dgerfsx.f

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*> \brief \b DGERFSX
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DGERFSX( TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV,
*                           R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
*                           ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
*                           WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS, EQUED
*       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
*      $                   N_ERR_BNDS
*       DOUBLE PRECISION   RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
*      $                   X( LDX , * ), WORK( * )
*       DOUBLE PRECISION   R( * ), C( * ), PARAMS( * ), BERR( * ),
*      $                   ERR_BNDS_NORM( NRHS, * ),
*      $                   ERR_BNDS_COMP( NRHS, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>    DGERFSX improves the computed solution to a system of linear
*>    equations and provides error bounds and backward error estimates
*>    for the solution.  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.
*>
*>    The original system of linear equations may have been equilibrated
*>    before calling this routine, as described by arguments EQUED, R
*>    and C below. In this case, the solution and error bounds returned
*>    are for the original unequilibrated system.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \verbatim
*>     Some optional parameters are bundled in the PARAMS array.  These
*>     settings determine how refinement is performed, but often the
*>     defaults are acceptable.  If the defaults are acceptable, users
*>     can pass NPARAMS = 0 which prevents the source code from accessing
*>     the PARAMS argument.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>     Specifies the form of the system of equations:
*>       = 'N':  A * X = B     (No transpose)
*>       = 'T':  A**T * X = B  (Transpose)
*>       = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] EQUED
*> \verbatim
*>          EQUED is CHARACTER*1
*>     Specifies the form of equilibration that was done to A
*>     before calling this routine. This is needed to compute
*>     the solution and error bounds correctly.
*>       = 'N':  No equilibration
*>       = 'R':  Row equilibration, i.e., A has been premultiplied by
*>               diag(R).
*>       = 'C':  Column equilibration, i.e., A has been postmultiplied
*>               by diag(C).
*>       = 'B':  Both row and column equilibration, i.e., A has been
*>               replaced by diag(R) * A * diag(C).
*>               The right hand side B has been changed accordingly.
*> \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 right hand sides, i.e., the number of columns
*>     of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA,N)
*>     The original 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 DGETRF; for 1<=i<=N, row i of the
*>     matrix was interchanged with row IPIV(i).
*> \endverbatim
*>
*> \param[in] R
*> \verbatim
*>          R is DOUBLE PRECISION array, dimension (N)
*>     The row scale factors for A.  If EQUED = 'R' or 'B', A is
*>     multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
*>     is not accessed.
*>     If R is accessed, each element of R 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] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (N)
*>     The column scale factors for A.  If EQUED = 'C' or 'B', A is
*>     multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
*>     is not accessed.
*>     If C is accessed, 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] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>     On entry, the solution matrix X, as computed by DGETRS.
*>     On exit, the improved 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] 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[out] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>     Componentwise relative backward error.  This is the
*>     componentwise relative backward error of each solution vector X(j)
*>     (i.e., the smallest relative change in any element of A or B that
*>     makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[in] N_ERR_BNDS
*> \verbatim
*>          N_ERR_BNDS is INTEGER
*>     Number of error bounds to return for each right hand side
*>     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
*>     ERR_BNDS_COMP below.
*> \endverbatim
*>
*> \param[out] ERR_BNDS_NORM
*> \verbatim
*>          ERR_BNDS_NORM 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 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) * dlamch('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) * dlamch('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) * dlamch('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.
*>
*>     See Lapack Working Note 165 for further details and extra
*>     cautions.
*> \endverbatim
*>
*> \param[out] ERR_BNDS_COMP
*> \verbatim
*>          ERR_BNDS_COMP 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
*>     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 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) * dlamch('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) * dlamch('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) * dlamch('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.
*>
*>     See Lapack Working Note 165 for further details and extra
*>     cautions.
*> \endverbatim
*>
*> \param[in] NPARAMS
*> \verbatim
*>          NPARAMS is INTEGER
*>     Specifies the number of parameters set in PARAMS.  If <= 0, the
*>     PARAMS array is never referenced and default values are used.
*> \endverbatim
*>
*> \param[in,out] PARAMS
*> \verbatim
*>          PARAMS is DOUBLE PRECISION array, dimension (NPARAMS)
*>     Specifies algorithm parameters.  If an entry is < 0.0, then
*>     that entry will be filled with default value used for that
*>     parameter.  Only positions up to NPARAMS are accessed; defaults
*>     are used for higher-numbered parameters.
*>
*>       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
*>            refinement or not.
*>         Default: 1.0D+0
*>            = 0.0:  No refinement is performed, and no error bounds are
*>                    computed.
*>            = 1.0:  Use the double-precision refinement algorithm,
*>                    possibly with doubled-single computations if the
*>                    compilation environment does not support DOUBLE
*>                    PRECISION.
*>              (other values are reserved for future use)
*>
*>       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
*>            computations allowed for refinement.
*>         Default: 10
*>         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.
*>
*>       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
*>            will attempt to find a solution with small componentwise
*>            relative error in the double-precision algorithm.  Positive
*>            is true, 0.0 is false.
*>         Default: 1.0 (attempt componentwise convergence)
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>       = 0:  Successful exit. The solution to every right-hand side is
*>         guaranteed.
*>       < 0:  If INFO = -i, the i-th argument had an illegal value
*>       > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
*>         has been completed, but the factor U is exactly singular, so
*>         the solution and error bounds could not be computed. RCOND = 0
*>         is returned.
*>       = N+J: The solution corresponding to the Jth right-hand side is
*>         not guaranteed. The solutions corresponding to other right-
*>         hand sides K with K > J may not be guaranteed as well, but
*>         only the first such right-hand side is reported. If a small
*>         componentwise error is not requested (PARAMS(3) = 0.0) then
*>         the Jth right-hand side is the first with a normwise error
*>         bound that is not guaranteed (the smallest J such
*>         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
*>         the Jth right-hand side is the first with either a normwise or
*>         componentwise error bound that is not guaranteed (the smallest
*>         J such that either ERR_BNDS_NORM(J,1) = 0.0 or
*>         ERR_BNDS_COMP(J,1) = 0.0). See the definition of
*>         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
*>         about all of the right-hand sides check ERR_BNDS_NORM or
*>         ERR_BNDS_COMP.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup doubleGEcomputational
*
*  =====================================================================
      SUBROUTINE dgerfsx( TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV,
     $                    R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
     $                    ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
     $                    WORK, IWORK, INFO )
*
*  -- LAPACK computational routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS, EQUED
      INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
     $                   N_ERR_BNDS
      DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
     $                   X( LDX , * ), WORK( * )
      DOUBLE PRECISION   R( * ), C( * ), PARAMS( * ), BERR( * ),
     $                   err_bnds_norm( nrhs, * ),
     $                   err_bnds_comp( nrhs, * )
*     ..
*
*  ==================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0d+0, one = 1.0d+0 )
      DOUBLE PRECISION   ITREF_DEFAULT, ITHRESH_DEFAULT
      DOUBLE PRECISION   COMPONENTWISE_DEFAULT, RTHRESH_DEFAULT
      DOUBLE PRECISION   DZTHRESH_DEFAULT
      parameter( itref_default = 1.0d+0 )
      parameter( ithresh_default = 10.0d+0 )
      parameter( componentwise_default = 1.0d+0 )
      parameter( rthresh_default = 0.5d+0 )
      parameter( dzthresh_default = 0.25d+0 )
      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 )
*     ..
*     .. Local Scalars ..
      CHARACTER(1)       NORM
      LOGICAL            ROWEQU, COLEQU, NOTRAN
      INTEGER            J, TRANS_TYPE, PREC_TYPE, REF_TYPE
      INTEGER            N_NORMS
      DOUBLE PRECISION   ANORM, RCOND_TMP
      DOUBLE PRECISION   ILLRCOND_THRESH, ERR_LBND, CWISE_WRONG
      LOGICAL            IGNORE_CWISE
      INTEGER            ITHRESH
      DOUBLE PRECISION   RTHRESH, UNSTABLE_THRESH
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, dgecon, dla_gerfsx_extended
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, sqrt
*     ..
*     .. External Functions ..
      EXTERNAL           lsame, ilatrans, ilaprec
      EXTERNAL           dlamch, dlange, dla_gercond
      DOUBLE PRECISION   DLAMCH, DLANGE, DLA_GERCOND
      LOGICAL            LSAME
      INTEGER            ILATRANS, ILAPREC
*     ..
*     .. Executable Statements ..
*
*     Check the input parameters.
*
      info = 0
      trans_type = ilatrans( trans )
      ref_type = int( itref_default )
      IF ( nparams .GE. la_linrx_itref_i ) THEN
         IF ( params( la_linrx_itref_i ) .LT. 0.0d+0 ) THEN
            params( la_linrx_itref_i ) = itref_default
         ELSE
            ref_type = params( la_linrx_itref_i )
         END IF
      END IF
*
*     Set default parameters.
*
      illrcond_thresh = dble( n ) * dlamch( 'Epsilon' )
      ithresh = int( ithresh_default )
      rthresh = rthresh_default
      unstable_thresh = dzthresh_default
      ignore_cwise = componentwise_default .EQ. 0.0d+0
*
      IF ( nparams.GE.la_linrx_ithresh_i ) THEN
         IF ( params( la_linrx_ithresh_i ).LT.0.0d+0 ) THEN
            params( la_linrx_ithresh_i ) = ithresh
         ELSE
            ithresh = int( params( la_linrx_ithresh_i ) )
         END IF
      END IF
      IF ( nparams.GE.la_linrx_cwise_i ) THEN
         IF ( params( la_linrx_cwise_i ).LT.0.0d+0 ) THEN
            IF ( ignore_cwise ) THEN
               params( la_linrx_cwise_i ) = 0.0d+0
            ELSE
               params( la_linrx_cwise_i ) = 1.0d+0
            END IF
         ELSE
            ignore_cwise = params( la_linrx_cwise_i ) .EQ. 0.0d+0
         END IF
      END IF
      IF ( ref_type .EQ. 0 .OR. n_err_bnds .EQ. 0 ) THEN
         n_norms = 0
      ELSE IF ( ignore_cwise ) THEN
         n_norms = 1
      ELSE
         n_norms = 2
      END IF
*
      notran = lsame( trans, 'N' )
      rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
      colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
*
*     Test input parameters.
*
      IF( trans_type.EQ.-1 ) THEN
        info = -1
      ELSE IF( .NOT.rowequ .AND. .NOT.colequ .AND.
     $         .NOT.lsame( equed, 'N' ) ) THEN
        info = -2
      ELSE IF( n.LT.0 ) THEN
        info = -3
      ELSE IF( nrhs.LT.0 ) THEN
        info = -4
      ELSE IF( lda.LT.max( 1, n ) ) THEN
        info = -6
      ELSE IF( ldaf.LT.max( 1, n ) ) THEN
        info = -8
      ELSE IF( ldb.LT.max( 1, n ) ) THEN
        info = -13
      ELSE IF( ldx.LT.max( 1, n ) ) THEN
        info = -15
      END IF
      IF( info.NE.0 ) THEN
        CALL xerbla( 'DGERFSX', -info )
        RETURN
      END IF
*
*     Quick return if possible.
*
      IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
         rcond = 1.0d+0
         DO j = 1, nrhs
            berr( j ) = 0.0d+0
            IF ( n_err_bnds .GE. 1 ) THEN
               err_bnds_norm( j, la_linrx_trust_i) = 1.0d+0
               err_bnds_comp( j, la_linrx_trust_i ) = 1.0d+0
            END IF
            IF ( n_err_bnds .GE. 2 ) THEN
               err_bnds_norm( j, la_linrx_err_i) = 0.0d+0
               err_bnds_comp( j, la_linrx_err_i ) = 0.0d+0
            END IF
            IF ( n_err_bnds .GE. 3 ) THEN
               err_bnds_norm( j, la_linrx_rcond_i) = 1.0d+0
               err_bnds_comp( j, la_linrx_rcond_i ) = 1.0d+0
            END IF
         END DO
         RETURN
      END IF
*
*     Default to failure.
*
      rcond = 0.0d+0
      DO j = 1, nrhs
         berr( j ) = 1.0d+0
         IF ( n_err_bnds .GE. 1 ) THEN
            err_bnds_norm( j, la_linrx_trust_i ) = 1.0d+0
            err_bnds_comp( j, la_linrx_trust_i ) = 1.0d+0
         END IF
         IF ( n_err_bnds .GE. 2 ) THEN
            err_bnds_norm( j, la_linrx_err_i ) = 1.0d+0
            err_bnds_comp( j, la_linrx_err_i ) = 1.0d+0
         END IF
         IF ( n_err_bnds .GE. 3 ) THEN
            err_bnds_norm( j, la_linrx_rcond_i ) = 0.0d+0
            err_bnds_comp( j, la_linrx_rcond_i ) = 0.0d+0
         END IF
      END DO
*
*     Compute the norm of A and the reciprocal of the condition
*     number of A.
*
      IF( notran ) THEN
         norm = 'I'
      ELSE
         norm = '1'
      END IF
      anorm = dlange( norm, n, n, a, lda, work )
      CALL dgecon( norm, n, af, ldaf, anorm, rcond, work, iwork, info )
*
*     Perform refinement on each right-hand side
*
      IF ( ref_type .NE. 0 ) THEN
 
         prec_type = ilaprec( 'E' )
 
         IF ( notran ) THEN
            CALL dla_gerfsx_extended( prec_type, trans_type,  n,
     $           nrhs, a, lda, af, ldaf, ipiv, colequ, c, b,
     $           ldb, x, ldx, berr, n_norms, err_bnds_norm,
     $           err_bnds_comp, work(n+1), work(1), work(2*n+1),
     $           work(1), rcond, ithresh, rthresh, unstable_thresh,
     $           ignore_cwise, info )
         ELSE
            CALL dla_gerfsx_extended( prec_type, trans_type,  n,
     $           nrhs, a, lda, af, ldaf, ipiv, rowequ, r, b,
     $           ldb, x, ldx, berr, n_norms, err_bnds_norm,
     $           err_bnds_comp, work(n+1), work(1), work(2*n+1),
     $           work(1), rcond, ithresh, rthresh, unstable_thresh,
     $           ignore_cwise, info )
         END IF
      END IF
 
      err_lbnd = max( 10.0d+0, sqrt( dble( n ) ) ) * dlamch( 'Epsilon' )
      IF ( n_err_bnds .GE. 1 .AND. n_norms .GE. 1 ) THEN
*
*     Compute scaled normwise condition number cond(A*C).
*
         IF ( colequ .AND. notran ) THEN
            rcond_tmp = dla_gercond( trans, n, a, lda, af, ldaf, ipiv,
     $           -1, c, info, work, iwork )
         ELSE IF ( rowequ .AND. .NOT. notran ) THEN
            rcond_tmp = dla_gercond( trans, n, a, lda, af, ldaf, ipiv,
     $           -1, r, info, work, iwork )
         ELSE
            rcond_tmp = dla_gercond( trans, n, a, lda, af, ldaf, ipiv,
     $           0, r, info, work, iwork )
         END IF
         DO j = 1, nrhs
*
*     Cap the error at 1.0.
*
            IF ( n_err_bnds .GE. la_linrx_err_i
     $           .AND. err_bnds_norm( j, la_linrx_err_i ) .GT. 1.0d+0 )
     $           err_bnds_norm( j, la_linrx_err_i ) = 1.0d+0
*
*     Threshold the error (see LAWN).
*
            IF ( rcond_tmp .LT. illrcond_thresh ) THEN
               err_bnds_norm( j, la_linrx_err_i ) = 1.0d+0
               err_bnds_norm( j, la_linrx_trust_i ) = 0.0d+0
               IF ( info .LE. n ) info = n + j
            ELSE IF ( err_bnds_norm( j, la_linrx_err_i ) .LT. err_lbnd )
     $     THEN
               err_bnds_norm( j, la_linrx_err_i ) = err_lbnd
               err_bnds_norm( j, la_linrx_trust_i ) = 1.0d+0
            END IF
*
*     Save the condition number.
*
            IF ( n_err_bnds .GE. la_linrx_rcond_i ) THEN
               err_bnds_norm( j, la_linrx_rcond_i ) = rcond_tmp
            END IF
         END DO
      END IF
 
      IF ( n_err_bnds .GE. 1 .AND. n_norms .GE. 2 ) THEN
*
*     Compute componentwise condition number cond(A*diag(Y(:,J))) for
*     each right-hand side using the current solution as an estimate of
*     the true solution.  If the componentwise error estimate is too
*     large, then the solution is a lousy estimate of truth and the
*     estimated RCOND may be too optimistic.  To avoid misleading users,
*     the inverse condition number is set to 0.0 when the estimated
*     cwise error is at least CWISE_WRONG.
*
         cwise_wrong = sqrt( dlamch( 'Epsilon' ) )
         DO j = 1, nrhs
            IF ( err_bnds_comp( j, la_linrx_err_i ) .LT. cwise_wrong )
     $           THEN
               rcond_tmp = dla_gercond( trans, n, a, lda, af, ldaf,
     $              ipiv, 1, x(1,j), info, work, iwork )
            ELSE
               rcond_tmp = 0.0d+0
            END IF
*
*     Cap the error at 1.0.
*
            IF ( n_err_bnds .GE. la_linrx_err_i
     $           .AND. err_bnds_comp( j, la_linrx_err_i ) .GT. 1.0d+0 )
     $           err_bnds_comp( j, la_linrx_err_i ) = 1.0d+0
*
*     Threshold the error (see LAWN).
*
            IF ( rcond_tmp .LT. illrcond_thresh ) THEN
               err_bnds_comp( j, la_linrx_err_i ) = 1.0d+0
               err_bnds_comp( j, la_linrx_trust_i ) = 0.0d+0
               IF ( params( la_linrx_cwise_i ) .EQ. 1.0d+0
     $              .AND. info.LT.n + j ) info = n + j
            ELSE IF ( err_bnds_comp( j, la_linrx_err_i )
     $              .LT. err_lbnd ) THEN
               err_bnds_comp( j, la_linrx_err_i ) = err_lbnd
               err_bnds_comp( j, la_linrx_trust_i ) = 1.0d+0
            END IF
*
*     Save the condition number.
*
            IF ( n_err_bnds .GE. la_linrx_rcond_i ) THEN
               err_bnds_comp( j, la_linrx_rcond_i ) = rcond_tmp
            END IF
         END DO
      END IF
*
      RETURN
*
*     End of DGERFSX
*
      END