forked from Reference-LAPACK/lapack
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathdlaqr3.f
690 lines (690 loc) · 20.8 KB
/
dlaqr3.f
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
*> \brief \b DLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR3 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr3.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr3.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr3.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
* IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
* LDT, NV, WV, LDWV, WORK, LWORK )
*
* .. Scalar Arguments ..
* INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
* $ LDZ, LWORK, N, ND, NH, NS, NV, NW
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
* $ V( LDV, * ), WORK( * ), WV( LDWV, * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Aggressive early deflation:
*>
*> DLAQR3 accepts as input an upper Hessenberg matrix
*> H and performs an orthogonal similarity transformation
*> designed to detect and deflate fully converged eigenvalues from
*> a trailing principal submatrix. On output H has been over-
*> written by a new Hessenberg matrix that is a perturbation of
*> an orthogonal similarity transformation of H. It is to be
*> hoped that the final version of H has many zero subdiagonal
*> entries.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTT
*> \verbatim
*> WANTT is LOGICAL
*> If .TRUE., then the Hessenberg matrix H is fully updated
*> so that the quasi-triangular Schur factor may be
*> computed (in cooperation with the calling subroutine).
*> If .FALSE., then only enough of H is updated to preserve
*> the eigenvalues.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> If .TRUE., then the orthogonal matrix Z is updated so
*> so that the orthogonal Schur factor may be computed
*> (in cooperation with the calling subroutine).
*> If .FALSE., then Z is not referenced.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H and (if WANTZ is .TRUE.) the
*> order of the orthogonal matrix Z.
*> \endverbatim
*>
*> \param[in] KTOP
*> \verbatim
*> KTOP is INTEGER
*> It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
*> KBOT and KTOP together determine an isolated block
*> along the diagonal of the Hessenberg matrix.
*> \endverbatim
*>
*> \param[in] KBOT
*> \verbatim
*> KBOT is INTEGER
*> It is assumed without a check that either
*> KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
*> determine an isolated block along the diagonal of the
*> Hessenberg matrix.
*> \endverbatim
*>
*> \param[in] NW
*> \verbatim
*> NW is INTEGER
*> Deflation window size. 1 <= NW <= (KBOT-KTOP+1).
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> On input the initial N-by-N section of H stores the
*> Hessenberg matrix undergoing aggressive early deflation.
*> On output H has been transformed by an orthogonal
*> similarity transformation, perturbed, and the returned
*> to Hessenberg form that (it is to be hoped) has some
*> zero subdiagonal entries.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> Leading dimension of H just as declared in the calling
*> subroutine. N <= LDH
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*> ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,N)
*> IF WANTZ is .TRUE., then on output, the orthogonal
*> similarity transformation mentioned above has been
*> accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
*> If WANTZ is .FALSE., then Z is unreferenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of Z just as declared in the
*> calling subroutine. 1 <= LDZ.
*> \endverbatim
*>
*> \param[out] NS
*> \verbatim
*> NS is INTEGER
*> The number of unconverged (ie approximate) eigenvalues
*> returned in SR and SI that may be used as shifts by the
*> calling subroutine.
*> \endverbatim
*>
*> \param[out] ND
*> \verbatim
*> ND is INTEGER
*> The number of converged eigenvalues uncovered by this
*> subroutine.
*> \endverbatim
*>
*> \param[out] SR
*> \verbatim
*> SR is DOUBLE PRECISION array, dimension (KBOT)
*> \endverbatim
*>
*> \param[out] SI
*> \verbatim
*> SI is DOUBLE PRECISION array, dimension (KBOT)
*> On output, the real and imaginary parts of approximate
*> eigenvalues that may be used for shifts are stored in
*> SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
*> SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
*> The real and imaginary parts of converged eigenvalues
*> are stored in SR(KBOT-ND+1) through SR(KBOT) and
*> SI(KBOT-ND+1) through SI(KBOT), respectively.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,NW)
*> An NW-by-NW work array.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of V just as declared in the
*> calling subroutine. NW <= LDV
*> \endverbatim
*>
*> \param[in] NH
*> \verbatim
*> NH is INTEGER
*> The number of columns of T. NH >= NW.
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,NW)
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of T just as declared in the
*> calling subroutine. NW <= LDT
*> \endverbatim
*>
*> \param[in] NV
*> \verbatim
*> NV is INTEGER
*> The number of rows of work array WV available for
*> workspace. NV >= NW.
*> \endverbatim
*>
*> \param[out] WV
*> \verbatim
*> WV is DOUBLE PRECISION array, dimension (LDWV,NW)
*> \endverbatim
*>
*> \param[in] LDWV
*> \verbatim
*> LDWV is INTEGER
*> The leading dimension of W just as declared in the
*> calling subroutine. NW <= LDV
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (LWORK)
*> On exit, WORK(1) is set to an estimate of the optimal value
*> of LWORK for the given values of N, NW, KTOP and KBOT.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the work array WORK. LWORK = 2*NW
*> suffices, but greater efficiency may result from larger
*> values of LWORK.
*>
*> If LWORK = -1, then a workspace query is assumed; DLAQR3
*> only estimates the optimal workspace size for the given
*> values of N, NW, KTOP and KBOT. The estimate is returned
*> in WORK(1). No error message related to LWORK is issued
*> by XERBLA. Neither H nor Z are accessed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*>
* =====================================================================
SUBROUTINE DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
$ IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T,
$ LDT, NV, WV, LDWV, WORK, LWORK )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
$ LDZ, LWORK, N, ND, NH, NS, NV, NW
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), T( LDT, * ),
$ V( LDV, * ), WORK( * ), WV( LDWV, * ),
$ Z( LDZ, * )
* ..
*
* ================================================================
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AA, BB, BETA, CC, CS, DD, EVI, EVK, FOO, S,
$ SAFMAX, SAFMIN, SMLNUM, SN, TAU, ULP
INTEGER I, IFST, ILST, INFO, INFQR, J, JW, K, KCOL,
$ KEND, KLN, KROW, KWTOP, LTOP, LWK1, LWK2, LWK3,
$ LWKOPT, NMIN
LOGICAL BULGE, SORTED
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
INTEGER ILAENV
EXTERNAL DLAMCH, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEHRD, DGEMM, DLACPY, DLAHQR, DLANV2,
$ DLAQR4, DLARF, DLARFG, DLASET, DORMHR, DTREXC
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* ==== Estimate optimal workspace. ====
*
JW = MIN( NW, KBOT-KTOP+1 )
IF( JW.LE.2 ) THEN
LWKOPT = 1
ELSE
*
* ==== Workspace query call to DGEHRD ====
*
CALL DGEHRD( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
LWK1 = INT( WORK( 1 ) )
*
* ==== Workspace query call to DORMHR ====
*
CALL DORMHR( 'R', 'N', JW, JW, 1, JW-1, T, LDT, WORK, V, LDV,
$ WORK, -1, INFO )
LWK2 = INT( WORK( 1 ) )
*
* ==== Workspace query call to DLAQR4 ====
*
CALL DLAQR4( .true., .true., JW, 1, JW, T, LDT, SR, SI, 1, JW,
$ V, LDV, WORK, -1, INFQR )
LWK3 = INT( WORK( 1 ) )
*
* ==== Optimal workspace ====
*
LWKOPT = MAX( JW+MAX( LWK1, LWK2 ), LWK3 )
END IF
*
* ==== Quick return in case of workspace query. ====
*
IF( LWORK.EQ.-1 ) THEN
WORK( 1 ) = DBLE( LWKOPT )
RETURN
END IF
*
* ==== Nothing to do ...
* ... for an empty active block ... ====
NS = 0
ND = 0
WORK( 1 ) = ONE
IF( KTOP.GT.KBOT )
$ RETURN
* ... nor for an empty deflation window. ====
IF( NW.LT.1 )
$ RETURN
*
* ==== Machine constants ====
*
SAFMIN = DLAMCH( 'SAFE MINIMUM' )
SAFMAX = ONE / SAFMIN
ULP = DLAMCH( 'PRECISION' )
SMLNUM = SAFMIN*( DBLE( N ) / ULP )
*
* ==== Setup deflation window ====
*
JW = MIN( NW, KBOT-KTOP+1 )
KWTOP = KBOT - JW + 1
IF( KWTOP.EQ.KTOP ) THEN
S = ZERO
ELSE
S = H( KWTOP, KWTOP-1 )
END IF
*
IF( KBOT.EQ.KWTOP ) THEN
*
* ==== 1-by-1 deflation window: not much to do ====
*
SR( KWTOP ) = H( KWTOP, KWTOP )
SI( KWTOP ) = ZERO
NS = 1
ND = 0
IF( ABS( S ).LE.MAX( SMLNUM, ULP*ABS( H( KWTOP, KWTOP ) ) ) )
$ THEN
NS = 0
ND = 1
IF( KWTOP.GT.KTOP )
$ H( KWTOP, KWTOP-1 ) = ZERO
END IF
WORK( 1 ) = ONE
RETURN
END IF
*
* ==== Convert to spike-triangular form. (In case of a
* . rare QR failure, this routine continues to do
* . aggressive early deflation using that part of
* . the deflation window that converged using INFQR
* . here and there to keep track.) ====
*
CALL DLACPY( 'U', JW, JW, H( KWTOP, KWTOP ), LDH, T, LDT )
CALL DCOPY( JW-1, H( KWTOP+1, KWTOP ), LDH+1, T( 2, 1 ), LDT+1 )
*
CALL DLASET( 'A', JW, JW, ZERO, ONE, V, LDV )
NMIN = ILAENV( 12, 'DLAQR3', 'SV', JW, 1, JW, LWORK )
IF( JW.GT.NMIN ) THEN
CALL DLAQR4( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
$ SI( KWTOP ), 1, JW, V, LDV, WORK, LWORK, INFQR )
ELSE
CALL DLAHQR( .true., .true., JW, 1, JW, T, LDT, SR( KWTOP ),
$ SI( KWTOP ), 1, JW, V, LDV, INFQR )
END IF
*
* ==== DTREXC needs a clean margin near the diagonal ====
*
DO 10 J = 1, JW - 3
T( J+2, J ) = ZERO
T( J+3, J ) = ZERO
10 CONTINUE
IF( JW.GT.2 )
$ T( JW, JW-2 ) = ZERO
*
* ==== Deflation detection loop ====
*
NS = JW
ILST = INFQR + 1
20 CONTINUE
IF( ILST.LE.NS ) THEN
IF( NS.EQ.1 ) THEN
BULGE = .FALSE.
ELSE
BULGE = T( NS, NS-1 ).NE.ZERO
END IF
*
* ==== Small spike tip test for deflation ====
*
IF( .NOT. BULGE ) THEN
*
* ==== Real eigenvalue ====
*
FOO = ABS( T( NS, NS ) )
IF( FOO.EQ.ZERO )
$ FOO = ABS( S )
IF( ABS( S*V( 1, NS ) ).LE.MAX( SMLNUM, ULP*FOO ) ) THEN
*
* ==== Deflatable ====
*
NS = NS - 1
ELSE
*
* ==== Undeflatable. Move it up out of the way.
* . (DTREXC can not fail in this case.) ====
*
IFST = NS
CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
$ INFO )
ILST = ILST + 1
END IF
ELSE
*
* ==== Complex conjugate pair ====
*
FOO = ABS( T( NS, NS ) ) + SQRT( ABS( T( NS, NS-1 ) ) )*
$ SQRT( ABS( T( NS-1, NS ) ) )
IF( FOO.EQ.ZERO )
$ FOO = ABS( S )
IF( MAX( ABS( S*V( 1, NS ) ), ABS( S*V( 1, NS-1 ) ) ).LE.
$ MAX( SMLNUM, ULP*FOO ) ) THEN
*
* ==== Deflatable ====
*
NS = NS - 2
ELSE
*
* ==== Undeflatable. Move them up out of the way.
* . Fortunately, DTREXC does the right thing with
* . ILST in case of a rare exchange failure. ====
*
IFST = NS
CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
$ INFO )
ILST = ILST + 2
END IF
END IF
*
* ==== End deflation detection loop ====
*
GO TO 20
END IF
*
* ==== Return to Hessenberg form ====
*
IF( NS.EQ.0 )
$ S = ZERO
*
IF( NS.LT.JW ) THEN
*
* ==== sorting diagonal blocks of T improves accuracy for
* . graded matrices. Bubble sort deals well with
* . exchange failures. ====
*
SORTED = .false.
I = NS + 1
30 CONTINUE
IF( SORTED )
$ GO TO 50
SORTED = .true.
*
KEND = I - 1
I = INFQR + 1
IF( I.EQ.NS ) THEN
K = I + 1
ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
K = I + 1
ELSE
K = I + 2
END IF
40 CONTINUE
IF( K.LE.KEND ) THEN
IF( K.EQ.I+1 ) THEN
EVI = ABS( T( I, I ) )
ELSE
EVI = ABS( T( I, I ) ) + SQRT( ABS( T( I+1, I ) ) )*
$ SQRT( ABS( T( I, I+1 ) ) )
END IF
*
IF( K.EQ.KEND ) THEN
EVK = ABS( T( K, K ) )
ELSE IF( T( K+1, K ).EQ.ZERO ) THEN
EVK = ABS( T( K, K ) )
ELSE
EVK = ABS( T( K, K ) ) + SQRT( ABS( T( K+1, K ) ) )*
$ SQRT( ABS( T( K, K+1 ) ) )
END IF
*
IF( EVI.GE.EVK ) THEN
I = K
ELSE
SORTED = .false.
IFST = I
ILST = K
CALL DTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, WORK,
$ INFO )
IF( INFO.EQ.0 ) THEN
I = ILST
ELSE
I = K
END IF
END IF
IF( I.EQ.KEND ) THEN
K = I + 1
ELSE IF( T( I+1, I ).EQ.ZERO ) THEN
K = I + 1
ELSE
K = I + 2
END IF
GO TO 40
END IF
GO TO 30
50 CONTINUE
END IF
*
* ==== Restore shift/eigenvalue array from T ====
*
I = JW
60 CONTINUE
IF( I.GE.INFQR+1 ) THEN
IF( I.EQ.INFQR+1 ) THEN
SR( KWTOP+I-1 ) = T( I, I )
SI( KWTOP+I-1 ) = ZERO
I = I - 1
ELSE IF( T( I, I-1 ).EQ.ZERO ) THEN
SR( KWTOP+I-1 ) = T( I, I )
SI( KWTOP+I-1 ) = ZERO
I = I - 1
ELSE
AA = T( I-1, I-1 )
CC = T( I, I-1 )
BB = T( I-1, I )
DD = T( I, I )
CALL DLANV2( AA, BB, CC, DD, SR( KWTOP+I-2 ),
$ SI( KWTOP+I-2 ), SR( KWTOP+I-1 ),
$ SI( KWTOP+I-1 ), CS, SN )
I = I - 2
END IF
GO TO 60
END IF
*
IF( NS.LT.JW .OR. S.EQ.ZERO ) THEN
IF( NS.GT.1 .AND. S.NE.ZERO ) THEN
*
* ==== Reflect spike back into lower triangle ====
*
CALL DCOPY( NS, V, LDV, WORK, 1 )
BETA = WORK( 1 )
CALL DLARFG( NS, BETA, WORK( 2 ), 1, TAU )
WORK( 1 ) = ONE
*
CALL DLASET( 'L', JW-2, JW-2, ZERO, ZERO, T( 3, 1 ), LDT )
*
CALL DLARF( 'L', NS, JW, WORK, 1, TAU, T, LDT,
$ WORK( JW+1 ) )
CALL DLARF( 'R', NS, NS, WORK, 1, TAU, T, LDT,
$ WORK( JW+1 ) )
CALL DLARF( 'R', JW, NS, WORK, 1, TAU, V, LDV,
$ WORK( JW+1 ) )
*
CALL DGEHRD( JW, 1, NS, T, LDT, WORK, WORK( JW+1 ),
$ LWORK-JW, INFO )
END IF
*
* ==== Copy updated reduced window into place ====
*
IF( KWTOP.GT.1 )
$ H( KWTOP, KWTOP-1 ) = S*V( 1, 1 )
CALL DLACPY( 'U', JW, JW, T, LDT, H( KWTOP, KWTOP ), LDH )
CALL DCOPY( JW-1, T( 2, 1 ), LDT+1, H( KWTOP+1, KWTOP ),
$ LDH+1 )
*
* ==== Accumulate orthogonal matrix in order update
* . H and Z, if requested. ====
*
IF( NS.GT.1 .AND. S.NE.ZERO )
$ CALL DORMHR( 'R', 'N', JW, NS, 1, NS, T, LDT, WORK, V, LDV,
$ WORK( JW+1 ), LWORK-JW, INFO )
*
* ==== Update vertical slab in H ====
*
IF( WANTT ) THEN
LTOP = 1
ELSE
LTOP = KTOP
END IF
DO 70 KROW = LTOP, KWTOP - 1, NV
KLN = MIN( NV, KWTOP-KROW )
CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, H( KROW, KWTOP ),
$ LDH, V, LDV, ZERO, WV, LDWV )
CALL DLACPY( 'A', KLN, JW, WV, LDWV, H( KROW, KWTOP ), LDH )
70 CONTINUE
*
* ==== Update horizontal slab in H ====
*
IF( WANTT ) THEN
DO 80 KCOL = KBOT + 1, N, NH
KLN = MIN( NH, N-KCOL+1 )
CALL DGEMM( 'C', 'N', JW, KLN, JW, ONE, V, LDV,
$ H( KWTOP, KCOL ), LDH, ZERO, T, LDT )
CALL DLACPY( 'A', JW, KLN, T, LDT, H( KWTOP, KCOL ),
$ LDH )
80 CONTINUE
END IF
*
* ==== Update vertical slab in Z ====
*
IF( WANTZ ) THEN
DO 90 KROW = ILOZ, IHIZ, NV
KLN = MIN( NV, IHIZ-KROW+1 )
CALL DGEMM( 'N', 'N', KLN, JW, JW, ONE, Z( KROW, KWTOP ),
$ LDZ, V, LDV, ZERO, WV, LDWV )
CALL DLACPY( 'A', KLN, JW, WV, LDWV, Z( KROW, KWTOP ),
$ LDZ )
90 CONTINUE
END IF
END IF
*
* ==== Return the number of deflations ... ====
*
ND = JW - NS
*
* ==== ... and the number of shifts. (Subtracting
* . INFQR from the spike length takes care
* . of the case of a rare QR failure while
* . calculating eigenvalues of the deflation
* . window.) ====
*
NS = NS - INFQR
*
* ==== Return optimal workspace. ====
*
WORK( 1 ) = DBLE( LWKOPT )
*
* ==== End of DLAQR3 ====
*
END