LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
dla_gbrfsx_extended.f
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00001 *> \brief \b DLA_GBRFSX_EXTENDED
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download DLA_GBRFSX_EXTENDED + dependencies 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dla_gbrfsx_extended.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE DLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, KL, KU,
00022 *                                       NRHS, AB, LDAB, AFB, LDAFB, IPIV,
00023 *                                       COLEQU, C, B, LDB, Y, LDY,
00024 *                                       BERR_OUT, N_NORMS, ERR_BNDS_NORM,
00025 *                                       ERR_BNDS_COMP, RES, AYB, DY,
00026 *                                       Y_TAIL, RCOND, ITHRESH, RTHRESH,
00027 *                                       DZ_UB, IGNORE_CWISE, INFO )
00028 * 
00029 *       .. Scalar Arguments ..
00030 *       INTEGER            INFO, LDAB, LDAFB, LDB, LDY, N, KL, KU, NRHS,
00031 *      $                   PREC_TYPE, TRANS_TYPE, N_NORMS, ITHRESH
00032 *       LOGICAL            COLEQU, IGNORE_CWISE
00033 *       DOUBLE PRECISION   RTHRESH, DZ_UB
00034 *       ..
00035 *       .. Array Arguments ..
00036 *       INTEGER            IPIV( * )
00037 *       DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
00038 *      $                   Y( LDY, * ), RES(*), DY(*), Y_TAIL(*)
00039 *       DOUBLE PRECISION   C( * ), AYB(*), RCOND, BERR_OUT(*),
00040 *      $                   ERR_BNDS_NORM( NRHS, * ),
00041 *      $                   ERR_BNDS_COMP( NRHS, * )
00042 *       ..
00043 *  
00044 *
00045 *> \par Purpose:
00046 *  =============
00047 *>
00048 *> \verbatim
00049 *>
00050 *> 
00051 *> DLA_GBRFSX_EXTENDED improves the computed solution to a system of
00052 *> linear equations by performing extra-precise iterative refinement
00053 *> and provides error bounds and backward error estimates for the solution.
00054 *> This subroutine is called by DGBRFSX to perform iterative refinement.
00055 *> In addition to normwise error bound, the code provides maximum
00056 *> componentwise error bound if possible. See comments for ERR_BNDS_NORM
00057 *> and ERR_BNDS_COMP for details of the error bounds. Note that this
00058 *> subroutine is only resonsible for setting the second fields of
00059 *> ERR_BNDS_NORM and ERR_BNDS_COMP.
00060 *> \endverbatim
00061 *
00062 *  Arguments:
00063 *  ==========
00064 *
00065 *> \param[in] PREC_TYPE
00066 *> \verbatim
00067 *>          PREC_TYPE is INTEGER
00068 *>     Specifies the intermediate precision to be used in refinement.
00069 *>     The value is defined by ILAPREC(P) where P is a CHARACTER and
00070 *>     P    = 'S':  Single
00071 *>          = 'D':  Double
00072 *>          = 'I':  Indigenous
00073 *>          = 'X', 'E':  Extra
00074 *> \endverbatim
00075 *>
00076 *> \param[in] TRANS_TYPE
00077 *> \verbatim
00078 *>          TRANS_TYPE is INTEGER
00079 *>     Specifies the transposition operation on A.
00080 *>     The value is defined by ILATRANS(T) where T is a CHARACTER and
00081 *>     T    = 'N':  No transpose
00082 *>          = 'T':  Transpose
00083 *>          = 'C':  Conjugate transpose
00084 *> \endverbatim
00085 *>
00086 *> \param[in] N
00087 *> \verbatim
00088 *>          N is INTEGER
00089 *>     The number of linear equations, i.e., the order of the
00090 *>     matrix A.  N >= 0.
00091 *> \endverbatim
00092 *>
00093 *> \param[in] KL
00094 *> \verbatim
00095 *>          KL is INTEGER
00096 *>     The number of subdiagonals within the band of A.  KL >= 0.
00097 *> \endverbatim
00098 *>
00099 *> \param[in] KU
00100 *> \verbatim
00101 *>          KU is INTEGER
00102 *>     The number of superdiagonals within the band of A.  KU >= 0
00103 *> \endverbatim
00104 *>
00105 *> \param[in] NRHS
00106 *> \verbatim
00107 *>          NRHS is INTEGER
00108 *>     The number of right-hand-sides, i.e., the number of columns of the
00109 *>     matrix B.
00110 *> \endverbatim
00111 *>
00112 *> \param[in] AB
00113 *> \verbatim
00114 *>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
00115 *>          On entry, the N-by-N matrix AB.
00116 *> \endverbatim
00117 *>
00118 *> \param[in] LDAB
00119 *> \verbatim
00120 *>          LDAB is INTEGER
00121 *>          The leading dimension of the array AB.  LDBA >= max(1,N).
00122 *> \endverbatim
00123 *>
00124 *> \param[in] AFB
00125 *> \verbatim
00126 *>          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
00127 *>     The factors L and U from the factorization
00128 *>     A = P*L*U as computed by DGBTRF.
00129 *> \endverbatim
00130 *>
00131 *> \param[in] LDAFB
00132 *> \verbatim
00133 *>          LDAFB is INTEGER
00134 *>     The leading dimension of the array AF.  LDAFB >= max(1,N).
00135 *> \endverbatim
00136 *>
00137 *> \param[in] IPIV
00138 *> \verbatim
00139 *>          IPIV is INTEGER array, dimension (N)
00140 *>     The pivot indices from the factorization A = P*L*U
00141 *>     as computed by DGBTRF; row i of the matrix was interchanged
00142 *>     with row IPIV(i).
00143 *> \endverbatim
00144 *>
00145 *> \param[in] COLEQU
00146 *> \verbatim
00147 *>          COLEQU is LOGICAL
00148 *>     If .TRUE. then column equilibration was done to A before calling
00149 *>     this routine. This is needed to compute the solution and error
00150 *>     bounds correctly.
00151 *> \endverbatim
00152 *>
00153 *> \param[in] C
00154 *> \verbatim
00155 *>          C is DOUBLE PRECISION array, dimension (N)
00156 *>     The column scale factors for A. If COLEQU = .FALSE., C
00157 *>     is not accessed. If C is input, each element of C should be a power
00158 *>     of the radix to ensure a reliable solution and error estimates.
00159 *>     Scaling by powers of the radix does not cause rounding errors unless
00160 *>     the result underflows or overflows. Rounding errors during scaling
00161 *>     lead to refining with a matrix that is not equivalent to the
00162 *>     input matrix, producing error estimates that may not be
00163 *>     reliable.
00164 *> \endverbatim
00165 *>
00166 *> \param[in] B
00167 *> \verbatim
00168 *>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
00169 *>     The right-hand-side matrix B.
00170 *> \endverbatim
00171 *>
00172 *> \param[in] LDB
00173 *> \verbatim
00174 *>          LDB is INTEGER
00175 *>     The leading dimension of the array B.  LDB >= max(1,N).
00176 *> \endverbatim
00177 *>
00178 *> \param[in,out] Y
00179 *> \verbatim
00180 *>          Y is DOUBLE PRECISION array, dimension
00181 *>                    (LDY,NRHS)
00182 *>     On entry, the solution matrix X, as computed by DGBTRS.
00183 *>     On exit, the improved solution matrix Y.
00184 *> \endverbatim
00185 *>
00186 *> \param[in] LDY
00187 *> \verbatim
00188 *>          LDY is INTEGER
00189 *>     The leading dimension of the array Y.  LDY >= max(1,N).
00190 *> \endverbatim
00191 *>
00192 *> \param[out] BERR_OUT
00193 *> \verbatim
00194 *>          BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
00195 *>     On exit, BERR_OUT(j) contains the componentwise relative backward
00196 *>     error for right-hand-side j from the formula
00197 *>         max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
00198 *>     where abs(Z) is the componentwise absolute value of the matrix
00199 *>     or vector Z. This is computed by DLA_LIN_BERR.
00200 *> \endverbatim
00201 *>
00202 *> \param[in] N_NORMS
00203 *> \verbatim
00204 *>          N_NORMS is INTEGER
00205 *>     Determines which error bounds to return (see ERR_BNDS_NORM
00206 *>     and ERR_BNDS_COMP).
00207 *>     If N_NORMS >= 1 return normwise error bounds.
00208 *>     If N_NORMS >= 2 return componentwise error bounds.
00209 *> \endverbatim
00210 *>
00211 *> \param[in,out] ERR_BNDS_NORM
00212 *> \verbatim
00213 *>          ERR_BNDS_NORM is DOUBLE PRECISION array, dimension
00214 *>                    (NRHS, N_ERR_BNDS)
00215 *>     For each right-hand side, this array contains information about
00216 *>     various error bounds and condition numbers corresponding to the
00217 *>     normwise relative error, which is defined as follows:
00218 *>
00219 *>     Normwise relative error in the ith solution vector:
00220 *>             max_j (abs(XTRUE(j,i) - X(j,i)))
00221 *>            ------------------------------
00222 *>                  max_j abs(X(j,i))
00223 *>
00224 *>     The array is indexed by the type of error information as described
00225 *>     below. There currently are up to three pieces of information
00226 *>     returned.
00227 *>
00228 *>     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
00229 *>     right-hand side.
00230 *>
00231 *>     The second index in ERR_BNDS_NORM(:,err) contains the following
00232 *>     three fields:
00233 *>     err = 1 "Trust/don't trust" boolean. Trust the answer if the
00234 *>              reciprocal condition number is less than the threshold
00235 *>              sqrt(n) * slamch('Epsilon').
00236 *>
00237 *>     err = 2 "Guaranteed" error bound: The estimated forward error,
00238 *>              almost certainly within a factor of 10 of the true error
00239 *>              so long as the next entry is greater than the threshold
00240 *>              sqrt(n) * slamch('Epsilon'). This error bound should only
00241 *>              be trusted if the previous boolean is true.
00242 *>
00243 *>     err = 3  Reciprocal condition number: Estimated normwise
00244 *>              reciprocal condition number.  Compared with the threshold
00245 *>              sqrt(n) * slamch('Epsilon') to determine if the error
00246 *>              estimate is "guaranteed". These reciprocal condition
00247 *>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
00248 *>              appropriately scaled matrix Z.
00249 *>              Let Z = S*A, where S scales each row by a power of the
00250 *>              radix so all absolute row sums of Z are approximately 1.
00251 *>
00252 *>     This subroutine is only responsible for setting the second field
00253 *>     above.
00254 *>     See Lapack Working Note 165 for further details and extra
00255 *>     cautions.
00256 *> \endverbatim
00257 *>
00258 *> \param[in,out] ERR_BNDS_COMP
00259 *> \verbatim
00260 *>          ERR_BNDS_COMP is DOUBLE PRECISION array, dimension
00261 *>                    (NRHS, N_ERR_BNDS)
00262 *>     For each right-hand side, this array contains information about
00263 *>     various error bounds and condition numbers corresponding to the
00264 *>     componentwise relative error, which is defined as follows:
00265 *>
00266 *>     Componentwise relative error in the ith solution vector:
00267 *>                    abs(XTRUE(j,i) - X(j,i))
00268 *>             max_j ----------------------
00269 *>                         abs(X(j,i))
00270 *>
00271 *>     The array is indexed by the right-hand side i (on which the
00272 *>     componentwise relative error depends), and the type of error
00273 *>     information as described below. There currently are up to three
00274 *>     pieces of information returned for each right-hand side. If
00275 *>     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
00276 *>     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS .LT. 3, then at most
00277 *>     the first (:,N_ERR_BNDS) entries are returned.
00278 *>
00279 *>     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
00280 *>     right-hand side.
00281 *>
00282 *>     The second index in ERR_BNDS_COMP(:,err) contains the following
00283 *>     three fields:
00284 *>     err = 1 "Trust/don't trust" boolean. Trust the answer if the
00285 *>              reciprocal condition number is less than the threshold
00286 *>              sqrt(n) * slamch('Epsilon').
00287 *>
00288 *>     err = 2 "Guaranteed" error bound: The estimated forward error,
00289 *>              almost certainly within a factor of 10 of the true error
00290 *>              so long as the next entry is greater than the threshold
00291 *>              sqrt(n) * slamch('Epsilon'). This error bound should only
00292 *>              be trusted if the previous boolean is true.
00293 *>
00294 *>     err = 3  Reciprocal condition number: Estimated componentwise
00295 *>              reciprocal condition number.  Compared with the threshold
00296 *>              sqrt(n) * slamch('Epsilon') to determine if the error
00297 *>              estimate is "guaranteed". These reciprocal condition
00298 *>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
00299 *>              appropriately scaled matrix Z.
00300 *>              Let Z = S*(A*diag(x)), where x is the solution for the
00301 *>              current right-hand side and S scales each row of
00302 *>              A*diag(x) by a power of the radix so all absolute row
00303 *>              sums of Z are approximately 1.
00304 *>
00305 *>     This subroutine is only responsible for setting the second field
00306 *>     above.
00307 *>     See Lapack Working Note 165 for further details and extra
00308 *>     cautions.
00309 *> \endverbatim
00310 *>
00311 *> \param[in] RES
00312 *> \verbatim
00313 *>          RES is DOUBLE PRECISION array, dimension (N)
00314 *>     Workspace to hold the intermediate residual.
00315 *> \endverbatim
00316 *>
00317 *> \param[in] AYB
00318 *> \verbatim
00319 *>          AYB is DOUBLE PRECISION array, dimension (N)
00320 *>     Workspace. This can be the same workspace passed for Y_TAIL.
00321 *> \endverbatim
00322 *>
00323 *> \param[in] DY
00324 *> \verbatim
00325 *>          DY is DOUBLE PRECISION array, dimension (N)
00326 *>     Workspace to hold the intermediate solution.
00327 *> \endverbatim
00328 *>
00329 *> \param[in] Y_TAIL
00330 *> \verbatim
00331 *>          Y_TAIL is DOUBLE PRECISION array, dimension (N)
00332 *>     Workspace to hold the trailing bits of the intermediate solution.
00333 *> \endverbatim
00334 *>
00335 *> \param[in] RCOND
00336 *> \verbatim
00337 *>          RCOND is DOUBLE PRECISION
00338 *>     Reciprocal scaled condition number.  This is an estimate of the
00339 *>     reciprocal Skeel condition number of the matrix A after
00340 *>     equilibration (if done).  If this is less than the machine
00341 *>     precision (in particular, if it is zero), the matrix is singular
00342 *>     to working precision.  Note that the error may still be small even
00343 *>     if this number is very small and the matrix appears ill-
00344 *>     conditioned.
00345 *> \endverbatim
00346 *>
00347 *> \param[in] ITHRESH
00348 *> \verbatim
00349 *>          ITHRESH is INTEGER
00350 *>     The maximum number of residual computations allowed for
00351 *>     refinement. The default is 10. For 'aggressive' set to 100 to
00352 *>     permit convergence using approximate factorizations or
00353 *>     factorizations other than LU. If the factorization uses a
00354 *>     technique other than Gaussian elimination, the guarantees in
00355 *>     ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.
00356 *> \endverbatim
00357 *>
00358 *> \param[in] RTHRESH
00359 *> \verbatim
00360 *>          RTHRESH is DOUBLE PRECISION
00361 *>     Determines when to stop refinement if the error estimate stops
00362 *>     decreasing. Refinement will stop when the next solution no longer
00363 *>     satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
00364 *>     the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
00365 *>     default value is 0.5. For 'aggressive' set to 0.9 to permit
00366 *>     convergence on extremely ill-conditioned matrices. See LAWN 165
00367 *>     for more details.
00368 *> \endverbatim
00369 *>
00370 *> \param[in] DZ_UB
00371 *> \verbatim
00372 *>          DZ_UB is DOUBLE PRECISION
00373 *>     Determines when to start considering componentwise convergence.
00374 *>     Componentwise convergence is only considered after each component
00375 *>     of the solution Y is stable, which we definte as the relative
00376 *>     change in each component being less than DZ_UB. The default value
00377 *>     is 0.25, requiring the first bit to be stable. See LAWN 165 for
00378 *>     more details.
00379 *> \endverbatim
00380 *>
00381 *> \param[in] IGNORE_CWISE
00382 *> \verbatim
00383 *>          IGNORE_CWISE is LOGICAL
00384 *>     If .TRUE. then ignore componentwise convergence. Default value
00385 *>     is .FALSE..
00386 *> \endverbatim
00387 *>
00388 *> \param[out] INFO
00389 *> \verbatim
00390 *>          INFO is INTEGER
00391 *>       = 0:  Successful exit.
00392 *>       < 0:  if INFO = -i, the ith argument to DGBTRS had an illegal
00393 *>             value
00394 *> \endverbatim
00395 *
00396 *  Authors:
00397 *  ========
00398 *
00399 *> \author Univ. of Tennessee 
00400 *> \author Univ. of California Berkeley 
00401 *> \author Univ. of Colorado Denver 
00402 *> \author NAG Ltd. 
00403 *
00404 *> \date November 2011
00405 *
00406 *> \ingroup doubleGBcomputational
00407 *
00408 *  =====================================================================
00409       SUBROUTINE DLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, KL, KU,
00410      $                                NRHS, AB, LDAB, AFB, LDAFB, IPIV,
00411      $                                COLEQU, C, B, LDB, Y, LDY,
00412      $                                BERR_OUT, N_NORMS, ERR_BNDS_NORM,
00413      $                                ERR_BNDS_COMP, RES, AYB, DY,
00414      $                                Y_TAIL, RCOND, ITHRESH, RTHRESH,
00415      $                                DZ_UB, IGNORE_CWISE, INFO )
00416 *
00417 *  -- LAPACK computational routine (version 3.4.0) --
00418 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00419 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00420 *     November 2011
00421 *
00422 *     .. Scalar Arguments ..
00423       INTEGER            INFO, LDAB, LDAFB, LDB, LDY, N, KL, KU, NRHS,
00424      $                   PREC_TYPE, TRANS_TYPE, N_NORMS, ITHRESH
00425       LOGICAL            COLEQU, IGNORE_CWISE
00426       DOUBLE PRECISION   RTHRESH, DZ_UB
00427 *     ..
00428 *     .. Array Arguments ..
00429       INTEGER            IPIV( * )
00430       DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
00431      $                   Y( LDY, * ), RES(*), DY(*), Y_TAIL(*)
00432       DOUBLE PRECISION   C( * ), AYB(*), RCOND, BERR_OUT(*),
00433      $                   ERR_BNDS_NORM( NRHS, * ),
00434      $                   ERR_BNDS_COMP( NRHS, * )
00435 *     ..
00436 *
00437 *  =====================================================================
00438 *
00439 *     .. Local Scalars ..
00440       CHARACTER          TRANS
00441       INTEGER            CNT, I, J, M, X_STATE, Z_STATE, Y_PREC_STATE
00442       DOUBLE PRECISION   YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
00443      $                   DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
00444      $                   DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
00445      $                   EPS, HUGEVAL, INCR_THRESH
00446       LOGICAL            INCR_PREC
00447 *     ..
00448 *     .. Parameters ..
00449       INTEGER            UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
00450      $                   NOPROG_STATE, BASE_RESIDUAL, EXTRA_RESIDUAL,
00451      $                   EXTRA_Y
00452       PARAMETER          ( UNSTABLE_STATE = 0, WORKING_STATE = 1,
00453      $                   CONV_STATE = 2, NOPROG_STATE = 3 )
00454       PARAMETER          ( BASE_RESIDUAL = 0, EXTRA_RESIDUAL = 1,
00455      $                   EXTRA_Y = 2 )
00456       INTEGER            FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
00457       INTEGER            RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
00458       INTEGER            CMP_ERR_I, PIV_GROWTH_I
00459       PARAMETER          ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
00460      $                   BERR_I = 3 )
00461       PARAMETER          ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
00462       PARAMETER          ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
00463      $                   PIV_GROWTH_I = 9 )
00464       INTEGER            LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
00465      $                   LA_LINRX_CWISE_I
00466       PARAMETER          ( LA_LINRX_ITREF_I = 1,
00467      $                   LA_LINRX_ITHRESH_I = 2 )
00468       PARAMETER          ( LA_LINRX_CWISE_I = 3 )
00469       INTEGER            LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
00470      $                   LA_LINRX_RCOND_I
00471       PARAMETER          ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 )
00472       PARAMETER          ( LA_LINRX_RCOND_I = 3 )
00473 *     ..
00474 *     .. External Subroutines ..
00475       EXTERNAL           DAXPY, DCOPY, DGBTRS, DGBMV, BLAS_DGBMV_X,
00476      $                   BLAS_DGBMV2_X, DLA_GBAMV, DLA_WWADDW, DLAMCH,
00477      $                   CHLA_TRANSTYPE, DLA_LIN_BERR
00478       DOUBLE PRECISION   DLAMCH
00479       CHARACTER          CHLA_TRANSTYPE
00480 *     ..
00481 *     .. Intrinsic Functions ..
00482       INTRINSIC          ABS, MAX, MIN
00483 *     ..
00484 *     .. Executable Statements ..
00485 *
00486       IF (INFO.NE.0) RETURN
00487       TRANS = CHLA_TRANSTYPE(TRANS_TYPE)
00488       EPS = DLAMCH( 'Epsilon' )
00489       HUGEVAL = DLAMCH( 'Overflow' )
00490 *     Force HUGEVAL to Inf
00491       HUGEVAL = HUGEVAL * HUGEVAL
00492 *     Using HUGEVAL may lead to spurious underflows.
00493       INCR_THRESH = DBLE( N ) * EPS
00494       M = KL+KU+1
00495 
00496       DO J = 1, NRHS
00497          Y_PREC_STATE = EXTRA_RESIDUAL
00498          IF ( Y_PREC_STATE .EQ. EXTRA_Y ) THEN
00499             DO I = 1, N
00500                Y_TAIL( I ) = 0.0D+0
00501             END DO
00502          END IF
00503 
00504          DXRAT = 0.0D+0
00505          DXRATMAX = 0.0D+0
00506          DZRAT = 0.0D+0
00507          DZRATMAX = 0.0D+0
00508          FINAL_DX_X = HUGEVAL
00509          FINAL_DZ_Z = HUGEVAL
00510          PREVNORMDX = HUGEVAL
00511          PREV_DZ_Z = HUGEVAL
00512          DZ_Z = HUGEVAL
00513          DX_X = HUGEVAL
00514 
00515          X_STATE = WORKING_STATE
00516          Z_STATE = UNSTABLE_STATE
00517          INCR_PREC = .FALSE.
00518 
00519          DO CNT = 1, ITHRESH
00520 *
00521 *        Compute residual RES = B_s - op(A_s) * Y,
00522 *            op(A) = A, A**T, or A**H depending on TRANS (and type).
00523 *
00524             CALL DCOPY( N, B( 1, J ), 1, RES, 1 )
00525             IF ( Y_PREC_STATE .EQ. BASE_RESIDUAL ) THEN
00526                CALL DGBMV( TRANS, M, N, KL, KU, -1.0D+0, AB, LDAB,
00527      $              Y( 1, J ), 1, 1.0D+0, RES, 1 )
00528             ELSE IF ( Y_PREC_STATE .EQ. EXTRA_RESIDUAL ) THEN
00529                CALL BLAS_DGBMV_X( TRANS_TYPE, N, N, KL, KU,
00530      $              -1.0D+0, AB, LDAB, Y( 1, J ), 1, 1.0D+0, RES, 1,
00531      $              PREC_TYPE )
00532             ELSE
00533                CALL BLAS_DGBMV2_X( TRANS_TYPE, N, N, KL, KU, -1.0D+0,
00534      $              AB, LDAB, Y( 1, J ), Y_TAIL, 1, 1.0D+0, RES, 1,
00535      $              PREC_TYPE )
00536             END IF
00537 
00538 !        XXX: RES is no longer needed.
00539             CALL DCOPY( N, RES, 1, DY, 1 )
00540             CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV, DY, N,
00541      $           INFO )
00542 *
00543 *         Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
00544 *
00545             NORMX = 0.0D+0
00546             NORMY = 0.0D+0
00547             NORMDX = 0.0D+0
00548             DZ_Z = 0.0D+0
00549             YMIN = HUGEVAL
00550 
00551             DO I = 1, N
00552                YK = ABS( Y( I, J ) )
00553                DYK = ABS( DY( I ) )
00554 
00555                IF ( YK .NE. 0.0D+0 ) THEN
00556                   DZ_Z = MAX( DZ_Z, DYK / YK )
00557                ELSE IF ( DYK .NE. 0.0D+0 ) THEN
00558                   DZ_Z = HUGEVAL
00559                END IF
00560 
00561                YMIN = MIN( YMIN, YK )
00562 
00563                NORMY = MAX( NORMY, YK )
00564 
00565                IF ( COLEQU ) THEN
00566                   NORMX = MAX( NORMX, YK * C( I ) )
00567                   NORMDX = MAX( NORMDX, DYK * C( I ) )
00568                ELSE
00569                   NORMX = NORMY
00570                   NORMDX = MAX( NORMDX, DYK )
00571                END IF
00572             END DO
00573 
00574             IF ( NORMX .NE. 0.0D+0 ) THEN
00575                DX_X = NORMDX / NORMX
00576             ELSE IF ( NORMDX .EQ. 0.0D+0 ) THEN
00577                DX_X = 0.0D+0
00578             ELSE
00579                DX_X = HUGEVAL
00580             END IF
00581 
00582             DXRAT = NORMDX / PREVNORMDX
00583             DZRAT = DZ_Z / PREV_DZ_Z
00584 *
00585 *         Check termination criteria.
00586 *
00587             IF ( .NOT.IGNORE_CWISE
00588      $           .AND. YMIN*RCOND .LT. INCR_THRESH*NORMY
00589      $           .AND. Y_PREC_STATE .LT. EXTRA_Y )
00590      $           INCR_PREC = .TRUE.
00591 
00592             IF ( X_STATE .EQ. NOPROG_STATE .AND. DXRAT .LE. RTHRESH )
00593      $           X_STATE = WORKING_STATE
00594             IF ( X_STATE .EQ. WORKING_STATE ) THEN
00595                IF ( DX_X .LE. EPS ) THEN
00596                   X_STATE = CONV_STATE
00597                ELSE IF ( DXRAT .GT. RTHRESH ) THEN
00598                   IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN
00599                      INCR_PREC = .TRUE.
00600                   ELSE
00601                      X_STATE = NOPROG_STATE
00602                   END IF
00603                ELSE
00604                   IF ( DXRAT .GT. DXRATMAX ) DXRATMAX = DXRAT
00605                END IF
00606                IF ( X_STATE .GT. WORKING_STATE ) FINAL_DX_X = DX_X
00607             END IF
00608 
00609             IF ( Z_STATE .EQ. UNSTABLE_STATE .AND. DZ_Z .LE. DZ_UB )
00610      $           Z_STATE = WORKING_STATE
00611             IF ( Z_STATE .EQ. NOPROG_STATE .AND. DZRAT .LE. RTHRESH )
00612      $           Z_STATE = WORKING_STATE
00613             IF ( Z_STATE .EQ. WORKING_STATE ) THEN
00614                IF ( DZ_Z .LE. EPS ) THEN
00615                   Z_STATE = CONV_STATE
00616                ELSE IF ( DZ_Z .GT. DZ_UB ) THEN
00617                   Z_STATE = UNSTABLE_STATE
00618                   DZRATMAX = 0.0D+0
00619                   FINAL_DZ_Z = HUGEVAL
00620                ELSE IF ( DZRAT .GT. RTHRESH ) THEN
00621                   IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN
00622                      INCR_PREC = .TRUE.
00623                   ELSE
00624                      Z_STATE = NOPROG_STATE
00625                   END IF
00626                ELSE
00627                   IF ( DZRAT .GT. DZRATMAX ) DZRATMAX = DZRAT
00628                END IF
00629                IF ( Z_STATE .GT. WORKING_STATE ) FINAL_DZ_Z = DZ_Z
00630             END IF
00631 *
00632 *           Exit if both normwise and componentwise stopped working,
00633 *           but if componentwise is unstable, let it go at least two
00634 *           iterations.
00635 *
00636             IF ( X_STATE.NE.WORKING_STATE ) THEN
00637                IF ( IGNORE_CWISE ) GOTO 666
00638                IF ( Z_STATE.EQ.NOPROG_STATE .OR. Z_STATE.EQ.CONV_STATE )
00639      $              GOTO 666
00640                IF ( Z_STATE.EQ.UNSTABLE_STATE .AND. CNT.GT.1 ) GOTO 666
00641             END IF
00642 
00643             IF ( INCR_PREC ) THEN
00644                INCR_PREC = .FALSE.
00645                Y_PREC_STATE = Y_PREC_STATE + 1
00646                DO I = 1, N
00647                   Y_TAIL( I ) = 0.0D+0
00648                END DO
00649             END IF
00650 
00651             PREVNORMDX = NORMDX
00652             PREV_DZ_Z = DZ_Z
00653 *
00654 *           Update soluton.
00655 *
00656             IF (Y_PREC_STATE .LT. EXTRA_Y) THEN
00657                CALL DAXPY( N, 1.0D+0, DY, 1, Y(1,J), 1 )
00658             ELSE
00659                CALL DLA_WWADDW( N, Y(1,J), Y_TAIL, DY )
00660             END IF
00661 
00662          END DO
00663 *        Target of "IF (Z_STOP .AND. X_STOP)".  Sun's f77 won't EXIT.
00664  666     CONTINUE
00665 *
00666 *     Set final_* when cnt hits ithresh.
00667 *
00668          IF ( X_STATE .EQ. WORKING_STATE ) FINAL_DX_X = DX_X
00669          IF ( Z_STATE .EQ. WORKING_STATE ) FINAL_DZ_Z = DZ_Z
00670 *
00671 *     Compute error bounds.
00672 *
00673          IF ( N_NORMS .GE. 1 ) THEN
00674             ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) =
00675      $           FINAL_DX_X / (1 - DXRATMAX)
00676          END IF
00677          IF (N_NORMS .GE. 2) THEN
00678             ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) =
00679      $           FINAL_DZ_Z / (1 - DZRATMAX)
00680          END IF
00681 *
00682 *     Compute componentwise relative backward error from formula
00683 *         max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
00684 *     where abs(Z) is the componentwise absolute value of the matrix
00685 *     or vector Z.
00686 *
00687 *        Compute residual RES = B_s - op(A_s) * Y,
00688 *            op(A) = A, A**T, or A**H depending on TRANS (and type).
00689 *
00690          CALL DCOPY( N, B( 1, J ), 1, RES, 1 )
00691          CALL DGBMV(TRANS, N, N, KL, KU, -1.0D+0, AB, LDAB, Y(1,J),
00692      $        1, 1.0D+0, RES, 1 )
00693 
00694          DO I = 1, N
00695             AYB( I ) = ABS( B( I, J ) )
00696          END DO
00697 *
00698 *     Compute abs(op(A_s))*abs(Y) + abs(B_s).
00699 *
00700         CALL DLA_GBAMV( TRANS_TYPE, N, N, KL, KU, 1.0D+0,
00701      $        AB, LDAB, Y(1, J), 1, 1.0D+0, AYB, 1 )
00702 
00703          CALL DLA_LIN_BERR( N, N, 1, RES, AYB, BERR_OUT( J ) )
00704 *
00705 *     End of loop for each RHS
00706 *
00707       END DO
00708 *
00709       RETURN
00710       END
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