1: *> \brief \b ZHERFSX
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZHERFSX + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zherfsx.f">
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 ..
455: EXTERNAL XERBLA, ZHECON, ZLA_HERFSX_EXTENDED
456: * ..
457: * .. Intrinsic Functions ..
458: INTRINSIC MAX, SQRT, TRANSFER
459: * ..
460: * .. External Functions ..
461: EXTERNAL LSAME, ILAPREC
462: EXTERNAL DLAMCH, ZLANHE, ZLA_HERCOND_X, ZLA_HERCOND_C
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
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