da/d2b/dla__gerfsx__extended_8f_source.html

 *> \brief \b DLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general 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
 *> Download DLA_GERFSX_EXTENDED + dependencies
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 *> [TGZ]</a>
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 *> [ZIP]</a>
 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dla_gerfsx_extended.f">
 *> [TXT]</a>
 *> \endhtmlonly
 *
 *  Definition:
 *  ===========
 *
 *       SUBROUTINE DLA_GERFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, NRHS, A,
 *                                       LDA, AF, LDAF, IPIV, COLEQU, C, B,
 *                                       LDB, Y, LDY, BERR_OUT, N_NORMS,
 *                                       ERRS_N, ERRS_C, RES, AYB, DY,
 *                                       Y_TAIL, RCOND, ITHRESH, RTHRESH,
 *                                       DZ_UB, IGNORE_CWISE, INFO )
 *
 *       .. Scalar Arguments ..
 *       INTEGER            INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
 *      $                   TRANS_TYPE, N_NORMS, ITHRESH
 *       LOGICAL            COLEQU, IGNORE_CWISE
 *       DOUBLE PRECISION   RTHRESH, DZ_UB
 *       ..
 *       .. Array Arguments ..
 *       INTEGER            IPIV( * )
 *       DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
 *      $                   Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * )
 *       DOUBLE PRECISION   C( * ), AYB( * ), RCOND, BERR_OUT( * ),
 *      $                   ERRS_N( NRHS, * ), ERRS_C( NRHS, * )
 *       ..
 *
 *
 *> \par Purpose:
 *  =============
 *>
 *> \verbatim
 *>
 *>
 *> DLA_GERFSX_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 DGERFSX to perform iterative refinement.
 *> In addition to normwise error bound, the code provides maximum
 *> componentwise error bound if possible. See comments for ERRS_N
 *> and ERRS_C for details of the error bounds. Note that this
 *> subroutine is only resonsible for setting the second fields of
 *> ERRS_N and ERRS_C.
 *> \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] NRHS
 *> \verbatim
 *>          NRHS is INTEGER
 *>     The number of right-hand-sides, i.e., the number of columns of the
 *>     matrix B.
 *> \endverbatim
 *>
 *> \param[in] A
 *> \verbatim
 *>          A is DOUBLE PRECISION array, dimension (LDA,N)
 *>     On entry, the N-by-N matrix A.
 *> \endverbatim
 *>
 *> \param[in] LDA
 *> \verbatim
 *>          LDA is INTEGER
 *>     The leading dimension of the array A.  LDA >= max(1,N).
 *> \endverbatim
 *>
 *> \param[in] AF
 *> \verbatim
 *>          AF is DOUBLE PRECISION array, dimension (LDAF,N)
 *>     The factors L and U from the factorization
 *>     A = P*L*U as computed by DGETRF.
 *> \endverbatim
 *>
 *> \param[in] LDAF
 *> \verbatim
 *>          LDAF is INTEGER
 *>     The leading dimension of the array AF.  LDAF >= 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 DGETRF; 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDY,NRHS)
 *>     On entry, the solution matrix X, as computed by DGETRS.
 *>     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 DOUBLE PRECISION 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 DLA_LIN_BERR.
 *> \endverbatim
 *>
 *> \param[in] N_NORMS
 *> \verbatim
 *>          N_NORMS is INTEGER
 *>     Determines which error bounds to return (see ERRS_N
 *>     and ERRS_C).
 *>     If N_NORMS >= 1 return normwise error bounds.
 *>     If N_NORMS >= 2 return componentwise error bounds.
 *> \endverbatim
 *>
 *> \param[in,out] ERRS_N
 *> \verbatim
 *>          ERRS_N is DOUBLE PRECISION 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 ERRS_N(i,:) corresponds to the ith
 *>     right-hand side.
 *>
 *>     The second index in ERRS_N(:,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] ERRS_C
 *> \verbatim
 *>          ERRS_C is DOUBLE PRECISION 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
 *>     ERRS_C is not accessed.  If N_ERR_BNDS .LT. 3, then at most
 *>     the first (:,N_ERR_BNDS) entries are returned.
 *>
 *>     The first index in ERRS_C(i,:) corresponds to the ith
 *>     right-hand side.
 *>
 *>     The second index in ERRS_C(:,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 DOUBLE PRECISION array, dimension (N)
 *>     Workspace to hold the intermediate residual.
 *> \endverbatim
 *>
 *> \param[in] AYB
 *> \verbatim
 *>          AYB is DOUBLE PRECISION array, dimension (N)
 *>     Workspace. This can be the same workspace passed for Y_TAIL.
 *> \endverbatim
 *>
 *> \param[in] DY
 *> \verbatim
 *>          DY is DOUBLE PRECISION array, dimension (N)
 *>     Workspace to hold the intermediate solution.
 *> \endverbatim
 *>
 *> \param[in] Y_TAIL
 *> \verbatim
 *>          Y_TAIL is DOUBLE PRECISION array, dimension (N)
 *>     Workspace to hold the trailing bits of the intermediate solution.
 *> \endverbatim
 *>
 *> \param[in] RCOND
 *> \verbatim
 *>          RCOND is DOUBLE PRECISION
 *>     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
 *>     ERRS_N and ERRS_C may no longer be trustworthy.
 *> \endverbatim
 *>
 *> \param[in] RTHRESH
 *> \verbatim
 *>          RTHRESH is DOUBLE PRECISION
 *>     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 DOUBLE PRECISION
 *>     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 DGETRS had an illegal
 *>             value
 *> \endverbatim
 *
 *  Authors:
 *  ========
 *
 *> \author Univ. of Tennessee
 *> \author Univ. of California Berkeley
 *> \author Univ. of Colorado Denver
 *> \author NAG Ltd.
 *
 *> \date June 2017
 *
 *> \ingroup doubleGEcomputational
 *
 *  =====================================================================
       SUBROUTINE dla_gerfsx_extended( PREC_TYPE, TRANS_TYPE, N, NRHS, A,
      $                                LDA, AF, LDAF, IPIV, COLEQU, C, B,
      $                                LDB, Y, LDY, BERR_OUT, N_NORMS,
      $                                ERRS_N, ERRS_C, RES, AYB, DY,
      $                                Y_TAIL, RCOND, ITHRESH, RTHRESH,
      $                                DZ_UB, IGNORE_CWISE, INFO )
 *
 *  -- LAPACK computational routine (version 3.7.1) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     June 2017
 *
 *     .. Scalar Arguments ..
       INTEGER            INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
      $                   trans_type, n_norms, ithresh
       LOGICAL            COLEQU, IGNORE_CWISE
       DOUBLE PRECISION   RTHRESH, DZ_UB
 *     ..
 *     .. Array Arguments ..
       INTEGER            IPIV( * )
       DOUBLE PRECISION   A( lda, * ), AF( ldaf, * ), B( ldb, * ),
      $                   y( ldy, * ), res( * ), dy( * ), y_tail( * )
       DOUBLE PRECISION   C( * ), AYB( * ), RCOND, BERR_OUT( * ),
      $                   errs_n( nrhs, * ), errs_c( nrhs, * )
 *     ..
 *
 *  =====================================================================
 *
 *     .. Local Scalars ..
       CHARACTER          TRANS
       INTEGER            CNT, I, J, X_STATE, Z_STATE, Y_PREC_STATE
       DOUBLE PRECISION   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           daxpy, dcopy, dgetrs, dgemv, blas_dgemv_x,
      $                   blas_dgemv2_x, dla_geamv, dla_wwaddw, dlamch,
      $                   chla_transtype, dla_lin_berr
       DOUBLE PRECISION   DLAMCH
       CHARACTER          CHLA_TRANSTYPE
 *     ..
 *     .. Intrinsic Functions ..
       INTRINSIC          abs, max, min
 *     ..
 *     .. Executable Statements ..
 *
       IF ( info.NE.0 ) RETURN
       trans = chla_transtype(trans_type)
       eps = dlamch( 'Epsilon' )
       hugeval = dlamch( 'Overflow' )
 *     Force HUGEVAL to Inf
       hugeval = hugeval * hugeval
 *     Using HUGEVAL may lead to spurious underflows.
       incr_thresh = dble( n ) * eps
 *
       DO j = 1, nrhs
          y_prec_state = extra_residual
          IF ( y_prec_state .EQ. extra_y ) THEN
             DO i = 1, n
                y_tail( i ) = 0.0d+0
             END DO
          END IF

          dxrat = 0.0d+0
          dxratmax = 0.0d+0
          dzrat = 0.0d+0
          dzratmax = 0.0d+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 dcopy( n, b( 1, j ), 1, res, 1 )
             IF ( y_prec_state .EQ. base_residual ) THEN
                CALL dgemv( trans, n, n, -1.0d+0, a, lda, y( 1, j ), 1,
      $              1.0d+0, res, 1 )
             ELSE IF ( y_prec_state .EQ. extra_residual ) THEN
                CALL blas_dgemv_x( trans_type, n, n, -1.0d+0, a, lda,
      $              y( 1, j ), 1, 1.0d+0, res, 1, prec_type )
             ELSE
                CALL blas_dgemv2_x( trans_type, n, n, -1.0d+0, a, lda,
      $              y( 1, j ), y_tail, 1, 1.0d+0, res, 1, prec_type )
             END IF

 !        XXX: RES is no longer needed.
             CALL dcopy( n, res, 1, dy, 1 )
             CALL dgetrs( trans, n, 1, af, ldaf, ipiv, dy, n, info )
 *
 *         Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
 *
             normx = 0.0d+0
             normy = 0.0d+0
             normdx = 0.0d+0
             dz_z = 0.0d+0
             ymin = hugeval
 *
             DO i = 1, n
                yk = abs( y( i, j ) )
                dyk = abs( dy( i ) )

                IF ( yk .NE. 0.0d+0 ) THEN
                   dz_z = max( dz_z, dyk / yk )
                ELSE IF ( dyk .NE. 0.0d+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. 0.0d+0 ) THEN
                dx_x = normdx / normx
             ELSE IF ( normdx .EQ. 0.0d+0 ) THEN
                dx_x = 0.0d+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 = 0.0d+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 ) = 0.0d+0
                END DO
             END IF

             prevnormdx = normdx
             prev_dz_z = dz_z
 *
 *           Update soluton.
 *
             IF ( y_prec_state .LT. extra_y ) THEN
                CALL daxpy( n, 1.0d+0, dy, 1, y( 1, j ), 1 )
             ELSE
                CALL dla_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
             errs_n( j, la_linrx_err_i ) = final_dx_x / (1 - dxratmax)
          END IF
          IF ( n_norms .GE. 2 ) THEN
             errs_c( 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 dcopy( n, b( 1, j ), 1, res, 1 )
          CALL dgemv( trans, n, n, -1.0d+0, a, lda, y(1,j), 1, 1.0d+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 dla_geamv ( trans_type, n, n, 1.0d+0,
      $        a, lda, y(1, j), 1, 1.0d+0, ayb, 1 )

          CALL dla_lin_berr ( n, n, 1, res, ayb, berr_out( j ) )
 *
 *     End of loop for each RHS.
 *
       END DO
 *
       RETURN
       END
dlamch
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65

dcopy
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:84

dgemv
subroutine dgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DGEMV
Definition: dgemv.f:158

daxpy
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:91

dgetrs
subroutine dgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
DGETRS
Definition: dgetrs.f:123

dla_wwaddw
subroutine dla_wwaddw(N, X, Y, W)
DLA_WWADDW adds a vector into a doubled-single vector.
Definition: dla_wwaddw.f:83

dla_geamv
subroutine dla_geamv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds...
Definition: dla_geamv.f:176

dla_gerfsx_extended
subroutine dla_gerfsx_extended(PREC_TYPE, TRANS_TYPE, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERRS_N, ERRS_C, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
DLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matric...
Definition: dla_gerfsx_extended.f:398

dla_lin_berr
subroutine dla_lin_berr(N, NZ, NRHS, RES, AYB, BERR)
DLA_LIN_BERR computes a component-wise relative backward error.
Definition: dla_lin_berr.f:103

chla_transtype
character *1 function chla_transtype(TRANS)
CHLA_TRANSTYPE
Definition: chla_transtype.f:60