cgelss.f

Go to the documentation of this file.

*> \brief <b> CGELSS solves overdetermined or underdetermined systems for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CGELSS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgelss.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgelss.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgelss.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE CGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
*                          WORK, LWORK, RWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
*       REAL               RCOND
*       ..
*       .. Array Arguments ..
*       REAL               RWORK( * ), S( * )
*       COMPLEX            A( LDA, * ), B( LDB, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> CGELSS computes the minimum norm solution to a complex 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 COMPLEX 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 COMPLEX 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 the modulus 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,M,N).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is REAL 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 REAL
*>          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 COMPLEX 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 >=  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] RWORK
*> \verbatim
*>          RWORK is REAL array, dimension (5*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:  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 complexGEsolve
*
*  =====================================================================
      SUBROUTINE cgelss( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
     $                   WORK, LWORK, RWORK, 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
      REAL               RCOND
*     ..
*     .. Array Arguments ..
      REAL               RWORK( * ), S( * )
      COMPLEX            A( LDA, * ), B( LDB, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter( zero = 0.0e+0, one = 1.0e+0 )
      COMPLEX            CZERO, CONE
      parameter( czero = ( 0.0e+0, 0.0e+0 ),
     $                   cone = ( 1.0e+0, 0.0e+0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            BL, CHUNK, I, IASCL, IBSCL, IE, IL, IRWORK,
     $                   itau, itaup, itauq, iwork, ldwork, maxmn,
     $                   maxwrk, minmn, minwrk, mm, mnthr
      INTEGER            LWORK_CGEQRF, LWORK_CUNMQR, LWORK_CGEBRD,
     $                   lwork_cunmbr, lwork_cungbr, lwork_cunmlq,
     $                   lwork_cgelqf
      REAL               ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM, THR
*     ..
*     .. Local Arrays ..
      COMPLEX            DUM( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           cbdsqr, ccopy, cgebrd, cgelqf, cgemm, cgemv,
     $                   cgeqrf, clacpy, clascl, claset, csrscl, cungbr,
     $                   cunmbr, cunmlq, cunmqr, slabad, slascl, slaset,
     $                   xerbla
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      REAL               CLANGE, SLAMCH
      EXTERNAL           ilaenv, clange, slamch
*     ..
*     .. 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.
*       CWorkspace refers to complex workspace, and RWorkspace refers
*       to real workspace. 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, 'CGELSS', ' ', 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 CGEQRF
               CALL cgeqrf( m, n, a, lda, dum(1), dum(1), -1, info )
               lwork_cgeqrf = real( dum(1) )
*              Compute space needed for CUNMQR
               CALL cunmqr( 'L', 'C', m, nrhs, n, a, lda, dum(1), b,
     $                   ldb, dum(1), -1, info )
               lwork_cunmqr = real( dum(1) )
               mm = n
               maxwrk = max( maxwrk, n + n*ilaenv( 1, 'CGEQRF', ' ', m,
     $                       n, -1, -1 ) )
               maxwrk = max( maxwrk, n + nrhs*ilaenv( 1, 'CUNMQR', 'LC',
     $                       m, nrhs, n, -1 ) )
            END IF
            IF( m.GE.n ) THEN
*
*              Path 1 - overdetermined or exactly determined
*
*              Compute space needed for CGEBRD
               CALL cgebrd( mm, n, a, lda, s, s, dum(1), dum(1), dum(1),
     $                      -1, info )
               lwork_cgebrd = real( dum(1) )
*              Compute space needed for CUNMBR
               CALL cunmbr( 'Q', 'L', 'C', mm, nrhs, n, a, lda, dum(1),
     $                b, ldb, dum(1), -1, info )
               lwork_cunmbr = real( dum(1) )
*              Compute space needed for CUNGBR
               CALL cungbr( 'P', n, n, n, a, lda, dum(1),
     $                   dum(1), -1, info )
               lwork_cungbr = real( dum(1) )
*              Compute total workspace needed
               maxwrk = max( maxwrk, 2*n + lwork_cgebrd )
               maxwrk = max( maxwrk, 2*n + lwork_cunmbr )
               maxwrk = max( maxwrk, 2*n + lwork_cungbr )
               maxwrk = max( maxwrk, n*nrhs )
               minwrk = 2*n + max( nrhs, m )
            END IF
            IF( n.GT.m ) THEN
               minwrk = 2*m + max( nrhs, n )
               IF( n.GE.mnthr ) THEN
*
*                 Path 2a - underdetermined, with many more columns
*                 than rows
*
*                 Compute space needed for CGELQF
                  CALL cgelqf( m, n, a, lda, dum(1), dum(1),
     $                -1, info )
                  lwork_cgelqf = real( dum(1) )
*                 Compute space needed for CGEBRD
                  CALL cgebrd( m, m, a, lda, s, s, dum(1), dum(1),
     $                         dum(1), -1, info )
                  lwork_cgebrd = real( dum(1) )
*                 Compute space needed for CUNMBR
                  CALL cunmbr( 'Q', 'L', 'C', m, nrhs, n, a, lda,
     $                dum(1), b, ldb, dum(1), -1, info )
                  lwork_cunmbr = real( dum(1) )
*                 Compute space needed for CUNGBR
                  CALL cungbr( 'P', m, m, m, a, lda, dum(1),
     $                   dum(1), -1, info )
                  lwork_cungbr = real( dum(1) )
*                 Compute space needed for CUNMLQ
                  CALL cunmlq( 'L', 'C', n, nrhs, m, a, lda, dum(1),
     $                 b, ldb, dum(1), -1, info )
                  lwork_cunmlq = real( dum(1) )
*                 Compute total workspace needed
                  maxwrk = m + lwork_cgelqf
                  maxwrk = max( maxwrk, 3*m + m*m + lwork_cgebrd )
                  maxwrk = max( maxwrk, 3*m + m*m + lwork_cunmbr )
                  maxwrk = max( maxwrk, 3*m + m*m + lwork_cungbr )
                  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_cunmlq )
               ELSE
*
*                 Path 2 - underdetermined
*
*                 Compute space needed for CGEBRD
                  CALL cgebrd( m, n, a, lda, s, s, dum(1), dum(1),
     $                         dum(1), -1, info )
                  lwork_cgebrd = real( dum(1) )
*                 Compute space needed for CUNMBR
                  CALL cunmbr( 'Q', 'L', 'C', m, nrhs, m, a, lda,
     $                dum(1), b, ldb, dum(1), -1, info )
                  lwork_cunmbr = real( dum(1) )
*                 Compute space needed for CUNGBR
                  CALL cungbr( 'P', m, n, m, a, lda, dum(1),
     $                   dum(1), -1, info )
                  lwork_cungbr = real( dum(1) )
                  maxwrk = 2*m + lwork_cgebrd
                  maxwrk = max( maxwrk, 2*m + lwork_cunmbr )
                  maxwrk = max( maxwrk, 2*m + lwork_cungbr )
                  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( 'CGELSS', -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 = slamch( 'P' )
      sfmin = slamch( 'S' )
      smlnum = sfmin / eps
      bignum = one / smlnum
      CALL slabad( smlnum, bignum )
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      anrm = clange( 'M', m, n, a, lda, rwork )
      iascl = 0
      IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL clascl( '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 clascl( '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 claset( 'F', max( m, n ), nrhs, czero, czero, b, ldb )
         CALL slaset( 'F', minmn, 1, zero, zero, s, minmn )
         rank = 0
         GO TO 70
      END IF
*
*     Scale B if max element outside range [SMLNUM,BIGNUM]
*
      bnrm = clange( 'm', M, NRHS, B, LDB, RWORK )
      IBSCL = 0
.GT..AND..LT.      IF( BNRMZERO  BNRMSMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL CLASCL( 'g', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
         IBSCL = 1
.GT.      ELSE IF( BNRMBIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL CLASCL( 'g', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
         IBSCL = 2
      END IF
*
*     Overdetermined case
*
.GE.      IF( MN ) THEN
*
*        Path 1 - overdetermined or exactly determined
*
         MM = M
.GE.         IF( MMNTHR ) THEN
*
*           Path 1a - overdetermined, with many more rows than columns
*
            MM = N
            ITAU = 1
            IWORK = ITAU + N
*
*           Compute A=Q*R
*           (CWorkspace: need 2*N, prefer N+N*NB)
*           (RWorkspace: none)
*
            CALL CGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
     $                   LWORK-IWORK+1, INFO )
*
*           Multiply B by transpose(Q)
*           (CWorkspace: need N+NRHS, prefer N+NRHS*NB)
*           (RWorkspace: none)
*
            CALL CUNMQR( 'l', 'c', M, NRHS, N, A, LDA, WORK( ITAU ), B,
     $                   LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
*
*           Zero out below R
*
.GT.            IF( N1 )
     $         CALL CLASET( 'l', N-1, N-1, CZERO, CZERO, A( 2, 1 ),
     $                      LDA )
         END IF
*
         IE = 1
         ITAUQ = 1
         ITAUP = ITAUQ + N
         IWORK = ITAUP + N
*
*        Bidiagonalize R in A
*        (CWorkspace: need 2*N+MM, prefer 2*N+(MM+N)*NB)
*        (RWorkspace: need N)
*
         CALL CGEBRD( MM, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
     $                WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
     $                INFO )
*
*        Multiply B by transpose of left bidiagonalizing vectors of R
*        (CWorkspace: need 2*N+NRHS, prefer 2*N+NRHS*NB)
*        (RWorkspace: none)
*
         CALL CUNMBR( 'q', 'l', 'c', MM, NRHS, N, A, LDA, WORK( ITAUQ ),
     $                B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
*
*        Generate right bidiagonalizing vectors of R in A
*        (CWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*        (RWorkspace: none)
*
         CALL CUNGBR( 'p', N, N, N, A, LDA, WORK( ITAUP ),
     $                WORK( IWORK ), LWORK-IWORK+1, INFO )
         IRWORK = IE + N
*
*        Perform bidiagonal QR iteration
*          multiply B by transpose of left singular vectors
*          compute right singular vectors in A
*        (CWorkspace: none)
*        (RWorkspace: need BDSPAC)
*
         CALL CBDSQR( 'u', N, N, 0, NRHS, S, RWORK( IE ), A, LDA, DUM,
     $                1, B, LDB, RWORK( IRWORK ), INFO )
.NE.         IF( INFO0 )
     $      GO TO 70
*
*        Multiply B by reciprocals of singular values
*
         THR = MAX( RCOND*S( 1 ), SFMIN )
.LT.         IF( RCONDZERO )
     $      THR = MAX( EPS*S( 1 ), SFMIN )
         RANK = 0
         DO 10 I = 1, N
.GT.            IF( S( I )THR ) THEN
               CALL CSRSCL( NRHS, S( I ), B( I, 1 ), LDB )
               RANK = RANK + 1
            ELSE
               CALL CLASET( 'f', 1, NRHS, CZERO, CZERO, B( I, 1 ), LDB )
            END IF
   10    CONTINUE
*
*        Multiply B by right singular vectors
*        (CWorkspace: need N, prefer N*NRHS)
*        (RWorkspace: none)
*
.GE..AND..GT.         IF( LWORKLDB*NRHS  NRHS1 ) THEN
            CALL CGEMM( 'c', 'n', N, NRHS, N, CONE, A, LDA, B, LDB,
     $                  CZERO, WORK, LDB )
            CALL CLACPY( 'g', N, NRHS, WORK, LDB, B, LDB )
.GT.         ELSE IF( NRHS1 ) THEN
            CHUNK = LWORK / N
            DO 20 I = 1, NRHS, CHUNK
               BL = MIN( NRHS-I+1, CHUNK )
               CALL CGEMM( 'c', 'n', N, BL, N, CONE, A, LDA, B( 1, I ),
     $                     LDB, CZERO, WORK, N )
               CALL CLACPY( 'g', N, BL, WORK, N, B( 1, I ), LDB )
   20       CONTINUE
         ELSE
            CALL CGEMV( 'c', N, N, CONE, A, LDA, B, 1, CZERO, WORK, 1 )
            CALL CCOPY( N, WORK, 1, B, 1 )
         END IF
*
.GE..AND..GE.      ELSE IF( NMNTHR  LWORK3*M+M*M+MAX( M, NRHS, N-2*M ) )
     $          THEN
*
*        Underdetermined case, M much less than N
*
*        Path 2a - underdetermined, with many more columns than rows
*        and sufficient workspace for an efficient algorithm
*
         LDWORK = M
.GE.         IF( LWORK3*M+M*LDA+MAX( M, NRHS, N-2*M ) )
     $      LDWORK = LDA
         ITAU = 1
         IWORK = M + 1
*
*        Compute A=L*Q
*        (CWorkspace: need 2*M, prefer M+M*NB)
*        (RWorkspace: none)
*
         CALL CGELQF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
     $                LWORK-IWORK+1, INFO )
         IL = IWORK
*
*        Copy L to WORK(IL), zeroing out above it
*
         CALL CLACPY( 'l', M, M, A, LDA, WORK( IL ), LDWORK )
         CALL CLASET( 'u', M-1, M-1, CZERO, CZERO, WORK( IL+LDWORK ),
     $                LDWORK )
         IE = 1
         ITAUQ = IL + LDWORK*M
         ITAUP = ITAUQ + M
         IWORK = ITAUP + M
*
*        Bidiagonalize L in WORK(IL)
*        (CWorkspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
*        (RWorkspace: need M)
*
         CALL CGEBRD( M, M, WORK( IL ), LDWORK, S, RWORK( IE ),
     $                WORK( ITAUQ ), WORK( ITAUP ), WORK( IWORK ),
     $                LWORK-IWORK+1, INFO )
*
*        Multiply B by transpose of left bidiagonalizing vectors of L
*        (CWorkspace: need M*M+3*M+NRHS, prefer M*M+3*M+NRHS*NB)
*        (RWorkspace: none)
*
         CALL CUNMBR( 'q', 'l', 'c', 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)
*        (CWorkspace: need M*M+4*M-1, prefer M*M+3*M+(M-1)*NB)
*        (RWorkspace: none)
*
         CALL CUNGBR( 'p', M, M, M, WORK( IL ), LDWORK, WORK( ITAUP ),
     $                WORK( IWORK ), LWORK-IWORK+1, INFO )
         IRWORK = IE + M
*
*        Perform bidiagonal QR iteration, computing right singular
*        vectors of L in WORK(IL) and multiplying B by transpose of
*        left singular vectors
*        (CWorkspace: need M*M)
*        (RWorkspace: need BDSPAC)
*
         CALL CBDSQR( 'u', M, M, 0, NRHS, S, RWORK( IE ), WORK( IL ),
     $                LDWORK, A, LDA, B, LDB, RWORK( IRWORK ), INFO )
.NE.         IF( INFO0 )
     $      GO TO 70
*
*        Multiply B by reciprocals of singular values
*
         THR = MAX( RCOND*S( 1 ), SFMIN )
.LT.         IF( RCONDZERO )
     $      THR = MAX( EPS*S( 1 ), SFMIN )
         RANK = 0
         DO 30 I = 1, M
.GT.            IF( S( I )THR ) THEN
               CALL CSRSCL( NRHS, S( I ), B( I, 1 ), LDB )
               RANK = RANK + 1
            ELSE
               CALL CLASET( 'f', 1, NRHS, CZERO, CZERO, B( I, 1 ), LDB )
            END IF
   30    CONTINUE
         IWORK = IL + M*LDWORK
*
*        Multiply B by right singular vectors of L in WORK(IL)
*        (CWorkspace: need M*M+2*M, prefer M*M+M+M*NRHS)
*        (RWorkspace: none)
*
.GE..AND..GT.         IF( LWORKLDB*NRHS+IWORK-1  NRHS1 ) THEN
            CALL CGEMM( 'c', 'n', M, NRHS, M, CONE, WORK( IL ), LDWORK,
     $                  B, LDB, CZERO, WORK( IWORK ), LDB )
            CALL CLACPY( 'g', M, NRHS, WORK( IWORK ), LDB, B, LDB )
.GT.         ELSE IF( NRHS1 ) THEN
            CHUNK = ( LWORK-IWORK+1 ) / M
            DO 40 I = 1, NRHS, CHUNK
               BL = MIN( NRHS-I+1, CHUNK )
               CALL CGEMM( 'c', 'n', M, BL, M, CONE, WORK( IL ), LDWORK,
     $                     B( 1, I ), LDB, CZERO, WORK( IWORK ), M )
               CALL CLACPY( 'g', M, BL, WORK( IWORK ), M, B( 1, I ),
     $                      LDB )
   40       CONTINUE
         ELSE
            CALL CGEMV( 'c', M, M, CONE, WORK( IL ), LDWORK, B( 1, 1 ),
     $                  1, CZERO, WORK( IWORK ), 1 )
            CALL CCOPY( M, WORK( IWORK ), 1, B( 1, 1 ), 1 )
         END IF
*
*        Zero out below first M rows of B
*
         CALL CLASET( 'f', N-M, NRHS, CZERO, CZERO, B( M+1, 1 ), LDB )
         IWORK = ITAU + M
*
*        Multiply transpose(Q) by B
*        (CWorkspace: need M+NRHS, prefer M+NHRS*NB)
*        (RWorkspace: none)
*
         CALL CUNMLQ( 'l', 'c', n, nrhs, m, a, lda, work( itau ), b,
     $                ldb, work( iwork ), lwork-iwork+1, info )
*
      ELSE
*
*        Path 2 - remaining underdetermined cases
*
         ie = 1
         itauq = 1
         itaup = itauq + m
         iwork = itaup + m
*
*        Bidiagonalize A
*        (CWorkspace: need 3*M, prefer 2*M+(M+N)*NB)
*        (RWorkspace: need N)
*
         CALL cgebrd( m, n, a, lda, s, rwork( ie ), work( itauq ),
     $                work( itaup ), work( iwork ), lwork-iwork+1,
     $                info )
*
*        Multiply B by transpose of left bidiagonalizing vectors
*        (CWorkspace: need 2*M+NRHS, prefer 2*M+NRHS*NB)
*        (RWorkspace: none)
*
         CALL cunmbr( 'Q', 'L', 'C', m, nrhs, n, a, lda, work( itauq ),
     $                b, ldb, work( iwork ), lwork-iwork+1, info )
*
*        Generate right bidiagonalizing vectors in A
*        (CWorkspace: need 3*M, prefer 2*M+M*NB)
*        (RWorkspace: none)
*
         CALL cungbr( 'P', m, n, m, a, lda, work( itaup ),
     $                work( iwork ), lwork-iwork+1, info )
         irwork = ie + m
*
*        Perform bidiagonal QR iteration,
*           computing right singular vectors of A in A and
*           multiplying B by transpose of left singular vectors
*        (CWorkspace: none)
*        (RWorkspace: need BDSPAC)
*
         CALL cbdsqr( 'L', m, n, 0, nrhs, s, rwork( ie ), a, lda, dum,
     $                1, b, ldb, rwork( irwork ), 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 csrscl( nrhs, s( i ), b( i, 1 ), ldb )
               rank = rank + 1
            ELSE
               CALL claset( 'F', 1, nrhs, czero, czero, b( i, 1 ), ldb )
            END IF
   50    CONTINUE
*
*        Multiply B by right singular vectors of A
*        (CWorkspace: need N, prefer N*NRHS)
*        (RWorkspace: none)
*
         IF( lwork.GE.ldb*nrhs .AND. nrhs.GT.1 ) THEN
            CALL cgemm( 'C', 'N', n, nrhs, m, cone, a, lda, b, ldb,
     $                  czero, work, ldb )
            CALL clacpy( 'G', 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 cgemm( 'C', 'N', n, bl, m, cone, a, lda, b( 1, i ),
     $                     ldb, czero, work, n )
               CALL clacpy( 'F', n, bl, work, n, b( 1, i ), ldb )
   60       CONTINUE
         ELSE
            CALL cgemv( 'C', m, n, cone, a, lda, b, 1, czero, work, 1 )
            CALL ccopy( n, work, 1, b, 1 )
         END IF
      END IF
*
*     Undo scaling
*
      IF( iascl.EQ.1 ) THEN
         CALL clascl( 'G', 0, 0, anrm, smlnum, n, nrhs, b, ldb, info )
         CALL slascl( 'G', 0, 0, smlnum, anrm, minmn, 1, s, minmn,
     $                info )
      ELSE IF( iascl.EQ.2 ) THEN
         CALL clascl( 'G', 0, 0, anrm, bignum, n, nrhs, b, ldb, info )
         CALL slascl( 'G', 0, 0, bignum, anrm, minmn, 1, s, minmn,
     $                info )
      END IF
      IF( ibscl.EQ.1 ) THEN
         CALL clascl( 'G', 0, 0, smlnum, bnrm, n, nrhs, b, ldb, info )
      ELSE IF( ibscl.EQ.2 ) THEN
         CALL clascl( 'G', 0, 0, bignum, bnrm, n, nrhs, b, ldb, info )
      END IF
   70 CONTINUE
      work( 1 ) = maxwrk
      RETURN
*
*     End of CGELSS
*
      END