dgelss.f

          SUBROUTINE DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
         $                   WORK, LWORK, INFO )
    *
    *  -- LAPACK driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
          DOUBLE PRECISION   RCOND
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  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.
    *
    *  Arguments
    *  =========
    *
    *  M       (input) INTEGER
    *          The number of rows of the matrix A. M >= 0.
    *
    *  N       (input) INTEGER
    *          The number of columns of the matrix A. N >= 0.
    *
    *  NRHS    (input) INTEGER
    *          The number of right hand sides, i.e., the number of columns
    *          of the matrices B and X. NRHS >= 0.
    *
    *  A       (input/output) 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.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A.  LDA >= max(1,M).
    *
    *  B       (input/output) 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.
    *
    *  LDB     (input) INTEGER
    *          The leading dimension of the array B. LDB >= max(1,max(M,N)).
    *
    *  S       (output) 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)).
    *
    *  RCOND   (input) 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.
    *
    *  RANK    (output) INTEGER
    *          The effective rank of A, i.e., the number of singular values
    *          which are greater than RCOND*S(1).
    *
    *  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, 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.
    *
    *  INFO    (output) 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.
    *
    *  =====================================================================
    *
    *     .. 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
          DOUBLE PRECISION   ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM, THR
    *     ..
    *     .. Local Arrays ..
          DOUBLE PRECISION   VDUM( 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
    *
                   MM = N
                   MAXWRK = MAX( MAXWRK, N + N*ILAENV( 1, 'DGEQRF', ' ', M,
         $                       N, -1, -1 ) )
                   MAXWRK = MAX( MAXWRK, N + NRHS*ILAENV( 1, 'DORMQR', 'LT',
         $                       M, NRHS, N, -1 ) )
                END IF
                IF( M.GE.N ) THEN
    *
    *              Path 1 - overdetermined or exactly determined
    *
    *              Compute workspace needed for DBDSQR
    *
                   BDSPAC = MAX( 1, 5*N )
                   MAXWRK = MAX( MAXWRK, 3*N + ( MM + N )*ILAENV( 1,
         $                       'DGEBRD', ' ', MM, N, -1, -1 ) )
                   MAXWRK = MAX( MAXWRK, 3*N + NRHS*ILAENV( 1, 'DORMBR',
         $                       'QLT', MM, NRHS, N, -1 ) )
                   MAXWRK = MAX( MAXWRK, 3*N + ( N - 1 )*ILAENV( 1,
         $                       'DORGBR', 'P', N, N, N, -1 ) )
                   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
    *
                      MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1,
         $                                  -1 )
                      MAXWRK = MAX( MAXWRK, M*M + 4*M + 2*M*ILAENV( 1,
         $                          'DGEBRD', ' ', M, M, -1, -1 ) )
                      MAXWRK = MAX( MAXWRK, M*M + 4*M + NRHS*ILAENV( 1,
         $                          'DORMBR', 'QLT', M, NRHS, M, -1 ) )
                      MAXWRK = MAX( MAXWRK, M*M + 4*M +
         $                          ( M - 1 )*ILAENV( 1, 'DORGBR', 'P', M,
         $                          M, M, -1 ) )
                      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 + NRHS*ILAENV( 1, 'DORMLQ',
         $                          'LT', N, NRHS, M, -1 ) )
                   ELSE
    *
    *                 Path 2 - underdetermined
    *
                      MAXWRK = 3*M + ( N + M )*ILAENV( 1, 'DGEBRD', ' ', M,
         $                     N, -1, -1 )
                      MAXWRK = MAX( MAXWRK, 3*M + NRHS*ILAENV( 1, 'DORMBR',
         $                          'QLT', M, NRHS, M, -1 ) )
                      MAXWRK = MAX( MAXWRK, 3*M + M*ILAENV( 1, 'DORGBR',
         $                          'P', M, N, M, -1 ) )
                      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, 1 )
             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, VDUM,
         $                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, VDUM,
         $                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