LAPACK  3.7.1
LAPACK: Linear Algebra PACKage
dsyrfsx.f
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1 *> \brief \b DSYRFSX
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
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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/dsyrfsx.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DSYRFSX( 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, IWORK, 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( * ), IWORK( * )
34 * DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
35 * $ X( LDX, * ), WORK( * )
36 * DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ),
37 * $ ERR_BNDS_NORM( NRHS, * ),
38 * $ ERR_BNDS_COMP( NRHS, * )
39 * ..
40 *
41 *
42 *> \par Purpose:
43 * =============
44 *>
45 *> \verbatim
46 *>
47 *> DSYRFSX improves the computed solution to a system of linear
48 *> equations when the coefficient matrix is symmetric indefinite, and
49 *> provides error bounds and backward error estimates for the
50 *> solution. In addition to normwise error bound, the code provides
51 *> maximum componentwise error bound if possible. See comments for
52 *> ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds.
53 *>
54 *> The original system of linear equations may have been equilibrated
55 *> before calling this routine, as described by arguments EQUED and S
56 *> below. In this case, the solution and error bounds returned are
57 *> for the original unequilibrated system.
58 *> \endverbatim
59 *
60 * Arguments:
61 * ==========
62 *
63 *> \verbatim
64 *> Some optional parameters are bundled in the PARAMS array. These
65 *> settings determine how refinement is performed, but often the
66 *> defaults are acceptable. If the defaults are acceptable, users
67 *> can pass NPARAMS = 0 which prevents the source code from accessing
68 *> the PARAMS argument.
69 *> \endverbatim
70 *>
71 *> \param[in] UPLO
72 *> \verbatim
73 *> UPLO is CHARACTER*1
74 *> = 'U': Upper triangle of A is stored;
75 *> = 'L': Lower triangle of A is stored.
76 *> \endverbatim
77 *>
78 *> \param[in] EQUED
79 *> \verbatim
80 *> EQUED is CHARACTER*1
81 *> Specifies the form of equilibration that was done to A
82 *> before calling this routine. This is needed to compute
83 *> the solution and error bounds correctly.
84 *> = 'N': No equilibration
85 *> = 'Y': Both row and column equilibration, i.e., A has been
86 *> replaced by diag(S) * A * diag(S).
87 *> The right hand side B has been changed accordingly.
88 *> \endverbatim
89 *>
90 *> \param[in] N
91 *> \verbatim
92 *> N is INTEGER
93 *> The order of the matrix A. N >= 0.
94 *> \endverbatim
95 *>
96 *> \param[in] NRHS
97 *> \verbatim
98 *> NRHS is INTEGER
99 *> The number of right hand sides, i.e., the number of columns
100 *> of the matrices B and X. NRHS >= 0.
101 *> \endverbatim
102 *>
103 *> \param[in] A
104 *> \verbatim
105 *> A is DOUBLE PRECISION array, dimension (LDA,N)
106 *> The symmetric matrix A. If UPLO = 'U', the leading N-by-N
107 *> upper triangular part of A contains the upper triangular
108 *> part of the matrix A, and the strictly lower triangular
109 *> part of A is not referenced. If UPLO = 'L', the leading
110 *> N-by-N lower triangular part of A contains the lower
111 *> triangular part of the matrix A, and the strictly upper
112 *> triangular part of A is not referenced.
113 *> \endverbatim
114 *>
115 *> \param[in] LDA
116 *> \verbatim
117 *> LDA is INTEGER
118 *> The leading dimension of the array A. LDA >= max(1,N).
119 *> \endverbatim
120 *>
121 *> \param[in] AF
122 *> \verbatim
123 *> AF is DOUBLE PRECISION array, dimension (LDAF,N)
124 *> The factored form of the matrix A. AF contains the block
125 *> diagonal matrix D and the multipliers used to obtain the
126 *> factor U or L from the factorization A = U*D*U**T or A =
127 *> L*D*L**T as computed by DSYTRF.
128 *> \endverbatim
129 *>
130 *> \param[in] LDAF
131 *> \verbatim
132 *> LDAF is INTEGER
133 *> The leading dimension of the array AF. LDAF >= max(1,N).
134 *> \endverbatim
135 *>
136 *> \param[in] IPIV
137 *> \verbatim
138 *> IPIV is INTEGER array, dimension (N)
139 *> Details of the interchanges and the block structure of D
140 *> as determined by DSYTRF.
141 *> \endverbatim
142 *>
143 *> \param[in,out] S
144 *> \verbatim
145 *> S is DOUBLE PRECISION array, dimension (N)
146 *> The scale factors for A. If EQUED = 'Y', A is multiplied on
147 *> the left and right by diag(S). S is an input argument if FACT =
148 *> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
149 *> = 'Y', each element of S must be positive. If S is output, each
150 *> element of S is a power of the radix. If S is input, each element
151 *> of S should be a power of the radix to ensure a reliable solution
152 *> and error estimates. Scaling by powers of the radix does not cause
153 *> rounding errors unless the result underflows or overflows.
154 *> Rounding errors during scaling lead to refining with a matrix that
155 *> is not equivalent to the input matrix, producing error estimates
156 *> that may not be reliable.
157 *> \endverbatim
158 *>
159 *> \param[in] B
160 *> \verbatim
161 *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
162 *> The right hand side matrix B.
163 *> \endverbatim
164 *>
165 *> \param[in] LDB
166 *> \verbatim
167 *> LDB is INTEGER
168 *> The leading dimension of the array B. LDB >= max(1,N).
169 *> \endverbatim
170 *>
171 *> \param[in,out] X
172 *> \verbatim
173 *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
174 *> On entry, the solution matrix X, as computed by DGETRS.
175 *> On exit, the improved solution matrix X.
176 *> \endverbatim
177 *>
178 *> \param[in] LDX
179 *> \verbatim
180 *> LDX is INTEGER
181 *> The leading dimension of the array X. LDX >= max(1,N).
182 *> \endverbatim
183 *>
184 *> \param[out] RCOND
185 *> \verbatim
186 *> RCOND is DOUBLE PRECISION
187 *> Reciprocal scaled condition number. This is an estimate of the
188 *> reciprocal Skeel condition number of the matrix A after
189 *> equilibration (if done). If this is less than the machine
190 *> precision (in particular, if it is zero), the matrix is singular
191 *> to working precision. Note that the error may still be small even
192 *> if this number is very small and the matrix appears ill-
193 *> conditioned.
194 *> \endverbatim
195 *>
196 *> \param[out] BERR
197 *> \verbatim
198 *> BERR is DOUBLE PRECISION array, dimension (NRHS)
199 *> Componentwise relative backward error. This is the
200 *> componentwise relative backward error of each solution vector X(j)
201 *> (i.e., the smallest relative change in any element of A or B that
202 *> makes X(j) an exact solution).
203 *> \endverbatim
204 *>
205 *> \param[in] N_ERR_BNDS
206 *> \verbatim
207 *> N_ERR_BNDS is INTEGER
208 *> Number of error bounds to return for each right hand side
209 *> and each type (normwise or componentwise). See ERR_BNDS_NORM and
210 *> ERR_BNDS_COMP below.
211 *> \endverbatim
212 *>
213 *> \param[out] ERR_BNDS_NORM
214 *> \verbatim
215 *> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
216 *> For each right-hand side, this array contains information about
217 *> various error bounds and condition numbers corresponding to the
218 *> normwise relative error, which is defined as follows:
219 *>
220 *> Normwise relative error in the ith solution vector:
221 *> max_j (abs(XTRUE(j,i) - X(j,i)))
222 *> ------------------------------
223 *> max_j abs(X(j,i))
224 *>
225 *> The array is indexed by the type of error information as described
226 *> below. There currently are up to three pieces of information
227 *> returned.
228 *>
229 *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
230 *> right-hand side.
231 *>
232 *> The second index in ERR_BNDS_NORM(:,err) contains the following
233 *> three fields:
234 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
235 *> reciprocal condition number is less than the threshold
236 *> sqrt(n) * dlamch('Epsilon').
237 *>
238 *> err = 2 "Guaranteed" error bound: The estimated forward error,
239 *> almost certainly within a factor of 10 of the true error
240 *> so long as the next entry is greater than the threshold
241 *> sqrt(n) * dlamch('Epsilon'). This error bound should only
242 *> be trusted if the previous boolean is true.
243 *>
244 *> err = 3 Reciprocal condition number: Estimated normwise
245 *> reciprocal condition number. Compared with the threshold
246 *> sqrt(n) * dlamch('Epsilon') to determine if the error
247 *> estimate is "guaranteed". These reciprocal condition
248 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
249 *> appropriately scaled matrix Z.
250 *> Let Z = S*A, where S scales each row by a power of the
251 *> radix so all absolute row sums of Z are approximately 1.
252 *>
253 *> See Lapack Working Note 165 for further details and extra
254 *> cautions.
255 *> \endverbatim
256 *>
257 *> \param[out] ERR_BNDS_COMP
258 *> \verbatim
259 *> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
260 *> For each right-hand side, this array contains information about
261 *> various error bounds and condition numbers corresponding to the
262 *> componentwise relative error, which is defined as follows:
263 *>
264 *> Componentwise relative error in the ith solution vector:
265 *> abs(XTRUE(j,i) - X(j,i))
266 *> max_j ----------------------
267 *> abs(X(j,i))
268 *>
269 *> The array is indexed by the right-hand side i (on which the
270 *> componentwise relative error depends), and the type of error
271 *> information as described below. There currently are up to three
272 *> pieces of information returned for each right-hand side. If
273 *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
274 *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
275 *> the first (:,N_ERR_BNDS) entries are returned.
276 *>
277 *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
278 *> right-hand side.
279 *>
280 *> The second index in ERR_BNDS_COMP(:,err) contains the following
281 *> three fields:
282 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
283 *> reciprocal condition number is less than the threshold
284 *> sqrt(n) * dlamch('Epsilon').
285 *>
286 *> err = 2 "Guaranteed" error bound: The estimated forward error,
287 *> almost certainly within a factor of 10 of the true error
288 *> so long as the next entry is greater than the threshold
289 *> sqrt(n) * dlamch('Epsilon'). This error bound should only
290 *> be trusted if the previous boolean is true.
291 *>
292 *> err = 3 Reciprocal condition number: Estimated componentwise
293 *> reciprocal condition number. Compared with the threshold
294 *> sqrt(n) * dlamch('Epsilon') to determine if the error
295 *> estimate is "guaranteed". These reciprocal condition
296 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
297 *> appropriately scaled matrix Z.
298 *> Let Z = S*(A*diag(x)), where x is the solution for the
299 *> current right-hand side and S scales each row of
300 *> A*diag(x) by a power of the radix so all absolute row
301 *> sums of Z are approximately 1.
302 *>
303 *> See Lapack Working Note 165 for further details and extra
304 *> cautions.
305 *> \endverbatim
306 *>
307 *> \param[in] NPARAMS
308 *> \verbatim
309 *> NPARAMS is INTEGER
310 *> Specifies the number of parameters set in PARAMS. If .LE. 0, the
311 *> PARAMS array is never referenced and default values are used.
312 *> \endverbatim
313 *>
314 *> \param[in,out] PARAMS
315 *> \verbatim
316 *> PARAMS is DOUBLE PRECISION array, dimension (NPARAMS)
317 *> Specifies algorithm parameters. If an entry is .LT. 0.0, then
318 *> that entry will be filled with default value used for that
319 *> parameter. Only positions up to NPARAMS are accessed; defaults
320 *> are used for higher-numbered parameters.
321 *>
322 *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
323 *> refinement or not.
324 *> Default: 1.0D+0
325 *> = 0.0 : No refinement is performed, and no error bounds are
326 *> computed.
327 *> = 1.0 : Use the double-precision refinement algorithm,
328 *> possibly with doubled-single computations if the
329 *> compilation environment does not support DOUBLE
330 *> PRECISION.
331 *> (other values are reserved for future use)
332 *>
333 *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
334 *> computations allowed for refinement.
335 *> Default: 10
336 *> Aggressive: Set to 100 to permit convergence using approximate
337 *> factorizations or factorizations other than LU. If
338 *> the factorization uses a technique other than
339 *> Gaussian elimination, the guarantees in
340 *> err_bnds_norm and err_bnds_comp may no longer be
341 *> trustworthy.
342 *>
343 *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
344 *> will attempt to find a solution with small componentwise
345 *> relative error in the double-precision algorithm. Positive
346 *> is true, 0.0 is false.
347 *> Default: 1.0 (attempt componentwise convergence)
348 *> \endverbatim
349 *>
350 *> \param[out] WORK
351 *> \verbatim
352 *> WORK is DOUBLE PRECISION array, dimension (4*N)
353 *> \endverbatim
354 *>
355 *> \param[out] IWORK
356 *> \verbatim
357 *> IWORK is INTEGER array, dimension (N)
358 *> \endverbatim
359 *>
360 *> \param[out] INFO
361 *> \verbatim
362 *> INFO is INTEGER
363 *> = 0: Successful exit. The solution to every right-hand side is
364 *> guaranteed.
365 *> < 0: If INFO = -i, the i-th argument had an illegal value
366 *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
367 *> has been completed, but the factor U is exactly singular, so
368 *> the solution and error bounds could not be computed. RCOND = 0
369 *> is returned.
370 *> = N+J: The solution corresponding to the Jth right-hand side is
371 *> not guaranteed. The solutions corresponding to other right-
372 *> hand sides K with K > J may not be guaranteed as well, but
373 *> only the first such right-hand side is reported. If a small
374 *> componentwise error is not requested (PARAMS(3) = 0.0) then
375 *> the Jth right-hand side is the first with a normwise error
376 *> bound that is not guaranteed (the smallest J such
377 *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
378 *> the Jth right-hand side is the first with either a normwise or
379 *> componentwise error bound that is not guaranteed (the smallest
380 *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
381 *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
382 *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
383 *> about all of the right-hand sides check ERR_BNDS_NORM or
384 *> ERR_BNDS_COMP.
385 *> \endverbatim
386 *
387 * Authors:
388 * ========
389 *
390 *> \author Univ. of Tennessee
391 *> \author Univ. of California Berkeley
392 *> \author Univ. of Colorado Denver
393 *> \author NAG Ltd.
394 *
395 *> \date April 2012
396 *
397 *> \ingroup doubleSYcomputational
398 *
399 * =====================================================================
400  SUBROUTINE dsyrfsx( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV,
401  $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
402  $ ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
403  $ WORK, IWORK, INFO )
404 *
405 * -- LAPACK computational routine (version 3.7.0) --
406 * -- LAPACK is a software package provided by Univ. of Tennessee, --
407 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
408 * April 2012
409 *
410 * .. Scalar Arguments ..
411  CHARACTER UPLO, EQUED
412  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
413  $ n_err_bnds
414  DOUBLE PRECISION RCOND
415 * ..
416 * .. Array Arguments ..
417  INTEGER IPIV( * ), IWORK( * )
418  DOUBLE PRECISION A( lda, * ), AF( ldaf, * ), B( ldb, * ),
419  $ x( ldx, * ), work( * )
420  DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ),
421  $ err_bnds_norm( nrhs, * ),
422  $ err_bnds_comp( nrhs, * )
423 * ..
424 *
425 * ==================================================================
426 *
427 * .. Parameters ..
428  DOUBLE PRECISION ZERO, ONE
429  parameter( zero = 0.0d+0, one = 1.0d+0 )
430  DOUBLE PRECISION ITREF_DEFAULT, ITHRESH_DEFAULT
431  DOUBLE PRECISION COMPONENTWISE_DEFAULT, RTHRESH_DEFAULT
432  DOUBLE PRECISION DZTHRESH_DEFAULT
433  parameter( itref_default = 1.0d+0 )
434  parameter( ithresh_default = 10.0d+0 )
435  parameter( componentwise_default = 1.0d+0 )
436  parameter( rthresh_default = 0.5d+0 )
437  parameter( dzthresh_default = 0.25d+0 )
438  INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
439  $ la_linrx_cwise_i
440  parameter( la_linrx_itref_i = 1,
441  $ la_linrx_ithresh_i = 2 )
442  parameter( la_linrx_cwise_i = 3 )
443  INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
444  $ la_linrx_rcond_i
445  parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
446  parameter( la_linrx_rcond_i = 3 )
447 * ..
448 * .. Local Scalars ..
449  CHARACTER(1) NORM
450  LOGICAL RCEQU
451  INTEGER J, PREC_TYPE, REF_TYPE, N_NORMS
452  DOUBLE PRECISION ANORM, RCOND_TMP
453  DOUBLE PRECISION ILLRCOND_THRESH, ERR_LBND, CWISE_WRONG
454  LOGICAL IGNORE_CWISE
455  INTEGER ITHRESH
456  DOUBLE PRECISION RTHRESH, UNSTABLE_THRESH
457 * ..
458 * .. External Subroutines ..
460 * ..
461 * .. Intrinsic Functions ..
462  INTRINSIC max, sqrt
463 * ..
464 * .. External Functions ..
465  EXTERNAL lsame, ilaprec
466  EXTERNAL dlamch, dlansy, dla_syrcond
467  DOUBLE PRECISION DLAMCH, DLANSY, DLA_SYRCOND
468  LOGICAL LSAME
469  INTEGER ILAPREC
470 * ..
471 * .. Executable Statements ..
472 *
473 * Check the input parameters.
474 *
475  info = 0
476  ref_type = int( itref_default )
477  IF ( nparams .GE. la_linrx_itref_i ) THEN
478  IF ( params( la_linrx_itref_i ) .LT. 0.0d+0 ) THEN
479  params( la_linrx_itref_i ) = itref_default
480  ELSE
481  ref_type = params( la_linrx_itref_i )
482  END IF
483  END IF
484 *
485 * Set default parameters.
486 *
487  illrcond_thresh = dble( n )*dlamch( 'Epsilon' )
488  ithresh = int( ithresh_default )
489  rthresh = rthresh_default
490  unstable_thresh = dzthresh_default
491  ignore_cwise = componentwise_default .EQ. 0.0d+0
492 *
493  IF ( nparams.GE.la_linrx_ithresh_i ) THEN
494  IF ( params( la_linrx_ithresh_i ).LT.0.0d+0 ) THEN
495  params( la_linrx_ithresh_i ) = ithresh
496  ELSE
497  ithresh = int( params( la_linrx_ithresh_i ) )
498  END IF
499  END IF
500  IF ( nparams.GE.la_linrx_cwise_i ) THEN
501  IF ( params( la_linrx_cwise_i ).LT.0.0d+0 ) THEN
502  IF ( ignore_cwise ) THEN
503  params( la_linrx_cwise_i ) = 0.0d+0
504  ELSE
505  params( la_linrx_cwise_i ) = 1.0d+0
506  END IF
507  ELSE
508  ignore_cwise = params( la_linrx_cwise_i ) .EQ. 0.0d+0
509  END IF
510  END IF
511  IF ( ref_type .EQ. 0 .OR. n_err_bnds .EQ. 0 ) THEN
512  n_norms = 0
513  ELSE IF ( ignore_cwise ) THEN
514  n_norms = 1
515  ELSE
516  n_norms = 2
517  END IF
518 *
519  rcequ = lsame( equed, 'Y' )
520 *
521 * Test input parameters.
522 *
523  IF ( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
524  info = -1
525  ELSE IF( .NOT.rcequ .AND. .NOT.lsame( equed, 'N' ) ) THEN
526  info = -2
527  ELSE IF( n.LT.0 ) THEN
528  info = -3
529  ELSE IF( nrhs.LT.0 ) THEN
530  info = -4
531  ELSE IF( lda.LT.max( 1, n ) ) THEN
532  info = -6
533  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
534  info = -8
535  ELSE IF( ldb.LT.max( 1, n ) ) THEN
536  info = -12
537  ELSE IF( ldx.LT.max( 1, n ) ) THEN
538  info = -14
539  END IF
540  IF( info.NE.0 ) THEN
541  CALL xerbla( 'DSYRFSX', -info )
542  RETURN
543  END IF
544 *
545 * Quick return if possible.
546 *
547  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
548  rcond = 1.0d+0
549  DO j = 1, nrhs
550  berr( j ) = 0.0d+0
551  IF ( n_err_bnds .GE. 1 ) THEN
552  err_bnds_norm( j, la_linrx_trust_i ) = 1.0d+0
553  err_bnds_comp( j, la_linrx_trust_i ) = 1.0d+0
554  END IF
555  IF ( n_err_bnds .GE. 2 ) THEN
556  err_bnds_norm( j, la_linrx_err_i ) = 0.0d+0
557  err_bnds_comp( j, la_linrx_err_i ) = 0.0d+0
558  END IF
559  IF ( n_err_bnds .GE. 3 ) THEN
560  err_bnds_norm( j, la_linrx_rcond_i ) = 1.0d+0
561  err_bnds_comp( j, la_linrx_rcond_i ) = 1.0d+0
562  END IF
563  END DO
564  RETURN
565  END IF
566 *
567 * Default to failure.
568 *
569  rcond = 0.0d+0
570  DO j = 1, nrhs
571  berr( j ) = 1.0d+0
572  IF ( n_err_bnds .GE. 1 ) THEN
573  err_bnds_norm( j, la_linrx_trust_i ) = 1.0d+0
574  err_bnds_comp( j, la_linrx_trust_i ) = 1.0d+0
575  END IF
576  IF ( n_err_bnds .GE. 2 ) THEN
577  err_bnds_norm( j, la_linrx_err_i ) = 1.0d+0
578  err_bnds_comp( j, la_linrx_err_i ) = 1.0d+0
579  END IF
580  IF ( n_err_bnds .GE. 3 ) THEN
581  err_bnds_norm( j, la_linrx_rcond_i ) = 0.0d+0
582  err_bnds_comp( j, la_linrx_rcond_i ) = 0.0d+0
583  END IF
584  END DO
585 *
586 * Compute the norm of A and the reciprocal of the condition
587 * number of A.
588 *
589  norm = 'I'
590  anorm = dlansy( norm, uplo, n, a, lda, work )
591  CALL dsycon( uplo, n, af, ldaf, ipiv, anorm, rcond, work,
592  $ iwork, info )
593 *
594 * Perform refinement on each right-hand side
595 *
596  IF ( ref_type .NE. 0 ) THEN
597 
598  prec_type = ilaprec( 'E' )
599 
600  CALL dla_syrfsx_extended( prec_type, uplo, n,
601  $ nrhs, a, lda, af, ldaf, ipiv, rcequ, s, b,
602  $ ldb, x, ldx, berr, n_norms, err_bnds_norm, err_bnds_comp,
603  $ work( n+1 ), work( 1 ), work( 2*n+1 ), work( 1 ), rcond,
604  $ ithresh, rthresh, unstable_thresh, ignore_cwise,
605  $ info )
606  END IF
607 
608  err_lbnd = max( 10.0d+0, sqrt( dble( n ) ) )*dlamch( 'Epsilon' )
609  IF (n_err_bnds .GE. 1 .AND. n_norms .GE. 1) THEN
610 *
611 * Compute scaled normwise condition number cond(A*C).
612 *
613  IF ( rcequ ) THEN
614  rcond_tmp = dla_syrcond( uplo, n, a, lda, af, ldaf, ipiv,
615  $ -1, s, info, work, iwork )
616  ELSE
617  rcond_tmp = dla_syrcond( uplo, n, a, lda, af, ldaf, ipiv,
618  $ 0, s, info, work, iwork )
619  END IF
620  DO j = 1, nrhs
621 *
622 * Cap the error at 1.0.
623 *
624  IF (n_err_bnds .GE. la_linrx_err_i
625  $ .AND. err_bnds_norm( j, la_linrx_err_i ) .GT. 1.0d+0)
626  $ err_bnds_norm( j, la_linrx_err_i ) = 1.0d+0
627 *
628 * Threshold the error (see LAWN).
629 *
630  IF ( rcond_tmp .LT. illrcond_thresh ) THEN
631  err_bnds_norm( j, la_linrx_err_i ) = 1.0d+0
632  err_bnds_norm( j, la_linrx_trust_i ) = 0.0d+0
633  IF ( info .LE. n ) info = n + j
634  ELSE IF (err_bnds_norm( j, la_linrx_err_i ) .LT. err_lbnd)
635  $ THEN
636  err_bnds_norm( j, la_linrx_err_i ) = err_lbnd
637  err_bnds_norm( j, la_linrx_trust_i ) = 1.0d+0
638  END IF
639 *
640 * Save the condition number.
641 *
642  IF (n_err_bnds .GE. la_linrx_rcond_i) THEN
643  err_bnds_norm( j, la_linrx_rcond_i ) = rcond_tmp
644  END IF
645  END DO
646  END IF
647 
648  IF ( n_err_bnds .GE. 1 .AND. n_norms .GE. 2 ) THEN
649 *
650 * Compute componentwise condition number cond(A*diag(Y(:,J))) for
651 * each right-hand side using the current solution as an estimate of
652 * the true solution. If the componentwise error estimate is too
653 * large, then the solution is a lousy estimate of truth and the
654 * estimated RCOND may be too optimistic. To avoid misleading users,
655 * the inverse condition number is set to 0.0 when the estimated
656 * cwise error is at least CWISE_WRONG.
657 *
658  cwise_wrong = sqrt( dlamch( 'Epsilon' ) )
659  DO j = 1, nrhs
660  IF ( err_bnds_comp( j, la_linrx_err_i ) .LT. cwise_wrong )
661  $ THEN
662  rcond_tmp = dla_syrcond( uplo, n, a, lda, af, ldaf, ipiv,
663  $ 1, x(1,j), info, work, iwork )
664  ELSE
665  rcond_tmp = 0.0d+0
666  END IF
667 *
668 * Cap the error at 1.0.
669 *
670  IF ( n_err_bnds .GE. la_linrx_err_i
671  $ .AND. err_bnds_comp( j, la_linrx_err_i ) .GT. 1.0d+0 )
672  $ err_bnds_comp( j, la_linrx_err_i ) = 1.0d+0
673 *
674 * Threshold the error (see LAWN).
675 *
676  IF ( rcond_tmp .LT. illrcond_thresh ) THEN
677  err_bnds_comp( j, la_linrx_err_i ) = 1.0d+0
678  err_bnds_comp( j, la_linrx_trust_i ) = 0.0d+0
679  IF ( .NOT. ignore_cwise
680  $ .AND. info.LT.n + j ) info = n + j
681  ELSE IF ( err_bnds_comp( j, la_linrx_err_i )
682  $ .LT. err_lbnd ) THEN
683  err_bnds_comp( j, la_linrx_err_i ) = err_lbnd
684  err_bnds_comp( j, la_linrx_trust_i ) = 1.0d+0
685  END IF
686 *
687 * Save the condition number.
688 *
689  IF ( n_err_bnds .GE. la_linrx_rcond_i ) THEN
690  err_bnds_comp( j, la_linrx_rcond_i ) = rcond_tmp
691  END IF
692 
693  END DO
694  END IF
695 *
696  RETURN
697 *
698 * End of DSYRFSX
699 *
700  END
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
double precision function dlansy(NORM, UPLO, N, A, LDA, WORK)
DLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix.
Definition: dlansy.f:124
subroutine dla_syrfsx_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)
DLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric inde...
double precision function dla_syrcond(UPLO, N, A, LDA, AF, LDAF, IPIV, CMODE, C, INFO, WORK, IWORK)
DLA_SYRCOND estimates the Skeel condition number for a symmetric indefinite matrix.
Definition: dla_syrcond.f:150
subroutine dsycon(UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
DSYCON
Definition: dsycon.f:132
integer function ilaprec(PREC)
ILAPREC
Definition: ilaprec.f:60
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dsyrfsx(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, IWORK, INFO)
DSYRFSX
Definition: dsyrfsx.f:404