LAPACK 3.3.0

sgbrfsx.f

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00001       SUBROUTINE SGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB,
00002      $                    LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND,
00003      $                    BERR, N_ERR_BNDS, ERR_BNDS_NORM,
00004      $                    ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK,
00005      $                    INFO )
00006 *
00007 *     -- LAPACK routine (version 3.2.2)                                 --
00008 *     -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
00009 *     -- Jason Riedy of Univ. of California Berkeley.                 --
00010 *     -- June 2010                                                    --
00011 *
00012 *     -- LAPACK is a software package provided by Univ. of Tennessee, --
00013 *     -- Univ. of California Berkeley and NAG Ltd.                    --
00014 *
00015       IMPLICIT NONE
00016 *     ..
00017 *     .. Scalar Arguments ..
00018       CHARACTER          TRANS, EQUED
00019       INTEGER            INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS,
00020      $                   NPARAMS, N_ERR_BNDS
00021       REAL               RCOND
00022 *     ..
00023 *     .. Array Arguments ..
00024       INTEGER            IPIV( * ), IWORK( * )
00025       REAL               AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
00026      $                   X( LDX , * ),WORK( * )
00027       REAL               R( * ), C( * ), PARAMS( * ), BERR( * ),
00028      $                   ERR_BNDS_NORM( NRHS, * ),
00029      $                   ERR_BNDS_COMP( NRHS, * )
00030 *     ..
00031 *
00032 *     Purpose
00033 *     =======
00034 *
00035 *     SGBRFSX improves the computed solution to a system of linear
00036 *     equations and provides error bounds and backward error estimates
00037 *     for the solution.  In addition to normwise error bound, the code
00038 *     provides maximum componentwise error bound if possible.  See
00039 *     comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
00040 *     error bounds.
00041 *
00042 *     The original system of linear equations may have been equilibrated
00043 *     before calling this routine, as described by arguments EQUED, R
00044 *     and C below. In this case, the solution and error bounds returned
00045 *     are for the original unequilibrated system.
00046 *
00047 *     Arguments
00048 *     =========
00049 *
00050 *     Some optional parameters are bundled in the PARAMS array.  These
00051 *     settings determine how refinement is performed, but often the
00052 *     defaults are acceptable.  If the defaults are acceptable, users
00053 *     can pass NPARAMS = 0 which prevents the source code from accessing
00054 *     the PARAMS argument.
00055 *
00056 *     TRANS   (input) CHARACTER*1
00057 *     Specifies the form of the system of equations:
00058 *       = 'N':  A * X = B     (No transpose)
00059 *       = 'T':  A**T * X = B  (Transpose)
00060 *       = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
00061 *
00062 *     EQUED   (input) CHARACTER*1
00063 *     Specifies the form of equilibration that was done to A
00064 *     before calling this routine. This is needed to compute
00065 *     the solution and error bounds correctly.
00066 *       = 'N':  No equilibration
00067 *       = 'R':  Row equilibration, i.e., A has been premultiplied by
00068 *               diag(R).
00069 *       = 'C':  Column equilibration, i.e., A has been postmultiplied
00070 *               by diag(C).
00071 *       = 'B':  Both row and column equilibration, i.e., A has been
00072 *               replaced by diag(R) * A * diag(C).
00073 *               The right hand side B has been changed accordingly.
00074 *
00075 *     N       (input) INTEGER
00076 *     The order of the matrix A.  N >= 0.
00077 *
00078 *     KL      (input) INTEGER
00079 *     The number of subdiagonals within the band of A.  KL >= 0.
00080 *
00081 *     KU      (input) INTEGER
00082 *     The number of superdiagonals within the band of A.  KU >= 0.
00083 *
00084 *     NRHS    (input) INTEGER
00085 *     The number of right hand sides, i.e., the number of columns
00086 *     of the matrices B and X.  NRHS >= 0.
00087 *
00088 *     AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
00089 *     The original band matrix A, stored in rows 1 to KL+KU+1.
00090 *     The j-th column of A is stored in the j-th column of the
00091 *     array AB as follows:
00092 *     AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
00093 *
00094 *     LDAB    (input) INTEGER
00095 *     The leading dimension of the array AB.  LDAB >= KL+KU+1.
00096 *
00097 *     AFB     (input) DOUBLE PRECISION array, dimension (LDAFB,N)
00098 *     Details of the LU factorization of the band matrix A, as
00099 *     computed by DGBTRF.  U is stored as an upper triangular band
00100 *     matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
00101 *     the multipliers used during the factorization are stored in
00102 *     rows KL+KU+2 to 2*KL+KU+1.
00103 *
00104 *     LDAFB   (input) INTEGER
00105 *     The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.
00106 *
00107 *     IPIV    (input) INTEGER array, dimension (N)
00108 *     The pivot indices from SGETRF; for 1<=i<=N, row i of the
00109 *     matrix was interchanged with row IPIV(i).
00110 *
00111 *     R       (input or output) REAL array, dimension (N)
00112 *     The row scale factors for A.  If EQUED = 'R' or 'B', A is
00113 *     multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
00114 *     is not accessed.  R is an input argument if FACT = 'F';
00115 *     otherwise, R is an output argument.  If FACT = 'F' and
00116 *     EQUED = 'R' or 'B', each element of R must be positive.
00117 *     If R is output, each element of R is a power of the radix.
00118 *     If R is input, each element of R should be a power of the radix
00119 *     to ensure a reliable solution and error estimates. Scaling by
00120 *     powers of the radix does not cause rounding errors unless the
00121 *     result underflows or overflows. Rounding errors during scaling
00122 *     lead to refining with a matrix that is not equivalent to the
00123 *     input matrix, producing error estimates that may not be
00124 *     reliable.
00125 *
00126 *     C       (input or output) REAL array, dimension (N)
00127 *     The column scale factors for A.  If EQUED = 'C' or 'B', A is
00128 *     multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
00129 *     is not accessed.  C is an input argument if FACT = 'F';
00130 *     otherwise, C is an output argument.  If FACT = 'F' and
00131 *     EQUED = 'C' or 'B', each element of C must be positive.
00132 *     If C is output, each element of C is a power of the radix.
00133 *     If C is input, each element of C should be a power of the radix
00134 *     to ensure a reliable solution and error estimates. Scaling by
00135 *     powers of the radix does not cause rounding errors unless the
00136 *     result underflows or overflows. Rounding errors during scaling
00137 *     lead to refining with a matrix that is not equivalent to the
00138 *     input matrix, producing error estimates that may not be
00139 *     reliable.
00140 *
00141 *     B       (input) REAL array, dimension (LDB,NRHS)
00142 *     The right hand side matrix B.
00143 *
00144 *     LDB     (input) INTEGER
00145 *     The leading dimension of the array B.  LDB >= max(1,N).
00146 *
00147 *     X       (input/output) REAL array, dimension (LDX,NRHS)
00148 *     On entry, the solution matrix X, as computed by SGETRS.
00149 *     On exit, the improved solution matrix X.
00150 *
00151 *     LDX     (input) INTEGER
00152 *     The leading dimension of the array X.  LDX >= max(1,N).
00153 *
00154 *     RCOND   (output) REAL
00155 *     Reciprocal scaled condition number.  This is an estimate of the
00156 *     reciprocal Skeel condition number of the matrix A after
00157 *     equilibration (if done).  If this is less than the machine
00158 *     precision (in particular, if it is zero), the matrix is singular
00159 *     to working precision.  Note that the error may still be small even
00160 *     if this number is very small and the matrix appears ill-
00161 *     conditioned.
00162 *
00163 *     BERR    (output) REAL array, dimension (NRHS)
00164 *     Componentwise relative backward error.  This is the
00165 *     componentwise relative backward error of each solution vector X(j)
00166 *     (i.e., the smallest relative change in any element of A or B that
00167 *     makes X(j) an exact solution).
00168 *
00169 *     N_ERR_BNDS (input) INTEGER
00170 *     Number of error bounds to return for each right hand side
00171 *     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
00172 *     ERR_BNDS_COMP below.
00173 *
00174 *     ERR_BNDS_NORM  (output) REAL array, dimension (NRHS, N_ERR_BNDS)
00175 *     For each right-hand side, this array contains information about
00176 *     various error bounds and condition numbers corresponding to the
00177 *     normwise relative error, which is defined as follows:
00178 *
00179 *     Normwise relative error in the ith solution vector:
00180 *             max_j (abs(XTRUE(j,i) - X(j,i)))
00181 *            ------------------------------
00182 *                  max_j abs(X(j,i))
00183 *
00184 *     The array is indexed by the type of error information as described
00185 *     below. There currently are up to three pieces of information
00186 *     returned.
00187 *
00188 *     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
00189 *     right-hand side.
00190 *
00191 *     The second index in ERR_BNDS_NORM(:,err) contains the following
00192 *     three fields:
00193 *     err = 1 "Trust/don't trust" boolean. Trust the answer if the
00194 *              reciprocal condition number is less than the threshold
00195 *              sqrt(n) * slamch('Epsilon').
00196 *
00197 *     err = 2 "Guaranteed" error bound: The estimated forward error,
00198 *              almost certainly within a factor of 10 of the true error
00199 *              so long as the next entry is greater than the threshold
00200 *              sqrt(n) * slamch('Epsilon'). This error bound should only
00201 *              be trusted if the previous boolean is true.
00202 *
00203 *     err = 3  Reciprocal condition number: Estimated normwise
00204 *              reciprocal condition number.  Compared with the threshold
00205 *              sqrt(n) * slamch('Epsilon') to determine if the error
00206 *              estimate is "guaranteed". These reciprocal condition
00207 *              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
00208 *              appropriately scaled matrix Z.
00209 *              Let Z = S*A, where S scales each row by a power of the
00210 *              radix so all absolute row sums of Z are approximately 1.
00211 *
00212 *     See Lapack Working Note 165 for further details and extra
00213 *     cautions.
00214 *
00215 *     ERR_BNDS_COMP  (output) REAL array, dimension (NRHS, N_ERR_BNDS)
00216 *     For each right-hand side, this array contains information about
00217 *     various error bounds and condition numbers corresponding to the
00218 *     componentwise relative error, which is defined as follows:
00219 *
00220 *     Componentwise relative error in the ith solution vector:
00221 *                    abs(XTRUE(j,i) - X(j,i))
00222 *             max_j ----------------------
00223 *                         abs(X(j,i))
00224 *
00225 *     The array is indexed by the right-hand side i (on which the
00226 *     componentwise relative error depends), and the type of error
00227 *     information as described below. There currently are up to three
00228 *     pieces of information returned for each right-hand side. If
00229 *     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
00230 *     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS .LT. 3, then at most
00231 *     the first (:,N_ERR_BNDS) entries are returned.
00232 *
00233 *     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
00234 *     right-hand side.
00235 *
00236 *     The second index in ERR_BNDS_COMP(:,err) contains the following
00237 *     three fields:
00238 *     err = 1 "Trust/don't trust" boolean. Trust the answer if the
00239 *              reciprocal condition number is less than the threshold
00240 *              sqrt(n) * slamch('Epsilon').
00241 *
00242 *     err = 2 "Guaranteed" error bound: The estimated forward error,
00243 *              almost certainly within a factor of 10 of the true error
00244 *              so long as the next entry is greater than the threshold
00245 *              sqrt(n) * slamch('Epsilon'). This error bound should only
00246 *              be trusted if the previous boolean is true.
00247 *
00248 *     err = 3  Reciprocal condition number: Estimated componentwise
00249 *              reciprocal condition number.  Compared with the threshold
00250 *              sqrt(n) * slamch('Epsilon') to determine if the error
00251 *              estimate is "guaranteed". These reciprocal condition
00252 *              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
00253 *              appropriately scaled matrix Z.
00254 *              Let Z = S*(A*diag(x)), where x is the solution for the
00255 *              current right-hand side and S scales each row of
00256 *              A*diag(x) by a power of the radix so all absolute row
00257 *              sums of Z are approximately 1.
00258 *
00259 *     See Lapack Working Note 165 for further details and extra
00260 *     cautions.
00261 *
00262 *     NPARAMS (input) INTEGER
00263 *     Specifies the number of parameters set in PARAMS.  If .LE. 0, the
00264 *     PARAMS array is never referenced and default values are used.
00265 *
00266 *     PARAMS  (input / output) REAL array, dimension NPARAMS
00267 *     Specifies algorithm parameters.  If an entry is .LT. 0.0, then
00268 *     that entry will be filled with default value used for that
00269 *     parameter.  Only positions up to NPARAMS are accessed; defaults
00270 *     are used for higher-numbered parameters.
00271 *
00272 *       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
00273 *            refinement or not.
00274 *         Default: 1.0
00275 *            = 0.0 : No refinement is performed, and no error bounds are
00276 *                    computed.
00277 *            = 1.0 : Use the double-precision refinement algorithm,
00278 *                    possibly with doubled-single computations if the
00279 *                    compilation environment does not support DOUBLE
00280 *                    PRECISION.
00281 *              (other values are reserved for future use)
00282 *
00283 *       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
00284 *            computations allowed for refinement.
00285 *         Default: 10
00286 *         Aggressive: Set to 100 to permit convergence using approximate
00287 *                     factorizations or factorizations other than LU. If
00288 *                     the factorization uses a technique other than
00289 *                     Gaussian elimination, the guarantees in
00290 *                     err_bnds_norm and err_bnds_comp may no longer be
00291 *                     trustworthy.
00292 *
00293 *       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
00294 *            will attempt to find a solution with small componentwise
00295 *            relative error in the double-precision algorithm.  Positive
00296 *            is true, 0.0 is false.
00297 *         Default: 1.0 (attempt componentwise convergence)
00298 *
00299 *     WORK    (workspace) REAL array, dimension (4*N)
00300 *
00301 *     IWORK   (workspace) INTEGER array, dimension (N)
00302 *
00303 *     INFO    (output) INTEGER
00304 *       = 0:  Successful exit. The solution to every right-hand side is
00305 *         guaranteed.
00306 *       < 0:  If INFO = -i, the i-th argument had an illegal value
00307 *       > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
00308 *         has been completed, but the factor U is exactly singular, so
00309 *         the solution and error bounds could not be computed. RCOND = 0
00310 *         is returned.
00311 *       = N+J: The solution corresponding to the Jth right-hand side is
00312 *         not guaranteed. The solutions corresponding to other right-
00313 *         hand sides K with K > J may not be guaranteed as well, but
00314 *         only the first such right-hand side is reported. If a small
00315 *         componentwise error is not requested (PARAMS(3) = 0.0) then
00316 *         the Jth right-hand side is the first with a normwise error
00317 *         bound that is not guaranteed (the smallest J such
00318 *         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
00319 *         the Jth right-hand side is the first with either a normwise or
00320 *         componentwise error bound that is not guaranteed (the smallest
00321 *         J such that either ERR_BNDS_NORM(J,1) = 0.0 or
00322 *         ERR_BNDS_COMP(J,1) = 0.0). See the definition of
00323 *         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
00324 *         about all of the right-hand sides check ERR_BNDS_NORM or
00325 *         ERR_BNDS_COMP.
00326 *
00327 *     ==================================================================
00328 *
00329 *     .. Parameters ..
00330       REAL               ZERO, ONE
00331       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
00332       REAL               ITREF_DEFAULT, ITHRESH_DEFAULT,
00333      $                   COMPONENTWISE_DEFAULT
00334       REAL               RTHRESH_DEFAULT, DZTHRESH_DEFAULT
00335       PARAMETER          ( ITREF_DEFAULT = 1.0 )
00336       PARAMETER          ( ITHRESH_DEFAULT = 10.0 )
00337       PARAMETER          ( COMPONENTWISE_DEFAULT = 1.0 )
00338       PARAMETER          ( RTHRESH_DEFAULT = 0.5 )
00339       PARAMETER          ( DZTHRESH_DEFAULT = 0.25 )
00340       INTEGER            LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
00341      $                   LA_LINRX_CWISE_I
00342       PARAMETER          ( LA_LINRX_ITREF_I = 1,
00343      $                   LA_LINRX_ITHRESH_I = 2 )
00344       PARAMETER          ( LA_LINRX_CWISE_I = 3 )
00345       INTEGER            LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
00346      $                   LA_LINRX_RCOND_I
00347       PARAMETER          ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 )
00348       PARAMETER          ( LA_LINRX_RCOND_I = 3 )
00349 *     ..
00350 *     .. Local Scalars ..
00351       CHARACTER(1)       NORM
00352       LOGICAL            ROWEQU, COLEQU, NOTRAN
00353       INTEGER            J, TRANS_TYPE, PREC_TYPE, REF_TYPE
00354       INTEGER            N_NORMS
00355       REAL               ANORM, RCOND_TMP
00356       REAL               ILLRCOND_THRESH, ERR_LBND, CWISE_WRONG
00357       LOGICAL            IGNORE_CWISE
00358       INTEGER            ITHRESH
00359       REAL               RTHRESH, UNSTABLE_THRESH
00360 *     ..
00361 *     .. External Subroutines ..
00362       EXTERNAL           XERBLA, SGBCON
00363       EXTERNAL           SLA_GBRFSX_EXTENDED
00364 *     ..
00365 *     .. Intrinsic Functions ..
00366       INTRINSIC          MAX, SQRT
00367 *     ..
00368 *     .. External Functions ..
00369       EXTERNAL           LSAME, BLAS_FPINFO_X, ILATRANS, ILAPREC
00370       EXTERNAL           SLAMCH, SLANGB, SLA_GBRCOND
00371       REAL               SLAMCH, SLANGB, SLA_GBRCOND
00372       LOGICAL            LSAME
00373       INTEGER            BLAS_FPINFO_X
00374       INTEGER            ILATRANS, ILAPREC
00375 *     ..
00376 *     .. Executable Statements ..
00377 *
00378 *     Check the input parameters.
00379 *
00380       INFO = 0
00381       TRANS_TYPE = ILATRANS( TRANS )
00382       REF_TYPE = INT( ITREF_DEFAULT )
00383       IF ( NPARAMS .GE. LA_LINRX_ITREF_I ) THEN
00384          IF ( PARAMS( LA_LINRX_ITREF_I ) .LT. 0.0 ) THEN
00385             PARAMS( LA_LINRX_ITREF_I ) = ITREF_DEFAULT
00386          ELSE
00387             REF_TYPE = PARAMS( LA_LINRX_ITREF_I )
00388          END IF
00389       END IF
00390 *
00391 *     Set default parameters.
00392 *
00393       ILLRCOND_THRESH = REAL( N ) * SLAMCH( 'Epsilon' )
00394       ITHRESH = INT( ITHRESH_DEFAULT )
00395       RTHRESH = RTHRESH_DEFAULT
00396       UNSTABLE_THRESH = DZTHRESH_DEFAULT
00397       IGNORE_CWISE = COMPONENTWISE_DEFAULT .EQ. 0.0
00398 *
00399       IF ( NPARAMS.GE.LA_LINRX_ITHRESH_I ) THEN
00400          IF ( PARAMS( LA_LINRX_ITHRESH_I ).LT.0.0 ) THEN
00401             PARAMS( LA_LINRX_ITHRESH_I ) = ITHRESH
00402          ELSE
00403             ITHRESH = INT( PARAMS( LA_LINRX_ITHRESH_I ) )
00404          END IF
00405       END IF
00406       IF ( NPARAMS.GE.LA_LINRX_CWISE_I ) THEN
00407          IF ( PARAMS( LA_LINRX_CWISE_I ).LT.0.0 ) THEN
00408             IF ( IGNORE_CWISE ) THEN
00409                PARAMS( LA_LINRX_CWISE_I ) = 0.0
00410             ELSE
00411                PARAMS( LA_LINRX_CWISE_I ) = 1.0
00412             END IF
00413          ELSE
00414             IGNORE_CWISE = PARAMS( LA_LINRX_CWISE_I ) .EQ. 0.0
00415          END IF
00416       END IF
00417       IF ( REF_TYPE .EQ. 0 .OR. N_ERR_BNDS .EQ. 0 ) THEN
00418          N_NORMS = 0
00419       ELSE IF ( IGNORE_CWISE ) THEN
00420          N_NORMS = 1
00421       ELSE
00422          N_NORMS = 2
00423       END IF
00424 *
00425       NOTRAN = LSAME( TRANS, 'N' )
00426       ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
00427       COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
00428 *
00429 *     Test input parameters.
00430 *
00431       IF( TRANS_TYPE.EQ.-1 ) THEN
00432         INFO = -1
00433       ELSE IF( .NOT.ROWEQU .AND. .NOT.COLEQU .AND.
00434      $         .NOT.LSAME( EQUED, 'N' ) ) THEN
00435         INFO = -2
00436       ELSE IF( N.LT.0 ) THEN
00437         INFO = -3
00438       ELSE IF( KL.LT.0 ) THEN
00439         INFO = -4
00440       ELSE IF( KU.LT.0 ) THEN
00441         INFO = -5
00442       ELSE IF( NRHS.LT.0 ) THEN
00443         INFO = -6
00444       ELSE IF( LDAB.LT.KL+KU+1 ) THEN
00445         INFO = -8
00446       ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
00447         INFO = -10
00448       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00449         INFO = -13
00450       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00451         INFO = -15
00452       END IF
00453       IF( INFO.NE.0 ) THEN
00454         CALL XERBLA( 'SGBRFSX', -INFO )
00455         RETURN
00456       END IF
00457 *
00458 *     Quick return if possible.
00459 *
00460       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
00461          RCOND = 1.0
00462          DO J = 1, NRHS
00463             BERR( J ) = 0.0
00464             IF ( N_ERR_BNDS .GE. 1 ) THEN
00465                ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0
00466                ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0
00467             END IF
00468             IF ( N_ERR_BNDS .GE. 2 ) THEN
00469                ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 0.0
00470                ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 0.0
00471             END IF
00472             IF ( N_ERR_BNDS .GE. 3 ) THEN
00473                ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = 1.0
00474                ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = 1.0
00475             END IF
00476          END DO
00477          RETURN
00478       END IF
00479 *
00480 *     Default to failure.
00481 *
00482       RCOND = 0.0
00483       DO J = 1, NRHS
00484          BERR( J ) = 1.0
00485          IF ( N_ERR_BNDS .GE. 1 ) THEN
00486             ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0
00487             ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0
00488          END IF
00489          IF ( N_ERR_BNDS .GE. 2 ) THEN
00490             ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0
00491             ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0
00492          END IF
00493          IF ( N_ERR_BNDS .GE. 3 ) THEN
00494             ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = 0.0
00495             ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = 0.0
00496          END IF
00497       END DO
00498 *
00499 *     Compute the norm of A and the reciprocal of the condition
00500 *     number of A.
00501 *
00502       IF( NOTRAN ) THEN
00503          NORM = 'I'
00504       ELSE
00505          NORM = '1'
00506       END IF
00507       ANORM = SLANGB( NORM, N, KL, KU, AB, LDAB, WORK )
00508       CALL SGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
00509      $     WORK, IWORK, INFO )
00510 *
00511 *     Perform refinement on each right-hand side
00512 *
00513       IF (REF_TYPE .NE. 0) THEN
00514 
00515          PREC_TYPE = ILAPREC( 'D' )
00516 
00517          IF ( NOTRAN ) THEN
00518             CALL SLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE,  N, KL, KU,
00519      $           NRHS, AB, LDAB, AFB, LDAFB, IPIV, COLEQU, C, B,
00520      $           LDB, X, LDX, BERR, N_NORMS, ERR_BNDS_NORM,
00521      $           ERR_BNDS_COMP, WORK( N+1 ), WORK( 1 ), WORK( 2*N+1 ),
00522      $           WORK( 1 ), RCOND, ITHRESH, RTHRESH, UNSTABLE_THRESH,
00523      $           IGNORE_CWISE, INFO )
00524          ELSE
00525             CALL SLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE,  N, KL, KU,
00526      $           NRHS, AB, LDAB, AFB, LDAFB, IPIV, ROWEQU, R, B,
00527      $           LDB, X, LDX, BERR, N_NORMS, ERR_BNDS_NORM,
00528      $           ERR_BNDS_COMP, WORK( N+1 ), WORK( 1 ), WORK( 2*N+1 ),
00529      $           WORK( 1 ), RCOND, ITHRESH, RTHRESH, UNSTABLE_THRESH,
00530      $           IGNORE_CWISE, INFO )
00531          END IF
00532       END IF
00533 
00534       ERR_LBND = MAX( 10.0, SQRT( REAL( N ) ) ) * SLAMCH( 'Epsilon' )
00535       IF ( N_ERR_BNDS .GE. 1 .AND. N_NORMS .GE. 1 ) THEN
00536 *
00537 *     Compute scaled normwise condition number cond(A*C).
00538 *
00539          IF ( COLEQU .AND. NOTRAN ) THEN
00540             RCOND_TMP = SLA_GBRCOND( TRANS, N, KL, KU, AB, LDAB, AFB,
00541      $           LDAFB, IPIV, -1, C, INFO, WORK, IWORK )
00542          ELSE IF ( ROWEQU .AND. .NOT. NOTRAN ) THEN
00543             RCOND_TMP = SLA_GBRCOND( TRANS, N, KL, KU, AB, LDAB, AFB,
00544      $           LDAFB, IPIV, -1, R, INFO, WORK, IWORK )
00545          ELSE
00546             RCOND_TMP = SLA_GBRCOND( TRANS, N, KL, KU, AB, LDAB, AFB,
00547      $           LDAFB, IPIV, 0, R, INFO, WORK, IWORK )
00548          END IF
00549          DO J = 1, NRHS
00550 *
00551 *     Cap the error at 1.0.
00552 *
00553             IF ( N_ERR_BNDS .GE. LA_LINRX_ERR_I
00554      $           .AND. ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) .GT. 1.0 )
00555      $           ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0
00556 *
00557 *     Threshold the error (see LAWN).
00558 *
00559             IF ( RCOND_TMP .LT. ILLRCOND_THRESH ) THEN
00560                ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0
00561                ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 0.0
00562                IF ( INFO .LE. N ) INFO = N + J
00563             ELSE IF ( ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) .LT. ERR_LBND )
00564      $     THEN
00565                ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = ERR_LBND
00566                ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0
00567             END IF
00568 *
00569 *     Save the condition number.
00570 *
00571             IF ( N_ERR_BNDS .GE. LA_LINRX_RCOND_I ) THEN
00572                ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = RCOND_TMP
00573             END IF
00574 
00575          END DO
00576       END IF
00577 
00578       IF (N_ERR_BNDS .GE. 1 .AND. N_NORMS .GE. 2) THEN
00579 *
00580 *     Compute componentwise condition number cond(A*diag(Y(:,J))) for
00581 *     each right-hand side using the current solution as an estimate of
00582 *     the true solution.  If the componentwise error estimate is too
00583 *     large, then the solution is a lousy estimate of truth and the
00584 *     estimated RCOND may be too optimistic.  To avoid misleading users,
00585 *     the inverse condition number is set to 0.0 when the estimated
00586 *     cwise error is at least CWISE_WRONG.
00587 *
00588          CWISE_WRONG = SQRT( SLAMCH( 'Epsilon' ) )
00589          DO J = 1, NRHS
00590             IF ( ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) .LT. CWISE_WRONG )
00591      $     THEN
00592                RCOND_TMP = SLA_GBRCOND( TRANS, N, KL, KU, AB, LDAB, AFB,
00593      $              LDAFB, IPIV, 1, X( 1, J ), INFO, WORK, IWORK )
00594             ELSE
00595                RCOND_TMP = 0.0
00596             END IF
00597 *
00598 *     Cap the error at 1.0.
00599 *
00600             IF ( N_ERR_BNDS .GE. LA_LINRX_ERR_I
00601      $           .AND. ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) .GT. 1.0 )
00602      $           ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0
00603 *
00604 *     Threshold the error (see LAWN).
00605 *
00606             IF ( RCOND_TMP .LT. ILLRCOND_THRESH ) THEN
00607                ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0
00608                ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 0.0
00609                IF ( PARAMS( LA_LINRX_CWISE_I ) .EQ. 1.0
00610      $              .AND. INFO.LT.N + J ) INFO = N + J
00611             ELSE IF ( ERR_BNDS_COMP( J, LA_LINRX_ERR_I )
00612      $              .LT. ERR_LBND ) THEN
00613                ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = ERR_LBND
00614                ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0
00615             END IF
00616 *
00617 *     Save the condition number.
00618 *
00619             IF ( N_ERR_BNDS .GE. LA_LINRX_RCOND_I ) THEN
00620                ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = RCOND_TMP
00621             END IF
00622 
00623          END DO
00624       END IF
00625 *
00626       RETURN
00627 *
00628 *     End of SGBRFSX
00629 *
00630       END
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