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
|
SUBROUTINE SGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
$ LDV, WORK, LWORK, 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.
*
IMPLICIT NONE
* ..
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDV, LWORK, M, MV, N
CHARACTER*1 JOBA, JOBU, JOBV
* ..
* .. Array Arguments ..
REAL A( LDA, * ), SVA( N ), V( LDV, * ),
$ WORK( LWORK )
* ..
*
* Purpose
* =======
*
* SGESVJ computes the singular value decomposition (SVD) of a real
* M-by-N matrix A, where M >= N. The SVD of A is written as
* [++] [xx] [x0] [xx]
* A = U * SIGMA * V^t, [++] = [xx] * [ox] * [xx]
* [++] [xx]
* where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
* matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
* of SIGMA are the singular values of A. The columns of U and V are the
* left and the right singular vectors of A, respectively.
*
* Further Details
* ~~~~~~~~~~~~~~~
* The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
* rotations. The rotations are implemented as fast scaled rotations of
* Anda and Park [1]. In the case of underflow of the Jacobi angle, a
* modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses
* column interchanges of de Rijk [2]. The relative accuracy of the computed
* singular values and the accuracy of the computed singular vectors (in
* angle metric) is as guaranteed by the theory of Demmel and Veselic [3].
* The condition number that determines the accuracy in the full rank case
* is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
* spectral condition number. The best performance of this Jacobi SVD
* procedure is achieved if used in an accelerated version of Drmac and
* Veselic [5,6], and it is the kernel routine in the SIGMA library [7].
* Some tunning parameters (marked with [TP]) are available for the
* implementer.
* The computational range for the nonzero singular values is the machine
* number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
* denormalized singular values can be computed with the corresponding
* gradual loss of accurate digits.
*
* Contributors
* ~~~~~~~~~~~~
* Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
*
* References
* ~~~~~~~~~~
* [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.
* SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.
* [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
* singular value decomposition on a vector computer.
* SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
* [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
* [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular
* value computation in floating point arithmetic.
* SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
* [5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
* SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
* LAPACK Working note 169.
* [6] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
* SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
* LAPACK Working note 170.
* [7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
* QSVD, (H,K)-SVD computations.
* Department of Mathematics, University of Zagreb, 2008.
*
* Bugs, Examples and Comments
* ~~~~~~~~~~~~~~~~~~~~~~~~~~~
* Please report all bugs and send interesting test examples and comments to
* drmac@math.hr. Thank you.
*
* Arguments
* =========
*
* JOBA (input) CHARACTER* 1
* Specifies the structure of A.
* = 'L': The input matrix A is lower triangular;
* = 'U': The input matrix A is upper triangular;
* = 'G': The input matrix A is general M-by-N matrix, M >= N.
*
* JOBU (input) CHARACTER*1
* Specifies whether to compute the left singular vectors
* (columns of U):
* = 'U': The left singular vectors corresponding to the nonzero
* singular values are computed and returned in the leading
* columns of A. See more details in the description of A.
* The default numerical orthogonality threshold is set to
* approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=SLAMCH('E').
* = 'C': Analogous to JOBU='U', except that user can control the
* level of numerical orthogonality of the computed left
* singular vectors. TOL can be set to TOL = CTOL*EPS, where
* CTOL is given on input in the array WORK.
* No CTOL smaller than ONE is allowed. CTOL greater
* than 1 / EPS is meaningless. The option 'C'
* can be used if M*EPS is satisfactory orthogonality
* of the computed left singular vectors, so CTOL=M could
* save few sweeps of Jacobi rotations.
* See the descriptions of A and WORK(1).
* = 'N': The matrix U is not computed. However, see the
* description of A.
*
* JOBV (input) CHARACTER*1
* Specifies whether to compute the right singular vectors, that
* is, the matrix V:
* = 'V' : the matrix V is computed and returned in the array V
* = 'A' : the Jacobi rotations are applied to the MV-by-N
* array V. In other words, the right singular vector
* matrix V is not computed explicitly; instead it is
* applied to an MV-by-N matrix initially stored in the
* first MV rows of V.
* = 'N' : the matrix V is not computed and the array V is not
* referenced
*
* M (input) INTEGER
* The number of rows of the input matrix A. 1/SLAMCH('E') > M >= 0.
*
* N (input) INTEGER
* The number of columns of the input matrix A.
* M >= N >= 0.
*
* A (input/output) REAL array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit,
* If JOBU .EQ. 'U' .OR. JOBU .EQ. 'C':
* If INFO .EQ. 0 :
* RANKA orthonormal columns of U are returned in the
* leading RANKA columns of the array A. Here RANKA <= N
* is the number of computed singular values of A that are
* above the underflow threshold SLAMCH('S'). The singular
* vectors corresponding to underflowed or zero singular
* values are not computed. The value of RANKA is returned
* in the array WORK as RANKA=NINT(WORK(2)). Also see the
* descriptions of SVA and WORK. The computed columns of U
* are mutually numerically orthogonal up to approximately
* TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU.EQ.'C'),
* see the description of JOBU.
* If INFO .GT. 0,
* the procedure SGESVJ did not converge in the given number
* of iterations (sweeps). In that case, the computed
* columns of U may not be orthogonal up to TOL. The output
* U (stored in A), SIGMA (given by the computed singular
* values in SVA(1:N)) and V is still a decomposition of the
* input matrix A in the sense that the residual
* ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small.
* If JOBU .EQ. 'N':
* If INFO .EQ. 0 :
* Note that the left singular vectors are 'for free' in the
* one-sided Jacobi SVD algorithm. However, if only the
* singular values are needed, the level of numerical
* orthogonality of U is not an issue and iterations are
* stopped when the columns of the iterated matrix are
* numerically orthogonal up to approximately M*EPS. Thus,
* on exit, A contains the columns of U scaled with the
* corresponding singular values.
* If INFO .GT. 0 :
* the procedure SGESVJ did not converge in the given number
* of iterations (sweeps).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* SVA (workspace/output) REAL array, dimension (N)
* On exit,
* If INFO .EQ. 0 :
* depending on the value SCALE = WORK(1), we have:
* If SCALE .EQ. ONE:
* SVA(1:N) contains the computed singular values of A.
* During the computation SVA contains the Euclidean column
* norms of the iterated matrices in the array A.
* If SCALE .NE. ONE:
* The singular values of A are SCALE*SVA(1:N), and this
* factored representation is due to the fact that some of the
* singular values of A might underflow or overflow.
*
* If INFO .GT. 0 :
* the procedure SGESVJ did not converge in the given number of
* iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
*
* MV (input) INTEGER
* If JOBV .EQ. 'A', then the product of Jacobi rotations in SGESVJ
* is applied to the first MV rows of V. See the description of JOBV.
*
* V (input/output) REAL array, dimension (LDV,N)
* If JOBV = 'V', then V contains on exit the N-by-N matrix of
* the right singular vectors;
* If JOBV = 'A', then V contains the product of the computed right
* singular vector matrix and the initial matrix in
* the array V.
* If JOBV = 'N', then V is not referenced.
*
* LDV (input) INTEGER
* The leading dimension of the array V, LDV .GE. 1.
* If JOBV .EQ. 'V', then LDV .GE. max(1,N).
* If JOBV .EQ. 'A', then LDV .GE. max(1,MV) .
*
* WORK (input/workspace/output) REAL array, dimension max(4,M+N).
* On entry,
* If JOBU .EQ. 'C' :
* WORK(1) = CTOL, where CTOL defines the threshold for convergence.
* The process stops if all columns of A are mutually
* orthogonal up to CTOL*EPS, EPS=SLAMCH('E').
* It is required that CTOL >= ONE, i.e. it is not
* allowed to force the routine to obtain orthogonality
* below EPSILON.
* On exit,
* WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
* are the computed singular vcalues of A.
* (See description of SVA().)
* WORK(2) = NINT(WORK(2)) is the number of the computed nonzero
* singular values.
* WORK(3) = NINT(WORK(3)) is the number of the computed singular
* values that are larger than the underflow threshold.
* WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi
* rotations needed for numerical convergence.
* WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
* This is useful information in cases when SGESVJ did
* not converge, as it can be used to estimate whether
* the output is stil useful and for post festum analysis.
* WORK(6) = the largest absolute value over all sines of the
* Jacobi rotation angles in the last sweep. It can be
* useful for a post festum analysis.
*
* LWORK (input) INTEGER
* length of WORK, WORK >= MAX(6,M+N)
*
* INFO (output) INTEGER
* = 0 : successful exit.
* < 0 : if INFO = -i, then the i-th argument had an illegal value
* > 0 : SGESVJ did not converge in the maximal allowed number (30)
* of sweeps. The output may still be useful. See the
* description of WORK.
*
* =====================================================================
*
* .. Local Parameters ..
REAL ZERO, HALF, ONE, TWO
PARAMETER ( ZERO = 0.0E0, HALF = 0.5E0, ONE = 1.0E0,
$ TWO = 2.0E0 )
INTEGER NSWEEP
PARAMETER ( NSWEEP = 30 )
* ..
* .. Local Scalars ..
REAL AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
$ BIGTHETA, CS, CTOL, EPSLN, LARGE, MXAAPQ,
$ MXSINJ, ROOTBIG, ROOTEPS, ROOTSFMIN, ROOTTOL,
$ SKL, SFMIN, SMALL, SN, T, TEMP1, THETA,
$ THSIGN, TOL
INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1,
$ ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, N2, N34,
$ N4, NBL, NOTROT, p, PSKIPPED, q, ROWSKIP,
$ SWBAND
LOGICAL APPLV, GOSCALE, LOWER, LSVEC, NOSCALE, ROTOK,
$ RSVEC, UCTOL, UPPER
* ..
* .. Local Arrays ..
REAL FASTR( 5 )
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AMAX1, AMIN1, FLOAT, MIN0, SIGN, SQRT
* ..
* .. External Functions ..
* ..
* from BLAS
REAL SDOT, SNRM2
EXTERNAL SDOT, SNRM2
INTEGER ISAMAX
EXTERNAL ISAMAX
* from LAPACK
REAL SLAMCH
EXTERNAL SLAMCH
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
* ..
* from BLAS
EXTERNAL SAXPY, SCOPY, SROTM, SSCAL, SSWAP
* from LAPACK
EXTERNAL SLASCL, SLASET, SLASSQ, XERBLA
*
EXTERNAL SGSVJ0, SGSVJ1
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
LSVEC = LSAME( JOBU, 'U' )
UCTOL = LSAME( JOBU, 'C' )
RSVEC = LSAME( JOBV, 'V' )
APPLV = LSAME( JOBV, 'A' )
UPPER = LSAME( JOBA, 'U' )
LOWER = LSAME( JOBA, 'L' )
*
IF( .NOT.( UPPER .OR. LOWER .OR. LSAME( JOBA, 'G' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( LSVEC .OR. UCTOL .OR. LSAME( JOBU, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
INFO = -5
ELSE IF( LDA.LT.M ) THEN
INFO = -7
ELSE IF( MV.LT.0 ) THEN
INFO = -9
ELSE IF( ( RSVEC .AND. ( LDV.LT.N ) ) .OR.
$ ( APPLV .AND. ( LDV.LT.MV ) ) ) THEN
INFO = -11
ELSE IF( UCTOL .AND. ( WORK( 1 ).LE.ONE ) ) THEN
INFO = -12
ELSE IF( LWORK.LT.MAX0( M+N, 6 ) ) THEN
INFO = -13
ELSE
INFO = 0
END IF
*
* #:(
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGESVJ', -INFO )
RETURN
END IF
*
* #:) Quick return for void matrix
*
IF( ( M.EQ.0 ) .OR. ( N.EQ.0 ) )RETURN
*
* Set numerical parameters
* The stopping criterion for Jacobi rotations is
*
* max_{i<>j}|A(:,i)^T * A(:,j)|/(||A(:,i)||*||A(:,j)||) < CTOL*EPS
*
* where EPS is the round-off and CTOL is defined as follows:
*
IF( UCTOL ) THEN
* ... user controlled
CTOL = WORK( 1 )
ELSE
* ... default
IF( LSVEC .OR. RSVEC .OR. APPLV ) THEN
CTOL = SQRT( FLOAT( M ) )
ELSE
CTOL = FLOAT( M )
END IF
END IF
* ... and the machine dependent parameters are
*[!] (Make sure that SLAMCH() works properly on the target machine.)
*
EPSLN = SLAMCH( 'Epsilon' )
ROOTEPS = SQRT( EPSLN )
SFMIN = SLAMCH( 'SafeMinimum' )
ROOTSFMIN = SQRT( SFMIN )
SMALL = SFMIN / EPSLN
BIG = SLAMCH( 'Overflow' )
* BIG = ONE / SFMIN
ROOTBIG = ONE / ROOTSFMIN
LARGE = BIG / SQRT( FLOAT( M*N ) )
BIGTHETA = ONE / ROOTEPS
*
TOL = CTOL*EPSLN
ROOTTOL = SQRT( TOL )
*
IF( FLOAT( M )*EPSLN.GE.ONE ) THEN
INFO = -4
CALL XERBLA( 'SGESVJ', -INFO )
RETURN
END IF
*
* Initialize the right singular vector matrix.
*
IF( RSVEC ) THEN
MVL = N
CALL SLASET( 'A', MVL, N, ZERO, ONE, V, LDV )
ELSE IF( APPLV ) THEN
MVL = MV
END IF
RSVEC = RSVEC .OR. APPLV
*
* Initialize SVA( 1:N ) = ( ||A e_i||_2, i = 1:N )
*(!) If necessary, scale A to protect the largest singular value
* from overflow. It is possible that saving the largest singular
* value destroys the information about the small ones.
* This initial scaling is almost minimal in the sense that the
* goal is to make sure that no column norm overflows, and that
* SQRT(N)*max_i SVA(i) does not overflow. If INFinite entries
* in A are detected, the procedure returns with INFO=-6.
*
SKL = ONE / SQRT( FLOAT( M )*FLOAT( N ) )
NOSCALE = .TRUE.
GOSCALE = .TRUE.
*
IF( LOWER ) THEN
* the input matrix is M-by-N lower triangular (trapezoidal)
DO 1874 p = 1, N
AAPP = ZERO
AAQQ = ONE
CALL SLASSQ( M-p+1, A( p, p ), 1, AAPP, AAQQ )
IF( AAPP.GT.BIG ) THEN
INFO = -6
CALL XERBLA( 'SGESVJ', -INFO )
RETURN
END IF
AAQQ = SQRT( AAQQ )
IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN
SVA( p ) = AAPP*AAQQ
ELSE
NOSCALE = .FALSE.
SVA( p ) = AAPP*( AAQQ*SKL )
IF( GOSCALE ) THEN
GOSCALE = .FALSE.
DO 1873 q = 1, p - 1
SVA( q ) = SVA( q )*SKL
1873 CONTINUE
END IF
END IF
1874 CONTINUE
ELSE IF( UPPER ) THEN
* the input matrix is M-by-N upper triangular (trapezoidal)
DO 2874 p = 1, N
AAPP = ZERO
AAQQ = ONE
CALL SLASSQ( p, A( 1, p ), 1, AAPP, AAQQ )
IF( AAPP.GT.BIG ) THEN
INFO = -6
CALL XERBLA( 'SGESVJ', -INFO )
RETURN
END IF
AAQQ = SQRT( AAQQ )
IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN
SVA( p ) = AAPP*AAQQ
ELSE
NOSCALE = .FALSE.
SVA( p ) = AAPP*( AAQQ*SKL )
IF( GOSCALE ) THEN
GOSCALE = .FALSE.
DO 2873 q = 1, p - 1
SVA( q ) = SVA( q )*SKL
2873 CONTINUE
END IF
END IF
2874 CONTINUE
ELSE
* the input matrix is M-by-N general dense
DO 3874 p = 1, N
AAPP = ZERO
AAQQ = ONE
CALL SLASSQ( M, A( 1, p ), 1, AAPP, AAQQ )
IF( AAPP.GT.BIG ) THEN
INFO = -6
CALL XERBLA( 'SGESVJ', -INFO )
RETURN
END IF
AAQQ = SQRT( AAQQ )
IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN
SVA( p ) = AAPP*AAQQ
ELSE
NOSCALE = .FALSE.
SVA( p ) = AAPP*( AAQQ*SKL )
IF( GOSCALE ) THEN
GOSCALE = .FALSE.
DO 3873 q = 1, p - 1
SVA( q ) = SVA( q )*SKL
3873 CONTINUE
END IF
END IF
3874 CONTINUE
END IF
*
IF( NOSCALE )SKL = ONE
*
* Move the smaller part of the spectrum from the underflow threshold
*(!) Start by determining the position of the nonzero entries of the
* array SVA() relative to ( SFMIN, BIG ).
*
AAPP = ZERO
AAQQ = BIG
DO 4781 p = 1, N
IF( SVA( p ).NE.ZERO )AAQQ = AMIN1( AAQQ, SVA( p ) )
AAPP = AMAX1( AAPP, SVA( p ) )
4781 CONTINUE
*
* #:) Quick return for zero matrix
*
IF( AAPP.EQ.ZERO ) THEN
IF( LSVEC )CALL SLASET( 'G', M, N, ZERO, ONE, A, LDA )
WORK( 1 ) = ONE
WORK( 2 ) = ZERO
WORK( 3 ) = ZERO
WORK( 4 ) = ZERO
WORK( 5 ) = ZERO
WORK( 6 ) = ZERO
RETURN
END IF
*
* #:) Quick return for one-column matrix
*
IF( N.EQ.1 ) THEN
IF( LSVEC )CALL SLASCL( 'G', 0, 0, SVA( 1 ), SKL, M, 1,
$ A( 1, 1 ), LDA, IERR )
WORK( 1 ) = ONE / SKL
IF( SVA( 1 ).GE.SFMIN ) THEN
WORK( 2 ) = ONE
ELSE
WORK( 2 ) = ZERO
END IF
WORK( 3 ) = ZERO
WORK( 4 ) = ZERO
WORK( 5 ) = ZERO
WORK( 6 ) = ZERO
RETURN
END IF
*
* Protect small singular values from underflow, and try to
* avoid underflows/overflows in computing Jacobi rotations.
*
SN = SQRT( SFMIN / EPSLN )
TEMP1 = SQRT( BIG / FLOAT( N ) )
IF( ( AAPP.LE.SN ) .OR. ( AAQQ.GE.TEMP1 ) .OR.
$ ( ( SN.LE.AAQQ ) .AND. ( AAPP.LE.TEMP1 ) ) ) THEN
TEMP1 = AMIN1( BIG, TEMP1 / AAPP )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.LE.TEMP1 ) ) THEN
TEMP1 = AMIN1( SN / AAQQ, BIG / ( AAPP*SQRT( FLOAT( N ) ) ) )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE IF( ( AAQQ.GE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN
TEMP1 = AMAX1( SN / AAQQ, TEMP1 / AAPP )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN
TEMP1 = AMIN1( SN / AAQQ, BIG / ( SQRT( FLOAT( N ) )*AAPP ) )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE
TEMP1 = ONE
END IF
*
* Scale, if necessary
*
IF( TEMP1.NE.ONE ) THEN
CALL SLASCL( 'G', 0, 0, ONE, TEMP1, N, 1, SVA, N, IERR )
END IF
SKL = TEMP1*SKL
IF( SKL.NE.ONE ) THEN
CALL SLASCL( JOBA, 0, 0, ONE, SKL, M, N, A, LDA, IERR )
SKL = ONE / SKL
END IF
*
* Row-cyclic Jacobi SVD algorithm with column pivoting
*
EMPTSW = ( N*( N-1 ) ) / 2
NOTROT = 0
FASTR( 1 ) = ZERO
*
* A is represented in factored form A = A * diag(WORK), where diag(WORK)
* is initialized to identity. WORK is updated during fast scaled
* rotations.
*
DO 1868 q = 1, N
WORK( q ) = ONE
1868 CONTINUE
*
*
SWBAND = 3
*[TP] SWBAND is a tuning parameter [TP]. It is meaningful and effective
* if SGESVJ is used as a computational routine in the preconditioned
* Jacobi SVD algorithm SGESVJ. For sweeps i=1:SWBAND the procedure
* works on pivots inside a band-like region around the diagonal.
* The boundaries are determined dynamically, based on the number of
* pivots above a threshold.
*
KBL = MIN0( 8, N )
*[TP] KBL is a tuning parameter that defines the tile size in the
* tiling of the p-q loops of pivot pairs. In general, an optimal
* value of KBL depends on the matrix dimensions and on the
* parameters of the computer's memory.
*
NBL = N / KBL
IF( ( NBL*KBL ).NE.N )NBL = NBL + 1
*
BLSKIP = KBL**2
*[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
*
ROWSKIP = MIN0( 5, KBL )
*[TP] ROWSKIP is a tuning parameter.
*
LKAHEAD = 1
*[TP] LKAHEAD is a tuning parameter.
*
* Quasi block transformations, using the lower (upper) triangular
* structure of the input matrix. The quasi-block-cycling usually
* invokes cubic convergence. Big part of this cycle is done inside
* canonical subspaces of dimensions less than M.
*
IF( ( LOWER .OR. UPPER ) .AND. ( N.GT.MAX0( 64, 4*KBL ) ) ) THEN
*[TP] The number of partition levels and the actual partition are
* tuning parameters.
N4 = N / 4
N2 = N / 2
N34 = 3*N4
IF( APPLV ) THEN
q = 0
ELSE
q = 1
END IF
*
IF( LOWER ) THEN
*
* This works very well on lower triangular matrices, in particular
* in the framework of the preconditioned Jacobi SVD (xGEJSV).
* The idea is simple:
* [+ 0 0 0] Note that Jacobi transformations of [0 0]
* [+ + 0 0] [0 0]
* [+ + x 0] actually work on [x 0] [x 0]
* [+ + x x] [x x]. [x x]
*
CALL SGSVJ0( JOBV, M-N34, N-N34, A( N34+1, N34+1 ), LDA,
$ WORK( N34+1 ), SVA( N34+1 ), MVL,
$ V( N34*q+1, N34+1 ), LDV, EPSLN, SFMIN, TOL,
$ 2, WORK( N+1 ), LWORK-N, IERR )
*
CALL SGSVJ0( JOBV, M-N2, N34-N2, A( N2+1, N2+1 ), LDA,
$ WORK( N2+1 ), SVA( N2+1 ), MVL,
$ V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 2,
$ WORK( N+1 ), LWORK-N, IERR )
*
CALL SGSVJ1( JOBV, M-N2, N-N2, N4, A( N2+1, N2+1 ), LDA,
$ WORK( N2+1 ), SVA( N2+1 ), MVL,
$ V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 1,
$ WORK( N+1 ), LWORK-N, IERR )
*
CALL SGSVJ0( JOBV, M-N4, N2-N4, A( N4+1, N4+1 ), LDA,
$ WORK( N4+1 ), SVA( N4+1 ), MVL,
$ V( N4*q+1, N4+1 ), LDV, EPSLN, SFMIN, TOL, 1,
$ WORK( N+1 ), LWORK-N, IERR )
*
CALL SGSVJ0( JOBV, M, N4, A, LDA, WORK, SVA, MVL, V, LDV,
$ EPSLN, SFMIN, TOL, 1, WORK( N+1 ), LWORK-N,
$ IERR )
*
CALL SGSVJ1( JOBV, M, N2, N4, A, LDA, WORK, SVA, MVL, V,
$ LDV, EPSLN, SFMIN, TOL, 1, WORK( N+1 ),
$ LWORK-N, IERR )
*
*
ELSE IF( UPPER ) THEN
*
*
CALL SGSVJ0( JOBV, N4, N4, A, LDA, WORK, SVA, MVL, V, LDV,
$ EPSLN, SFMIN, TOL, 2, WORK( N+1 ), LWORK-N,
$ IERR )
*
CALL SGSVJ0( JOBV, N2, N4, A( 1, N4+1 ), LDA, WORK( N4+1 ),
$ SVA( N4+1 ), MVL, V( N4*q+1, N4+1 ), LDV,
$ EPSLN, SFMIN, TOL, 1, WORK( N+1 ), LWORK-N,
$ IERR )
*
CALL SGSVJ1( JOBV, N2, N2, N4, A, LDA, WORK, SVA, MVL, V,
$ LDV, EPSLN, SFMIN, TOL, 1, WORK( N+1 ),
$ LWORK-N, IERR )
*
CALL SGSVJ0( JOBV, N2+N4, N4, A( 1, N2+1 ), LDA,
$ WORK( N2+1 ), SVA( N2+1 ), MVL,
$ V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 1,
$ WORK( N+1 ), LWORK-N, IERR )
END IF
*
END IF
*
* .. Row-cyclic pivot strategy with de Rijk's pivoting ..
*
DO 1993 i = 1, NSWEEP
*
* .. go go go ...
*
MXAAPQ = ZERO
MXSINJ = ZERO
ISWROT = 0
*
NOTROT = 0
PSKIPPED = 0
*
* Each sweep is unrolled using KBL-by-KBL tiles over the pivot pairs
* 1 <= p < q <= N. This is the first step toward a blocked implementation
* of the rotations. New implementation, based on block transformations,
* is under development.
*
DO 2000 ibr = 1, NBL
*
igl = ( ibr-1 )*KBL + 1
*
DO 1002 ir1 = 0, MIN0( LKAHEAD, NBL-ibr )
*
igl = igl + ir1*KBL
*
DO 2001 p = igl, MIN0( igl+KBL-1, N-1 )
*
* .. de Rijk's pivoting
*
q = ISAMAX( N-p+1, SVA( p ), 1 ) + p - 1
IF( p.NE.q ) THEN
CALL SSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
IF( RSVEC )CALL SSWAP( MVL, V( 1, p ), 1,
$ V( 1, q ), 1 )
TEMP1 = SVA( p )
SVA( p ) = SVA( q )
SVA( q ) = TEMP1
TEMP1 = WORK( p )
WORK( p ) = WORK( q )
WORK( q ) = TEMP1
END IF
*
IF( ir1.EQ.0 ) THEN
*
* Column norms are periodically updated by explicit
* norm computation.
* Caveat:
* Unfortunately, some BLAS implementations compute SNRM2(M,A(1,p),1)
* as SQRT(SDOT(M,A(1,p),1,A(1,p),1)), which may cause the result to
* overflow for ||A(:,p)||_2 > SQRT(overflow_threshold), and to
* underflow for ||A(:,p)||_2 < SQRT(underflow_threshold).
* Hence, SNRM2 cannot be trusted, not even in the case when
* the true norm is far from the under(over)flow boundaries.
* If properly implemented SNRM2 is available, the IF-THEN-ELSE
* below should read "AAPP = SNRM2( M, A(1,p), 1 ) * WORK(p)".
*
IF( ( SVA( p ).LT.ROOTBIG ) .AND.
$ ( SVA( p ).GT.ROOTSFMIN ) ) THEN
SVA( p ) = SNRM2( M, A( 1, p ), 1 )*WORK( p )
ELSE
TEMP1 = ZERO
AAPP = ONE
CALL SLASSQ( M, A( 1, p ), 1, TEMP1, AAPP )
SVA( p ) = TEMP1*SQRT( AAPP )*WORK( p )
END IF
AAPP = SVA( p )
ELSE
AAPP = SVA( p )
END IF
*
IF( AAPP.GT.ZERO ) THEN
*
PSKIPPED = 0
*
DO 2002 q = p + 1, MIN0( igl+KBL-1, N )
*
AAQQ = SVA( q )
*
IF( AAQQ.GT.ZERO ) THEN
*
AAPP0 = AAPP
IF( AAQQ.GE.ONE ) THEN
ROTOK = ( SMALL*AAPP ).LE.AAQQ
IF( AAPP.LT.( BIG / AAQQ ) ) THEN
AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1,
$ q ), 1 )*WORK( p )*WORK( q ) /
$ AAQQ ) / AAPP
ELSE
CALL SCOPY( M, A( 1, p ), 1,
$ WORK( N+1 ), 1 )
CALL SLASCL( 'G', 0, 0, AAPP,
$ WORK( p ), M, 1,
$ WORK( N+1 ), LDA, IERR )
AAPQ = SDOT( M, WORK( N+1 ), 1,
$ A( 1, q ), 1 )*WORK( q ) / AAQQ
END IF
ELSE
ROTOK = AAPP.LE.( AAQQ / SMALL )
IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1,
$ q ), 1 )*WORK( p )*WORK( q ) /
$ AAQQ ) / AAPP
ELSE
CALL SCOPY( M, A( 1, q ), 1,
$ WORK( N+1 ), 1 )
CALL SLASCL( 'G', 0, 0, AAQQ,
$ WORK( q ), M, 1,
$ WORK( N+1 ), LDA, IERR )
AAPQ = SDOT( M, WORK( N+1 ), 1,
$ A( 1, p ), 1 )*WORK( 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
*
* .. rotate
*[RTD] ROTATED = ROTATED + ONE
*
IF( ir1.EQ.0 ) THEN
NOTROT = 0
PSKIPPED = 0
ISWROT = ISWROT + 1
END IF
*
IF( ROTOK ) THEN
*
AQOAP = AAQQ / AAPP
APOAQ = AAPP / AAQQ
THETA = -HALF*ABS( AQOAP-APOAQ ) / AAPQ
*
IF( ABS( THETA ).GT.BIGTHETA ) THEN
*
T = HALF / THETA
FASTR( 3 ) = T*WORK( p ) / WORK( q )
FASTR( 4 ) = -T*WORK( q ) /
$ WORK( 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 )
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 = WORK( p ) / WORK( q )
AQOAP = WORK( q ) / WORK( p )
IF( WORK( p ).GE.ONE ) THEN
IF( WORK( q ).GE.ONE ) THEN
FASTR( 3 ) = T*APOAQ
FASTR( 4 ) = -T*AQOAP
WORK( p ) = WORK( p )*CS
WORK( q ) = WORK( 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 )
WORK( p ) = WORK( p )*CS
WORK( q ) = WORK( 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
END IF
ELSE
IF( WORK( 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 )
WORK( p ) = WORK( p ) / CS
WORK( q ) = WORK( 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
ELSE
IF( WORK( p ).GE.WORK( 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 )
WORK( p ) = WORK( p )*CS
WORK( q ) = WORK( 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 )
WORK( p ) = WORK( p ) / CS
WORK( q ) = WORK( 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
* .. have to use modified Gram-Schmidt like transformation
CALL SCOPY( M, A( 1, p ), 1,
$ WORK( N+1 ), 1 )
CALL SLASCL( 'G', 0, 0, AAPP, ONE, M,
$ 1, WORK( N+1 ), LDA,
$ IERR )
CALL SLASCL( 'G', 0, 0, AAQQ, ONE, M,
$ 1, A( 1, q ), LDA, IERR )
TEMP1 = -AAPQ*WORK( p ) / WORK( q )
CALL SAXPY( M, TEMP1, WORK( N+1 ), 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 )
END IF
* END IF ROTOK THEN ... ELSE
*
* In the case of cancellation in updating SVA(q), SVA(p)
* recompute SVA(q), SVA(p).
*
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 )*
$ WORK( q )
ELSE
T = ZERO
AAQQ = ONE
CALL SLASSQ( M, A( 1, q ), 1, T,
$ AAQQ )
SVA( q ) = T*SQRT( AAQQ )*WORK( q )
END IF
END IF
IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN
IF( ( AAPP.LT.ROOTBIG ) .AND.
$ ( AAPP.GT.ROOTSFMIN ) ) THEN
AAPP = SNRM2( M, A( 1, p ), 1 )*
$ WORK( p )
ELSE
T = ZERO
AAPP = ONE
CALL SLASSQ( M, A( 1, p ), 1, T,
$ AAPP )
AAPP = T*SQRT( AAPP )*WORK( p )
END IF
SVA( p ) = AAPP
END IF
*
ELSE
* A(:,p) and A(:,q) already numerically orthogonal
IF( ir1.EQ.0 )NOTROT = NOTROT + 1
*[RTD] SKIPPED = SKIPPED + 1
PSKIPPED = PSKIPPED + 1
END IF
ELSE
* A(:,q) is zero column
IF( ir1.EQ.0 )NOTROT = NOTROT + 1
PSKIPPED = PSKIPPED + 1
END IF
*
IF( ( i.LE.SWBAND ) .AND.
$ ( PSKIPPED.GT.ROWSKIP ) ) THEN
IF( ir1.EQ.0 )AAPP = -AAPP
NOTROT = 0
GO TO 2103
END IF
*
2002 CONTINUE
* END q-LOOP
*
2103 CONTINUE
* bailed out of q-loop
*
SVA( p ) = AAPP
*
ELSE
SVA( p ) = AAPP
IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) )
$ NOTROT = NOTROT + MIN0( igl+KBL-1, N ) - p
END IF
*
2001 CONTINUE
* end of the p-loop
* end of doing the block ( ibr, ibr )
1002 CONTINUE
* end of ir1-loop
*
* ... go to the off diagonal blocks
*
igl = ( ibr-1 )*KBL + 1
*
DO 2010 jbc = ibr + 1, NBL
*
jgl = ( jbc-1 )*KBL + 1
*
* doing the block at ( ibr, jbc )
*
IJBLSK = 0
DO 2100 p = igl, MIN0( igl+KBL-1, N )
*
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 )*WORK( p )*WORK( q ) /
$ AAQQ ) / AAPP
ELSE
CALL SCOPY( M, A( 1, p ), 1,
$ WORK( N+1 ), 1 )
CALL SLASCL( 'G', 0, 0, AAPP,
$ WORK( p ), M, 1,
$ WORK( N+1 ), LDA, IERR )
AAPQ = SDOT( M, WORK( N+1 ), 1,
$ A( 1, q ), 1 )*WORK( 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 )*WORK( p )*WORK( q ) /
$ AAQQ ) / AAPP
ELSE
CALL SCOPY( M, A( 1, q ), 1,
$ WORK( N+1 ), 1 )
CALL SLASCL( 'G', 0, 0, AAQQ,
$ WORK( q ), M, 1,
$ WORK( N+1 ), LDA, IERR )
AAPQ = SDOT( M, WORK( N+1 ), 1,
$ A( 1, p ), 1 )*WORK( 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
*[RTD] 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*WORK( p ) / WORK( q )
FASTR( 4 ) = -T*WORK( q ) /
$ WORK( 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 = WORK( p ) / WORK( q )
AQOAP = WORK( q ) / WORK( p )
IF( WORK( p ).GE.ONE ) THEN
*
IF( WORK( q ).GE.ONE ) THEN
FASTR( 3 ) = T*APOAQ
FASTR( 4 ) = -T*AQOAP
WORK( p ) = WORK( p )*CS
WORK( q ) = WORK( 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
WORK( p ) = WORK( p )*CS
WORK( q ) = WORK( q ) / CS
END IF
ELSE
IF( WORK( 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
WORK( p ) = WORK( p ) / CS
WORK( q ) = WORK( q )*CS
ELSE
IF( WORK( p ).GE.WORK( 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 )
WORK( p ) = WORK( p )*CS
WORK( q ) = WORK( 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 )
WORK( p ) = WORK( p ) / CS
WORK( q ) = WORK( 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( N+1 ), 1 )
CALL SLASCL( 'G', 0, 0, AAPP, ONE,
$ M, 1, WORK( N+1 ), LDA,
$ IERR )
CALL SLASCL( 'G', 0, 0, AAQQ, ONE,
$ M, 1, A( 1, q ), LDA,
$ IERR )
TEMP1 = -AAPQ*WORK( p ) / WORK( q )
CALL SAXPY( M, TEMP1, WORK( N+1 ),
$ 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( N+1 ), 1 )
CALL SLASCL( 'G', 0, 0, AAQQ, ONE,
$ M, 1, WORK( N+1 ), LDA,
$ IERR )
CALL SLASCL( 'G', 0, 0, AAPP, ONE,
$ M, 1, A( 1, p ), LDA,
$ IERR )
TEMP1 = -AAPQ*WORK( q ) / WORK( p )
CALL SAXPY( M, TEMP1, WORK( N+1 ),
$ 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 )*
$ WORK( q )
ELSE
T = ZERO
AAQQ = ONE
CALL SLASSQ( M, A( 1, q ), 1, T,
$ AAQQ )
SVA( q ) = T*SQRT( AAQQ )*WORK( 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 )*
$ WORK( p )
ELSE
T = ZERO
AAPP = ONE
CALL SLASSQ( M, A( 1, p ), 1, T,
$ AAPP )
AAPP = T*SQRT( AAPP )*WORK( p )
END IF
SVA( p ) = AAPP
END IF
* end of OK rotation
ELSE
NOTROT = NOTROT + 1
*[RTD] SKIPPED = SKIPPED + 1
PSKIPPED = PSKIPPED + 1
IJBLSK = IJBLSK + 1
END IF
ELSE
NOTROT = NOTROT + 1
PSKIPPED = PSKIPPED + 1
IJBLSK = IJBLSK + 1
END IF
*
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
*
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
***
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 )*WORK( N )
ELSE
T = ZERO
AAPP = ONE
CALL SLASSQ( M, A( 1, N ), 1, T, AAPP )
SVA( N ) = T*SQRT( AAPP )*WORK( 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.SQRT( 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 not converged.
INFO = NSWEEP - 1
GO TO 1995
*
1994 CONTINUE
* #:) Reaching this point means numerical convergence after the i-th
* sweep.
*
INFO = 0
* #:) INFO = 0 confirms successful iterations.
1995 CONTINUE
*
* Sort the singular values and find how many are above
* the underflow threshold.
*
N2 = 0
N4 = 0
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 = WORK( p )
WORK( p ) = WORK( q )
WORK( 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
IF( SVA( p ).NE.ZERO ) THEN
N4 = N4 + 1
IF( SVA( p )*SKL.GT.SFMIN )N2 = N2 + 1
END IF
5991 CONTINUE
IF( SVA( N ).NE.ZERO ) THEN
N4 = N4 + 1
IF( SVA( N )*SKL.GT.SFMIN )N2 = N2 + 1
END IF
*
* Normalize the left singular vectors.
*
IF( LSVEC .OR. UCTOL ) THEN
DO 1998 p = 1, N2
CALL SSCAL( M, WORK( p ) / SVA( p ), A( 1, p ), 1 )
1998 CONTINUE
END IF
*
* Scale the product of Jacobi rotations (assemble the fast rotations).
*
IF( RSVEC ) THEN
IF( APPLV ) THEN
DO 2398 p = 1, N
CALL SSCAL( MVL, WORK( p ), V( 1, p ), 1 )
2398 CONTINUE
ELSE
DO 2399 p = 1, N
TEMP1 = ONE / SNRM2( MVL, V( 1, p ), 1 )
CALL SSCAL( MVL, TEMP1, V( 1, p ), 1 )
2399 CONTINUE
END IF
END IF
*
* Undo scaling, if necessary (and possible).
IF( ( ( SKL.GT.ONE ) .AND. ( SVA( 1 ).LT.( BIG /
$ SKL ) ) ) .OR. ( ( SKL.LT.ONE ) .AND. ( SVA( N2 ).GT.
$ ( SFMIN / SKL ) ) ) ) THEN
DO 2400 p = 1, N
SVA( p ) = SKL*SVA( p )
2400 CONTINUE
SKL = ONE
END IF
*
WORK( 1 ) = SKL
* The singular values of A are SKL*SVA(1:N). If SKL.NE.ONE
* then some of the singular values may overflow or underflow and
* the spectrum is given in this factored representation.
*
WORK( 2 ) = FLOAT( N4 )
* N4 is the number of computed nonzero singular values of A.
*
WORK( 3 ) = FLOAT( N2 )
* N2 is the number of singular values of A greater than SFMIN.
* If N2<N, SVA(N2:N) contains ZEROS and/or denormalized numbers
* that may carry some information.
*
WORK( 4 ) = FLOAT( i )
* i is the index of the last sweep before declaring convergence.
*
WORK( 5 ) = MXAAPQ
* MXAAPQ is the largest absolute value of scaled pivots in the
* last sweep
*
WORK( 6 ) = MXSINJ
* MXSINJ is the largest absolute value of the sines of Jacobi angles
* in the last sweep
*
RETURN
* ..
* .. END OF SGESVJ
* ..
END
|