path: root/SRC/sla_gbrfsx_extended.f

*> \brief \b SLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at 
*            http://www.netlib.org/lapack/explore-html/ 
*
*> \htmlonly
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*> [TXT]</a>
*> \endhtmlonly 
*
*  Definition:
*  ===========
*
*       SUBROUTINE SLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, KL, KU,
*                                       NRHS, AB, LDAB, AFB, LDAFB, IPIV,
*                                       COLEQU, C, B, LDB, Y, LDY,
*                                       BERR_OUT, N_NORMS, ERR_BNDS_NORM,
*                                       ERR_BNDS_COMP, RES, AYB, DY,
*                                       Y_TAIL, RCOND, ITHRESH, RTHRESH,
*                                       DZ_UB, IGNORE_CWISE, INFO )
* 
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDAB, LDAFB, LDB, LDY, N, KL, KU, NRHS,
*      $                   PREC_TYPE, TRANS_TYPE, N_NORMS, ITHRESH
*       LOGICAL            COLEQU, IGNORE_CWISE
*       REAL               RTHRESH, DZ_UB
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       REAL               AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
*      $                   Y( LDY, * ), RES(*), DY(*), Y_TAIL(*)
*       REAL               C( * ), AYB(*), RCOND, BERR_OUT(*),
*      $                   ERR_BNDS_NORM( NRHS, * ),
*      $                   ERR_BNDS_COMP( NRHS, * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SLA_GBRFSX_EXTENDED improves the computed solution to a system of
*> linear equations by performing extra-precise iterative refinement
*> and provides error bounds and backward error estimates for the solution.
*> This subroutine is called by SGBRFSX to perform iterative refinement.
*> In addition to normwise error bound, the code provides maximum
*> componentwise error bound if possible. See comments for ERR_BNDS_NORM
*> and ERR_BNDS_COMP for details of the error bounds. Note that this
*> subroutine is only resonsible for setting the second fields of
*> ERR_BNDS_NORM and ERR_BNDS_COMP.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] PREC_TYPE
*> \verbatim
*>          PREC_TYPE is INTEGER
*>     Specifies the intermediate precision to be used in refinement.
*>     The value is defined by ILAPREC(P) where P is a CHARACTER and
*>     P    = 'S':  Single
*>          = 'D':  Double
*>          = 'I':  Indigenous
*>          = 'X', 'E':  Extra
*> \endverbatim
*>
*> \param[in] TRANS_TYPE
*> \verbatim
*>          TRANS_TYPE is INTEGER
*>     Specifies the transposition operation on A.
*>     The value is defined by ILATRANS(T) where T is a CHARACTER and
*>     T    = 'N':  No transpose
*>          = 'T':  Transpose
*>          = 'C':  Conjugate transpose
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>     The number of linear equations, i.e., the order of the
*>     matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*>          KL is INTEGER
*>     The number of subdiagonals within the band of A.  KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*>          KU is INTEGER
*>     The number of superdiagonals within the band of A.  KU >= 0
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>     The number of right-hand-sides, i.e., the number of columns of the
*>     matrix B.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is REAL array, dimension (LDAB,N)
*>     On entry, the N-by-N matrix AB.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>     The leading dimension of the array AB.  LDAB >= max(1,N).
*> \endverbatim
*>
*> \param[in] AFB
*> \verbatim
*>          AFB is REAL array, dimension (LDAFB,N)
*>     The factors L and U from the factorization
*>     A = P*L*U as computed by SGBTRF.
*> \endverbatim
*>
*> \param[in] LDAFB
*> \verbatim
*>          LDAFB is INTEGER
*>     The leading dimension of the array AF.  LDAFB >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>     The pivot indices from the factorization A = P*L*U
*>     as computed by SGBTRF; row i of the matrix was interchanged
*>     with row IPIV(i).
*> \endverbatim
*>
*> \param[in] COLEQU
*> \verbatim
*>          COLEQU is LOGICAL
*>     If .TRUE. then column equilibration was done to A before calling
*>     this routine. This is needed to compute the solution and error
*>     bounds correctly.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*>          C is REAL array, dimension (N)
*>     The column scale factors for A. If COLEQU = .FALSE., C
*>     is not accessed. If C is input, each element of C should be a power
*>     of the radix to ensure a reliable solution and error estimates.
*>     Scaling by powers of the radix does not cause rounding errors unless
*>     the result underflows or overflows. Rounding errors during scaling
*>     lead to refining with a matrix that is not equivalent to the
*>     input matrix, producing error estimates that may not be
*>     reliable.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is REAL array, dimension (LDB,NRHS)
*>     The right-hand-side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>     The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*>          Y is REAL array, dimension (LDY,NRHS)
*>     On entry, the solution matrix X, as computed by SGBTRS.
*>     On exit, the improved solution matrix Y.
*> \endverbatim
*>
*> \param[in] LDY
*> \verbatim
*>          LDY is INTEGER
*>     The leading dimension of the array Y.  LDY >= max(1,N).
*> \endverbatim
*>
*> \param[out] BERR_OUT
*> \verbatim
*>          BERR_OUT is REAL array, dimension (NRHS)
*>     On exit, BERR_OUT(j) contains the componentwise relative backward
*>     error for right-hand-side j from the formula
*>         max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
*>     where abs(Z) is the componentwise absolute value of the matrix
*>     or vector Z. This is computed by SLA_LIN_BERR.
*> \endverbatim
*>
*> \param[in] N_NORMS
*> \verbatim
*>          N_NORMS is INTEGER
*>     Determines which error bounds to return (see ERR_BNDS_NORM
*>     and ERR_BNDS_COMP).
*>     If N_NORMS >= 1 return normwise error bounds.
*>     If N_NORMS >= 2 return componentwise error bounds.
*> \endverbatim
*>
*> \param[in,out] ERR_BNDS_NORM
*> \verbatim
*>          ERR_BNDS_NORM is REAL array, dimension
*>                    (NRHS, N_ERR_BNDS)
*>     For each right-hand side, this array contains information about
*>     various error bounds and condition numbers corresponding to the
*>     normwise relative error, which is defined as follows:
*>
*>     Normwise relative error in the ith solution vector:
*>             max_j (abs(XTRUE(j,i) - X(j,i)))
*>            ------------------------------
*>                  max_j abs(X(j,i))
*>
*>     The array is indexed by the type of error information as described
*>     below. There currently are up to three pieces of information
*>     returned.
*>
*>     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
*>     right-hand side.
*>
*>     The second index in ERR_BNDS_NORM(:,err) contains the following
*>     three fields:
*>     err = 1 "Trust/don't trust" boolean. Trust the answer if the
*>              reciprocal condition number is less than the threshold
*>              sqrt(n) * slamch('Epsilon').
*>
*>     err = 2 "Guaranteed" error bound: The estimated forward error,
*>              almost certainly within a factor of 10 of the true error
*>              so long as the next entry is greater than the threshold
*>              sqrt(n) * slamch('Epsilon'). This error bound should only
*>              be trusted if the previous boolean is true.
*>
*>     err = 3  Reciprocal condition number: Estimated normwise
*>              reciprocal condition number.  Compared with the threshold
*>              sqrt(n) * slamch('Epsilon') to determine if the error
*>              estimate is "guaranteed". These reciprocal condition
*>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
*>              appropriately scaled matrix Z.
*>              Let Z = S*A, where S scales each row by a power of the
*>              radix so all absolute row sums of Z are approximately 1.
*>
*>     This subroutine is only responsible for setting the second field
*>     above.
*>     See Lapack Working Note 165 for further details and extra
*>     cautions.
*> \endverbatim
*>
*> \param[in,out] ERR_BNDS_COMP
*> \verbatim
*>          ERR_BNDS_COMP is REAL array, dimension
*>                    (NRHS, N_ERR_BNDS)
*>     For each right-hand side, this array contains information about
*>     various error bounds and condition numbers corresponding to the
*>     componentwise relative error, which is defined as follows:
*>
*>     Componentwise relative error in the ith solution vector:
*>                    abs(XTRUE(j,i) - X(j,i))
*>             max_j ----------------------
*>                         abs(X(j,i))
*>
*>     The array is indexed by the right-hand side i (on which the
*>     componentwise relative error depends), and the type of error
*>     information as described below. There currently are up to three
*>     pieces of information returned for each right-hand side. If
*>     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
*>     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS .LT. 3, then at most
*>     the first (:,N_ERR_BNDS) entries are returned.
*>
*>     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
*>     right-hand side.
*>
*>     The second index in ERR_BNDS_COMP(:,err) contains the following
*>     three fields:
*>     err = 1 "Trust/don't trust" boolean. Trust the answer if the
*>              reciprocal condition number is less than the threshold
*>              sqrt(n) * slamch('Epsilon').
*>
*>     err = 2 "Guaranteed" error bound: The estimated forward error,
*>              almost certainly within a factor of 10 of the true error
*>              so long as the next entry is greater than the threshold
*>              sqrt(n) * slamch('Epsilon'). This error bound should only
*>              be trusted if the previous boolean is true.
*>
*>     err = 3  Reciprocal condition number: Estimated componentwise
*>              reciprocal condition number.  Compared with the threshold
*>              sqrt(n) * slamch('Epsilon') to determine if the error
*>              estimate is "guaranteed". These reciprocal condition
*>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
*>              appropriately scaled matrix Z.
*>              Let Z = S*(A*diag(x)), where x is the solution for the
*>              current right-hand side and S scales each row of
*>              A*diag(x) by a power of the radix so all absolute row
*>              sums of Z are approximately 1.
*>
*>     This subroutine is only responsible for setting the second field
*>     above.
*>     See Lapack Working Note 165 for further details and extra
*>     cautions.
*> \endverbatim
*>
*> \param[in] RES
*> \verbatim
*>          RES is REAL array, dimension (N)
*>     Workspace to hold the intermediate residual.
*> \endverbatim
*>
*> \param[in] AYB
*> \verbatim
*>          AYB is REAL array, dimension (N)
*>     Workspace. This can be the same workspace passed for Y_TAIL.
*> \endverbatim
*>
*> \param[in] DY
*> \verbatim
*>          DY is REAL array, dimension (N)
*>     Workspace to hold the intermediate solution.
*> \endverbatim
*>
*> \param[in] Y_TAIL
*> \verbatim
*>          Y_TAIL is REAL array, dimension (N)
*>     Workspace to hold the trailing bits of the intermediate solution.
*> \endverbatim
*>
*> \param[in] RCOND
*> \verbatim
*>          RCOND is REAL
*>     Reciprocal scaled condition number.  This is an estimate of the
*>     reciprocal Skeel condition number of the matrix A after
*>     equilibration (if done).  If this is less than the machine
*>     precision (in particular, if it is zero), the matrix is singular
*>     to working precision.  Note that the error may still be small even
*>     if this number is very small and the matrix appears ill-
*>     conditioned.
*> \endverbatim
*>
*> \param[in] ITHRESH
*> \verbatim
*>          ITHRESH is INTEGER
*>     The maximum number of residual computations allowed for
*>     refinement. The default is 10. For 'aggressive' set to 100 to
*>     permit convergence using approximate factorizations or
*>     factorizations other than LU. If the factorization uses a
*>     technique other than Gaussian elimination, the guarantees in
*>     ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.
*> \endverbatim
*>
*> \param[in] RTHRESH
*> \verbatim
*>          RTHRESH is REAL
*>     Determines when to stop refinement if the error estimate stops
*>     decreasing. Refinement will stop when the next solution no longer
*>     satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
*>     the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
*>     default value is 0.5. For 'aggressive' set to 0.9 to permit
*>     convergence on extremely ill-conditioned matrices. See LAWN 165
*>     for more details.
*> \endverbatim
*>
*> \param[in] DZ_UB
*> \verbatim
*>          DZ_UB is REAL
*>     Determines when to start considering componentwise convergence.
*>     Componentwise convergence is only considered after each component
*>     of the solution Y is stable, which we definte as the relative
*>     change in each component being less than DZ_UB. The default value
*>     is 0.25, requiring the first bit to be stable. See LAWN 165 for
*>     more details.
*> \endverbatim
*>
*> \param[in] IGNORE_CWISE
*> \verbatim
*>          IGNORE_CWISE is LOGICAL
*>     If .TRUE. then ignore componentwise convergence. Default value
*>     is .FALSE..
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>       = 0:  Successful exit.
*>       < 0:  if INFO = -i, the ith argument to SGBTRS had an illegal
*>             value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date September 2012
*
*> \ingroup realGBcomputational
*
*  =====================================================================
      SUBROUTINE SLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, KL, KU,
     $                                NRHS, AB, LDAB, AFB, LDAFB, IPIV,
     $                                COLEQU, C, B, LDB, Y, LDY,
     $                                BERR_OUT, N_NORMS, ERR_BNDS_NORM,
     $                                ERR_BNDS_COMP, RES, AYB, DY,
     $                                Y_TAIL, RCOND, ITHRESH, RTHRESH,
     $                                DZ_UB, IGNORE_CWISE, INFO )
*
*  -- LAPACK computational routine (version 3.4.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     September 2012
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDAB, LDAFB, LDB, LDY, N, KL, KU, NRHS,
     $                   PREC_TYPE, TRANS_TYPE, N_NORMS, ITHRESH
      LOGICAL            COLEQU, IGNORE_CWISE
      REAL               RTHRESH, DZ_UB
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      REAL               AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
     $                   Y( LDY, * ), RES(*), DY(*), Y_TAIL(*)
      REAL               C( * ), AYB(*), RCOND, BERR_OUT(*),
     $                   ERR_BNDS_NORM( NRHS, * ),
     $                   ERR_BNDS_COMP( NRHS, * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      CHARACTER          TRANS
      INTEGER            CNT, I, J, M, X_STATE, Z_STATE, Y_PREC_STATE
      REAL               YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
     $                   DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
     $                   DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
     $                   EPS, HUGEVAL, INCR_THRESH
      LOGICAL            INCR_PREC
*     ..
*     .. Parameters ..
      INTEGER            UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
     $                   NOPROG_STATE, BASE_RESIDUAL, EXTRA_RESIDUAL,
     $                   EXTRA_Y
      PARAMETER          ( UNSTABLE_STATE = 0, WORKING_STATE = 1,
     $                   CONV_STATE = 2, NOPROG_STATE = 3 )
      PARAMETER          ( BASE_RESIDUAL = 0, EXTRA_RESIDUAL = 1,
     $                   EXTRA_Y = 2 )
      INTEGER            FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
      INTEGER            RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
      INTEGER            CMP_ERR_I, PIV_GROWTH_I
      PARAMETER          ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
     $                   BERR_I = 3 )
      PARAMETER          ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
      PARAMETER          ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
     $                   PIV_GROWTH_I = 9 )
      INTEGER            LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
     $                   LA_LINRX_CWISE_I
      PARAMETER          ( LA_LINRX_ITREF_I = 1,
     $                   LA_LINRX_ITHRESH_I = 2 )
      PARAMETER          ( LA_LINRX_CWISE_I = 3 )
      INTEGER            LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
     $                   LA_LINRX_RCOND_I
      PARAMETER          ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 )
      PARAMETER          ( LA_LINRX_RCOND_I = 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           SAXPY, SCOPY, SGBTRS, SGBMV, BLAS_SGBMV_X,
     $                   BLAS_SGBMV2_X, SLA_GBAMV, SLA_WWADDW, SLAMCH,
     $                   CHLA_TRANSTYPE, SLA_LIN_BERR
      REAL               SLAMCH
      CHARACTER          CHLA_TRANSTYPE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
      IF (INFO.NE.0) RETURN
      TRANS = CHLA_TRANSTYPE(TRANS_TYPE)
      EPS = SLAMCH( 'Epsilon' )
      HUGEVAL = SLAMCH( 'Overflow' )
*     Force HUGEVAL to Inf
      HUGEVAL = HUGEVAL * HUGEVAL
*     Using HUGEVAL may lead to spurious underflows.
      INCR_THRESH = REAL( N ) * EPS
      M = KL+KU+1

      DO J = 1, NRHS
         Y_PREC_STATE = EXTRA_RESIDUAL
         IF ( Y_PREC_STATE .EQ. EXTRA_Y ) THEN
            DO I = 1, N
               Y_TAIL( I ) = 0.0
            END DO
         END IF

         DXRAT = 0.0
         DXRATMAX = 0.0
         DZRAT = 0.0
         DZRATMAX = 0.0
         FINAL_DX_X = HUGEVAL
         FINAL_DZ_Z = HUGEVAL
         PREVNORMDX = HUGEVAL
         PREV_DZ_Z = HUGEVAL
         DZ_Z = HUGEVAL
         DX_X = HUGEVAL

         X_STATE = WORKING_STATE
         Z_STATE = UNSTABLE_STATE
         INCR_PREC = .FALSE.

         DO CNT = 1, ITHRESH
*
*        Compute residual RES = B_s - op(A_s) * Y,
*            op(A) = A, A**T, or A**H depending on TRANS (and type).
*
            CALL SCOPY( N, B( 1, J ), 1, RES, 1 )
            IF ( Y_PREC_STATE .EQ. BASE_RESIDUAL ) THEN
               CALL SGBMV( TRANS, M, N, KL, KU, -1.0, AB, LDAB,
     $              Y( 1, J ), 1, 1.0, RES, 1 )
            ELSE IF ( Y_PREC_STATE .EQ. EXTRA_RESIDUAL ) THEN
               CALL BLAS_SGBMV_X( TRANS_TYPE, N, N, KL, KU,
     $              -1.0, AB, LDAB, Y( 1, J ), 1, 1.0, RES, 1,
     $              PREC_TYPE )
            ELSE
               CALL BLAS_SGBMV2_X( TRANS_TYPE, N, N, KL, KU, -1.0,
     $              AB, LDAB, Y( 1, J ), Y_TAIL, 1, 1.0, RES, 1,
     $              PREC_TYPE )
            END IF

!        XXX: RES is no longer needed.
            CALL SCOPY( N, RES, 1, DY, 1 )
            CALL SGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV, DY, N,
     $           INFO )
*
*         Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
*
            NORMX = 0.0
            NORMY = 0.0
            NORMDX = 0.0
            DZ_Z = 0.0
            YMIN = HUGEVAL

            DO I = 1, N
               YK = ABS( Y( I, J ) )
               DYK = ABS( DY( I ) )

               IF ( YK .NE. 0.0 ) THEN
                  DZ_Z = MAX( DZ_Z, DYK / YK )
               ELSE IF ( DYK .NE. 0.0 ) THEN
                  DZ_Z = HUGEVAL
               END IF

               YMIN = MIN( YMIN, YK )

               NORMY = MAX( NORMY, YK )

               IF ( COLEQU ) THEN
                  NORMX = MAX( NORMX, YK * C( I ) )
                  NORMDX = MAX( NORMDX, DYK * C( I ) )
               ELSE
                  NORMX = NORMY
                  NORMDX = MAX( NORMDX, DYK )
               END IF
            END DO

            IF ( NORMX .NE. 0.0 ) THEN
               DX_X = NORMDX / NORMX
            ELSE IF ( NORMDX .EQ. 0.0 ) THEN
               DX_X = 0.0
            ELSE
               DX_X = HUGEVAL
            END IF

            DXRAT = NORMDX / PREVNORMDX
            DZRAT = DZ_Z / PREV_DZ_Z
*
*         Check termination criteria.
*
            IF ( .NOT.IGNORE_CWISE
     $           .AND. YMIN*RCOND .LT. INCR_THRESH*NORMY
     $           .AND. Y_PREC_STATE .LT. EXTRA_Y )
     $           INCR_PREC = .TRUE.

            IF ( X_STATE .EQ. NOPROG_STATE .AND. DXRAT .LE. RTHRESH )
     $           X_STATE = WORKING_STATE
            IF ( X_STATE .EQ. WORKING_STATE ) THEN
               IF ( DX_X .LE. EPS ) THEN
                  X_STATE = CONV_STATE
               ELSE IF ( DXRAT .GT. RTHRESH ) THEN
                  IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN
                     INCR_PREC = .TRUE.
                  ELSE
                     X_STATE = NOPROG_STATE
                  END IF
               ELSE
                  IF ( DXRAT .GT. DXRATMAX ) DXRATMAX = DXRAT
               END IF
               IF ( X_STATE .GT. WORKING_STATE ) FINAL_DX_X = DX_X
            END IF

            IF ( Z_STATE .EQ. UNSTABLE_STATE .AND. DZ_Z .LE. DZ_UB )
     $           Z_STATE = WORKING_STATE
            IF ( Z_STATE .EQ. NOPROG_STATE .AND. DZRAT .LE. RTHRESH )
     $           Z_STATE = WORKING_STATE
            IF ( Z_STATE .EQ. WORKING_STATE ) THEN
               IF ( DZ_Z .LE. EPS ) THEN
                  Z_STATE = CONV_STATE
               ELSE IF ( DZ_Z .GT. DZ_UB ) THEN
                  Z_STATE = UNSTABLE_STATE
                  DZRATMAX = 0.0
                  FINAL_DZ_Z = HUGEVAL
               ELSE IF ( DZRAT .GT. RTHRESH ) THEN
                  IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN
                     INCR_PREC = .TRUE.
                  ELSE
                     Z_STATE = NOPROG_STATE
                  END IF
               ELSE
                  IF ( DZRAT .GT. DZRATMAX ) DZRATMAX = DZRAT
               END IF
               IF ( Z_STATE .GT. WORKING_STATE ) FINAL_DZ_Z = DZ_Z
            END IF
*
*           Exit if both normwise and componentwise stopped working,
*           but if componentwise is unstable, let it go at least two
*           iterations.
*
            IF ( X_STATE.NE.WORKING_STATE ) THEN
               IF ( IGNORE_CWISE ) GOTO 666
               IF ( Z_STATE.EQ.NOPROG_STATE .OR. Z_STATE.EQ.CONV_STATE )
     $              GOTO 666
               IF ( Z_STATE.EQ.UNSTABLE_STATE .AND. CNT.GT.1 ) GOTO 666
            END IF

            IF ( INCR_PREC ) THEN
               INCR_PREC = .FALSE.
               Y_PREC_STATE = Y_PREC_STATE + 1
               DO I = 1, N
                  Y_TAIL( I ) = 0.0
               END DO
            END IF

            PREVNORMDX = NORMDX
            PREV_DZ_Z = DZ_Z
*
*           Update soluton.
*
            IF (Y_PREC_STATE .LT. EXTRA_Y) THEN
               CALL SAXPY( N, 1.0, DY, 1, Y(1,J), 1 )
            ELSE
               CALL SLA_WWADDW( N, Y(1,J), Y_TAIL, DY )
            END IF

         END DO
*        Target of "IF (Z_STOP .AND. X_STOP)".  Sun's f77 won't EXIT.
 666     CONTINUE
*
*     Set final_* when cnt hits ithresh.
*
         IF ( X_STATE .EQ. WORKING_STATE ) FINAL_DX_X = DX_X
         IF ( Z_STATE .EQ. WORKING_STATE ) FINAL_DZ_Z = DZ_Z
*
*     Compute error bounds.
*
         IF ( N_NORMS .GE. 1 ) THEN
            ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) =
     $           FINAL_DX_X / (1 - DXRATMAX)
         END IF
         IF (N_NORMS .GE. 2) THEN
            ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) =
     $           FINAL_DZ_Z / (1 - DZRATMAX)
         END IF
*
*     Compute componentwise relative backward error from formula
*         max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
*     where abs(Z) is the componentwise absolute value of the matrix
*     or vector Z.
*
*        Compute residual RES = B_s - op(A_s) * Y,
*            op(A) = A, A**T, or A**H depending on TRANS (and type).
*
         CALL SCOPY( N, B( 1, J ), 1, RES, 1 )
         CALL SGBMV(TRANS, N, N, KL, KU, -1.0, AB, LDAB, Y(1,J),
     $        1, 1.0, RES, 1 )

         DO I = 1, N
            AYB( I ) = ABS( B( I, J ) )
         END DO
*
*     Compute abs(op(A_s))*abs(Y) + abs(B_s).
*
        CALL SLA_GBAMV( TRANS_TYPE, N, N, KL, KU, 1.0,
     $        AB, LDAB, Y(1, J), 1, 1.0, AYB, 1 )

         CALL SLA_LIN_BERR( N, N, 1, RES, AYB, BERR_OUT( J ) )
*
*     End of loop for each RHS
*
      END DO
*
      RETURN
      END