LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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sla_porfsx_extended.f
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1*> \brief \b SLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download SLA_PORFSX_EXTENDED + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sla_porfsx_extended.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sla_porfsx_extended.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sla_porfsx_extended.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE SLA_PORFSX_EXTENDED( PREC_TYPE, UPLO, N, NRHS, A, LDA,
22* AF, LDAF, COLEQU, C, B, LDB, Y,
23* LDY, BERR_OUT, N_NORMS,
24* ERR_BNDS_NORM, ERR_BNDS_COMP, RES,
25* AYB, DY, Y_TAIL, RCOND, ITHRESH,
26* RTHRESH, DZ_UB, IGNORE_CWISE,
27* INFO )
28*
29* .. Scalar Arguments ..
30* INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
31* $ N_NORMS, ITHRESH
32* CHARACTER UPLO
33* LOGICAL COLEQU, IGNORE_CWISE
34* REAL RTHRESH, DZ_UB
35* ..
36* .. Array Arguments ..
37* REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
38* $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * )
39* REAL C( * ), AYB(*), RCOND, BERR_OUT( * ),
40* $ ERR_BNDS_NORM( NRHS, * ),
41* $ ERR_BNDS_COMP( NRHS, * )
42* ..
43*
44*
45*> \par Purpose:
46* =============
47*>
48*> \verbatim
49*>
50*> SLA_PORFSX_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 SPORFSX to perform iterative refinement.
54*> In addition to normwise error bound, the code provides maximum
55*> componentwise error bound if possible. See comments for ERR_BNDS_NORM
56*> and ERR_BNDS_COMP for details of the error bounds. Note that this
57*> subroutine is only responsible for setting the second fields of
58*> ERR_BNDS_NORM and ERR_BNDS_COMP.
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 P
69*> = 'S': Single
70*> = 'D': Double
71*> = 'I': Indigenous
72*> = 'X' or 'E': Extra
73*> \endverbatim
74*>
75*> \param[in] UPLO
76*> \verbatim
77*> UPLO is CHARACTER*1
78*> = 'U': Upper triangle of A is stored;
79*> = 'L': Lower triangle of A is stored.
80*> \endverbatim
81*>
82*> \param[in] N
83*> \verbatim
84*> N is INTEGER
85*> The number of linear equations, i.e., the order of the
86*> matrix A. N >= 0.
87*> \endverbatim
88*>
89*> \param[in] NRHS
90*> \verbatim
91*> NRHS is INTEGER
92*> The number of right-hand-sides, i.e., the number of columns of the
93*> matrix B.
94*> \endverbatim
95*>
96*> \param[in] A
97*> \verbatim
98*> A is REAL array, dimension (LDA,N)
99*> On entry, the N-by-N matrix A.
100*> \endverbatim
101*>
102*> \param[in] LDA
103*> \verbatim
104*> LDA is INTEGER
105*> The leading dimension of the array A. LDA >= max(1,N).
106*> \endverbatim
107*>
108*> \param[in] AF
109*> \verbatim
110*> AF is REAL array, dimension (LDAF,N)
111*> The triangular factor U or L from the Cholesky factorization
112*> A = U**T*U or A = L*L**T, as computed by SPOTRF.
113*> \endverbatim
114*>
115*> \param[in] LDAF
116*> \verbatim
117*> LDAF is INTEGER
118*> The leading dimension of the array AF. LDAF >= max(1,N).
119*> \endverbatim
120*>
121*> \param[in] COLEQU
122*> \verbatim
123*> COLEQU is LOGICAL
124*> If .TRUE. then column equilibration was done to A before calling
125*> this routine. This is needed to compute the solution and error
126*> bounds correctly.
127*> \endverbatim
128*>
129*> \param[in] C
130*> \verbatim
131*> C is REAL array, dimension (N)
132*> The column scale factors for A. If COLEQU = .FALSE., C
133*> is not accessed. If C is input, each element of C should be a power
134*> of the radix to ensure a reliable solution and error estimates.
135*> Scaling by powers of the radix does not cause rounding errors unless
136*> the result underflows or overflows. Rounding errors during scaling
137*> lead to refining with a matrix that is not equivalent to the
138*> input matrix, producing error estimates that may not be
139*> reliable.
140*> \endverbatim
141*>
142*> \param[in] B
143*> \verbatim
144*> B is REAL array, dimension (LDB,NRHS)
145*> The right-hand-side matrix B.
146*> \endverbatim
147*>
148*> \param[in] LDB
149*> \verbatim
150*> LDB is INTEGER
151*> The leading dimension of the array B. LDB >= max(1,N).
152*> \endverbatim
153*>
154*> \param[in,out] Y
155*> \verbatim
156*> Y is REAL array, dimension (LDY,NRHS)
157*> On entry, the solution matrix X, as computed by SPOTRS.
158*> On exit, the improved solution matrix Y.
159*> \endverbatim
160*>
161*> \param[in] LDY
162*> \verbatim
163*> LDY is INTEGER
164*> The leading dimension of the array Y. LDY >= max(1,N).
165*> \endverbatim
166*>
167*> \param[out] BERR_OUT
168*> \verbatim
169*> BERR_OUT is REAL array, dimension (NRHS)
170*> On exit, BERR_OUT(j) contains the componentwise relative backward
171*> error for right-hand-side j from the formula
172*> max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
173*> where abs(Z) is the componentwise absolute value of the matrix
174*> or vector Z. This is computed by SLA_LIN_BERR.
175*> \endverbatim
176*>
177*> \param[in] N_NORMS
178*> \verbatim
179*> N_NORMS is INTEGER
180*> Determines which error bounds to return (see ERR_BNDS_NORM
181*> and ERR_BNDS_COMP).
182*> If N_NORMS >= 1 return normwise error bounds.
183*> If N_NORMS >= 2 return componentwise error bounds.
184*> \endverbatim
185*>
186*> \param[in,out] ERR_BNDS_NORM
187*> \verbatim
188*> ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
189*> For each right-hand side, this array contains information about
190*> various error bounds and condition numbers corresponding to the
191*> normwise relative error, which is defined as follows:
192*>
193*> Normwise relative error in the ith solution vector:
194*> max_j (abs(XTRUE(j,i) - X(j,i)))
195*> ------------------------------
196*> max_j abs(X(j,i))
197*>
198*> The array is indexed by the type of error information as described
199*> below. There currently are up to three pieces of information
200*> returned.
201*>
202*> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
203*> right-hand side.
204*>
205*> The second index in ERR_BNDS_NORM(:,err) contains the following
206*> three fields:
207*> err = 1 "Trust/don't trust" boolean. Trust the answer if the
208*> reciprocal condition number is less than the threshold
209*> sqrt(n) * slamch('Epsilon').
210*>
211*> err = 2 "Guaranteed" error bound: The estimated forward error,
212*> almost certainly within a factor of 10 of the true error
213*> so long as the next entry is greater than the threshold
214*> sqrt(n) * slamch('Epsilon'). This error bound should only
215*> be trusted if the previous boolean is true.
216*>
217*> err = 3 Reciprocal condition number: Estimated normwise
218*> reciprocal condition number. Compared with the threshold
219*> sqrt(n) * slamch('Epsilon') to determine if the error
220*> estimate is "guaranteed". These reciprocal condition
221*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
222*> appropriately scaled matrix Z.
223*> Let Z = S*A, where S scales each row by a power of the
224*> radix so all absolute row sums of Z are approximately 1.
225*>
226*> This subroutine is only responsible for setting the second field
227*> above.
228*> See Lapack Working Note 165 for further details and extra
229*> cautions.
230*> \endverbatim
231*>
232*> \param[in,out] ERR_BNDS_COMP
233*> \verbatim
234*> ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
235*> For each right-hand side, this array contains information about
236*> various error bounds and condition numbers corresponding to the
237*> componentwise relative error, which is defined as follows:
238*>
239*> Componentwise relative error in the ith solution vector:
240*> abs(XTRUE(j,i) - X(j,i))
241*> max_j ----------------------
242*> abs(X(j,i))
243*>
244*> The array is indexed by the right-hand side i (on which the
245*> componentwise relative error depends), and the type of error
246*> information as described below. There currently are up to three
247*> pieces of information returned for each right-hand side. If
248*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
249*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
250*> the first (:,N_ERR_BNDS) entries are returned.
251*>
252*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
253*> right-hand side.
254*>
255*> The second index in ERR_BNDS_COMP(:,err) contains the following
256*> three fields:
257*> err = 1 "Trust/don't trust" boolean. Trust the answer if the
258*> reciprocal condition number is less than the threshold
259*> sqrt(n) * slamch('Epsilon').
260*>
261*> err = 2 "Guaranteed" error bound: The estimated forward error,
262*> almost certainly within a factor of 10 of the true error
263*> so long as the next entry is greater than the threshold
264*> sqrt(n) * slamch('Epsilon'). This error bound should only
265*> be trusted if the previous boolean is true.
266*>
267*> err = 3 Reciprocal condition number: Estimated componentwise
268*> reciprocal condition number. Compared with the threshold
269*> sqrt(n) * slamch('Epsilon') to determine if the error
270*> estimate is "guaranteed". These reciprocal condition
271*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
272*> appropriately scaled matrix Z.
273*> Let Z = S*(A*diag(x)), where x is the solution for the
274*> current right-hand side and S scales each row of
275*> A*diag(x) by a power of the radix so all absolute row
276*> sums of Z are approximately 1.
277*>
278*> This subroutine is only responsible for setting the second field
279*> above.
280*> See Lapack Working Note 165 for further details and extra
281*> cautions.
282*> \endverbatim
283*>
284*> \param[in] RES
285*> \verbatim
286*> RES is REAL array, dimension (N)
287*> Workspace to hold the intermediate residual.
288*> \endverbatim
289*>
290*> \param[in] AYB
291*> \verbatim
292*> AYB is REAL array, dimension (N)
293*> Workspace. This can be the same workspace passed for Y_TAIL.
294*> \endverbatim
295*>
296*> \param[in] DY
297*> \verbatim
298*> DY is REAL array, dimension (N)
299*> Workspace to hold the intermediate solution.
300*> \endverbatim
301*>
302*> \param[in] Y_TAIL
303*> \verbatim
304*> Y_TAIL is REAL array, dimension (N)
305*> Workspace to hold the trailing bits of the intermediate solution.
306*> \endverbatim
307*>
308*> \param[in] RCOND
309*> \verbatim
310*> RCOND is REAL
311*> Reciprocal scaled condition number. This is an estimate of the
312*> reciprocal Skeel condition number of the matrix A after
313*> equilibration (if done). If this is less than the machine
314*> precision (in particular, if it is zero), the matrix is singular
315*> to working precision. Note that the error may still be small even
316*> if this number is very small and the matrix appears ill-
317*> conditioned.
318*> \endverbatim
319*>
320*> \param[in] ITHRESH
321*> \verbatim
322*> ITHRESH is INTEGER
323*> The maximum number of residual computations allowed for
324*> refinement. The default is 10. For 'aggressive' set to 100 to
325*> permit convergence using approximate factorizations or
326*> factorizations other than LU. If the factorization uses a
327*> technique other than Gaussian elimination, the guarantees in
328*> ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.
329*> \endverbatim
330*>
331*> \param[in] RTHRESH
332*> \verbatim
333*> RTHRESH is REAL
334*> Determines when to stop refinement if the error estimate stops
335*> decreasing. Refinement will stop when the next solution no longer
336*> satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
337*> the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
338*> default value is 0.5. For 'aggressive' set to 0.9 to permit
339*> convergence on extremely ill-conditioned matrices. See LAWN 165
340*> for more details.
341*> \endverbatim
342*>
343*> \param[in] DZ_UB
344*> \verbatim
345*> DZ_UB is REAL
346*> Determines when to start considering componentwise convergence.
347*> Componentwise convergence is only considered after each component
348*> of the solution Y is stable, which we define as the relative
349*> change in each component being less than DZ_UB. The default value
350*> is 0.25, requiring the first bit to be stable. See LAWN 165 for
351*> more details.
352*> \endverbatim
353*>
354*> \param[in] IGNORE_CWISE
355*> \verbatim
356*> IGNORE_CWISE is LOGICAL
357*> If .TRUE. then ignore componentwise convergence. Default value
358*> is .FALSE..
359*> \endverbatim
360*>
361*> \param[out] INFO
362*> \verbatim
363*> INFO is INTEGER
364*> = 0: Successful exit.
365*> < 0: if INFO = -i, the ith argument to SPOTRS had an illegal
366*> value
367*> \endverbatim
368*
369* Authors:
370* ========
371*
372*> \author Univ. of Tennessee
373*> \author Univ. of California Berkeley
374*> \author Univ. of Colorado Denver
375*> \author NAG Ltd.
376*
377*> \ingroup la_porfsx_extended
378*
379* =====================================================================
380 SUBROUTINE sla_porfsx_extended( PREC_TYPE, UPLO, N, NRHS, A, LDA,
381 $ AF, LDAF, COLEQU, C, B, LDB, Y,
382 $ LDY, BERR_OUT, N_NORMS,
383 $ ERR_BNDS_NORM, ERR_BNDS_COMP, RES,
384 $ AYB, DY, Y_TAIL, RCOND, ITHRESH,
385 $ RTHRESH, DZ_UB, IGNORE_CWISE,
386 $ INFO )
387*
388* -- LAPACK computational routine --
389* -- LAPACK is a software package provided by Univ. of Tennessee, --
390* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
391*
392* .. Scalar Arguments ..
393 INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
394 $ N_NORMS, ITHRESH
395 CHARACTER UPLO
396 LOGICAL COLEQU, IGNORE_CWISE
397 REAL RTHRESH, DZ_UB
398* ..
399* .. Array Arguments ..
400 REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
401 $ y( ldy, * ), res( * ), dy( * ), y_tail( * )
402 REAL C( * ), AYB(*), RCOND, BERR_OUT( * ),
403 $ err_bnds_norm( nrhs, * ),
404 $ err_bnds_comp( nrhs, * )
405* ..
406*
407* =====================================================================
408*
409* .. Local Scalars ..
410 INTEGER UPLO2, CNT, I, J, X_STATE, Z_STATE
411 REAL YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
412 $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
413 $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
414 $ EPS, HUGEVAL, INCR_THRESH
415 LOGICAL INCR_PREC
416* ..
417* .. Parameters ..
418 INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
419 $ NOPROG_STATE, Y_PREC_STATE, BASE_RESIDUAL,
420 $ EXTRA_RESIDUAL, EXTRA_Y
421 parameter( unstable_state = 0, working_state = 1,
422 $ conv_state = 2, noprog_state = 3 )
423 parameter( base_residual = 0, extra_residual = 1,
424 $ extra_y = 2 )
425 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
426 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
427 INTEGER CMP_ERR_I, PIV_GROWTH_I
428 PARAMETER ( FINAL_NRM_ERR_I = 1, final_cmp_err_i = 2,
429 $ berr_i = 3 )
430 parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
431 parameter( cmp_rcond_i = 7, cmp_err_i = 8,
432 $ piv_growth_i = 9 )
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* .. External Functions ..
444 LOGICAL LSAME
445 EXTERNAL ILAUPLO
446 INTEGER ILAUPLO
447* ..
448* .. External Subroutines ..
449 EXTERNAL saxpy, scopy, spotrs, ssymv, blas_ssymv_x,
450 $ blas_ssymv2_x, sla_syamv, sla_wwaddw,
452 REAL SLAMCH
453* ..
454* .. Intrinsic Functions ..
455 INTRINSIC abs, max, min
456* ..
457* .. Executable Statements ..
458*
459 IF (info.NE.0) RETURN
460 eps = slamch( 'Epsilon' )
461 hugeval = slamch( 'Overflow' )
462* Force HUGEVAL to Inf
463 hugeval = hugeval * hugeval
464* Using HUGEVAL may lead to spurious underflows.
465 incr_thresh = real( n ) * eps
466
467 IF ( lsame( uplo, 'L' ) ) THEN
468 uplo2 = ilauplo( 'L' )
469 ELSE
470 uplo2 = ilauplo( 'U' )
471 ENDIF
472
473 DO j = 1, nrhs
474 y_prec_state = extra_residual
475 IF ( y_prec_state .EQ. extra_y ) THEN
476 DO i = 1, n
477 y_tail( i ) = 0.0
478 END DO
479 END IF
480
481 dxrat = 0.0
482 dxratmax = 0.0
483 dzrat = 0.0
484 dzratmax = 0.0
485 final_dx_x = hugeval
486 final_dz_z = hugeval
487 prevnormdx = hugeval
488 prev_dz_z = hugeval
489 dz_z = hugeval
490 dx_x = hugeval
491
492 x_state = working_state
493 z_state = unstable_state
494 incr_prec = .false.
495
496 DO cnt = 1, ithresh
497*
498* Compute residual RES = B_s - op(A_s) * Y,
499* op(A) = A, A**T, or A**H depending on TRANS (and type).
500*
501 CALL scopy( n, b( 1, j ), 1, res, 1 )
502 IF ( y_prec_state .EQ. base_residual ) THEN
503 CALL ssymv( uplo, n, -1.0, a, lda, y(1,j), 1,
504 $ 1.0, res, 1 )
505 ELSE IF ( y_prec_state .EQ. extra_residual ) THEN
506 CALL blas_ssymv_x( uplo2, n, -1.0, a, lda,
507 $ y( 1, j ), 1, 1.0, res, 1, prec_type )
508 ELSE
509 CALL blas_ssymv2_x(uplo2, n, -1.0, a, lda,
510 $ y(1, j), y_tail, 1, 1.0, res, 1, prec_type)
511 END IF
512
513! XXX: RES is no longer needed.
514 CALL scopy( n, res, 1, dy, 1 )
515 CALL spotrs( uplo, n, 1, af, ldaf, dy, n, info )
516*
517* Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
518*
519 normx = 0.0
520 normy = 0.0
521 normdx = 0.0
522 dz_z = 0.0
523 ymin = hugeval
524
525 DO i = 1, n
526 yk = abs( y( i, j ) )
527 dyk = abs( dy( i ) )
528
529 IF ( yk .NE. 0.0 ) THEN
530 dz_z = max( dz_z, dyk / yk )
531 ELSE IF ( dyk .NE. 0.0 ) THEN
532 dz_z = hugeval
533 END IF
534
535 ymin = min( ymin, yk )
536
537 normy = max( normy, yk )
538
539 IF ( colequ ) THEN
540 normx = max( normx, yk * c( i ) )
541 normdx = max( normdx, dyk * c( i ) )
542 ELSE
543 normx = normy
544 normdx = max( normdx, dyk )
545 END IF
546 END DO
547
548 IF ( normx .NE. 0.0 ) THEN
549 dx_x = normdx / normx
550 ELSE IF ( normdx .EQ. 0.0 ) THEN
551 dx_x = 0.0
552 ELSE
553 dx_x = hugeval
554 END IF
555
556 dxrat = normdx / prevnormdx
557 dzrat = dz_z / prev_dz_z
558*
559* Check termination criteria.
560*
561 IF ( ymin*rcond .LT. incr_thresh*normy
562 $ .AND. y_prec_state .LT. extra_y )
563 $ incr_prec = .true.
564
565 IF ( x_state .EQ. noprog_state .AND. dxrat .LE. rthresh )
566 $ x_state = working_state
567 IF ( x_state .EQ. working_state ) THEN
568 IF ( dx_x .LE. eps ) THEN
569 x_state = conv_state
570 ELSE IF ( dxrat .GT. rthresh ) THEN
571 IF ( y_prec_state .NE. extra_y ) THEN
572 incr_prec = .true.
573 ELSE
574 x_state = noprog_state
575 END IF
576 ELSE
577 IF ( dxrat .GT. dxratmax ) dxratmax = dxrat
578 END IF
579 IF ( x_state .GT. working_state ) final_dx_x = dx_x
580 END IF
581
582 IF ( z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub )
583 $ z_state = working_state
584 IF ( z_state .EQ. noprog_state .AND. dzrat .LE. rthresh )
585 $ z_state = working_state
586 IF ( z_state .EQ. working_state ) THEN
587 IF ( dz_z .LE. eps ) THEN
588 z_state = conv_state
589 ELSE IF ( dz_z .GT. dz_ub ) THEN
590 z_state = unstable_state
591 dzratmax = 0.0
592 final_dz_z = hugeval
593 ELSE IF ( dzrat .GT. rthresh ) THEN
594 IF ( y_prec_state .NE. extra_y ) THEN
595 incr_prec = .true.
596 ELSE
597 z_state = noprog_state
598 END IF
599 ELSE
600 IF ( dzrat .GT. dzratmax ) dzratmax = dzrat
601 END IF
602 IF ( z_state .GT. working_state ) final_dz_z = dz_z
603 END IF
604
605 IF ( x_state.NE.working_state.AND.
606 $ ( ignore_cwise.OR.z_state.NE.working_state ) )
607 $ GOTO 666
608
609 IF ( incr_prec ) THEN
610 incr_prec = .false.
611 y_prec_state = y_prec_state + 1
612 DO i = 1, n
613 y_tail( i ) = 0.0
614 END DO
615 END IF
616
617 prevnormdx = normdx
618 prev_dz_z = dz_z
619*
620* Update solution.
621*
622 IF (y_prec_state .LT. extra_y) THEN
623 CALL saxpy( n, 1.0, dy, 1, y(1,j), 1 )
624 ELSE
625 CALL sla_wwaddw( n, y( 1, j ), y_tail, dy )
626 END IF
627
628 END DO
629* Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
630 666 CONTINUE
631*
632* Set final_* when cnt hits ithresh.
633*
634 IF ( x_state .EQ. working_state ) final_dx_x = dx_x
635 IF ( z_state .EQ. working_state ) final_dz_z = dz_z
636*
637* Compute error bounds.
638*
639 IF ( n_norms .GE. 1 ) THEN
640 err_bnds_norm( j, la_linrx_err_i ) =
641 $ final_dx_x / (1 - dxratmax)
642 END IF
643 IF ( n_norms .GE. 2 ) THEN
644 err_bnds_comp( j, la_linrx_err_i ) =
645 $ final_dz_z / (1 - dzratmax)
646 END IF
647*
648* Compute componentwise relative backward error from formula
649* max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
650* where abs(Z) is the componentwise absolute value of the matrix
651* or vector Z.
652*
653* Compute residual RES = B_s - op(A_s) * Y,
654* op(A) = A, A**T, or A**H depending on TRANS (and type).
655*
656 CALL scopy( n, b( 1, j ), 1, res, 1 )
657 CALL ssymv( uplo, n, -1.0, a, lda, y(1,j), 1, 1.0, res, 1 )
658
659 DO i = 1, n
660 ayb( i ) = abs( b( i, j ) )
661 END DO
662*
663* Compute abs(op(A_s))*abs(Y) + abs(B_s).
664*
665 CALL sla_syamv( uplo2, n, 1.0,
666 $ a, lda, y(1, j), 1, 1.0, ayb, 1 )
667
668 CALL sla_lin_berr( n, n, 1, res, ayb, berr_out( j ) )
669*
670* End of loop for each RHS.
671*
672 END DO
673*
674 RETURN
675*
676* End of SLA_PORFSX_EXTENDED
677*
678 END
subroutine saxpy(n, sa, sx, incx, sy, incy)
SAXPY
Definition saxpy.f:89
subroutine scopy(n, sx, incx, sy, incy)
SCOPY
Definition scopy.f:82
subroutine ssymv(uplo, n, alpha, a, lda, x, incx, beta, y, incy)
SSYMV
Definition ssymv.f:152
subroutine sla_syamv(uplo, n, alpha, a, lda, x, incx, beta, y, incy)
SLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bou...
Definition sla_syamv.f:177
subroutine sla_lin_berr(n, nz, nrhs, res, ayb, berr)
SLA_LIN_BERR computes a component-wise relative backward error.
subroutine sla_porfsx_extended(prec_type, uplo, n, nrhs, a, lda, af, ldaf, 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)
SLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or H...
subroutine sla_wwaddw(n, x, y, w)
SLA_WWADDW adds a vector into a doubled-single vector.
Definition sla_wwaddw.f:81
subroutine spotrs(uplo, n, nrhs, a, lda, b, ldb, info)
SPOTRS
Definition spotrs.f:110