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
|
SUBROUTINE DTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
$ LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL,
$ IWORK, PQ, INFO )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N,
$ PQ
DOUBLE PRECISION RDSCAL, RDSUM, SCALE
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), E( LDE, * ), F( LDF, * )
* ..
*
* Purpose
* =======
*
* DTGSY2 solves the generalized Sylvester equation:
*
* A * R - L * B = scale * C (1)
* D * R - L * E = scale * F,
*
* using Level 1 and 2 BLAS. where R and L are unknown M-by-N matrices,
* (A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,
* N-by-N and M-by-N, respectively, with real entries. (A, D) and (B, E)
* must be in generalized Schur canonical form, i.e. A, B are upper
* quasi triangular and D, E are upper triangular. The solution (R, L)
* overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor
* chosen to avoid overflow.
*
* In matrix notation solving equation (1) corresponds to solve
* Z*x = scale*b, where Z is defined as
*
* Z = [ kron(In, A) -kron(B', Im) ] (2)
* [ kron(In, D) -kron(E', Im) ],
*
* Ik is the identity matrix of size k and X' is the transpose of X.
* kron(X, Y) is the Kronecker product between the matrices X and Y.
* In the process of solving (1), we solve a number of such systems
* where Dim(In), Dim(In) = 1 or 2.
*
* If TRANS = 'T', solve the transposed system Z'*y = scale*b for y,
* which is equivalent to solve for R and L in
*
* A' * R + D' * L = scale * C (3)
* R * B' + L * E' = scale * -F
*
* This case is used to compute an estimate of Dif[(A, D), (B, E)] =
* sigma_min(Z) using reverse communicaton with DLACON.
*
* DTGSY2 also (IJOB >= 1) contributes to the computation in STGSYL
* of an upper bound on the separation between to matrix pairs. Then
* the input (A, D), (B, E) are sub-pencils of the matrix pair in
* DTGSYL. See STGSYL for details.
*
* Arguments
* =========
*
* TRANS (input) CHARACTER*1
* = 'N', solve the generalized Sylvester equation (1).
* = 'T': solve the 'transposed' system (3).
*
* IJOB (input) INTEGER
* Specifies what kind of functionality to be performed.
* = 0: solve (1) only.
* = 1: A contribution from this subsystem to a Frobenius
* norm-based estimate of the separation between two matrix
* pairs is computed. (look ahead strategy is used).
* = 2: A contribution from this subsystem to a Frobenius
* norm-based estimate of the separation between two matrix
* pairs is computed. (DGECON on sub-systems is used.)
* Not referenced if TRANS = 'T'.
*
* M (input) INTEGER
* On entry, M specifies the order of A and D, and the row
* dimension of C, F, R and L.
*
* N (input) INTEGER
* On entry, N specifies the order of B and E, and the column
* dimension of C, F, R and L.
*
* A (input) DOUBLE PRECISION array, dimension (LDA, M)
* On entry, A contains an upper quasi triangular matrix.
*
* LDA (input) INTEGER
* The leading dimension of the matrix A. LDA >= max(1, M).
*
* B (input) DOUBLE PRECISION array, dimension (LDB, N)
* On entry, B contains an upper quasi triangular matrix.
*
* LDB (input) INTEGER
* The leading dimension of the matrix B. LDB >= max(1, N).
*
* C (input/output) DOUBLE PRECISION array, dimension (LDC, N)
* On entry, C contains the right-hand-side of the first matrix
* equation in (1).
* On exit, if IJOB = 0, C has been overwritten by the
* solution R.
*
* LDC (input) INTEGER
* The leading dimension of the matrix C. LDC >= max(1, M).
*
* D (input) DOUBLE PRECISION array, dimension (LDD, M)
* On entry, D contains an upper triangular matrix.
*
* LDD (input) INTEGER
* The leading dimension of the matrix D. LDD >= max(1, M).
*
* E (input) DOUBLE PRECISION array, dimension (LDE, N)
* On entry, E contains an upper triangular matrix.
*
* LDE (input) INTEGER
* The leading dimension of the matrix E. LDE >= max(1, N).
*
* F (input/output) DOUBLE PRECISION array, dimension (LDF, N)
* On entry, F contains the right-hand-side of the second matrix
* equation in (1).
* On exit, if IJOB = 0, F has been overwritten by the
* solution L.
*
* LDF (input) INTEGER
* The leading dimension of the matrix F. LDF >= max(1, M).
*
* SCALE (output) DOUBLE PRECISION
* On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions
* R and L (C and F on entry) will hold the solutions to a
* slightly perturbed system but the input matrices A, B, D and
* E have not been changed. If SCALE = 0, R and L will hold the
* solutions to the homogeneous system with C = F = 0. Normally,
* SCALE = 1.
*
* RDSUM (input/output) DOUBLE PRECISION
* On entry, the sum of squares of computed contributions to
* the Dif-estimate under computation by DTGSYL, where the
* scaling factor RDSCAL (see below) has been factored out.
* On exit, the corresponding sum of squares updated with the
* contributions from the current sub-system.
* If TRANS = 'T' RDSUM is not touched.
* NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.
*
* RDSCAL (input/output) DOUBLE PRECISION
* On entry, scaling factor used to prevent overflow in RDSUM.
* On exit, RDSCAL is updated w.r.t. the current contributions
* in RDSUM.
* If TRANS = 'T', RDSCAL is not touched.
* NOTE: RDSCAL only makes sense when DTGSY2 is called by
* DTGSYL.
*
* IWORK (workspace) INTEGER array, dimension (M+N+2)
*
* PQ (output) INTEGER
* On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
* 8-by-8) solved by this routine.
*
* INFO (output) INTEGER
* On exit, if INFO is set to
* =0: Successful exit
* <0: If INFO = -i, the i-th argument had an illegal value.
* >0: The matrix pairs (A, D) and (B, E) have common or very
* close eigenvalues.
*
* Further Details
* ===============
*
* Based on contributions by
* Bo Kagstrom and Peter Poromaa, Department of Computing Science,
* Umea University, S-901 87 Umea, Sweden.
*
* =====================================================================
* Replaced various illegal calls to DCOPY by calls to DLASET.
* Sven Hammarling, 27/5/02.
*
* .. Parameters ..
INTEGER LDZ
PARAMETER ( LDZ = 8 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN
INTEGER I, IE, IERR, II, IS, ISP1, J, JE, JJ, JS, JSP1,
$ K, MB, NB, P, Q, ZDIM
DOUBLE PRECISION ALPHA, SCALOC
* ..
* .. Local Arrays ..
INTEGER IPIV( LDZ ), JPIV( LDZ )
DOUBLE PRECISION RHS( LDZ ), Z( LDZ, LDZ )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMM, DGEMV, DGER, DGESC2,
$ DGETC2, DLASET, DLATDF, DSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Decode and test input parameters
*
INFO = 0
IERR = 0
NOTRAN = LSAME( TRANS, 'N' )
IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -1
ELSE IF( NOTRAN ) THEN
IF( ( IJOB.LT.0 ) .OR. ( IJOB.GT.2 ) ) THEN
INFO = -2
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( M.LE.0 ) THEN
INFO = -3
ELSE IF( N.LE.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LDD.LT.MAX( 1, M ) ) THEN
INFO = -12
ELSE IF( LDE.LT.MAX( 1, N ) ) THEN
INFO = -14
ELSE IF( LDF.LT.MAX( 1, M ) ) THEN
INFO = -16
END IF
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTGSY2', -INFO )
RETURN
END IF
*
* Determine block structure of A
*
PQ = 0
P = 0
I = 1
10 CONTINUE
IF( I.GT.M )
$ GO TO 20
P = P + 1
IWORK( P ) = I
IF( I.EQ.M )
$ GO TO 20
IF( A( I+1, I ).NE.ZERO ) THEN
I = I + 2
ELSE
I = I + 1
END IF
GO TO 10
20 CONTINUE
IWORK( P+1 ) = M + 1
*
* Determine block structure of B
*
Q = P + 1
J = 1
30 CONTINUE
IF( J.GT.N )
$ GO TO 40
Q = Q + 1
IWORK( Q ) = J
IF( J.EQ.N )
$ GO TO 40
IF( B( J+1, J ).NE.ZERO ) THEN
J = J + 2
ELSE
J = J + 1
END IF
GO TO 30
40 CONTINUE
IWORK( Q+1 ) = N + 1
PQ = P*( Q-P-1 )
*
IF( NOTRAN ) THEN
*
* Solve (I, J) - subsystem
* A(I, I) * R(I, J) - L(I, J) * B(J, J) = C(I, J)
* D(I, I) * R(I, J) - L(I, J) * E(J, J) = F(I, J)
* for I = P, P - 1, ..., 1; J = 1, 2, ..., Q
*
SCALE = ONE
SCALOC = ONE
DO 120 J = P + 2, Q
JS = IWORK( J )
JSP1 = JS + 1
JE = IWORK( J+1 ) - 1
NB = JE - JS + 1
DO 110 I = P, 1, -1
*
IS = IWORK( I )
ISP1 = IS + 1
IE = IWORK( I+1 ) - 1
MB = IE - IS + 1
ZDIM = MB*NB*2
*
IF( ( MB.EQ.1 ) .AND. ( NB.EQ.1 ) ) THEN
*
* Build a 2-by-2 system Z * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = D( IS, IS )
Z( 1, 2 ) = -B( JS, JS )
Z( 2, 2 ) = -E( JS, JS )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = F( IS, JS )
*
* Solve Z * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
*
IF( IJOB.EQ.0 ) THEN
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
$ SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 50 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
50 CONTINUE
SCALE = SCALE*SCALOC
END IF
ELSE
CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
$ RDSCAL, IPIV, JPIV )
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
F( IS, JS ) = RHS( 2 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( I.GT.1 ) THEN
ALPHA = -RHS( 1 )
CALL DAXPY( IS-1, ALPHA, A( 1, IS ), 1, C( 1, JS ),
$ 1 )
CALL DAXPY( IS-1, ALPHA, D( 1, IS ), 1, F( 1, JS ),
$ 1 )
END IF
IF( J.LT.Q ) THEN
CALL DAXPY( N-JE, RHS( 2 ), B( JS, JE+1 ), LDB,
$ C( IS, JE+1 ), LDC )
CALL DAXPY( N-JE, RHS( 2 ), E( JS, JE+1 ), LDE,
$ F( IS, JE+1 ), LDF )
END IF
*
ELSE IF( ( MB.EQ.1 ) .AND. ( NB.EQ.2 ) ) THEN
*
* Build a 4-by-4 system Z * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = ZERO
Z( 3, 1 ) = D( IS, IS )
Z( 4, 1 ) = ZERO
*
Z( 1, 2 ) = ZERO
Z( 2, 2 ) = A( IS, IS )
Z( 3, 2 ) = ZERO
Z( 4, 2 ) = D( IS, IS )
*
Z( 1, 3 ) = -B( JS, JS )
Z( 2, 3 ) = -B( JS, JSP1 )
Z( 3, 3 ) = -E( JS, JS )
Z( 4, 3 ) = -E( JS, JSP1 )
*
Z( 1, 4 ) = -B( JSP1, JS )
Z( 2, 4 ) = -B( JSP1, JSP1 )
Z( 3, 4 ) = ZERO
Z( 4, 4 ) = -E( JSP1, JSP1 )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = C( IS, JSP1 )
RHS( 3 ) = F( IS, JS )
RHS( 4 ) = F( IS, JSP1 )
*
* Solve Z * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
*
IF( IJOB.EQ.0 ) THEN
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
$ SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 60 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
60 CONTINUE
SCALE = SCALE*SCALOC
END IF
ELSE
CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
$ RDSCAL, IPIV, JPIV )
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
C( IS, JSP1 ) = RHS( 2 )
F( IS, JS ) = RHS( 3 )
F( IS, JSP1 ) = RHS( 4 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( I.GT.1 ) THEN
CALL DGER( IS-1, NB, -ONE, A( 1, IS ), 1, RHS( 1 ),
$ 1, C( 1, JS ), LDC )
CALL DGER( IS-1, NB, -ONE, D( 1, IS ), 1, RHS( 1 ),
$ 1, F( 1, JS ), LDF )
END IF
IF( J.LT.Q ) THEN
CALL DAXPY( N-JE, RHS( 3 ), B( JS, JE+1 ), LDB,
$ C( IS, JE+1 ), LDC )
CALL DAXPY( N-JE, RHS( 3 ), E( JS, JE+1 ), LDE,
$ F( IS, JE+1 ), LDF )
CALL DAXPY( N-JE, RHS( 4 ), B( JSP1, JE+1 ), LDB,
$ C( IS, JE+1 ), LDC )
CALL DAXPY( N-JE, RHS( 4 ), E( JSP1, JE+1 ), LDE,
$ F( IS, JE+1 ), LDF )
END IF
*
ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.1 ) ) THEN
*
* Build a 4-by-4 system Z * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = A( ISP1, IS )
Z( 3, 1 ) = D( IS, IS )
Z( 4, 1 ) = ZERO
*
Z( 1, 2 ) = A( IS, ISP1 )
Z( 2, 2 ) = A( ISP1, ISP1 )
Z( 3, 2 ) = D( IS, ISP1 )
Z( 4, 2 ) = D( ISP1, ISP1 )
*
Z( 1, 3 ) = -B( JS, JS )
Z( 2, 3 ) = ZERO
Z( 3, 3 ) = -E( JS, JS )
Z( 4, 3 ) = ZERO
*
Z( 1, 4 ) = ZERO
Z( 2, 4 ) = -B( JS, JS )
Z( 3, 4 ) = ZERO
Z( 4, 4 ) = -E( JS, JS )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = C( ISP1, JS )
RHS( 3 ) = F( IS, JS )
RHS( 4 ) = F( ISP1, JS )
*
* Solve Z * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
IF( IJOB.EQ.0 ) THEN
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
$ SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 70 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
70 CONTINUE
SCALE = SCALE*SCALOC
END IF
ELSE
CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
$ RDSCAL, IPIV, JPIV )
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
C( ISP1, JS ) = RHS( 2 )
F( IS, JS ) = RHS( 3 )
F( ISP1, JS ) = RHS( 4 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( I.GT.1 ) THEN
CALL DGEMV( 'N', IS-1, MB, -ONE, A( 1, IS ), LDA,
$ RHS( 1 ), 1, ONE, C( 1, JS ), 1 )
CALL DGEMV( 'N', IS-1, MB, -ONE, D( 1, IS ), LDD,
$ RHS( 1 ), 1, ONE, F( 1, JS ), 1 )
END IF
IF( J.LT.Q ) THEN
CALL DGER( MB, N-JE, ONE, RHS( 3 ), 1,
$ B( JS, JE+1 ), LDB, C( IS, JE+1 ), LDC )
CALL DGER( MB, N-JE, ONE, RHS( 3 ), 1,
$ E( JS, JE+1 ), LDB, F( IS, JE+1 ), LDC )
END IF
*
ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.2 ) ) THEN
*
* Build an 8-by-8 system Z * x = RHS
*
CALL DLASET( 'F', LDZ, LDZ, ZERO, ZERO, Z, LDZ )
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = A( ISP1, IS )
Z( 5, 1 ) = D( IS, IS )
*
Z( 1, 2 ) = A( IS, ISP1 )
Z( 2, 2 ) = A( ISP1, ISP1 )
Z( 5, 2 ) = D( IS, ISP1 )
Z( 6, 2 ) = D( ISP1, ISP1 )
*
Z( 3, 3 ) = A( IS, IS )
Z( 4, 3 ) = A( ISP1, IS )
Z( 7, 3 ) = D( IS, IS )
*
Z( 3, 4 ) = A( IS, ISP1 )
Z( 4, 4 ) = A( ISP1, ISP1 )
Z( 7, 4 ) = D( IS, ISP1 )
Z( 8, 4 ) = D( ISP1, ISP1 )
*
Z( 1, 5 ) = -B( JS, JS )
Z( 3, 5 ) = -B( JS, JSP1 )
Z( 5, 5 ) = -E( JS, JS )
Z( 7, 5 ) = -E( JS, JSP1 )
*
Z( 2, 6 ) = -B( JS, JS )
Z( 4, 6 ) = -B( JS, JSP1 )
Z( 6, 6 ) = -E( JS, JS )
Z( 8, 6 ) = -E( JS, JSP1 )
*
Z( 1, 7 ) = -B( JSP1, JS )
Z( 3, 7 ) = -B( JSP1, JSP1 )
Z( 7, 7 ) = -E( JSP1, JSP1 )
*
Z( 2, 8 ) = -B( JSP1, JS )
Z( 4, 8 ) = -B( JSP1, JSP1 )
Z( 8, 8 ) = -E( JSP1, JSP1 )
*
* Set up right hand side(s)
*
K = 1
II = MB*NB + 1
DO 80 JJ = 0, NB - 1
CALL DCOPY( MB, C( IS, JS+JJ ), 1, RHS( K ), 1 )
CALL DCOPY( MB, F( IS, JS+JJ ), 1, RHS( II ), 1 )
K = K + MB
II = II + MB
80 CONTINUE
*
* Solve Z * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
IF( IJOB.EQ.0 ) THEN
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV,
$ SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 90 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
90 CONTINUE
SCALE = SCALE*SCALOC
END IF
ELSE
CALL DLATDF( IJOB, ZDIM, Z, LDZ, RHS, RDSUM,
$ RDSCAL, IPIV, JPIV )
END IF
*
* Unpack solution vector(s)
*
K = 1
II = MB*NB + 1
DO 100 JJ = 0, NB - 1
CALL DCOPY( MB, RHS( K ), 1, C( IS, JS+JJ ), 1 )
CALL DCOPY( MB, RHS( II ), 1, F( IS, JS+JJ ), 1 )
K = K + MB
II = II + MB
100 CONTINUE
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( I.GT.1 ) THEN
CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
$ A( 1, IS ), LDA, RHS( 1 ), MB, ONE,
$ C( 1, JS ), LDC )
CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
$ D( 1, IS ), LDD, RHS( 1 ), MB, ONE,
$ F( 1, JS ), LDF )
END IF
IF( J.LT.Q ) THEN
K = MB*NB + 1
CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE, RHS( K ),
$ MB, B( JS, JE+1 ), LDB, ONE,
$ C( IS, JE+1 ), LDC )
CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE, RHS( K ),
$ MB, E( JS, JE+1 ), LDE, ONE,
$ F( IS, JE+1 ), LDF )
END IF
*
END IF
*
110 CONTINUE
120 CONTINUE
ELSE
*
* Solve (I, J) - subsystem
* A(I, I)' * R(I, J) + D(I, I)' * L(J, J) = C(I, J)
* R(I, I) * B(J, J) + L(I, J) * E(J, J) = -F(I, J)
* for I = 1, 2, ..., P, J = Q, Q - 1, ..., 1
*
SCALE = ONE
SCALOC = ONE
DO 200 I = 1, P
*
IS = IWORK( I )
ISP1 = IS + 1
IE = ( I+1 ) - 1
MB = IE - IS + 1
DO 190 J = Q, P + 2, -1
*
JS = IWORK( J )
JSP1 = JS + 1
JE = IWORK( J+1 ) - 1
NB = JE - JS + 1
ZDIM = MB*NB*2
IF( ( MB.EQ.1 ) .AND. ( NB.EQ.1 ) ) THEN
*
* Build a 2-by-2 system Z' * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = -B( JS, JS )
Z( 1, 2 ) = D( IS, IS )
Z( 2, 2 ) = -E( JS, JS )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = F( IS, JS )
*
* Solve Z' * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
*
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 130 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
130 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
F( IS, JS ) = RHS( 2 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( J.GT.P+2 ) THEN
ALPHA = RHS( 1 )
CALL DAXPY( JS-1, ALPHA, B( 1, JS ), 1, F( IS, 1 ),
$ LDF )
ALPHA = RHS( 2 )
CALL DAXPY( JS-1, ALPHA, E( 1, JS ), 1, F( IS, 1 ),
$ LDF )
END IF
IF( I.LT.P ) THEN
ALPHA = -RHS( 1 )
CALL DAXPY( M-IE, ALPHA, A( IS, IE+1 ), LDA,
$ C( IE+1, JS ), 1 )
ALPHA = -RHS( 2 )
CALL DAXPY( M-IE, ALPHA, D( IS, IE+1 ), LDD,
$ C( IE+1, JS ), 1 )
END IF
*
ELSE IF( ( MB.EQ.1 ) .AND. ( NB.EQ.2 ) ) THEN
*
* Build a 4-by-4 system Z' * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = ZERO
Z( 3, 1 ) = -B( JS, JS )
Z( 4, 1 ) = -B( JSP1, JS )
*
Z( 1, 2 ) = ZERO
Z( 2, 2 ) = A( IS, IS )
Z( 3, 2 ) = -B( JS, JSP1 )
Z( 4, 2 ) = -B( JSP1, JSP1 )
*
Z( 1, 3 ) = D( IS, IS )
Z( 2, 3 ) = ZERO
Z( 3, 3 ) = -E( JS, JS )
Z( 4, 3 ) = ZERO
*
Z( 1, 4 ) = ZERO
Z( 2, 4 ) = D( IS, IS )
Z( 3, 4 ) = -E( JS, JSP1 )
Z( 4, 4 ) = -E( JSP1, JSP1 )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = C( IS, JSP1 )
RHS( 3 ) = F( IS, JS )
RHS( 4 ) = F( IS, JSP1 )
*
* Solve Z' * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 140 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
140 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
C( IS, JSP1 ) = RHS( 2 )
F( IS, JS ) = RHS( 3 )
F( IS, JSP1 ) = RHS( 4 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( J.GT.P+2 ) THEN
CALL DAXPY( JS-1, RHS( 1 ), B( 1, JS ), 1,
$ F( IS, 1 ), LDF )
CALL DAXPY( JS-1, RHS( 2 ), B( 1, JSP1 ), 1,
$ F( IS, 1 ), LDF )
CALL DAXPY( JS-1, RHS( 3 ), E( 1, JS ), 1,
$ F( IS, 1 ), LDF )
CALL DAXPY( JS-1, RHS( 4 ), E( 1, JSP1 ), 1,
$ F( IS, 1 ), LDF )
END IF
IF( I.LT.P ) THEN
CALL DGER( M-IE, NB, -ONE, A( IS, IE+1 ), LDA,
$ RHS( 1 ), 1, C( IE+1, JS ), LDC )
CALL DGER( M-IE, NB, -ONE, D( IS, IE+1 ), LDD,
$ RHS( 3 ), 1, C( IE+1, JS ), LDC )
END IF
*
ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.1 ) ) THEN
*
* Build a 4-by-4 system Z' * x = RHS
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = A( IS, ISP1 )
Z( 3, 1 ) = -B( JS, JS )
Z( 4, 1 ) = ZERO
*
Z( 1, 2 ) = A( ISP1, IS )
Z( 2, 2 ) = A( ISP1, ISP1 )
Z( 3, 2 ) = ZERO
Z( 4, 2 ) = -B( JS, JS )
*
Z( 1, 3 ) = D( IS, IS )
Z( 2, 3 ) = D( IS, ISP1 )
Z( 3, 3 ) = -E( JS, JS )
Z( 4, 3 ) = ZERO
*
Z( 1, 4 ) = ZERO
Z( 2, 4 ) = D( ISP1, ISP1 )
Z( 3, 4 ) = ZERO
Z( 4, 4 ) = -E( JS, JS )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( IS, JS )
RHS( 2 ) = C( ISP1, JS )
RHS( 3 ) = F( IS, JS )
RHS( 4 ) = F( ISP1, JS )
*
* Solve Z' * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
*
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 150 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
150 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* Unpack solution vector(s)
*
C( IS, JS ) = RHS( 1 )
C( ISP1, JS ) = RHS( 2 )
F( IS, JS ) = RHS( 3 )
F( ISP1, JS ) = RHS( 4 )
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( J.GT.P+2 ) THEN
CALL DGER( MB, JS-1, ONE, RHS( 1 ), 1, B( 1, JS ),
$ 1, F( IS, 1 ), LDF )
CALL DGER( MB, JS-1, ONE, RHS( 3 ), 1, E( 1, JS ),
$ 1, F( IS, 1 ), LDF )
END IF
IF( I.LT.P ) THEN
CALL DGEMV( 'T', MB, M-IE, -ONE, A( IS, IE+1 ),
$ LDA, RHS( 1 ), 1, ONE, C( IE+1, JS ),
$ 1 )
CALL DGEMV( 'T', MB, M-IE, -ONE, D( IS, IE+1 ),
$ LDD, RHS( 3 ), 1, ONE, C( IE+1, JS ),
$ 1 )
END IF
*
ELSE IF( ( MB.EQ.2 ) .AND. ( NB.EQ.2 ) ) THEN
*
* Build an 8-by-8 system Z' * x = RHS
*
CALL DLASET( 'F', LDZ, LDZ, ZERO, ZERO, Z, LDZ )
*
Z( 1, 1 ) = A( IS, IS )
Z( 2, 1 ) = A( IS, ISP1 )
Z( 5, 1 ) = -B( JS, JS )
Z( 7, 1 ) = -B( JSP1, JS )
*
Z( 1, 2 ) = A( ISP1, IS )
Z( 2, 2 ) = A( ISP1, ISP1 )
Z( 6, 2 ) = -B( JS, JS )
Z( 8, 2 ) = -B( JSP1, JS )
*
Z( 3, 3 ) = A( IS, IS )
Z( 4, 3 ) = A( IS, ISP1 )
Z( 5, 3 ) = -B( JS, JSP1 )
Z( 7, 3 ) = -B( JSP1, JSP1 )
*
Z( 3, 4 ) = A( ISP1, IS )
Z( 4, 4 ) = A( ISP1, ISP1 )
Z( 6, 4 ) = -B( JS, JSP1 )
Z( 8, 4 ) = -B( JSP1, JSP1 )
*
Z( 1, 5 ) = D( IS, IS )
Z( 2, 5 ) = D( IS, ISP1 )
Z( 5, 5 ) = -E( JS, JS )
*
Z( 2, 6 ) = D( ISP1, ISP1 )
Z( 6, 6 ) = -E( JS, JS )
*
Z( 3, 7 ) = D( IS, IS )
Z( 4, 7 ) = D( IS, ISP1 )
Z( 5, 7 ) = -E( JS, JSP1 )
Z( 7, 7 ) = -E( JSP1, JSP1 )
*
Z( 4, 8 ) = D( ISP1, ISP1 )
Z( 6, 8 ) = -E( JS, JSP1 )
Z( 8, 8 ) = -E( JSP1, JSP1 )
*
* Set up right hand side(s)
*
K = 1
II = MB*NB + 1
DO 160 JJ = 0, NB - 1
CALL DCOPY( MB, C( IS, JS+JJ ), 1, RHS( K ), 1 )
CALL DCOPY( MB, F( IS, JS+JJ ), 1, RHS( II ), 1 )
K = K + MB
II = II + MB
160 CONTINUE
*
*
* Solve Z' * x = RHS
*
CALL DGETC2( ZDIM, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
*
CALL DGESC2( ZDIM, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 170 K = 1, N
CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
170 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* Unpack solution vector(s)
*
K = 1
II = MB*NB + 1
DO 180 JJ = 0, NB - 1
CALL DCOPY( MB, RHS( K ), 1, C( IS, JS+JJ ), 1 )
CALL DCOPY( MB, RHS( II ), 1, F( IS, JS+JJ ), 1 )
K = K + MB
II = II + MB
180 CONTINUE
*
* Substitute R(I, J) and L(I, J) into remaining
* equation.
*
IF( J.GT.P+2 ) THEN
CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE,
$ C( IS, JS ), LDC, B( 1, JS ), LDB, ONE,
$ F( IS, 1 ), LDF )
CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE,
$ F( IS, JS ), LDF, E( 1, JS ), LDE, ONE,
$ F( IS, 1 ), LDF )
END IF
IF( I.LT.P ) THEN
CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
$ A( IS, IE+1 ), LDA, C( IS, JS ), LDC,
$ ONE, C( IE+1, JS ), LDC )
CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
$ D( IS, IE+1 ), LDD, F( IS, JS ), LDF,
$ ONE, C( IE+1, JS ), LDC )
END IF
*
END IF
*
190 CONTINUE
200 CONTINUE
*
END IF
RETURN
*
* End of DTGSY2
*
END
|
