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
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
|
/*
* Copyright (c) 2012, Michael Lehn
*
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
*
* 1) Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2) Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in
* the documentation and/or other materials provided with the
* distribution.
* 3) Neither the name of the FLENS development group nor the names of
* its contributors may be used to endorse or promote products derived
* from this software without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
* "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
* LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
* A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
* OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
* LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
/* Based on
*
SUBROUTINE DGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
$ M, N, A, LDA, SVA, U, LDU, V, LDV,
$ WORK, LWORK, IWORK, INFO )
*
* -- LAPACK routine (version 3.3.1) --
*
* -- Contributed by Zlatko Drmac of the University of Zagreb and --
* -- Kresimir Veselic of the Fernuniversitaet Hagen --
* -- April 2011 --
*
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* This routine is also part of SIGMA (version 1.23, October 23. 2008.)
* SIGMA is a library of algorithms for highly accurate algorithms for
* computation of SVD, PSVD, QSVD, (H,K)-SVD, and for solution of the
* eigenvalue problems Hx = lambda M x, H M x = lambda x with H, M > 0.
*
*/
#ifndef FLENS_LAPACK_GE_JSV_TCC
#define FLENS_LAPACK_GE_JSV_TCC 1
#include <flens/blas/blas.h>
#include <flens/lapack/lapack.h>
namespace flens { namespace lapack {
//== generic lapack implementation =============================================
template <typename MA, typename VSVA, typename MU, typename MV,
typename VWORK, typename VIWORK>
typename GeMatrix<MA>::IndexType
jsv_generic(JSV::Accuracy ,
JSV::JobU ,
JSV::JobV ,
bool ,
bool ,
bool ,
GeMatrix<MA> &,
DenseVector<VSVA> &,
GeMatrix<MU> &,
GeMatrix<MV> &,
DenseVector<VWORK> &,
DenseVector<VIWORK> &)
{
// This implementation needs to be fixed. But the original Fortran code is
// horrible. Unlike any other Lapack function. So it is not fun.
// More likely: We remove all of this!
ASSERT(0);
return -1;
}
/*
template <typename MA, typename VSVA, typename MU, typename MV,
typename VWORK, typename VIWORK>
typename GeMatrix<MA>::IndexType
jsv_generic(JSV::Accuracy accuracy,
JSV::JobU jobU,
JSV::JobV jobV,
bool restrictedRange,
bool considerTransA,
bool perturb,
GeMatrix<MA> &A,
DenseVector<VSVA> &sva,
GeMatrix<MU> &U,
GeMatrix<MV> &V,
DenseVector<VWORK> &work,
DenseVector<VIWORK> &iwork)
{
using std::abs;
using std::max;
using std::min;
using std::sqrt;
typedef typename GeMatrix<MA>::ElementType ElementType;
typedef typename GeMatrix<MA>::IndexType IndexType;
const ElementType Zero(0), One(1);
const Underscore<IndexType> _;
const bool lsvec = (jobU==ComputeU) || (jobU==FullsetU);
const bool jracc = (jobV=='J');
const bool rsvec = (jobV=='V') || jracc;
const bool rowpiv = (jobA=='F') || (jobA=='G');
const bool l2rank = (jobA=='R');
const bool l2aber = (jobA=='A');
const bool errest = (jobA=='E') || (jobA=='G');
const bool l2tran = (jobT=='T');
const bool l2kill = (jobR=='R');
const bool defr = (jobR=='N');
const bool l2pert = (jobP=='P');
IndexType info = 0;
//
// Quick return for void matrix (Y3K safe)
//#:)
if (m==0 || n==0) {
return info;
}
//
// Set numerical parameters
//
//! NOTE: Make sure DLAMCH() does not fail on the target architecture.
//
const ElementType eps = lamch<ElementType>(Eps);
const ElementType safeMin = lamch<ElementType>(SafeMin);
const ElementType small = safeMin / eps;
const ElementType big = lamch<ElementType>(OverflowThreshold);
//
// Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N
//
//! If necessary, scale SVA() to protect the largest norm from
// overflow. It is possible that this scaling pushes the smallest
// column norm left from the underflow threshold (extreme case).
//
ElementType scaleM = One / sqrt(ElementType(m)*ElementType(n));
bool noScale = true;
bool goScale = true;
for (IndexType p=1; p<=n; ++p) {
aapp = Zero;
aaqq = One;
lassq(A(_,p), aapp, aaqq);
if (aapp>big) {
return -9;
}
aaqq = sqrt(aaqq);
if (aapp<(big/aaqq) && noscal) {
sva(p) = aapp * aaqq;
} else {
noscal = false;
sva(p) = aapp * ( aaqq * scalem )
if (goScale) {
goScale = false;
sva(_(1,p)) *= scaleM;
}
}
}
if (noScale) {
scaleM = One;
}
aapp = Zero;
aaqq = big;
for (IndexType p=1; p<=n) {
aapp = max(aapp, sva(p));
if (sva(p)!=Zero) {
aaqq = min(aaqq, sva(p));
}
}
//
// Quick return for zero M x N matrix
//#:)
if (aapp==Zero) {
if (lsvec) {
U = Zero;
U.diag(0) = One;
}
if (rsvec) {
V = Zero;
V.diag(0) = One;
}
work(1) = One;
work(2) = One;
if (errest) {
work(3) = One;
}
if (lsvec && rsvec) {
work(4) = One;
work(5) = One;
}
if (l2tran) {
work(6) = Zero;
work(7) = Zero;
}
iwork(1) = 0;
iwork(2) = 0;
iwork(3) = 0;
return info;
}
//
// Issue warning if denormalized column norms detected. Override the
// high relative accuracy request. Issue licence to kill columns
// (set them to zero) whose norm is less than sigma_max / BIG (roughly).
//#:(
warning = 0
if (aaqq<=SFMIN) {
l2rank = true;
l2kill = true;
warning = 1;
}
//
// Quick return for one-column matrix
//#:)
auto U1 = U(_,_(1,n));
if (n==1) {
if (lsvec) {
lascl(LASCL::FullMatrix, 0, 0, sva(1), scaleM, A);
U1 = A;
// computing all M left singular vectors of the M x 1 matrix
if (nu!=n) {
auto tau = work(_(1,n));
auto work_ = work(_(n+1,lWork));
qrf(U1, tau, work_);
orgqr(1, U, tau, work_);
U1 = A;
}
}
if (rsvec) {
V(1,1) = One
}
if (sva(1)<big*scaleM) {
sva(1) /= scaleM;
scalem = One;
}
work(1) = One / scaleM;
work(2) = One;
if (sva(1)!=Zero) {
iwork(1) = 1;
if (sva(1)/scaleM>=safeMin) {
iwork(2) = 1;
} else {
iwork(2) = 0;
}
} else {
iwork(1) = 0;
iwork(2) = 0;
}
if (errest) {
work(3) = One;
}
if (lsvec && rsvec) {
work(4) = One;
work(5) = One;
}
if (l2tran) {
work(6) = Zero;
work(7) = Zero;
}
return info;
}
bool transp = false;
bool l2tran = l2tran && m==n;
ElementType aatMax = -One;
ElementType aatMin = big;
if (rowpiv || l2tran) {
//
// Compute the row norms, needed to determine row pivoting sequence
// (in the case of heavily row weighted A, row pivoting is strongly
// advised) and to collect information needed to compare the
// structures of A * A^t and A^t * A (in the case L2TRAN.EQ.true).
//
if (l2tran) {
for (IndexType p=1; p<=m; ++p) {
xsc = Zero;
tmp = One;
lassq(A(p,_), xsc, tmp);
// DLASSQ gets both the ell_2 and the ell_infinity norm
// in one pass through the vector
work(m+n+p) = xsc * scaleM;
work(n+p) = xsc * (scaleN*sqrt(tmp));
aatMax = max(aatMax, work(n+p));
if (work(n+p)!=Zero) {
aatMin = min(aatMin, work(n+p));
}
}
} else {
for (IndexType p=1; p<=m; ++p) {
const IndexType jp = blas::iamax(A(p,_));
work(m+n+p) = scaleM*abs(A(p,jp));
aatMax = max(aatMax, work(m+n+p));
aatMin = min(aatMin, work(m+n+p));
}
}
}
//
// For square matrix A try to determine whether A^t would be better
// input for the preconditioned Jacobi SVD, with faster convergence.
// The decision is based on an O(N) function of the vector of column
// and row norms of A, based on the Shannon entropy. This should give
// the right choice in most cases when the difference actually matters.
// It may fail and pick the slower converging side.
//
entra = Zero
entrat = Zero
if (l2tran) {
xsc = Zero;
tmp = One;
lassq(sva, xsc, tmp);
tmp = One / tmp;
entra = Zero
for (IndexType p=1; p<=n; ++p) {
const ElementType big_ = pow(sva(p)/xsc, 2) * tmp;
if (big_!=Zero) {
entra += big_ * log(big_);
}
}
entra = - entra / log(ElementType(n));
//
// Now, SVA().^2/Trace(A^t * A) is a point in the probability simplex.
// It is derived from the diagonal of A^t * A. Do the same with the
// diagonal of A * A^t, compute the entropy of the corresponding
// probability distribution. Note that A * A^t and A^t * A have the
// same trace.
//
entrat = Zero
for (IndexType p=n+1; p<=n+m; ++p) {
const ElementType big_ = pow(work(p)/xsc, 2) * tmp;
if (big_!=Zero) {
entrat += big_ * log(big_);
}
}
entrat = -entrat / log(ElementType(m));
//
// Analyze the entropies and decide A or A^t. Smaller entropy
// usually means better input for the algorithm.
//
transp = entrat<entra;
//
// If A^t is better than A, transpose A.
//
if (transp) {
// In an optimal implementation, this trivial transpose
// should be replaced with faster transpose.
// TODO: in-place transpose:
// transpose(A);
for (IndexType p=1; p<=n-1; ++p) {
for (IndexType q=p+1; q<=n; ++q) {
swap(A(q,p), A(p,q));
}
}
for (IndexType p=1; p<=n; ++p) {
work(m+n+p) = sva(p);
sva(p) = work(n+p);
}
swap(aapp, aatMax);
swap(aaqq, aatMin);
swap(lsvec, rsvec);
if (lsvec) {
// Lehn: transposing A is only considered if A is square,
// so in this case m==n and therefore nu==n. Or am I
// wrong??
ASSERT(nu==n);
}
rowpiv = true;
}
}
//
// Scale the matrix so that its maximal singular value remains less
// than DSQRT(BIG) -- the matrix is scaled so that its maximal column
// has Euclidean norm equal to DSQRT(BIG/N). The only reason to keep
// DSQRT(BIG) instead of BIG is the fact that DGEJSV uses LAPACK and
// BLAS routines that, in some implementations, are not capable of
// working in the full interval [SFMIN,BIG] and that they may provoke
// overflows in the intermediate results. If the singular values spread
// from SFMIN to BIG, then DGESVJ will compute them. So, in that case,
// one should use DGESVJ instead of DGEJSV.
//
const ElementType bigRoot = sqrt(big);
tmp = sqrt(big/ElementType(n));
lascl(LASCL::FullMatrix, 0, 0, aapp, tmp, sva);
if (aaqq>aapp*safeMin) {
aaqq = (aaqq/aapp) * tmp;
} else {
aaqq = (aaqq*tmp) / aapp;
}
tmp *= scaleM;
lascl(LASCL::FullMatrix, 0, 0, aapp, tmp, A);
//
// To undo scaling at the end of this procedure, multiply the
// computed singular values with USCAL2 / USCAL1.
//
uScale1 = tmp;
uScale2 = aapp;
if (l2kill) {
// L2KILL enforces computation of nonzero singular values in
// the restricted range of condition number of the initial A,
// sigma_max(A) / sigma_min(A) approx. DSQRT(BIG)/DSQRT(SFMIN).
xsc = sqrt(safeMin);
} else {
xsc = small;
//
// Now, if the condition number of A is too big,
// sigma_max(A) / sigma_min(A)>DSQRT(BIG/N) * EPSLN / SFMIN,
// as a precaution measure, the full SVD is computed using DGESVJ
// with accumulated Jacobi rotations. This provides numerically
// more robust computation, at the cost of slightly increased run
// time. Depending on the concrete implementation of BLAS and LAPACK
// (i.e. how they behave in presence of extreme ill-conditioning) the
// implementor may decide to remove this switch.
if (aaqq<sqrt(safeMin) && lsvec && rsvec ) {
jracc = true;
}
}
if (aaqq<xsc) {
for (IndexType p=1; p<=n; ++p) {
if (sva(p)<xsc) {
A(_,p) = Zero;
sva(p) = Zero;
}
}
}
//
// Preconditioning using QR factorization with pivoting
//
if (rowpiv) {
// Optional row permutation (Bjoerck row pivoting):
// A result by Cox and Higham shows that the Bjoerck's
// row pivoting combined with standard column pivoting
// has similar effect as Powell-Reid complete pivoting.
// The ell-infinity norms of A are made nonincreasing.
for (IndexType p=1; p<=m-1; ++p) {
const IndexType q = blas::iamax(work(_(m+n+p,2*m+n))) + p - 1;
iwork(2*n+p) = q;
if (p!=q) {
swap(work(m+n+p), work(m+n+q));
}
}
laswp(A, iwork(_(2*n+1, 2*n+m-1)));
}
//
// End of the preparation phase (scaling, optional sorting and
// transposing, optional flushing of small columns).
//
// Preconditioning
//
// If the full SVD is needed, the right singular vectors are computed
// from a matrix equation, and for that we need theoretical analysis
// of the Businger-Golub pivoting. So we use DGEQP3 as the first RR QRF.
// In all other cases the first RR QRF can be chosen by other criteria
// (eg speed by replacing global with restricted window pivoting, such
// as in SGEQPX from TOMS # 782). Good results will be obtained using
// SGEQPX with properly (!) chosen numerical parameters.
// Any improvement of DGEQP3 improves overal performance of DGEJSV.
//
// A * P1 = Q1 * [ R1^t 0]^t:
auto tau_ = work(_(1,n));
auto work_ = work(_(n+1,lWork));
auto iwork_ = iwork(_(1,n));
// .. all columns are free columns
iwork_ = 0;
qp3(A, iwork_, tau_, work_);
//
// The upper triangular matrix R1 from the first QRF is inspected for
// rank deficiency and possibilities for deflation, or possible
// ill-conditioning. Depending on the user specified flag L2RANK,
// the procedure explores possibilities to reduce the numerical
// rank by inspecting the computed upper triangular factor. If
// L2RANK or l2aber are up, then DGEJSV will compute the SVD of
// A + dA, where ||dA|| <= f(M,N)*EPSLN.
//
IndexType nr = 1;
if (l2aber) {
// Standard absolute error bound suffices. All sigma_i with
// sigma_i < N*EPSLN*||A|| are flushed to zero. This is an
// agressive enforcement of lower numerical rank by introducing a
// backward error of the order of N*EPSLN*||A||.
tmp = sqrt(ElementType(n))*eps;
for (IndexType p=2; p<=n; ++p) {
if (abs(A(p,p))>=tmp*abs(A(1,1))) {
++nr;
} else {
break;
}
}
} else if (l2rank) {
// .. similarly as above, only slightly more gentle (less agressive).
// Sudden drop on the diagonal of R1 is used as the criterion for
// close-to-rank-defficient.
tmp = sqrt(safeMin);
for (IndexType p=2; p<=n; ++p) {
if (abs(A(p,p))<eps*abs(A(p-1,p-1)) || abs(A(p,p))<small
|| (l2kill && abs(A(p,p))<tmp))
{
break;
}
++nr;
}
} else {
// The goal is high relative accuracy. However, if the matrix
// has high scaled condition number the relative accuracy is in
// general not feasible. Later on, a condition number estimator
// will be deployed to estimate the scaled condition number.
// Here we just remove the underflowed part of the triangular
// factor. This prevents the situation in which the code is
// working hard to get the accuracy not warranted by the data.
tmp = sqrt(safeMin);
for (IndexType p=2; p<=n; ++p) {
if (abs(A(p,p))<small || (l2kill && abs(A(p,p))<tmp)) {
break;
}
++nr;
}
}
bool almort = false;
if (nr==n) {
ElementType maxprj = One;
for (IndexType p=2; p<=n; ++p) {
maxprj = min(maxprj, abs(A(p,p))/sva(iwork(p)));
}
if (pow(maxprj,2)>=One-ElementType(n)*eps) {
almort = true;
}
}
sconda = -One;
condr1 = -One;
condr2 = -One;
if (errest) {
if (n==nr) {
if (rsvec) {
// .. V is available as workspace
V.upper() = A(_(1,n),_).upper();
for (IndexType p=1; p<=n; ++p) {
tmp = sva(iwork(p));
V(_(1,p),p) *= One/tmp;
}
auto work_ = work(_(n+1, 4*n));
auto iwork_ = iwork(_(2*n+m+1, 3*n+m));
pocon(V.upper(), One, tmp, work_, iwork_);
} else if (lsvec) {
// .. U is available as workspace
auto U_ = U(_(1,n),_(1,n));
U_.upper() = A(_(1,n),_).upper();
for (IndexType p=1; p<=n; ++p) {
tmp = sva(iwork(p));
U(_(1,p),p) *= One/tmp;
}
auto work_ = work(_(n+1, 4*n));
auto iwork_ = iwork(_(2*n+m+1, 3*n+m));
pocon(U_.upper(), One, tmp, work_, iwork_);
} else {
auto work1_ = work(_(n+1,n+n*n));
auto work2_ = work(_(n+n*n+1,n+n*n+3*n));
auto iwork_ = work(_(2*n+m+1,3*n+m));
GeMatrixView<ElementType> Work(n, n, work1_, n);
Work.upper() = A(_(1,n),_).upper();
for (IndexType p=1; p<=n; ++p) {
tmp = sva(iwork(p));
Work(_(1,p),p) *= One/tmp;
}
// .. the columns of R are scaled to have unit Euclidean lengths.
pocon(Work.upper(), One, tmp, work2_, iwork_);
}
sconda = One/sqrt(tmp);
// SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1).
// N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
} else {
sconda = -One;
}
}
l2pert = l2pert && abs(A(1,1)/A(nr,nr))>sqrt(bigRoot);
// If there is no violent scaling, artificial perturbation is not needed.
//
// Phase 3:
//
if (!(rsvec || lsvec)) {
//
// Singular Values only
//
// .. transpose A(1:NR,1:N)
for (IndexType p=1; p<=min(n-1, nr); ++p) {
A(p,_(p+1,n)) = A(_(p+1,n),p);
}
//
// The following two DO-loops introduce small relative perturbation
// into the strict upper triangle of the lower triangular matrix.
// Small entries below the main diagonal are also changed.
// This modification is useful if the computing environment does not
// provide/allow FLUSH TO Zero underflow, for it prevents many
// annoying denormalized numbers in case of strongly scaled matrices.
// The perturbation is structured so that it does not introduce any
// new perturbation of the singular values, and it does not destroy
// the job done by the preconditioner.
// The licence for this perturbation is in the variable L2PERT, which
// should be false if FLUSH TO Zero underflow is active.
//
if (! almort) {
if (l2pert) {
// XSC = DSQRT(SMALL)
xsc = eps / ElementType(n);
for (IndexType q=1; q<=nr; ++q) {
tmp = xsc*abs(A(q,q));
for (IndexType p=1; p<=n, ++p) {
if ((p>q && abs(A(p,q))<=tmp) || p<q) {
A(p,q) = sign(tmp, A(p,q));
}
}
}
} else {
A(_(1,nr),_(1,nr)).strictUpper() = Zero;
}
//
// .. second preconditioning using the QR factorization
//
auto A_ = A(_(1,n),_(1,nr));
auto tau_ = work(_(1,nr));
auto work_ = work(_(n+1,lWork));
qrf(A_, tau_, work_);
//
// .. and transpose upper to lower triangular
for (IndexType p=1; p<=nr-1; ++p) {
A(p,_(p+1,nr)) = A(_(p+1,nr),p);
}
}
//
// Row-cyclic Jacobi SVD algorithm with column pivoting
//
// .. again some perturbation (a "background noise") is added
// to drown denormals
if (l2pert) {
// XSC = DSQRT(SMALL)
xsc = eps / ElementType(n);
for (IndexType q=1; q<=nr; ++q) {
ElementType tmp = xsc*abs(A(q,q));
for (IndexType p=1; p<=nr; ++p) {
if (((p>q) && abs(A(p,q))<=tmp) || p<q) {
A(p,q) = sign(tmp, A(p,q));
}
}
}
} else {
A(_(1,nr),_(1,nr)).strictUpper() = Zero;
}
//
// .. and one-sided Jacobi rotations are started on a lower
// triangular matrix (plus perturbation which is ignored in
// the part which destroys triangular form (confusing?!))
//
auto A_ = A(_(1,nr),_(1,nr));
auto sva_ = sva(_(1,nr));
svj(SVJ::Lower, SVJ::NoU, SVJ::NoV, A_, sva_, V, work);
scaleM = work(1);
numRank = nint(work(2));
} else if (rsvec && !lsvec) {
//
// -> Singular Values and Right Singular Vectors <-
//
if (almort) {
//
// .. in this case NR equals N
ASSERT(nr==n);
for (IndexType p=1; p<=nr; ++p) {
V(_(p,n),p) = A(p,_(p,n));
}
auto V_ = V(_,_(1,nr));
V_.strictUpper() = Zero;
svj(SVJ::Lower, SVJ::ComputeU, SVJ::NoV, V_, sva, A, work);
scaleM = work(1);
numRank = nint(work(2));
} else {
//
// .. two more QR factorizations ( one QRF is not enough, two require
// accumulated product of Jacobi rotations, three are perfect )
//
A(_(1,nr),_(1,nr)).strictLower() = Zero;
auto A_ = A(_(1,nr),_);
auto tau1_ = work(_(1,nr));
auto work1_ = work(_(n+1,lWork));
lqf(A_, tau1_, work1_);
auto V_ = V(_(1,nr),_(1,nr))
V_.lower() = A_(_,_(1,nr)).lower();
V_.strictUpper() = Zero;
auto tau2_ = work(_(n+1,n+nr));
auto work2_ = work(_(2*n+1,lWork));
qrf(V_, tau2_, work2_);
for (IndexType p=1; p<=nr; ++p) {
V(_(p,nr),p) = V(p,_(p,nr));
}
V_.strictUpper() = Zero;
auto sva_ = sva(_(1,nr));
svj(SVJ::Lower, SVJ::ComputeU, SVJ::NoV, V_, sva_, U, work1_);
scaleM = work(n+1);
numRank = nint(work(n+2));
if (nr<n) {
V(_(nr+1,n),_(1,nr)) = Zero;
V(_(1,nr),_(nr+1,n)) = Zero;
V(_(nr+1,n),_(nr+1,n)) = Zero;
V(_(nr+1,n),_(nr+1,n)).diag(0) = One;
}
ormlq(Left, Trans, A_, tau1_, V, work1_);
}
for (IndexType p=1; p<=n; ++p) {
A(iwork(p),_) = V(p,_);
}
V = A;
if (transp) {
U = V;
}
} else if (lsvec && !rsvec) {
//
// .. Singular Values and Left Singular Vectors ..
//
// .. second preconditioning step to avoid need to accumulate
// Jacobi rotations in the Jacobi iterations.
auto U_ = U(_(1,nr),_(1,nr));
auto tau_ = work(_(n+1,n+nr));
auto sva_ = sva(_(1,nr));
auto work1_ = work(_(n+1,lWork));
auto work2_ = work(_(2*n+1,lWork));
for (IndexType p=1; p<=nr; ++p) {
A(p,_(p,nr)) = U(_(p,nr),p);
}
U_.strictUpper() = Zero;
qrf(U(_(1,n),_(1,nr)), tau_, work2_);
for (IndexType p=1; p<=nr-1; ++p) {
U(p,_(p+1,nr)) = U(_(p+1,nr),p);
}
U_.strictUpper() = Zero;
svj(SVJ::Lower, SVJ::ComputeU, SVJ::NoV, U_, sva_, A, work1_);
scaleM = work(n+1);
numRank = nint(work(n+2));
if (nr<m) {
U(_(nr+1,m),_(1,nr)) = Zero;
if (nr<nu) {
U(_(1,nr),_(nr+1,nu)) = Zero;
U(_(nr+1,m),_(nr+1,nu)) = Zero;
U(_(nr+1,m),_(nr+1,nu)).diag(0) = One;
}
}
ormqr(Left, NoTrans, A, work, U);
if (rowpiv) {
auto piv = iwork(_(2*n+1,2*n+m-1));
laswp(U, piv.reverse());
}
for (IndexType p=1; p<=nu; ++p) {
xsc = One / blas::nrm2(U(_,p));
U(_,p) *= xsc;
}
if (transp) {
V = U;
}
//
} else {
//
// .. Full SVD ..
//
if (!jracc) {
if (!almort) {
//
// Second Preconditioning Step (QRF [with pivoting])
// Note that the composition of TRANSPOSE, QRF and TRANSPOSE is
// equivalent to an LQF CALL. Since in many libraries the QRF
// seems to be better optimized than the LQF, we do explicit
// transpose and use the QRF. This is subject to changes in an
// optimized implementation of DGEJSV.
//
for (IndexType p=1; p<=nr; ++p) {
V(_(p,n),p) = A(p,_(p,n));
}
//
// .. the following two loops perturb small entries to avoid
// denormals in the second QR factorization, where they are
// as good as zeros. This is done to avoid painfully slow
// computation with denormals. The relative size of the
// perturbation is a parameter that can be changed by the
// implementer. This perturbation device will be obsolete on
// machines with properly implemented arithmetic.
// To switch it off, set L2PERT=false To remove it from the
// code, remove the action under L2PERT=true, leave the ELSE
// part. The following two loops should be blocked and fused with
// the transposed copy above.
//
if (l2pert) {
xsc = sqrt(small);
for (IndexType q=1; q<=nr; ++q) {
tmp = xsc*abs(V(q,q));
for (IndexType p=1; p<=n; ++p) {
if (p>q && abs(V(p,q))<=tmp || p<q) {
V(p,q) = sign(tmp, V(p,q));
}
if (p<q) {
V(p,q) = -V(p,q);
}
}
}
} else {
V(_(1,nr),_(1,nr)).strictUpper() = Zero;
}
//
// Estimate the row scaled condition number of R1
// (If R1 is rectangular, N > NR, then the condition number
// of the leading NR x NR submatrix is estimated.)
//
auto work1_ = work(_(2*n+1,2*n+nr*nr));
auto work2_ = work(_(2*n+nr*nr+1, 2*n+nr*nr+3*nr));
auto iwork_ = iwork(_(m+2*n+1,m+2*n+nr));
GeMatrixView<ElementType> Work(nr, nr, work1_, nr);
Work.lower() = V.lower();
for (IndexType p=1; p<=nr; ++p) {
tmp = blas::nrm2(Work(_(p,nr),p));
Work(_(p,nr),p) *= One/tmp;
}
pocon(Work, One, tmp, work2_, iwork_);
condr1 = One / sqrt(tmp);
// .. here need a second oppinion on the condition number
// .. then assume worst case scenario
// R1 is OK for inverse <=> condr1 < DBLE(N)
// more conservative <=> condr1 < DSQRT(DBLE(N))
//
cond_ok = sqrt(ElementType(nr));
//[TP] COND_OK is a tuning parameter.
if (condr1<cond_ok) {
// .. the second QRF without pivoting. Note: in an optimized
// implementation, this QRF should be implemented as the QRF
// of a lower triangular matrix.
// R1^t = Q2 * R2
auto tau = work(_(n+1,n+nr));
auto work_ = work(_(2*n+1,lWork));
qrf(V(_,_(1,nr)), tau, work_);
if (l2pert) {
xsc = sqrt(small) / Eps;
for (IndexType p=2; p<=nr; ++p) {
for (IndexType q=1; q<=p-1; ++q) {
tmp = xsc*min(abs(V(p,p)), abs(V(q,q)));
if (abs(V(q,p))<=tmp) {
V(q,p) = sign(tmp, V(q,p));
}
}
}
}
//
if (nr!=n) {
auto work_ = work(_(2*n+1, 2*n+n*nr));
GeMatrixView<ElementType> Work(n, nr, work_, n);
Work = V(_,_(1,nr));
}
// .. save ...
//
// .. this transposed copy should be better than naive
// TODO: auto V_ = V(_(1,nr),_(1,nr));
// V_.lower() = transpose(V_.upper());
//
for (IndexType p=1; p<=nr-1; ++p) {
V(_(p+1,nr),p) = V(p,_(p+1,nr));
}
condr2 = condr1;
} else {
//
// .. ill-conditioned case: second QRF with pivoting
// Note that windowed pivoting would be equaly good
// numerically, and more run-time efficient. So, in
// an optimal implementation, the next call to DGEQP3
// should be replaced with eg. CALL SGEQPX (ACM TOMS #782)
// with properly (carefully) chosen parameters.
//
// R1^t * P2 = Q2 * R2
auto V_ = V(_,_(1,nr));
auto piv = iwork(_(n+1,n+nr));
auto tau = work(_(n+1,,n+nr));
auto work_ = work(_(2*n+1, lWork));
piv = 0;
qp3(V_, piv, tau, work_);
if (l2pert) {
xsc = sqrt(small);
for (IndexType p=2; p<=nr; ++p) {
for (IndexType q=1; q<=p-1; ++q) {
tmp = xsc*min(abs(V(p,p)), abs(V(q,q)));
if (abs(V(q,p))<=tmp) {
V(q,p) = sign(tmp, V(q,p));
}
}
}
}
auto work1_ = work(_(2*n+1, 2*n+n*nr));
GeMatrixView<ElementType> Work1(n, nr, work1_, n);
Work1 = V(_,_(1,nr));
if (l2pert) {
xsc = sqrt(small);
for (IndexType p=2; p<=nr; ++p) {
for (IndexType q=1; q<=p-1; ++q) {
tmp = xsc * min(abs(V(p,p)), abs(V(q,q)));
V(p,q) = -sign(tmp, V(q,p));
}
}
} else {
V(_(1,nr),_(1,nr)).strictLower() = Zero;
}
// Now, compute R2 = L3 * Q3, the LQ factorization.
auto V_ = V(_(1,nr),_(1,nr));
auto tau = work(_(2*n+n*nr+1,2*n+n*nr+nr));
auto work2_ = work(_(2*n+n*nr+nr+1,lWork));
lqf(V_, tau, work2_);
// .. and estimate the condition number
auto work3_ = work(_(2*n+n*nr+nr+1,2*n+n*nr+nr+nr*nr));
GeMatrixView<ElementType> Work3(nr, nr, work3_, nr);
Work3.lower() = V_.lower();
for (IndexType p=1; p<=nr; ++p) {
tmp = blas::nrm2(Work3(p,_(1,p)));
Work3(p,_(1,p)) *= One/tmp;
}
auto work4_ = work(_(2*n+n*nr+nr+nr*nr+1,
2*n+n*nr+nr+nr*nr+3*nr));
auto iwork_ = iwork(_(m+2*n+1, m+2*n+nr));
pocon(Work3.lower(), One, tmp, work4_, iwork_);
condr2 = One / sqrt(tmp);
if (condr2>=cond_ok) {
// .. save the Householder vectors used for Q3
// (this overwrittes the copy of R2, as it will not be
// needed in this branch, but it does not overwritte the
// Huseholder vectors of Q2.).
Work1(_(1,nr),_(1,nr)).upper() = V.upper();
// .. and the rest of the information on Q3 is in
// WORK(2*N+N*NR+1:2*N+N*NR+N)
}
}
if (l2pert) {
xsc = sqrt(small);
for (IndexType q=2; q<=nr; ++q) {
tmp = xsc * V(q,q);
for (IndexType p=1; p<=q-1; ++p) {
// V(p,q) = - DSIGN( TEMP1, V(q,p) )
V(p,q) = -sign(tmp, V(p,q));
}
}
} else {
V(_(1,nr),_(1,nr)).strictLower();
}
//
// Second preconditioning finished; continue with Jacobi SVD
// The input matrix is lower triangular.
//
// Recover the right singular vectors as solution of a well
// conditioned triangular matrix equation.
//
if (condr1<cond_ok) {
auto U_ = U(_(1,nr),_(1,nr));
auto V_ = V(_(1,nr),_(1,nr));
auto sva_ = sva(_(1,nr));
auto work_ = work(_(2*n+n*nr+nr+1,lWork));
svj(SVJ::Lower, SVJ::ComputeU, SVJ::NoV,
V_, sva_, U, work_);
scaleM = work_(1);
numRank = nint(work_(2));
for (IndexType p=1; p<=nr; ++p) {
U_(_,p) = V_(_,p);
V_(_,p) *= sva(p);
}
// .. pick the right matrix equation and solve it
//
if (nr==n) {
//:)) .. best case, R1 is inverted. The solution of this
// matrix equation is Q2*V2 = the product of the Jacobi
// rotations used in DGESVJ, premultiplied with the
// orthogonal matrix from the second QR factorization.
const auto A_ = A(_(1,nr),_(1,nr));
blas::sm(Left, NoTrans, One, A_.upper(), V_);
} else {
// .. R1 is well conditioned, but non-square. Transpose(R2)
// is inverted to get the product of the Jacobi rotations
// used in DGESVJ. The Q-factor from the second QR
// factorization is then built in explicitly.
auto work_ = work(_(2*n+1, 2*n+n*nr));
GeMatrixView<ElementType> Work1(nr, nr, work_, n);
GeMatrixView<ElementType> Work(n, nr, work_, n);
blas::sm(Left, NoTrans, One, Work1.upper(), V_);
if (nr<n) {
V(_(nr+1,n),_(1,nr)) = Zero;
V(_(1,nr),_(nr+1,n)) = Zero;
V(_(nr+1,n),_(nr+1,n)) = Zero;
V(_(nr+1,n),_(nr+1,n)).diag(0) = One;
}
auto tau = work(_(n+1,n+nr));
auto work_ormqr_ = work(_(2*n+n*nr+nr+1,lWork));
ormqr(Left, NoTrans, Work, tau, V, work_ormqr_);
}
//
} else if (condr2<cond_ok) {
//
//:) .. the input matrix A is very likely a relative of
// the Kahan matrix :)
// The matrix R2 is inverted. The solution of the matrix
// equation is Q3^T*V3 = the product of the Jacobi rotations
// (appplied to the lower triangular L3 from the LQ
// factorization of R2=L3*Q3), pre-multiplied with the
// transposed Q3.
CALL DGESVJ( 'L', 'U', 'N', NR, NR, V, LDV, SVA, NR, U,
$ LDU, work(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )
SCALEM = work(2*N+N*NR+NR+1)
numRank = IDNINT(work(2*N+N*NR+NR+2))
DO 3870 p = 1, NR
CALL DCOPY( NR, V(1,p), 1, U(1,p), 1 )
CALL DSCAL( NR, SVA(p), U(1,p), 1 )
3870 CONTINUE
CALL DTRSM('L','U','N','N',NR,NR,One,work(2*N+1),N,U,LDU)
// .. apply the permutation from the second QR factorization
DO 873 q = 1, NR
DO 872 p = 1, NR
work(2*N+N*NR+NR+iwork(N+p)) = U(p,q)
872 CONTINUE
DO 874 p = 1, NR
U(p,q) = work(2*N+N*NR+NR+p)
874 CONTINUE
873 CONTINUE
if ( NR < N ) {
CALL DLASET( 'A',N-NR,NR,Zero,Zero,V(NR+1,1),LDV )
CALL DLASET( 'A',NR,N-NR,Zero,Zero,V(1,NR+1),LDV )
CALL DLASET( 'A',N-NR,N-NR,Zero,One,V(NR+1,NR+1),LDV )
}
CALL DORMQR( 'L','N',N,N,NR,work(2*N+1),N,work(N+1),
$ V,LDV,work(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
} else {
// Last line of defense.
//#:( This is a rather pathological case: no scaled condition
// improvement after two pivoted QR factorizations. Other
// possibility is that the rank revealing QR factorization
// or the condition estimator has failed, or the COND_OK
// is set very close to One (which is unnecessary). Normally,
// this branch should never be executed, but in rare cases of
// failure of the RRQR or condition estimator, the last line of
// defense ensures that DGEJSV completes the task.
// Compute the full SVD of L3 using DGESVJ with explicit
// accumulation of Jacobi rotations.
CALL DGESVJ( 'L', 'U', 'V', NR, NR, V, LDV, SVA, NR, U,
$ LDU, work(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )
SCALEM = work(2*N+N*NR+NR+1)
numRank = IDNINT(work(2*N+N*NR+NR+2))
if ( NR < N ) {
CALL DLASET( 'A',N-NR,NR,Zero,Zero,V(NR+1,1),LDV )
CALL DLASET( 'A',NR,N-NR,Zero,Zero,V(1,NR+1),LDV )
CALL DLASET( 'A',N-NR,N-NR,Zero,One,V(NR+1,NR+1),LDV )
}
CALL DORMQR( 'L','N',N,N,NR,work(2*N+1),N,work(N+1),
$ V,LDV,work(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
//
CALL DORMLQ( 'L', 'T', NR, NR, NR, work(2*N+1), N,
$ work(2*N+N*NR+1), U, LDU, work(2*N+N*NR+NR+1),
$ LWORK-2*N-N*NR-NR, IERR )
DO 773 q = 1, NR
DO 772 p = 1, NR
work(2*N+N*NR+NR+iwork(N+p)) = U(p,q)
772 CONTINUE
DO 774 p = 1, NR
U(p,q) = work(2*N+N*NR+NR+p)
774 CONTINUE
773 CONTINUE
//
}
//
// Permute the rows of V using the (column) permutation from the
// first QRF. Also, scale the columns to make them unit in
// Euclidean norm. This applies to all cases.
//
TEMP1 = DSQRT(DBLE(N)) * EPSLN
DO 1972 q = 1, N
DO 972 p = 1, N
work(2*N+N*NR+NR+iwork(p)) = V(p,q)
972 CONTINUE
DO 973 p = 1, N
V(p,q) = work(2*N+N*NR+NR+p)
973 CONTINUE
XSC = One / DNRM2( N, V(1,q), 1 )
if ( (XSC < (One-TEMP1)) || (XSC>(One+TEMP1)) )
$ CALL DSCAL( N, XSC, V(1,q), 1 )
1972 CONTINUE
// At this moment, V contains the right singular vectors of A.
// Next, assemble the left singular vector matrix U (M x N).
if ( NR < M ) {
CALL DLASET( 'A', M-NR, NR, Zero, Zero, U(NR+1,1), LDU )
if ( NR < N1 ) {
CALL DLASET('A',NR,N1-NR,Zero,Zero,U(1,NR+1),LDU)
CALL DLASET('A',M-NR,N1-NR,Zero,One,U(NR+1,NR+1),LDU)
}
}
//
// The Q matrix from the first QRF is built into the left singular
// matrix U. This applies to all cases.
//
CALL DORMQR( 'Left', 'No_Tr', M, N1, N, A, LDA, work, U,
$ LDU, work(N+1), LWORK-N, IERR )
// The columns of U are normalized. The cost is O(M*N) flops.
TEMP1 = DSQRT(DBLE(M)) * EPSLN
DO 1973 p = 1, NR
XSC = One / DNRM2( M, U(1,p), 1 )
if ( (XSC < (One-TEMP1)) || (XSC>(One+TEMP1)) )
$ CALL DSCAL( M, XSC, U(1,p), 1 )
1973 CONTINUE
//
// If the initial QRF is computed with row pivoting, the left
// singular vectors must be adjusted.
//
if ( ROWPIV )
$ CALL DLASWP( N1, U, LDU, 1, M-1, iwork(2*N+1), -1 )
//
} else {
//
// .. the initial matrix A has almost orthogonal columns and
// the second QRF is not needed
//
CALL DLACPY( 'Upper', N, N, A, LDA, work(N+1), N )
if ( L2PERT ) {
XSC = DSQRT(SMALL)
DO 5970 p = 2, N
TEMP1 = XSC * work( N + (p-1)*N + p )
DO 5971 q = 1, p - 1
work(N+(q-1)*N+p)=-DSIGN(TEMP1,work(N+(p-1)*N+q))
5971 CONTINUE
5970 CONTINUE
} else {
CALL DLASET( 'Lower',N-1,N-1,Zero,Zero,work(N+2),N )
}
//
CALL DGESVJ( 'Upper', 'U', 'N', N, N, work(N+1), N, SVA,
$ N, U, LDU, work(N+N*N+1), LWORK-N-N*N, INFO )
//
SCALEM = work(N+N*N+1)
numRank = IDNINT(work(N+N*N+2))
DO 6970 p = 1, N
CALL DCOPY( N, work(N+(p-1)*N+1), 1, U(1,p), 1 )
CALL DSCAL( N, SVA(p), work(N+(p-1)*N+1), 1 )
6970 CONTINUE
//
CALL DTRSM( 'Left', 'Upper', 'NoTrans', 'No UD', N, N,
$ One, A, LDA, work(N+1), N )
DO 6972 p = 1, N
CALL DCOPY( N, work(N+p), N, V(iwork(p),1), LDV )
6972 CONTINUE
TEMP1 = DSQRT(DBLE(N))*EPSLN
DO 6971 p = 1, N
XSC = One / DNRM2( N, V(1,p), 1 )
if ( (XSC < (One-TEMP1)) || (XSC>(One+TEMP1)) )
$ CALL DSCAL( N, XSC, V(1,p), 1 )
6971 CONTINUE
//
// Assemble the left singular vector matrix U (M x N).
//
if ( N < M ) {
CALL DLASET( 'A', M-N, N, Zero, Zero, U(N+1,1), LDU )
if ( N < N1 ) {
CALL DLASET( 'A',N, N1-N, Zero, Zero, U(1,N+1),LDU )
CALL DLASET( 'A',M-N,N1-N, Zero, One,U(N+1,N+1),LDU )
}
}
CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, work, U,
$ LDU, work(N+1), LWORK-N, IERR )
TEMP1 = DSQRT(DBLE(M))*EPSLN
DO 6973 p = 1, N1
XSC = One / DNRM2( M, U(1,p), 1 )
if ( (XSC < (One-TEMP1)) || (XSC>(One+TEMP1)) )
$ CALL DSCAL( M, XSC, U(1,p), 1 )
6973 CONTINUE
//
if ( ROWPIV )
$ CALL DLASWP( N1, U, LDU, 1, M-1, iwork(2*N+1), -1 )
//
}
//
// end of the >> almost orthogonal case << in the full SVD
//
} else {
//
// This branch deploys a preconditioned Jacobi SVD with explicitly
// accumulated rotations. It is included as optional, mainly for
// experimental purposes. It does perfom well, and can also be used.
// In this implementation, this branch will be automatically activated
// if the condition number sigma_max(A) / sigma_min(A) is predicted
// to be greater than the overflow threshold. This is because the
// a posteriori computation of the singular vectors assumes robust
// implementation of BLAS and some LAPACK procedures, capable of
// working in presence of extreme values. Since that is not always
// the case, ...
//
DO 7968 p = 1, NR
CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
7968 CONTINUE
//
if ( L2PERT ) {
XSC = DSQRT(SMALL/EPSLN)
for (IndexType q=1; q<=nr; ++q) {
tmp = xsc*abs(V(q,q));
for (IndexType p=1; p<=n; ++p) {
if (p>q && abs(V(p,q))<=tmp || p<q) {
V(p,q) = sign(tmp, V(p,q));
}
if (p<q) {
V(p,q) = - V(p,q);
}
}
}
} else {
CALL DLASET( 'U', NR-1, NR-1, Zero, Zero, V(1,2), LDV )
}
CALL DGEQRF( N, NR, V, LDV, work(N+1), work(2*N+1),
$ LWORK-2*N, IERR )
CALL DLACPY( 'L', N, NR, V, LDV, work(2*N+1), N )
//
DO 7969 p = 1, NR
CALL DCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 )
7969 CONTINUE
if ( L2PERT ) {
XSC = DSQRT(SMALL/EPSLN)
DO 9970 q = 2, NR
DO 9971 p = 1, q - 1
TEMP1 = XSC * DMIN1(DABS(U(p,p)),DABS(U(q,q)))
U(p,q) = - DSIGN( TEMP1, U(q,p) )
9971 CONTINUE
9970 CONTINUE
} else {
CALL DLASET('U', NR-1, NR-1, Zero, Zero, U(1,2), LDU )
}
CALL DGESVJ( 'G', 'U', 'V', NR, NR, U, LDU, SVA,
$ N, V, LDV, work(2*N+N*NR+1), LWORK-2*N-N*NR, INFO )
SCALEM = work(2*N+N*NR+1)
numRank = IDNINT(work(2*N+N*NR+2))
if ( NR < N ) {
CALL DLASET( 'A',N-NR,NR,Zero,Zero,V(NR+1,1),LDV )
CALL DLASET( 'A',NR,N-NR,Zero,Zero,V(1,NR+1),LDV )
CALL DLASET( 'A',N-NR,N-NR,Zero,One,V(NR+1,NR+1),LDV )
}
CALL DORMQR( 'L','N',N,N,NR,work(2*N+1),N,work(N+1),
$ V,LDV,work(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
//
// Permute the rows of V using the (column) permutation from the
// first QRF. Also, scale the columns to make them unit in
// Euclidean norm. This applies to all cases.
//
TEMP1 = DSQRT(DBLE(N)) * EPSLN
DO 7972 q = 1, N
DO 8972 p = 1, N
work(2*N+N*NR+NR+iwork(p)) = V(p,q)
8972 CONTINUE
DO 8973 p = 1, N
V(p,q) = work(2*N+N*NR+NR+p)
8973 CONTINUE
XSC = One / DNRM2( N, V(1,q), 1 )
if ( (XSC < (One-TEMP1)) || (XSC>(One+TEMP1)) )
$ CALL DSCAL( N, XSC, V(1,q), 1 )
7972 CONTINUE
//
// At this moment, V contains the right singular vectors of A.
// Next, assemble the left singular vector matrix U (M x N).
//
if (NR<M) {
CALL DLASET( 'A', M-NR, NR, Zero, Zero, U(NR+1,1), LDU )
if (NR<N1) {
CALL DLASET( 'A',NR, N1-NR, Zero, Zero, U(1,NR+1),LDU )
CALL DLASET( 'A',M-NR,N1-NR, Zero, One,U(NR+1,NR+1),LDU )
}
}
CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, work, U,
$ LDU, work(N+1), LWORK-N, IERR )
if (ROWPIV)
$ CALL DLASWP( N1, U, LDU, 1, M-1, iwork(2*N+1), -1 )
}
if ( TRANSP ) {
// .. swap U and V because the procedure worked on A^t
for (IndexType p=1; p<=n; ++p) {
blas::swap(U(_,p),V(_,p));
}
}
}
// end of the full SVD
//
// Undo scaling, if necessary (and possible)
//
if ( USCAL2 <= (BIG/SVA(1))*USCAL1 ) {
CALL DLASCL( 'G', 0, 0, USCAL1, USCAL2, NR, 1, SVA, N, IERR )
USCAL1 = One
USCAL2 = One
}
if ( NR < N ) {
sva(_(nr+1,n)) = Zero;
}
work(1) = USCAL2 * SCALEM
work(2) = USCAL1
if ( errest ) work(3) = SCONDA
if ( lsvec && rsvec ) {
work(4) = condr1
work(5) = condr2
}
if ( L2TRAN ) {
work(6) = entra
work(7) = entrat
}
iwork(1) = NR
iwork(2) = numRank
iwork(3) = warning
}
*/
//== interface for native lapack ===========================================
#ifdef USE_CXXLAPACK
namespace external {
template <typename MA, typename VSVA, typename MU, typename MV,
typename VWORK, typename VIWORK>
typename GeMatrix<MA>::IndexType
jsv_impl(JSV::Accuracy accuracy,
JSV::JobU jobU,
JSV::JobV jobV,
bool restrictedRange,
bool considerTransA,
bool perturb,
GeMatrix<MA> &A,
DenseVector<VSVA> &sva,
GeMatrix<MU> &U,
GeMatrix<MV> &V,
DenseVector<VWORK> &work,
DenseVector<VIWORK> &iwork)
{
typedef typename GeMatrix<MA>::IndexType IndexType;
IndexType info;
info = cxxlapack::gejsv<IndexType>(getF77Char(accuracy),
getF77Char(jobU),
getF77Char(jobV),
restrictedRange ? 'R' : 'N',
considerTransA ? 'T' : 'N',
perturb ? 'P' : 'N',
A.numRows(),
A.numCols(),
A.data(),
A.leadingDimension(),
sva.data(),
U.data(),
U.leadingDimension(),
V.data(),
V.leadingDimension(),
work.data(),
work.length(),
iwork.data());
return info;
}
} // namespace external
#endif // USE_CXXLAPACK
//== public interface ==========================================================
template <typename MA, typename VSVA, typename MU, typename MV,
typename VWORK, typename VIWORK>
typename GeMatrix<MA>::IndexType
jsv(JSV::Accuracy accuracy,
JSV::JobU jobU,
JSV::JobV jobV,
bool restrictedRange,
bool considerTransA,
bool perturb,
GeMatrix<MA> &A,
DenseVector<VSVA> &sva,
GeMatrix<MU> &U,
GeMatrix<MV> &V,
DenseVector<VWORK> &work,
DenseVector<VIWORK> &iwork)
{
typedef typename GeMatrix<MA>::IndexType IndexType;
//
// Test the input parameters
//
# ifndef NDEBUG
ASSERT(A.firstRow()==1);
ASSERT(A.firstCol()==1);
const IndexType m = A.numRows();
const IndexType n = A.numCols();
ASSERT(m>=n);
ASSERT(sva.firstIndex()==1);
ASSERT(sva.length()==n);
ASSERT(U.firstRow()==1);
ASSERT(U.firstCol()==1);
ASSERT(V.firstRow()==1);
ASSERT(V.firstCol()==1);
ASSERT(iwork.length()==m+3*n);
# endif
//
// Make copies of output arguments
//
# ifdef CHECK_CXXLAPACK
typename GeMatrix<MA>::NoView A_org = A;
typename DenseVector<VSVA>::NoView sva_org = sva;
typename GeMatrix<MU>::NoView U_org = U;
typename GeMatrix<MV>::NoView V_org = V;
typename DenseVector<VWORK>::NoView work_org = work;
# endif
//
// Call implementation
//
IndexType info = jsv_generic(accuracy, jobU, jobV,
restrictedRange, considerTransA, perturb,
A, sva, U, V, work, iwork);
# ifdef CHECK_CXXLAPACK
//
// Make copies of results computed by the generic implementation
//
typename GeMatrix<MA>::NoView A_generic = A;
typename DenseVector<VSVA>::NoView sva_generic = sva;
typename GeMatrix<MU>::NoView U_generic = U;
typename GeMatrix<MV>::NoView V_generic = V;
typename DenseVector<VWORK>::NoView work_generic = work;
//
// restore output arguments
//
A = A_org;
sva = sva_org;
U = U_org;
V = V_org;
work = work_org;
//
// Compare generic results with results from the native implementation
//
IndexType info_ = external::jsv_impl(accuracy, jobU, jobV, restrictedRange,
considerTransA, perturb,
A, sva, U, V, work, iwork);
bool failed = false;
if (! isIdentical(A_generic, A, "A_generic", "A")) {
std::cerr << "CXXLAPACK: A_generic = " << A_generic << std::endl;
std::cerr << "F77LAPACK: A = " << A << std::endl;
failed = true;
}
if (! isIdentical(sva_generic, sva, "sva_generic", "sva")) {
std::cerr << "CXXLAPACK: sva_generic = " << sva_generic << std::endl;
std::cerr << "F77LAPACK: sva = " << sva << std::endl;
failed = true;
}
if (! isIdentical(U_generic, U, "U_generic", "U")) {
std::cerr << "CXXLAPACK: U_generic = " << U_generic << std::endl;
std::cerr << "F77LAPACK: U = " << U << std::endl;
failed = true;
}
if (! isIdentical(V_generic, V, "V_generic", "V")) {
std::cerr << "CXXLAPACK: V_generic = " << V_generic << std::endl;
std::cerr << "F77LAPACK: V = " << V << std::endl;
failed = true;
}
if (! isIdentical(work_generic, work, "work_generic", "work")) {
std::cerr << "CXXLAPACK: work_generic = " << work_generic << std::endl;
std::cerr << "F77LAPACK: work = " << work << std::endl;
failed = true;
}
if (! isIdentical(info, info_, "info", "info_")) {
std::cerr << "CXXLAPACK: info = " << info << std::endl;
std::cerr << "F77LAPACK: info_ = " << info_ << std::endl;
failed = true;
}
if (failed) {
std::cerr << "error in: jsv.tcc " << std::endl;
ASSERT(0);
} else {
std::cerr << "passed: jsv.tcc " << std::endl;
}
# endif
return info;
}
//-- forwarding ----------------------------------------------------------------
template <typename MA, typename VSVA, typename MU, typename MV,
typename VWORK, typename VIWORK>
typename MA::IndexType
jsv(JSV::Accuracy accuracy,
JSV::JobU jobU,
JSV::JobV jobV,
bool restrictedRange,
bool considerTransA,
bool perturb,
MA &&A,
VSVA &&sva,
MU &&U,
MV &&V,
VWORK &&work,
VIWORK &&iwork)
{
typename MA::IndexType info;
CHECKPOINT_ENTER;
info = jsv(accuracy, jobU, jobV,
restrictedRange, considerTransA, perturb,
A, sva, U, V, work, iwork);
CHECKPOINT_LEAVE;
return info;
}
} } // namespace lapack, flens
#endif // FLENS_LAPACK_GE_JSV_TCC
|