*              Workspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
*
               CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Copy L to WORK(IL), zeroing about above it
*
               CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL )
               CALL DLASET( 'U', M - 1, M - 1, ZERO, ZERO,
     $                      WORK( IL + LDWRKL ), LDWRKL )
*
*              Generate Q in A
*              Workspace: need   M*M [VT] + M*M [L] + M [tau] + M    [work]
*              Workspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
*
               CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
               IE = ITAU
               ITAUQ = IE + M
               ITAUP = ITAUQ + M
               NWORK = ITAUP + M
*
*              Bidiagonalize L in WORK(IL)
*              Workspace: need   M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M      [work]
*              Workspace: prefer M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
               CALL DGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ),
     $                      WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U, and computing right singular
*              vectors of bidiagonal matrix in WORK(IVT)
*              Workspace: need   M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + BDSPAC
*
               CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU,
     $                      WORK( IVT ), M, DUM, IDUM, WORK( NWORK ),
     $                      IWORK, INFO )
*
*              Overwrite U by left singular vectors of L and WORK(IVT)
*              by right singular vectors of L
*              Workspace: need   M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M    [work]
*              Workspace: prefer M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ), LDWRKL,
     $                      WORK( ITAUP ), WORK( IVT ), M,
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
*              Multiply right singular vectors of L in WORK(IVT) by Q
*              in A, storing result in WORK(IL) and copying to A
*              Workspace: need   M*M [VT] + M*M [L]
*              Workspace: prefer M*M [VT] + M*N [L]
*              At this point, L is resized as M by chunk.
*
               DO 30 I = 1, N, CHUNK
                  BLK = MIN( N - I + 1, CHUNK )
                  CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IVT ), M,
     $                        A( 1, I ), LDA, ZERO, WORK( IL ), LDWRKL )
                  CALL DLACPY( 'F', M, BLK, WORK( IL ), LDWRKL,
     $                         A( 1, I ), LDA )
   30          CONTINUE
*
            ELSE IF( WNTQS ) THEN
*
*              Path 3t (N >> M, JOBZ='S')
*              M right singular vectors to be computed in VT and
*              M left singular vectors to be computed in U
*
               IL = 1
*
*              WORK(IL) is M by M
*
               LDWRKL = M
               ITAU = IL + LDWRKL*M
               NWORK = ITAU + M
*
*              Compute A=L*Q
*              Workspace: need   M*M [L] + M [tau] + M    [work]
*              Workspace: prefer M*M [L] + M [tau] + M*NB [work]
*
               CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Copy L to WORK(IL), zeroing out above it
*
               CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL )
               CALL DLASET( 'U', M - 1, M - 1, ZERO, ZERO,
     $                      WORK( IL + LDWRKL ), LDWRKL )
*
*              Generate Q in A
*              Workspace: need   M*M [L] + M [tau] + M    [work]
*              Workspace: prefer M*M [L] + M [tau] + M*NB [work]
*
               CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
               IE = ITAU
               ITAUQ = IE + M
               ITAUP = ITAUQ + M
               NWORK = ITAUP + M
*
*              Bidiagonalize L in WORK(IU).
*              Workspace: need   M*M [L] + 3*M [e, tauq, taup] + M      [work]
*              Workspace: prefer M*M [L] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
               CALL DGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ),
     $                      WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U and computing right singular
*              vectors of bidiagonal matrix in VT
*              Workspace: need   M*M [L] + 3*M [e, tauq, taup] + BDSPAC
*
               CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU, VT,
     $                      LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
     $                      INFO )
*
*              Overwrite U by left singular vectors of L and VT
*              by right singular vectors of L
*              Workspace: need   M*M [L] + 3*M [e, tauq, taup] + M    [work]
*              Workspace: prefer M*M [L] + 3*M [e, tauq, taup] + M*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ), LDWRKL,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
*              Multiply right singular vectors of L in WORK(IL) by
*              Q in A, storing result in VT
*              Workspace: need   M*M [L]
*
               CALL DLACPY( 'F', M, M, VT, LDVT, WORK( IL ), LDWRKL )
               CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IL ), LDWRKL,
     $                     A, LDA, ZERO, VT, LDVT )
*
            ELSE IF( WNTQA ) THEN
*
*              Path 4t (N >> M, JOBZ='A')
*              N right singular vectors to be computed in VT and
*              M left singular vectors to be computed in U
*
               IVT = 1
*
*              WORK(IVT) is M by M
*
               LDWKVT = M
               ITAU = IVT + LDWKVT*M
               NWORK = ITAU + M
*
*              Compute A=L*Q, copying result to VT
*              Workspace: need   M*M [VT] + M [tau] + M    [work]
*              Workspace: prefer M*M [VT] + M [tau] + M*NB [work]
*
               CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
*              Generate Q in VT
*              Workspace: need   M*M [VT] + M [tau] + N    [work]
*              Workspace: prefer M*M [VT] + M [tau] + N*NB [work]
*
               CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
*              Produce L in A, zeroing out other entries
*
               CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ), LDA )
               IE = ITAU
               ITAUQ = IE + M
               ITAUP = ITAUQ + M
               NWORK = ITAUP + M
*
*              Bidiagonalize L in A
*              Workspace: need   M*M [VT] + 3*M [e, tauq, taup] + M      [work]
*              Workspace: prefer M*M [VT] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
               CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
     $                      WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
     $                      IERR )
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U and computing right singular
*              vectors of bidiagonal matrix in WORK(IVT)
*              Workspace: need   M*M [VT] + 3*M [e, tauq, taup] + BDSPAC
*
               CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU,
     $                      WORK( IVT ), LDWKVT, DUM, IDUM,
     $                      WORK( NWORK ), IWORK, INFO )
*
*              Overwrite U by left singular vectors of L and WORK(IVT)
*              by right singular vectors of L
*              Workspace: need   M*M [VT] + 3*M [e, tauq, taup]+ M    [work]
*              Workspace: prefer M*M [VT] + 3*M [e, tauq, taup]+ M*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', M, M, M, A, LDA,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', M, M, M, A, LDA,
     $                      WORK( ITAUP ), WORK( IVT ), LDWKVT,
     $                      WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
*              Multiply right singular vectors of L in WORK(IVT) by
*              Q in VT, storing result in A
*              Workspace: need   M*M [VT]
*
               CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IVT ), LDWKVT,
     $                     VT, LDVT, ZERO, A, LDA )
*
*              Copy right singular vectors of A from A to VT
*
               CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT )
*
            END IF
*
         ELSE
*
*           N .LT. MNTHR
*
*           Path 5t (N > M, but not much larger)
*           Reduce to bidiagonal form without LQ decomposition
*
            IE = 1
            ITAUQ = IE + M
            ITAUP = ITAUQ + M
            NWORK = ITAUP + M
*
*           Bidiagonalize A
*           Workspace: need   3*M [e, tauq, taup] + N        [work]
*           Workspace: prefer 3*M [e, tauq, taup] + (M+N)*NB [work]
*
            CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
     $                   WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
     $                   IERR )
            IF( WNTQN ) THEN
*
*              Path 5tn (N > M, JOBZ='N')
*              Perform bidiagonal SVD, only computing singular values
*              Workspace: need   3*M [e, tauq, taup] + BDSPAC
*
               CALL DBDSDC( 'L', 'N', M, S, WORK( IE ), DUM, 1, DUM, 1,
     $                      DUM, IDUM, WORK( NWORK ), IWORK, INFO )
            ELSE IF( WNTQO ) THEN
*              Path 5to (N > M, JOBZ='O')
               LDWKVT = M
               IVT = NWORK
               IF( LWORK .GE. M*N + 3*M + BDSPAC ) THEN
*
*                 WORK( IVT ) is M by N
*
                  CALL DLASET( 'F', M, N, ZERO, ZERO, WORK( IVT ),
     $                         LDWKVT )
                  NWORK = IVT + LDWKVT*N
*                 IL is unused; silence compile warnings
                  IL = -1
               ELSE
*
*                 WORK( IVT ) is M by M
*
                  NWORK = IVT + LDWKVT*M
                  IL = NWORK
*
*                 WORK(IL) is M by CHUNK
*
                  CHUNK = ( LWORK - M*M - 3*M ) / M
               END IF
*
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U and computing right singular
*              vectors of bidiagonal matrix in WORK(IVT)
*              Workspace: need   3*M [e, tauq, taup] + M*M [VT] + BDSPAC
*
               CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU,
     $                      WORK( IVT ), LDWKVT, DUM, IDUM,
     $                      WORK( NWORK ), IWORK, INFO )
*
*              Overwrite U by left singular vectors of A
*              Workspace: need   3*M [e, tauq, taup] + M*M [VT] + M    [work]
*              Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
*
               IF( LWORK .GE. M*N + 3*M + BDSPAC ) THEN
*
*                 Path 5to-fast
*                 Overwrite WORK(IVT) by left singular vectors of A
*                 Workspace: need   3*M [e, tauq, taup] + M*N [VT] + M    [work]
*                 Workspace: prefer 3*M [e, tauq, taup] + M*N [VT] + M*NB [work]
*
                  CALL DORMBR( 'P', 'R', 'T', M, N, M, A, LDA,
     $                         WORK( ITAUP ), WORK( IVT ), LDWKVT,
     $                         WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
*                 Copy right singular vectors of A from WORK(IVT) to A
*
                  CALL DLACPY( 'F', M, N, WORK( IVT ), LDWKVT, A, LDA )
               ELSE
*
*                 Path 5to-slow
*                 Generate P**T in A
*                 Workspace: need   3*M [e, tauq, taup] + M*M [VT] + M    [work]
*                 Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*NB [work]
*
                  CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
     $                         WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
*                 Multiply Q in A by right singular vectors of
*                 bidiagonal matrix in WORK(IVT), storing result in
*                 WORK(IL) and copying to A
*                 Workspace: need   3*M [e, tauq, taup] + M*M [VT] + M*NB [L]
*                 Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*N  [L]
*
                  DO 40 I = 1, N, CHUNK
                     BLK = MIN( N - I + 1, CHUNK )
                     CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IVT ),
     $                           LDWKVT, A( 1, I ), LDA, ZERO,
     $                           WORK( IL ), M )
                     CALL DLACPY( 'F', M, BLK, WORK( IL ), M, A( 1, I ),
     $                            LDA )
   40             CONTINUE
               END IF
            ELSE IF( WNTQS ) THEN
*
*              Path 5ts (N > M, JOBZ='S')
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U and computing right singular
*              vectors of bidiagonal matrix in VT
*              Workspace: need   3*M [e, tauq, taup] + BDSPAC
*
               CALL DLASET( 'F', M, N, ZERO, ZERO, VT, LDVT )
               CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT,
     $                      LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
     $                      INFO )
*
*              Overwrite U by left singular vectors of A and VT
*              by right singular vectors of A
*              Workspace: need   3*M [e, tauq, taup] + M    [work]
*              Workspace: prefer 3*M [e, tauq, taup] + M*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', M, N, M, A, LDA,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
            ELSE IF( WNTQA ) THEN
*
*              Path 5ta (N > M, JOBZ='A')
*              Perform bidiagonal SVD, computing left singular vectors
*              of bidiagonal matrix in U and computing right singular
*              vectors of bidiagonal matrix in VT
*              Workspace: need   3*M [e, tauq, taup] + BDSPAC
*
               CALL DLASET( 'F', N, N, ZERO, ZERO, VT, LDVT )
               CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT,
     $                      LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
     $                      INFO )
*
*              Set the right corner of VT to identity matrix
*
               IF( N.GT.M ) THEN
                  CALL DLASET( 'F', N-M, N-M, ZERO, ONE, VT(M+1,M+1),
     $                         LDVT )
               END IF
*
*              Overwrite U by left singular vectors of A and VT
*              by right singular vectors of A
*              Workspace: need   3*M [e, tauq, taup] + N    [work]
*              Workspace: prefer 3*M [e, tauq, taup] + N*NB [work]
*
               CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
     $                      WORK( ITAUQ ), U, LDU, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
               CALL DORMBR( 'P', 'R', 'T', N, N, M, A, LDA,
     $                      WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
     $                      LWORK - NWORK + 1, IERR )
            END IF
*
         END IF
*
      END IF
*
*     Undo scaling if necessary
*
      IF( ISCL.EQ.1 ) THEN
         IF( ANRM.GT.BIGNUM )
     $      CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
     $                   IERR )
         IF( ANRM.LT.SMLNUM )
     $      CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
     $                   IERR )
      END IF
*
*     Return optimal workspace in WORK(1)
*
      WORK( 1 ) = MAXWRK
*
      RETURN
*
*     End of DGESDD
*
      END