LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
zherfsx.f
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1 *> \brief \b ZHERFSX
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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11 *> [TGZ]</a>
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13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zherfsx.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZHERFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV,
22 * S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
23 * ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
24 * WORK, RWORK, INFO )
25 *
26 * .. Scalar Arguments ..
27 * CHARACTER UPLO, EQUED
28 * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
29 * $ N_ERR_BNDS
30 * DOUBLE PRECISION RCOND
31 * ..
32 * .. Array Arguments ..
33 * INTEGER IPIV( * )
34 * COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
35 * $ X( LDX, * ), WORK( * )
36 * DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
37 * $ ERR_BNDS_NORM( NRHS, * ),
38 * $ ERR_BNDS_COMP( NRHS, * )
39 *
40 *
41 *> \par Purpose:
42 * =============
43 *>
44 *> \verbatim
45 *>
46 *> ZHERFSX improves the computed solution to a system of linear
47 *> equations when the coefficient matrix is Hermitian indefinite, and
48 *> provides error bounds and backward error estimates for the
49 *> solution. In addition to normwise error bound, the code provides
50 *> maximum componentwise error bound if possible. See comments for
51 *> ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds.
52 *>
53 *> The original system of linear equations may have been equilibrated
54 *> before calling this routine, as described by arguments EQUED and S
55 *> below. In this case, the solution and error bounds returned are
56 *> for the original unequilibrated system.
57 *> \endverbatim
58 *
59 * Arguments:
60 * ==========
61 *
62 *> \verbatim
63 *> Some optional parameters are bundled in the PARAMS array. These
64 *> settings determine how refinement is performed, but often the
65 *> defaults are acceptable. If the defaults are acceptable, users
66 *> can pass NPARAMS = 0 which prevents the source code from accessing
67 *> the PARAMS argument.
68 *> \endverbatim
69 *>
70 *> \param[in] UPLO
71 *> \verbatim
72 *> UPLO is CHARACTER*1
73 *> = 'U': Upper triangle of A is stored;
74 *> = 'L': Lower triangle of A is stored.
75 *> \endverbatim
76 *>
77 *> \param[in] EQUED
78 *> \verbatim
79 *> EQUED is CHARACTER*1
80 *> Specifies the form of equilibration that was done to A
81 *> before calling this routine. This is needed to compute
82 *> the solution and error bounds correctly.
83 *> = 'N': No equilibration
84 *> = 'Y': Both row and column equilibration, i.e., A has been
85 *> replaced by diag(S) * A * diag(S).
86 *> The right hand side B has been changed accordingly.
87 *> \endverbatim
88 *>
89 *> \param[in] N
90 *> \verbatim
91 *> N is INTEGER
92 *> The order of the matrix A. N >= 0.
93 *> \endverbatim
94 *>
95 *> \param[in] NRHS
96 *> \verbatim
97 *> NRHS is INTEGER
98 *> The number of right hand sides, i.e., the number of columns
99 *> of the matrices B and X. NRHS >= 0.
100 *> \endverbatim
101 *>
102 *> \param[in] A
103 *> \verbatim
104 *> A is COMPLEX*16 array, dimension (LDA,N)
105 *> The Hermitian matrix A. If UPLO = 'U', the leading N-by-N
106 *> upper triangular part of A contains the upper triangular
107 *> part of the matrix A, and the strictly lower triangular
108 *> part of A is not referenced. If UPLO = 'L', the leading
109 *> N-by-N lower triangular part of A contains the lower
110 *> triangular part of the matrix A, and the strictly upper
111 *> triangular part of A is not referenced.
112 *> \endverbatim
113 *>
114 *> \param[in] LDA
115 *> \verbatim
116 *> LDA is INTEGER
117 *> The leading dimension of the array A. LDA >= max(1,N).
118 *> \endverbatim
119 *>
120 *> \param[in] AF
121 *> \verbatim
122 *> AF is COMPLEX*16 array, dimension (LDAF,N)
123 *> The factored form of the matrix A. AF contains the block
124 *> diagonal matrix D and the multipliers used to obtain the
125 *> factor U or L from the factorization A = U*D*U**H or A =
126 *> L*D*L**H as computed by ZHETRF.
127 *> \endverbatim
128 *>
129 *> \param[in] LDAF
130 *> \verbatim
131 *> LDAF is INTEGER
132 *> The leading dimension of the array AF. LDAF >= max(1,N).
133 *> \endverbatim
134 *>
135 *> \param[in] IPIV
136 *> \verbatim
137 *> IPIV is INTEGER array, dimension (N)
138 *> Details of the interchanges and the block structure of D
139 *> as determined by ZHETRF.
140 *> \endverbatim
141 *>
142 *> \param[in,out] S
143 *> \verbatim
144 *> S is DOUBLE PRECISION array, dimension (N)
145 *> The scale factors for A. If EQUED = 'Y', A is multiplied on
146 *> the left and right by diag(S). S is an input argument if FACT =
147 *> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
148 *> = 'Y', each element of S must be positive. If S is output, each
149 *> element of S is a power of the radix. If S is input, each element
150 *> of S should be a power of the radix to ensure a reliable solution
151 *> and error estimates. Scaling by powers of the radix does not cause
152 *> rounding errors unless the result underflows or overflows.
153 *> Rounding errors during scaling lead to refining with a matrix that
154 *> is not equivalent to the input matrix, producing error estimates
155 *> that may not be reliable.
156 *> \endverbatim
157 *>
158 *> \param[in] B
159 *> \verbatim
160 *> B is COMPLEX*16 array, dimension (LDB,NRHS)
161 *> The right hand side matrix B.
162 *> \endverbatim
163 *>
164 *> \param[in] LDB
165 *> \verbatim
166 *> LDB is INTEGER
167 *> The leading dimension of the array B. LDB >= max(1,N).
168 *> \endverbatim
169 *>
170 *> \param[in,out] X
171 *> \verbatim
172 *> X is COMPLEX*16 array, dimension (LDX,NRHS)
173 *> On entry, the solution matrix X, as computed by ZHETRS.
174 *> On exit, the improved solution matrix X.
175 *> \endverbatim
176 *>
177 *> \param[in] LDX
178 *> \verbatim
179 *> LDX is INTEGER
180 *> The leading dimension of the array X. LDX >= max(1,N).
181 *> \endverbatim
182 *>
183 *> \param[out] RCOND
184 *> \verbatim
185 *> RCOND is DOUBLE PRECISION
186 *> Reciprocal scaled condition number. This is an estimate of the
187 *> reciprocal Skeel condition number of the matrix A after
188 *> equilibration (if done). If this is less than the machine
189 *> precision (in particular, if it is zero), the matrix is singular
190 *> to working precision. Note that the error may still be small even
191 *> if this number is very small and the matrix appears ill-
192 *> conditioned.
193 *> \endverbatim
194 *>
195 *> \param[out] BERR
196 *> \verbatim
197 *> BERR is DOUBLE PRECISION array, dimension (NRHS)
198 *> Componentwise relative backward error. This is the
199 *> componentwise relative backward error of each solution vector X(j)
200 *> (i.e., the smallest relative change in any element of A or B that
201 *> makes X(j) an exact solution).
202 *> \endverbatim
203 *>
204 *> \param[in] N_ERR_BNDS
205 *> \verbatim
206 *> N_ERR_BNDS is INTEGER
207 *> Number of error bounds to return for each right hand side
208 *> and each type (normwise or componentwise). See ERR_BNDS_NORM and
209 *> ERR_BNDS_COMP below.
210 *> \endverbatim
211 *>
212 *> \param[out] ERR_BNDS_NORM
213 *> \verbatim
214 *> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
215 *> For each right-hand side, this array contains information about
216 *> various error bounds and condition numbers corresponding to the
217 *> normwise relative error, which is defined as follows:
218 *>
219 *> Normwise relative error in the ith solution vector:
220 *> max_j (abs(XTRUE(j,i) - X(j,i)))
221 *> ------------------------------
222 *> max_j abs(X(j,i))
223 *>
224 *> The array is indexed by the type of error information as described
225 *> below. There currently are up to three pieces of information
226 *> returned.
227 *>
228 *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
229 *> right-hand side.
230 *>
231 *> The second index in ERR_BNDS_NORM(:,err) contains the following
232 *> three fields:
233 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
234 *> reciprocal condition number is less than the threshold
235 *> sqrt(n) * dlamch('Epsilon').
236 *>
237 *> err = 2 "Guaranteed" error bound: The estimated forward error,
238 *> almost certainly within a factor of 10 of the true error
239 *> so long as the next entry is greater than the threshold
240 *> sqrt(n) * dlamch('Epsilon'). This error bound should only
241 *> be trusted if the previous boolean is true.
242 *>
243 *> err = 3 Reciprocal condition number: Estimated normwise
244 *> reciprocal condition number. Compared with the threshold
245 *> sqrt(n) * dlamch('Epsilon') to determine if the error
246 *> estimate is "guaranteed". These reciprocal condition
247 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
248 *> appropriately scaled matrix Z.
249 *> Let Z = S*A, where S scales each row by a power of the
250 *> radix so all absolute row sums of Z are approximately 1.
251 *>
252 *> See Lapack Working Note 165 for further details and extra
253 *> cautions.
254 *> \endverbatim
255 *>
256 *> \param[out] ERR_BNDS_COMP
257 *> \verbatim
258 *> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
259 *> For each right-hand side, this array contains information about
260 *> various error bounds and condition numbers corresponding to the
261 *> componentwise relative error, which is defined as follows:
262 *>
263 *> Componentwise relative error in the ith solution vector:
264 *> abs(XTRUE(j,i) - X(j,i))
265 *> max_j ----------------------
266 *> abs(X(j,i))
267 *>
268 *> The array is indexed by the right-hand side i (on which the
269 *> componentwise relative error depends), and the type of error
270 *> information as described below. There currently are up to three
271 *> pieces of information returned for each right-hand side. If
272 *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
273 *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
274 *> the first (:,N_ERR_BNDS) entries are returned.
275 *>
276 *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
277 *> right-hand side.
278 *>
279 *> The second index in ERR_BNDS_COMP(:,err) contains the following
280 *> three fields:
281 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
282 *> reciprocal condition number is less than the threshold
283 *> sqrt(n) * dlamch('Epsilon').
284 *>
285 *> err = 2 "Guaranteed" error bound: The estimated forward error,
286 *> almost certainly within a factor of 10 of the true error
287 *> so long as the next entry is greater than the threshold
288 *> sqrt(n) * dlamch('Epsilon'). This error bound should only
289 *> be trusted if the previous boolean is true.
290 *>
291 *> err = 3 Reciprocal condition number: Estimated componentwise
292 *> reciprocal condition number. Compared with the threshold
293 *> sqrt(n) * dlamch('Epsilon') to determine if the error
294 *> estimate is "guaranteed". These reciprocal condition
295 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
296 *> appropriately scaled matrix Z.
297 *> Let Z = S*(A*diag(x)), where x is the solution for the
298 *> current right-hand side and S scales each row of
299 *> A*diag(x) by a power of the radix so all absolute row
300 *> sums of Z are approximately 1.
301 *>
302 *> See Lapack Working Note 165 for further details and extra
303 *> cautions.
304 *> \endverbatim
305 *>
306 *> \param[in] NPARAMS
307 *> \verbatim
308 *> NPARAMS is INTEGER
309 *> Specifies the number of parameters set in PARAMS. If <= 0, the
310 *> PARAMS array is never referenced and default values are used.
311 *> \endverbatim
312 *>
313 *> \param[in,out] PARAMS
314 *> \verbatim
315 *> PARAMS is DOUBLE PRECISION array, dimension NPARAMS
316 *> Specifies algorithm parameters. If an entry is < 0.0, then
317 *> that entry will be filled with default value used for that
318 *> parameter. Only positions up to NPARAMS are accessed; defaults
319 *> are used for higher-numbered parameters.
320 *>
321 *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
322 *> refinement or not.
323 *> Default: 1.0D+0
324 *> = 0.0: No refinement is performed, and no error bounds are
325 *> computed.
326 *> = 1.0: Use the double-precision refinement algorithm,
327 *> possibly with doubled-single computations if the
328 *> compilation environment does not support DOUBLE
329 *> PRECISION.
330 *> (other values are reserved for future use)
331 *>
332 *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
333 *> computations allowed for refinement.
334 *> Default: 10
335 *> Aggressive: Set to 100 to permit convergence using approximate
336 *> factorizations or factorizations other than LU. If
337 *> the factorization uses a technique other than
338 *> Gaussian elimination, the guarantees in
339 *> err_bnds_norm and err_bnds_comp may no longer be
340 *> trustworthy.
341 *>
342 *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
343 *> will attempt to find a solution with small componentwise
344 *> relative error in the double-precision algorithm. Positive
345 *> is true, 0.0 is false.
346 *> Default: 1.0 (attempt componentwise convergence)
347 *> \endverbatim
348 *>
349 *> \param[out] WORK
350 *> \verbatim
351 *> WORK is COMPLEX*16 array, dimension (2*N)
352 *> \endverbatim
353 *>
354 *> \param[out] RWORK
355 *> \verbatim
356 *> RWORK is DOUBLE PRECISION array, dimension (2*N)
357 *> \endverbatim
358 *>
359 *> \param[out] INFO
360 *> \verbatim
361 *> INFO is INTEGER
362 *> = 0: Successful exit. The solution to every right-hand side is
363 *> guaranteed.
364 *> < 0: If INFO = -i, the i-th argument had an illegal value
365 *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
366 *> has been completed, but the factor U is exactly singular, so
367 *> the solution and error bounds could not be computed. RCOND = 0
368 *> is returned.
369 *> = N+J: The solution corresponding to the Jth right-hand side is
370 *> not guaranteed. The solutions corresponding to other right-
371 *> hand sides K with K > J may not be guaranteed as well, but
372 *> only the first such right-hand side is reported. If a small
373 *> componentwise error is not requested (PARAMS(3) = 0.0) then
374 *> the Jth right-hand side is the first with a normwise error
375 *> bound that is not guaranteed (the smallest J such
376 *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
377 *> the Jth right-hand side is the first with either a normwise or
378 *> componentwise error bound that is not guaranteed (the smallest
379 *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
380 *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
381 *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
382 *> about all of the right-hand sides check ERR_BNDS_NORM or
383 *> ERR_BNDS_COMP.
384 *> \endverbatim
385 *
386 * Authors:
387 * ========
388 *
389 *> \author Univ. of Tennessee
390 *> \author Univ. of California Berkeley
391 *> \author Univ. of Colorado Denver
392 *> \author NAG Ltd.
393 *
394 *> \ingroup complex16HEcomputational
395 *
396 * =====================================================================
397  SUBROUTINE zherfsx( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV,
398  $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
399  $ ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
400  $ WORK, RWORK, INFO )
401 *
402 * -- LAPACK computational routine --
403 * -- LAPACK is a software package provided by Univ. of Tennessee, --
404 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
405 *
406 * .. Scalar Arguments ..
407  CHARACTER UPLO, EQUED
408  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
409  $ N_ERR_BNDS
410  DOUBLE PRECISION RCOND
411 * ..
412 * .. Array Arguments ..
413  INTEGER IPIV( * )
414  COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
415  $ X( LDX, * ), WORK( * )
416  DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
417  $ err_bnds_norm( nrhs, * ),
418  $ err_bnds_comp( nrhs, * )
419 *
420 * ==================================================================
421 *
422 * .. Parameters ..
423  DOUBLE PRECISION ZERO, ONE
424  PARAMETER ( ZERO = 0.0d+0, one = 1.0d+0 )
425  DOUBLE PRECISION ITREF_DEFAULT, ITHRESH_DEFAULT
426  DOUBLE PRECISION COMPONENTWISE_DEFAULT, RTHRESH_DEFAULT
427  DOUBLE PRECISION DZTHRESH_DEFAULT
428  parameter( itref_default = 1.0d+0 )
429  parameter( ithresh_default = 10.0d+0 )
430  parameter( componentwise_default = 1.0d+0 )
431  parameter( rthresh_default = 0.5d+0 )
432  parameter( dzthresh_default = 0.25d+0 )
433  INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
434  $ la_linrx_cwise_i
435  parameter( la_linrx_itref_i = 1,
436  $ la_linrx_ithresh_i = 2 )
437  parameter( la_linrx_cwise_i = 3 )
438  INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
439  $ LA_LINRX_RCOND_I
440  parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
441  parameter( la_linrx_rcond_i = 3 )
442 * ..
443 * .. Local Scalars ..
444  CHARACTER(1) NORM
445  LOGICAL RCEQU
446  INTEGER J, PREC_TYPE, REF_TYPE
447  INTEGER N_NORMS
448  DOUBLE PRECISION ANORM, RCOND_TMP
449  DOUBLE PRECISION ILLRCOND_THRESH, ERR_LBND, CWISE_WRONG
450  LOGICAL IGNORE_CWISE
451  INTEGER ITHRESH
452  DOUBLE PRECISION RTHRESH, UNSTABLE_THRESH
453 * ..
454 * .. External Subroutines ..
456 * ..
457 * .. Intrinsic Functions ..
458  INTRINSIC max, sqrt, transfer
459 * ..
460 * .. External Functions ..
461  EXTERNAL lsame, ilaprec
463  DOUBLE PRECISION DLAMCH, ZLANHE, ZLA_HERCOND_X, ZLA_HERCOND_C
464  LOGICAL LSAME
465  INTEGER ILAPREC
466 * ..
467 * .. Executable Statements ..
468 *
469 * Check the input parameters.
470 *
471  info = 0
472  ref_type = int( itref_default )
473  IF ( nparams .GE. la_linrx_itref_i ) THEN
474  IF ( params( la_linrx_itref_i ) .LT. 0.0d+0 ) THEN
475  params( la_linrx_itref_i ) = itref_default
476  ELSE
477  ref_type = params( la_linrx_itref_i )
478  END IF
479  END IF
480 *
481 * Set default parameters.
482 *
483  illrcond_thresh = dble( n ) * dlamch( 'Epsilon' )
484  ithresh = int( ithresh_default )
485  rthresh = rthresh_default
486  unstable_thresh = dzthresh_default
487  ignore_cwise = componentwise_default .EQ. 0.0d+0
488 *
489  IF ( nparams.GE.la_linrx_ithresh_i ) THEN
490  IF ( params( la_linrx_ithresh_i ).LT.0.0d+0 ) THEN
491  params( la_linrx_ithresh_i ) = ithresh
492  ELSE
493  ithresh = int( params( la_linrx_ithresh_i ) )
494  END IF
495  END IF
496  IF ( nparams.GE.la_linrx_cwise_i ) THEN
497  IF ( params(la_linrx_cwise_i ).LT.0.0d+0 ) THEN
498  IF ( ignore_cwise ) THEN
499  params( la_linrx_cwise_i ) = 0.0d+0
500  ELSE
501  params( la_linrx_cwise_i ) = 1.0d+0
502  END IF
503  ELSE
504  ignore_cwise = params( la_linrx_cwise_i ) .EQ. 0.0d+0
505  END IF
506  END IF
507  IF ( ref_type .EQ. 0 .OR. n_err_bnds .EQ. 0 ) THEN
508  n_norms = 0
509  ELSE IF ( ignore_cwise ) THEN
510  n_norms = 1
511  ELSE
512  n_norms = 2
513  END IF
514 *
515  rcequ = lsame( equed, 'Y' )
516 *
517 * Test input parameters.
518 *
519  IF (.NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
520  info = -1
521  ELSE IF( .NOT.rcequ .AND. .NOT.lsame( equed, 'N' ) ) THEN
522  info = -2
523  ELSE IF( n.LT.0 ) THEN
524  info = -3
525  ELSE IF( nrhs.LT.0 ) THEN
526  info = -4
527  ELSE IF( lda.LT.max( 1, n ) ) THEN
528  info = -6
529  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
530  info = -8
531  ELSE IF( ldb.LT.max( 1, n ) ) THEN
532  info = -12
533  ELSE IF( ldx.LT.max( 1, n ) ) THEN
534  info = -14
535  END IF
536  IF( info.NE.0 ) THEN
537  CALL xerbla( 'ZHERFSX', -info )
538  RETURN
539  END IF
540 *
541 * Quick return if possible.
542 *
543  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
544  rcond = 1.0d+0
545  DO j = 1, nrhs
546  berr( j ) = 0.0d+0
547  IF ( n_err_bnds .GE. 1 ) THEN
548  err_bnds_norm( j, la_linrx_trust_i ) = 1.0d+0
549  err_bnds_comp( j, la_linrx_trust_i ) = 1.0d+0
550  END IF
551  IF ( n_err_bnds .GE. 2 ) THEN
552  err_bnds_norm( j, la_linrx_err_i ) = 0.0d+0
553  err_bnds_comp( j, la_linrx_err_i ) = 0.0d+0
554  END IF
555  IF ( n_err_bnds .GE. 3 ) THEN
556  err_bnds_norm( j, la_linrx_rcond_i ) = 1.0d+0
557  err_bnds_comp( j, la_linrx_rcond_i ) = 1.0d+0
558  END IF
559  END DO
560  RETURN
561  END IF
562 *
563 * Default to failure.
564 *
565  rcond = 0.0d+0
566  DO j = 1, nrhs
567  berr( j ) = 1.0d+0
568  IF ( n_err_bnds .GE. 1 ) THEN
569  err_bnds_norm( j, la_linrx_trust_i ) = 1.0d+0
570  err_bnds_comp( j, la_linrx_trust_i ) = 1.0d+0
571  END IF
572  IF ( n_err_bnds .GE. 2 ) THEN
573  err_bnds_norm( j, la_linrx_err_i ) = 1.0d+0
574  err_bnds_comp( j, la_linrx_err_i ) = 1.0d+0
575  END IF
576  IF ( n_err_bnds .GE. 3 ) THEN
577  err_bnds_norm( j, la_linrx_rcond_i ) = 0.0d+0
578  err_bnds_comp( j, la_linrx_rcond_i ) = 0.0d+0
579  END IF
580  END DO
581 *
582 * Compute the norm of A and the reciprocal of the condition
583 * number of A.
584 *
585  norm = 'I'
586  anorm = zlanhe( norm, uplo, n, a, lda, rwork )
587  CALL zhecon( uplo, n, af, ldaf, ipiv, anorm, rcond, work,
588  $ info )
589 *
590 * Perform refinement on each right-hand side
591 *
592  IF ( ref_type .NE. 0 ) THEN
593 
594  prec_type = ilaprec( 'E' )
595 
596  CALL zla_herfsx_extended( prec_type, uplo, n,
597  $ nrhs, a, lda, af, ldaf, ipiv, rcequ, s, b,
598  $ ldb, x, ldx, berr, n_norms, err_bnds_norm, err_bnds_comp,
599  $ work, rwork, work(n+1),
600  $ transfer(rwork(1:2*n), (/ (zero, zero) /), n), rcond,
601  $ ithresh, rthresh, unstable_thresh, ignore_cwise,
602  $ info )
603  END IF
604 
605  err_lbnd = max( 10.0d+0, sqrt( dble( n ) ) ) * dlamch( 'Epsilon' )
606  IF ( n_err_bnds .GE. 1 .AND. n_norms .GE. 1 ) THEN
607 *
608 * Compute scaled normwise condition number cond(A*C).
609 *
610  IF ( rcequ ) THEN
611  rcond_tmp = zla_hercond_c( uplo, n, a, lda, af, ldaf, ipiv,
612  $ s, .true., info, work, rwork )
613  ELSE
614  rcond_tmp = zla_hercond_c( uplo, n, a, lda, af, ldaf, ipiv,
615  $ s, .false., info, work, rwork )
616  END IF
617  DO j = 1, nrhs
618 *
619 * Cap the error at 1.0.
620 *
621  IF ( n_err_bnds .GE. la_linrx_err_i
622  $ .AND. err_bnds_norm( j, la_linrx_err_i ) .GT. 1.0d+0 )
623  $ err_bnds_norm( j, la_linrx_err_i ) = 1.0d+0
624 *
625 * Threshold the error (see LAWN).
626 *
627  IF (rcond_tmp .LT. illrcond_thresh) THEN
628  err_bnds_norm( j, la_linrx_err_i ) = 1.0d+0
629  err_bnds_norm( j, la_linrx_trust_i ) = 0.0d+0
630  IF ( info .LE. n ) info = n + j
631  ELSE IF ( err_bnds_norm( j, la_linrx_err_i ) .LT. err_lbnd )
632  $ THEN
633  err_bnds_norm( j, la_linrx_err_i ) = err_lbnd
634  err_bnds_norm( j, la_linrx_trust_i ) = 1.0d+0
635  END IF
636 *
637 * Save the condition number.
638 *
639  IF ( n_err_bnds .GE. la_linrx_rcond_i ) THEN
640  err_bnds_norm( j, la_linrx_rcond_i ) = rcond_tmp
641  END IF
642  END DO
643  END IF
644 
645  IF ( n_err_bnds .GE. 1 .AND. n_norms .GE. 2 ) THEN
646 *
647 * Compute componentwise condition number cond(A*diag(Y(:,J))) for
648 * each right-hand side using the current solution as an estimate of
649 * the true solution. If the componentwise error estimate is too
650 * large, then the solution is a lousy estimate of truth and the
651 * estimated RCOND may be too optimistic. To avoid misleading users,
652 * the inverse condition number is set to 0.0 when the estimated
653 * cwise error is at least CWISE_WRONG.
654 *
655  cwise_wrong = sqrt( dlamch( 'Epsilon' ) )
656  DO j = 1, nrhs
657  IF ( err_bnds_comp( j, la_linrx_err_i ) .LT. cwise_wrong )
658  $ THEN
659  rcond_tmp = zla_hercond_x( uplo, n, a, lda, af, ldaf,
660  $ ipiv, x( 1, j ), info, work, rwork )
661  ELSE
662  rcond_tmp = 0.0d+0
663  END IF
664 *
665 * Cap the error at 1.0.
666 *
667  IF ( n_err_bnds .GE. la_linrx_err_i
668  $ .AND. err_bnds_comp( j, la_linrx_err_i ) .GT. 1.0d+0 )
669  $ err_bnds_comp( j, la_linrx_err_i ) = 1.0d+0
670 *
671 * Threshold the error (see LAWN).
672 *
673  IF ( rcond_tmp .LT. illrcond_thresh ) THEN
674  err_bnds_comp( j, la_linrx_err_i ) = 1.0d+0
675  err_bnds_comp( j, la_linrx_trust_i ) = 0.0d+0
676  IF ( .NOT. ignore_cwise
677  $ .AND. info.LT.n + j ) info = n + j
678  ELSE IF ( err_bnds_comp( j, la_linrx_err_i )
679  $ .LT. err_lbnd ) THEN
680  err_bnds_comp( j, la_linrx_err_i ) = err_lbnd
681  err_bnds_comp( j, la_linrx_trust_i ) = 1.0d+0
682  END IF
683 *
684 * Save the condition number.
685 *
686  IF ( n_err_bnds .GE. la_linrx_rcond_i ) THEN
687  err_bnds_comp( j, la_linrx_rcond_i ) = rcond_tmp
688  END IF
689 
690  END DO
691  END IF
692 *
693  RETURN
694 *
695 * End of ZHERFSX
696 *
697  END
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
integer function ilaprec(PREC)
ILAPREC
Definition: ilaprec.f:58
double precision function zlanhe(NORM, UPLO, N, A, LDA, WORK)
ZLANHE returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: zlanhe.f:124
double precision function zla_hercond_c(UPLO, N, A, LDA, AF, LDAF, IPIV, C, CAPPLY, INFO, WORK, RWORK)
ZLA_HERCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian indefin...
double precision function zla_hercond_x(UPLO, N, A, LDA, AF, LDAF, IPIV, X, INFO, WORK, RWORK)
ZLA_HERCOND_X computes the infinity norm condition number of op(A)*diag(x) for Hermitian indefinite m...
subroutine zhecon(UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)
ZHECON
Definition: zhecon.f:125
subroutine zherfsx(UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
ZHERFSX
Definition: zherfsx.f:401
subroutine zla_herfsx_extended(PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
ZLA_HERFSX_EXTENDED improves the computed solution to a system of linear equations for Hermitian inde...