summaryrefslogtreecommitdiff
path: root/SRC/sgesvxx.f
blob: 281f198d5caad5a543231dc94dd1e650eaa338a3 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
*> \brief <b> SGESVXX computes the solution to system of linear equations A * X = B for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SGESVXX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgesvxx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgesvxx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgesvxx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE SGESVXX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
*                           EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW,
*                           BERR, N_ERR_BNDS, ERR_BNDS_NORM,
*                           ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK,
*                           INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          EQUED, FACT, TRANS
*       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
*      $                   N_ERR_BNDS
*       REAL               RCOND, RPVGRW
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * ), IWORK( * )
*       REAL               A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
*      $                   X( LDX , * ),WORK( * )
*       REAL               R( * ), C( * ), PARAMS( * ), BERR( * ),
*      $                   ERR_BNDS_NORM( NRHS, * ),
*      $                   ERR_BNDS_COMP( NRHS, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>    SGESVXX uses the LU factorization to compute the solution to a
*>    real system of linear equations  A * X = B,  where A is an
*>    N-by-N matrix and X and B are N-by-NRHS matrices.
*>
*>    If requested, both normwise and maximum componentwise error bounds
*>    are returned. SGESVXX will return a solution with a tiny
*>    guaranteed error (O(eps) where eps is the working machine
*>    precision) unless the matrix is very ill-conditioned, in which
*>    case a warning is returned. Relevant condition numbers also are
*>    calculated and returned.
*>
*>    SGESVXX accepts user-provided factorizations and equilibration
*>    factors; see the definitions of the FACT and EQUED options.
*>    Solving with refinement and using a factorization from a previous
*>    SGESVXX call will also produce a solution with either O(eps)
*>    errors or warnings, but we cannot make that claim for general
*>    user-provided factorizations and equilibration factors if they
*>    differ from what SGESVXX would itself produce.
*> \endverbatim
*
*> \par Description:
*  =================
*>
*> \verbatim
*>
*>    The following steps are performed:
*>
*>    1. If FACT = 'E', real scaling factors are computed to equilibrate
*>    the system:
*>
*>      TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
*>      TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
*>      TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
*>
*>    Whether or not the system will be equilibrated depends on the
*>    scaling of the matrix A, but if equilibration is used, A is
*>    overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
*>    or diag(C)*B (if TRANS = 'T' or 'C').
*>
*>    2. If FACT = 'N' or 'E', the LU decomposition is used to factor
*>    the matrix A (after equilibration if FACT = 'E') as
*>
*>      A = P * L * U,
*>
*>    where P is a permutation matrix, L is a unit lower triangular
*>    matrix, and U is upper triangular.
*>
*>    3. If some U(i,i)=0, so that U is exactly singular, then the
*>    routine returns with INFO = i. Otherwise, the factored form of A
*>    is used to estimate the condition number of the matrix A (see
*>    argument RCOND). If the reciprocal of the condition number is less
*>    than machine precision, the routine still goes on to solve for X
*>    and compute error bounds as described below.
*>
*>    4. The system of equations is solved for X using the factored form
*>    of A.
*>
*>    5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
*>    the routine will use iterative refinement to try to get a small
*>    error and error bounds.  Refinement calculates the residual to at
*>    least twice the working precision.
*>
*>    6. If equilibration was used, the matrix X is premultiplied by
*>    diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
*>    that it solves the original system before equilibration.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \verbatim
*>     Some optional parameters are bundled in the PARAMS array.  These
*>     settings determine how refinement is performed, but often the
*>     defaults are acceptable.  If the defaults are acceptable, users
*>     can pass NPARAMS = 0 which prevents the source code from accessing
*>     the PARAMS argument.
*> \endverbatim
*>
*> \param[in] FACT
*> \verbatim
*>          FACT is CHARACTER*1
*>     Specifies whether or not the factored form of the matrix A is
*>     supplied on entry, and if not, whether the matrix A should be
*>     equilibrated before it is factored.
*>       = 'F':  On entry, AF and IPIV contain the factored form of A.
*>               If EQUED is not 'N', the matrix A has been
*>               equilibrated with scaling factors given by R and C.
*>               A, AF, and IPIV are not modified.
*>       = 'N':  The matrix A will be copied to AF and factored.
*>       = 'E':  The matrix A will be equilibrated if necessary, then
*>               copied to AF and factored.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>     Specifies the form of the system of equations:
*>       = 'N':  A * X = B     (No transpose)
*>       = 'T':  A**T * X = B  (Transpose)
*>       = 'C':  A**H * X = B  (Conjugate Transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>     The number of linear equations, i.e., the order of the
*>     matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>     The number of right hand sides, i.e., the number of columns
*>     of the matrices B and X.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is REAL array, dimension (LDA,N)
*>     On entry, the N-by-N matrix A.  If FACT = 'F' and EQUED is
*>     not 'N', then A must have been equilibrated by the scaling
*>     factors in R and/or C.  A is not modified if FACT = 'F' or
*>     'N', or if FACT = 'E' and EQUED = 'N' on exit.
*>
*>     On exit, if EQUED .ne. 'N', A is scaled as follows:
*>     EQUED = 'R':  A := diag(R) * A
*>     EQUED = 'C':  A := A * diag(C)
*>     EQUED = 'B':  A := diag(R) * A * diag(C).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>     The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] AF
*> \verbatim
*>          AF is REAL array, dimension (LDAF,N)
*>     If FACT = 'F', then AF is an input argument and on entry
*>     contains the factors L and U from the factorization
*>     A = P*L*U as computed by SGETRF.  If EQUED .ne. 'N', then
*>     AF is the factored form of the equilibrated matrix A.
*>
*>     If FACT = 'N', then AF is an output argument and on exit
*>     returns the factors L and U from the factorization A = P*L*U
*>     of the original matrix A.
*>
*>     If FACT = 'E', then AF is an output argument and on exit
*>     returns the factors L and U from the factorization A = P*L*U
*>     of the equilibrated matrix A (see the description of A for
*>     the form of the equilibrated matrix).
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*>          LDAF is INTEGER
*>     The leading dimension of the array AF.  LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>     If FACT = 'F', then IPIV is an input argument and on entry
*>     contains the pivot indices from the factorization A = P*L*U
*>     as computed by SGETRF; row i of the matrix was interchanged
*>     with row IPIV(i).
*>
*>     If FACT = 'N', then IPIV is an output argument and on exit
*>     contains the pivot indices from the factorization A = P*L*U
*>     of the original matrix A.
*>
*>     If FACT = 'E', then IPIV is an output argument and on exit
*>     contains the pivot indices from the factorization A = P*L*U
*>     of the equilibrated matrix A.
*> \endverbatim
*>
*> \param[in,out] EQUED
*> \verbatim
*>          EQUED is CHARACTER*1
*>     Specifies the form of equilibration that was done.
*>       = 'N':  No equilibration (always true if FACT = 'N').
*>       = 'R':  Row equilibration, i.e., A has been premultiplied by
*>               diag(R).
*>       = 'C':  Column equilibration, i.e., A has been postmultiplied
*>               by diag(C).
*>       = 'B':  Both row and column equilibration, i.e., A has been
*>               replaced by diag(R) * A * diag(C).
*>     EQUED is an input argument if FACT = 'F'; otherwise, it is an
*>     output argument.
*> \endverbatim
*>
*> \param[in,out] R
*> \verbatim
*>          R is REAL array, dimension (N)
*>     The row scale factors for A.  If EQUED = 'R' or 'B', A is
*>     multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
*>     is not accessed.  R is an input argument if FACT = 'F';
*>     otherwise, R is an output argument.  If FACT = 'F' and
*>     EQUED = 'R' or 'B', each element of R must be positive.
*>     If R is output, each element of R is a power of the radix.
*>     If R is input, each element of R should be a power of the radix
*>     to ensure a reliable solution and error estimates. Scaling by
*>     powers of the radix does not cause rounding errors unless the
*>     result underflows or overflows. Rounding errors during scaling
*>     lead to refining with a matrix that is not equivalent to the
*>     input matrix, producing error estimates that may not be
*>     reliable.
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*>          C is REAL array, dimension (N)
*>     The column scale factors for A.  If EQUED = 'C' or 'B', A is
*>     multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
*>     is not accessed.  C is an input argument if FACT = 'F';
*>     otherwise, C is an output argument.  If FACT = 'F' and
*>     EQUED = 'C' or 'B', each element of C must be positive.
*>     If C is output, each element of C is a power of the radix.
*>     If C is input, each element of C should be a power of the radix
*>     to ensure a reliable solution and error estimates. Scaling by
*>     powers of the radix does not cause rounding errors unless the
*>     result underflows or overflows. Rounding errors during scaling
*>     lead to refining with a matrix that is not equivalent to the
*>     input matrix, producing error estimates that may not be
*>     reliable.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is REAL array, dimension (LDB,NRHS)
*>     On entry, the N-by-NRHS right hand side matrix B.
*>     On exit,
*>     if EQUED = 'N', B is not modified;
*>     if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
*>        diag(R)*B;
*>     if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
*>        overwritten by diag(C)*B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>     The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is REAL array, dimension (LDX,NRHS)
*>     If INFO = 0, the N-by-NRHS solution matrix X to the original
*>     system of equations.  Note that A and B are modified on exit
*>     if EQUED .ne. 'N', and the solution to the equilibrated system is
*>     inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
*>     inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>     The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*>          RCOND is REAL
*>     Reciprocal scaled condition number.  This is an estimate of the
*>     reciprocal Skeel condition number of the matrix A after
*>     equilibration (if done).  If this is less than the machine
*>     precision (in particular, if it is zero), the matrix is singular
*>     to working precision.  Note that the error may still be small even
*>     if this number is very small and the matrix appears ill-
*>     conditioned.
*> \endverbatim
*>
*> \param[out] RPVGRW
*> \verbatim
*>          RPVGRW is REAL
*>     Reciprocal pivot growth.  On exit, this contains the reciprocal
*>     pivot growth factor norm(A)/norm(U). The "max absolute element"
*>     norm is used.  If this is much less than 1, then the stability of
*>     the LU factorization of the (equilibrated) matrix A could be poor.
*>     This also means that the solution X, estimated condition numbers,
*>     and error bounds could be unreliable. If factorization fails with
*>     0<INFO<=N, then this contains the reciprocal pivot growth factor
*>     for the leading INFO columns of A.  In SGESVX, this quantity is
*>     returned in WORK(1).
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*>          BERR is REAL array, dimension (NRHS)
*>     Componentwise relative backward error.  This is the
*>     componentwise relative backward error of each solution vector X(j)
*>     (i.e., the smallest relative change in any element of A or B that
*>     makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[in] N_ERR_BNDS
*> \verbatim
*>          N_ERR_BNDS is INTEGER
*>     Number of error bounds to return for each right hand side
*>     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
*>     ERR_BNDS_COMP below.
*> \endverbatim
*>
*> \param[out] ERR_BNDS_NORM
*> \verbatim
*>          ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
*>     For each right-hand side, this array contains information about
*>     various error bounds and condition numbers corresponding to the
*>     normwise relative error, which is defined as follows:
*>
*>     Normwise relative error in the ith solution vector:
*>             max_j (abs(XTRUE(j,i) - X(j,i)))
*>            ------------------------------
*>                  max_j abs(X(j,i))
*>
*>     The array is indexed by the type of error information as described
*>     below. There currently are up to three pieces of information
*>     returned.
*>
*>     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
*>     right-hand side.
*>
*>     The second index in ERR_BNDS_NORM(:,err) contains the following
*>     three fields:
*>     err = 1 "Trust/don't trust" boolean. Trust the answer if the
*>              reciprocal condition number is less than the threshold
*>              sqrt(n) * slamch('Epsilon').
*>
*>     err = 2 "Guaranteed" error bound: The estimated forward error,
*>              almost certainly within a factor of 10 of the true error
*>              so long as the next entry is greater than the threshold
*>              sqrt(n) * slamch('Epsilon'). This error bound should only
*>              be trusted if the previous boolean is true.
*>
*>     err = 3  Reciprocal condition number: Estimated normwise
*>              reciprocal condition number.  Compared with the threshold
*>              sqrt(n) * slamch('Epsilon') to determine if the error
*>              estimate is "guaranteed". These reciprocal condition
*>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
*>              appropriately scaled matrix Z.
*>              Let Z = S*A, where S scales each row by a power of the
*>              radix so all absolute row sums of Z are approximately 1.
*>
*>     See Lapack Working Note 165 for further details and extra
*>     cautions.
*> \endverbatim
*>
*> \param[out] ERR_BNDS_COMP
*> \verbatim
*>          ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
*>     For each right-hand side, this array contains information about
*>     various error bounds and condition numbers corresponding to the
*>     componentwise relative error, which is defined as follows:
*>
*>     Componentwise relative error in the ith solution vector:
*>                    abs(XTRUE(j,i) - X(j,i))
*>             max_j ----------------------
*>                         abs(X(j,i))
*>
*>     The array is indexed by the right-hand side i (on which the
*>     componentwise relative error depends), and the type of error
*>     information as described below. There currently are up to three
*>     pieces of information returned for each right-hand side. If
*>     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
*>     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS .LT. 3, then at most
*>     the first (:,N_ERR_BNDS) entries are returned.
*>
*>     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
*>     right-hand side.
*>
*>     The second index in ERR_BNDS_COMP(:,err) contains the following
*>     three fields:
*>     err = 1 "Trust/don't trust" boolean. Trust the answer if the
*>              reciprocal condition number is less than the threshold
*>              sqrt(n) * slamch('Epsilon').
*>
*>     err = 2 "Guaranteed" error bound: The estimated forward error,
*>              almost certainly within a factor of 10 of the true error
*>              so long as the next entry is greater than the threshold
*>              sqrt(n) * slamch('Epsilon'). This error bound should only
*>              be trusted if the previous boolean is true.
*>
*>     err = 3  Reciprocal condition number: Estimated componentwise
*>              reciprocal condition number.  Compared with the threshold
*>              sqrt(n) * slamch('Epsilon') to determine if the error
*>              estimate is "guaranteed". These reciprocal condition
*>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
*>              appropriately scaled matrix Z.
*>              Let Z = S*(A*diag(x)), where x is the solution for the
*>              current right-hand side and S scales each row of
*>              A*diag(x) by a power of the radix so all absolute row
*>              sums of Z are approximately 1.
*>
*>     See Lapack Working Note 165 for further details and extra
*>     cautions.
*> \endverbatim
*>
*> \param[in] NPARAMS
*> \verbatim
*>          NPARAMS is INTEGER
*>     Specifies the number of parameters set in PARAMS.  If .LE. 0, the
*>     PARAMS array is never referenced and default values are used.
*> \endverbatim
*>
*> \param[in,out] PARAMS
*> \verbatim
*>          PARAMS is REAL array, dimension NPARAMS
*>     Specifies algorithm parameters.  If an entry is .LT. 0.0, then
*>     that entry will be filled with default value used for that
*>     parameter.  Only positions up to NPARAMS are accessed; defaults
*>     are used for higher-numbered parameters.
*>
*>       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
*>            refinement or not.
*>         Default: 1.0
*>            = 0.0 : No refinement is performed, and no error bounds are
*>                    computed.
*>            = 1.0 : Use the double-precision refinement algorithm,
*>                    possibly with doubled-single computations if the
*>                    compilation environment does not support DOUBLE
*>                    PRECISION.
*>              (other values are reserved for future use)
*>
*>       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
*>            computations allowed for refinement.
*>         Default: 10
*>         Aggressive: Set to 100 to permit convergence using approximate
*>                     factorizations or factorizations other than LU. If
*>                     the factorization uses a technique other than
*>                     Gaussian elimination, the guarantees in
*>                     err_bnds_norm and err_bnds_comp may no longer be
*>                     trustworthy.
*>
*>       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
*>            will attempt to find a solution with small componentwise
*>            relative error in the double-precision algorithm.  Positive
*>            is true, 0.0 is false.
*>         Default: 1.0 (attempt componentwise convergence)
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>       = 0:  Successful exit. The solution to every right-hand side is
*>         guaranteed.
*>       < 0:  If INFO = -i, the i-th argument had an illegal value
*>       > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
*>         has been completed, but the factor U is exactly singular, so
*>         the solution and error bounds could not be computed. RCOND = 0
*>         is returned.
*>       = N+J: The solution corresponding to the Jth right-hand side is
*>         not guaranteed. The solutions corresponding to other right-
*>         hand sides K with K > J may not be guaranteed as well, but
*>         only the first such right-hand side is reported. If a small
*>         componentwise error is not requested (PARAMS(3) = 0.0) then
*>         the Jth right-hand side is the first with a normwise error
*>         bound that is not guaranteed (the smallest J such
*>         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
*>         the Jth right-hand side is the first with either a normwise or
*>         componentwise error bound that is not guaranteed (the smallest
*>         J such that either ERR_BNDS_NORM(J,1) = 0.0 or
*>         ERR_BNDS_COMP(J,1) = 0.0). See the definition of
*>         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
*>         about all of the right-hand sides check ERR_BNDS_NORM or
*>         ERR_BNDS_COMP.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date April 2012
*
*> \ingroup realGEsolve
*
*  =====================================================================
      SUBROUTINE SGESVXX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
     $                    EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW,
     $                    BERR, N_ERR_BNDS, ERR_BNDS_NORM,
     $                    ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK,
     $                    INFO )
*
*  -- LAPACK driver routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     April 2012
*
*     .. Scalar Arguments ..
      CHARACTER          EQUED, FACT, TRANS
      INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
     $                   N_ERR_BNDS
      REAL               RCOND, RPVGRW
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * ), IWORK( * )
      REAL               A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
     $                   X( LDX , * ),WORK( * )
      REAL               R( * ), C( * ), PARAMS( * ), BERR( * ),
     $                   ERR_BNDS_NORM( NRHS, * ),
     $                   ERR_BNDS_COMP( NRHS, * )
*     ..
*
*  ==================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
      INTEGER            FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
      INTEGER            RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
      INTEGER            CMP_ERR_I, PIV_GROWTH_I
      PARAMETER          ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
     $                   BERR_I = 3 )
      PARAMETER          ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
      PARAMETER          ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
     $                   PIV_GROWTH_I = 9 )
*     ..
*     .. Local Scalars ..
      LOGICAL            COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
      INTEGER            INFEQU, J
      REAL               AMAX, BIGNUM, COLCND, RCMAX, RCMIN, ROWCND,
     $                   SMLNUM
*     ..
*     .. External Functions ..
      EXTERNAL           LSAME, SLAMCH, SLA_GERPVGRW
      LOGICAL            LSAME
      REAL               SLAMCH, SLA_GERPVGRW
*     ..
*     .. External Subroutines ..
      EXTERNAL           SGEEQUB, SGETRF, SGETRS, SLACPY, SLAQGE,
     $                   XERBLA, SLASCL2, SGERFSX
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      NOFACT = LSAME( FACT, 'N' )
      EQUIL = LSAME( FACT, 'E' )
      NOTRAN = LSAME( TRANS, 'N' )
      SMLNUM = SLAMCH( 'Safe minimum' )
      BIGNUM = ONE / SMLNUM
      IF( NOFACT .OR. EQUIL ) THEN
         EQUED = 'N'
         ROWEQU = .FALSE.
         COLEQU = .FALSE.
      ELSE
         ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
         COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
      END IF
*
*     Default is failure.  If an input parameter is wrong or
*     factorization fails, make everything look horrible.  Only the
*     pivot growth is set here, the rest is initialized in SGERFSX.
*
      RPVGRW = ZERO
*
*     Test the input parameters.  PARAMS is not tested until SGERFSX.
*
      IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
     $     LSAME( FACT, 'F' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
     $        LSAME( TRANS, 'C' ) ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
         INFO = -8
      ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
     $        ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
         INFO = -10
      ELSE
         IF( ROWEQU ) THEN
            RCMIN = BIGNUM
            RCMAX = ZERO
            DO 10 J = 1, N
               RCMIN = MIN( RCMIN, R( J ) )
               RCMAX = MAX( RCMAX, R( J ) )
 10         CONTINUE
            IF( RCMIN.LE.ZERO ) THEN
               INFO = -11
            ELSE IF( N.GT.0 ) THEN
               ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
            ELSE
               ROWCND = ONE
            END IF
         END IF
         IF( COLEQU .AND. INFO.EQ.0 ) THEN
            RCMIN = BIGNUM
            RCMAX = ZERO
            DO 20 J = 1, N
               RCMIN = MIN( RCMIN, C( J ) )
               RCMAX = MAX( RCMAX, C( J ) )
 20         CONTINUE
            IF( RCMIN.LE.ZERO ) THEN
               INFO = -12
            ELSE IF( N.GT.0 ) THEN
               COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
            ELSE
               COLCND = ONE
            END IF
         END IF
         IF( INFO.EQ.0 ) THEN
            IF( LDB.LT.MAX( 1, N ) ) THEN
               INFO = -14
            ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
               INFO = -16
            END IF
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'SGESVXX', -INFO )
         RETURN
      END IF
*
      IF( EQUIL ) THEN
*
*     Compute row and column scalings to equilibrate the matrix A.
*
         CALL SGEEQUB( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
     $        INFEQU )
         IF( INFEQU.EQ.0 ) THEN
*
*     Equilibrate the matrix.
*
            CALL SLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
     $           EQUED )
            ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
            COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
         END IF
*
*     If the scaling factors are not applied, set them to 1.0.
*
         IF ( .NOT.ROWEQU ) THEN
            DO J = 1, N
               R( J ) = 1.0
            END DO
         END IF
         IF ( .NOT.COLEQU ) THEN
            DO J = 1, N
               C( J ) = 1.0
            END DO
         END IF
      END IF
*
*     Scale the right-hand side.
*
      IF( NOTRAN ) THEN
         IF( ROWEQU ) CALL SLASCL2( N, NRHS, R, B, LDB )
      ELSE
         IF( COLEQU ) CALL SLASCL2( N, NRHS, C, B, LDB )
      END IF
*
      IF( NOFACT .OR. EQUIL ) THEN
*
*        Compute the LU factorization of A.
*
         CALL SLACPY( 'Full', N, N, A, LDA, AF, LDAF )
         CALL SGETRF( N, N, AF, LDAF, IPIV, INFO )
*
*        Return if INFO is non-zero.
*
         IF( INFO.GT.0 ) THEN
*
*           Pivot in column INFO is exactly 0
*           Compute the reciprocal pivot growth factor of the
*           leading rank-deficient INFO columns of A.
*
            RPVGRW = SLA_GERPVGRW( N, INFO, A, LDA, AF, LDAF )
            RETURN
         END IF
      END IF
*
*     Compute the reciprocal pivot growth factor RPVGRW.
*
      RPVGRW = SLA_GERPVGRW( N, N, A, LDA, AF, LDAF )
*
*     Compute the solution matrix X.
*
      CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
      CALL SGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
*
*     Use iterative refinement to improve the computed solution and
*     compute error bounds and backward error estimates for it.
*
      CALL SGERFSX( TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF,
     $     IPIV, R, C, B, LDB, X, LDX, RCOND, BERR,
     $     N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
     $     WORK, IWORK, INFO )
*
*     Scale solutions.
*
      IF ( COLEQU .AND. NOTRAN ) THEN
         CALL SLASCL2 ( N, NRHS, C, X, LDX )
      ELSE IF ( ROWEQU .AND. .NOT.NOTRAN ) THEN
         CALL SLASCL2 ( N, NRHS, R, X, LDX )
      END IF
*
      RETURN
*
*     End of SGESVXX

      END