forked from Reference-LAPACK/lapack
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathdlasd4.f
1061 lines (1060 loc) · 32.6 KB
/
dlasd4.f
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
*> \brief \b DLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by dbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD4 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd4.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd4.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd4.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER I, INFO, N
* DOUBLE PRECISION RHO, SIGMA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), DELTA( * ), WORK( * ), Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This subroutine computes the square root of the I-th updated
*> eigenvalue of a positive symmetric rank-one modification to
*> a positive diagonal matrix whose entries are given as the squares
*> of the corresponding entries in the array d, and that
*>
*> 0 <= D(i) < D(j) for i < j
*>
*> and that RHO > 0. This is arranged by the calling routine, and is
*> no loss in generality. The rank-one modified system is thus
*>
*> diag( D ) * diag( D ) + RHO * Z * Z_transpose.
*>
*> where we assume the Euclidean norm of Z is 1.
*>
*> The method consists of approximating the rational functions in the
*> secular equation by simpler interpolating rational functions.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The length of all arrays.
*> \endverbatim
*>
*> \param[in] I
*> \verbatim
*> I is INTEGER
*> The index of the eigenvalue to be computed. 1 <= I <= N.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension ( N )
*> The original eigenvalues. It is assumed that they are in
*> order, 0 <= D(I) < D(J) for I < J.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( N )
*> The components of the updating vector.
*> \endverbatim
*>
*> \param[out] DELTA
*> \verbatim
*> DELTA is DOUBLE PRECISION array, dimension ( N )
*> If N .ne. 1, DELTA contains (D(j) - sigma_I) in its j-th
*> component. If N = 1, then DELTA(1) = 1. The vector DELTA
*> contains the information necessary to construct the
*> (singular) eigenvectors.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> The scalar in the symmetric updating formula.
*> \endverbatim
*>
*> \param[out] SIGMA
*> \verbatim
*> SIGMA is DOUBLE PRECISION
*> The computed sigma_I, the I-th updated eigenvalue.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension ( N )
*> If N .ne. 1, WORK contains (D(j) + sigma_I) in its j-th
*> component. If N = 1, then WORK( 1 ) = 1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> > 0: if INFO = 1, the updating process failed.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> Logical variable ORGATI (origin-at-i?) is used for distinguishing
*> whether D(i) or D(i+1) is treated as the origin.
*>
*> ORGATI = .true. origin at i
*> ORGATI = .false. origin at i+1
*>
*> Logical variable SWTCH3 (switch-for-3-poles?) is for noting
*> if we are working with THREE poles!
*>
*> MAXIT is the maximum number of iterations allowed for each
*> eigenvalue.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup OTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Ren-Cang Li, Computer Science Division, University of California
*> at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
*
* -- LAPACK auxiliary routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
INTEGER I, INFO, N
DOUBLE PRECISION RHO, SIGMA
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), DELTA( * ), WORK( * ), Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER MAXIT
PARAMETER ( MAXIT = 400 )
DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, EIGHT, TEN
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
$ THREE = 3.0D+0, FOUR = 4.0D+0, EIGHT = 8.0D+0,
$ TEN = 10.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL ORGATI, SWTCH, SWTCH3, GEOMAVG
INTEGER II, IIM1, IIP1, IP1, ITER, J, NITER
DOUBLE PRECISION A, B, C, DELSQ, DELSQ2, SQ2, DPHI, DPSI, DTIIM,
$ DTIIP, DTIPSQ, DTISQ, DTNSQ, DTNSQ1, DW, EPS,
$ ERRETM, ETA, PHI, PREW, PSI, RHOINV, SGLB,
$ SGUB, TAU, TAU2, TEMP, TEMP1, TEMP2, W
* ..
* .. Local Arrays ..
DOUBLE PRECISION DD( 3 ), ZZ( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DLAED6, DLASD5
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Since this routine is called in an inner loop, we do no argument
* checking.
*
* Quick return for N=1 and 2.
*
INFO = 0
IF( N.EQ.1 ) THEN
*
* Presumably, I=1 upon entry
*
SIGMA = SQRT( D( 1 )*D( 1 )+RHO*Z( 1 )*Z( 1 ) )
DELTA( 1 ) = ONE
WORK( 1 ) = ONE
RETURN
END IF
IF( N.EQ.2 ) THEN
CALL DLASD5( I, D, Z, DELTA, RHO, SIGMA, WORK )
RETURN
END IF
*
* Compute machine epsilon
*
EPS = DLAMCH( 'Epsilon' )
RHOINV = ONE / RHO
TAU2= ZERO
*
* The case I = N
*
IF( I.EQ.N ) THEN
*
* Initialize some basic variables
*
II = N - 1
NITER = 1
*
* Calculate initial guess
*
TEMP = RHO / TWO
*
* If ||Z||_2 is not one, then TEMP should be set to
* RHO * ||Z||_2^2 / TWO
*
TEMP1 = TEMP / ( D( N )+SQRT( D( N )*D( N )+TEMP ) )
DO 10 J = 1, N
WORK( J ) = D( J ) + D( N ) + TEMP1
DELTA( J ) = ( D( J )-D( N ) ) - TEMP1
10 CONTINUE
*
PSI = ZERO
DO 20 J = 1, N - 2
PSI = PSI + Z( J )*Z( J ) / ( DELTA( J )*WORK( J ) )
20 CONTINUE
*
C = RHOINV + PSI
W = C + Z( II )*Z( II ) / ( DELTA( II )*WORK( II ) ) +
$ Z( N )*Z( N ) / ( DELTA( N )*WORK( N ) )
*
IF( W.LE.ZERO ) THEN
TEMP1 = SQRT( D( N )*D( N )+RHO )
TEMP = Z( N-1 )*Z( N-1 ) / ( ( D( N-1 )+TEMP1 )*
$ ( D( N )-D( N-1 )+RHO / ( D( N )+TEMP1 ) ) ) +
$ Z( N )*Z( N ) / RHO
*
* The following TAU2 is to approximate
* SIGMA_n^2 - D( N )*D( N )
*
IF( C.LE.TEMP ) THEN
TAU = RHO
ELSE
DELSQ = ( D( N )-D( N-1 ) )*( D( N )+D( N-1 ) )
A = -C*DELSQ + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N )
B = Z( N )*Z( N )*DELSQ
IF( A.LT.ZERO ) THEN
TAU2 = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
ELSE
TAU2 = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
END IF
TAU = TAU2 / ( D( N )+SQRT( D( N )*D( N )+TAU2 ) )
END IF
*
* It can be proved that
* D(N)^2+RHO/2 <= SIGMA_n^2 < D(N)^2+TAU2 <= D(N)^2+RHO
*
ELSE
DELSQ = ( D( N )-D( N-1 ) )*( D( N )+D( N-1 ) )
A = -C*DELSQ + Z( N-1 )*Z( N-1 ) + Z( N )*Z( N )
B = Z( N )*Z( N )*DELSQ
*
* The following TAU2 is to approximate
* SIGMA_n^2 - D( N )*D( N )
*
IF( A.LT.ZERO ) THEN
TAU2 = TWO*B / ( SQRT( A*A+FOUR*B*C )-A )
ELSE
TAU2 = ( A+SQRT( A*A+FOUR*B*C ) ) / ( TWO*C )
END IF
TAU = TAU2 / ( D( N )+SQRT( D( N )*D( N )+TAU2 ) )
*
* It can be proved that
* D(N)^2 < D(N)^2+TAU2 < SIGMA(N)^2 < D(N)^2+RHO/2
*
END IF
*
* The following TAU is to approximate SIGMA_n - D( N )
*
* TAU = TAU2 / ( D( N )+SQRT( D( N )*D( N )+TAU2 ) )
*
SIGMA = D( N ) + TAU
DO 30 J = 1, N
DELTA( J ) = ( D( J )-D( N ) ) - TAU
WORK( J ) = D( J ) + D( N ) + TAU
30 CONTINUE
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 40 J = 1, II
TEMP = Z( J ) / ( DELTA( J )*WORK( J ) )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
40 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
TEMP = Z( N ) / ( DELTA( N )*WORK( N ) )
PHI = Z( N )*TEMP
DPHI = TEMP*TEMP
ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV
* $ + ABS( TAU2 )*( DPSI+DPHI )
*
W = RHOINV + PHI + PSI
*
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
GO TO 240
END IF
*
* Calculate the new step
*
NITER = NITER + 1
DTNSQ1 = WORK( N-1 )*DELTA( N-1 )
DTNSQ = WORK( N )*DELTA( N )
C = W - DTNSQ1*DPSI - DTNSQ*DPHI
A = ( DTNSQ+DTNSQ1 )*W - DTNSQ*DTNSQ1*( DPSI+DPHI )
B = DTNSQ*DTNSQ1*W
IF( C.LT.ZERO )
$ C = ABS( C )
IF( C.EQ.ZERO ) THEN
ETA = RHO - SIGMA*SIGMA
ELSE IF( A.GE.ZERO ) THEN
ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
*
* Note, eta should be positive if w is negative, and
* eta should be negative otherwise. However,
* if for some reason caused by roundoff, eta*w > 0,
* we simply use one Newton step instead. This way
* will guarantee eta*w < 0.
*
IF( W*ETA.GT.ZERO )
$ ETA = -W / ( DPSI+DPHI )
TEMP = ETA - DTNSQ
IF( TEMP.GT.RHO )
$ ETA = RHO + DTNSQ
*
ETA = ETA / ( SIGMA+SQRT( ETA+SIGMA*SIGMA ) )
TAU = TAU + ETA
SIGMA = SIGMA + ETA
*
DO 50 J = 1, N
DELTA( J ) = DELTA( J ) - ETA
WORK( J ) = WORK( J ) + ETA
50 CONTINUE
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 60 J = 1, II
TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
60 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
TAU2 = WORK( N )*DELTA( N )
TEMP = Z( N ) / TAU2
PHI = Z( N )*TEMP
DPHI = TEMP*TEMP
ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV
* $ + ABS( TAU2 )*( DPSI+DPHI )
*
W = RHOINV + PHI + PSI
*
* Main loop to update the values of the array DELTA
*
ITER = NITER + 1
*
DO 90 NITER = ITER, MAXIT
*
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
GO TO 240
END IF
*
* Calculate the new step
*
DTNSQ1 = WORK( N-1 )*DELTA( N-1 )
DTNSQ = WORK( N )*DELTA( N )
C = W - DTNSQ1*DPSI - DTNSQ*DPHI
A = ( DTNSQ+DTNSQ1 )*W - DTNSQ1*DTNSQ*( DPSI+DPHI )
B = DTNSQ1*DTNSQ*W
IF( A.GE.ZERO ) THEN
ETA = ( A+SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A-SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
*
* Note, eta should be positive if w is negative, and
* eta should be negative otherwise. However,
* if for some reason caused by roundoff, eta*w > 0,
* we simply use one Newton step instead. This way
* will guarantee eta*w < 0.
*
IF( W*ETA.GT.ZERO )
$ ETA = -W / ( DPSI+DPHI )
TEMP = ETA - DTNSQ
IF( TEMP.LE.ZERO )
$ ETA = ETA / TWO
*
ETA = ETA / ( SIGMA+SQRT( ETA+SIGMA*SIGMA ) )
TAU = TAU + ETA
SIGMA = SIGMA + ETA
*
DO 70 J = 1, N
DELTA( J ) = DELTA( J ) - ETA
WORK( J ) = WORK( J ) + ETA
70 CONTINUE
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 80 J = 1, II
TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
80 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
TAU2 = WORK( N )*DELTA( N )
TEMP = Z( N ) / TAU2
PHI = Z( N )*TEMP
DPHI = TEMP*TEMP
ERRETM = EIGHT*( -PHI-PSI ) + ERRETM - PHI + RHOINV
* $ + ABS( TAU2 )*( DPSI+DPHI )
*
W = RHOINV + PHI + PSI
90 CONTINUE
*
* Return with INFO = 1, NITER = MAXIT and not converged
*
INFO = 1
GO TO 240
*
* End for the case I = N
*
ELSE
*
* The case for I < N
*
NITER = 1
IP1 = I + 1
*
* Calculate initial guess
*
DELSQ = ( D( IP1 )-D( I ) )*( D( IP1 )+D( I ) )
DELSQ2 = DELSQ / TWO
SQ2=SQRT( ( D( I )*D( I )+D( IP1 )*D( IP1 ) ) / TWO )
TEMP = DELSQ2 / ( D( I )+SQ2 )
DO 100 J = 1, N
WORK( J ) = D( J ) + D( I ) + TEMP
DELTA( J ) = ( D( J )-D( I ) ) - TEMP
100 CONTINUE
*
PSI = ZERO
DO 110 J = 1, I - 1
PSI = PSI + Z( J )*Z( J ) / ( WORK( J )*DELTA( J ) )
110 CONTINUE
*
PHI = ZERO
DO 120 J = N, I + 2, -1
PHI = PHI + Z( J )*Z( J ) / ( WORK( J )*DELTA( J ) )
120 CONTINUE
C = RHOINV + PSI + PHI
W = C + Z( I )*Z( I ) / ( WORK( I )*DELTA( I ) ) +
$ Z( IP1 )*Z( IP1 ) / ( WORK( IP1 )*DELTA( IP1 ) )
*
GEOMAVG = .FALSE.
IF( W.GT.ZERO ) THEN
*
* d(i)^2 < the ith sigma^2 < (d(i)^2+d(i+1)^2)/2
*
* We choose d(i) as origin.
*
ORGATI = .TRUE.
II = I
SGLB = ZERO
SGUB = DELSQ2 / ( D( I )+SQ2 )
A = C*DELSQ + Z( I )*Z( I ) + Z( IP1 )*Z( IP1 )
B = Z( I )*Z( I )*DELSQ
IF( A.GT.ZERO ) THEN
TAU2 = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
ELSE
TAU2 = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
END IF
*
* TAU2 now is an estimation of SIGMA^2 - D( I )^2. The
* following, however, is the corresponding estimation of
* SIGMA - D( I ).
*
TAU = TAU2 / ( D( I )+SQRT( D( I )*D( I )+TAU2 ) )
TEMP = SQRT(EPS)
IF( (D(I).LE.TEMP*D(IP1)).AND.(ABS(Z(I)).LE.TEMP)
$ .AND.(D(I).GT.ZERO) ) THEN
TAU = MIN( TEN*D(I), SGUB )
GEOMAVG = .TRUE.
END IF
ELSE
*
* (d(i)^2+d(i+1)^2)/2 <= the ith sigma^2 < d(i+1)^2/2
*
* We choose d(i+1) as origin.
*
ORGATI = .FALSE.
II = IP1
SGLB = -DELSQ2 / ( D( II )+SQ2 )
SGUB = ZERO
A = C*DELSQ - Z( I )*Z( I ) - Z( IP1 )*Z( IP1 )
B = Z( IP1 )*Z( IP1 )*DELSQ
IF( A.LT.ZERO ) THEN
TAU2 = TWO*B / ( A-SQRT( ABS( A*A+FOUR*B*C ) ) )
ELSE
TAU2 = -( A+SQRT( ABS( A*A+FOUR*B*C ) ) ) / ( TWO*C )
END IF
*
* TAU2 now is an estimation of SIGMA^2 - D( IP1 )^2. The
* following, however, is the corresponding estimation of
* SIGMA - D( IP1 ).
*
TAU = TAU2 / ( D( IP1 )+SQRT( ABS( D( IP1 )*D( IP1 )+
$ TAU2 ) ) )
END IF
*
SIGMA = D( II ) + TAU
DO 130 J = 1, N
WORK( J ) = D( J ) + D( II ) + TAU
DELTA( J ) = ( D( J )-D( II ) ) - TAU
130 CONTINUE
IIM1 = II - 1
IIP1 = II + 1
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 150 J = 1, IIM1
TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
150 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
DPHI = ZERO
PHI = ZERO
DO 160 J = N, IIP1, -1
TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
PHI = PHI + Z( J )*TEMP
DPHI = DPHI + TEMP*TEMP
ERRETM = ERRETM + PHI
160 CONTINUE
*
W = RHOINV + PHI + PSI
*
* W is the value of the secular function with
* its ii-th element removed.
*
SWTCH3 = .FALSE.
IF( ORGATI ) THEN
IF( W.LT.ZERO )
$ SWTCH3 = .TRUE.
ELSE
IF( W.GT.ZERO )
$ SWTCH3 = .TRUE.
END IF
IF( II.EQ.1 .OR. II.EQ.N )
$ SWTCH3 = .FALSE.
*
TEMP = Z( II ) / ( WORK( II )*DELTA( II ) )
DW = DPSI + DPHI + TEMP*TEMP
TEMP = Z( II )*TEMP
W = W + TEMP
ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV
$ + THREE*ABS( TEMP )
* $ + ABS( TAU2 )*DW
*
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
GO TO 240
END IF
*
IF( W.LE.ZERO ) THEN
SGLB = MAX( SGLB, TAU )
ELSE
SGUB = MIN( SGUB, TAU )
END IF
*
* Calculate the new step
*
NITER = NITER + 1
IF( .NOT.SWTCH3 ) THEN
DTIPSQ = WORK( IP1 )*DELTA( IP1 )
DTISQ = WORK( I )*DELTA( I )
IF( ORGATI ) THEN
C = W - DTIPSQ*DW + DELSQ*( Z( I ) / DTISQ )**2
ELSE
C = W - DTISQ*DW - DELSQ*( Z( IP1 ) / DTIPSQ )**2
END IF
A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW
B = DTIPSQ*DTISQ*W
IF( C.EQ.ZERO ) THEN
IF( A.EQ.ZERO ) THEN
IF( ORGATI ) THEN
A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*( DPSI+DPHI )
ELSE
A = Z( IP1 )*Z( IP1 ) + DTISQ*DTISQ*( DPSI+DPHI )
END IF
END IF
ETA = B / A
ELSE IF( A.LE.ZERO ) THEN
ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
ELSE
*
* Interpolation using THREE most relevant poles
*
DTIIM = WORK( IIM1 )*DELTA( IIM1 )
DTIIP = WORK( IIP1 )*DELTA( IIP1 )
TEMP = RHOINV + PSI + PHI
IF( ORGATI ) THEN
TEMP1 = Z( IIM1 ) / DTIIM
TEMP1 = TEMP1*TEMP1
C = ( TEMP - DTIIP*( DPSI+DPHI ) ) -
$ ( D( IIM1 )-D( IIP1 ) )*( D( IIM1 )+D( IIP1 ) )*TEMP1
ZZ( 1 ) = Z( IIM1 )*Z( IIM1 )
IF( DPSI.LT.TEMP1 ) THEN
ZZ( 3 ) = DTIIP*DTIIP*DPHI
ELSE
ZZ( 3 ) = DTIIP*DTIIP*( ( DPSI-TEMP1 )+DPHI )
END IF
ELSE
TEMP1 = Z( IIP1 ) / DTIIP
TEMP1 = TEMP1*TEMP1
C = ( TEMP - DTIIM*( DPSI+DPHI ) ) -
$ ( D( IIP1 )-D( IIM1 ) )*( D( IIM1 )+D( IIP1 ) )*TEMP1
IF( DPHI.LT.TEMP1 ) THEN
ZZ( 1 ) = DTIIM*DTIIM*DPSI
ELSE
ZZ( 1 ) = DTIIM*DTIIM*( DPSI+( DPHI-TEMP1 ) )
END IF
ZZ( 3 ) = Z( IIP1 )*Z( IIP1 )
END IF
ZZ( 2 ) = Z( II )*Z( II )
DD( 1 ) = DTIIM
DD( 2 ) = DELTA( II )*WORK( II )
DD( 3 ) = DTIIP
CALL DLAED6( NITER, ORGATI, C, DD, ZZ, W, ETA, INFO )
*
IF( INFO.NE.0 ) THEN
*
* If INFO is not 0, i.e., DLAED6 failed, switch back
* to 2 pole interpolation.
*
SWTCH3 = .FALSE.
INFO = 0
DTIPSQ = WORK( IP1 )*DELTA( IP1 )
DTISQ = WORK( I )*DELTA( I )
IF( ORGATI ) THEN
C = W - DTIPSQ*DW + DELSQ*( Z( I ) / DTISQ )**2
ELSE
C = W - DTISQ*DW - DELSQ*( Z( IP1 ) / DTIPSQ )**2
END IF
A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW
B = DTIPSQ*DTISQ*W
IF( C.EQ.ZERO ) THEN
IF( A.EQ.ZERO ) THEN
IF( ORGATI ) THEN
A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*( DPSI+DPHI )
ELSE
A = Z( IP1 )*Z( IP1 ) + DTISQ*DTISQ*( DPSI+DPHI)
END IF
END IF
ETA = B / A
ELSE IF( A.LE.ZERO ) THEN
ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
END IF
END IF
*
* Note, eta should be positive if w is negative, and
* eta should be negative otherwise. However,
* if for some reason caused by roundoff, eta*w > 0,
* we simply use one Newton step instead. This way
* will guarantee eta*w < 0.
*
IF( W*ETA.GE.ZERO )
$ ETA = -W / DW
*
ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) )
TEMP = TAU + ETA
IF( TEMP.GT.SGUB .OR. TEMP.LT.SGLB ) THEN
IF( W.LT.ZERO ) THEN
ETA = ( SGUB-TAU ) / TWO
ELSE
ETA = ( SGLB-TAU ) / TWO
END IF
IF( GEOMAVG ) THEN
IF( W .LT. ZERO ) THEN
IF( TAU .GT. ZERO ) THEN
ETA = SQRT(SGUB*TAU)-TAU
END IF
ELSE
IF( SGLB .GT. ZERO ) THEN
ETA = SQRT(SGLB*TAU)-TAU
END IF
END IF
END IF
END IF
*
PREW = W
*
TAU = TAU + ETA
SIGMA = SIGMA + ETA
*
DO 170 J = 1, N
WORK( J ) = WORK( J ) + ETA
DELTA( J ) = DELTA( J ) - ETA
170 CONTINUE
*
* Evaluate PSI and the derivative DPSI
*
DPSI = ZERO
PSI = ZERO
ERRETM = ZERO
DO 180 J = 1, IIM1
TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
PSI = PSI + Z( J )*TEMP
DPSI = DPSI + TEMP*TEMP
ERRETM = ERRETM + PSI
180 CONTINUE
ERRETM = ABS( ERRETM )
*
* Evaluate PHI and the derivative DPHI
*
DPHI = ZERO
PHI = ZERO
DO 190 J = N, IIP1, -1
TEMP = Z( J ) / ( WORK( J )*DELTA( J ) )
PHI = PHI + Z( J )*TEMP
DPHI = DPHI + TEMP*TEMP
ERRETM = ERRETM + PHI
190 CONTINUE
*
TAU2 = WORK( II )*DELTA( II )
TEMP = Z( II ) / TAU2
DW = DPSI + DPHI + TEMP*TEMP
TEMP = Z( II )*TEMP
W = RHOINV + PHI + PSI + TEMP
ERRETM = EIGHT*( PHI-PSI ) + ERRETM + TWO*RHOINV
$ + THREE*ABS( TEMP )
* $ + ABS( TAU2 )*DW
*
SWTCH = .FALSE.
IF( ORGATI ) THEN
IF( -W.GT.ABS( PREW ) / TEN )
$ SWTCH = .TRUE.
ELSE
IF( W.GT.ABS( PREW ) / TEN )
$ SWTCH = .TRUE.
END IF
*
* Main loop to update the values of the array DELTA and WORK
*
ITER = NITER + 1
*
DO 230 NITER = ITER, MAXIT
*
* Test for convergence
*
IF( ABS( W ).LE.EPS*ERRETM ) THEN
* $ .OR. (SGUB-SGLB).LE.EIGHT*ABS(SGUB+SGLB) ) THEN
GO TO 240
END IF
*
IF( W.LE.ZERO ) THEN
SGLB = MAX( SGLB, TAU )
ELSE
SGUB = MIN( SGUB, TAU )
END IF
*
* Calculate the new step
*
IF( .NOT.SWTCH3 ) THEN
DTIPSQ = WORK( IP1 )*DELTA( IP1 )
DTISQ = WORK( I )*DELTA( I )
IF( .NOT.SWTCH ) THEN
IF( ORGATI ) THEN
C = W - DTIPSQ*DW + DELSQ*( Z( I ) / DTISQ )**2
ELSE
C = W - DTISQ*DW - DELSQ*( Z( IP1 ) / DTIPSQ )**2
END IF
ELSE
TEMP = Z( II ) / ( WORK( II )*DELTA( II ) )
IF( ORGATI ) THEN
DPSI = DPSI + TEMP*TEMP
ELSE
DPHI = DPHI + TEMP*TEMP
END IF
C = W - DTISQ*DPSI - DTIPSQ*DPHI
END IF
A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW
B = DTIPSQ*DTISQ*W
IF( C.EQ.ZERO ) THEN
IF( A.EQ.ZERO ) THEN
IF( .NOT.SWTCH ) THEN
IF( ORGATI ) THEN
A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*
$ ( DPSI+DPHI )
ELSE
A = Z( IP1 )*Z( IP1 ) +
$ DTISQ*DTISQ*( DPSI+DPHI )
END IF
ELSE
A = DTISQ*DTISQ*DPSI + DTIPSQ*DTIPSQ*DPHI
END IF
END IF
ETA = B / A
ELSE IF( A.LE.ZERO ) THEN
ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
ELSE
*
* Interpolation using THREE most relevant poles
*
DTIIM = WORK( IIM1 )*DELTA( IIM1 )
DTIIP = WORK( IIP1 )*DELTA( IIP1 )
TEMP = RHOINV + PSI + PHI
IF( SWTCH ) THEN
C = TEMP - DTIIM*DPSI - DTIIP*DPHI
ZZ( 1 ) = DTIIM*DTIIM*DPSI
ZZ( 3 ) = DTIIP*DTIIP*DPHI
ELSE
IF( ORGATI ) THEN
TEMP1 = Z( IIM1 ) / DTIIM
TEMP1 = TEMP1*TEMP1
TEMP2 = ( D( IIM1 )-D( IIP1 ) )*
$ ( D( IIM1 )+D( IIP1 ) )*TEMP1
C = TEMP - DTIIP*( DPSI+DPHI ) - TEMP2
ZZ( 1 ) = Z( IIM1 )*Z( IIM1 )
IF( DPSI.LT.TEMP1 ) THEN
ZZ( 3 ) = DTIIP*DTIIP*DPHI
ELSE
ZZ( 3 ) = DTIIP*DTIIP*( ( DPSI-TEMP1 )+DPHI )
END IF
ELSE
TEMP1 = Z( IIP1 ) / DTIIP
TEMP1 = TEMP1*TEMP1
TEMP2 = ( D( IIP1 )-D( IIM1 ) )*
$ ( D( IIM1 )+D( IIP1 ) )*TEMP1
C = TEMP - DTIIM*( DPSI+DPHI ) - TEMP2
IF( DPHI.LT.TEMP1 ) THEN
ZZ( 1 ) = DTIIM*DTIIM*DPSI
ELSE
ZZ( 1 ) = DTIIM*DTIIM*( DPSI+( DPHI-TEMP1 ) )
END IF
ZZ( 3 ) = Z( IIP1 )*Z( IIP1 )
END IF
END IF
DD( 1 ) = DTIIM
DD( 2 ) = DELTA( II )*WORK( II )
DD( 3 ) = DTIIP
CALL DLAED6( NITER, ORGATI, C, DD, ZZ, W, ETA, INFO )
*
IF( INFO.NE.0 ) THEN
*
* If INFO is not 0, i.e., DLAED6 failed, switch
* back to two pole interpolation
*
SWTCH3 = .FALSE.
INFO = 0
DTIPSQ = WORK( IP1 )*DELTA( IP1 )
DTISQ = WORK( I )*DELTA( I )
IF( .NOT.SWTCH ) THEN
IF( ORGATI ) THEN
C = W - DTIPSQ*DW + DELSQ*( Z( I )/DTISQ )**2
ELSE
C = W - DTISQ*DW - DELSQ*( Z( IP1 )/DTIPSQ )**2
END IF
ELSE
TEMP = Z( II ) / ( WORK( II )*DELTA( II ) )
IF( ORGATI ) THEN
DPSI = DPSI + TEMP*TEMP
ELSE
DPHI = DPHI + TEMP*TEMP
END IF
C = W - DTISQ*DPSI - DTIPSQ*DPHI
END IF
A = ( DTIPSQ+DTISQ )*W - DTIPSQ*DTISQ*DW
B = DTIPSQ*DTISQ*W
IF( C.EQ.ZERO ) THEN
IF( A.EQ.ZERO ) THEN
IF( .NOT.SWTCH ) THEN
IF( ORGATI ) THEN
A = Z( I )*Z( I ) + DTIPSQ*DTIPSQ*
$ ( DPSI+DPHI )
ELSE
A = Z( IP1 )*Z( IP1 ) +
$ DTISQ*DTISQ*( DPSI+DPHI )
END IF
ELSE
A = DTISQ*DTISQ*DPSI + DTIPSQ*DTIPSQ*DPHI
END IF
END IF
ETA = B / A
ELSE IF( A.LE.ZERO ) THEN
ETA = ( A-SQRT( ABS( A*A-FOUR*B*C ) ) ) / ( TWO*C )
ELSE
ETA = TWO*B / ( A+SQRT( ABS( A*A-FOUR*B*C ) ) )
END IF
END IF
END IF
*
* Note, eta should be positive if w is negative, and
* eta should be negative otherwise. However,
* if for some reason caused by roundoff, eta*w > 0,
* we simply use one Newton step instead. This way
* will guarantee eta*w < 0.
*
IF( W*ETA.GE.ZERO )
$ ETA = -W / DW
*
ETA = ETA / ( SIGMA+SQRT( SIGMA*SIGMA+ETA ) )
TEMP=TAU+ETA
IF( TEMP.GT.SGUB .OR. TEMP.LT.SGLB ) THEN
IF( W.LT.ZERO ) THEN
ETA = ( SGUB-TAU ) / TWO
ELSE
ETA = ( SGLB-TAU ) / TWO
END IF
IF( GEOMAVG ) THEN
IF( W .LT. ZERO ) THEN
IF( TAU .GT. ZERO ) THEN
ETA = SQRT(SGUB*TAU)-TAU
END IF
ELSE
IF( SGLB .GT. ZERO ) THEN
ETA = SQRT(SGLB*TAU)-TAU
END IF
END IF
END IF
END IF