summaryrefslogtreecommitdiff
path: root/SRC/cunbdb.f
blob: b1f8c1acbf8bbd16254c9d3dec971cd31509cf66 (plain)
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
*> \brief \b CUNBDB
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CUNBDB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cunbdb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cunbdb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cunbdb.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE CUNBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12,
*                          X21, LDX21, X22, LDX22, THETA, PHI, TAUP1,
*                          TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          SIGNS, TRANS
*       INTEGER            INFO, LDX11, LDX12, LDX21, LDX22, LWORK, M, P,
*      $                   Q
*       ..
*       .. Array Arguments ..
*       REAL               PHI( * ), THETA( * )
*       COMPLEX            TAUP1( * ), TAUP2( * ), TAUQ1( * ), TAUQ2( * ),
*      $                   WORK( * ), X11( LDX11, * ), X12( LDX12, * ),
*      $                   X21( LDX21, * ), X22( LDX22, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> CUNBDB simultaneously bidiagonalizes the blocks of an M-by-M
*> partitioned unitary matrix X:
*>
*>                                 [ B11 | B12 0  0 ]
*>     [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**H
*> X = [-----------] = [---------] [----------------] [---------]   .
*>     [ X21 | X22 ]   [    | P2 ] [ B21 | B22 0  0 ] [    | Q2 ]
*>                                 [  0  |  0  0  I ]
*>
*> X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
*> not the case, then X must be transposed and/or permuted. This can be
*> done in constant time using the TRANS and SIGNS options. See CUNCSD
*> for details.)
*>
*> The unitary matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
*> (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
*> represented implicitly by Householder vectors.
*>
*> B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
*> implicitly by angles THETA, PHI.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER
*>          = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
*>                      order;
*>          otherwise:  X, U1, U2, V1T, and V2T are stored in column-
*>                      major order.
*> \endverbatim
*>
*> \param[in] SIGNS
*> \verbatim
*>          SIGNS is CHARACTER
*>          = 'O':      The lower-left block is made nonpositive (the
*>                      "other" convention);
*>          otherwise:  The upper-right block is made nonpositive (the
*>                      "default" convention).
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows and columns in X.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*>          P is INTEGER
*>          The number of rows in X11 and X12. 0 <= P <= M.
*> \endverbatim
*>
*> \param[in] Q
*> \verbatim
*>          Q is INTEGER
*>          The number of columns in X11 and X21. 0 <= Q <=
*>          MIN(P,M-P,M-Q).
*> \endverbatim
*>
*> \param[in,out] X11
*> \verbatim
*>          X11 is COMPLEX array, dimension (LDX11,Q)
*>          On entry, the top-left block of the unitary matrix to be
*>          reduced. On exit, the form depends on TRANS:
*>          If TRANS = 'N', then
*>             the columns of tril(X11) specify reflectors for P1,
*>             the rows of triu(X11,1) specify reflectors for Q1;
*>          else TRANS = 'T', and
*>             the rows of triu(X11) specify reflectors for P1,
*>             the columns of tril(X11,-1) specify reflectors for Q1.
*> \endverbatim
*>
*> \param[in] LDX11
*> \verbatim
*>          LDX11 is INTEGER
*>          The leading dimension of X11. If TRANS = 'N', then LDX11 >=
*>          P; else LDX11 >= Q.
*> \endverbatim
*>
*> \param[in,out] X12
*> \verbatim
*>          X12 is COMPLEX array, dimension (LDX12,M-Q)
*>          On entry, the top-right block of the unitary matrix to
*>          be reduced. On exit, the form depends on TRANS:
*>          If TRANS = 'N', then
*>             the rows of triu(X12) specify the first P reflectors for
*>             Q2;
*>          else TRANS = 'T', and
*>             the columns of tril(X12) specify the first P reflectors
*>             for Q2.
*> \endverbatim
*>
*> \param[in] LDX12
*> \verbatim
*>          LDX12 is INTEGER
*>          The leading dimension of X12. If TRANS = 'N', then LDX12 >=
*>          P; else LDX11 >= M-Q.
*> \endverbatim
*>
*> \param[in,out] X21
*> \verbatim
*>          X21 is COMPLEX array, dimension (LDX21,Q)
*>          On entry, the bottom-left block of the unitary matrix to
*>          be reduced. On exit, the form depends on TRANS:
*>          If TRANS = 'N', then
*>             the columns of tril(X21) specify reflectors for P2;
*>          else TRANS = 'T', and
*>             the rows of triu(X21) specify reflectors for P2.
*> \endverbatim
*>
*> \param[in] LDX21
*> \verbatim
*>          LDX21 is INTEGER
*>          The leading dimension of X21. If TRANS = 'N', then LDX21 >=
*>          M-P; else LDX21 >= Q.
*> \endverbatim
*>
*> \param[in,out] X22
*> \verbatim
*>          X22 is COMPLEX array, dimension (LDX22,M-Q)
*>          On entry, the bottom-right block of the unitary matrix to
*>          be reduced. On exit, the form depends on TRANS:
*>          If TRANS = 'N', then
*>             the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
*>             M-P-Q reflectors for Q2,
*>          else TRANS = 'T', and
*>             the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
*>             M-P-Q reflectors for P2.
*> \endverbatim
*>
*> \param[in] LDX22
*> \verbatim
*>          LDX22 is INTEGER
*>          The leading dimension of X22. If TRANS = 'N', then LDX22 >=
*>          M-P; else LDX22 >= M-Q.
*> \endverbatim
*>
*> \param[out] THETA
*> \verbatim
*>          THETA is REAL array, dimension (Q)
*>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
*>          be computed from the angles THETA and PHI. See Further
*>          Details.
*> \endverbatim
*>
*> \param[out] PHI
*> \verbatim
*>          PHI is REAL array, dimension (Q-1)
*>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
*>          be computed from the angles THETA and PHI. See Further
*>          Details.
*> \endverbatim
*>
*> \param[out] TAUP1
*> \verbatim
*>          TAUP1 is COMPLEX array, dimension (P)
*>          The scalar factors of the elementary reflectors that define
*>          P1.
*> \endverbatim
*>
*> \param[out] TAUP2
*> \verbatim
*>          TAUP2 is COMPLEX array, dimension (M-P)
*>          The scalar factors of the elementary reflectors that define
*>          P2.
*> \endverbatim
*>
*> \param[out] TAUQ1
*> \verbatim
*>          TAUQ1 is COMPLEX array, dimension (Q)
*>          The scalar factors of the elementary reflectors that define
*>          Q1.
*> \endverbatim
*>
*> \param[out] TAUQ2
*> \verbatim
*>          TAUQ2 is COMPLEX array, dimension (M-Q)
*>          The scalar factors of the elementary reflectors that define
*>          Q2.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK. LWORK >= M-Q.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal size of the WORK array, returns
*>          this value as the first entry of the WORK array, and no error
*>          message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2013
*
*> \ingroup complexOTHERcomputational
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The bidiagonal blocks B11, B12, B21, and B22 are represented
*>  implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
*>  PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
*>  lower bidiagonal. Every entry in each bidiagonal band is a product
*>  of a sine or cosine of a THETA with a sine or cosine of a PHI. See
*>  [1] or CUNCSD for details.
*>
*>  P1, P2, Q1, and Q2 are represented as products of elementary
*>  reflectors. See CUNCSD for details on generating P1, P2, Q1, and Q2
*>  using CUNGQR and CUNGLQ.
*> \endverbatim
*
*> \par References:
*  ================
*>
*>  [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
*>      Algorithms, 50(1):33-65, 2009.
*>
*  =====================================================================
      SUBROUTINE CUNBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12,
     $                   X21, LDX21, X22, LDX22, THETA, PHI, TAUP1,
     $                   TAUP2, TAUQ1, TAUQ2, WORK, LWORK, 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          SIGNS, TRANS
      INTEGER            INFO, LDX11, LDX12, LDX21, LDX22, LWORK, M, P,
     $                   Q
*     ..
*     .. Array Arguments ..
      REAL               PHI( * ), THETA( * )
      COMPLEX            TAUP1( * ), TAUP2( * ), TAUQ1( * ), TAUQ2( * ),
     $                   WORK( * ), X11( LDX11, * ), X12( LDX12, * ),
     $                   X21( LDX21, * ), X22( LDX22, * )
*     ..
*
*  ====================================================================
*
*     .. Parameters ..
      REAL               REALONE
      PARAMETER          ( REALONE = 1.0E0 )
      COMPLEX            ONE
      PARAMETER          ( ONE = (1.0E0,0.0E0) )
*     ..
*     .. Local Scalars ..
      LOGICAL            COLMAJOR, LQUERY
      INTEGER            I, LWORKMIN, LWORKOPT
      REAL               Z1, Z2, Z3, Z4
*     ..
*     .. External Subroutines ..
      EXTERNAL           CAXPY, CLARF, CLARFGP, CSCAL, XERBLA
      EXTERNAL           CLACGV
*
*     ..
*     .. External Functions ..
      REAL               SCNRM2
      LOGICAL            LSAME
      EXTERNAL           SCNRM2, LSAME
*     ..
*     .. Intrinsic Functions
      INTRINSIC          ATAN2, COS, MAX, MIN, SIN
      INTRINSIC          CMPLX, CONJG
*     ..
*     .. Executable Statements ..
*
*     Test input arguments
*
      INFO = 0
      COLMAJOR = .NOT. LSAME( TRANS, 'T' )
      IF( .NOT. LSAME( SIGNS, 'O' ) ) THEN
         Z1 = REALONE
         Z2 = REALONE
         Z3 = REALONE
         Z4 = REALONE
      ELSE
         Z1 = REALONE
         Z2 = -REALONE
         Z3 = REALONE
         Z4 = -REALONE
      END IF
      LQUERY = LWORK .EQ. -1
*
      IF( M .LT. 0 ) THEN
         INFO = -3
      ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN
         INFO = -4
      ELSE IF( Q .LT. 0 .OR. Q .GT. P .OR. Q .GT. M-P .OR.
     $         Q .GT. M-Q ) THEN
         INFO = -5
      ELSE IF( COLMAJOR .AND. LDX11 .LT. MAX( 1, P ) ) THEN
         INFO = -7
      ELSE IF( .NOT.COLMAJOR .AND. LDX11 .LT. MAX( 1, Q ) ) THEN
         INFO = -7
      ELSE IF( COLMAJOR .AND. LDX12 .LT. MAX( 1, P ) ) THEN
         INFO = -9
      ELSE IF( .NOT.COLMAJOR .AND. LDX12 .LT. MAX( 1, M-Q ) ) THEN
         INFO = -9
      ELSE IF( COLMAJOR .AND. LDX21 .LT. MAX( 1, M-P ) ) THEN
         INFO = -11
      ELSE IF( .NOT.COLMAJOR .AND. LDX21 .LT. MAX( 1, Q ) ) THEN
         INFO = -11
      ELSE IF( COLMAJOR .AND. LDX22 .LT. MAX( 1, M-P ) ) THEN
         INFO = -13
      ELSE IF( .NOT.COLMAJOR .AND. LDX22 .LT. MAX( 1, M-Q ) ) THEN
         INFO = -13
      END IF
*
*     Compute workspace
*
      IF( INFO .EQ. 0 ) THEN
         LWORKOPT = M - Q
         LWORKMIN = M - Q
         WORK(1) = LWORKOPT
         IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN
            INFO = -21
         END IF
      END IF
      IF( INFO .NE. 0 ) THEN
         CALL XERBLA( 'xORBDB', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Handle column-major and row-major separately
*
      IF( COLMAJOR ) THEN
*
*        Reduce columns 1, ..., Q of X11, X12, X21, and X22
*
         DO I = 1, Q
*
            IF( I .EQ. 1 ) THEN
               CALL CSCAL( P-I+1, CMPLX( Z1, 0.0E0 ), X11(I,I), 1 )
            ELSE
               CALL CSCAL( P-I+1, CMPLX( Z1*COS(PHI(I-1)), 0.0E0 ),
     $                     X11(I,I), 1 )
               CALL CAXPY( P-I+1, CMPLX( -Z1*Z3*Z4*SIN(PHI(I-1)),
     $                     0.0E0 ), X12(I,I-1), 1, X11(I,I), 1 )
            END IF
            IF( I .EQ. 1 ) THEN
               CALL CSCAL( M-P-I+1, CMPLX( Z2, 0.0E0 ), X21(I,I), 1 )
            ELSE
               CALL CSCAL( M-P-I+1, CMPLX( Z2*COS(PHI(I-1)), 0.0E0 ),
     $                     X21(I,I), 1 )
               CALL CAXPY( M-P-I+1, CMPLX( -Z2*Z3*Z4*SIN(PHI(I-1)),
     $                     0.0E0 ), X22(I,I-1), 1, X21(I,I), 1 )
            END IF
*
            THETA(I) = ATAN2( SCNRM2( M-P-I+1, X21(I,I), 1 ),
     $                 SCNRM2( P-I+1, X11(I,I), 1 ) )
*
            IF( P .GT. I ) THEN
               CALL CLARFGP( P-I+1, X11(I,I), X11(I+1,I), 1, TAUP1(I) )
            ELSE IF ( P .EQ. I ) THEN
               CALL CLARFGP( P-I+1, X11(I,I), X11(I,I), 1, TAUP1(I) )
            END IF
            X11(I,I) = ONE
            IF ( M-P .GT. I ) THEN
               CALL CLARFGP( M-P-I+1, X21(I,I), X21(I+1,I), 1,
     $                       TAUP2(I) )
            ELSE IF ( M-P .EQ. I ) THEN
               CALL CLARFGP( M-P-I+1, X21(I,I), X21(I,I), 1,
     $                       TAUP2(I) )
            END IF
            X21(I,I) = ONE
*
            IF ( Q .GT. I ) THEN
               CALL CLARF( 'L', P-I+1, Q-I, X11(I,I), 1,
     $                     CONJG(TAUP1(I)), X11(I,I+1), LDX11, WORK )
               CALL CLARF( 'L', M-P-I+1, Q-I, X21(I,I), 1,
     $                     CONJG(TAUP2(I)), X21(I,I+1), LDX21, WORK )
            END IF
            IF ( M-Q+1 .GT. I ) THEN
               CALL CLARF( 'L', P-I+1, M-Q-I+1, X11(I,I), 1,
     $                     CONJG(TAUP1(I)), X12(I,I), LDX12, WORK )
               CALL CLARF( 'L', M-P-I+1, M-Q-I+1, X21(I,I), 1,
     $                     CONJG(TAUP2(I)), X22(I,I), LDX22, WORK )
            END IF
*
            IF( I .LT. Q ) THEN
               CALL CSCAL( Q-I, CMPLX( -Z1*Z3*SIN(THETA(I)), 0.0E0 ),
     $                     X11(I,I+1), LDX11 )
               CALL CAXPY( Q-I, CMPLX( Z2*Z3*COS(THETA(I)), 0.0E0 ),
     $                     X21(I,I+1), LDX21, X11(I,I+1), LDX11 )
            END IF
            CALL CSCAL( M-Q-I+1, CMPLX( -Z1*Z4*SIN(THETA(I)), 0.0E0 ),
     $                  X12(I,I), LDX12 )
            CALL CAXPY( M-Q-I+1, CMPLX( Z2*Z4*COS(THETA(I)), 0.0E0 ),
     $                  X22(I,I), LDX22, X12(I,I), LDX12 )
*
            IF( I .LT. Q )
     $         PHI(I) = ATAN2( SCNRM2( Q-I, X11(I,I+1), LDX11 ),
     $                  SCNRM2( M-Q-I+1, X12(I,I), LDX12 ) )
*
            IF( I .LT. Q ) THEN
               CALL CLACGV( Q-I, X11(I,I+1), LDX11 )
               IF ( I .EQ. Q-1 ) THEN
                  CALL CLARFGP( Q-I, X11(I,I+1), X11(I,I+1), LDX11,
     $                          TAUQ1(I) )
               ELSE
                  CALL CLARFGP( Q-I, X11(I,I+1), X11(I,I+2), LDX11,
     $                          TAUQ1(I) )
               END IF
               X11(I,I+1) = ONE
            END IF
            IF ( M-Q+1 .GT. I ) THEN
               CALL CLACGV( M-Q-I+1, X12(I,I), LDX12 )
               IF ( M-Q .EQ. I ) THEN
                  CALL CLARFGP( M-Q-I+1, X12(I,I), X12(I,I), LDX12,
     $                          TAUQ2(I) )
               ELSE
                  CALL CLARFGP( M-Q-I+1, X12(I,I), X12(I,I+1), LDX12,
     $                          TAUQ2(I) )
               END IF
            END IF
            X12(I,I) = ONE
*
            IF( I .LT. Q ) THEN
               CALL CLARF( 'R', P-I, Q-I, X11(I,I+1), LDX11, TAUQ1(I),
     $                     X11(I+1,I+1), LDX11, WORK )
               CALL CLARF( 'R', M-P-I, Q-I, X11(I,I+1), LDX11, TAUQ1(I),
     $                     X21(I+1,I+1), LDX21, WORK )
            END IF
            IF ( P .GT. I ) THEN
               CALL CLARF( 'R', P-I, M-Q-I+1, X12(I,I), LDX12, TAUQ2(I),
     $                     X12(I+1,I), LDX12, WORK )
            END IF
            IF ( M-P .GT. I ) THEN
               CALL CLARF( 'R', M-P-I, M-Q-I+1, X12(I,I), LDX12,
     $                     TAUQ2(I), X22(I+1,I), LDX22, WORK )
            END IF
*
            IF( I .LT. Q )
     $         CALL CLACGV( Q-I, X11(I,I+1), LDX11 )
            CALL CLACGV( M-Q-I+1, X12(I,I), LDX12 )
*
         END DO
*
*        Reduce columns Q + 1, ..., P of X12, X22
*
         DO I = Q + 1, P
*
            CALL CSCAL( M-Q-I+1, CMPLX( -Z1*Z4, 0.0E0 ), X12(I,I),
     $                  LDX12 )
            CALL CLACGV( M-Q-I+1, X12(I,I), LDX12 )
            IF ( I .GE. M-Q ) THEN
               CALL CLARFGP( M-Q-I+1, X12(I,I), X12(I,I), LDX12,
     $                       TAUQ2(I) )
            ELSE
               CALL CLARFGP( M-Q-I+1, X12(I,I), X12(I,I+1), LDX12,
     $                       TAUQ2(I) )
            END IF
            X12(I,I) = ONE
*
            IF ( P .GT. I ) THEN
               CALL CLARF( 'R', P-I, M-Q-I+1, X12(I,I), LDX12, TAUQ2(I),
     $                     X12(I+1,I), LDX12, WORK )
            END IF
            IF( M-P-Q .GE. 1 )
     $         CALL CLARF( 'R', M-P-Q, M-Q-I+1, X12(I,I), LDX12,
     $                     TAUQ2(I), X22(Q+1,I), LDX22, WORK )
*
            CALL CLACGV( M-Q-I+1, X12(I,I), LDX12 )
*
         END DO
*
*        Reduce columns P + 1, ..., M - Q of X12, X22
*
         DO I = 1, M - P - Q
*
            CALL CSCAL( M-P-Q-I+1, CMPLX( Z2*Z4, 0.0E0 ),
     $                  X22(Q+I,P+I), LDX22 )
            CALL CLACGV( M-P-Q-I+1, X22(Q+I,P+I), LDX22 )
            CALL CLARFGP( M-P-Q-I+1, X22(Q+I,P+I), X22(Q+I,P+I+1),
     $                    LDX22, TAUQ2(P+I) )
            X22(Q+I,P+I) = ONE
            CALL CLARF( 'R', M-P-Q-I, M-P-Q-I+1, X22(Q+I,P+I), LDX22,
     $                  TAUQ2(P+I), X22(Q+I+1,P+I), LDX22, WORK )
*
            CALL CLACGV( M-P-Q-I+1, X22(Q+I,P+I), LDX22 )
*
         END DO
*
      ELSE
*
*        Reduce columns 1, ..., Q of X11, X12, X21, X22
*
         DO I = 1, Q
*
            IF( I .EQ. 1 ) THEN
               CALL CSCAL( P-I+1, CMPLX( Z1, 0.0E0 ), X11(I,I),
     $                     LDX11 )
            ELSE
               CALL CSCAL( P-I+1, CMPLX( Z1*COS(PHI(I-1)), 0.0E0 ),
     $                     X11(I,I), LDX11 )
               CALL CAXPY( P-I+1, CMPLX( -Z1*Z3*Z4*SIN(PHI(I-1)),
     $                     0.0E0 ), X12(I-1,I), LDX12, X11(I,I), LDX11 )
            END IF
            IF( I .EQ. 1 ) THEN
               CALL CSCAL( M-P-I+1, CMPLX( Z2, 0.0E0 ), X21(I,I),
     $                     LDX21 )
            ELSE
               CALL CSCAL( M-P-I+1, CMPLX( Z2*COS(PHI(I-1)), 0.0E0 ),
     $                     X21(I,I), LDX21 )
               CALL CAXPY( M-P-I+1, CMPLX( -Z2*Z3*Z4*SIN(PHI(I-1)),
     $                     0.0E0 ), X22(I-1,I), LDX22, X21(I,I), LDX21 )
            END IF
*
            THETA(I) = ATAN2( SCNRM2( M-P-I+1, X21(I,I), LDX21 ),
     $                 SCNRM2( P-I+1, X11(I,I), LDX11 ) )
*
            CALL CLACGV( P-I+1, X11(I,I), LDX11 )
            CALL CLACGV( M-P-I+1, X21(I,I), LDX21 )
*
            CALL CLARFGP( P-I+1, X11(I,I), X11(I,I+1), LDX11, TAUP1(I) )
            X11(I,I) = ONE
            IF ( I .EQ. M-P ) THEN
               CALL CLARFGP( M-P-I+1, X21(I,I), X21(I,I), LDX21,
     $                       TAUP2(I) )
            ELSE
               CALL CLARFGP( M-P-I+1, X21(I,I), X21(I,I+1), LDX21,
     $                       TAUP2(I) )
            END IF
            X21(I,I) = ONE
*
            CALL CLARF( 'R', Q-I, P-I+1, X11(I,I), LDX11, TAUP1(I),
     $                  X11(I+1,I), LDX11, WORK )
            CALL CLARF( 'R', M-Q-I+1, P-I+1, X11(I,I), LDX11, TAUP1(I),
     $                  X12(I,I), LDX12, WORK )
            CALL CLARF( 'R', Q-I, M-P-I+1, X21(I,I), LDX21, TAUP2(I),
     $                  X21(I+1,I), LDX21, WORK )
            CALL CLARF( 'R', M-Q-I+1, M-P-I+1, X21(I,I), LDX21,
     $                  TAUP2(I), X22(I,I), LDX22, WORK )
*
            CALL CLACGV( P-I+1, X11(I,I), LDX11 )
            CALL CLACGV( M-P-I+1, X21(I,I), LDX21 )
*
            IF( I .LT. Q ) THEN
               CALL CSCAL( Q-I, CMPLX( -Z1*Z3*SIN(THETA(I)), 0.0E0 ),
     $                     X11(I+1,I), 1 )
               CALL CAXPY( Q-I, CMPLX( Z2*Z3*COS(THETA(I)), 0.0E0 ),
     $                     X21(I+1,I), 1, X11(I+1,I), 1 )
            END IF
            CALL CSCAL( M-Q-I+1, CMPLX( -Z1*Z4*SIN(THETA(I)), 0.0E0 ),
     $                  X12(I,I), 1 )
            CALL CAXPY( M-Q-I+1, CMPLX( Z2*Z4*COS(THETA(I)), 0.0E0 ),
     $                  X22(I,I), 1, X12(I,I), 1 )
*
            IF( I .LT. Q )
     $         PHI(I) = ATAN2( SCNRM2( Q-I, X11(I+1,I), 1 ),
     $                  SCNRM2( M-Q-I+1, X12(I,I), 1 ) )
*
            IF( I .LT. Q ) THEN
               CALL CLARFGP( Q-I, X11(I+1,I), X11(I+2,I), 1, TAUQ1(I) )
               X11(I+1,I) = ONE
            END IF
            CALL CLARFGP( M-Q-I+1, X12(I,I), X12(I+1,I), 1, TAUQ2(I) )
            X12(I,I) = ONE
*
            IF( I .LT. Q ) THEN
               CALL CLARF( 'L', Q-I, P-I, X11(I+1,I), 1,
     $                     CONJG(TAUQ1(I)), X11(I+1,I+1), LDX11, WORK )
               CALL CLARF( 'L', Q-I, M-P-I, X11(I+1,I), 1,
     $                     CONJG(TAUQ1(I)), X21(I+1,I+1), LDX21, WORK )
            END IF
            CALL CLARF( 'L', M-Q-I+1, P-I, X12(I,I), 1, CONJG(TAUQ2(I)),
     $                  X12(I,I+1), LDX12, WORK )

            IF ( M-P .GT. I ) THEN
               CALL CLARF( 'L', M-Q-I+1, M-P-I, X12(I,I), 1,
     $                     CONJG(TAUQ2(I)), X22(I,I+1), LDX22, WORK )
            END IF
         END DO
*
*        Reduce columns Q + 1, ..., P of X12, X22
*
         DO I = Q + 1, P
*
            CALL CSCAL( M-Q-I+1, CMPLX( -Z1*Z4, 0.0E0 ), X12(I,I), 1 )
            CALL CLARFGP( M-Q-I+1, X12(I,I), X12(I+1,I), 1, TAUQ2(I) )
            X12(I,I) = ONE
*
            IF ( P .GT. I ) THEN
               CALL CLARF( 'L', M-Q-I+1, P-I, X12(I,I), 1,
     $                     CONJG(TAUQ2(I)), X12(I,I+1), LDX12, WORK )
            END IF
            IF( M-P-Q .GE. 1 )
     $         CALL CLARF( 'L', M-Q-I+1, M-P-Q, X12(I,I), 1,
     $                     CONJG(TAUQ2(I)), X22(I,Q+1), LDX22, WORK )
*
         END DO
*
*        Reduce columns P + 1, ..., M - Q of X12, X22
*
         DO I = 1, M - P - Q
*
            CALL CSCAL( M-P-Q-I+1, CMPLX( Z2*Z4, 0.0E0 ),
     $                  X22(P+I,Q+I), 1 )
            CALL CLARFGP( M-P-Q-I+1, X22(P+I,Q+I), X22(P+I+1,Q+I), 1,
     $                    TAUQ2(P+I) )
            X22(P+I,Q+I) = ONE
            IF ( M-P-Q .NE. I ) THEN
               CALL CLARF( 'L', M-P-Q-I+1, M-P-Q-I, X22(P+I,Q+I), 1,
     $                     CONJG(TAUQ2(P+I)), X22(P+I,Q+I+1), LDX22,
     $                     WORK )
            END IF
         END DO
*
      END IF
*
      RETURN
*
*     End of CUNBDB
*
      END