sgelss.f

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         T1 = SECOND( )
         CALL SORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ),
     $                B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
         T2 = SECOND( )
         TIMNG( ORMBR ) = TIMNG( ORMBR ) + ( T2-T1 )
*
*        Generate right bidiagonalizing vectors of R in A
*        (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
*
         NB = ILAENV( 1, 'SORGBR', 'P', N, N, N, -1 )
         OPCNT( ORGBR ) = OPCNT( ORGBR ) +
     $                    SOPLA2( 'SORGBR', 'P', N, N, N, 0, NB )
         T1 = SECOND( )
         CALL SORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
     $                WORK( IWORK ), LWORK-IWORK+1, INFO )
         T2 = SECOND( )
         TIMNG( ORGBR ) = TIMNG( ORGBR ) + ( T2-T1 )
         IWORK = IE + N
*
*        Perform bidiagonal QR iteration
*          multiply B by transpose of left singular vectors
*          compute right singular vectors in A
*        (Workspace: need BDSPAC)
*
         OPS = 0
         T1 = SECOND( )
         CALL SBDSQR( 'U', N, N, 0, NRHS, S, WORK( IE ), A, LDA, VDUM,
     $                1, B, LDB, WORK( IWORK ), INFO )
         T2 = SECOND( )
         TIMNG( BDSQR ) = TIMNG( BDSQR ) + ( T2-T1 )
         OPCNT( BDSQR ) = OPCNT( BDSQR ) + OPS
         IF( INFO.NE.0 )
     $      GO TO 70
*
*        Multiply B by reciprocals of singular values
*
         OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( 1 )
         THR = MAX( RCOND*S( 1 ), SFMIN )
         IF( RCOND.LT.ZERO ) THEN
            OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( 1 )
            THR = MAX( EPS*S( 1 ), SFMIN )
         END IF
         RANK = 0
         DO 10 I = 1, N
            IF( S( I ).GT.THR ) THEN
               OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( NRHS + 3 )
               CALL SRSCL( NRHS, S( I ), B( I, 1 ), LDB )
               RANK = RANK + 1
            ELSE
               CALL SLASET( '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
            OPCNT( GEMM ) = OPCNT( GEMM ) +
     $                      SOPBL3( 'SGEMM ', N, NRHS, N ) 
            T1 = SECOND( )
            CALL SGEMM( 'T', 'N', N, NRHS, N, ONE, A, LDA, B, LDB, ZERO,
     $                  WORK, LDB )
            T2 = SECOND( )
            TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 )
            CALL SLACPY( '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 )
               OPCNT( GEMM ) = OPCNT( GEMM ) +
     $                         SOPBL3( 'SGEMM ', N, BL, N ) 
               T1 = SECOND( )
               CALL SGEMM( 'T', 'N', N, BL, N, ONE, A, LDA, B( 1, I ),
     $                     LDB, ZERO, WORK, N )
               T2 = SECOND( )
               TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 )
               CALL SLACPY( 'G', N, BL, WORK, N, B( 1, I ), LDB )
   20       CONTINUE
         ELSE
            OPCNT( GEMV ) = OPCNT( GEMV ) +
     $                      SOPBL2( 'SGEMV ', N, N, 0, 0 ) 
            T1 = SECOND( )
            CALL SGEMV( 'T', N, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 )
            T2 = SECOND( )
            TIMNG( GEMV ) = TIMNG( GEMV ) + ( T2-T1 )
            CALL SCOPY( 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)
*
         NB = ILAENV( 1, 'SGELQF', ' ', M, N, -1, -1 )
         OPCNT( GELQF ) = OPCNT( GELQF ) +
     $                    SOPLA( 'SGELQF', M, N, 0, 0, NB )
         T1 = SECOND( )
         CALL SGELQF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
     $                LWORK-IWORK+1, INFO )
         T2 = SECOND( )
         TIMNG( GELQF ) = TIMNG( GELQF ) + ( T2-T1 )
         IL = IWORK
*
*        Copy L to WORK(IL), zeroing out above it
*
         CALL SLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
         CALL SLASET( '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)
*
         NB = ILAENV( 1, 'SGEBRD', ' ', M, M, -1, -1 )
         OPCNT( GEBRD ) = OPCNT( GEBRD ) +
     $                    SOPLA( 'SGEBRD', M, M, 0, 0, NB )
         T1 = SECOND( )
         CALL SGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ),
     $                WORK( ITAUQ ), WORK( ITAUP ), WORK( IWORK ),
     $                LWORK-IWORK+1, INFO )
         T2 = SECOND( )
         TIMNG( GEBRD ) = TIMNG( GEBRD ) + ( T2-T1 )
*
*        Multiply B by transpose of left bidiagonalizing vectors of L
*        (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
*
         NB = ILAENV( 1, 'SORMBR', 'QLT', M, NRHS, M, -1 )
         OPCNT( ORMBR ) = OPCNT( ORMBR ) +
     $                    SOPLA2( 'SORMBR', 'QLT', M, NRHS, M, 0, NB )
         T1 = SECOND( )
         CALL SORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK,
     $                WORK( ITAUQ ), B, LDB, WORK( IWORK ),
     $                LWORK-IWORK+1, INFO )
         T2 = SECOND( )
         TIMNG( ORMBR ) = TIMNG( ORMBR ) + ( T2-T1 )
*
*        Generate right bidiagonalizing vectors of R in WORK(IL)
*        (Workspace: need M*M+5*M-1, prefer M*M+4*M+(M-1)*NB)
*
         NB = ILAENV( 1, 'SORGBR', 'P', M, M, M, -1 )
         OPCNT( ORGBR ) = OPCNT( ORGBR ) +
     $                    SOPLA2( 'SORGBR', 'P', M, M, M, 0, NB )
         T1 = SECOND( )
         CALL SORGBR( 'P', M, M, M, WORK( IL ), LDWORK, WORK( ITAUP ),
     $                WORK( IWORK ), LWORK-IWORK+1, INFO )
         T2 = SECOND( )
         TIMNG( ORGBR ) = TIMNG( ORGBR ) + ( T2-T1 )
         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)
*
         OPS = 0
         T1 = SECOND( )
         CALL SBDSQR( 'U', M, M, 0, NRHS, S, WORK( IE ), WORK( IL ),
     $                LDWORK, A, LDA, B, LDB, WORK( IWORK ), INFO )
         T2 = SECOND( )
         TIMNG( BDSQR ) = TIMNG( BDSQR ) + ( T2-T1 )
         OPCNT( BDSQR ) = OPCNT( BDSQR ) + OPS
         IF( INFO.NE.0 )
     $      GO TO 70
*
*        Multiply B by reciprocals of singular values
*
         OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( 1 )
         THR = MAX( RCOND*S( 1 ), SFMIN )
         IF( RCOND.LT.ZERO ) THEN
            OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( 1 )
            THR = MAX( EPS*S( 1 ), SFMIN )
         END IF
         RANK = 0
         DO 30 I = 1, M
            IF( S( I ).GT.THR ) THEN
               OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( NRHS + 3 )
               CALL SRSCL( NRHS, S( I ), B( I, 1 ), LDB )
               RANK = RANK + 1
            ELSE
               CALL SLASET( '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
            OPCNT( GEMM ) = OPCNT( GEMM ) +
     $                      SOPBL3( 'SGEMM ', M, NRHS, M ) 
            T1 = SECOND( )
            CALL SGEMM( 'T', 'N', M, NRHS, M, ONE, WORK( IL ), LDWORK,
     $                  B, LDB, ZERO, WORK( IWORK ), LDB )
            T2 = SECOND( )
            TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 )
            CALL SLACPY( '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 )
               OPCNT( GEMM ) = OPCNT( GEMM ) +
     $                         SOPBL3( 'SGEMM ', M, BL, M ) 
               T1 = SECOND( )
               CALL SGEMM( 'T', 'N', M, BL, M, ONE, WORK( IL ), LDWORK,
     $                     B( 1, I ), LDB, ZERO, WORK( IWORK ), N )
               T2 = SECOND( )
               TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 )
               CALL SLACPY( 'G', M, BL, WORK( IWORK ), N, B( 1, I ),
     $                      LDB )
   40       CONTINUE
         ELSE
            OPCNT( GEMV ) = OPCNT( GEMV ) +
     $                      SOPBL2( 'SGEMV ', M, M, 0, 0 ) 
            T1 = SECOND( )
            CALL SGEMV( 'T', M, M, ONE, WORK( IL ), LDWORK, B( 1, 1 ),
     $                  1, ZERO, WORK( IWORK ), 1 )
            T2 = SECOND( )
            TIMNG( GEMV ) = TIMNG( GEMV ) + ( T2-T1 )
            CALL SCOPY( M, WORK( IWORK ), 1, B( 1, 1 ), 1 )
         END IF
*
*        Zero out below first M rows of B
*
         CALL SLASET( '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)
*
         NB = ILAENV( 1, 'SORMLQ', 'LT', N, NRHS, M, -1 )
         OPCNT( ORMLQ ) = OPCNT( ORMLQ ) +
     $                    SOPLA( 'SORMLQ', N, NRHS, M, 0, NB )
         T1 = SECOND( )
         CALL SORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B,
     $                LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
         T2 = SECOND( )
         TIMNG( ORMLQ ) = TIMNG( ORMLQ ) + ( T2-T1 )
*
      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)
*
         NB = ILAENV( 1, 'SGEBRD', ' ', M, N, -1, -1 )
         OPCNT( GEBRD ) = OPCNT( GEBRD ) +
     $                    SOPLA( 'SGEBRD', M, N, 0, 0, NB )
         T1 = SECOND( )
         CALL SGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
     $                WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
     $                INFO )
         T2 = SECOND( )
         TIMNG( GEBRD ) = TIMNG( GEBRD ) + ( T2-T1 )
*
*        Multiply B by transpose of left bidiagonalizing vectors
*        (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
*
         NB = ILAENV( 1, 'SORMBR', 'QLT', M, NRHS, N, -1 )
         OPCNT( ORMBR ) = OPCNT( ORMBR ) +
     $                    SOPLA2( 'SORMBR', 'QLT', M, NRHS, N, 0, NB )
         T1 = SECOND( )
         CALL SORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ),
     $                B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
         T2 = SECOND( )
         TIMNG( ORMBR ) = TIMNG( ORMBR ) + ( T2-T1 )
*
*        Generate right bidiagonalizing vectors in A
*        (Workspace: need 4*M, prefer 3*M+M*NB)
*
         NB = ILAENV( 1, 'SORGBR', 'P', M, N, M, -1 )
         OPCNT( ORGBR ) = OPCNT( ORGBR ) +
     $                    SOPLA2( 'SORGBR', 'P', M, N, M, 0, NB )
         T1 = SECOND( )
         CALL SORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
     $                WORK( IWORK ), LWORK-IWORK+1, INFO )
         T2 = SECOND( )
         TIMNG( ORGBR ) = TIMNG( ORGBR ) + ( T2-T1 )
         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)
*
         OPS = 0
         T1 = SECOND( )
         CALL SBDSQR( 'L', M, N, 0, NRHS, S, WORK( IE ), A, LDA, VDUM,
     $                1, B, LDB, WORK( IWORK ), INFO )
         T2 = SECOND( )
         TIMNG( BDSQR ) = TIMNG( BDSQR ) + ( T2-T1 )
         OPCNT( BDSQR ) = OPCNT( BDSQR ) + OPS
         IF( INFO.NE.0 )
     $      GO TO 70
*
*        Multiply B by reciprocals of singular values
*
         OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( 1 )
         THR = MAX( RCOND*S( 1 ), SFMIN )
         IF( RCOND.LT.ZERO ) THEN
            OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( 1 )
            THR = MAX( EPS*S( 1 ), SFMIN )
         END IF
         RANK = 0
         DO 50 I = 1, M
            IF( S( I ).GT.THR ) THEN
               OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( NRHS + 3 )
               CALL SRSCL( NRHS, S( I ), B( I, 1 ), LDB )
               RANK = RANK + 1
            ELSE
               CALL SLASET( '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
            OPCNT( GEMM ) = OPCNT( GEMM ) +
     $                      SOPBL3( 'SGEMM ', N, NRHS, M ) 
            T1 = SECOND( )
            CALL SGEMM( 'T', 'N', N, NRHS, M, ONE, A, LDA, B, LDB, ZERO,
     $                  WORK, LDB )
            T2 = SECOND( )
            TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 )
            CALL SLACPY( '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 )
               OPCNT( GEMM ) = OPCNT( GEMM ) +
     $                         SOPBL3( 'SGEMM ', N, BL, M ) 
               T1 = SECOND( )
               CALL SGEMM( 'T', 'N', N, BL, M, ONE, A, LDA, B( 1, I ),
     $                     LDB, ZERO, WORK, N )
               T2 = SECOND( )
               TIMNG( GEMM ) = TIMNG( GEMM ) + ( T2-T1 )
               CALL SLACPY( 'F', N, BL, WORK, N, B( 1, I ), LDB )
   60       CONTINUE
         ELSE
            OPCNT( GEMV ) = OPCNT( GEMV ) +
     $                      SOPBL2( 'SGEMV ', M, N, 0, 0 ) 
            T1 = SECOND( )
            CALL SGEMV( 'T', M, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 )
            T2 = SECOND( )
            TIMNG( GEMV ) = TIMNG( GEMV ) + ( T2-T1 )
            CALL SCOPY( N, WORK, 1, B, 1 )
         END IF
      END IF
*
*     Undo scaling
*
      IF( IASCL.EQ.1 ) THEN
         OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( N*NRHS + MINMN )
         CALL SLASCL( '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
         OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( N*NRHS + MINMN )
         CALL SLASCL( '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
         OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( N*NRHS )
         CALL SLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
      ELSE IF( IBSCL.EQ.2 ) THEN
         OPCNT( GELSS ) = OPCNT( GELSS ) + REAL( N*NRHS )
         CALL SLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
      END IF
*
   70 CONTINUE
      WORK( 1 ) = MAXWRK
      RETURN
*
*     End of SGELSS
*
      END