dgesdd.f

          SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
         $                   LWORK, IWORK, INFO )
    *
    *  -- LAPACK driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. 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( * )
    *     ..
    *
    *  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)*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 = -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 ..
          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
          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
          INTEGER            ILAENV
          DOUBLE PRECISION   DLAMCH, DLANGE
          EXTERNAL           DLAMCH, DLANGE, ILAENV, 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 needed 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
             IF( M.GE.N .AND. MINMN.GT.0 ) THEN
    *
    *           Compute space needed for DBDSDC
    *
                MNTHR = INT( MINMN*11.0D0 / 6.0D0 )
                IF( WNTQN ) THEN
                   BDSPAC = 7*N
                ELSE
                   BDSPAC = 3*N*N + 4*N
                END IF
                IF( M.GE.MNTHR ) THEN
                   IF( WNTQN ) THEN
    *
    *                 Path 1 (M much larger than N, JOBZ='N')
    *
                      WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1,
         $                    -1 )
                      WRKBL = MAX( WRKBL, 3*N+2*N*
         $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                      MAXWRK = MAX( WRKBL, BDSPAC+N )
                      MINWRK = BDSPAC + N
                   ELSE IF( WNTQO ) THEN
    *
    *                 Path 2 (M much larger than N, JOBZ='O')
    *
                      WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
                      WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
         $                    N, N, -1 ) )
                      WRKBL = MAX( WRKBL, 3*N+2*N*
         $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                      WRKBL = MAX( WRKBL, 3*N+N*
         $                    ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) )
                      WRKBL = MAX( WRKBL, 3*N+N*
         $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
                      WRKBL = MAX( WRKBL, BDSPAC+3*N )
                      MAXWRK = WRKBL + 2*N*N
                      MINWRK = BDSPAC + 2*N*N + 3*N
                   ELSE IF( WNTQS ) THEN
    *
    *                 Path 3 (M much larger than N, JOBZ='S')
    *
                      WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
                      WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
         $                    N, N, -1 ) )
                      WRKBL = MAX( WRKBL, 3*N+2*N*
         $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                      WRKBL = MAX( WRKBL, 3*N+N*
         $                    ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) )
                      WRKBL = MAX( WRKBL, 3*N+N*
         $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
                      WRKBL = MAX( WRKBL, BDSPAC+3*N )
                      MAXWRK = WRKBL + N*N
                      MINWRK = BDSPAC + N*N + 3*N
                   ELSE IF( WNTQA ) THEN
    *
    *                 Path 4 (M much larger than N, JOBZ='A')
    *
                      WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
                      WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'DORGQR', ' ', M,
         $                    M, N, -1 ) )
                      WRKBL = MAX( WRKBL, 3*N+2*N*
         $                    ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
                      WRKBL = MAX( WRKBL, 3*N+N*
         $                    ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) )
                      WRKBL = MAX( WRKBL, 3*N+N*
         $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
                      WRKBL = MAX( WRKBL, BDSPAC+3*N )
                      MAXWRK = WRKBL + N*N
                      MINWRK = BDSPAC + N*N + 3*N
                   END IF
                ELSE
    *
    *              Path 5 (M at least N, but not much larger)
    *
                   WRKBL = 3*N + ( M+N )*ILAENV( 1, 'DGEBRD', ' ', M, N, -1,
         $                 -1 )
                   IF( WNTQN ) THEN
                      MAXWRK = MAX( WRKBL, BDSPAC+3*N )
                      MINWRK = 3*N + MAX( M, BDSPAC )
                   ELSE IF( WNTQO ) THEN
                      WRKBL = MAX( WRKBL, 3*N+N*
         $                    ILAENV( 1, 'DORMBR', 'QLN', M, N, N, -1 ) )
                      WRKBL = MAX( WRKBL, 3*N+N*
         $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
                      WRKBL = MAX( WRKBL, BDSPAC+3*N )
                      MAXWRK = WRKBL + M*N
                      MINWRK = 3*N + MAX( M, N*N+BDSPAC )
                   ELSE IF( WNTQS ) THEN
                      WRKBL = MAX( WRKBL, 3*N+N*
         $                    ILAENV( 1, 'DORMBR', 'QLN', M, N, N, -1 ) )
                      WRKBL = MAX( WRKBL, 3*N+N*
         $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
                      MAXWRK = MAX( WRKBL, BDSPAC+3*N )
                      MINWRK = 3*N + MAX( M, BDSPAC )
                   ELSE IF( WNTQA ) THEN
                      WRKBL = MAX( WRKBL, 3*N+M*
         $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) )
                      WRKBL = MAX( WRKBL, 3*N+N*
         $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
                      MAXWRK = MAX( MAXWRK, BDSPAC+3*N )
                      MINWRK = 3*N + MAX( M, BDSPAC )
                   END IF
                END IF
             ELSE IF( MINMN.GT.0 ) THEN
    *
    *           Compute space needed for DBDSDC
    *
                MNTHR = INT( MINMN*11.0D0 / 6.0D0 )
                IF( WNTQN ) THEN
                   BDSPAC = 7*M
                ELSE
                   BDSPAC = 3*M*M + 4*M
                END IF
                IF( N.GE.MNTHR ) THEN
                   IF( WNTQN ) THEN
    *
    *                 Path 1t (N much larger than M, JOBZ='N')
    *
                      WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1,
         $                    -1 )
                      WRKBL = MAX( WRKBL, 3*M+2*M*
         $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                      MAXWRK = MAX( WRKBL, BDSPAC+M )
                      MINWRK = BDSPAC + M
                   ELSE IF( WNTQO ) THEN
    *
    *                 Path 2t (N much larger than M, JOBZ='O')
    *
                      WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
                      WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
         $                    N, M, -1 ) )
                      WRKBL = MAX( WRKBL, 3*M+2*M*
         $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                      WRKBL = MAX( WRKBL, 3*M+M*
         $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, M, -1 ) )
                      WRKBL = MAX( WRKBL, 3*M+M*
         $                    ILAENV( 1, 'DORMBR', 'PRT', M, M, M, -1 ) )
                      WRKBL = MAX( WRKBL, BDSPAC+3*M )
                      MAXWRK = WRKBL + 2*M*M
                      MINWRK = BDSPAC + 2*M*M + 3*M
                   ELSE IF( WNTQS ) THEN
    *
    *                 Path 3t (N much larger than M, JOBZ='S')
    *
                      WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
                      WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
         $                    N, M, -1 ) )
                      WRKBL = MAX( WRKBL, 3*M+2*M*
         $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                      WRKBL = MAX( WRKBL, 3*M+M*
         $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, M, -1 ) )
                      WRKBL = MAX( WRKBL, 3*M+M*
         $                    ILAENV( 1, 'DORMBR', 'PRT', M, M, M, -1 ) )
                      WRKBL = MAX( WRKBL, BDSPAC+3*M )
                      MAXWRK = WRKBL + M*M
                      MINWRK = BDSPAC + M*M + 3*M
                   ELSE IF( WNTQA ) THEN
    *
    *                 Path 4t (N much larger than M, JOBZ='A')
    *
                      WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
                      WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'DORGLQ', ' ', N,
         $                    N, M, -1 ) )
                      WRKBL = MAX( WRKBL, 3*M+2*M*
         $                    ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                      WRKBL = MAX( WRKBL, 3*M+M*
         $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, M, -1 ) )
                      WRKBL = MAX( WRKBL, 3*M+M*
         $                    ILAENV( 1, 'DORMBR', 'PRT', M, M, M, -1 ) )
                      WRKBL = MAX( WRKBL, BDSPAC+3*M )
                      MAXWRK = WRKBL + M*M
                      MINWRK = BDSPAC + M*M + 3*M
                   END IF
                ELSE
    *
    *              Path 5t (N greater than M, but not much larger)
    *
                   WRKBL = 3*M + ( M+N )*ILAENV( 1, 'DGEBRD', ' ', M, N, -1,
         $                 -1 )
                   IF( WNTQN ) THEN
                      MAXWRK = MAX( WRKBL, BDSPAC+3*M )
                      MINWRK = 3*M + MAX( N, BDSPAC )
                   ELSE IF( WNTQO ) THEN
                      WRKBL = MAX( WRKBL, 3*M+M*
         $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) )
                      WRKBL = MAX( WRKBL, 3*M+M*
         $                    ILAENV( 1, 'DORMBR', 'PRT', M, N, M, -1 ) )
                      WRKBL = MAX( WRKBL, BDSPAC+3*M )
                      MAXWRK = WRKBL + M*N
                      MINWRK = 3*M + MAX( N, M*M+BDSPAC )
                   ELSE IF( WNTQS ) THEN
                      WRKBL = MAX( WRKBL, 3*M+M*
         $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) )
                      WRKBL = MAX( WRKBL, 3*M+M*
         $                    ILAENV( 1, 'DORMBR', 'PRT', M, N, M, -1 ) )
                      MAXWRK = MAX( WRKBL, BDSPAC+3*M )
                      MINWRK = 3*M + MAX( N, BDSPAC )
                   ELSE IF( WNTQA ) THEN
                      WRKBL = MAX( WRKBL, 3*M+M*
         $                    ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) )
                      WRKBL = MAX( WRKBL, 3*M+M*
         $                    ILAENV( 1, 'DORMBR', 'PRT', N, N, M, -1 ) )
                      MAXWRK = MAX( WRKBL, BDSPAC+3*M )
                      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 much larger than N, JOBZ='N')
    *              No singular vectors to be computed
    *
                   ITAU = 1
                   NWORK = ITAU + N
    *
    *              Compute A=Q*R
    *              (Workspace: need 2*N, prefer N+N*NB)
    *
                   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 4*N, prefer 3*N+2*N*NB)
    *
                   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+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 much larger than 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+2*N, prefer N*N+N+N*NB)
    *
                   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+2*N, prefer N*N+N+N*NB)
    *
                   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 VT, copying result to WORK(IR)
    *              (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
    *
                   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*N+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 2*N*N+3*N, prefer 2*N*N+2*N+N*NB)
    *
                   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 2*N*N, prefer N*N+M*N)
    *
                   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 much larger than 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+2*N, prefer N*N+N+N*NB)
    *
                   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+2*N, prefer N*N+N+N*NB)
    *
                   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+4*N, prefer N*N+3*N+2*N*NB)
    *
                   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+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+3*N, prefer N*N+2*N+N*NB)
    *
                   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)
    *
                   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 much larger than 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+2*N, prefer N*N+N+N*NB)
    *
                   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+2*N, prefer N*N+N+N*NB)
                   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+4*N, prefer N*N+3*N+2*N*NB)
    *
                   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*N+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+3*N, prefer N*N+2*N+N*NB)
    *
                   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)
    *
                   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 at least 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+M, prefer 3*N+(M+N)*NB)
    *
                CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
         $                   WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
         $                   IERR )
                IF( WNTQN ) THEN
    *
    *              Perform bidiagonal SVD, only computing singular values
    *              (Workspace: need N+BDSPAC)
    *
                   CALL DBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM, 1,
         $                      DUM, IDUM, WORK( NWORK ), IWORK, INFO )
                ELSE IF( WNTQO ) THEN
                   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 )
                   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 N+N*N+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 N*N+2*N, prefer N*N+N+N*NB)
    *
                   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
    *
    *                 Overwrite WORK(IU) by left singular vectors of A
    *                 (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
    *
                      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
    *
    *                 Generate Q in A
    *                 (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
    *
                      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 2*N*N, prefer N*N+M*N)
    *
                      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
    *
    *              Perform bidiagonal SVD, computing left singular vectors
    *              of bidiagonal matrix in U and computing right singular
    *              vectors of bidiagonal matrix in VT
    *              (Workspace: need N+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, prefer 2*N+N*NB)
    *
                   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
    *
    *              Perform bidiagonal SVD, computing left singular vectors
    *              of bidiagonal matrix in U and computing right singular
    *              vectors of bidiagonal matrix in VT
    *              (Workspace: need N+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 N*N+2*N+M, prefer N*N+2*N+M*NB)
    *
                   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 much larger than M, JOBZ='N')
    *              No singular vectors to be computed
    *
                   ITAU = 1
                   NWORK = ITAU + M
    *
    *              Compute A=L*Q
    *              (Workspace: need 2*M, prefer M+M*NB)
    *
                   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 4*M, prefer 3*M+2*M*NB)
    *
                   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+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 much larger than M, JOBZ='O')
    *              M right singular vectors to be overwritten on A and
    *              M left singular vectors to be computed in U
    *
                   IVT = 1
    *
    *              IVT is M by M
    *
                   IL = IVT + M*M
                   IF( LWORK.GE.M*N+M*M+3*M+BDSPAC ) THEN
    *
    *                 WORK(IL) is M by N
    *
                      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+2*M, prefer M*M+M+M*NB)
    *
                   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+2*M, prefer M*M+M+M*NB)
    *
                   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+4*M, prefer M*M+3*M+2*M*NB)
    *
                   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*M+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 2*M*M+3*M, prefer 2*M*M+2*M+M*NB)
    *
                   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 2*M*M, prefer M*M+M*N)
    *
                   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 much larger than 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+2*M, prefer M*M+M+M*NB)
    *
                   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+2*M, prefer M*M+M+M*NB)
    *
                   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), copying result to U
    *              (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
    *
                   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+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+3*M, prefer M*M+2*M+M*NB)
    *
                   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)
    *
                   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 much larger than 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+2*M, prefer M*M+M+M*NB)
    *
                   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+2*M, prefer M*M+M+M*NB)
    *
                   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+4*M, prefer M*M+3*M+2*M*NB)
    *
                   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*M+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+3*M, prefer M*M+2*M+M*NB)
    *
                   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)
    *
                   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 greater than 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+N, prefer 3*M+(M+N)*NB)
    *
                CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
         $                   WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
         $                   IERR )
                IF( WNTQN ) THEN
    *
    *              Perform bidiagonal SVD, only computing singular values
    *              (Workspace: need M+BDSPAC)
    *
                   CALL DBDSDC( 'L', 'N', M, S, WORK( IE ), DUM, 1, DUM, 1,
         $                      DUM, IDUM, WORK( NWORK ), IWORK, INFO )
                ELSE IF( WNTQO ) THEN
                   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
                   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 M*M+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 M*M+2*M, prefer M*M+M+M*NB)
    *
                   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
    *
    *                 Overwrite WORK(IVT) by left singular vectors of A
    *                 (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
    *
                      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
    *
    *                 Generate P**T in A
    *                 (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
    *
                      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 2*M*M, prefer M*M+M*N)
    *
                      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
    *
    *              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+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, prefer 2*M+M*NB)
    *
                   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
    *
    *              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+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 2*M+N, prefer 2*M+N*NB)
    *
                   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