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
|
*> \brief \b ZUNBDB
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> Download ZUNBDB + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zunbdb.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zunbdb.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zunbdb.f">
*> [TXT]</a>
*
* Definition
* ==========
*
* SUBROUTINE ZUNBDB( 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 ..
* DOUBLE PRECISION PHI( * ), THETA( * )
* COMPLEX*16 TAUP1( * ), TAUP2( * ), TAUQ1( * ), TAUQ2( * ),
* $ WORK( * ), X11( LDX11, * ), X12( LDX12, * ),
* $ X21( LDX21, * ), X22( LDX22, * )
* ..
*
* Purpose
* =======
*
*>\details \b Purpose:
*>\verbatim
*>
*> ZUNBDB 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 ZUNCSD
*> 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*16 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*16 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*16 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*16 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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*16 array, dimension (P)
*> The scalar factors of the elementary reflectors that define
*> P1.
*> \endverbatim
*>
*> \param[out] TAUP2
*> \verbatim
*> TAUP2 is COMPLEX*16 array, dimension (M-P)
*> The scalar factors of the elementary reflectors that define
*> P2.
*> \endverbatim
*>
*> \param[out] TAUQ1
*> \verbatim
*> TAUQ1 is COMPLEX*16 array, dimension (Q)
*> The scalar factors of the elementary reflectors that define
*> Q1.
*> \endverbatim
*>
*> \param[out] TAUQ2
*> \verbatim
*> TAUQ2 is COMPLEX*16 array, dimension (M-Q)
*> The scalar factors of the elementary reflectors that define
*> Q2.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX*16 array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= M-Q.
*> \endverbatim
*> \verbatim
*> 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 2011
*
*> \ingroup complex16OTHERcomputational
*
*
* Further Details
* ===============
*>\details \b Further \b 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 ZUNCSD for details.
*>
*> P1, P2, Q1, and Q2 are represented as products of elementary
*> reflectors. See ZUNCSD for details on generating P1, P2, Q1, and Q2
*> using ZUNGQR and ZUNGLQ.
*>
*> Reference
*> =========
*>
*> [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
*> Algorithms, 50(1):33-65, 2009.
*>
*> \endverbatim
*>
* =====================================================================
SUBROUTINE ZUNBDB( 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 3.3.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER SIGNS, TRANS
INTEGER INFO, LDX11, LDX12, LDX21, LDX22, LWORK, M, P,
$ Q
* ..
* .. Array Arguments ..
DOUBLE PRECISION PHI( * ), THETA( * )
COMPLEX*16 TAUP1( * ), TAUP2( * ), TAUQ1( * ), TAUQ2( * ),
$ WORK( * ), X11( LDX11, * ), X12( LDX12, * ),
$ X21( LDX21, * ), X22( LDX22, * )
* ..
*
* ====================================================================
*
* .. Parameters ..
DOUBLE PRECISION REALONE
PARAMETER ( REALONE = 1.0D0 )
COMPLEX*16 NEGONE, ONE
PARAMETER ( NEGONE = (-1.0D0,0.0D0),
$ ONE = (1.0D0,0.0D0) )
* ..
* .. Local Scalars ..
LOGICAL COLMAJOR, LQUERY
INTEGER I, LWORKMIN, LWORKOPT
DOUBLE PRECISION Z1, Z2, Z3, Z4
* ..
* .. External Subroutines ..
EXTERNAL ZAXPY, ZLARF, ZLARFGP, ZSCAL, XERBLA
EXTERNAL ZLACGV
*
* ..
* .. External Functions ..
DOUBLE PRECISION DZNRM2
LOGICAL LSAME
EXTERNAL DZNRM2, LSAME
* ..
* .. Intrinsic Functions
INTRINSIC ATAN2, COS, MAX, MIN, SIN
INTRINSIC DCMPLX, DCONJG
* ..
* .. 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 ZSCAL( P-I+1, DCMPLX( Z1, 0.0D0 ), X11(I,I), 1 )
ELSE
CALL ZSCAL( P-I+1, DCMPLX( Z1*COS(PHI(I-1)), 0.0D0 ),
$ X11(I,I), 1 )
CALL ZAXPY( P-I+1, DCMPLX( -Z1*Z3*Z4*SIN(PHI(I-1)),
$ 0.0D0 ), X12(I,I-1), 1, X11(I,I), 1 )
END IF
IF( I .EQ. 1 ) THEN
CALL ZSCAL( M-P-I+1, DCMPLX( Z2, 0.0D0 ), X21(I,I), 1 )
ELSE
CALL ZSCAL( M-P-I+1, DCMPLX( Z2*COS(PHI(I-1)), 0.0D0 ),
$ X21(I,I), 1 )
CALL ZAXPY( M-P-I+1, DCMPLX( -Z2*Z3*Z4*SIN(PHI(I-1)),
$ 0.0D0 ), X22(I,I-1), 1, X21(I,I), 1 )
END IF
*
THETA(I) = ATAN2( DZNRM2( M-P-I+1, X21(I,I), 1 ),
$ DZNRM2( P-I+1, X11(I,I), 1 ) )
*
CALL ZLARFGP( P-I+1, X11(I,I), X11(I+1,I), 1, TAUP1(I) )
X11(I,I) = ONE
CALL ZLARFGP( M-P-I+1, X21(I,I), X21(I+1,I), 1, TAUP2(I) )
X21(I,I) = ONE
*
CALL ZLARF( 'L', P-I+1, Q-I, X11(I,I), 1, DCONJG(TAUP1(I)),
$ X11(I,I+1), LDX11, WORK )
CALL ZLARF( 'L', P-I+1, M-Q-I+1, X11(I,I), 1,
$ DCONJG(TAUP1(I)), X12(I,I), LDX12, WORK )
CALL ZLARF( 'L', M-P-I+1, Q-I, X21(I,I), 1,
$ DCONJG(TAUP2(I)), X21(I,I+1), LDX21, WORK )
CALL ZLARF( 'L', M-P-I+1, M-Q-I+1, X21(I,I), 1,
$ DCONJG(TAUP2(I)), X22(I,I), LDX22, WORK )
*
IF( I .LT. Q ) THEN
CALL ZSCAL( Q-I, DCMPLX( -Z1*Z3*SIN(THETA(I)), 0.0D0 ),
$ X11(I,I+1), LDX11 )
CALL ZAXPY( Q-I, DCMPLX( Z2*Z3*COS(THETA(I)), 0.0D0 ),
$ X21(I,I+1), LDX21, X11(I,I+1), LDX11 )
END IF
CALL ZSCAL( M-Q-I+1, DCMPLX( -Z1*Z4*SIN(THETA(I)), 0.0D0 ),
$ X12(I,I), LDX12 )
CALL ZAXPY( M-Q-I+1, DCMPLX( Z2*Z4*COS(THETA(I)), 0.0D0 ),
$ X22(I,I), LDX22, X12(I,I), LDX12 )
*
IF( I .LT. Q )
$ PHI(I) = ATAN2( DZNRM2( Q-I, X11(I,I+1), LDX11 ),
$ DZNRM2( M-Q-I+1, X12(I,I), LDX12 ) )
*
IF( I .LT. Q ) THEN
CALL ZLACGV( Q-I, X11(I,I+1), LDX11 )
CALL ZLARFGP( Q-I, X11(I,I+1), X11(I,I+2), LDX11,
$ TAUQ1(I) )
X11(I,I+1) = ONE
END IF
CALL ZLACGV( M-Q-I+1, X12(I,I), LDX12 )
CALL ZLARFGP( M-Q-I+1, X12(I,I), X12(I,I+1), LDX12,
$ TAUQ2(I) )
X12(I,I) = ONE
*
IF( I .LT. Q ) THEN
CALL ZLARF( 'R', P-I, Q-I, X11(I,I+1), LDX11, TAUQ1(I),
$ X11(I+1,I+1), LDX11, WORK )
CALL ZLARF( 'R', M-P-I, Q-I, X11(I,I+1), LDX11, TAUQ1(I),
$ X21(I+1,I+1), LDX21, WORK )
END IF
CALL ZLARF( 'R', P-I, M-Q-I+1, X12(I,I), LDX12, TAUQ2(I),
$ X12(I+1,I), LDX12, WORK )
CALL ZLARF( 'R', M-P-I, M-Q-I+1, X12(I,I), LDX12, TAUQ2(I),
$ X22(I+1,I), LDX22, WORK )
*
IF( I .LT. Q )
$ CALL ZLACGV( Q-I, X11(I,I+1), LDX11 )
CALL ZLACGV( M-Q-I+1, X12(I,I), LDX12 )
*
END DO
*
* Reduce columns Q + 1, ..., P of X12, X22
*
DO I = Q + 1, P
*
CALL ZSCAL( M-Q-I+1, DCMPLX( -Z1*Z4, 0.0D0 ), X12(I,I),
$ LDX12 )
CALL ZLACGV( M-Q-I+1, X12(I,I), LDX12 )
CALL ZLARFGP( M-Q-I+1, X12(I,I), X12(I,I+1), LDX12,
$ TAUQ2(I) )
X12(I,I) = ONE
*
CALL ZLARF( 'R', P-I, M-Q-I+1, X12(I,I), LDX12, TAUQ2(I),
$ X12(I+1,I), LDX12, WORK )
IF( M-P-Q .GE. 1 )
$ CALL ZLARF( 'R', M-P-Q, M-Q-I+1, X12(I,I), LDX12,
$ TAUQ2(I), X22(Q+1,I), LDX22, WORK )
*
CALL ZLACGV( 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 ZSCAL( M-P-Q-I+1, DCMPLX( Z2*Z4, 0.0D0 ),
$ X22(Q+I,P+I), LDX22 )
CALL ZLACGV( M-P-Q-I+1, X22(Q+I,P+I), LDX22 )
CALL ZLARFGP( 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 ZLARF( '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 ZLACGV( 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 ZSCAL( P-I+1, DCMPLX( Z1, 0.0D0 ), X11(I,I),
$ LDX11 )
ELSE
CALL ZSCAL( P-I+1, DCMPLX( Z1*COS(PHI(I-1)), 0.0D0 ),
$ X11(I,I), LDX11 )
CALL ZAXPY( P-I+1, DCMPLX( -Z1*Z3*Z4*SIN(PHI(I-1)),
$ 0.0D0 ), X12(I-1,I), LDX12, X11(I,I), LDX11 )
END IF
IF( I .EQ. 1 ) THEN
CALL ZSCAL( M-P-I+1, DCMPLX( Z2, 0.0D0 ), X21(I,I),
$ LDX21 )
ELSE
CALL ZSCAL( M-P-I+1, DCMPLX( Z2*COS(PHI(I-1)), 0.0D0 ),
$ X21(I,I), LDX21 )
CALL ZAXPY( M-P-I+1, DCMPLX( -Z2*Z3*Z4*SIN(PHI(I-1)),
$ 0.0D0 ), X22(I-1,I), LDX22, X21(I,I), LDX21 )
END IF
*
THETA(I) = ATAN2( DZNRM2( M-P-I+1, X21(I,I), LDX21 ),
$ DZNRM2( P-I+1, X11(I,I), LDX11 ) )
*
CALL ZLACGV( P-I+1, X11(I,I), LDX11 )
CALL ZLACGV( M-P-I+1, X21(I,I), LDX21 )
*
CALL ZLARFGP( P-I+1, X11(I,I), X11(I,I+1), LDX11, TAUP1(I) )
X11(I,I) = ONE
CALL ZLARFGP( M-P-I+1, X21(I,I), X21(I,I+1), LDX21,
$ TAUP2(I) )
X21(I,I) = ONE
*
CALL ZLARF( 'R', Q-I, P-I+1, X11(I,I), LDX11, TAUP1(I),
$ X11(I+1,I), LDX11, WORK )
CALL ZLARF( 'R', M-Q-I+1, P-I+1, X11(I,I), LDX11, TAUP1(I),
$ X12(I,I), LDX12, WORK )
CALL ZLARF( 'R', Q-I, M-P-I+1, X21(I,I), LDX21, TAUP2(I),
$ X21(I+1,I), LDX21, WORK )
CALL ZLARF( 'R', M-Q-I+1, M-P-I+1, X21(I,I), LDX21,
$ TAUP2(I), X22(I,I), LDX22, WORK )
*
CALL ZLACGV( P-I+1, X11(I,I), LDX11 )
CALL ZLACGV( M-P-I+1, X21(I,I), LDX21 )
*
IF( I .LT. Q ) THEN
CALL ZSCAL( Q-I, DCMPLX( -Z1*Z3*SIN(THETA(I)), 0.0D0 ),
$ X11(I+1,I), 1 )
CALL ZAXPY( Q-I, DCMPLX( Z2*Z3*COS(THETA(I)), 0.0D0 ),
$ X21(I+1,I), 1, X11(I+1,I), 1 )
END IF
CALL ZSCAL( M-Q-I+1, DCMPLX( -Z1*Z4*SIN(THETA(I)), 0.0D0 ),
$ X12(I,I), 1 )
CALL ZAXPY( M-Q-I+1, DCMPLX( Z2*Z4*COS(THETA(I)), 0.0D0 ),
$ X22(I,I), 1, X12(I,I), 1 )
*
IF( I .LT. Q )
$ PHI(I) = ATAN2( DZNRM2( Q-I, X11(I+1,I), 1 ),
$ DZNRM2( M-Q-I+1, X12(I,I), 1 ) )
*
IF( I .LT. Q ) THEN
CALL ZLARFGP( Q-I, X11(I+1,I), X11(I+2,I), 1, TAUQ1(I) )
X11(I+1,I) = ONE
END IF
CALL ZLARFGP( M-Q-I+1, X12(I,I), X12(I+1,I), 1, TAUQ2(I) )
X12(I,I) = ONE
*
IF( I .LT. Q ) THEN
CALL ZLARF( 'L', Q-I, P-I, X11(I+1,I), 1,
$ DCONJG(TAUQ1(I)), X11(I+1,I+1), LDX11, WORK )
CALL ZLARF( 'L', Q-I, M-P-I, X11(I+1,I), 1,
$ DCONJG(TAUQ1(I)), X21(I+1,I+1), LDX21, WORK )
END IF
CALL ZLARF( 'L', M-Q-I+1, P-I, X12(I,I), 1,
$ DCONJG(TAUQ2(I)), X12(I,I+1), LDX12, WORK )
CALL ZLARF( 'L', M-Q-I+1, M-P-I, X12(I,I), 1,
$ DCONJG(TAUQ2(I)), X22(I,I+1), LDX22, WORK )
*
END DO
*
* Reduce columns Q + 1, ..., P of X12, X22
*
DO I = Q + 1, P
*
CALL ZSCAL( M-Q-I+1, DCMPLX( -Z1*Z4, 0.0D0 ), X12(I,I), 1 )
CALL ZLARFGP( M-Q-I+1, X12(I,I), X12(I+1,I), 1, TAUQ2(I) )
X12(I,I) = ONE
*
CALL ZLARF( 'L', M-Q-I+1, P-I, X12(I,I), 1,
$ DCONJG(TAUQ2(I)), X12(I,I+1), LDX12, WORK )
IF( M-P-Q .GE. 1 )
$ CALL ZLARF( 'L', M-Q-I+1, M-P-Q, X12(I,I), 1,
$ DCONJG(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 ZSCAL( M-P-Q-I+1, DCMPLX( Z2*Z4, 0.0D0 ),
$ X22(P+I,Q+I), 1 )
CALL ZLARFGP( 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
*
CALL ZLARF( 'L', M-P-Q-I+1, M-P-Q-I, X22(P+I,Q+I), 1,
$ DCONJG(TAUQ2(P+I)), X22(P+I,Q+I+1), LDX22,
$ WORK )
*
END DO
*
END IF
*
RETURN
*
* End of ZUNBDB
*
END
|
