          SUBROUTINE SLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U,
         $                   LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR,
         $                   GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK,
         $                   IWORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS,
         $                   SMLSIZ
    *     ..
    *     .. Array Arguments ..
          INTEGER            GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
         $                   K( * ), PERM( LDGCOL, * )
          REAL               B( LDB, * ), BX( LDBX, * ), C( * ),
         $                   DIFL( LDU, * ), DIFR( LDU, * ),
         $                   GIVNUM( LDU, * ), POLES( LDU, * ), S( * ),
         $                   U( LDU, * ), VT( LDU, * ), WORK( * ),
         $                   Z( LDU, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SLALSA is an itermediate step in solving the least squares problem
    *  by computing the SVD of the coefficient matrix in compact form (The
    *  singular vectors are computed as products of simple orthorgonal
    *  matrices.).
    *
    *  If ICOMPQ = 0, SLALSA applies the inverse of the left singular vector
    *  matrix of an upper bidiagonal matrix to the right hand side; and if
    *  ICOMPQ = 1, SLALSA applies the right singular vector matrix to the
    *  right hand side. The singular vector matrices were generated in
    *  compact form by SLALSA.
    *
    *  Arguments
    *  =========
    *
    *
    *  ICOMPQ (input) INTEGER
    *         Specifies whether the left or the right singular vector
    *         matrix is involved.
    *         = 0: Left singular vector matrix
    *         = 1: Right singular vector matrix
    *
    *  SMLSIZ (input) INTEGER
    *         The maximum size of the subproblems at the bottom of the
    *         computation tree.
    *
    *  N      (input) INTEGER
    *         The row and column dimensions of the upper bidiagonal matrix.
    *
    *  NRHS   (input) INTEGER
    *         The number of columns of B and BX. NRHS must be at least 1.
    *
    *  B      (input/output) REAL array, dimension ( LDB, NRHS )
    *         On input, B contains the right hand sides of the least
    *         squares problem in rows 1 through M.
    *         On output, B contains the solution X in rows 1 through N.
    *
    *  LDB    (input) INTEGER
    *         The leading dimension of B in the calling subprogram.
    *         LDB must be at least max(1,MAX( M, N ) ).
    *
    *  BX     (output) REAL array, dimension ( LDBX, NRHS )
    *         On exit, the result of applying the left or right singular
    *         vector matrix to B.
    *
    *  LDBX   (input) INTEGER
    *         The leading dimension of BX.
    *
    *  U      (input) REAL array, dimension ( LDU, SMLSIZ ).
    *         On entry, U contains the left singular vector matrices of all
    *         subproblems at the bottom level.
    *
    *  LDU    (input) INTEGER, LDU = > N.
    *         The leading dimension of arrays U, VT, DIFL, DIFR,
    *         POLES, GIVNUM, and Z.
    *
    *  VT     (input) REAL array, dimension ( LDU, SMLSIZ+1 ).
    *         On entry, VT' contains the right singular vector matrices of
    *         all subproblems at the bottom level.
    *
    *  K      (input) INTEGER array, dimension ( N ).
    *
    *  DIFL   (input) REAL array, dimension ( LDU, NLVL ).
    *         where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
    *
    *  DIFR   (input) REAL array, dimension ( LDU, 2 * NLVL ).
    *         On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
    *         distances between singular values on the I-th level and
    *         singular values on the (I -1)-th level, and DIFR(*, 2 * I)
    *         record the normalizing factors of the right singular vectors
    *         matrices of subproblems on I-th level.
    *
    *  Z      (input) REAL array, dimension ( LDU, NLVL ).
    *         On entry, Z(1, I) contains the components of the deflation-
    *         adjusted updating row vector for subproblems on the I-th
    *         level.
    *
    *  POLES  (input) REAL array, dimension ( LDU, 2 * NLVL ).
    *         On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
    *         singular values involved in the secular equations on the I-th
    *         level.
    *
    *  GIVPTR (input) INTEGER array, dimension ( N ).
    *         On entry, GIVPTR( I ) records the number of Givens
    *         rotations performed on the I-th problem on the computation
    *         tree.
    *
    *  GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
    *         On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the
    *         locations of Givens rotations performed on the I-th level on
    *         the computation tree.
    *
    *  LDGCOL (input) INTEGER, LDGCOL = > N.
    *         The leading dimension of arrays GIVCOL and PERM.
    *
    *  PERM   (input) INTEGER array, dimension ( LDGCOL, NLVL ).
    *         On entry, PERM(*, I) records permutations done on the I-th
    *         level of the computation tree.
    *
    *  GIVNUM (input) REAL array, dimension ( LDU, 2 * NLVL ).
    *         On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
    *         values of Givens rotations performed on the I-th level on the
    *         computation tree.
    *
    *  C      (input) REAL array, dimension ( N ).
    *         On entry, if the I-th subproblem is not square,
    *         C( I ) contains the C-value of a Givens rotation related to
    *         the right null space of the I-th subproblem.
    *
    *  S      (input) REAL array, dimension ( N ).
    *         On entry, if the I-th subproblem is not square,
    *         S( I ) contains the S-value of a Givens rotation related to
    *         the right null space of the I-th subproblem.
    *
    *  WORK   (workspace) REAL array.
    *         The dimension must be at least N.
    *
    *  IWORK  (workspace) INTEGER array.
    *         The dimension must be at least 3 * N
    *
    *  INFO   (output) INTEGER
    *          = 0:  successful exit.
    *          < 0:  if INFO = -i, the i-th argument had an illegal value.
    *
    *  Further Details
    *  ===============
    *
    *  Based on contributions by
    *     Ming Gu and Ren-Cang Li, Computer Science Division, University of
    *       California at Berkeley, USA
    *     Osni Marques, LBNL/NERSC, USA
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          REAL               ZERO, ONE
          PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0 )
    *     ..
    *     .. Local Scalars ..
          INTEGER            I, I1, IC, IM1, INODE, J, LF, LL, LVL, LVL2,
         $                   ND, NDB1, NDIML, NDIMR, NL, NLF, NLP1, NLVL,
         $                   NR, NRF, NRP1, SQRE
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SCOPY, SGEMM, SLALS0, SLASDT, XERBLA
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters.
    *
          INFO = 0
    *
          IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
             INFO = -1
          ELSE IF( SMLSIZ.LT.3 ) THEN
             INFO = -2
          ELSE IF( N.LT.SMLSIZ ) THEN
             INFO = -3
          ELSE IF( NRHS.LT.1 ) THEN
             INFO = -4
          ELSE IF( LDB.LT.N ) THEN
             INFO = -6
          ELSE IF( LDBX.LT.N ) THEN
             INFO = -8
          ELSE IF( LDU.LT.N ) THEN
             INFO = -10
          ELSE IF( LDGCOL.LT.N ) THEN
             INFO = -19
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'SLALSA', -INFO )
             RETURN
          END IF
    *
    *     Book-keeping and  setting up the computation tree.
    *
          INODE = 1
          NDIML = INODE + N
          NDIMR = NDIML + N
    *
          CALL SLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ),
         $             IWORK( NDIMR ), SMLSIZ )
    *
    *     The following code applies back the left singular vector factors.
    *     For applying back the right singular vector factors, go to 50.
    *
          IF( ICOMPQ.EQ.1 ) THEN
             GO TO 50
          END IF
    *
    *     The nodes on the bottom level of the tree were solved
    *     by SLASDQ. The corresponding left and right singular vector
    *     matrices are in explicit form. First apply back the left
    *     singular vector matrices.
    *
          NDB1 = ( ND+1 ) / 2
          DO 10 I = NDB1, ND
    *
    *        IC : center row of each node
    *        NL : number of rows of left  subproblem
    *        NR : number of rows of right subproblem
    *        NLF: starting row of the left   subproblem
    *        NRF: starting row of the right  subproblem
    *
             I1 = I - 1
             IC = IWORK( INODE+I1 )
             NL = IWORK( NDIML+I1 )
             NR = IWORK( NDIMR+I1 )
             NLF = IC - NL
             NRF = IC + 1
             CALL SGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU,
         $               B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX )
             CALL SGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU,
         $               B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX )
       10 CONTINUE
    *
    *     Next copy the rows of B that correspond to unchanged rows
    *     in the bidiagonal matrix to BX.
    *
          DO 20 I = 1, ND
             IC = IWORK( INODE+I-1 )
             CALL SCOPY( NRHS, B( IC, 1 ), LDB, BX( IC, 1 ), LDBX )
       20 CONTINUE
    *
    *     Finally go through the left singular vector matrices of all
    *     the other subproblems bottom-up on the tree.
    *
          J = 2**NLVL
          SQRE = 0
    *
          DO 40 LVL = NLVL, 1, -1
             LVL2 = 2*LVL - 1
    *
    *        find the first node LF and last node LL on
    *        the current level LVL
    *
             IF( LVL.EQ.1 ) THEN
                LF = 1
                LL = 1
             ELSE
                LF = 2**( LVL-1 )
                LL = 2*LF - 1
             END IF
             DO 30 I = LF, LL
                IM1 = I - 1
                IC = IWORK( INODE+IM1 )
                NL = IWORK( NDIML+IM1 )
                NR = IWORK( NDIMR+IM1 )
                NLF = IC - NL
                NRF = IC + 1
                J = J - 1
                CALL SLALS0( ICOMPQ, NL, NR, SQRE, NRHS, BX( NLF, 1 ), LDBX,
         $                   B( NLF, 1 ), LDB, PERM( NLF, LVL ),
         $                   GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL,
         $                   GIVNUM( NLF, LVL2 ), LDU, POLES( NLF, LVL2 ),
         $                   DIFL( NLF, LVL ), DIFR( NLF, LVL2 ),
         $                   Z( NLF, LVL ), K( J ), C( J ), S( J ), WORK,
         $                   INFO )
       30    CONTINUE
       40 CONTINUE
          GO TO 90
    *
    *     ICOMPQ = 1: applying back the right singular vector factors.
    *
       50 CONTINUE
    *
    *     First now go through the right singular vector matrices of all
    *     the tree nodes top-down.
    *
          J = 0
          DO 70 LVL = 1, NLVL
             LVL2 = 2*LVL - 1
    *
    *        Find the first node LF and last node LL on
    *        the current level LVL.
    *
             IF( LVL.EQ.1 ) THEN
                LF = 1
                LL = 1
             ELSE
                LF = 2**( LVL-1 )
                LL = 2*LF - 1
             END IF
             DO 60 I = LL, LF, -1
                IM1 = I - 1
                IC = IWORK( INODE+IM1 )
                NL = IWORK( NDIML+IM1 )
                NR = IWORK( NDIMR+IM1 )
                NLF = IC - NL
                NRF = IC + 1
                IF( I.EQ.LL ) THEN
                   SQRE = 0
                ELSE
                   SQRE = 1
                END IF
                J = J + 1
                CALL SLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B( NLF, 1 ), LDB,
         $                   BX( NLF, 1 ), LDBX, PERM( NLF, LVL ),
         $                   GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL,
         $                   GIVNUM( NLF, LVL2 ), LDU, POLES( NLF, LVL2 ),
         $                   DIFL( NLF, LVL ), DIFR( NLF, LVL2 ),
         $                   Z( NLF, LVL ), K( J ), C( J ), S( J ), WORK,
         $                   INFO )
       60    CONTINUE
       70 CONTINUE
    *
    *     The nodes on the bottom level of the tree were solved
    *     by SLASDQ. The corresponding right singular vector
    *     matrices are in explicit form. Apply them back.
    *
          NDB1 = ( ND+1 ) / 2
          DO 80 I = NDB1, ND
             I1 = I - 1
             IC = IWORK( INODE+I1 )
             NL = IWORK( NDIML+I1 )
             NR = IWORK( NDIMR+I1 )
             NLP1 = NL + 1
             IF( I.EQ.ND ) THEN
                NRP1 = NR
             ELSE
                NRP1 = NR + 1
             END IF
             NLF = IC - NL
             NRF = IC + 1
             CALL SGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ), LDU,
         $               B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX )
             CALL SGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ), LDU,
         $               B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX )
       80 CONTINUE
    *
       90 CONTINUE
    *
          RETURN
    *
    *     End of SLALSA
    *
          END