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ssyrfsx.f
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1 *> \brief \b SSYRFSX
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download SSYRFSX + dependencies
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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/ssyrfsx.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SSYRFSX( 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 * REAL RCOND
31 * ..
32 * .. Array Arguments ..
33 * INTEGER IPIV( * ), IWORK( * )
34 * REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
35 * $ X( LDX, * ), WORK( * )
36 * REAL 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 *> SSYRFSX 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 REAL 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 REAL 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 SSYTRF.
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 SSYTRF.
141 *> \endverbatim
142 *>
143 *> \param[in,out] S
144 *> \verbatim
145 *> S is REAL 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 REAL 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 REAL array, dimension (LDX,NRHS)
174 *> On entry, the solution matrix X, as computed by SGETRS.
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 REAL
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 REAL 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 REAL 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) * slamch('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) * slamch('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) * slamch('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 REAL 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) * slamch('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) * slamch('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) * slamch('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 REAL 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: 1.0
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 REAL 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 realSYcomputational
398 *
399 * =====================================================================
400  SUBROUTINE ssyrfsx( 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 3.4.1) --
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  REAL rcond
415 * ..
416 * .. Array Arguments ..
417  INTEGER ipiv( * ), iwork( * )
418  REAL a( lda, * ), af( ldaf, * ), b( ldb, * ),
419  $ x( ldx, * ), work( * )
420  REAL s( * ), params( * ), berr( * ),
421  $ err_bnds_norm( nrhs, * ),
422  $ err_bnds_comp( nrhs, * )
423 * ..
424 *
425 * ==================================================================
426 *
427 * .. Parameters ..
428  REAL zero, one
429  parameter( zero = 0.0e+0, one = 1.0e+0 )
430  REAL itref_default, ithresh_default,
431  $ componentwise_default
432  REAL rthresh_default, dzthresh_default
433  parameter( itref_default = 1.0 )
434  parameter( ithresh_default = 10.0 )
435  parameter( componentwise_default = 1.0 )
436  parameter( rthresh_default = 0.5 )
437  parameter( dzthresh_default = 0.25 )
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  REAL anorm, rcond_tmp
453  REAL illrcond_thresh, err_lbnd, cwise_wrong
454  LOGICAL ignore_cwise
455  INTEGER ithresh
456  REAL rthresh, unstable_thresh
457 * ..
458 * .. External Subroutines ..
460 * ..
461 * .. Intrinsic Functions ..
462  INTRINSIC max, sqrt
463 * ..
464 * .. External Functions ..
465  EXTERNAL lsame, blas_fpinfo_x, ilatrans, ilaprec
466  EXTERNAL slamch, slansy, sla_syrcond
467  REAL slamch, slansy, sla_syrcond
468  LOGICAL lsame
469  INTEGER blas_fpinfo_x
470  INTEGER ilatrans, ilaprec
471 * ..
472 * .. Executable Statements ..
473 *
474 * Check the input parameters.
475 *
476  info = 0
477  ref_type = int( itref_default )
478  IF ( nparams .GE. la_linrx_itref_i ) THEN
479  IF ( params( la_linrx_itref_i ) .LT. 0.0 ) THEN
480  params( la_linrx_itref_i ) = itref_default
481  ELSE
482  ref_type = params( la_linrx_itref_i )
483  END IF
484  END IF
485 *
486 * Set default parameters.
487 *
488  illrcond_thresh = REAL( n )*slamch( 'Epsilon' )
489  ithresh = int( ithresh_default )
490  rthresh = rthresh_default
491  unstable_thresh = dzthresh_default
492  ignore_cwise = componentwise_default .EQ. 0.0
493 *
494  IF ( nparams.GE.la_linrx_ithresh_i ) THEN
495  IF ( params( la_linrx_ithresh_i ).LT.0.0 ) THEN
496  params( la_linrx_ithresh_i ) = ithresh
497  ELSE
498  ithresh = int( params( la_linrx_ithresh_i ) )
499  END IF
500  END IF
501  IF ( nparams.GE.la_linrx_cwise_i ) THEN
502  IF ( params( la_linrx_cwise_i ).LT.0.0 ) THEN
503  IF ( ignore_cwise ) THEN
504  params( la_linrx_cwise_i ) = 0.0
505  ELSE
506  params( la_linrx_cwise_i ) = 1.0
507  END IF
508  ELSE
509  ignore_cwise = params( la_linrx_cwise_i ) .EQ. 0.0
510  END IF
511  END IF
512  IF ( ref_type .EQ. 0 .OR. n_err_bnds .EQ. 0 ) THEN
513  n_norms = 0
514  ELSE IF ( ignore_cwise ) THEN
515  n_norms = 1
516  ELSE
517  n_norms = 2
518  END IF
519 *
520  rcequ = lsame( equed, 'Y' )
521 *
522 * Test input parameters.
523 *
524  IF ( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
525  info = -1
526  ELSE IF( .NOT.rcequ .AND. .NOT.lsame( equed, 'N' ) ) THEN
527  info = -2
528  ELSE IF( n.LT.0 ) THEN
529  info = -3
530  ELSE IF( nrhs.LT.0 ) THEN
531  info = -4
532  ELSE IF( lda.LT.max( 1, n ) ) THEN
533  info = -6
534  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
535  info = -8
536  ELSE IF( ldb.LT.max( 1, n ) ) THEN
537  info = -12
538  ELSE IF( ldx.LT.max( 1, n ) ) THEN
539  info = -14
540  END IF
541  IF( info.NE.0 ) THEN
542  CALL xerbla( 'SSYRFSX', -info )
543  RETURN
544  END IF
545 *
546 * Quick return if possible.
547 *
548  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
549  rcond = 1.0
550  DO j = 1, nrhs
551  berr( j ) = 0.0
552  IF ( n_err_bnds .GE. 1 ) THEN
553  err_bnds_norm( j, la_linrx_trust_i ) = 1.0
554  err_bnds_comp( j, la_linrx_trust_i ) = 1.0
555  END IF
556  IF ( n_err_bnds .GE. 2 ) THEN
557  err_bnds_norm( j, la_linrx_err_i ) = 0.0
558  err_bnds_comp( j, la_linrx_err_i ) = 0.0
559  END IF
560  IF ( n_err_bnds .GE. 3 ) THEN
561  err_bnds_norm( j, la_linrx_rcond_i ) = 1.0
562  err_bnds_comp( j, la_linrx_rcond_i ) = 1.0
563  END IF
564  END DO
565  RETURN
566  END IF
567 *
568 * Default to failure.
569 *
570  rcond = 0.0
571  DO j = 1, nrhs
572  berr( j ) = 1.0
573  IF ( n_err_bnds .GE. 1 ) THEN
574  err_bnds_norm( j, la_linrx_trust_i ) = 1.0
575  err_bnds_comp( j, la_linrx_trust_i ) = 1.0
576  END IF
577  IF ( n_err_bnds .GE. 2 ) THEN
578  err_bnds_norm( j, la_linrx_err_i ) = 1.0
579  err_bnds_comp( j, la_linrx_err_i ) = 1.0
580  END IF
581  IF ( n_err_bnds .GE. 3 ) THEN
582  err_bnds_norm( j, la_linrx_rcond_i ) = 0.0
583  err_bnds_comp( j, la_linrx_rcond_i ) = 0.0
584  END IF
585  END DO
586 *
587 * Compute the norm of A and the reciprocal of the condition
588 * number of A.
589 *
590  norm = 'I'
591  anorm = slansy( norm, uplo, n, a, lda, work )
592  CALL ssycon( uplo, n, af, ldaf, ipiv, anorm, rcond, work,
593  $ iwork, info )
594 *
595 * Perform refinement on each right-hand side
596 *
597  IF ( ref_type .NE. 0 ) THEN
598 
599  prec_type = ilaprec( 'D' )
600 
601  CALL sla_syrfsx_extended( prec_type, uplo, n,
602  $ nrhs, a, lda, af, ldaf, ipiv, rcequ, s, b,
603  $ ldb, x, ldx, berr, n_norms, err_bnds_norm, err_bnds_comp,
604  $ work( n+1 ), work( 1 ), work( 2*n+1 ), work( 1 ), rcond,
605  $ ithresh, rthresh, unstable_thresh, ignore_cwise,
606  $ info )
607  END IF
608 
609  err_lbnd = max( 10.0, sqrt( REAL( N ) ) )*slamch( 'Epsilon' )
610  IF (n_err_bnds .GE. 1 .AND. n_norms .GE. 1) THEN
611 *
612 * Compute scaled normwise condition number cond(A*C).
613 *
614  IF ( rcequ ) THEN
615  rcond_tmp = sla_syrcond( uplo, n, a, lda, af, ldaf, ipiv,
616  $ -1, s, info, work, iwork )
617  ELSE
618  rcond_tmp = sla_syrcond( uplo, n, a, lda, af, ldaf, ipiv,
619  $ 0, s, info, work, iwork )
620  END IF
621  DO j = 1, nrhs
622 *
623 * Cap the error at 1.0.
624 *
625  IF (n_err_bnds .GE. la_linrx_err_i
626  $ .AND. err_bnds_norm( j, la_linrx_err_i ) .GT. 1.0)
627  $ err_bnds_norm( j, la_linrx_err_i ) = 1.0
628 *
629 * Threshold the error (see LAWN).
630 *
631  IF ( rcond_tmp .LT. illrcond_thresh ) THEN
632  err_bnds_norm( j, la_linrx_err_i ) = 1.0
633  err_bnds_norm( j, la_linrx_trust_i ) = 0.0
634  IF ( info .LE. n ) info = n + j
635  ELSE IF (err_bnds_norm( j, la_linrx_err_i ) .LT. err_lbnd)
636  $ THEN
637  err_bnds_norm( j, la_linrx_err_i ) = err_lbnd
638  err_bnds_norm( j, la_linrx_trust_i ) = 1.0
639  END IF
640 *
641 * Save the condition number.
642 *
643  IF (n_err_bnds .GE. la_linrx_rcond_i) THEN
644  err_bnds_norm( j, la_linrx_rcond_i ) = rcond_tmp
645  END IF
646  END DO
647  END IF
648 
649  IF ( n_err_bnds .GE. 1 .AND. n_norms .GE. 2 ) THEN
650 *
651 * Compute componentwise condition number cond(A*diag(Y(:,J))) for
652 * each right-hand side using the current solution as an estimate of
653 * the true solution. If the componentwise error estimate is too
654 * large, then the solution is a lousy estimate of truth and the
655 * estimated RCOND may be too optimistic. To avoid misleading users,
656 * the inverse condition number is set to 0.0 when the estimated
657 * cwise error is at least CWISE_WRONG.
658 *
659  cwise_wrong = sqrt( slamch( 'Epsilon' ) )
660  DO j = 1, nrhs
661  IF ( err_bnds_comp( j, la_linrx_err_i ) .LT. cwise_wrong )
662  $ THEN
663  rcond_tmp = sla_syrcond( uplo, n, a, lda, af, ldaf, ipiv,
664  $ 1, x(1,j), info, work, iwork )
665  ELSE
666  rcond_tmp = 0.0
667  END IF
668 *
669 * Cap the error at 1.0.
670 *
671  IF ( n_err_bnds .GE. la_linrx_err_i
672  $ .AND. err_bnds_comp( j, la_linrx_err_i ) .GT. 1.0 )
673  $ err_bnds_comp( j, la_linrx_err_i ) = 1.0
674 *
675 * Threshold the error (see LAWN).
676 *
677  IF ( rcond_tmp .LT. illrcond_thresh ) THEN
678  err_bnds_comp( j, la_linrx_err_i ) = 1.0
679  err_bnds_comp( j, la_linrx_trust_i ) = 0.0
680  IF ( .NOT. ignore_cwise
681  $ .AND. info.LT.n + j ) info = n + j
682  ELSE IF ( err_bnds_comp( j, la_linrx_err_i )
683  $ .LT. err_lbnd ) THEN
684  err_bnds_comp( j, la_linrx_err_i ) = err_lbnd
685  err_bnds_comp( j, la_linrx_trust_i ) = 1.0
686  END IF
687 *
688 * Save the condition number.
689 *
690  IF ( n_err_bnds .GE. la_linrx_rcond_i ) THEN
691  err_bnds_comp( j, la_linrx_rcond_i ) = rcond_tmp
692  END IF
693 
694  END DO
695  END IF
696 *
697  RETURN
698 *
699 * End of SSYRFSX
700 *
701  END