forked from Reference-LAPACK/lapack
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathdlasyf_rook.f
892 lines (891 loc) · 26.7 KB
/
dlasyf_rook.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
*> \brief \b DLASYF_ROOK *> DLASYF_ROOK computes a partial factorization of a real symmetric matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASYF_ROOK + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasyf_rook.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasyf_rook.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasyf_rook.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASYF_ROOK( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, KB, LDA, LDW, N, NB
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * ), W( LDW, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASYF_ROOK computes a partial factorization of a real symmetric
*> matrix A using the bounded Bunch-Kaufman ("rook") diagonal
*> pivoting method. The partial factorization has the form:
*>
*> A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
*> ( 0 U22 ) ( 0 D ) ( U12**T U22**T )
*>
*> A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L'
*> ( L21 I ) ( 0 A22 ) ( 0 I )
*>
*> where the order of D is at most NB. The actual order is returned in
*> the argument KB, and is either NB or NB-1, or N if N <= NB.
*>
*> DLASYF_ROOK is an auxiliary routine called by DSYTRF_ROOK. It uses
*> blocked code (calling Level 3 BLAS) to update the submatrix
*> A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric 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] NB
*> \verbatim
*> NB is INTEGER
*> The maximum number of columns of the matrix A that should be
*> factored. NB should be at least 2 to allow for 2-by-2 pivot
*> blocks.
*> \endverbatim
*>
*> \param[out] KB
*> \verbatim
*> KB is INTEGER
*> The number of columns of A that were actually factored.
*> KB is either NB-1 or NB, or N if N <= NB.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric 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, A contains details of the partial factorization.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D.
*>
*> If UPLO = 'U':
*> Only the last KB elements of IPIV are set.
*>
*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*> interchanged and D(k,k) is a 1-by-1 diagonal block.
*>
*> If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
*> columns k and -IPIV(k) were interchanged and rows and
*> columns k-1 and -IPIV(k-1) were inerchaged,
*> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
*>
*> If UPLO = 'L':
*> Only the first KB elements of IPIV are set.
*>
*> If IPIV(k) > 0, then rows and columns k and IPIV(k)
*> were interchanged and D(k,k) is a 1-by-1 diagonal block.
*>
*> If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
*> columns k and -IPIV(k) were interchanged and rows and
*> columns k+1 and -IPIV(k+1) were inerchaged,
*> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (LDW,NB)
*> \endverbatim
*>
*> \param[in] LDW
*> \verbatim
*> LDW is INTEGER
*> The leading dimension of the array W. LDW >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
*> has been completed, but the block diagonal matrix D is
*> exactly singular.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2013
*
*> \ingroup doubleSYcomputational
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*>
*> November 2013, Igor Kozachenko,
*> Computer Science Division,
*> University of California, Berkeley
*>
*> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
*> School of Mathematics,
*> University of Manchester
*>
*> \endverbatim
*
* =====================================================================
SUBROUTINE DLASYF_ROOK( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW,
$ INFO )
*
* -- LAPACK computational routine (version 3.5.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2013
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, KB, LDA, LDW, N, NB
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), W( LDW, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION EIGHT, SEVTEN
PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL DONE
INTEGER IMAX, ITEMP, J, JB, JJ, JMAX, JP1, JP2, K, KK,
$ KW, KKW, KP, KSTEP, P, II
DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
$ DTEMP, R1, ROWMAX, T, SFMIN
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, IDAMAX, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DGEMV, DSCAL, DSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Initialize ALPHA for use in choosing pivot block size.
*
ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
*
* Compute machine safe minimum
*
SFMIN = DLAMCH( 'S' )
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Factorize the trailing columns of A using the upper triangle
* of A and working backwards, and compute the matrix W = U12*D
* for use in updating A11
*
* K is the main loop index, decreasing from N in steps of 1 or 2
*
K = N
10 CONTINUE
*
* KW is the column of W which corresponds to column K of A
*
KW = NB + K - N
*
* Exit from loop
*
IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
$ GO TO 30
*
KSTEP = 1
P = K
*
* Copy column K of A to column KW of W and update it
*
CALL DCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
IF( K.LT.N )
$ CALL DGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ),
$ LDA, W( K, KW+1 ), LDW, ONE, W( 1, KW ), 1 )
*
* Determine rows and columns to be interchanged and whether
* a 1-by-1 or 2-by-2 pivot block will be used
*
ABSAKK = ABS( W( K, KW ) )
*
* 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 = IDAMAX( K-1, W( 1, KW ), 1 )
COLMAX = ABS( W( IMAX, KW ) )
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
CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
ELSE
*
* ============================================================
*
* Test for interchange
*
* 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
*
*
* Copy column IMAX to column KW-1 of W and update it
*
CALL DCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
CALL DCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
$ W( IMAX+1, KW-1 ), 1 )
*
IF( K.LT.N )
$ CALL DGEMV( 'No transpose', K, N-K, -ONE,
$ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
$ ONE, W( 1, KW-1 ), 1 )
*
* 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 + IDAMAX( K-IMAX, W( IMAX+1, KW-1 ),
$ 1 )
ROWMAX = ABS( W( JMAX, KW-1 ) )
ELSE
ROWMAX = ZERO
END IF
*
IF( IMAX.GT.1 ) THEN
ITEMP = IDAMAX( IMAX-1, W( 1, KW-1 ), 1 )
DTEMP = ABS( W( ITEMP, KW-1 ) )
IF( DTEMP.GT.ROWMAX ) THEN
ROWMAX = DTEMP
JMAX = ITEMP
END IF
END IF
*
* Equivalent to testing for
* ABS( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX
* (used to handle NaN and Inf)
*
IF( .NOT.(ABS( W( IMAX, KW-1 ) ).LT.ALPHA*ROWMAX ) )
$ THEN
*
* interchange rows and columns K and IMAX,
* use 1-by-1 pivot block
*
KP = IMAX
*
* copy column KW-1 of W to column KW of W
*
CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
*
DONE = .TRUE.
*
* 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.
ELSE
*
* Pivot not found: set params and repeat
*
P = IMAX
COLMAX = ROWMAX
IMAX = JMAX
*
* Copy updated JMAXth (next IMAXth) column to Kth of W
*
CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
*
END IF
*
* End pivot search loop body
*
IF( .NOT. DONE ) GOTO 12
*
END IF
*
* ============================================================
*
KK = K - KSTEP + 1
*
* KKW is the column of W which corresponds to column KK of A
*
KKW = NB + KK - N
*
IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
*
* Copy non-updated column K to column P
*
CALL DCOPY( K-P, A( P+1, K ), 1, A( P, P+1 ), LDA )
CALL DCOPY( P, A( 1, K ), 1, A( 1, P ), 1 )
*
* Interchange rows K and P in last N-K+1 columns of A
* and last N-K+2 columns of W
*
CALL DSWAP( N-K+1, A( K, K ), LDA, A( P, K ), LDA )
CALL DSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ), LDW )
END IF
*
* Updated column KP is already stored in column KKW of W
*
IF( KP.NE.KK ) THEN
*
* Copy non-updated column KK to column KP
*
A( KP, K ) = A( KK, K )
CALL DCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
$ LDA )
CALL DCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 )
*
* Interchange rows KK and KP in last N-KK+1 columns
* of A and W
*
CALL DSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA )
CALL DSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
$ LDW )
END IF
*
IF( KSTEP.EQ.1 ) THEN
*
* 1-by-1 pivot block D(k): column KW of W now holds
*
* W(k) = U(k)*D(k)
*
* where U(k) is the k-th column of U
*
* Store U(k) in column k of A
*
CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
IF( K.GT.1 ) THEN
IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
R1 = ONE / A( K, K )
CALL DSCAL( K-1, R1, A( 1, K ), 1 )
ELSE IF( A( K, K ).NE.ZERO ) THEN
DO 14 II = 1, K - 1
A( II, K ) = A( II, K ) / A( K, K )
14 CONTINUE
END IF
END IF
*
ELSE
*
* 2-by-2 pivot block D(k): columns KW and KW-1 of W 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
*
IF( K.GT.2 ) THEN
*
* Store U(k) and U(k-1) in columns k and k-1 of A
*
D12 = W( K-1, KW )
D11 = W( K, KW ) / D12
D22 = W( K-1, KW-1 ) / D12
T = ONE / ( D11*D22-ONE )
DO 20 J = 1, K - 2
A( J, K-1 ) = T*( (D11*W( J, KW-1 )-W( J, KW ) ) /
$ D12 )
A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
$ D12 )
20 CONTINUE
END IF
*
* Copy D(k) to A
*
A( K-1, K-1 ) = W( K-1, KW-1 )
A( K-1, K ) = W( K-1, KW )
A( K, K ) = W( K, KW )
END IF
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
*
30 CONTINUE
*
* Update the upper triangle of A11 (= A(1:k,1:k)) as
*
* A11 := A11 - U12*D*U12**T = A11 - U12*W**T
*
* computing blocks of NB columns at a time
*
DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
JB = MIN( NB, K-J+1 )
*
* Update the upper triangle of the diagonal block
*
DO 40 JJ = J, J + JB - 1
CALL DGEMV( 'No transpose', JJ-J+1, N-K, -ONE,
$ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, ONE,
$ A( J, JJ ), 1 )
40 CONTINUE
*
* Update the rectangular superdiagonal block
*
IF( J.GE.2 )
$ CALL DGEMM( 'No transpose', 'Transpose', J-1, JB,
$ N-K, -ONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
$ ONE, A( 1, J ), LDA )
50 CONTINUE
*
* Put U12 in standard form by partially undoing the interchanges
* in columns k+1:n
*
J = K + 1
60 CONTINUE
*
KSTEP = 1
JP1 = 1
JJ = J
JP2 = IPIV( J )
IF( JP2.LT.0 ) THEN
JP2 = -JP2
J = J + 1
JP1 = -IPIV( J )
KSTEP = 2
END IF
*
J = J + 1
IF( JP2.NE.JJ .AND. J.LE.N )
$ CALL DSWAP( N-J+1, A( JP2, J ), LDA, A( JJ, J ), LDA )
JJ = J - 1
IF( JP1.NE.JJ .AND. KSTEP.EQ.2 )
$ CALL DSWAP( N-J+1, A( JP1, J ), LDA, A( JJ, J ), LDA )
IF( J.LE.N )
$ GO TO 60
*
* Set KB to the number of columns factorized
*
KB = N - K
*
ELSE
*
* Factorize the leading columns of A using the lower triangle
* of A and working forwards, and compute the matrix W = L21*D
* for use in updating A22
*
* K is the main loop index, increasing from 1 in steps of 1 or 2
*
K = 1
70 CONTINUE
*
* Exit from loop
*
IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
$ GO TO 90
*
KSTEP = 1
P = K
*
* Copy column K of A to column K of W and update it
*
CALL DCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
IF( K.GT.1 )
$ CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ),
$ LDA, W( K, 1 ), LDW, ONE, W( K, K ), 1 )
*
* Determine rows and columns to be interchanged and whether
* a 1-by-1 or 2-by-2 pivot block will be used
*
ABSAKK = ABS( W( 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 + IDAMAX( N-K, W( K+1, K ), 1 )
COLMAX = ABS( W( 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
CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
ELSE
*
* ============================================================
*
* Test for interchange
*
* 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
*
72 CONTINUE
*
* Begin pivot search loop body
*
*
* Copy column IMAX to column K+1 of W and update it
*
CALL DCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
CALL DCOPY( N-IMAX+1, A( IMAX, IMAX ), 1,
$ W( IMAX, K+1 ), 1 )
IF( K.GT.1 )
$ CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE,
$ A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
$ ONE, W( K, K+1 ), 1 )
*
* 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 + IDAMAX( IMAX-K, W( K, K+1 ), 1 )
ROWMAX = ABS( W( JMAX, K+1 ) )
ELSE
ROWMAX = ZERO
END IF
*
IF( IMAX.LT.N ) THEN
ITEMP = IMAX + IDAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
DTEMP = ABS( W( ITEMP, K+1 ) )
IF( DTEMP.GT.ROWMAX ) THEN
ROWMAX = DTEMP
JMAX = ITEMP
END IF
END IF
*
* Equivalent to testing for
* ABS( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX
* (used to handle NaN and Inf)
*
IF( .NOT.( ABS( W( IMAX, K+1 ) ).LT.ALPHA*ROWMAX ) )
$ THEN
*
* interchange rows and columns K and IMAX,
* use 1-by-1 pivot block
*
KP = IMAX
*
* copy column K+1 of W to column K of W
*
CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
*
DONE = .TRUE.
*
* 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.
ELSE
*
* Pivot not found: set params and repeat
*
P = IMAX
COLMAX = ROWMAX
IMAX = JMAX
*
* Copy updated JMAXth (next IMAXth) column to Kth of W
*
CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
*
END IF
*
* End pivot search loop body
*
IF( .NOT. DONE ) GOTO 72
*
END IF
*
* ============================================================
*
KK = K + KSTEP - 1
*
IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
*
* Copy non-updated column K to column P
*
CALL DCOPY( P-K, A( K, K ), 1, A( P, K ), LDA )
CALL DCOPY( N-P+1, A( P, K ), 1, A( P, P ), 1 )
*
* Interchange rows K and P in first K columns of A
* and first K+1 columns of W
*
CALL DSWAP( K, A( K, 1 ), LDA, A( P, 1 ), LDA )
CALL DSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
END IF
*
* Updated column KP is already stored in column KK of W
*
IF( KP.NE.KK ) THEN
*
* Copy non-updated column KK to column KP
*
A( KP, K ) = A( KK, K )
CALL DCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA )
CALL DCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 )
*
* Interchange rows KK and KP in first KK columns of A and W
*
CALL DSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
CALL DSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
END IF
*
IF( KSTEP.EQ.1 ) THEN
*
* 1-by-1 pivot block D(k): column k of W now holds
*
* W(k) = L(k)*D(k)
*
* where L(k) is the k-th column of L
*
* Store L(k) in column k of A
*
CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
IF( K.LT.N ) THEN
IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
R1 = ONE / A( K, K )
CALL DSCAL( N-K, R1, A( K+1, K ), 1 )
ELSE IF( A( K, K ).NE.ZERO ) THEN
DO 74 II = K + 1, N
A( II, K ) = A( II, K ) / A( K, K )
74 CONTINUE
END IF
END IF
*
ELSE
*
* 2-by-2 pivot block D(k): columns k and k+1 of W 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
*
IF( K.LT.N-1 ) THEN
*
* Store L(k) and L(k+1) in columns k and k+1 of A
*
D21 = W( K+1, K )
D11 = W( K+1, K+1 ) / D21
D22 = W( K, K ) / D21
T = ONE / ( D11*D22-ONE )
DO 80 J = K + 2, N
A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
$ D21 )
A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
$ D21 )
80 CONTINUE
END IF
*
* Copy D(k) to A
*
A( K, K ) = W( K, K )
A( K+1, K ) = W( K+1, K )
A( K+1, K+1 ) = W( K+1, K+1 )
END IF
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
*
* Increase K and return to the start of the main loop
*
K = K + KSTEP
GO TO 70
*
90 CONTINUE
*
* Update the lower triangle of A22 (= A(k:n,k:n)) as
*
* A22 := A22 - L21*D*L21**T = A22 - L21*W**T
*
* computing blocks of NB columns at a time
*
DO 110 J = K, N, NB
JB = MIN( NB, N-J+1 )
*
* Update the lower triangle of the diagonal block
*
DO 100 JJ = J, J + JB - 1
CALL DGEMV( 'No transpose', J+JB-JJ, K-1, -ONE,
$ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, ONE,
$ A( JJ, JJ ), 1 )
100 CONTINUE
*
* Update the rectangular subdiagonal block
*
IF( J+JB.LE.N )
$ CALL DGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
$ K-1, -ONE, A( J+JB, 1 ), LDA, W( J, 1 ), LDW,
$ ONE, A( J+JB, J ), LDA )
110 CONTINUE
*
* Put L21 in standard form by partially undoing the interchanges
* in columns 1:k-1
*
J = K - 1
120 CONTINUE
*
KSTEP = 1
JP1 = 1
JJ = J
JP2 = IPIV( J )
IF( JP2.LT.0 ) THEN
JP2 = -JP2
J = J - 1
JP1 = -IPIV( J )
KSTEP = 2
END IF
*
J = J - 1
IF( JP2.NE.JJ .AND. J.GE.1 )
$ CALL DSWAP( J, A( JP2, 1 ), LDA, A( JJ, 1 ), LDA )
JJ = J + 1
IF( JP1.NE.JJ .AND. KSTEP.EQ.2 )
$ CALL DSWAP( J, A( JP1, 1 ), LDA, A( JJ, 1 ), LDA )
IF( J.GE.1 )
$ GO TO 120
*
* Set KB to the number of columns factorized
*
KB = K - 1
*
END IF
RETURN
*
* End of DLASYF_ROOK
*
END