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
773
774
775
776
777
778
779
780
781
782
783
|
*> \brief \b SGSVJ1 pre-processor for the routine sgesvj, applies Jacobi rotations targeting only particular pivots.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SGSVJ1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgsvj1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgsvj1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgsvj1.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SGSVJ1( JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV,
* EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* REAL EPS, SFMIN, TOL
* INTEGER INFO, LDA, LDV, LWORK, M, MV, N, N1, NSWEEP
* CHARACTER*1 JOBV
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), D( N ), SVA( N ), V( LDV, * ),
* $ WORK( LWORK )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SGSVJ1 is called from SGESVJ as a pre-processor and that is its main
*> purpose. It applies Jacobi rotations in the same way as SGESVJ does, but
*> it targets only particular pivots and it does not check convergence
*> (stopping criterion). Few tunning parameters (marked by [TP]) are
*> available for the implementer.
*>
*> Further Details
*> ~~~~~~~~~~~~~~~
*> SGSVJ1 applies few sweeps of Jacobi rotations in the column space of
*> the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
*> off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
*> block-entries (tiles) of the (1,2) off-diagonal block are marked by the
*> [x]'s in the following scheme:
*>
*> | * * * [x] [x] [x]|
*> | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
*> | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
*> |[x] [x] [x] * * * |
*> |[x] [x] [x] * * * |
*> |[x] [x] [x] * * * |
*>
*> In terms of the columns of A, the first N1 columns are rotated 'against'
*> the remaining N-N1 columns, trying to increase the angle between the
*> corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
*> tiled using quadratic tiles of side KBL. Here, KBL is a tunning parmeter.
*> The number of sweeps is given in NSWEEP and the orthogonality threshold
*> is given in TOL.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBV
*> \verbatim
*> JOBV is CHARACTER*1
*> Specifies whether the output from this procedure is used
*> to compute the matrix V:
*> = 'V': the product of the Jacobi rotations is accumulated
*> by postmulyiplying the N-by-N array V.
*> (See the description of V.)
*> = 'A': the product of the Jacobi rotations is accumulated
*> by postmulyiplying the MV-by-N array V.
*> (See the descriptions of MV and V.)
*> = 'N': the Jacobi rotations are not accumulated.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the input matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the input matrix A.
*> M >= N >= 0.
*> \endverbatim
*>
*> \param[in] N1
*> \verbatim
*> N1 is INTEGER
*> N1 specifies the 2 x 2 block partition, the first N1 columns are
*> rotated 'against' the remaining N-N1 columns of A.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is REAL array, dimension (LDA,N)
*> On entry, M-by-N matrix A, such that A*diag(D) represents
*> the input matrix.
*> On exit,
*> A_onexit * D_onexit represents the input matrix A*diag(D)
*> post-multiplied by a sequence of Jacobi rotations, where the
*> rotation threshold and the total number of sweeps are given in
*> TOL and NSWEEP, respectively.
*> (See the descriptions of N1, D, TOL and NSWEEP.)
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is REAL array, dimension (N)
*> The array D accumulates the scaling factors from the fast scaled
*> Jacobi rotations.
*> On entry, A*diag(D) represents the input matrix.
*> On exit, A_onexit*diag(D_onexit) represents the input matrix
*> post-multiplied by a sequence of Jacobi rotations, where the
*> rotation threshold and the total number of sweeps are given in
*> TOL and NSWEEP, respectively.
*> (See the descriptions of N1, A, TOL and NSWEEP.)
*> \endverbatim
*>
*> \param[in,out] SVA
*> \verbatim
*> SVA is REAL array, dimension (N)
*> On entry, SVA contains the Euclidean norms of the columns of
*> the matrix A*diag(D).
*> On exit, SVA contains the Euclidean norms of the columns of
*> the matrix onexit*diag(D_onexit).
*> \endverbatim
*>
*> \param[in] MV
*> \verbatim
*> MV is INTEGER
*> If JOBV .EQ. 'A', then MV rows of V are post-multipled by a
*> sequence of Jacobi rotations.
*> If JOBV = 'N', then MV is not referenced.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*> V is REAL array, dimension (LDV,N)
*> If JOBV .EQ. 'V' then N rows of V are post-multipled by a
*> sequence of Jacobi rotations.
*> If JOBV .EQ. 'A' then MV rows of V are post-multipled by a
*> sequence of Jacobi rotations.
*> If JOBV = 'N', then V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V, LDV >= 1.
*> If JOBV = 'V', LDV .GE. N.
*> If JOBV = 'A', LDV .GE. MV.
*> \endverbatim
*>
*> \param[in] EPS
*> \verbatim
*> EPS is REAL
*> EPS = SLAMCH('Epsilon')
*> \endverbatim
*>
*> \param[in] SFMIN
*> \verbatim
*> SFMIN is REAL
*> SFMIN = SLAMCH('Safe Minimum')
*> \endverbatim
*>
*> \param[in] TOL
*> \verbatim
*> TOL is REAL
*> TOL is the threshold for Jacobi rotations. For a pair
*> A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
*> applied only if ABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.
*> \endverbatim
*>
*> \param[in] NSWEEP
*> \verbatim
*> NSWEEP is INTEGER
*> NSWEEP is the number of sweeps of Jacobi rotations to be
*> performed.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> LWORK is the dimension of WORK. LWORK .GE. M.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0 : successful exit.
*> < 0 : if INFO = -i, then the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup realOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
*
* =====================================================================
SUBROUTINE SGSVJ1( JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV,
$ EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
REAL EPS, SFMIN, TOL
INTEGER INFO, LDA, LDV, LWORK, M, MV, N, N1, NSWEEP
CHARACTER*1 JOBV
* ..
* .. Array Arguments ..
REAL A( LDA, * ), D( N ), SVA( N ), V( LDV, * ),
$ WORK( LWORK )
* ..
*
* =====================================================================
*
* .. Local Parameters ..
REAL ZERO, HALF, ONE
PARAMETER ( ZERO = 0.0E0, HALF = 0.5E0, ONE = 1.0E0)
* ..
* .. Local Scalars ..
REAL AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
$ BIGTHETA, CS, LARGE, MXAAPQ, MXSINJ, ROOTBIG,
$ ROOTEPS, ROOTSFMIN, ROOTTOL, SMALL, SN, T,
$ TEMP1, THETA, THSIGN
INTEGER BLSKIP, EMPTSW, i, ibr, igl, IERR, IJBLSK,
$ ISWROT, jbc, jgl, KBL, MVL, NOTROT, nblc, nblr,
$ p, PSKIPPED, q, ROWSKIP, SWBAND
LOGICAL APPLV, ROTOK, RSVEC
* ..
* .. Local Arrays ..
REAL FASTR( 5 )
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AMAX1, FLOAT, MIN0, SIGN, SQRT
* ..
* .. External Functions ..
REAL SDOT, SNRM2
INTEGER ISAMAX
LOGICAL LSAME
EXTERNAL ISAMAX, LSAME, SDOT, SNRM2
* ..
* .. External Subroutines ..
EXTERNAL SAXPY, SCOPY, SLASCL, SLASSQ, SROTM, SSWAP
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
APPLV = LSAME( JOBV, 'A' )
RSVEC = LSAME( JOBV, 'V' )
IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
INFO = -3
ELSE IF( N1.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.M ) THEN
INFO = -6
ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN
INFO = -9
ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR.
$ ( APPLV.AND.( LDV.LT.MV ) ) ) THEN
INFO = -11
ELSE IF( TOL.LE.EPS ) THEN
INFO = -14
ELSE IF( NSWEEP.LT.0 ) THEN
INFO = -15
ELSE IF( LWORK.LT.M ) THEN
INFO = -17
ELSE
INFO = 0
END IF
*
* #:(
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGSVJ1', -INFO )
RETURN
END IF
*
IF( RSVEC ) THEN
MVL = N
ELSE IF( APPLV ) THEN
MVL = MV
END IF
RSVEC = RSVEC .OR. APPLV
ROOTEPS = SQRT( EPS )
ROOTSFMIN = SQRT( SFMIN )
SMALL = SFMIN / EPS
BIG = ONE / SFMIN
ROOTBIG = ONE / ROOTSFMIN
LARGE = BIG / SQRT( FLOAT( M*N ) )
BIGTHETA = ONE / ROOTEPS
ROOTTOL = SQRT( TOL )
*
* .. Initialize the right singular vector matrix ..
*
* RSVEC = LSAME( JOBV, 'Y' )
*
EMPTSW = N1*( N-N1 )
NOTROT = 0
FASTR( 1 ) = ZERO
*
* .. Row-cyclic pivot strategy with de Rijk's pivoting ..
*
KBL = MIN0( 8, N )
NBLR = N1 / KBL
IF( ( NBLR*KBL ).NE.N1 )NBLR = NBLR + 1
* .. the tiling is nblr-by-nblc [tiles]
NBLC = ( N-N1 ) / KBL
IF( ( NBLC*KBL ).NE.( N-N1 ) )NBLC = NBLC + 1
BLSKIP = ( KBL**2 ) + 1
*[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
ROWSKIP = MIN0( 5, KBL )
*[TP] ROWSKIP is a tuning parameter.
SWBAND = 0
*[TP] SWBAND is a tuning parameter. It is meaningful and effective
* if SGESVJ is used as a computational routine in the preconditioned
* Jacobi SVD algorithm SGESVJ.
*
*
* | * * * [x] [x] [x]|
* | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
* | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
* |[x] [x] [x] * * * |
* |[x] [x] [x] * * * |
* |[x] [x] [x] * * * |
*
*
DO 1993 i = 1, NSWEEP
* .. go go go ...
*
MXAAPQ = ZERO
MXSINJ = ZERO
ISWROT = 0
*
NOTROT = 0
PSKIPPED = 0
*
DO 2000 ibr = 1, NBLR
igl = ( ibr-1 )*KBL + 1
*
*
*........................................................
* ... go to the off diagonal blocks
igl = ( ibr-1 )*KBL + 1
DO 2010 jbc = 1, NBLC
jgl = N1 + ( jbc-1 )*KBL + 1
* doing the block at ( ibr, jbc )
IJBLSK = 0
DO 2100 p = igl, MIN0( igl+KBL-1, N1 )
AAPP = SVA( p )
IF( AAPP.GT.ZERO ) THEN
PSKIPPED = 0
DO 2200 q = jgl, MIN0( jgl+KBL-1, N )
*
AAQQ = SVA( q )
IF( AAQQ.GT.ZERO ) THEN
AAPP0 = AAPP
*
* .. M x 2 Jacobi SVD ..
*
* .. Safe Gram matrix computation ..
*
IF( AAQQ.GE.ONE ) THEN
IF( AAPP.GE.AAQQ ) THEN
ROTOK = ( SMALL*AAPP ).LE.AAQQ
ELSE
ROTOK = ( SMALL*AAQQ ).LE.AAPP
END IF
IF( AAPP.LT.( BIG / AAQQ ) ) THEN
AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1,
$ q ), 1 )*D( p )*D( q ) / AAQQ )
$ / AAPP
ELSE
CALL SCOPY( M, A( 1, p ), 1, WORK, 1 )
CALL SLASCL( 'G', 0, 0, AAPP, D( p ),
$ M, 1, WORK, LDA, IERR )
AAPQ = SDOT( M, WORK, 1, A( 1, q ),
$ 1 )*D( q ) / AAQQ
END IF
ELSE
IF( AAPP.GE.AAQQ ) THEN
ROTOK = AAPP.LE.( AAQQ / SMALL )
ELSE
ROTOK = AAQQ.LE.( AAPP / SMALL )
END IF
IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1,
$ q ), 1 )*D( p )*D( q ) / AAQQ )
$ / AAPP
ELSE
CALL SCOPY( M, A( 1, q ), 1, WORK, 1 )
CALL SLASCL( 'G', 0, 0, AAQQ, D( q ),
$ M, 1, WORK, LDA, IERR )
AAPQ = SDOT( M, WORK, 1, A( 1, p ),
$ 1 )*D( p ) / AAPP
END IF
END IF
MXAAPQ = AMAX1( MXAAPQ, ABS( AAPQ ) )
* TO rotate or NOT to rotate, THAT is the question ...
*
IF( ABS( AAPQ ).GT.TOL ) THEN
NOTROT = 0
* ROTATED = ROTATED + 1
PSKIPPED = 0
ISWROT = ISWROT + 1
*
IF( ROTOK ) THEN
*
AQOAP = AAQQ / AAPP
APOAQ = AAPP / AAQQ
THETA = -HALF*ABS( AQOAP-APOAQ ) / AAPQ
IF( AAQQ.GT.AAPP0 )THETA = -THETA
IF( ABS( THETA ).GT.BIGTHETA ) THEN
T = HALF / THETA
FASTR( 3 ) = T*D( p ) / D( q )
FASTR( 4 ) = -T*D( q ) / D( p )
CALL SROTM( M, A( 1, p ), 1,
$ A( 1, q ), 1, FASTR )
IF( RSVEC )CALL SROTM( MVL,
$ V( 1, p ), 1,
$ V( 1, q ), 1,
$ FASTR )
SVA( q ) = AAQQ*SQRT( AMAX1( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
AAPP = AAPP*SQRT( AMAX1( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
MXSINJ = AMAX1( MXSINJ, ABS( T ) )
ELSE
*
* .. choose correct signum for THETA and rotate
*
THSIGN = -SIGN( ONE, AAPQ )
IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
T = ONE / ( THETA+THSIGN*
$ SQRT( ONE+THETA*THETA ) )
CS = SQRT( ONE / ( ONE+T*T ) )
SN = T*CS
MXSINJ = AMAX1( MXSINJ, ABS( SN ) )
SVA( q ) = AAQQ*SQRT( AMAX1( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
AAPP = AAPP*SQRT( AMAX1( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
APOAQ = D( p ) / D( q )
AQOAP = D( q ) / D( p )
IF( D( p ).GE.ONE ) THEN
*
IF( D( q ).GE.ONE ) THEN
FASTR( 3 ) = T*APOAQ
FASTR( 4 ) = -T*AQOAP
D( p ) = D( p )*CS
D( q ) = D( q )*CS
CALL SROTM( M, A( 1, p ), 1,
$ A( 1, q ), 1,
$ FASTR )
IF( RSVEC )CALL SROTM( MVL,
$ V( 1, p ), 1, V( 1, q ),
$ 1, FASTR )
ELSE
CALL SAXPY( M, -T*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
CALL SAXPY( M, CS*SN*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
IF( RSVEC ) THEN
CALL SAXPY( MVL, -T*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
CALL SAXPY( MVL,
$ CS*SN*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
END IF
D( p ) = D( p )*CS
D( q ) = D( q ) / CS
END IF
ELSE
IF( D( q ).GE.ONE ) THEN
CALL SAXPY( M, T*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
CALL SAXPY( M, -CS*SN*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
IF( RSVEC ) THEN
CALL SAXPY( MVL, T*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
CALL SAXPY( MVL,
$ -CS*SN*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
END IF
D( p ) = D( p ) / CS
D( q ) = D( q )*CS
ELSE
IF( D( p ).GE.D( q ) ) THEN
CALL SAXPY( M, -T*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
CALL SAXPY( M, CS*SN*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
D( p ) = D( p )*CS
D( q ) = D( q ) / CS
IF( RSVEC ) THEN
CALL SAXPY( MVL,
$ -T*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
CALL SAXPY( MVL,
$ CS*SN*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
END IF
ELSE
CALL SAXPY( M, T*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
CALL SAXPY( M,
$ -CS*SN*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
D( p ) = D( p ) / CS
D( q ) = D( q )*CS
IF( RSVEC ) THEN
CALL SAXPY( MVL,
$ T*APOAQ, V( 1, p ),
$ 1, V( 1, q ), 1 )
CALL SAXPY( MVL,
$ -CS*SN*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
END IF
END IF
END IF
END IF
END IF
ELSE
IF( AAPP.GT.AAQQ ) THEN
CALL SCOPY( M, A( 1, p ), 1, WORK,
$ 1 )
CALL SLASCL( 'G', 0, 0, AAPP, ONE,
$ M, 1, WORK, LDA, IERR )
CALL SLASCL( 'G', 0, 0, AAQQ, ONE,
$ M, 1, A( 1, q ), LDA,
$ IERR )
TEMP1 = -AAPQ*D( p ) / D( q )
CALL SAXPY( M, TEMP1, WORK, 1,
$ A( 1, q ), 1 )
CALL SLASCL( 'G', 0, 0, ONE, AAQQ,
$ M, 1, A( 1, q ), LDA,
$ IERR )
SVA( q ) = AAQQ*SQRT( AMAX1( ZERO,
$ ONE-AAPQ*AAPQ ) )
MXSINJ = AMAX1( MXSINJ, SFMIN )
ELSE
CALL SCOPY( M, A( 1, q ), 1, WORK,
$ 1 )
CALL SLASCL( 'G', 0, 0, AAQQ, ONE,
$ M, 1, WORK, LDA, IERR )
CALL SLASCL( 'G', 0, 0, AAPP, ONE,
$ M, 1, A( 1, p ), LDA,
$ IERR )
TEMP1 = -AAPQ*D( q ) / D( p )
CALL SAXPY( M, TEMP1, WORK, 1,
$ A( 1, p ), 1 )
CALL SLASCL( 'G', 0, 0, ONE, AAPP,
$ M, 1, A( 1, p ), LDA,
$ IERR )
SVA( p ) = AAPP*SQRT( AMAX1( ZERO,
$ ONE-AAPQ*AAPQ ) )
MXSINJ = AMAX1( MXSINJ, SFMIN )
END IF
END IF
* END IF ROTOK THEN ... ELSE
*
* In the case of cancellation in updating SVA(q)
* .. recompute SVA(q)
IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
$ THEN
IF( ( AAQQ.LT.ROOTBIG ) .AND.
$ ( AAQQ.GT.ROOTSFMIN ) ) THEN
SVA( q ) = SNRM2( M, A( 1, q ), 1 )*
$ D( q )
ELSE
T = ZERO
AAQQ = ONE
CALL SLASSQ( M, A( 1, q ), 1, T,
$ AAQQ )
SVA( q ) = T*SQRT( AAQQ )*D( q )
END IF
END IF
IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
IF( ( AAPP.LT.ROOTBIG ) .AND.
$ ( AAPP.GT.ROOTSFMIN ) ) THEN
AAPP = SNRM2( M, A( 1, p ), 1 )*
$ D( p )
ELSE
T = ZERO
AAPP = ONE
CALL SLASSQ( M, A( 1, p ), 1, T,
$ AAPP )
AAPP = T*SQRT( AAPP )*D( p )
END IF
SVA( p ) = AAPP
END IF
* end of OK rotation
ELSE
NOTROT = NOTROT + 1
* SKIPPED = SKIPPED + 1
PSKIPPED = PSKIPPED + 1
IJBLSK = IJBLSK + 1
END IF
ELSE
NOTROT = NOTROT + 1
PSKIPPED = PSKIPPED + 1
IJBLSK = IJBLSK + 1
END IF
* IF ( NOTROT .GE. EMPTSW ) GO TO 2011
IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) )
$ THEN
SVA( p ) = AAPP
NOTROT = 0
GO TO 2011
END IF
IF( ( i.LE.SWBAND ) .AND.
$ ( PSKIPPED.GT.ROWSKIP ) ) THEN
AAPP = -AAPP
NOTROT = 0
GO TO 2203
END IF
*
2200 CONTINUE
* end of the q-loop
2203 CONTINUE
SVA( p ) = AAPP
*
ELSE
IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
$ MIN0( jgl+KBL-1, N ) - jgl + 1
IF( AAPP.LT.ZERO )NOTROT = 0
*** IF ( NOTROT .GE. EMPTSW ) GO TO 2011
END IF
2100 CONTINUE
* end of the p-loop
2010 CONTINUE
* end of the jbc-loop
2011 CONTINUE
*2011 bailed out of the jbc-loop
DO 2012 p = igl, MIN0( igl+KBL-1, N )
SVA( p ) = ABS( SVA( p ) )
2012 CONTINUE
*** IF ( NOTROT .GE. EMPTSW ) GO TO 1994
2000 CONTINUE
*2000 :: end of the ibr-loop
*
* .. update SVA(N)
IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) )
$ THEN
SVA( N ) = SNRM2( M, A( 1, N ), 1 )*D( N )
ELSE
T = ZERO
AAPP = ONE
CALL SLASSQ( M, A( 1, N ), 1, T, AAPP )
SVA( N ) = T*SQRT( AAPP )*D( N )
END IF
*
* Additional steering devices
*
IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR.
$ ( ISWROT.LE.N ) ) )SWBAND = i
IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.FLOAT( N )*TOL ) .AND.
$ ( FLOAT( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN
GO TO 1994
END IF
*
IF( NOTROT.GE.EMPTSW )GO TO 1994
1993 CONTINUE
* end i=1:NSWEEP loop
* #:) Reaching this point means that the procedure has completed the given
* number of sweeps.
INFO = NSWEEP - 1
GO TO 1995
1994 CONTINUE
* #:) Reaching this point means that during the i-th sweep all pivots were
* below the given threshold, causing early exit.
INFO = 0
* #:) INFO = 0 confirms successful iterations.
1995 CONTINUE
*
* Sort the vector D
*
DO 5991 p = 1, N - 1
q = ISAMAX( N-p+1, SVA( p ), 1 ) + p - 1
IF( p.NE.q ) THEN
TEMP1 = SVA( p )
SVA( p ) = SVA( q )
SVA( q ) = TEMP1
TEMP1 = D( p )
D( p ) = D( q )
D( q ) = TEMP1
CALL SSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
IF( RSVEC )CALL SSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
END IF
5991 CONTINUE
*
RETURN
* ..
* .. END OF SGSVJ1
* ..
END
|
