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cla_gerfsx_extended.f
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1 *> \brief \b CLA_GERFSX_EXTENDED
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download CLA_GERFSX_EXTENDED + 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/cla_gerfsx_extended.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cla_gerfsx_extended.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CLA_GERFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, NRHS, A,
22 * LDA, AF, LDAF, IPIV, COLEQU, C, B,
23 * LDB, Y, LDY, BERR_OUT, N_NORMS,
24 * ERRS_N, ERRS_C, RES, AYB, DY,
25 * Y_TAIL, RCOND, ITHRESH, RTHRESH,
26 * DZ_UB, IGNORE_CWISE, INFO )
27 *
28 * .. Scalar Arguments ..
29 * INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
30 * $ TRANS_TYPE, N_NORMS
31 * LOGICAL COLEQU, IGNORE_CWISE
32 * INTEGER ITHRESH
33 * REAL RTHRESH, DZ_UB
34 * ..
35 * .. Array Arguments
36 * INTEGER IPIV( * )
37 * COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
38 * $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * )
39 * REAL C( * ), AYB( * ), RCOND, BERR_OUT( * ),
40 * $ ERRS_N( NRHS, * ), ERRS_C( NRHS, * )
41 * ..
42 *
43 *
44 *> \par Purpose:
45 * =============
46 *>
47 *> \verbatim
48 *>
49 *>
50 *> CLA_GERFSX_EXTENDED improves the computed solution to a system of
51 *> linear equations by performing extra-precise iterative refinement
52 *> and provides error bounds and backward error estimates for the solution.
53 *> This subroutine is called by CGERFSX to perform iterative refinement.
54 *> In addition to normwise error bound, the code provides maximum
55 *> componentwise error bound if possible. See comments for ERRS_N
56 *> and ERRS_C for details of the error bounds. Note that this
57 *> subroutine is only resonsible for setting the second fields of
58 *> ERRS_N and ERRS_C.
59 *> \endverbatim
60 *
61 * Arguments:
62 * ==========
63 *
64 *> \param[in] PREC_TYPE
65 *> \verbatim
66 *> PREC_TYPE is INTEGER
67 *> Specifies the intermediate precision to be used in refinement.
68 *> The value is defined by ILAPREC(P) where P is a CHARACTER and
69 *> P = 'S': Single
70 *> = 'D': Double
71 *> = 'I': Indigenous
72 *> = 'X', 'E': Extra
73 *> \endverbatim
74 *>
75 *> \param[in] TRANS_TYPE
76 *> \verbatim
77 *> TRANS_TYPE is INTEGER
78 *> Specifies the transposition operation on A.
79 *> The value is defined by ILATRANS(T) where T is a CHARACTER and
80 *> T = 'N': No transpose
81 *> = 'T': Transpose
82 *> = 'C': Conjugate transpose
83 *> \endverbatim
84 *>
85 *> \param[in] N
86 *> \verbatim
87 *> N is INTEGER
88 *> The number of linear equations, i.e., the order of the
89 *> matrix A. N >= 0.
90 *> \endverbatim
91 *>
92 *> \param[in] NRHS
93 *> \verbatim
94 *> NRHS is INTEGER
95 *> The number of right-hand-sides, i.e., the number of columns of the
96 *> matrix B.
97 *> \endverbatim
98 *>
99 *> \param[in] A
100 *> \verbatim
101 *> A is COMPLEX array, dimension (LDA,N)
102 *> On entry, the N-by-N matrix A.
103 *> \endverbatim
104 *>
105 *> \param[in] LDA
106 *> \verbatim
107 *> LDA is INTEGER
108 *> The leading dimension of the array A. LDA >= max(1,N).
109 *> \endverbatim
110 *>
111 *> \param[in] AF
112 *> \verbatim
113 *> AF is COMPLEX array, dimension (LDAF,N)
114 *> The factors L and U from the factorization
115 *> A = P*L*U as computed by CGETRF.
116 *> \endverbatim
117 *>
118 *> \param[in] LDAF
119 *> \verbatim
120 *> LDAF is INTEGER
121 *> The leading dimension of the array AF. LDAF >= max(1,N).
122 *> \endverbatim
123 *>
124 *> \param[in] IPIV
125 *> \verbatim
126 *> IPIV is INTEGER array, dimension (N)
127 *> The pivot indices from the factorization A = P*L*U
128 *> as computed by CGETRF; row i of the matrix was interchanged
129 *> with row IPIV(i).
130 *> \endverbatim
131 *>
132 *> \param[in] COLEQU
133 *> \verbatim
134 *> COLEQU is LOGICAL
135 *> If .TRUE. then column equilibration was done to A before calling
136 *> this routine. This is needed to compute the solution and error
137 *> bounds correctly.
138 *> \endverbatim
139 *>
140 *> \param[in] C
141 *> \verbatim
142 *> C is REAL array, dimension (N)
143 *> The column scale factors for A. If COLEQU = .FALSE., C
144 *> is not accessed. If C is input, each element of C should be a power
145 *> of the radix to ensure a reliable solution and error estimates.
146 *> Scaling by powers of the radix does not cause rounding errors unless
147 *> the result underflows or overflows. Rounding errors during scaling
148 *> lead to refining with a matrix that is not equivalent to the
149 *> input matrix, producing error estimates that may not be
150 *> reliable.
151 *> \endverbatim
152 *>
153 *> \param[in] B
154 *> \verbatim
155 *> B is COMPLEX array, dimension (LDB,NRHS)
156 *> The right-hand-side matrix B.
157 *> \endverbatim
158 *>
159 *> \param[in] LDB
160 *> \verbatim
161 *> LDB is INTEGER
162 *> The leading dimension of the array B. LDB >= max(1,N).
163 *> \endverbatim
164 *>
165 *> \param[in,out] Y
166 *> \verbatim
167 *> Y is COMPLEX array, dimension (LDY,NRHS)
168 *> On entry, the solution matrix X, as computed by CGETRS.
169 *> On exit, the improved solution matrix Y.
170 *> \endverbatim
171 *>
172 *> \param[in] LDY
173 *> \verbatim
174 *> LDY is INTEGER
175 *> The leading dimension of the array Y. LDY >= max(1,N).
176 *> \endverbatim
177 *>
178 *> \param[out] BERR_OUT
179 *> \verbatim
180 *> BERR_OUT is REAL array, dimension (NRHS)
181 *> On exit, BERR_OUT(j) contains the componentwise relative backward
182 *> error for right-hand-side j from the formula
183 *> max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
184 *> where abs(Z) is the componentwise absolute value of the matrix
185 *> or vector Z. This is computed by CLA_LIN_BERR.
186 *> \endverbatim
187 *>
188 *> \param[in] N_NORMS
189 *> \verbatim
190 *> N_NORMS is INTEGER
191 *> Determines which error bounds to return (see ERRS_N
192 *> and ERRS_C).
193 *> If N_NORMS >= 1 return normwise error bounds.
194 *> If N_NORMS >= 2 return componentwise error bounds.
195 *> \endverbatim
196 *>
197 *> \param[in,out] ERRS_N
198 *> \verbatim
199 *> ERRS_N is REAL array, dimension (NRHS, N_ERR_BNDS)
200 *> For each right-hand side, this array contains information about
201 *> various error bounds and condition numbers corresponding to the
202 *> normwise relative error, which is defined as follows:
203 *>
204 *> Normwise relative error in the ith solution vector:
205 *> max_j (abs(XTRUE(j,i) - X(j,i)))
206 *> ------------------------------
207 *> max_j abs(X(j,i))
208 *>
209 *> The array is indexed by the type of error information as described
210 *> below. There currently are up to three pieces of information
211 *> returned.
212 *>
213 *> The first index in ERRS_N(i,:) corresponds to the ith
214 *> right-hand side.
215 *>
216 *> The second index in ERRS_N(:,err) contains the following
217 *> three fields:
218 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
219 *> reciprocal condition number is less than the threshold
220 *> sqrt(n) * slamch('Epsilon').
221 *>
222 *> err = 2 "Guaranteed" error bound: The estimated forward error,
223 *> almost certainly within a factor of 10 of the true error
224 *> so long as the next entry is greater than the threshold
225 *> sqrt(n) * slamch('Epsilon'). This error bound should only
226 *> be trusted if the previous boolean is true.
227 *>
228 *> err = 3 Reciprocal condition number: Estimated normwise
229 *> reciprocal condition number. Compared with the threshold
230 *> sqrt(n) * slamch('Epsilon') to determine if the error
231 *> estimate is "guaranteed". These reciprocal condition
232 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
233 *> appropriately scaled matrix Z.
234 *> Let Z = S*A, where S scales each row by a power of the
235 *> radix so all absolute row sums of Z are approximately 1.
236 *>
237 *> This subroutine is only responsible for setting the second field
238 *> above.
239 *> See Lapack Working Note 165 for further details and extra
240 *> cautions.
241 *> \endverbatim
242 *>
243 *> \param[in,out] ERRS_C
244 *> \verbatim
245 *> ERRS_C is REAL array, dimension (NRHS, N_ERR_BNDS)
246 *> For each right-hand side, this array contains information about
247 *> various error bounds and condition numbers corresponding to the
248 *> componentwise relative error, which is defined as follows:
249 *>
250 *> Componentwise relative error in the ith solution vector:
251 *> abs(XTRUE(j,i) - X(j,i))
252 *> max_j ----------------------
253 *> abs(X(j,i))
254 *>
255 *> The array is indexed by the right-hand side i (on which the
256 *> componentwise relative error depends), and the type of error
257 *> information as described below. There currently are up to three
258 *> pieces of information returned for each right-hand side. If
259 *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
260 *> ERRS_C is not accessed. If N_ERR_BNDS .LT. 3, then at most
261 *> the first (:,N_ERR_BNDS) entries are returned.
262 *>
263 *> The first index in ERRS_C(i,:) corresponds to the ith
264 *> right-hand side.
265 *>
266 *> The second index in ERRS_C(:,err) contains the following
267 *> three fields:
268 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
269 *> reciprocal condition number is less than the threshold
270 *> sqrt(n) * slamch('Epsilon').
271 *>
272 *> err = 2 "Guaranteed" error bound: The estimated forward error,
273 *> almost certainly within a factor of 10 of the true error
274 *> so long as the next entry is greater than the threshold
275 *> sqrt(n) * slamch('Epsilon'). This error bound should only
276 *> be trusted if the previous boolean is true.
277 *>
278 *> err = 3 Reciprocal condition number: Estimated componentwise
279 *> reciprocal condition number. Compared with the threshold
280 *> sqrt(n) * slamch('Epsilon') to determine if the error
281 *> estimate is "guaranteed". These reciprocal condition
282 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
283 *> appropriately scaled matrix Z.
284 *> Let Z = S*(A*diag(x)), where x is the solution for the
285 *> current right-hand side and S scales each row of
286 *> A*diag(x) by a power of the radix so all absolute row
287 *> sums of Z are approximately 1.
288 *>
289 *> This subroutine is only responsible for setting the second field
290 *> above.
291 *> See Lapack Working Note 165 for further details and extra
292 *> cautions.
293 *> \endverbatim
294 *>
295 *> \param[in] RES
296 *> \verbatim
297 *> RES is COMPLEX array, dimension (N)
298 *> Workspace to hold the intermediate residual.
299 *> \endverbatim
300 *>
301 *> \param[in] AYB
302 *> \verbatim
303 *> AYB is REAL array, dimension (N)
304 *> Workspace.
305 *> \endverbatim
306 *>
307 *> \param[in] DY
308 *> \verbatim
309 *> DY is COMPLEX array, dimension (N)
310 *> Workspace to hold the intermediate solution.
311 *> \endverbatim
312 *>
313 *> \param[in] Y_TAIL
314 *> \verbatim
315 *> Y_TAIL is COMPLEX array, dimension (N)
316 *> Workspace to hold the trailing bits of the intermediate solution.
317 *> \endverbatim
318 *>
319 *> \param[in] RCOND
320 *> \verbatim
321 *> RCOND is REAL
322 *> Reciprocal scaled condition number. This is an estimate of the
323 *> reciprocal Skeel condition number of the matrix A after
324 *> equilibration (if done). If this is less than the machine
325 *> precision (in particular, if it is zero), the matrix is singular
326 *> to working precision. Note that the error may still be small even
327 *> if this number is very small and the matrix appears ill-
328 *> conditioned.
329 *> \endverbatim
330 *>
331 *> \param[in] ITHRESH
332 *> \verbatim
333 *> ITHRESH is INTEGER
334 *> The maximum number of residual computations allowed for
335 *> refinement. The default is 10. For 'aggressive' set to 100 to
336 *> permit convergence using approximate factorizations or
337 *> factorizations other than LU. If the factorization uses a
338 *> technique other than Gaussian elimination, the guarantees in
339 *> ERRS_N and ERRS_C may no longer be trustworthy.
340 *> \endverbatim
341 *>
342 *> \param[in] RTHRESH
343 *> \verbatim
344 *> RTHRESH is REAL
345 *> Determines when to stop refinement if the error estimate stops
346 *> decreasing. Refinement will stop when the next solution no longer
347 *> satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
348 *> the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
349 *> default value is 0.5. For 'aggressive' set to 0.9 to permit
350 *> convergence on extremely ill-conditioned matrices. See LAWN 165
351 *> for more details.
352 *> \endverbatim
353 *>
354 *> \param[in] DZ_UB
355 *> \verbatim
356 *> DZ_UB is REAL
357 *> Determines when to start considering componentwise convergence.
358 *> Componentwise convergence is only considered after each component
359 *> of the solution Y is stable, which we definte as the relative
360 *> change in each component being less than DZ_UB. The default value
361 *> is 0.25, requiring the first bit to be stable. See LAWN 165 for
362 *> more details.
363 *> \endverbatim
364 *>
365 *> \param[in] IGNORE_CWISE
366 *> \verbatim
367 *> IGNORE_CWISE is LOGICAL
368 *> If .TRUE. then ignore componentwise convergence. Default value
369 *> is .FALSE..
370 *> \endverbatim
371 *>
372 *> \param[out] INFO
373 *> \verbatim
374 *> INFO is INTEGER
375 *> = 0: Successful exit.
376 *> < 0: if INFO = -i, the ith argument to CGETRS had an illegal
377 *> value
378 *> \endverbatim
379 *
380 * Authors:
381 * ========
382 *
383 *> \author Univ. of Tennessee
384 *> \author Univ. of California Berkeley
385 *> \author Univ. of Colorado Denver
386 *> \author NAG Ltd.
387 *
388 *> \date November 2011
389 *
390 *> \ingroup complexGEcomputational
391 *
392 * =====================================================================
393  SUBROUTINE cla_gerfsx_extended( PREC_TYPE, TRANS_TYPE, N, NRHS, A,
394  $ lda, af, ldaf, ipiv, colequ, c, b,
395  $ ldb, y, ldy, berr_out, n_norms,
396  $ errs_n, errs_c, res, ayb, dy,
397  $ y_tail, rcond, ithresh, rthresh,
398  $ dz_ub, ignore_cwise, info )
399 *
400 * -- LAPACK computational routine (version 3.4.0) --
401 * -- LAPACK is a software package provided by Univ. of Tennessee, --
402 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
403 * November 2011
404 *
405 * .. Scalar Arguments ..
406  INTEGER info, lda, ldaf, ldb, ldy, n, nrhs, prec_type,
407  $ trans_type, n_norms
408  LOGICAL colequ, ignore_cwise
409  INTEGER ithresh
410  REAL rthresh, dz_ub
411 * ..
412 * .. Array Arguments
413  INTEGER ipiv( * )
414  COMPLEX a( lda, * ), af( ldaf, * ), b( ldb, * ),
415  $ y( ldy, * ), res( * ), dy( * ), y_tail( * )
416  REAL c( * ), ayb( * ), rcond, berr_out( * ),
417  $ errs_n( nrhs, * ), errs_c( nrhs, * )
418 * ..
419 *
420 * =====================================================================
421 *
422 * .. Local Scalars ..
423  CHARACTER trans
424  INTEGER cnt, i, j, x_state, z_state, y_prec_state
425  REAL yk, dyk, ymin, normy, normx, normdx, dxrat,
426  $ dzrat, prevnormdx, prev_dz_z, dxratmax,
427  $ dzratmax, dx_x, dz_z, final_dx_x, final_dz_z,
428  $ eps, hugeval, incr_thresh
429  LOGICAL incr_prec
430  COMPLEX zdum
431 * ..
432 * .. Parameters ..
433  INTEGER unstable_state, working_state, conv_state,
434  $ noprog_state, base_residual, extra_residual,
435  $ extra_y
436  parameter( unstable_state = 0, working_state = 1,
437  $ conv_state = 2,
438  $ noprog_state = 3 )
439  parameter( base_residual = 0, extra_residual = 1,
440  $ extra_y = 2 )
441  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
442  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
443  INTEGER cmp_err_i, piv_growth_i
444  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
445  $ berr_i = 3 )
446  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
447  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
448  $ piv_growth_i = 9 )
449  INTEGER la_linrx_itref_i, la_linrx_ithresh_i,
450  $ la_linrx_cwise_i
451  parameter( la_linrx_itref_i = 1,
452  $ la_linrx_ithresh_i = 2 )
453  parameter( la_linrx_cwise_i = 3 )
454  INTEGER la_linrx_trust_i, la_linrx_err_i,
455  $ la_linrx_rcond_i
456  parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
457  parameter( la_linrx_rcond_i = 3 )
458 * ..
459 * .. External Subroutines ..
460  EXTERNAL caxpy, ccopy, cgetrs, cgemv, blas_cgemv_x,
461  $ blas_cgemv2_x, cla_geamv, cla_wwaddw, slamch,
463  REAL slamch
464  CHARACTER chla_transtype
465 * ..
466 * .. Intrinsic Functions ..
467  INTRINSIC abs, max, min
468 * ..
469 * .. Statement Functions ..
470  REAL cabs1
471 * ..
472 * .. Statement Function Definitions ..
473  cabs1( zdum ) = abs( REAL( ZDUM ) ) + abs( aimag( zdum ) )
474 * ..
475 * .. Executable Statements ..
476 *
477  IF ( info.NE.0 ) RETURN
478  trans = chla_transtype(trans_type)
479  eps = slamch( 'Epsilon' )
480  hugeval = slamch( 'Overflow' )
481 * Force HUGEVAL to Inf
482  hugeval = hugeval * hugeval
483 * Using HUGEVAL may lead to spurious underflows.
484  incr_thresh = REAL( N ) * eps
485 *
486  DO j = 1, nrhs
487  y_prec_state = extra_residual
488  IF ( y_prec_state .EQ. extra_y ) THEN
489  DO i = 1, n
490  y_tail( i ) = 0.0
491  END DO
492  END IF
493 
494  dxrat = 0.0
495  dxratmax = 0.0
496  dzrat = 0.0
497  dzratmax = 0.0
498  final_dx_x = hugeval
499  final_dz_z = hugeval
500  prevnormdx = hugeval
501  prev_dz_z = hugeval
502  dz_z = hugeval
503  dx_x = hugeval
504 
505  x_state = working_state
506  z_state = unstable_state
507  incr_prec = .false.
508 
509  DO cnt = 1, ithresh
510 *
511 * Compute residual RES = B_s - op(A_s) * Y,
512 * op(A) = A, A**T, or A**H depending on TRANS (and type).
513 *
514  CALL ccopy( n, b( 1, j ), 1, res, 1 )
515  IF ( y_prec_state .EQ. base_residual ) THEN
516  CALL cgemv( trans, n, n, (-1.0e+0,0.0e+0), a, lda,
517  $ y( 1, j ), 1, (1.0e+0,0.0e+0), res, 1)
518  ELSE IF (y_prec_state .EQ. extra_residual) THEN
519  CALL blas_cgemv_x( trans_type, n, n, (-1.0e+0,0.0e+0), a,
520  $ lda, y( 1, j ), 1, (1.0e+0,0.0e+0),
521  $ res, 1, prec_type )
522  ELSE
523  CALL blas_cgemv2_x( trans_type, n, n, (-1.0e+0,0.0e+0),
524  $ a, lda, y(1, j), y_tail, 1, (1.0e+0,0.0e+0), res, 1,
525  $ prec_type)
526  END IF
527 
528 ! XXX: RES is no longer needed.
529  CALL ccopy( n, res, 1, dy, 1 )
530  CALL cgetrs( trans, n, 1, af, ldaf, ipiv, dy, n, info )
531 *
532 * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
533 *
534  normx = 0.0e+0
535  normy = 0.0e+0
536  normdx = 0.0e+0
537  dz_z = 0.0e+0
538  ymin = hugeval
539 *
540  DO i = 1, n
541  yk = cabs1( y( i, j ) )
542  dyk = cabs1( dy( i ) )
543 
544  IF ( yk .NE. 0.0e+0 ) THEN
545  dz_z = max( dz_z, dyk / yk )
546  ELSE IF ( dyk .NE. 0.0 ) THEN
547  dz_z = hugeval
548  END IF
549 
550  ymin = min( ymin, yk )
551 
552  normy = max( normy, yk )
553 
554  IF ( colequ ) THEN
555  normx = max( normx, yk * c( i ) )
556  normdx = max( normdx, dyk * c( i ) )
557  ELSE
558  normx = normy
559  normdx = max(normdx, dyk)
560  END IF
561  END DO
562 
563  IF ( normx .NE. 0.0 ) THEN
564  dx_x = normdx / normx
565  ELSE IF ( normdx .EQ. 0.0 ) THEN
566  dx_x = 0.0
567  ELSE
568  dx_x = hugeval
569  END IF
570 
571  dxrat = normdx / prevnormdx
572  dzrat = dz_z / prev_dz_z
573 *
574 * Check termination criteria
575 *
576  IF (.NOT.ignore_cwise
577  $ .AND. ymin*rcond .LT. incr_thresh*normy
578  $ .AND. y_prec_state .LT. extra_y )
579  $ incr_prec = .true.
580 
581  IF ( x_state .EQ. noprog_state .AND. dxrat .LE. rthresh )
582  $ x_state = working_state
583  IF ( x_state .EQ. working_state ) THEN
584  IF (dx_x .LE. eps) THEN
585  x_state = conv_state
586  ELSE IF ( dxrat .GT. rthresh ) THEN
587  IF ( y_prec_state .NE. extra_y ) THEN
588  incr_prec = .true.
589  ELSE
590  x_state = noprog_state
591  END IF
592  ELSE
593  IF ( dxrat .GT. dxratmax ) dxratmax = dxrat
594  END IF
595  IF ( x_state .GT. working_state ) final_dx_x = dx_x
596  END IF
597 
598  IF ( z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub )
599  $ z_state = working_state
600  IF ( z_state .EQ. noprog_state .AND. dzrat .LE. rthresh )
601  $ z_state = working_state
602  IF ( z_state .EQ. working_state ) THEN
603  IF ( dz_z .LE. eps ) THEN
604  z_state = conv_state
605  ELSE IF ( dz_z .GT. dz_ub ) THEN
606  z_state = unstable_state
607  dzratmax = 0.0
608  final_dz_z = hugeval
609  ELSE IF ( dzrat .GT. rthresh ) THEN
610  IF ( y_prec_state .NE. extra_y ) THEN
611  incr_prec = .true.
612  ELSE
613  z_state = noprog_state
614  END IF
615  ELSE
616  IF ( dzrat .GT. dzratmax ) dzratmax = dzrat
617  END IF
618  IF ( z_state .GT. working_state ) final_dz_z = dz_z
619  END IF
620 *
621 * Exit if both normwise and componentwise stopped working,
622 * but if componentwise is unstable, let it go at least two
623 * iterations.
624 *
625  IF ( x_state.NE.working_state ) THEN
626  IF ( ignore_cwise ) goto 666
627  IF ( z_state.EQ.noprog_state .OR. z_state.EQ.conv_state )
628  $ goto 666
629  IF ( z_state.EQ.unstable_state .AND. cnt.GT.1 ) goto 666
630  END IF
631 
632  IF ( incr_prec ) THEN
633  incr_prec = .false.
634  y_prec_state = y_prec_state + 1
635  DO i = 1, n
636  y_tail( i ) = 0.0
637  END DO
638  END IF
639 
640  prevnormdx = normdx
641  prev_dz_z = dz_z
642 *
643 * Update soluton.
644 *
645  IF ( y_prec_state .LT. extra_y ) THEN
646  CALL caxpy( n, (1.0e+0,0.0e+0), dy, 1, y(1,j), 1 )
647  ELSE
648  CALL cla_wwaddw( n, y( 1, j ), y_tail, dy )
649  END IF
650 
651  END DO
652 * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
653  666 CONTINUE
654 *
655 * Set final_* when cnt hits ithresh
656 *
657  IF ( x_state .EQ. working_state ) final_dx_x = dx_x
658  IF ( z_state .EQ. working_state ) final_dz_z = dz_z
659 *
660 * Compute error bounds
661 *
662  IF (n_norms .GE. 1) THEN
663  errs_n( j, la_linrx_err_i ) = final_dx_x / (1 - dxratmax)
664 
665  END IF
666  IF ( n_norms .GE. 2 ) THEN
667  errs_c( j, la_linrx_err_i ) = final_dz_z / (1 - dzratmax)
668  END IF
669 *
670 * Compute componentwise relative backward error from formula
671 * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
672 * where abs(Z) is the componentwise absolute value of the matrix
673 * or vector Z.
674 *
675 * Compute residual RES = B_s - op(A_s) * Y,
676 * op(A) = A, A**T, or A**H depending on TRANS (and type).
677 *
678  CALL ccopy( n, b( 1, j ), 1, res, 1 )
679  CALL cgemv( trans, n, n, (-1.0e+0,0.0e+0), a, lda, y(1,j), 1,
680  $ (1.0e+0,0.0e+0), res, 1 )
681 
682  DO i = 1, n
683  ayb( i ) = cabs1( b( i, j ) )
684  END DO
685 *
686 * Compute abs(op(A_s))*abs(Y) + abs(B_s).
687 *
688  CALL cla_geamv( trans_type, n, n, 1.0e+0,
689  $ a, lda, y(1, j), 1, 1.0e+0, ayb, 1 )
690 
691  CALL cla_lin_berr( n, n, 1, res, ayb, berr_out( j ) )
692 *
693 * End of loop for each RHS.
694 *
695  END DO
696 *
697  RETURN
698  END