dgelss.f

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*> \brief <b> DGELSS solves overdetermined or underdetermined systems for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
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*            http://www.netlib.org/lapack/explore-html/
*
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*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
*                          WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGELSS computes the minimum norm solution to a real linear least
*> squares problem:
*>
*> Minimize 2-norm(| b - A*x |).
*>
*> using the singular value decomposition (SVD) of A. A is an M-by-N
*> matrix which may be rank-deficient.
*>
*> Several right hand side vectors b and solution vectors x can be
*> handled in a single call; they are stored as the columns of the
*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix
*> X.
*>
*> The effective rank of A is determined by treating as zero those
*> singular values which are less than RCOND times the largest singular
*> value.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices B and X. NRHS >= 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, the first min(m,n) rows of A are overwritten with
*>          its right singular vectors, stored rowwise.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          On entry, the M-by-NRHS right hand side matrix B.
*>          On exit, B is overwritten by the N-by-NRHS solution
*>          matrix X.  If m >= n and RANK = n, the residual
*>          sum-of-squares for the solution in the i-th column is given
*>          by the sum of squares of elements n+1:m in that column.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,max(M,N)).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (min(M,N))
*>          The singular values of A in decreasing order.
*>          The condition number of A in the 2-norm = S(1)/S(min(m,n)).
*> \endverbatim
*>
*> \param[in] RCOND
*> \verbatim
*>          RCOND is DOUBLE PRECISION
*>          RCOND is used to determine the effective rank of A.
*>          Singular values S(i) <= RCOND*S(1) are treated as zero.
*>          If RCOND < 0, machine precision is used instead.
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*>          RANK is INTEGER
*>          The effective rank of A, i.e., the number of singular values
*>          which are greater than RCOND*S(1).
*> \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, and also:
*>          LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )
*>          For good performance, LWORK should generally be larger.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  the algorithm for computing the SVD failed to converge;
*>                if INFO = i, i off-diagonal elements of an intermediate
*>                bidiagonal form did not converge to zero.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleGEsolve
*
*  =====================================================================
      SUBROUTINE dgelss( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
     $                   WORK, LWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      parameter( zero = 0.0d+0, one = 1.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            BDSPAC, BL, CHUNK, I, IASCL, IBSCL, IE, IL,
     $                   itau, itaup, itauq, iwork, ldwork, maxmn,
     $                   maxwrk, minmn, minwrk, mm, mnthr
      INTEGER            LWORK_DGEQRF, LWORK_DORMQR, LWORK_DGEBRD,
     $                   lwork_dormbr, lwork_dorgbr, lwork_dormlq,
     $                   lwork_dgelqf
      DOUBLE PRECISION   ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM, THR
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   DUM( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           dbdsqr, dcopy, dgebrd, dgelqf, dgemm, dgemv,
     $                   dgeqrf, dlabad, dlacpy, dlascl, dlaset, dorgbr,
     $                   dormbr, dormlq, dormqr, drscl, xerbla
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           ilaenv, dlamch, dlange
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      info = 0
      minmn = min( m, n )
      maxmn = max( m, n )
      lquery = ( lwork.EQ.-1 )
      IF( m.LT.0 ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( nrhs.LT.0 ) THEN
         info = -3
      ELSE IF( lda.LT.max( 1, m ) ) THEN
         info = -5
      ELSE IF( ldb.LT.max( 1, maxmn ) ) THEN
         info = -7
      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( minmn.GT.0 ) THEN
            mm = m
            mnthr = ilaenv( 6, 'DGELSS', ' ', m, n, nrhs, -1 )
            IF( m.GE.n .AND. m.GE.mnthr ) THEN
*
*              Path 1a - overdetermined, with many more rows than
*                        columns
*
*              Compute space needed for DGEQRF
               CALL dgeqrf( m, n, a, lda, dum(1), dum(1), -1, info )
               lwork_dgeqrf=dum(1)
*              Compute space needed for DORMQR
               CALL dormqr( 'L', 'T', m, nrhs, n, a, lda, dum(1), b,
     $                   ldb, dum(1), -1, info )
               lwork_dormqr=dum(1)
               mm = n
               maxwrk = max( maxwrk, n + lwork_dgeqrf )
               maxwrk = max( maxwrk, n + lwork_dormqr )
            END IF
            IF( m.GE.n ) THEN
*
*              Path 1 - overdetermined or exactly determined
*
*              Compute workspace needed for DBDSQR
*
               bdspac = max( 1, 5*n )
*              Compute space needed for DGEBRD
               CALL dgebrd( mm, n, a, lda, s, dum(1), dum(1),
     $                      dum(1), dum(1), -1, info )
               lwork_dgebrd=dum(1)
*              Compute space needed for DORMBR
               CALL dormbr( 'Q', 'L', 'T', mm, nrhs, n, a, lda, dum(1),
     $                b, ldb, dum(1), -1, info )
               lwork_dormbr=dum(1)
*              Compute space needed for DORGBR
               CALL dorgbr( 'P', n, n, n, a, lda, dum(1),
     $                   dum(1), -1, info )
               lwork_dorgbr=dum(1)
*              Compute total workspace needed
               maxwrk = max( maxwrk, 3*n + lwork_dgebrd )
               maxwrk = max( maxwrk, 3*n + lwork_dormbr )
               maxwrk = max( maxwrk, 3*n + lwork_dorgbr )
               maxwrk = max( maxwrk, bdspac )
               maxwrk = max( maxwrk, n*nrhs )
               minwrk = max( 3*n + mm, 3*n + nrhs, bdspac )
               maxwrk = max( minwrk, maxwrk )
            END IF
            IF( n.GT.m ) THEN
*
*              Compute workspace needed for DBDSQR
*
               bdspac = max( 1, 5*m )
               minwrk = max( 3*m+nrhs, 3*m+n, bdspac )
               IF( n.GE.mnthr ) THEN
*
*                 Path 2a - underdetermined, with many more columns
*                 than rows
*
*                 Compute space needed for DGELQF
                  CALL dgelqf( m, n, a, lda, dum(1), dum(1),
     $                -1, info )
                  lwork_dgelqf=dum(1)
*                 Compute space needed for DGEBRD
                  CALL dgebrd( m, m, a, lda, s, dum(1), dum(1),
     $                      dum(1), dum(1), -1, info )
                  lwork_dgebrd=dum(1)
*                 Compute space needed for DORMBR
                  CALL dormbr( 'Q', 'L', 'T', m, nrhs, n, a, lda,
     $                dum(1), b, ldb, dum(1), -1, info )
                  lwork_dormbr=dum(1)
*                 Compute space needed for DORGBR
                  CALL dorgbr( 'P', m, m, m, a, lda, dum(1),
     $                   dum(1), -1, info )
                  lwork_dorgbr=dum(1)
*                 Compute space needed for DORMLQ
                  CALL dormlq( 'L', 'T', n, nrhs, m, a, lda, dum(1),
     $                 b, ldb, dum(1), -1, info )
                  lwork_dormlq=dum(1)
*                 Compute total workspace needed
                  maxwrk = m + lwork_dgelqf
                  maxwrk = max( maxwrk, m*m + 4*m + lwork_dgebrd )
                  maxwrk = max( maxwrk, m*m + 4*m + lwork_dormbr )
                  maxwrk = max( maxwrk, m*m + 4*m + lwork_dorgbr )
                  maxwrk = max( maxwrk, m*m + m + bdspac )
                  IF( nrhs.GT.1 ) THEN
                     maxwrk = max( maxwrk, m*m + m + m*nrhs )
                  ELSE
                     maxwrk = max( maxwrk, m*m + 2*m )
                  END IF
                  maxwrk = max( maxwrk, m + lwork_dormlq )
               ELSE
*
*                 Path 2 - underdetermined
*
*                 Compute space needed for DGEBRD
                  CALL dgebrd( m, n, a, lda, s, dum(1), dum(1),
     $                      dum(1), dum(1), -1, info )
                  lwork_dgebrd=dum(1)
*                 Compute space needed for DORMBR
                  CALL dormbr( 'Q', 'L', 'T', m, nrhs, m, a, lda,
     $                dum(1), b, ldb, dum(1), -1, info )
                  lwork_dormbr=dum(1)
*                 Compute space needed for DORGBR
                  CALL dorgbr( 'P', m, n, m, a, lda, dum(1),
     $                   dum(1), -1, info )
                  lwork_dorgbr=dum(1)
                  maxwrk = 3*m + lwork_dgebrd
                  maxwrk = max( maxwrk, 3*m + lwork_dormbr )
                  maxwrk = max( maxwrk, 3*m + lwork_dorgbr )
                  maxwrk = max( maxwrk, bdspac )
                  maxwrk = max( maxwrk, n*nrhs )
               END IF
            END IF
            maxwrk = max( minwrk, maxwrk )
         END IF
         work( 1 ) = maxwrk
*
         IF( lwork.LT.minwrk .AND. .NOT.lquery )
     $      info = -12
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'DGELSS', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( m.EQ.0 .OR. n.EQ.0 ) THEN
         rank = 0
         RETURN
      END IF
*
*     Get machine parameters
*
      eps = dlamch( 'P' )
      sfmin = dlamch( 'S' )
      smlnum = sfmin / eps
      bignum = one / smlnum
      CALL dlabad( smlnum, bignum )
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      anrm = dlange( 'M', m, n, a, lda, work )
      iascl = 0
      IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL dlascl( 'G', 0, 0, anrm, smlnum, m, n, a, lda, info )
         iascl = 1
      ELSE IF( anrm.GT.bignum ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL dlascl( 'G', 0, 0, anrm, bignum, m, n, a, lda, info )
         iascl = 2
      ELSE IF( anrm.EQ.zero ) THEN
*
*        Matrix all zero. Return zero solution.
*
         CALL dlaset( 'F', max( m, n ), nrhs, zero, zero, b, ldb )
         CALL dlaset( 'F', minmn, 1, zero, zero, s, minmn )
         rank = 0
         GO TO 70
      END IF
*
*     Scale B if max element outside range [SMLNUM,BIGNUM]
*
      bnrm = dlange( 'M', m, nrhs, b, ldb, work )
      ibscl = 0
      IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL dlascl( 'G', 0, 0, bnrm, smlnum, m, nrhs, b, ldb, info )
         ibscl = 1
      ELSE IF( bnrm.GT.bignum ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL dlascl( 'G', 0, 0, bnrm, bignum, m, nrhs, b, ldb, info )
         ibscl = 2
      END IF
*
*     Overdetermined case
*
      IF( m.GE.n ) THEN
*
*        Path 1 - overdetermined or exactly determined
*
         mm = m
         IF( m.GE.mnthr ) THEN
*
*           Path 1a - overdetermined, with many more rows than columns
*
            mm = n
            itau = 1
            iwork = itau + n
*
*           Compute A=Q*R
*           (Workspace: need 2*N, prefer N+N*NB)
*
            CALL dgeqrf( m, n, a, lda, work( itau ), work( iwork ),
     $                   lwork-iwork+1, info )
*
*           Multiply B by transpose(Q)
*           (Workspace: need N+NRHS, prefer N+NRHS*NB)
*
            CALL dormqr( 'L', 'T', m, nrhs, n, a, lda, work( itau ), b,
     $                   ldb, work( iwork ), lwork-iwork+1, info )
*
*           Zero out below R
*
            IF( n.GT.1 )
     $         CALL dlaset( 'L', n-1, n-1, zero, zero, a( 2, 1 ), lda )
         END IF
*
         ie = 1
         itauq = ie + n
         itaup = itauq + n
         iwork = itaup + n
*
*        Bidiagonalize R in A
*        (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
*
         CALL dgebrd( mm, n, a, lda, s, work( ie ), work( itauq ),
     $                work( itaup ), work( iwork ), lwork-iwork+1,
     $                info )
*
*        Multiply B by transpose of left bidiagonalizing vectors of R
*        (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
*
         CALL dormbr( 'Q', 'L', 'T', mm, nrhs, n, a, lda, work( itauq ),
     $                b, ldb, work( iwork ), lwork-iwork+1, info )
*
*        Generate right bidiagonalizing vectors of R in A
*        (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
*
         CALL dorgbr( 'P', n, n, n, a, lda, work( itaup ),
     $                work( iwork ), lwork-iwork+1, info )
         iwork = ie + n
*
*        Perform bidiagonal QR iteration
*          multiply B by transpose of left singular vectors
*          compute right singular vectors in A
*        (Workspace: need BDSPAC)
*
         CALL dbdsqr( 'U', n, n, 0, nrhs, s, work( ie ), a, lda, dum,
     $                1, b, ldb, work( iwork ), info )
         IF( info.NE.0 )
     $      GO TO 70
*
*        Multiply B by reciprocals of singular values
*
         thr = max( rcond*s( 1 ), sfmin )
         IF( rcond.LT.zero )
     $      thr = max( eps*s( 1 ), sfmin )
         rank = 0
         DO 10 i = 1, n
            IF( s( i ).GT.thr ) THEN
               CALL drscl( nrhs, s( i ), b( i, 1 ), ldb )
               rank = rank + 1
            ELSE
               CALL dlaset( 'F', 1, nrhs, zero, zero, b( i, 1 ), ldb )
            END IF
   10    CONTINUE
*
*        Multiply B by right singular vectors
*        (Workspace: need N, prefer N*NRHS)
*
         IF( lwork.GE.ldb*nrhs .AND. nrhs.GT.1 ) THEN
            CALL dgemm( 'T', 'N', n, nrhs, n, one, a, lda, b, ldb, zero,
     $                  work, ldb )
            CALL dlacpy( 'G', n, nrhs, work, ldb, b, ldb )
         ELSE IF( nrhs.GT.1 ) THEN
            chunk = lwork / n
            DO 20 i = 1, nrhs, chunk
               bl = min( nrhs-i+1, chunk )
               CALL dgemm( 'T', 'N', n, bl, n, one, a, lda, b( 1, i ),
     $                     ldb, zero, work, n )
               CALL dlacpy( 'G', n, bl, work, n, b( 1, i ), ldb )
   20       CONTINUE
         ELSE
            CALL dgemv( 'T', n, n, one, a, lda, b, 1, zero, work, 1 )
            CALL dcopy( n, work, 1, b, 1 )
         END IF
*
      ELSE IF( n.GE.mnthr .AND. lwork.GE.4*m+m*m+
     $         max( m, 2*m-4, nrhs, n-3*m ) ) THEN
*
*        Path 2a - underdetermined, with many more columns than rows
*        and sufficient workspace for an efficient algorithm
*
         ldwork = m
         IF( lwork.GE.max( 4*m+m*lda+max( m, 2*m-4, nrhs, n-3*m ),
     $       m*lda+m+m*nrhs ) )ldwork = lda
         itau = 1
         iwork = m + 1
*
*        Compute A=L*Q
*        (Workspace: need 2*M, prefer M+M*NB)
*
         CALL dgelqf( m, n, a, lda, work( itau ), work( iwork ),
     $                lwork-iwork+1, info )
         il = iwork
*
*        Copy L to WORK(IL), zeroing out above it
*
         CALL dlacpy( 'L', m, m, a, lda, work( il ), ldwork )
         CALL dlaset( 'U', m-1, m-1, zero, zero, work( il+ldwork ),
     $                ldwork )
         ie = il + ldwork*m
         itauq = ie + m
         itaup = itauq + m
         iwork = itaup + m
*
*        Bidiagonalize L in WORK(IL)
*        (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
*
         CALL dgebrd( m, m, work( il ), ldwork, s, work( ie ),
     $                work( itauq ), work( itaup ), work( iwork ),
     $                lwork-iwork+1, info )
*
*        Multiply B by transpose of left bidiagonalizing vectors of L
*        (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
*
         CALL dormbr( 'Q', 'L', 'T', m, nrhs, m, work( il ), ldwork,
     $                work( itauq ), b, ldb, work( iwork ),
     $                lwork-iwork+1, info )
*
*        Generate right bidiagonalizing vectors of R in WORK(IL)
*        (Workspace: need M*M+5*M-1, prefer M*M+4*M+(M-1)*NB)
*
         CALL dorgbr( 'P', m, m, m, work( il ), ldwork, work( itaup ),
     $                work( iwork ), lwork-iwork+1, info )
         iwork = ie + m
*
*        Perform bidiagonal QR iteration,
*           computing right singular vectors of L in WORK(IL) and
*           multiplying B by transpose of left singular vectors
*        (Workspace: need M*M+M+BDSPAC)
*
         CALL dbdsqr( 'U', m, m, 0, nrhs, s, work( ie ), work( il ),
     $                ldwork, a, lda, b, ldb, work( iwork ), info )
         IF( info.NE.0 )
     $      GO TO 70
*
*        Multiply B by reciprocals of singular values
*
         thr = max( rcond*s( 1 ), sfmin )
         IF( rcond.LT.zero )
     $      thr = max( eps*s( 1 ), sfmin )
         rank = 0
         DO 30 i = 1, m
            IF( s( i ).GT.thr ) THEN
               CALL drscl( nrhs, s( i ), b( i, 1 ), ldb )
               rank = rank + 1
            ELSE
               CALL dlaset( 'F', 1, nrhs, zero, zero, b( i, 1 ), ldb )
            END IF
   30    CONTINUE
         iwork = ie
*
*        Multiply B by right singular vectors of L in WORK(IL)
*        (Workspace: need M*M+2*M, prefer M*M+M+M*NRHS)
*
         IF( lwork.GE.ldb*nrhs+iwork-1 .AND. nrhs.GT.1 ) THEN
            CALL dgemm( 'T', 'N', m, nrhs, m, one, work( il ), ldwork,
     $                  b, ldb, zero, work( iwork ), ldb )
            CALL dlacpy( 'G', m, nrhs, work( iwork ), ldb, b, ldb )
         ELSE IF( nrhs.GT.1 ) THEN
            chunk = ( lwork-iwork+1 ) / m
            DO 40 i = 1, nrhs, chunk
               bl = min( nrhs-i+1, chunk )
               CALL dgemm( 'T', 'N', m, bl, m, one, work( il ), ldwork,
     $                     b( 1, i ), ldb, zero, work( iwork ), m )
               CALL dlacpy( 'G', m, bl, work( iwork ), m, b( 1, i ),
     $                      ldb )
   40       CONTINUE
         ELSE
            CALL dgemv( 'T', m, m, one, work( il ), ldwork, b( 1, 1 ),
     $                  1, zero, work( iwork ), 1 )
            CALL dcopy( m, work( iwork ), 1, b( 1, 1 ), 1 )
         END IF
*
*        Zero out below first M rows of B
*
         CALL dlaset( 'F', n-m, nrhs, zero, zero, b( m+1, 1 ), ldb )
         iwork = itau + m
*
*        Multiply transpose(Q) by B
*        (Workspace: need M+NRHS, prefer M+NRHS*NB)
*
         CALL dormlq( 'L', 'T', n, nrhs, m, a, lda, work( itau ), b,
     $                ldb, work( iwork ), lwork-iwork+1, info )
*
      ELSE
*
*        Path 2 - remaining underdetermined cases
*
         ie = 1
         itauq = ie + m
         itaup = itauq + m
         iwork = 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( iwork ), lwork-iwork+1,
     $                info )
*
*        Multiply B by transpose of left bidiagonalizing vectors
*        (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
*
         CALL dormbr( 'Q', 'L', 'T', m, nrhs, n, a, lda, work( itauq ),
     $                b, ldb, work( iwork ), lwork-iwork+1, info )
*
*        Generate right bidiagonalizing vectors in A
*        (Workspace: need 4*M, prefer 3*M+M*NB)
*
         CALL dorgbr( 'P', m, n, m, a, lda, work( itaup ),
     $                work( iwork ), lwork-iwork+1, info )
         iwork = ie + m
*
*        Perform bidiagonal QR iteration,
*           computing right singular vectors of A in A and
*           multiplying B by transpose of left singular vectors
*        (Workspace: need BDSPAC)
*
         CALL dbdsqr( 'L', m, n, 0, nrhs, s, work( ie ), a, lda, dum,
     $                1, b, ldb, work( iwork ), info )
         IF( info.NE.0 )
     $      GO TO 70
*
*        Multiply B by reciprocals of singular values
*
         thr = max( rcond*s( 1 ), sfmin )
         IF( rcond.LT.zero )
     $      thr = max( eps*s( 1 ), sfmin )
         rank = 0
         DO 50 i = 1, m
            IF( s( i ).GT.thr ) THEN
               CALL drscl( nrhs, s( i ), b( i, 1 ), ldb )
               rank = rank + 1
            ELSE
               CALL dlaset( 'F', 1, nrhs, zero, zero, b( i, 1 ), ldb )
            END IF
   50    CONTINUE
*
*        Multiply B by right singular vectors of A
*        (Workspace: need N, prefer N*NRHS)
*
         IF( lwork.GE.ldb*nrhs .AND. nrhs.GT.1 ) THEN
            CALL dgemm( 'T', 'N', n, nrhs, m, one, a, lda, b, ldb, zero,
     $                  work, ldb )
            CALL dlacpy( 'F', n, nrhs, work, ldb, b, ldb )
         ELSE IF( nrhs.GT.1 ) THEN
            chunk = lwork / n
            DO 60 i = 1, nrhs, chunk
               bl = min( nrhs-i+1, chunk )
               CALL dgemm( 'T', 'N', n, bl, m, one, a, lda, b( 1, i ),
     $                     ldb, zero, work, n )
               CALL dlacpy( 'F', n, bl, work, n, b( 1, i ), ldb )
   60       CONTINUE
         ELSE
            CALL dgemv( 'T', m, n, one, a, lda, b, 1, zero, work, 1 )
            CALL dcopy( n, work, 1, b, 1 )
         END IF
      END IF
*
*     Undo scaling
*
      IF( iascl.EQ.1 ) THEN
         CALL dlascl( 'G', 0, 0, anrm, smlnum, n, nrhs, b, ldb, info )
         CALL dlascl( 'G', 0, 0, smlnum, anrm, minmn, 1, s, minmn,
     $                info )
      ELSE IF( iascl.EQ.2 ) THEN
         CALL dlascl( 'G', 0, 0, anrm, bignum, n, nrhs, b, ldb, info )
         CALL dlascl( 'G', 0, 0, bignum, anrm, minmn, 1, s, minmn,
     $                info )
      END IF
      IF( ibscl.EQ.1 ) THEN
         CALL dlascl( 'G', 0, 0, smlnum, bnrm, n, nrhs, b, ldb, info )
      ELSE IF( ibscl.EQ.2 ) THEN
         CALL dlascl( 'G', 0, 0, bignum, bnrm, n, nrhs, b, ldb, info )
      END IF
*
   70 CONTINUE
      work( 1 ) = maxwrk
      RETURN
*
*     End of DGELSS
*
      END