LAPACK 3.3.0

zherfsx.f

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00001       SUBROUTINE ZHERFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV,
00002      $                    S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
00003      $                    ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
00004      $                    WORK, RWORK, INFO )
00005 *
00006 *     -- LAPACK routine (version 3.2.2)                                 --
00007 *     -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
00008 *     -- Jason Riedy of Univ. of California Berkeley.                 --
00009 *     -- June 2010                                                    --
00010 *
00011 *     -- LAPACK is a software package provided by Univ. of Tennessee, --
00012 *     -- Univ. of California Berkeley and NAG Ltd.                    --
00013 *
00014       IMPLICIT NONE
00015 *     ..
00016 *     .. Scalar Arguments ..
00017       CHARACTER          UPLO, EQUED
00018       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
00019      $                   N_ERR_BNDS
00020       DOUBLE PRECISION   RCOND
00021 *     ..
00022 *     .. Array Arguments ..
00023       INTEGER            IPIV( * )
00024       COMPLEX*16         A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
00025      $                   X( LDX, * ), WORK( * )
00026       DOUBLE PRECISION   S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
00027      $                   ERR_BNDS_NORM( NRHS, * ),
00028      $                   ERR_BNDS_COMP( NRHS, * )
00029 *
00030 *     Purpose
00031 *     =======
00032 *
00033 *     ZHERFSX improves the computed solution to a system of linear
00034 *     equations when the coefficient matrix is Hermitian indefinite, and
00035 *     provides error bounds and backward error estimates for the
00036 *     solution.  In addition to normwise error bound, the code provides
00037 *     maximum componentwise error bound if possible.  See comments for
00038 *     ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds.
00039 *
00040 *     The original system of linear equations may have been equilibrated
00041 *     before calling this routine, as described by arguments EQUED and S
00042 *     below. In this case, the solution and error bounds returned are
00043 *     for the original unequilibrated system.
00044 *
00045 *     Arguments
00046 *     =========
00047 *
00048 *     Some optional parameters are bundled in the PARAMS array.  These
00049 *     settings determine how refinement is performed, but often the
00050 *     defaults are acceptable.  If the defaults are acceptable, users
00051 *     can pass NPARAMS = 0 which prevents the source code from accessing
00052 *     the PARAMS argument.
00053 *
00054 *     UPLO    (input) CHARACTER*1
00055 *       = 'U':  Upper triangle of A is stored;
00056 *       = 'L':  Lower triangle of A is stored.
00057 *
00058 *     EQUED   (input) CHARACTER*1
00059 *     Specifies the form of equilibration that was done to A
00060 *     before calling this routine. This is needed to compute
00061 *     the solution and error bounds correctly.
00062 *       = 'N':  No equilibration
00063 *       = 'Y':  Both row and column equilibration, i.e., A has been
00064 *               replaced by diag(S) * A * diag(S).
00065 *               The right hand side B has been changed accordingly.
00066 *
00067 *     N       (input) INTEGER
00068 *     The order of the matrix A.  N >= 0.
00069 *
00070 *     NRHS    (input) INTEGER
00071 *     The number of right hand sides, i.e., the number of columns
00072 *     of the matrices B and X.  NRHS >= 0.
00073 *
00074 *     A       (input) COMPLEX*16 array, dimension (LDA,N)
00075 *     The symmetric matrix A.  If UPLO = 'U', the leading N-by-N
00076 *     upper triangular part of A contains the upper triangular
00077 *     part of the matrix A, and the strictly lower triangular
00078 *     part of A is not referenced.  If UPLO = 'L', the leading
00079 *     N-by-N lower triangular part of A contains the lower
00080 *     triangular part of the matrix A, and the strictly upper
00081 *     triangular part of A is not referenced.
00082 *
00083 *     LDA     (input) INTEGER
00084 *     The leading dimension of the array A.  LDA >= max(1,N).
00085 *
00086 *     AF      (input) COMPLEX*16 array, dimension (LDAF,N)
00087 *     The factored form of the matrix A.  AF contains the block
00088 *     diagonal matrix D and the multipliers used to obtain the
00089 *     factor U or L from the factorization A = U*D*U**T or A =
00090 *     L*D*L**T as computed by DSYTRF.
00091 *
00092 *     LDAF    (input) INTEGER
00093 *     The leading dimension of the array AF.  LDAF >= max(1,N).
00094 *
00095 *     IPIV    (input) INTEGER array, dimension (N)
00096 *     Details of the interchanges and the block structure of D
00097 *     as determined by DSYTRF.
00098 *
00099 *     S       (input or output) DOUBLE PRECISION array, dimension (N)
00100 *     The scale factors for A.  If EQUED = 'Y', A is multiplied on
00101 *     the left and right by diag(S).  S is an input argument if FACT =
00102 *     'F'; otherwise, S is an output argument.  If FACT = 'F' and EQUED
00103 *     = 'Y', each element of S must be positive.  If S is output, each
00104 *     element of S is a power of the radix. If S is input, each element
00105 *     of S should be a power of the radix to ensure a reliable solution
00106 *     and error estimates. Scaling by powers of the radix does not cause
00107 *     rounding errors unless the result underflows or overflows.
00108 *     Rounding errors during scaling lead to refining with a matrix that
00109 *     is not equivalent to the input matrix, producing error estimates
00110 *     that may not be reliable.
00111 *
00112 *     B       (input) COMPLEX*16 array, dimension (LDB,NRHS)
00113 *     The right hand side matrix B.
00114 *
00115 *     LDB     (input) INTEGER
00116 *     The leading dimension of the array B.  LDB >= max(1,N).
00117 *
00118 *     X       (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
00119 *     On entry, the solution matrix X, as computed by DGETRS.
00120 *     On exit, the improved solution matrix X.
00121 *
00122 *     LDX     (input) INTEGER
00123 *     The leading dimension of the array X.  LDX >= max(1,N).
00124 *
00125 *     RCOND   (output) DOUBLE PRECISION
00126 *     Reciprocal scaled condition number.  This is an estimate of the
00127 *     reciprocal Skeel condition number of the matrix A after
00128 *     equilibration (if done).  If this is less than the machine
00129 *     precision (in particular, if it is zero), the matrix is singular
00130 *     to working precision.  Note that the error may still be small even
00131 *     if this number is very small and the matrix appears ill-
00132 *     conditioned.
00133 *
00134 *     BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
00135 *     Componentwise relative backward error.  This is the
00136 *     componentwise relative backward error of each solution vector X(j)
00137 *     (i.e., the smallest relative change in any element of A or B that
00138 *     makes X(j) an exact solution).
00139 *
00140 *     N_ERR_BNDS (input) INTEGER
00141 *     Number of error bounds to return for each right hand side
00142 *     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
00143 *     ERR_BNDS_COMP below.
00144 *
00145 *     ERR_BNDS_NORM  (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
00146 *     For each right-hand side, this array contains information about
00147 *     various error bounds and condition numbers corresponding to the
00148 *     normwise relative error, which is defined as follows:
00149 *
00150 *     Normwise relative error in the ith solution vector:
00151 *             max_j (abs(XTRUE(j,i) - X(j,i)))
00152 *            ------------------------------
00153 *                  max_j abs(X(j,i))
00154 *
00155 *     The array is indexed by the type of error information as described
00156 *     below. There currently are up to three pieces of information
00157 *     returned.
00158 *
00159 *     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
00160 *     right-hand side.
00161 *
00162 *     The second index in ERR_BNDS_NORM(:,err) contains the following
00163 *     three fields:
00164 *     err = 1 "Trust/don't trust" boolean. Trust the answer if the
00165 *              reciprocal condition number is less than the threshold
00166 *              sqrt(n) * dlamch('Epsilon').
00167 *
00168 *     err = 2 "Guaranteed" error bound: The estimated forward error,
00169 *              almost certainly within a factor of 10 of the true error
00170 *              so long as the next entry is greater than the threshold
00171 *              sqrt(n) * dlamch('Epsilon'). This error bound should only
00172 *              be trusted if the previous boolean is true.
00173 *
00174 *     err = 3  Reciprocal condition number: Estimated normwise
00175 *              reciprocal condition number.  Compared with the threshold
00176 *              sqrt(n) * dlamch('Epsilon') to determine if the error
00177 *              estimate is "guaranteed". These reciprocal condition
00178 *              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
00179 *              appropriately scaled matrix Z.
00180 *              Let Z = S*A, where S scales each row by a power of the
00181 *              radix so all absolute row sums of Z are approximately 1.
00182 *
00183 *     See Lapack Working Note 165 for further details and extra
00184 *     cautions.
00185 *
00186 *     ERR_BNDS_COMP  (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
00187 *     For each right-hand side, this array contains information about
00188 *     various error bounds and condition numbers corresponding to the
00189 *     componentwise relative error, which is defined as follows:
00190 *
00191 *     Componentwise relative error in the ith solution vector:
00192 *                    abs(XTRUE(j,i) - X(j,i))
00193 *             max_j ----------------------
00194 *                         abs(X(j,i))
00195 *
00196 *     The array is indexed by the right-hand side i (on which the
00197 *     componentwise relative error depends), and the type of error
00198 *     information as described below. There currently are up to three
00199 *     pieces of information returned for each right-hand side. If
00200 *     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
00201 *     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS .LT. 3, then at most
00202 *     the first (:,N_ERR_BNDS) entries are returned.
00203 *
00204 *     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
00205 *     right-hand side.
00206 *
00207 *     The second index in ERR_BNDS_COMP(:,err) contains the following
00208 *     three fields:
00209 *     err = 1 "Trust/don't trust" boolean. Trust the answer if the
00210 *              reciprocal condition number is less than the threshold
00211 *              sqrt(n) * dlamch('Epsilon').
00212 *
00213 *     err = 2 "Guaranteed" error bound: The estimated forward error,
00214 *              almost certainly within a factor of 10 of the true error
00215 *              so long as the next entry is greater than the threshold
00216 *              sqrt(n) * dlamch('Epsilon'). This error bound should only
00217 *              be trusted if the previous boolean is true.
00218 *
00219 *     err = 3  Reciprocal condition number: Estimated componentwise
00220 *              reciprocal condition number.  Compared with the threshold
00221 *              sqrt(n) * dlamch('Epsilon') to determine if the error
00222 *              estimate is "guaranteed". These reciprocal condition
00223 *              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
00224 *              appropriately scaled matrix Z.
00225 *              Let Z = S*(A*diag(x)), where x is the solution for the
00226 *              current right-hand side and S scales each row of
00227 *              A*diag(x) by a power of the radix so all absolute row
00228 *              sums of Z are approximately 1.
00229 *
00230 *     See Lapack Working Note 165 for further details and extra
00231 *     cautions.
00232 *
00233 *     NPARAMS (input) INTEGER
00234 *     Specifies the number of parameters set in PARAMS.  If .LE. 0, the
00235 *     PARAMS array is never referenced and default values are used.
00236 *
00237 *     PARAMS  (input / output) DOUBLE PRECISION array, dimension NPARAMS
00238 *     Specifies algorithm parameters.  If an entry is .LT. 0.0, then
00239 *     that entry will be filled with default value used for that
00240 *     parameter.  Only positions up to NPARAMS are accessed; defaults
00241 *     are used for higher-numbered parameters.
00242 *
00243 *       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
00244 *            refinement or not.
00245 *         Default: 1.0D+0
00246 *            = 0.0 : No refinement is performed, and no error bounds are
00247 *                    computed.
00248 *            = 1.0 : Use the double-precision refinement algorithm,
00249 *                    possibly with doubled-single computations if the
00250 *                    compilation environment does not support DOUBLE
00251 *                    PRECISION.
00252 *              (other values are reserved for future use)
00253 *
00254 *       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
00255 *            computations allowed for refinement.
00256 *         Default: 10
00257 *         Aggressive: Set to 100 to permit convergence using approximate
00258 *                     factorizations or factorizations other than LU. If
00259 *                     the factorization uses a technique other than
00260 *                     Gaussian elimination, the guarantees in
00261 *                     err_bnds_norm and err_bnds_comp may no longer be
00262 *                     trustworthy.
00263 *
00264 *       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
00265 *            will attempt to find a solution with small componentwise
00266 *            relative error in the double-precision algorithm.  Positive
00267 *            is true, 0.0 is false.
00268 *         Default: 1.0 (attempt componentwise convergence)
00269 *
00270 *     WORK    (workspace) COMPLEX*16 array, dimension (2*N)
00271 *
00272 *     RWORK   (workspace) DOUBLE PRECISION array, dimension (2*N)
00273 *
00274 *     INFO    (output) INTEGER
00275 *       = 0:  Successful exit. The solution to every right-hand side is
00276 *         guaranteed.
00277 *       < 0:  If INFO = -i, the i-th argument had an illegal value
00278 *       > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
00279 *         has been completed, but the factor U is exactly singular, so
00280 *         the solution and error bounds could not be computed. RCOND = 0
00281 *         is returned.
00282 *       = N+J: The solution corresponding to the Jth right-hand side is
00283 *         not guaranteed. The solutions corresponding to other right-
00284 *         hand sides K with K > J may not be guaranteed as well, but
00285 *         only the first such right-hand side is reported. If a small
00286 *         componentwise error is not requested (PARAMS(3) = 0.0) then
00287 *         the Jth right-hand side is the first with a normwise error
00288 *         bound that is not guaranteed (the smallest J such
00289 *         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
00290 *         the Jth right-hand side is the first with either a normwise or
00291 *         componentwise error bound that is not guaranteed (the smallest
00292 *         J such that either ERR_BNDS_NORM(J,1) = 0.0 or
00293 *         ERR_BNDS_COMP(J,1) = 0.0). See the definition of
00294 *         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
00295 *         about all of the right-hand sides check ERR_BNDS_NORM or
00296 *         ERR_BNDS_COMP.
00297 *
00298 *     ==================================================================
00299 *
00300 *     .. Parameters ..
00301       DOUBLE PRECISION   ZERO, ONE
00302       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00303       DOUBLE PRECISION   ITREF_DEFAULT, ITHRESH_DEFAULT
00304       DOUBLE PRECISION   COMPONENTWISE_DEFAULT, RTHRESH_DEFAULT
00305       DOUBLE PRECISION   DZTHRESH_DEFAULT
00306       PARAMETER          ( ITREF_DEFAULT = 1.0D+0 )
00307       PARAMETER          ( ITHRESH_DEFAULT = 10.0D+0 )
00308       PARAMETER          ( COMPONENTWISE_DEFAULT = 1.0D+0 )
00309       PARAMETER          ( RTHRESH_DEFAULT = 0.5D+0 )
00310       PARAMETER          ( DZTHRESH_DEFAULT = 0.25D+0 )
00311       INTEGER            LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
00312      $                   LA_LINRX_CWISE_I
00313       PARAMETER          ( LA_LINRX_ITREF_I = 1,
00314      $                   LA_LINRX_ITHRESH_I = 2 )
00315       PARAMETER          ( LA_LINRX_CWISE_I = 3 )
00316       INTEGER            LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
00317      $                   LA_LINRX_RCOND_I
00318       PARAMETER          ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 )
00319       PARAMETER          ( LA_LINRX_RCOND_I = 3 )
00320 *     ..
00321 *     .. Local Scalars ..
00322       CHARACTER(1)       NORM
00323       LOGICAL            RCEQU
00324       INTEGER            J, PREC_TYPE, REF_TYPE
00325       INTEGER            N_NORMS
00326       DOUBLE PRECISION   ANORM, RCOND_TMP
00327       DOUBLE PRECISION   ILLRCOND_THRESH, ERR_LBND, CWISE_WRONG
00328       LOGICAL            IGNORE_CWISE
00329       INTEGER            ITHRESH
00330       DOUBLE PRECISION   RTHRESH, UNSTABLE_THRESH
00331 *     ..
00332 *     .. External Subroutines ..
00333       EXTERNAL           XERBLA, ZHECON, ZLA_HERFSX_EXTENDED
00334 *     ..
00335 *     .. Intrinsic Functions ..
00336       INTRINSIC          MAX, SQRT, TRANSFER
00337 *     ..
00338 *     .. External Functions ..
00339       EXTERNAL           LSAME, BLAS_FPINFO_X, ILATRANS, ILAPREC
00340       EXTERNAL           DLAMCH, ZLANHE, ZLA_HERCOND_X, ZLA_HERCOND_C
00341       DOUBLE PRECISION   DLAMCH, ZLANHE, ZLA_HERCOND_X, ZLA_HERCOND_C
00342       LOGICAL            LSAME
00343       INTEGER            BLAS_FPINFO_X
00344       INTEGER            ILATRANS, ILAPREC
00345 *     ..
00346 *     .. Executable Statements ..
00347 *
00348 *     Check the input parameters.
00349 *
00350       INFO = 0
00351       REF_TYPE = INT( ITREF_DEFAULT )
00352       IF ( NPARAMS .GE. LA_LINRX_ITREF_I ) THEN
00353          IF ( PARAMS( LA_LINRX_ITREF_I ) .LT. 0.0D+0 ) THEN
00354             PARAMS( LA_LINRX_ITREF_I ) = ITREF_DEFAULT
00355          ELSE
00356             REF_TYPE = PARAMS( LA_LINRX_ITREF_I )
00357          END IF
00358       END IF
00359 *
00360 *     Set default parameters.
00361 *
00362       ILLRCOND_THRESH = DBLE( N ) * DLAMCH( 'Epsilon' )
00363       ITHRESH = INT( ITHRESH_DEFAULT )
00364       RTHRESH = RTHRESH_DEFAULT
00365       UNSTABLE_THRESH = DZTHRESH_DEFAULT
00366       IGNORE_CWISE = COMPONENTWISE_DEFAULT .EQ. 0.0D+0
00367 *
00368       IF ( NPARAMS.GE.LA_LINRX_ITHRESH_I ) THEN
00369          IF ( PARAMS( LA_LINRX_ITHRESH_I ).LT.0.0D+0 ) THEN
00370             PARAMS( LA_LINRX_ITHRESH_I ) = ITHRESH
00371          ELSE
00372             ITHRESH = INT( PARAMS( LA_LINRX_ITHRESH_I ) )
00373          END IF
00374       END IF
00375       IF ( NPARAMS.GE.LA_LINRX_CWISE_I ) THEN
00376          IF ( PARAMS(LA_LINRX_CWISE_I ).LT.0.0D+0 ) THEN
00377             IF ( IGNORE_CWISE ) THEN
00378                PARAMS( LA_LINRX_CWISE_I ) = 0.0D+0
00379             ELSE
00380                PARAMS( LA_LINRX_CWISE_I ) = 1.0D+0
00381             END IF
00382          ELSE
00383             IGNORE_CWISE = PARAMS( LA_LINRX_CWISE_I ) .EQ. 0.0D+0
00384          END IF
00385       END IF
00386       IF ( REF_TYPE .EQ. 0 .OR. N_ERR_BNDS .EQ. 0 ) THEN
00387          N_NORMS = 0
00388       ELSE IF ( IGNORE_CWISE ) THEN
00389          N_NORMS = 1
00390       ELSE
00391          N_NORMS = 2
00392       END IF
00393 *
00394       RCEQU = LSAME( EQUED, 'Y' )
00395 *
00396 *     Test input parameters.
00397 *
00398       IF (.NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00399         INFO = -1
00400       ELSE IF( .NOT.RCEQU .AND. .NOT.LSAME( EQUED, 'N' ) ) THEN
00401         INFO = -2
00402       ELSE IF( N.LT.0 ) THEN
00403         INFO = -3
00404       ELSE IF( NRHS.LT.0 ) THEN
00405         INFO = -4
00406       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00407         INFO = -6
00408       ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
00409         INFO = -8
00410       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00411         INFO = -11
00412       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00413         INFO = -13
00414       END IF
00415       IF( INFO.NE.0 ) THEN
00416         CALL XERBLA( 'ZHERFSX', -INFO )
00417         RETURN
00418       END IF
00419 *
00420 *     Quick return if possible.
00421 *
00422       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
00423          RCOND = 1.0D+0
00424          DO J = 1, NRHS
00425             BERR( J ) = 0.0D+0
00426             IF ( N_ERR_BNDS .GE. 1 ) THEN
00427                ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0D+0
00428                ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0D+0
00429             END IF
00430             IF ( N_ERR_BNDS .GE. 2 ) THEN
00431                ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 0.0D+0
00432                ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 0.0D+0
00433             END IF
00434             IF ( N_ERR_BNDS .GE. 3 ) THEN
00435                ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = 1.0D+0
00436                ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = 1.0D+0
00437             END IF
00438          END DO
00439          RETURN
00440       END IF
00441 *
00442 *     Default to failure.
00443 *
00444       RCOND = 0.0D+0
00445       DO J = 1, NRHS
00446          BERR( J ) = 1.0D+0
00447          IF ( N_ERR_BNDS .GE. 1 ) THEN
00448             ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0D+0
00449             ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0D+0
00450          END IF
00451          IF ( N_ERR_BNDS .GE. 2 ) THEN
00452             ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0D+0
00453             ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0D+0
00454          END IF
00455          IF ( N_ERR_BNDS .GE. 3 ) THEN
00456             ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = 0.0D+0
00457             ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = 0.0D+0
00458          END IF
00459       END DO
00460 *
00461 *     Compute the norm of A and the reciprocal of the condition
00462 *     number of A.
00463 *
00464       NORM = 'I'
00465       ANORM = ZLANHE( NORM, UPLO, N, A, LDA, RWORK )
00466       CALL ZHECON( UPLO, N, AF, LDAF, IPIV, ANORM, RCOND, WORK,
00467      $     INFO )
00468 *
00469 *     Perform refinement on each right-hand side
00470 *
00471       IF ( REF_TYPE .NE. 0 ) THEN
00472 
00473          PREC_TYPE = ILAPREC( 'E' )
00474 
00475          CALL ZLA_HERFSX_EXTENDED( PREC_TYPE, UPLO,  N,
00476      $        NRHS, A, LDA, AF, LDAF, IPIV, RCEQU, S, B,
00477      $        LDB, X, LDX, BERR, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP,
00478      $        WORK, RWORK, WORK(N+1),
00479      $        TRANSFER (RWORK(1:2*N), (/ (ZERO, ZERO) /), N), RCOND,
00480      $        ITHRESH, RTHRESH, UNSTABLE_THRESH, IGNORE_CWISE,
00481      $        INFO )
00482       END IF
00483 
00484       ERR_LBND = MAX( 10.0D+0, SQRT( DBLE( N ) ) ) * DLAMCH( 'Epsilon' )
00485       IF ( N_ERR_BNDS .GE. 1 .AND. N_NORMS .GE. 1 ) THEN
00486 *
00487 *     Compute scaled normwise condition number cond(A*C).
00488 *
00489          IF ( RCEQU ) THEN
00490             RCOND_TMP = ZLA_HERCOND_C( UPLO, N, A, LDA, AF, LDAF, IPIV,
00491      $           S, .TRUE., INFO, WORK, RWORK )
00492          ELSE
00493             RCOND_TMP = ZLA_HERCOND_C( UPLO, N, A, LDA, AF, LDAF, IPIV,
00494      $           S, .FALSE., INFO, WORK, RWORK )
00495          END IF
00496          DO J = 1, NRHS
00497 *
00498 *     Cap the error at 1.0.
00499 *
00500             IF ( N_ERR_BNDS .GE. LA_LINRX_ERR_I
00501      $           .AND. ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) .GT. 1.0D+0 )
00502      $           ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0D+0
00503 *
00504 *     Threshold the error (see LAWN).
00505 *
00506             IF (RCOND_TMP .LT. ILLRCOND_THRESH) THEN
00507                ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0D+0
00508                ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 0.0D+0
00509                IF ( INFO .LE. N ) INFO = N + J
00510             ELSE IF ( ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) .LT. ERR_LBND )
00511      $              THEN
00512                ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = ERR_LBND
00513                ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0D+0
00514             END IF
00515 *
00516 *     Save the condition number.
00517 *
00518             IF ( N_ERR_BNDS .GE. LA_LINRX_RCOND_I ) THEN
00519                ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = RCOND_TMP
00520             END IF
00521          END DO
00522       END IF
00523 
00524       IF ( N_ERR_BNDS .GE. 1 .AND. N_NORMS .GE. 2 ) THEN
00525 *
00526 *     Compute componentwise condition number cond(A*diag(Y(:,J))) for
00527 *     each right-hand side using the current solution as an estimate of
00528 *     the true solution.  If the componentwise error estimate is too
00529 *     large, then the solution is a lousy estimate of truth and the
00530 *     estimated RCOND may be too optimistic.  To avoid misleading users,
00531 *     the inverse condition number is set to 0.0 when the estimated
00532 *     cwise error is at least CWISE_WRONG.
00533 *
00534          CWISE_WRONG = SQRT( DLAMCH( 'Epsilon' ) )
00535          DO J = 1, NRHS
00536             IF ( ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) .LT. CWISE_WRONG )
00537      $     THEN
00538                RCOND_TMP = ZLA_HERCOND_X( UPLO, N, A, LDA, AF, LDAF,
00539      $         IPIV, X( 1, J ), INFO, WORK, RWORK )
00540             ELSE
00541                RCOND_TMP = 0.0D+0
00542             END IF
00543 *
00544 *     Cap the error at 1.0.
00545 *
00546             IF ( N_ERR_BNDS .GE. LA_LINRX_ERR_I
00547      $           .AND. ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) .GT. 1.0D+0 )
00548      $           ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0D+0
00549 *
00550 *     Threshold the error (see LAWN).
00551 *
00552             IF ( RCOND_TMP .LT. ILLRCOND_THRESH ) THEN
00553                ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0D+0
00554                ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 0.0D+0
00555                IF ( PARAMS( LA_LINRX_CWISE_I ) .EQ. 1.0D+0
00556      $              .AND. INFO.LT.N + J ) INFO = N + J
00557             ELSE IF ( ERR_BNDS_COMP( J, LA_LINRX_ERR_I )
00558      $              .LT. ERR_LBND ) THEN
00559                ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = ERR_LBND
00560                ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0D+0
00561             END IF
00562 *
00563 *     Save the condition number.
00564 *
00565             IF ( N_ERR_BNDS .GE. LA_LINRX_RCOND_I ) THEN
00566                ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = RCOND_TMP
00567             END IF
00568 
00569          END DO
00570       END IF
00571 *
00572       RETURN
00573 *
00574 *     End of ZHERFSX
00575 *
00576       END
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