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
|
SUBROUTINE TG01FZ( COMPQ, COMPZ, JOBA, L, N, M, P, A, LDA, E, LDE,
$ B, LDB, C, LDC, Q, LDQ, Z, LDZ, RANKE, RNKA22,
$ TOL, IWORK, DWORK, ZWORK, LZWORK, INFO )
C
C SLICOT RELEASE 5.0.
C
C Copyright (c) 2002-2009 NICONET e.V.
C
C This program is free software: you can redistribute it and/or
C modify it under the terms of the GNU General Public License as
C published by the Free Software Foundation, either version 2 of
C the License, or (at your option) any later version.
C
C This program is distributed in the hope that it will be useful,
C but WITHOUT ANY WARRANTY; without even the implied warranty of
C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
C GNU General Public License for more details.
C
C You should have received a copy of the GNU General Public License
C along with this program. If not, see
C <http://www.gnu.org/licenses/>.
C
C PURPOSE
C
C To compute for the descriptor system (A-lambda E,B,C)
C the unitary transformation matrices Q and Z such that the
C transformed system (Q'*A*Z-lambda Q'*E*Z, Q'*B, C*Z) is
C in a SVD-like coordinate form with
C
C ( A11 A12 ) ( Er 0 )
C Q'*A*Z = ( ) , Q'*E*Z = ( ) ,
C ( A21 A22 ) ( 0 0 )
C
C where Er is an upper triangular invertible matrix, and ' denotes
C the conjugate transpose. Optionally, the A22 matrix can be further
C reduced to the form
C
C ( Ar X )
C A22 = ( ) ,
C ( 0 0 )
C
C with Ar an upper triangular invertible matrix, and X either a full
C or a zero matrix.
C The left and/or right unitary transformations performed
C to reduce E and A22 can be optionally accumulated.
C
C ARGUMENTS
C
C Mode Parameters
C
C COMPQ CHARACTER*1
C = 'N': do not compute Q;
C = 'I': Q is initialized to the unit matrix, and the
C unitary matrix Q is returned;
C = 'U': Q must contain a unitary matrix Q1 on entry,
C and the product Q1*Q is returned.
C
C COMPZ CHARACTER*1
C = 'N': do not compute Z;
C = 'I': Z is initialized to the unit matrix, and the
C unitary matrix Z is returned;
C = 'U': Z must contain a unitary matrix Z1 on entry,
C and the product Z1*Z is returned.
C
C JOBA CHARACTER*1
C = 'N': do not reduce A22;
C = 'R': reduce A22 to a SVD-like upper triangular form.
C = 'T': reduce A22 to an upper trapezoidal form.
C
C Input/Output Parameters
C
C L (input) INTEGER
C The number of rows of matrices A, B, and E. L >= 0.
C
C N (input) INTEGER
C The number of columns of matrices A, E, and C. N >= 0.
C
C M (input) INTEGER
C The number of columns of matrix B. M >= 0.
C
C P (input) INTEGER
C The number of rows of matrix C. P >= 0.
C
C A (input/output) COMPLEX*16 array, dimension (LDA,N)
C On entry, the leading L-by-N part of this array must
C contain the state dynamics matrix A.
C On exit, the leading L-by-N part of this array contains
C the transformed matrix Q'*A*Z. If JOBA = 'T', this matrix
C is in the form
C
C ( A11 * * )
C Q'*A*Z = ( * Ar X ) ,
C ( * 0 0 )
C
C where A11 is a RANKE-by-RANKE matrix and Ar is a
C RNKA22-by-RNKA22 invertible upper triangular matrix.
C If JOBA = 'R' then A has the above form with X = 0.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,L).
C
C E (input/output) COMPLEX*16 array, dimension (LDE,N)
C On entry, the leading L-by-N part of this array must
C contain the descriptor matrix E.
C On exit, the leading L-by-N part of this array contains
C the transformed matrix Q'*E*Z.
C
C ( Er 0 )
C Q'*E*Z = ( ) ,
C ( 0 0 )
C
C where Er is a RANKE-by-RANKE upper triangular invertible
C matrix.
C
C LDE INTEGER
C The leading dimension of array E. LDE >= MAX(1,L).
C
C B (input/output) COMPLEX*16 array, dimension (LDB,M)
C On entry, the leading L-by-M part of this array must
C contain the input/state matrix B.
C On exit, the leading L-by-M part of this array contains
C the transformed matrix Q'*B.
C
C LDB INTEGER
C The leading dimension of array B.
C LDB >= MAX(1,L) if M > 0 or LDB >= 1 if M = 0.
C
C C (input/output) COMPLEX*16 array, dimension (LDC,N)
C On entry, the leading P-by-N part of this array must
C contain the state/output matrix C.
C On exit, the leading P-by-N part of this array contains
C the transformed matrix C*Z.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,P).
C
C Q (input/output) COMPLEX*16 array, dimension (LDQ,L)
C If COMPQ = 'N': Q is not referenced.
C If COMPQ = 'I': on entry, Q need not be set;
C on exit, the leading L-by-L part of this
C array contains the unitary matrix Q,
C where Q' is the product of Householder
C transformations which are applied to A,
C E, and B on the left.
C If COMPQ = 'U': on entry, the leading L-by-L part of this
C array must contain a unitary matrix Q1;
C on exit, the leading L-by-L part of this
C array contains the unitary matrix Q1*Q.
C
C LDQ INTEGER
C The leading dimension of array Q.
C LDQ >= 1, if COMPQ = 'N';
C LDQ >= MAX(1,L), if COMPQ = 'U' or 'I'.
C
C Z (input/output) COMPLEX*16 array, dimension (LDZ,N)
C If COMPZ = 'N': Z is not referenced.
C If COMPZ = 'I': on entry, Z need not be set;
C on exit, the leading N-by-N part of this
C array contains the unitary matrix Z,
C which is the product of Householder
C transformations applied to A, E, and C
C on the right.
C If COMPZ = 'U': on entry, the leading N-by-N part of this
C array must contain a unitary matrix Z1;
C on exit, the leading N-by-N part of this
C array contains the unitary matrix Z1*Z.
C
C LDZ INTEGER
C The leading dimension of array Z.
C LDZ >= 1, if COMPZ = 'N';
C LDZ >= MAX(1,N), if COMPZ = 'U' or 'I'.
C
C RANKE (output) INTEGER
C The estimated rank of matrix E, and thus also the order
C of the invertible upper triangular submatrix Er.
C
C RNKA22 (output) INTEGER
C If JOBA = 'R' or 'T', then RNKA22 is the estimated rank of
C matrix A22, and thus also the order of the invertible
C upper triangular submatrix Ar.
C If JOBA = 'N', then RNKA22 is not referenced.
C
C Tolerances
C
C TOL DOUBLE PRECISION
C The tolerance to be used in determining the rank of E
C and of A22. If the user sets TOL > 0, then the given
C value of TOL is used as a lower bound for the
C reciprocal condition numbers of leading submatrices
C of R or R22 in the QR decompositions E * P = Q * R of E
C or A22 * P22 = Q22 * R22 of A22.
C A submatrix whose estimated condition number is less than
C 1/TOL is considered to be of full rank. If the user sets
C TOL <= 0, then an implicitly computed, default tolerance,
C defined by TOLDEF = L*N*EPS, is used instead, where
C EPS is the machine precision (see LAPACK Library routine
C DLAMCH). TOL < 1.
C
C Workspace
C
C IWORK INTEGER array, dimension (N)
C
C DWORK DOUBLE PRECISION array, dimension (2*N)
C
C ZWORK DOUBLE PRECISION array, dimension (LZWORK)
C On exit, if INFO = 0, ZWORK(1) returns the optimal value
C of LZWORK.
C
C LZWORK INTEGER
C The length of the array ZWORK.
C LZWORK >= MAX( 1, N+P, MIN(L,N)+MAX(3*N-1,M,L) ).
C For optimal performance, LZWORK should be larger.
C
C If LZWORK = -1, then a workspace query is assumed;
C the routine only calculates the optimal size of the
C ZWORK array, returns this value as the first entry of
C the ZWORK array, and no error message related to LZWORK
C is issued by XERBLA.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value.
C
C METHOD
C
C The routine computes a truncated QR factorization with column
C pivoting of E, in the form
C
C ( E11 E12 )
C E * P = Q * ( )
C ( 0 E22 )
C
C and finds the largest RANKE-by-RANKE leading submatrix E11 whose
C estimated condition number is less than 1/TOL. RANKE defines thus
C the rank of matrix E. Further E22, being negligible, is set to
C zero, and a unitary matrix Y is determined such that
C
C ( E11 E12 ) = ( Er 0 ) * Y .
C
C The overal transformation matrix Z results as Z = P * Y' and the
C resulting transformed matrices Q'*A*Z and Q'*E*Z have the form
C
C ( Er 0 ) ( A11 A12 )
C E <- Q'* E * Z = ( ) , A <- Q' * A * Z = ( ) ,
C ( 0 0 ) ( A21 A22 )
C
C where Er is an upper triangular invertible matrix.
C If JOBA = 'R' the same reduction is performed on A22 to obtain it
C in the form
C
C ( Ar 0 )
C A22 = ( ) ,
C ( 0 0 )
C
C with Ar an upper triangular invertible matrix.
C If JOBA = 'T' then A22 is row compressed using the QR
C factorization with column pivoting to the form
C
C ( Ar X )
C A22 = ( )
C ( 0 0 )
C
C with Ar an upper triangular invertible matrix.
C
C The transformations are also applied to the rest of system
C matrices
C
C B <- Q' * B, C <- C * Z.
C
C NUMERICAL ASPECTS
C
C The algorithm is numerically backward stable and requires
C 0( L*L*N ) floating point operations.
C
C CONTRIBUTOR
C
C A. Varga, German Aerospace Center, DLR Oberpfaffenhofen.
C March 1999.
C Complex version: V. Sima, Research Institute for Informatics,
C Bucharest, Nov. 2008.
C
C REVISIONS
C
C -
C
C KEYWORDS
C
C Descriptor system, matrix algebra, matrix operations, unitary
C transformation.
C
C ******************************************************************
C
C .. Parameters ..
COMPLEX*16 ONE, ZERO
PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ),
$ ZERO = ( 0.0D+0, 0.0D+0 ) )
DOUBLE PRECISION DONE, DZERO
PARAMETER ( DONE = 1.0D+0, DZERO = 0.0D+0 )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, JOBA
INTEGER INFO, L, LDA, LDB, LDC, LDE, LDQ, LDZ, LZWORK,
$ M, N, P, RANKE, RNKA22
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ E( LDE, * ), Q( LDQ, * ), Z( LDZ, * ),
$ ZWORK( * )
DOUBLE PRECISION DWORK( * )
C .. Local Scalars ..
LOGICAL ILQ, ILZ, LQUERY, REDA, REDTR, WITHB, WITHC
INTEGER I, ICOMPQ, ICOMPZ, IR1, IRE1, J, K, KW, LA22,
$ LH, LN, LWR, NA22, NB, WRKOPT
DOUBLE PRECISION SVLMAX, TOLDEF
C .. Local Arrays ..
DOUBLE PRECISION SVAL(3)
C .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, ZLANGE
EXTERNAL DLAMCH, ILAENV, LSAME, ZLANGE
C .. External Subroutines ..
EXTERNAL MB3OYZ, XERBLA, ZLASET, ZSWAP, ZTZRZF, ZUNMQR,
$ ZUNMRZ
C .. Intrinsic Functions ..
INTRINSIC DBLE, INT, MAX, MIN
C
C .. Executable Statements ..
C
C Decode COMPQ.
C
IF( LSAME( COMPQ, 'N' ) ) THEN
ILQ = .FALSE.
ICOMPQ = 1
ELSE IF( LSAME( COMPQ, 'U' ) ) THEN
ILQ = .TRUE.
ICOMPQ = 2
ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
ILQ = .TRUE.
ICOMPQ = 3
ELSE
ICOMPQ = 0
END IF
C
C Decode COMPZ.
C
IF( LSAME( COMPZ, 'N' ) ) THEN
ILZ = .FALSE.
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'U' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 2
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 3
ELSE
ICOMPZ = 0
END IF
REDA = LSAME( JOBA, 'R' )
REDTR = LSAME( JOBA, 'T' )
WITHB = M.GT.0
WITHC = P.GT.0
LQUERY = ( LZWORK.EQ.-1 )
C
C Test the input parameters.
C
LN = MIN( L, N )
INFO = 0
WRKOPT = MAX( 1, N+P, LN + MAX( 3*N-1, M, L ) )
IF( ICOMPQ.LE.0 ) THEN
INFO = -1
ELSE IF( ICOMPZ.LE.0 ) THEN
INFO = -2
ELSE IF( .NOT.LSAME( JOBA, 'N' ) .AND. .NOT.REDA .AND.
$ .NOT.REDTR ) THEN
INFO = -3
ELSE IF( L.LT.0 ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( M.LT.0 ) THEN
INFO = -6
ELSE IF( P.LT.0 ) THEN
INFO = -7
ELSE IF( LDA.LT.MAX( 1, L ) ) THEN
INFO = -9
ELSE IF( LDE.LT.MAX( 1, L ) ) THEN
INFO = -11
ELSE IF( LDB.LT.1 .OR. ( WITHB .AND. LDB.LT.L ) ) THEN
INFO = -13
ELSE IF( LDC.LT.MAX( 1, P ) ) THEN
INFO = -15
ELSE IF( ( ILQ .AND. LDQ.LT.L ) .OR. LDQ.LT.1 ) THEN
INFO = -17
ELSE IF( ( ILZ .AND. LDZ.LT.N ) .OR. LDZ.LT.1 ) THEN
INFO = -19
ELSE IF( TOL.GE.DONE ) THEN
INFO = -22
ELSE
IF( LQUERY ) THEN
NB = MIN( 64, ILAENV( 1, 'ZUNMQR', 'LC', L, N, LN, -1 ) )
WRKOPT = MAX( WRKOPT, LN + N*NB )
IF( WITHB ) THEN
NB = MIN( 64, ILAENV( 1, 'ZUNMQR', 'LC', L, M, LN, -1 ) )
WRKOPT = MAX( WRKOPT, LN + M*NB )
END IF
IF( ILQ ) THEN
NB = MIN( 64, ILAENV( 1, 'ZUNMQR', 'RN', L, L, LN, -1 ) )
WRKOPT = MAX( WRKOPT, LN + L*NB )
END IF
NB = ILAENV( 1, 'ZGERQF', ' ', L, N, -1, -1 )
WRKOPT = MAX( WRKOPT, LN + N*NB )
NB = MIN( 64, ILAENV( 1, 'ZUNMRQ', 'RC', L, N, N, -1 ) )
WRKOPT = MAX( WRKOPT, N + MAX( 1, L )*NB )
IF( WITHC ) THEN
NB = MIN( 64, ILAENV( 1, 'ZUNMRQ', 'RC', P, N, N, -1 ) )
WRKOPT = MAX( WRKOPT, N + MAX( 1, P )*NB )
END IF
IF( ILZ ) THEN
NB = MIN( 64, ILAENV( 1, 'ZUNMRQ', 'RC', N, N, N, -1 ) )
WRKOPT = MAX( WRKOPT, N + MAX( 1, N )*NB )
END IF
ELSE IF( LZWORK.LT.WRKOPT ) THEN
INFO = -26
END IF
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'TG01FZ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
ZWORK(1) = WRKOPT
RETURN
END IF
C
C Initialize Q and Z if necessary.
C
IF( ICOMPQ.EQ.3 )
$ CALL ZLASET( 'Full', L, L, ZERO, ONE, Q, LDQ )
IF( ICOMPZ.EQ.3 )
$ CALL ZLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
C
C Quick return if possible.
C
IF( L.EQ.0 .OR. N.EQ.0 ) THEN
ZWORK(1) = ONE
RANKE = 0
IF( REDA .OR. REDTR ) RNKA22 = 0
RETURN
END IF
C
TOLDEF = TOL
IF( TOLDEF.LE.DZERO ) THEN
C
C Use the default tolerance for rank determination.
C
TOLDEF = DBLE( L*N )*DLAMCH( 'EPSILON' )
END IF
C
C Set the estimate of maximum singular value of E to
C max(||E||,||A||) to detect negligible A or E matrices.
C
SVLMAX = MAX( ZLANGE( 'F', L, N, E, LDE, DWORK ),
$ ZLANGE( 'F', L, N, A, LDA, DWORK ) )
C
C Compute the rank-revealing QR decomposition of E,
C
C ( E11 E12 )
C E * P = Qr * ( ) ,
C ( 0 E22 )
C
C and determine the rank of E using incremental condition
C estimation.
C Complex Workspace: MIN(L,N) + 3*N - 1.
C Real Workspace: 2*N.
C
LWR = LZWORK - LN
KW = LN + 1
C
CALL MB3OYZ( L, N, E, LDE, TOLDEF, SVLMAX, RANKE, SVAL, IWORK,
$ ZWORK, DWORK, ZWORK(KW), INFO )
C
C Apply transformation on the rest of matrices.
C
IF( RANKE.GT.0 ) THEN
C
C A <-- Qr' * A.
C Complex Workspace: need MIN(L,N) + N;
C prefer MIN(L,N) + N*NB.
C
CALL ZUNMQR( 'Left', 'ConjTranspose', L, N, RANKE, E, LDE,
$ ZWORK, A, LDA, ZWORK(KW), LWR, INFO )
WRKOPT = MAX( WRKOPT, LN + INT( ZWORK(KW) ) )
C
C B <-- Qr' * B.
C Complex Workspace: need MIN(L,N) + M;
C prefer MIN(L,N) + M*NB.
C
IF( WITHB ) THEN
CALL ZUNMQR( 'Left', 'ConjTranspose', L, M, RANKE, E, LDE,
$ ZWORK, B, LDB, ZWORK(KW), LWR, INFO )
WRKOPT = MAX( WRKOPT, LN + INT( ZWORK(KW) ) )
END IF
C
C Q <-- Q * Qr.
C Complex Workspace: need MIN(L,N) + L;
C prefer MIN(L,N) + L*NB.
C
IF( ILQ ) THEN
CALL ZUNMQR( 'Right', 'No Transpose', L, L, RANKE, E, LDE,
$ ZWORK, Q, LDQ, ZWORK(KW), LWR, INFO )
WRKOPT = MAX( WRKOPT, LN + INT( ZWORK(KW) ) )
END IF
C
C Set lower triangle of E to zero.
C
IF( L.GE.2 )
$ CALL ZLASET( 'Lower', L-1, RANKE, ZERO, ZERO, E(2,1), LDE )
C
C Compute A*P, C*P and Z*P by forward permuting the columns of
C A, C and Z based on information in IWORK.
C
DO 10 J = 1, N
IWORK(J) = -IWORK(J)
10 CONTINUE
DO 30 I = 1, N
IF( IWORK(I).LT.0 ) THEN
J = I
IWORK(J) = -IWORK(J)
20 CONTINUE
K = IWORK(J)
IF( IWORK(K).LT.0 ) THEN
CALL ZSWAP( L, A(1,J), 1, A(1,K), 1 )
IF( WITHC )
$ CALL ZSWAP( P, C(1,J), 1, C(1,K), 1 )
IF( ILZ )
$ CALL ZSWAP( N, Z(1,J), 1, Z(1,K), 1 )
IWORK(K) = -IWORK(K)
J = K
GO TO 20
END IF
END IF
30 CONTINUE
C
C Determine a unitary matrix Y such that
C
C ( E11 E12 ) = ( Er 0 ) * Y .
C
C Compute E <-- E*Y', A <-- A*Y', C <-- C*Y', Z <-- Z*Y'.
C
IF( RANKE.LT.N ) THEN
C
C Complex Workspace: need 2*N;
C prefer N + N*NB.
C
KW = RANKE + 1
CALL ZTZRZF( RANKE, N, E, LDE, ZWORK, ZWORK(KW),
$ LZWORK-KW+1, INFO )
WRKOPT = MAX( WRKOPT, INT( ZWORK(KW) ) + KW - 1 )
C
C Complex Workspace: need N + MAX(L,P,N);
C prefer N + MAX(L,P,N)*NB.
C
LH = N - RANKE
CALL ZUNMRZ( 'Right', 'Conjugate transpose', L, N, RANKE,
$ LH, E, LDE, ZWORK, A, LDA, ZWORK(KW),
$ LZWORK-KW+1, INFO )
WRKOPT = MAX( WRKOPT, INT( ZWORK(KW) ) + KW - 1 )
IF( WITHC ) THEN
CALL ZUNMRZ( 'Right', 'Conjugate transpose', P, N, RANKE,
$ LH, E, LDE, ZWORK, C, LDC, ZWORK(KW),
$ LZWORK-KW+1, INFO )
WRKOPT = MAX( WRKOPT, INT( ZWORK(KW) ) + KW - 1 )
END IF
IF( ILZ ) THEN
CALL ZUNMRZ( 'Right', 'Conjugate transpose', N, N, RANKE,
$ LH, E, LDE, ZWORK, Z, LDZ, ZWORK(KW),
$ LZWORK-KW+1, INFO )
WRKOPT = MAX( WRKOPT, INT( ZWORK(KW) ) + KW - 1 )
END IF
C
C Set E12 and E22 to zero.
C
CALL ZLASET( 'Full', L, LH, ZERO, ZERO, E(1,KW), LDE )
END IF
ELSE
CALL ZLASET( 'Full', L, N, ZERO, ZERO, E, LDE )
END IF
C
C Reduce A22 if necessary.
C
IF( REDA .OR. REDTR ) THEN
LA22 = L - RANKE
NA22 = N - RANKE
IF( MIN( LA22, NA22 ).EQ.0 ) THEN
RNKA22 = 0
ELSE
C
C Compute the rank-revealing QR decomposition of A22,
C
C ( R11 R12 )
C A22 * P2 = Q2 * ( ) ,
C ( 0 R22 )
C
C and determine the rank of A22 using incremental
C condition estimation.
C Complex Workspace: MIN(L,N) + 3*N - 1.
C Real Workspace: 2*N.
C
IR1 = RANKE + 1
CALL MB3OYZ( LA22, NA22, A(IR1,IR1), LDA, TOLDEF,
$ SVLMAX, RNKA22, SVAL, IWORK, ZWORK,
$ DWORK, ZWORK(KW), INFO )
C
C Apply transformation on the rest of matrices.
C
IF( RNKA22.GT.0 ) THEN
C
C A <-- diag(I, Q2') * A
C Complex Workspace: need MIN(L,N) + N;
C prefer MIN(L,N) + N*NB.
C
CALL ZUNMQR( 'Left', 'ConjTranspose', LA22, RANKE,
$ RNKA22, A(IR1,IR1), LDA, ZWORK, A(IR1,1),
$ LDA, ZWORK(KW), LWR, INFO )
C
C B <-- diag(I, Q2') * B
C Complex Workspace: need MIN(L,N) + M;
C prefer MIN(L,N) + M*NB.
C
IF ( WITHB )
$ CALL ZUNMQR( 'Left', 'ConjTranspose', LA22, M, RNKA22,
$ A(IR1,IR1), LDA, ZWORK, B(IR1,1), LDB,
$ ZWORK(KW), LWR, INFO )
C
C Q <-- Q * diag(I, Q2)
C Complex Workspace: need MIN(L,N) + L;
C prefer MIN(L,N) + L*NB.
C
IF( ILQ )
$ CALL ZUNMQR( 'Right', 'No transpose', L, LA22, RNKA22,
$ A(IR1,IR1), LDA, ZWORK, Q(1,IR1), LDQ,
$ ZWORK(KW), LWR, INFO )
C
C Set lower triangle of A22 to zero.
C
IF( LA22.GE.2 )
$ CALL ZLASET( 'Lower', LA22-1, RNKA22, ZERO, ZERO,
$ A(IR1+1,IR1), LDA )
C
C Compute A*diag(I,P2), C*diag(I,P2) and Z*diag(I,P2)
C by forward permuting the columns of A, C and Z based
C on information in IWORK.
C
DO 40 J = 1, NA22
IWORK(J) = -IWORK(J)
40 CONTINUE
DO 60 I = 1, NA22
IF( IWORK(I).LT.0 ) THEN
J = I
IWORK(J) = -IWORK(J)
50 CONTINUE
K = IWORK(J)
IF( IWORK(K).LT.0 ) THEN
CALL ZSWAP( RANKE, A(1,RANKE+J), 1,
$ A(1,RANKE+K), 1 )
IF( WITHC )
$ CALL ZSWAP( P, C(1,RANKE+J), 1,
$ C(1,RANKE+K), 1 )
IF( ILZ )
$ CALL ZSWAP( N, Z(1,RANKE+J), 1,
$ Z(1,RANKE+K), 1 )
IWORK(K) = -IWORK(K)
J = K
GO TO 50
END IF
END IF
60 CONTINUE
C
IF( REDA .AND. RNKA22.LT.NA22 ) THEN
C
C Determine a unitary matrix Y2 such that
C
C ( R11 R12 ) = ( Ar 0 ) * Y2 .
C
C Compute A <-- A*diag(I, Y2'), C <-- C*diag(I, Y2'),
C Z <-- Z*diag(I, Y2').
C
C Complex Workspace: need 2*N;
C prefer N + N*NB.
C
KW = RANKE + 1
CALL ZTZRZF( RNKA22, NA22, A(IR1,IR1), LDA, ZWORK,
$ ZWORK(KW), LZWORK-KW+1, INFO )
WRKOPT = MAX( WRKOPT, INT( ZWORK(KW) ) + KW - 1 )
C
C Complex Workspace: need N + MAX(P,N);
C prefer N + MAX(P,N)*NB.
C
LH = NA22 - RNKA22
IF( WITHC ) THEN
CALL ZUNMRZ( 'Right', 'Conjugate transpose', P, N,
$ RNKA22, LH, A(IR1,IR1), LDA, ZWORK, C,
$ LDC, ZWORK(KW), LZWORK-KW+1, INFO )
WRKOPT = MAX( WRKOPT, INT( ZWORK(KW) ) + KW - 1 )
END IF
IF( ILZ ) THEN
CALL ZUNMRZ( 'Right', 'Conjugate transpose', N, N,
$ RNKA22, LH, A(IR1,IR1), LDA, ZWORK, Z,
$ LDZ, ZWORK(KW), LZWORK-KW+1, INFO )
WRKOPT = MAX( WRKOPT, INT( ZWORK(KW) ) + KW - 1 )
END IF
IRE1 = RANKE + RNKA22 + 1
C
C Set R12 and R22 to zero.
C
CALL ZLASET( 'Full', LA22, LH, ZERO, ZERO,
$ A(IR1,IRE1), LDA )
END IF
ELSE
CALL ZLASET( 'Full', LA22, NA22, ZERO, ZERO,
$ A(IR1,IR1), LDA)
END IF
END IF
END IF
C
ZWORK(1) = WRKOPT
C
RETURN
C *** Last line of TG01FZ ***
END
|
