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
1.7.2