slatrs3.f

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*> \brief \b SLATRS3 solves a triangular system of equations with the scale factors set to prevent overflow.
*
*  Definition:
*  ===========
*
*      SUBROUTINE SLATRS3( UPLO, TRANS, DIAG, NORMIN, N, NRHS, A, LDA,
*                          X, LDX, SCALE, CNORM, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, NORMIN, TRANS, UPLO
*       INTEGER            INFO, LDA, LWORK, LDX, N, NRHS
*       ..
*       .. Array Arguments ..
*       REAL               A( LDA, * ), CNORM( * ), SCALE( * ), 
*                          WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SLATRS3 solves one of the triangular systems
*>
*>    A * X = B * diag(scale)  or  A**T * X = B * diag(scale)
*>
*> with scaling to prevent overflow.  Here A is an upper or lower
*> triangular matrix, A**T denotes the transpose of A. X and B are
*> n by nrhs matrices and scale is an nrhs element vector of scaling
*> factors. A scaling factor scale(j) is usually less than or equal
*> to 1, chosen such that X(:,j) is less than the overflow threshold.
*> If the matrix A is singular (A(j,j) = 0 for some j), then
*> a non-trivial solution to A*X = 0 is returned. If the system is
*> so badly scaled that the solution cannot be represented as
*> (1/scale(k))*X(:,k), then x(:,k) = 0 and scale(k) is returned.
*>
*> This is a BLAS-3 version of LATRS for solving several right
*> hand sides simultaneously.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the matrix A is upper or lower triangular.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the operation applied to A.
*>          = 'N':  Solve A * x = s*b  (No transpose)
*>          = 'T':  Solve A**T* x = s*b  (Transpose)
*>          = 'C':  Solve A**T* x = s*b  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          Specifies whether or not the matrix A is unit triangular.
*>          = 'N':  Non-unit triangular
*>          = 'U':  Unit triangular
*> \endverbatim
*>
*> \param[in] NORMIN
*> \verbatim
*>          NORMIN is CHARACTER*1
*>          Specifies whether CNORM has been set or not.
*>          = 'Y':  CNORM contains the column norms on entry
*>          = 'N':  CNORM is not set on entry.  On exit, the norms will
*>                  be computed and stored in CNORM.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of columns of X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is REAL array, dimension (LDA,N)
*>          The triangular matrix A.  If UPLO = 'U', the leading n by n
*>          upper triangular part of the array A contains the upper
*>          triangular matrix, and the strictly lower triangular part of
*>          A is not referenced.  If UPLO = 'L', the leading n by n lower
*>          triangular part of the array A contains the lower triangular
*>          matrix, and the strictly upper triangular part of A is not
*>          referenced.  If DIAG = 'U', the diagonal elements of A are
*>          also not referenced and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max (1,N).
*> \endverbatim
*>
*> \param[in,out] X
*> \verbatim
*>          X is REAL array, dimension (LDX,NRHS)
*>          On entry, the right hand side B of the triangular system.
*>          On exit, X is overwritten by the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max (1,N).
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*>          SCALE is REAL array, dimension (NRHS)
*>          The scaling factor s(k) is for the triangular system
*>          A * x(:,k) = s(k)*b(:,k)  or  A**T* x(:,k) = s(k)*b(:,k).
*>          If SCALE = 0, the matrix A is singular or badly scaled.
*>          If A(j,j) = 0 is encountered, a non-trivial vector x(:,k)
*>          that is an exact or approximate solution to A*x(:,k) = 0
*>          is returned. If the system so badly scaled that solution
*>          cannot be presented as x(:,k) * 1/s(k), then x(:,k) = 0
*>          is returned.
*> \endverbatim
*>
*> \param[in,out] CNORM
*> \verbatim
*>          CNORM is REAL array, dimension (N)
*>
*>          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
*>          contains the norm of the off-diagonal part of the j-th column
*>          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
*>          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
*>          must be greater than or equal to the 1-norm.
*>
*>          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
*>          returns the 1-norm of the offdiagonal part of the j-th column
*>          of A.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (LWORK).
*>          On exit, if INFO = 0, WORK(1) returns the optimal size of
*>          WORK.
*> \endverbatim
*>
*> \param[in] LWORK
*>          LWORK is INTEGER
*>          LWORK >= MAX(1, 2*NBA * MAX(NBA, MIN(NRHS, 32)), where
*>          NBA = (N + NB - 1)/NB and NB is the optimal block size.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal dimensions 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.
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -k, the k-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup latrs3
*> \par Further Details:
*  =====================
*  \verbatim
*  The algorithm follows the structure of a block triangular solve.
*  The diagonal block is solved with a call to the robust the triangular
*  solver LATRS for every right-hand side RHS = 1, ..., NRHS
*     op(A( J, J )) * x( J, RHS ) = SCALOC * b( J, RHS ),
*  where op( A ) = A or op( A ) = A**T.
*  The linear block updates operate on block columns of X,
*     B( I, K ) - op(A( I, J )) * X( J, K )
*  and use GEMM. To avoid overflow in the linear block update, the worst case
*  growth is estimated. For every RHS, a scale factor s <= 1.0 is computed
*  such that
*     || s * B( I, RHS )||_oo
*   + || op(A( I, J )) ||_oo * || s *  X( J, RHS ) ||_oo <= Overflow threshold
*
*  Once all columns of a block column have been rescaled (BLAS-1), the linear
*  update is executed with GEMM without overflow.
*
*  To limit rescaling, local scale factors track the scaling of column segments.
*  There is one local scale factor s( I, RHS ) per block row I = 1, ..., NBA
*  per right-hand side column RHS = 1, ..., NRHS. The global scale factor
*  SCALE( RHS ) is chosen as the smallest local scale factor s( I, RHS )
*  I = 1, ..., NBA.
*  A triangular solve op(A( J, J )) * x( J, RHS ) = SCALOC * b( J, RHS )
*  updates the local scale factor s( J, RHS ) := s( J, RHS ) * SCALOC. The
*  linear update of potentially inconsistently scaled vector segments
*     s( I, RHS ) * b( I, RHS ) - op(A( I, J )) * ( s( J, RHS )* x( J, RHS ) )
*  computes a consistent scaling SCAMIN = MIN( s(I, RHS ), s(J, RHS) ) and,
*  if necessary, rescales the blocks prior to calling GEMM.
*
*  \endverbatim
*  =====================================================================
*  References:
*  C. C. Kjelgaard Mikkelsen, A. B. Schwarz and L. Karlsson (2019).
*  Parallel robust solution of triangular linear systems. Concurrency
*  and Computation: Practice and Experience, 31(19), e5064.
*
*  Contributor:
*   Angelika Schwarz, Umea University, Sweden.
*
*  =====================================================================
      SUBROUTINE slatrs3( UPLO, TRANS, DIAG, NORMIN, N, NRHS, A, LDA,
     $                    X, LDX, SCALE, CNORM, WORK, LWORK, INFO )
      IMPLICIT NONE
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, TRANS, NORMIN, UPLO
      INTEGER            INFO, LDA, LWORK, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * ), CNORM( * ), X( LDX, * ),
     $                   scale( * ), work( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter( zero = 0.0e+0, one = 1.0e+0 )
      INTEGER            NBMAX, NBMIN, NBRHS, NRHSMIN
      parameter( nrhsmin = 2, nbrhs = 32 )
      parameter( nbmin = 8, nbmax = 64 )
*     ..
*     .. Local Arrays ..
      REAL               W( NBMAX ), XNRM( NBRHS )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, NOTRAN, NOUNIT, UPPER
      INTEGER            AWRK, I, IFIRST, IINC, ILAST, II, I1, I2, J,
     $                   jfirst, jinc, jlast, j1, j2, k, kk, k1, k2,
     $                   lanrm, lds, lscale, nb, nba, nbx, rhs
      REAL               ANRM, BIGNUM, BNRM, RSCAL, SCAL, SCALOC,
     $                   scamin, smlnum, tmax
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      REAL               SLAMCH, SLANGE, SLARMM
      EXTERNAL           ilaenv, lsame, slamch, slange, slarmm
*     ..
*     .. External Subroutines ..
      EXTERNAL           slatrs, sscal, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min
*     ..
*     .. Executable Statements ..
*
      info = 0
      upper = lsame( uplo, 'U' )
      notran = lsame( trans, 'N' )
      nounit = lsame( diag, 'N' )
      lquery = ( lwork.EQ.-1 )
*
*     Partition A and X into blocks.
*
      nb = max( 8, ilaenv( 1, 'SLATRS', '', n, n, -1, -1 ) )
      nb = min( nbmax, nb )
      nba = max( 1, (n + nb - 1) / nb )
      nbx = max( 1, (nrhs + nbrhs - 1) / nbrhs )
*
*     Compute the workspace
*
*     The workspace comprises two parts.
*     The first part stores the local scale factors. Each simultaneously
*     computed right-hand side requires one local scale factor per block
*     row. WORK( I + KK * LDS ) is the scale factor of the vector
*     segment associated with the I-th block row and the KK-th vector
*     in the block column.
      lscale = nba * max( nba, min( nrhs, nbrhs ) )
      lds = nba
*     The second part stores upper bounds of the triangular A. There are
*     a total of NBA x NBA blocks, of which only the upper triangular
*     part or the lower triangular part is referenced. The upper bound of
*     the block A( I, J ) is stored as WORK( AWRK + I + J * NBA ).
      lanrm = nba * nba
      awrk = lscale
      work( 1 ) = lscale + lanrm
*
*     Test the input parameters.
*
      IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
         info = -1
      ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
     $         lsame( trans, 'C' ) ) THEN
         info = -2
      ELSE IF( .NOT.nounit .AND. .NOT.lsame( diag, 'U' ) ) THEN
         info = -3
      ELSE IF( .NOT.lsame( normin, 'Y' ) .AND. .NOT.
     $         lsame( normin, 'N' ) ) THEN
         info = -4
      ELSE IF( n.LT.0 ) THEN
         info = -5
      ELSE IF( nrhs.LT.0 ) THEN
         info = -6
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -8
      ELSE IF( ldx.LT.max( 1, n ) ) THEN
         info = -10
      ELSE IF( .NOT.lquery .AND. lwork.LT.work( 1 ) ) THEN
         info = -14
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SLATRS3', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Initialize scaling factors
*
      DO kk = 1, nrhs
         scale( kk ) = one
      END DO
*
*     Quick return if possible
*
      IF( min( n, nrhs ).EQ.0 )
     $   RETURN
*
*     Determine machine dependent constant to control overflow.
*
      bignum = slamch( 'Overflow' )
      smlnum = slamch( 'Safe Minimum' )
*
*     Use unblocked code for small problems
*
      IF( nrhs.LT.nrhsmin ) THEN
         CALL slatrs( uplo, trans, diag, normin, n, a, lda, x( 1, 1),
     $                scale( 1 ), cnorm, info )
         DO k = 2, nrhs
            CALL slatrs( uplo, trans, diag, 'Y', n, a, lda, x( 1, k ),
     $                   scale( k ), cnorm, info )
         END DO
         RETURN
      END IF
*
*     Compute norms of blocks of A excluding diagonal blocks and find
*     the block with the largest norm TMAX.
*
      tmax = zero
      DO j = 1, nba
         j1 = (j-1)*nb + 1
         j2 = min( j*nb, n ) + 1
         IF ( upper ) THEN
            ifirst = 1
            ilast = j - 1
         ELSE
            ifirst = j + 1
            ilast = nba
         END IF
         DO i = ifirst, ilast
            i1 = (i-1)*nb + 1
            i2 = min( i*nb, n ) + 1
*
*           Compute upper bound of A( I1:I2-1, J1:J2-1 ).
*
            IF( notran ) THEN
               anrm = slange( 'I', i2-i1, j2-j1, a( i1, j1 ), lda, w )
               work( awrk + i+(j-1)*nba ) = anrm
            ELSE
               anrm = slange( '1', i2-i1, j2-j1, a( i1, j1 ), lda, w )
               work( awrk + j+(i-1)*nba ) = anrm
            END IF
            tmax = max( tmax, anrm )
         END DO
      END DO
*
      IF( .NOT. tmax.LE.slamch('Overflow') ) THEN
*
*        Some matrix entries have huge absolute value. At least one upper
*        bound norm( A(I1:I2-1, J1:J2-1), 'I') is not a valid floating-point
*        number, either due to overflow in LANGE or due to Inf in A.
*        Fall back to LATRS. Set normin = 'N' for every right-hand side to
*        force computation of TSCAL in LATRS to avoid the likely overflow
*        in the computation of the column norms CNORM.
*
         DO k = 1, nrhs
            CALL slatrs( uplo, trans, diag, 'N', n, a, lda, x( 1, k ),
     $                   scale( k ), cnorm, info )
         END DO
         RETURN
      END IF
*
*     Every right-hand side requires workspace to store NBA local scale
*     factors. To save workspace, X is computed successively in block columns
*     of width NBRHS, requiring a total of NBA x NBRHS space. If sufficient
*     workspace is available, larger values of NBRHS or NBRHS = NRHS are viable.
      DO k = 1, nbx
*        Loop over block columns (index = K) of X and, for column-wise scalings,
*        over individual columns (index = KK).
*        K1: column index of the first column in X( J, K )
*        K2: column index of the first column in X( J, K+1 )
*        so the K2 - K1 is the column count of the block X( J, K )
         k1 = (k-1)*nbrhs + 1
         k2 = min( k*nbrhs, nrhs ) + 1
*
*        Initialize local scaling factors of current block column X( J, K )
*
         DO kk = 1, k2 - k1
            DO i = 1, nba
               work( i+kk*lds ) = one
            END DO
         END DO
*
         IF( notran ) THEN
*
*           Solve A * X(:, K1:K2-1) = B * diag(scale(K1:K2-1))
*
            IF( upper ) THEN
               jfirst = nba
               jlast = 1
               jinc = -1
            ELSE
               jfirst = 1
               jlast = nba
               jinc = 1
            END IF
         ELSE
*
*           Solve A**T * X(:, K1:K2-1) = B * diag(scale(K1:K2-1))
*
            IF( upper ) THEN
               jfirst = 1
               jlast = nba
               jinc = 1
            ELSE
               jfirst = nba
               jlast = 1
               jinc = -1
            END IF
         END IF
*
         DO j = jfirst, jlast, jinc
*           J1: row index of the first row in A( J, J )
*           J2: row index of the first row in A( J+1, J+1 )
*           so that J2 - J1 is the row count of the block A( J, J )
            j1 = (j-1)*nb + 1
            j2 = min( j*nb, n ) + 1
*
*           Solve op(A( J, J )) * X( J, RHS ) = SCALOC * B( J, RHS )
*           for all right-hand sides in the current block column,
*           one RHS at a time.
*
            DO kk = 1, k2-k1
               rhs = k1 + kk - 1
               IF( kk.EQ.1 ) THEN
                  CALL slatrs( uplo, trans, diag, 'N', j2-j1,
     $                         a( j1, j1 ), lda, x( j1, rhs ),
     $                         scaloc, cnorm, info )
               ELSE
                  CALL slatrs( uplo, trans, diag, 'Y', j2-j1,
     $                         a( j1, j1 ), lda, x( j1, rhs ),
     $                         scaloc, cnorm, info )
               END IF
*              Find largest absolute value entry in the vector segment
*              X( J1:J2-1, RHS ) as an upper bound for the worst case
*              growth in the linear updates.
               xnrm( kk ) = slange( 'I', j2-j1, 1, x( j1, rhs ),
     $                              ldx, w )
*
               IF( scaloc .EQ. zero ) THEN
*                 LATRS found that A is singular through A(j,j) = 0.
*                 Reset the computation x(1:n) = 0, x(j) = 1, SCALE = 0
*                 and compute A*x = 0 (or A**T*x = 0). Note that
*                 X(J1:J2-1, KK) is set by LATRS.
                  scale( rhs ) = zero
                  DO ii = 1, j1-1
                     x( ii, kk ) = zero
                  END DO
                  DO ii = j2, n
                     x( ii, kk ) = zero
                  END DO
*                 Discard the local scale factors.
                  DO ii = 1, nba
                     work( ii+kk*lds ) = one
                  END DO
                  scaloc = one
               ELSE IF( scaloc*work( j+kk*lds ) .EQ. zero ) THEN
*                 LATRS computed a valid scale factor, but combined with
*                 the current scaling the solution does not have a
*                 scale factor > 0.
*
*                 Set WORK( J+KK*LDS ) to smallest valid scale
*                 factor and increase SCALOC accordingly.
                  scal = work( j+kk*lds ) / smlnum
                  scaloc = scaloc * scal
                  work( j+kk*lds ) = smlnum
*                 If LATRS overestimated the growth, x may be
*                 rescaled to preserve a valid combined scale
*                 factor WORK( J, KK ) > 0.
                  rscal = one / scaloc
                  IF( xnrm( kk )*rscal .LE. bignum ) THEN
                     xnrm( kk ) = xnrm( kk ) * rscal
                     CALL sscal( j2-j1, rscal, x( j1, rhs ), 1 )
                     scaloc = one
                  ELSE
*                    The system op(A) * x = b is badly scaled and its
*                    solution cannot be represented as (1/scale) * x.
*                    Set x to zero. This approach deviates from LATRS
*                    where a completely meaningless non-zero vector
*                    is returned that is not a solution to op(A) * x = b.
                     scale( rhs ) = zero
                     DO ii = 1, n
                        x( ii, kk ) = zero
                     END DO
*                    Discard the local scale factors.
                     DO ii = 1, nba
                        work( ii+kk*lds ) = one
                     END DO
                     scaloc = one
                  END IF
               END IF
               scaloc = scaloc * work( j+kk*lds )
               work( j+kk*lds ) = scaloc
            END DO
*
*           Linear block updates
*
            IF( notran ) THEN
               IF( upper ) THEN
                  ifirst = j - 1
                  ilast = 1
                  iinc = -1
               ELSE
                  ifirst = j + 1
                  ilast = nba
                  iinc = 1
               END IF
            ELSE
               IF( upper ) THEN
                  ifirst = j + 1
                  ilast = nba
                  iinc = 1
               ELSE
                  ifirst = j - 1
                  ilast = 1
                  iinc = -1
               END IF
            END IF
*
            DO i = ifirst, ilast, iinc
*              I1: row index of the first column in X( I, K )
*              I2: row index of the first column in X( I+1, K )
*              so the I2 - I1 is the row count of the block X( I, K )
               i1 = (i-1)*nb + 1
               i2 = min( i*nb, n ) + 1
*
*              Prepare the linear update to be executed with GEMM.
*              For each column, compute a consistent scaling, a
*              scaling factor to survive the linear update, and
*              rescale the column segments, if necessary. Then
*              the linear update is safely executed.
*
               DO kk = 1, k2-k1
                  rhs = k1 + kk - 1
*                 Compute consistent scaling
                  scamin = min( work( i+kk*lds), work( j+kk*lds ) )
*
*                 Compute scaling factor to survive the linear update
*                 simulating consistent scaling.
*
                  bnrm = slange( 'I', i2-i1, 1, x( i1, rhs ), ldx, w )
                  bnrm = bnrm*( scamin / work( i+kk*lds ) )
                  xnrm( kk ) = xnrm( kk )*(scamin / work( j+kk*lds ))
                  anrm = work( awrk + i+(j-1)*nba )
                  scaloc = slarmm( anrm, xnrm( kk ), bnrm )
*
*                 Simultaneously apply the robust update factor and the
*                 consistency scaling factor to B( I, KK ) and B( J, KK ).
*
                  scal = ( scamin / work( i+kk*lds) )*scaloc
                  IF( scal.NE.one ) THEN
                     CALL sscal( i2-i1, scal, x( i1, rhs ), 1 )
                     work( i+kk*lds ) = scamin*scaloc
                  END IF
*
                  scal = ( scamin / work( j+kk*lds ) )*scaloc
                  IF( scal.NE.one ) THEN
                     CALL sscal( j2-j1, scal, x( j1, rhs ), 1 )
                     work( j+kk*lds ) = scamin*scaloc
                  END IF
               END DO
*
               IF( notran ) THEN
*
*                 B( I, K ) := B( I, K ) - A( I, J ) * X( J, K )
*
                  CALL sgemm( 'N', 'N', i2-i1, k2-k1, j2-j1, -one,
     $                        a( i1, j1 ), lda, x( j1, k1 ), ldx,
     $                        one, x( i1, k1 ), ldx )
               ELSE
*
*                 B( I, K ) := B( I, K ) - A( I, J )**T * X( J, K )
*
                  CALL sgemm( 'T', 'N', i2-i1, k2-k1, j2-j1, -one,
     $                        a( j1, i1 ), lda, x( j1, k1 ), ldx,
     $                        one, x( i1, k1 ), ldx )
               END IF
            END DO
         END DO
*
*        Reduce local scaling factors
*
         DO kk = 1, k2-k1
            rhs = k1 + kk - 1
            DO i = 1, nba
               scale( rhs ) = min( scale( rhs ), work( i+kk*lds ) )
            END DO
         END DO
*
*        Realize consistent scaling
*
         DO kk = 1, k2-k1
            rhs = k1 + kk - 1
            IF( scale( rhs ).NE.one .AND. scale( rhs ).NE. zero ) THEN
               DO i = 1, nba
                  i1 = (i-1)*nb + 1
                  i2 = min( i*nb, n ) + 1
                  scal = scale( rhs ) / work( i+kk*lds )
                  IF( scal.NE.one )
     $               CALL sscal( i2-i1, scal, x( i1, rhs ), 1 )
               END DO
            END IF
         END DO
      END DO
      RETURN
*
*     End of SLATRS3
*
      END