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version 1.4, 2010/12/21 13:53:36 version 1.5, 2011/11/21 20:43:02
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   *> \brief \b DPORFSX
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DPORFSX + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dporfsx.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dporfsx.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dporfsx.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B,
   *                           LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
   *                           ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
   *                           WORK, IWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          UPLO, EQUED
   *       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
   *      $                   N_ERR_BNDS
   *       DOUBLE PRECISION   RCOND
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IWORK( * )
   *       DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
   *      $                   X( LDX, * ), WORK( * )
   *       DOUBLE PRECISION   S( * ), PARAMS( * ), BERR( * ),
   *      $                   ERR_BNDS_NORM( NRHS, * ),
   *      $                   ERR_BNDS_COMP( NRHS, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *>    DPORFSX improves the computed solution to a system of linear
   *>    equations when the coefficient matrix is symmetric positive
   *>    definite, 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 and S
   *>    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] UPLO
   *> \verbatim
   *>          UPLO is CHARACTER*1
   *>       = 'U':  Upper triangle of A is stored;
   *>       = 'L':  Lower triangle of A is stored.
   *> \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
   *>       = 'Y':  Both row and column equilibration, i.e., A has been
   *>               replaced by diag(S) * A * diag(S).
   *>               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 symmetric matrix A.  If UPLO = 'U', the leading N-by-N
   *>     upper triangular part of A contains the upper triangular part
   *>     of the matrix A, and the strictly lower triangular part of A
   *>     is not referenced.  If UPLO = 'L', the leading N-by-N lower
   *>     triangular part of A contains the lower triangular part of
   *>     the matrix A, and the strictly upper triangular part of A is
   *>     not referenced.
   *> \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 triangular factor U or L from the Cholesky factorization
   *>     A = U**T*U or A = L*L**T, as computed by DPOTRF.
   *> \endverbatim
   *>
   *> \param[in] LDAF
   *> \verbatim
   *>          LDAF is INTEGER
   *>     The leading dimension of the array AF.  LDAF >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in,out] S
   *> \verbatim
   *>          S is or output) DOUBLE PRECISION array, dimension (N)
   *>     The row scale factors for A.  If EQUED = 'Y', A is multiplied on
   *>     the left and right by diag(S).  S is an input argument if FACT =
   *>     'F'; otherwise, S is an output argument.  If FACT = 'F' and EQUED
   *>     = 'Y', each element of S must be positive.  If S is output, each
   *>     element of S is a power of the radix. If S is input, each element
   *>     of S 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 .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) * 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 .LE. 0, the
   *>     PARAMS array is never referenced and default values are used.
   *> \endverbatim
   *>
   *> \param[in,out] PARAMS
   *> \verbatim
   *>          PARAMS is / output) DOUBLE PRECISION array, dimension (NPARAMS)
   *>     Specifies algorithm parameters.  If an entry is .LT. 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 November 2011
   *
   *> \ingroup doublePOcomputational
   *
   *  =====================================================================
       SUBROUTINE DPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B,        SUBROUTINE DPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B,
      $                    LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,       $                    LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
      $                    ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,       $                    ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
      $                    WORK, IWORK, INFO )       $                    WORK, IWORK, INFO )
 *  *
 *     -- LAPACK routine (version 3.2.2)                                 --  *  -- LAPACK computational routine (version 3.4.0) --
 *     -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *     -- Jason Riedy of Univ. of California Berkeley.                 --  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     -- June 2010                                                    --  *     November 2011
 *  
 *     -- LAPACK is a software package provided by Univ. of Tennessee, --  
 *     -- Univ. of California Berkeley and NAG Ltd.                    --  
 *  *
       IMPLICIT NONE  
 *     ..  
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          UPLO, EQUED        CHARACTER          UPLO, EQUED
       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,        INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
Line 28 Line 414
      $                   ERR_BNDS_COMP( NRHS, * )       $                   ERR_BNDS_COMP( NRHS, * )
 *     ..  *     ..
 *  *
 *     Purpose  *  ==================================================================
 *     =======  
 *  
 *     DPORFSX improves the computed solution to a system of linear  
 *     equations when the coefficient matrix is symmetric positive  
 *     definite, 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 and S  
 *     below. In this case, the solution and error bounds returned are  
 *     for the original unequilibrated system.  
 *  
 *     Arguments  
 *     =========  
 *  
 *     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.  
 *  
 *     UPLO    (input) CHARACTER*1  
 *       = 'U':  Upper triangle of A is stored;  
 *       = 'L':  Lower triangle of A is stored.  
 *  
 *     EQUED   (input) 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  
 *       = 'Y':  Both row and column equilibration, i.e., A has been  
 *               replaced by diag(S) * A * diag(S).  
 *               The right hand side B has been changed accordingly.  
 *  
 *     N       (input) INTEGER  
 *     The order of the matrix A.  N >= 0.  
 *  
 *     NRHS    (input) INTEGER  
 *     The number of right hand sides, i.e., the number of columns  
 *     of the matrices B and X.  NRHS >= 0.  
 *  
 *     A       (input) DOUBLE PRECISION array, dimension (LDA,N)  
 *     The symmetric matrix A.  If UPLO = 'U', the leading N-by-N  
 *     upper triangular part of A contains the upper triangular part  
 *     of the matrix A, and the strictly lower triangular part of A  
 *     is not referenced.  If UPLO = 'L', the leading N-by-N lower  
 *     triangular part of A contains the lower triangular part of  
 *     the matrix A, and the strictly upper triangular part of A is  
 *     not referenced.  
 *  
 *     LDA     (input) INTEGER  
 *     The leading dimension of the array A.  LDA >= max(1,N).  
 *  
 *     AF      (input) DOUBLE PRECISION array, dimension (LDAF,N)  
 *     The triangular factor U or L from the Cholesky factorization  
 *     A = U**T*U or A = L*L**T, as computed by DPOTRF.  
 *  
 *     LDAF    (input) INTEGER  
 *     The leading dimension of the array AF.  LDAF >= max(1,N).  
 *  
 *     S       (input or output) DOUBLE PRECISION array, dimension (N)  
 *     The row scale factors for A.  If EQUED = 'Y', A is multiplied on  
 *     the left and right by diag(S).  S is an input argument if FACT =  
 *     'F'; otherwise, S is an output argument.  If FACT = 'F' and EQUED  
 *     = 'Y', each element of S must be positive.  If S is output, each  
 *     element of S is a power of the radix. If S is input, each element  
 *     of S 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.  
 *  
 *     B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)  
 *     The right hand side matrix B.  
 *  
 *     LDB     (input) INTEGER  
 *     The leading dimension of the array B.  LDB >= max(1,N).  
 *  
 *     X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)  
 *     On entry, the solution matrix X, as computed by DGETRS.  
 *     On exit, the improved solution matrix X.  
 *  
 *     LDX     (input) INTEGER  
 *     The leading dimension of the array X.  LDX >= max(1,N).  
 *  
 *     RCOND   (output) 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.  
 *  
 *     BERR    (output) 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).  
 *  
 *     N_ERR_BNDS (input) 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.  
 *  
 *     ERR_BNDS_NORM  (output) 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.  
 *  
 *     ERR_BNDS_COMP  (output) 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 .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) * 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.  
 *  
 *     NPARAMS (input) INTEGER  
 *     Specifies the number of parameters set in PARAMS.  If .LE. 0, the  
 *     PARAMS array is never referenced and default values are used.  
 *  
 *     PARAMS  (input / output) DOUBLE PRECISION array, dimension (NPARAMS)  
 *     Specifies algorithm parameters.  If an entry is .LT. 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)  
 *  
 *     WORK    (workspace) DOUBLE PRECISION array, dimension (4*N)  
 *  
 *     IWORK   (workspace) INTEGER array, dimension (N)  
 *  
 *     INFO    (output) 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.  
 *  
 *     ==================================================================  
 *  *
 *     .. Parameters ..  *     .. Parameters ..
       DOUBLE PRECISION   ZERO, ONE        DOUBLE PRECISION   ZERO, ONE
Removed from v.1.4  
changed lines
  Added in v.1.5

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