*> \brief \b DLATRS3 solves a triangular system of equations with the scale factors set to prevent overflow.
*
* Definition:
* ===========
*
* SUBROUTINE DLATRS3( 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 ..
* DOUBLE PRECISION A( LDA, * ), CNORM( * ), SCALE( * ),
* WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLATRS3 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (MAX(1,LWORK)).
*> On exit, if INFO = 0, WORK(1) returns the optimal size of
*> WORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*>
*> If MIN(N,NRHS) = 0, LWORK >= 1, else
*> 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.
*> \endverbatim
*>
*> \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 DLATRS3( 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 ..
DOUBLE PRECISION A( LDA, * ), CNORM( * ), X( LDX, * ),
$ SCALE( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
INTEGER NBMAX, NBMIN, NBRHS, NRHSMIN
PARAMETER ( NRHSMIN = 2, NBRHS = 32 )
PARAMETER ( NBMIN = 8, NBMAX = 64 )
* ..
* .. Local Arrays ..
DOUBLE PRECISION 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, LWMIN
DOUBLE PRECISION ANRM, BIGNUM, BNRM, RSCAL, SCAL, SCALOC,
$ SCAMIN, SMLNUM, TMAX
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE, DLARMM
EXTERNAL DLAMCH, DLANGE, DLARMM, ILAENV,
$ LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLATRS, DSCAL, 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, 'DLATRS', '', 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
*
IF( MIN( N, NRHS ).EQ.0 ) THEN
LWMIN = 1
ELSE
LWMIN = LSCALE + LANRM
END IF
WORK( 1 ) = LWMIN
*
* 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.LWMIN ) THEN
INFO = -14
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLATRS3', -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 = DLAMCH( 'Overflow' )
SMLNUM = DLAMCH( 'Safe Minimum' )
*
* Use unblocked code for small problems
*
IF( NRHS.LT.NRHSMIN ) THEN
CALL DLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X( 1, 1),
$ SCALE( 1 ), CNORM, INFO )
DO K = 2, NRHS
CALL DLATRS( 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 = DLANGE( 'I', I2-I1, J2-J1, A( I1, J1 ), LDA,
$ W )
WORK( AWRK + I+(J-1)*NBA ) = ANRM
ELSE
ANRM = DLANGE( '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.DLAMCH('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 DLATRS( 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 DLATRS( UPLO, TRANS, DIAG, 'N', J2-J1,
$ A( J1, J1 ), LDA, X( J1, RHS ),
$ SCALOC, CNORM, INFO )
ELSE
CALL DLATRS( 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 ) = DLANGE( '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 DSCAL( 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 = DLANGE( '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 = DLARMM( 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 DSCAL( 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 DSCAL( 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 DGEMM( '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( J, I )**T * X( J, K )
*
CALL DGEMM( '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 DSCAL( I2-I1, SCAL, X( I1, RHS ), 1 )
END DO
END IF
END DO
END DO
*
WORK( 1 ) = LWMIN
*
RETURN
*
* End of DLATRS3
*
END