Original Version

1 files changed, 617 insertions, 0 deletions

diff --git a/src/fortran/lapack/dgelss.f b/src/fortran/lapack/dgelss.f
new file mode 100644
index 0000000..f024e13
--- /dev/null
+++ b/src/fortran/lapack/dgelss.f
@@ -0,0 +1,617 @@
+      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