forked from Reference-LAPACK/lapack
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathchetf2_rk.f
1036 lines (1036 loc) · 32.8 KB
/
chetf2_rk.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 CHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CHETF2_RK + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chetf2_rk.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/chetf2_rk.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/chetf2_rk.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CHETF2_RK( UPLO, N, A, LDA, E, IPIV, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* COMPLEX A( LDA, * ), E ( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*> CHETF2_RK computes the factorization of a complex Hermitian matrix A
*> using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
*>
*> A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
*>
*> where U (or L) is unit upper (or lower) triangular matrix,
*> U**H (or L**H) is the conjugate of U (or L), P is a permutation
*> matrix, P**T is the transpose of P, and D is Hermitian and block
*> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
*>
*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
*> For more information see Further Details section.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> Hermitian matrix A is stored:
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA,N)
*> On entry, the Hermitian matrix A.
*> If UPLO = 'U': the leading N-by-N upper triangular part
*> of A contains the upper triangular part of the matrix A,
*> and the strictly lower triangular part of A is not
*> referenced.
*>
*> If UPLO = 'L': the leading N-by-N lower triangular part
*> of A contains the lower triangular part of the matrix A,
*> and the strictly upper triangular part of A is not
*> referenced.
*>
*> On exit, contains:
*> a) ONLY diagonal elements of the Hermitian block diagonal
*> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
*> (superdiagonal (or subdiagonal) elements of D
*> are stored on exit in array E), and
*> b) If UPLO = 'U': factor U in the superdiagonal part of A.
*> If UPLO = 'L': factor L in the subdiagonal part of A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is COMPLEX array, dimension (N)
*> On exit, contains the superdiagonal (or subdiagonal)
*> elements of the Hermitian block diagonal matrix D
*> with 1-by-1 or 2-by-2 diagonal blocks, where
*> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
*> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
*>
*> NOTE: For 1-by-1 diagonal block D(k), where
*> 1 <= k <= N, the element E(k) is set to 0 in both
*> UPLO = 'U' or UPLO = 'L' cases.
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> IPIV describes the permutation matrix P in the factorization
*> of matrix A as follows. The absolute value of IPIV(k)
*> represents the index of row and column that were
*> interchanged with the k-th row and column. The value of UPLO
*> describes the order in which the interchanges were applied.
*> Also, the sign of IPIV represents the block structure of
*> the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
*> diagonal blocks which correspond to 1 or 2 interchanges
*> at each factorization step. For more info see Further
*> Details section.
*>
*> If UPLO = 'U',
*> ( in factorization order, k decreases from N to 1 ):
*> a) A single positive entry IPIV(k) > 0 means:
*> D(k,k) is a 1-by-1 diagonal block.
*> If IPIV(k) != k, rows and columns k and IPIV(k) were
*> interchanged in the matrix A(1:N,1:N);
*> If IPIV(k) = k, no interchange occurred.
*>
*> b) A pair of consecutive negative entries
*> IPIV(k) < 0 and IPIV(k-1) < 0 means:
*> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
*> (NOTE: negative entries in IPIV appear ONLY in pairs).
*> 1) If -IPIV(k) != k, rows and columns
*> k and -IPIV(k) were interchanged
*> in the matrix A(1:N,1:N).
*> If -IPIV(k) = k, no interchange occurred.
*> 2) If -IPIV(k-1) != k-1, rows and columns
*> k-1 and -IPIV(k-1) were interchanged
*> in the matrix A(1:N,1:N).
*> If -IPIV(k-1) = k-1, no interchange occurred.
*>
*> c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
*>
*> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
*>
*> If UPLO = 'L',
*> ( in factorization order, k increases from 1 to N ):
*> a) A single positive entry IPIV(k) > 0 means:
*> D(k,k) is a 1-by-1 diagonal block.
*> If IPIV(k) != k, rows and columns k and IPIV(k) were
*> interchanged in the matrix A(1:N,1:N).
*> If IPIV(k) = k, no interchange occurred.
*>
*> b) A pair of consecutive negative entries
*> IPIV(k) < 0 and IPIV(k+1) < 0 means:
*> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*> (NOTE: negative entries in IPIV appear ONLY in pairs).
*> 1) If -IPIV(k) != k, rows and columns
*> k and -IPIV(k) were interchanged
*> in the matrix A(1:N,1:N).
*> If -IPIV(k) = k, no interchange occurred.
*> 2) If -IPIV(k+1) != k+1, rows and columns
*> k-1 and -IPIV(k-1) were interchanged
*> in the matrix A(1:N,1:N).
*> If -IPIV(k+1) = k+1, no interchange occurred.
*>
*> c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
*>
*> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*>
*> < 0: If INFO = -k, the k-th argument had an illegal value
*>
*> > 0: If INFO = k, the matrix A is singular, because:
*> If UPLO = 'U': column k in the upper
*> triangular part of A contains all zeros.
*> If UPLO = 'L': column k in the lower
*> triangular part of A contains all zeros.
*>
*> Therefore D(k,k) is exactly zero, and superdiagonal
*> elements of column k of U (or subdiagonal elements of
*> column k of L ) are all zeros. The factorization has
*> been completed, but the block diagonal matrix D is
*> exactly singular, and division by zero will occur if
*> it is used to solve a system of equations.
*>
*> NOTE: INFO only stores the first occurrence of
*> a singularity, any subsequent occurrence of singularity
*> is not stored in INFO even though the factorization
*> always completes.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complexHEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*> TODO: put further details
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*>
*> December 2016, Igor Kozachenko,
*> Computer Science Division,
*> University of California, Berkeley
*>
*> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
*> School of Mathematics,
*> University of Manchester
*>
*> 01-01-96 - Based on modifications by
*> J. Lewis, Boeing Computer Services Company
*> A. Petitet, Computer Science Dept.,
*> Univ. of Tenn., Knoxville abd , USA
*> \endverbatim
*
* =====================================================================
SUBROUTINE CHETF2_RK( UPLO, N, A, LDA, E, IPIV, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
COMPLEX A( LDA, * ), E( * )
* ..
*
* ======================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
REAL EIGHT, SEVTEN
PARAMETER ( EIGHT = 8.0E+0, SEVTEN = 17.0E+0 )
COMPLEX CZERO
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
LOGICAL DONE, UPPER
INTEGER I, II, IMAX, ITEMP, J, JMAX, K, KK, KP, KSTEP,
$ P
REAL ABSAKK, ALPHA, COLMAX, D, D11, D22, R1, STEMP,
$ ROWMAX, TT, SFMIN
COMPLEX D12, D21, T, WK, WKM1, WKP1, Z
* ..
* .. External Functions ..
*
LOGICAL LSAME
INTEGER ICAMAX
REAL SLAMCH, SLAPY2
EXTERNAL LSAME, ICAMAX, SLAMCH, SLAPY2
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, CSSCAL, CHER, CSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, CMPLX, CONJG, MAX, REAL, SQRT
* ..
* .. Statement Functions ..
REAL CABS1
* ..
* .. Statement Function definitions ..
CABS1( Z ) = ABS( REAL( Z ) ) + ABS( AIMAG( Z ) )
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CHETF2_RK', -INFO )
RETURN
END IF
*
* Initialize ALPHA for use in choosing pivot block size.
*
ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
*
* Compute machine safe minimum
*
SFMIN = SLAMCH( 'S' )
*
IF( UPPER ) THEN
*
* Factorize A as U*D*U**H using the upper triangle of A
*
* Initialize the first entry of array E, where superdiagonal
* elements of D are stored
*
E( 1 ) = CZERO
*
* K is the main loop index, decreasing from N to 1 in steps of
* 1 or 2
*
K = N
10 CONTINUE
*
* If K < 1, exit from loop
*
IF( K.LT.1 )
$ GO TO 34
KSTEP = 1
P = K
*
* Determine rows and columns to be interchanged and whether
* a 1-by-1 or 2-by-2 pivot block will be used
*
ABSAKK = ABS( REAL( A( K, K ) ) )
*
* IMAX is the row-index of the largest off-diagonal element in
* column K, and COLMAX is its absolute value.
* Determine both COLMAX and IMAX.
*
IF( K.GT.1 ) THEN
IMAX = ICAMAX( K-1, A( 1, K ), 1 )
COLMAX = CABS1( A( IMAX, K ) )
ELSE
COLMAX = ZERO
END IF
*
IF( ( MAX( ABSAKK, COLMAX ).EQ.ZERO ) ) THEN
*
* Column K is zero or underflow: set INFO and continue
*
IF( INFO.EQ.0 )
$ INFO = K
KP = K
A( K, K ) = REAL( A( K, K ) )
*
* Set E( K ) to zero
*
IF( K.GT.1 )
$ E( K ) = CZERO
*
ELSE
*
* ============================================================
*
* BEGIN pivot search
*
* Case(1)
* Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
* (used to handle NaN and Inf)
*
IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
*
ELSE
*
DONE = .FALSE.
*
* Loop until pivot found
*
12 CONTINUE
*
* BEGIN pivot search loop body
*
*
* JMAX is the column-index of the largest off-diagonal
* element in row IMAX, and ROWMAX is its absolute value.
* Determine both ROWMAX and JMAX.
*
IF( IMAX.NE.K ) THEN
JMAX = IMAX + ICAMAX( K-IMAX, A( IMAX, IMAX+1 ),
$ LDA )
ROWMAX = CABS1( A( IMAX, JMAX ) )
ELSE
ROWMAX = ZERO
END IF
*
IF( IMAX.GT.1 ) THEN
ITEMP = ICAMAX( IMAX-1, A( 1, IMAX ), 1 )
STEMP = CABS1( A( ITEMP, IMAX ) )
IF( STEMP.GT.ROWMAX ) THEN
ROWMAX = STEMP
JMAX = ITEMP
END IF
END IF
*
* Case(2)
* Equivalent to testing for
* ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX
* (used to handle NaN and Inf)
*
IF( .NOT.( ABS( REAL( A( IMAX, IMAX ) ) )
$ .LT.ALPHA*ROWMAX ) ) THEN
*
* interchange rows and columns K and IMAX,
* use 1-by-1 pivot block
*
KP = IMAX
DONE = .TRUE.
*
* Case(3)
* Equivalent to testing for ROWMAX.EQ.COLMAX,
* (used to handle NaN and Inf)
*
ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
$ THEN
*
* interchange rows and columns K-1 and IMAX,
* use 2-by-2 pivot block
*
KP = IMAX
KSTEP = 2
DONE = .TRUE.
*
* Case(4)
ELSE
*
* Pivot not found: set params and repeat
*
P = IMAX
COLMAX = ROWMAX
IMAX = JMAX
END IF
*
* END pivot search loop body
*
IF( .NOT.DONE ) GOTO 12
*
END IF
*
* END pivot search
*
* ============================================================
*
* KK is the column of A where pivoting step stopped
*
KK = K - KSTEP + 1
*
* For only a 2x2 pivot, interchange rows and columns K and P
* in the leading submatrix A(1:k,1:k)
*
IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
* (1) Swap columnar parts
IF( P.GT.1 )
$ CALL CSWAP( P-1, A( 1, K ), 1, A( 1, P ), 1 )
* (2) Swap and conjugate middle parts
DO 14 J = P + 1, K - 1
T = CONJG( A( J, K ) )
A( J, K ) = CONJG( A( P, J ) )
A( P, J ) = T
14 CONTINUE
* (3) Swap and conjugate corner elements at row-col intersection
A( P, K ) = CONJG( A( P, K ) )
* (4) Swap diagonal elements at row-col intersection
R1 = REAL( A( K, K ) )
A( K, K ) = REAL( A( P, P ) )
A( P, P ) = R1
*
* Convert upper triangle of A into U form by applying
* the interchanges in columns k+1:N.
*
IF( K.LT.N )
$ CALL CSWAP( N-K, A( K, K+1 ), LDA, A( P, K+1 ), LDA )
*
END IF
*
* For both 1x1 and 2x2 pivots, interchange rows and
* columns KK and KP in the leading submatrix A(1:k,1:k)
*
IF( KP.NE.KK ) THEN
* (1) Swap columnar parts
IF( KP.GT.1 )
$ CALL CSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
* (2) Swap and conjugate middle parts
DO 15 J = KP + 1, KK - 1
T = CONJG( A( J, KK ) )
A( J, KK ) = CONJG( A( KP, J ) )
A( KP, J ) = T
15 CONTINUE
* (3) Swap and conjugate corner elements at row-col intersection
A( KP, KK ) = CONJG( A( KP, KK ) )
* (4) Swap diagonal elements at row-col intersection
R1 = REAL( A( KK, KK ) )
A( KK, KK ) = REAL( A( KP, KP ) )
A( KP, KP ) = R1
*
IF( KSTEP.EQ.2 ) THEN
* (*) Make sure that diagonal element of pivot is real
A( K, K ) = REAL( A( K, K ) )
* (5) Swap row elements
T = A( K-1, K )
A( K-1, K ) = A( KP, K )
A( KP, K ) = T
END IF
*
* Convert upper triangle of A into U form by applying
* the interchanges in columns k+1:N.
*
IF( K.LT.N )
$ CALL CSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
$ LDA )
*
ELSE
* (*) Make sure that diagonal element of pivot is real
A( K, K ) = REAL( A( K, K ) )
IF( KSTEP.EQ.2 )
$ A( K-1, K-1 ) = REAL( A( K-1, K-1 ) )
END IF
*
* Update the leading submatrix
*
IF( KSTEP.EQ.1 ) THEN
*
* 1-by-1 pivot block D(k): column k now holds
*
* W(k) = U(k)*D(k)
*
* where U(k) is the k-th column of U
*
IF( K.GT.1 ) THEN
*
* Perform a rank-1 update of A(1:k-1,1:k-1) and
* store U(k) in column k
*
IF( ABS( REAL( A( K, K ) ) ).GE.SFMIN ) THEN
*
* Perform a rank-1 update of A(1:k-1,1:k-1) as
* A := A - U(k)*D(k)*U(k)**T
* = A - W(k)*1/D(k)*W(k)**T
*
D11 = ONE / REAL( A( K, K ) )
CALL CHER( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
*
* Store U(k) in column k
*
CALL CSSCAL( K-1, D11, A( 1, K ), 1 )
ELSE
*
* Store L(k) in column K
*
D11 = REAL( A( K, K ) )
DO 16 II = 1, K - 1
A( II, K ) = A( II, K ) / D11
16 CONTINUE
*
* Perform a rank-1 update of A(k+1:n,k+1:n) as
* A := A - U(k)*D(k)*U(k)**T
* = A - W(k)*(1/D(k))*W(k)**T
* = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
*
CALL CHER( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
END IF
*
* Store the superdiagonal element of D in array E
*
E( K ) = CZERO
*
END IF
*
ELSE
*
* 2-by-2 pivot block D(k): columns k and k-1 now hold
*
* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
*
* where U(k) and U(k-1) are the k-th and (k-1)-th columns
* of U
*
* Perform a rank-2 update of A(1:k-2,1:k-2) as
*
* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
* = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T
*
* and store L(k) and L(k+1) in columns k and k+1
*
IF( K.GT.2 ) THEN
* D = |A12|
D = SLAPY2( REAL( A( K-1, K ) ),
$ AIMAG( A( K-1, K ) ) )
D11 = REAL( A( K, K ) / D )
D22 = REAL( A( K-1, K-1 ) / D )
D12 = A( K-1, K ) / D
TT = ONE / ( D11*D22-ONE )
*
DO 30 J = K - 2, 1, -1
*
* Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
*
WKM1 = TT*( D11*A( J, K-1 )-CONJG( D12 )*
$ A( J, K ) )
WK = TT*( D22*A( J, K )-D12*A( J, K-1 ) )
*
* Perform a rank-2 update of A(1:k-2,1:k-2)
*
DO 20 I = J, 1, -1
A( I, J ) = A( I, J ) -
$ ( A( I, K ) / D )*CONJG( WK ) -
$ ( A( I, K-1 ) / D )*CONJG( WKM1 )
20 CONTINUE
*
* Store U(k) and U(k-1) in cols k and k-1 for row J
*
A( J, K ) = WK / D
A( J, K-1 ) = WKM1 / D
* (*) Make sure that diagonal element of pivot is real
A( J, J ) = CMPLX( REAL( A( J, J ) ), ZERO )
*
30 CONTINUE
*
END IF
*
* Copy superdiagonal elements of D(K) to E(K) and
* ZERO out superdiagonal entry of A
*
E( K ) = A( K-1, K )
E( K-1 ) = CZERO
A( K-1, K ) = CZERO
*
END IF
*
* End column K is nonsingular
*
END IF
*
* Store details of the interchanges in IPIV
*
IF( KSTEP.EQ.1 ) THEN
IPIV( K ) = KP
ELSE
IPIV( K ) = -P
IPIV( K-1 ) = -KP
END IF
*
* Decrease K and return to the start of the main loop
*
K = K - KSTEP
GO TO 10
*
34 CONTINUE
*
ELSE
*
* Factorize A as L*D*L**H using the lower triangle of A
*
* Initialize the unused last entry of the subdiagonal array E.
*
E( N ) = CZERO
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2
*
K = 1
40 CONTINUE
*
* If K > N, exit from loop
*
IF( K.GT.N )
$ GO TO 64
KSTEP = 1
P = K
*
* Determine rows and columns to be interchanged and whether
* a 1-by-1 or 2-by-2 pivot block will be used
*
ABSAKK = ABS( REAL( A( K, K ) ) )
*
* IMAX is the row-index of the largest off-diagonal element in
* column K, and COLMAX is its absolute value.
* Determine both COLMAX and IMAX.
*
IF( K.LT.N ) THEN
IMAX = K + ICAMAX( N-K, A( K+1, K ), 1 )
COLMAX = CABS1( A( IMAX, K ) )
ELSE
COLMAX = ZERO
END IF
*
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*
* Column K is zero or underflow: set INFO and continue
*
IF( INFO.EQ.0 )
$ INFO = K
KP = K
A( K, K ) = REAL( A( K, K ) )
*
* Set E( K ) to zero
*
IF( K.LT.N )
$ E( K ) = CZERO
*
ELSE
*
* ============================================================
*
* BEGIN pivot search
*
* Case(1)
* Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
* (used to handle NaN and Inf)
*
IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
*
* no interchange, use 1-by-1 pivot block
*
KP = K
*
ELSE
*
DONE = .FALSE.
*
* Loop until pivot found
*
42 CONTINUE
*
* BEGIN pivot search loop body
*
*
* JMAX is the column-index of the largest off-diagonal
* element in row IMAX, and ROWMAX is its absolute value.
* Determine both ROWMAX and JMAX.
*
IF( IMAX.NE.K ) THEN
JMAX = K - 1 + ICAMAX( IMAX-K, A( IMAX, K ), LDA )
ROWMAX = CABS1( A( IMAX, JMAX ) )
ELSE
ROWMAX = ZERO
END IF
*
IF( IMAX.LT.N ) THEN
ITEMP = IMAX + ICAMAX( N-IMAX, A( IMAX+1, IMAX ),
$ 1 )
STEMP = CABS1( A( ITEMP, IMAX ) )
IF( STEMP.GT.ROWMAX ) THEN
ROWMAX = STEMP
JMAX = ITEMP
END IF
END IF
*
* Case(2)
* Equivalent to testing for
* ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX
* (used to handle NaN and Inf)
*
IF( .NOT.( ABS( REAL( A( IMAX, IMAX ) ) )
$ .LT.ALPHA*ROWMAX ) ) THEN
*
* interchange rows and columns K and IMAX,
* use 1-by-1 pivot block
*
KP = IMAX
DONE = .TRUE.
*
* Case(3)
* Equivalent to testing for ROWMAX.EQ.COLMAX,
* (used to handle NaN and Inf)
*
ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
$ THEN
*
* interchange rows and columns K+1 and IMAX,
* use 2-by-2 pivot block
*
KP = IMAX
KSTEP = 2
DONE = .TRUE.
*
* Case(4)
ELSE
*
* Pivot not found: set params and repeat
*
P = IMAX
COLMAX = ROWMAX
IMAX = JMAX
END IF
*
*
* END pivot search loop body
*
IF( .NOT.DONE ) GOTO 42
*
END IF
*
* END pivot search
*
* ============================================================
*
* KK is the column of A where pivoting step stopped
*
KK = K + KSTEP - 1
*
* For only a 2x2 pivot, interchange rows and columns K and P
* in the trailing submatrix A(k:n,k:n)
*
IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
* (1) Swap columnar parts
IF( P.LT.N )
$ CALL CSWAP( N-P, A( P+1, K ), 1, A( P+1, P ), 1 )
* (2) Swap and conjugate middle parts
DO 44 J = K + 1, P - 1
T = CONJG( A( J, K ) )
A( J, K ) = CONJG( A( P, J ) )
A( P, J ) = T
44 CONTINUE
* (3) Swap and conjugate corner elements at row-col intersection
A( P, K ) = CONJG( A( P, K ) )
* (4) Swap diagonal elements at row-col intersection
R1 = REAL( A( K, K ) )
A( K, K ) = REAL( A( P, P ) )
A( P, P ) = R1
*
* Convert lower triangle of A into L form by applying
* the interchanges in columns 1:k-1.
*
IF ( K.GT.1 )
$ CALL CSWAP( K-1, A( K, 1 ), LDA, A( P, 1 ), LDA )
*
END IF
*
* For both 1x1 and 2x2 pivots, interchange rows and
* columns KK and KP in the trailing submatrix A(k:n,k:n)
*
IF( KP.NE.KK ) THEN
* (1) Swap columnar parts
IF( KP.LT.N )
$ CALL CSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
* (2) Swap and conjugate middle parts
DO 45 J = KK + 1, KP - 1
T = CONJG( A( J, KK ) )
A( J, KK ) = CONJG( A( KP, J ) )
A( KP, J ) = T
45 CONTINUE
* (3) Swap and conjugate corner elements at row-col intersection
A( KP, KK ) = CONJG( A( KP, KK ) )
* (4) Swap diagonal elements at row-col intersection
R1 = REAL( A( KK, KK ) )
A( KK, KK ) = REAL( A( KP, KP ) )
A( KP, KP ) = R1
*
IF( KSTEP.EQ.2 ) THEN
* (*) Make sure that diagonal element of pivot is real
A( K, K ) = REAL( A( K, K ) )
* (5) Swap row elements
T = A( K+1, K )
A( K+1, K ) = A( KP, K )
A( KP, K ) = T
END IF
*
* Convert lower triangle of A into L form by applying
* the interchanges in columns 1:k-1.
*
IF ( K.GT.1 )
$ CALL CSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
*
ELSE
* (*) Make sure that diagonal element of pivot is real
A( K, K ) = REAL( A( K, K ) )
IF( KSTEP.EQ.2 )
$ A( K+1, K+1 ) = REAL( A( K+1, K+1 ) )
END IF
*
* Update the trailing submatrix
*
IF( KSTEP.EQ.1 ) THEN
*
* 1-by-1 pivot block D(k): column k of A now holds
*
* W(k) = L(k)*D(k),
*
* where L(k) is the k-th column of L
*
IF( K.LT.N ) THEN
*
* Perform a rank-1 update of A(k+1:n,k+1:n) and
* store L(k) in column k
*
* Handle division by a small number
*
IF( ABS( REAL( A( K, K ) ) ).GE.SFMIN ) THEN
*
* Perform a rank-1 update of A(k+1:n,k+1:n) as
* A := A - L(k)*D(k)*L(k)**T
* = A - W(k)*(1/D(k))*W(k)**T
*
D11 = ONE / REAL( A( K, K ) )
CALL CHER( UPLO, N-K, -D11, A( K+1, K ), 1,
$ A( K+1, K+1 ), LDA )
*
* Store L(k) in column k
*
CALL CSSCAL( N-K, D11, A( K+1, K ), 1 )
ELSE
*
* Store L(k) in column k
*
D11 = REAL( A( K, K ) )
DO 46 II = K + 1, N
A( II, K ) = A( II, K ) / D11
46 CONTINUE
*
* Perform a rank-1 update of A(k+1:n,k+1:n) as
* A := A - L(k)*D(k)*L(k)**T
* = A - W(k)*(1/D(k))*W(k)**T
* = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
*
CALL CHER( UPLO, N-K, -D11, A( K+1, K ), 1,
$ A( K+1, K+1 ), LDA )
END IF
*
* Store the subdiagonal element of D in array E
*
E( K ) = CZERO
*
END IF
*
ELSE
*
* 2-by-2 pivot block D(k): columns k and k+1 now hold
*
* ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
*
* where L(k) and L(k+1) are the k-th and (k+1)-th columns
* of L
*
*
* Perform a rank-2 update of A(k+2:n,k+2:n) as
*
* A := A - ( L(k) L(k+1) ) * D(k) * ( L(k) L(k+1) )**T
* = A - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T
*
* and store L(k) and L(k+1) in columns k and k+1
*
IF( K.LT.N-1 ) THEN
* D = |A21|
D = SLAPY2( REAL( A( K+1, K ) ),
$ AIMAG( A( K+1, K ) ) )
D11 = REAL( A( K+1, K+1 ) ) / D
D22 = REAL( A( K, K ) ) / D
D21 = A( K+1, K ) / D
TT = ONE / ( D11*D22-ONE )
*
DO 60 J = K + 2, N
*
* Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
*
WK = TT*( D11*A( J, K )-D21*A( J, K+1 ) )
WKP1 = TT*( D22*A( J, K+1 )-CONJG( D21 )*
$ A( J, K ) )
*
* Perform a rank-2 update of A(k+2:n,k+2:n)
*
DO 50 I = J, N
A( I, J ) = A( I, J ) -
$ ( A( I, K ) / D )*CONJG( WK ) -
$ ( A( I, K+1 ) / D )*CONJG( WKP1 )
50 CONTINUE
*
* Store L(k) and L(k+1) in cols k and k+1 for row J
*
A( J, K ) = WK / D
A( J, K+1 ) = WKP1 / D
* (*) Make sure that diagonal element of pivot is real
A( J, J ) = CMPLX( REAL( A( J, J ) ), ZERO )
*
60 CONTINUE
*
END IF
*