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
|
SUBROUTINE PDHSEQR( JOB, COMPZ, N, ILO, IHI, H, DESCH, WR, WI, Z,
$ DESCZ, WORK, LWORK, IWORK, LIWORK, INFO )
*
* Contribution from the Department of Computing Science and HPC2N,
* Umea University, Sweden
*
* -- ScaLAPACK driver routine (version 2.0.1) --
* University of Tennessee, Knoxville, Oak Ridge National Laboratory,
* Univ. of Colorado Denver and University of California, Berkeley.
* January, 2012
*
IMPLICIT NONE
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LWORK, LIWORK, N
CHARACTER COMPZ, JOB
* ..
* .. Array Arguments ..
INTEGER DESCH( * ) , DESCZ( * ), IWORK( * )
DOUBLE PRECISION H( * ), WI( N ), WORK( * ), WR( N ), Z( * )
* ..
* Purpose
* =======
*
* PDHSEQR computes the eigenvalues of an upper Hessenberg matrix H
* and, optionally, the matrices T and Z from the Schur decomposition
* H = Z*T*Z**T, where T is an upper quasi-triangular matrix (the
* Schur form), and Z is the orthogonal matrix of Schur vectors.
*
* Optionally Z may be postmultiplied into an input orthogonal
* matrix Q so that this routine can give the Schur factorization
* of a matrix A which has been reduced to the Hessenberg form H
* by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
*
* Notes
* =====
*
* Each global data object is described by an associated description
* vector. This vector stores the information required to establish
* the mapping between an object element and its corresponding process
* and memory location.
*
* Let A be a generic term for any 2D block cyclicly distributed array.
* Such a global array has an associated description vector DESCA.
* In the following comments, the character _ should be read as
* "of the global array".
*
* NOTATION STORED IN EXPLANATION
* --------------- -------------- --------------------------------------
* DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case,
* DTYPE_A = 1.
* CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
* the BLACS process grid A is distribu-
* ted over. The context itself is glo-
* bal, but the handle (the integer
* value) may vary.
* M_A (global) DESCA( M_ ) The number of rows in the global
* array A.
* N_A (global) DESCA( N_ ) The number of columns in the global
* array A.
* MB_A (global) DESCA( MB_ ) The blocking factor used to distribute
* the rows of the array.
* NB_A (global) DESCA( NB_ ) The blocking factor used to distribute
* the columns of the array.
* RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
* row of the array A is distributed.
* CSRC_A (global) DESCA( CSRC_ ) The process column over which the
* first column of the array A is
* distributed.
* LLD_A (local) DESCA( LLD_ ) The leading dimension of the local
* array. LLD_A >= MAX(1,LOCr(M_A)).
*
* Let K be the number of rows or columns of a distributed matrix,
* and assume that its process grid has dimension p x q.
* LOCr( K ) denotes the number of elements of K that a process
* would receive if K were distributed over the p processes of its
* process column.
* Similarly, LOCc( K ) denotes the number of elements of K that a
* process would receive if K were distributed over the q processes of
* its process row.
* The values of LOCr() and LOCc() may be determined via a call to the
* ScaLAPACK tool function, NUMROC:
* LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
* LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).
* An upper bound for these quantities may be computed by:
* LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
* LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A
*
* Arguments
* =========
*
* JOB (global input) CHARACTER*1
* = 'E': compute eigenvalues only;
* = 'S': compute eigenvalues and the Schur form T.
*
* COMPZ (global input) CHARACTER*1
* = 'N': no Schur vectors are computed;
* = 'I': Z is initialized to the unit matrix and the matrix Z
* of Schur vectors of H is returned;
* = 'V': Z must contain an orthogonal matrix Q on entry, and
* the product Q*Z is returned.
*
* N (global input) INTEGER
* The order of the Hessenberg matrix H (and Z if WANTZ).
* N >= 0.
*
* ILO (global input) INTEGER
* IHI (global input) INTEGER
* It is assumed that H is already upper triangular in rows
* and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
* set by a previous call to PDGEBAL, and then passed to PDGEHRD
* when the matrix output by PDGEBAL is reduced to Hessenberg
* form. Otherwise ILO and IHI should be set to 1 and N
* respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
* If N = 0, then ILO = 1 and IHI = 0.
*
* H (global input/output) DOUBLE PRECISION array, dimension
* (DESCH(LLD_),*)
* On entry, the upper Hessenberg matrix H.
* On exit, if JOB = 'S', H is upper quasi-triangular in
* rows and columns ILO:IHI, with 1-by-1 and 2-by-2 blocks on
* the main diagonal. The 2-by-2 diagonal blocks (corresponding
* to complex conjugate pairs of eigenvalues) are returned in
* standard form, with H(i,i) = H(i+1,i+1) and
* H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and JOB = 'E', the
* contents of H are unspecified on exit.
*
* DESCH (global and local input) INTEGER array of dimension DLEN_.
* The array descriptor for the distributed matrix H.
*
* WR (global output) DOUBLE PRECISION array, dimension (N)
* WI (global output) DOUBLE PRECISION array, dimension (N)
* The real and imaginary parts, respectively, of the computed
* eigenvalues ILO to IHI are stored in the corresponding
* elements of WR and WI. If two eigenvalues are computed as a
* complex conjugate pair, they are stored in consecutive
* elements of WR and WI, say the i-th and (i+1)th, with
* WI(i) > 0 and WI(i+1) < 0. If JOB = 'S', the
* eigenvalues are stored in the same order as on the diagonal
* of the Schur form returned in H.
*
* Z (global input/output) DOUBLE PRECISION array.
* If COMPZ = 'V', on entry Z must contain the current
* matrix Z of accumulated transformations from, e.g., PDGEHRD,
* and on exit Z has been updated; transformations are applied
* only to the submatrix Z(ILO:IHI,ILO:IHI).
* If COMPZ = 'N', Z is not referenced.
* If COMPZ = 'I', on entry Z need not be set and on exit,
* if INFO = 0, Z contains the orthogonal matrix Z of the Schur
* vectors of H.
*
* DESCZ (global and local input) INTEGER array of dimension DLEN_.
* The array descriptor for the distributed matrix Z.
*
* WORK (local workspace) DOUBLE PRECISION array, dimension(LWORK)
*
* LWORK (local input) INTEGER
* The length of the workspace array WORK.
*
* IWORK (local workspace) INTEGER array, dimension (LIWORK)
*
* LIWORK (local input) INTEGER
* The length of the workspace array IWORK.
*
* INFO (output) INTEGER
* = 0: successful exit
* .LT. 0: if INFO = -i, the i-th argument had an illegal
* value (see also below for -7777 and -8888).
* .GT. 0: if INFO = i, PDHSEQR failed to compute all of
* the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
* and WI contain those eigenvalues which have been
* successfully computed. (Failures are rare.)
*
* If INFO .GT. 0 and JOB = 'E', then on exit, the
* remaining unconverged eigenvalues are the eigen-
* values of the upper Hessenberg matrix rows and
* columns ILO through INFO of the final, output
* value of H.
*
* If INFO .GT. 0 and JOB = 'S', then on exit
*
* (*) (initial value of H)*U = U*(final value of H)
*
* where U is an orthogonal matrix. The final
* value of H is upper Hessenberg and quasi-triangular
* in rows and columns INFO+1 through IHI.
*
* If INFO .GT. 0 and COMPZ = 'V', then on exit
*
* (final value of Z) = (initial value of Z)*U
*
* where U is the orthogonal matrix in (*) (regard-
* less of the value of JOB.)
*
* If INFO .GT. 0 and COMPZ = 'I', then on exit
* (final value of Z) = U
* where U is the orthogonal matrix in (*) (regard-
* less of the value of JOB.)
*
* If INFO .GT. 0 and COMPZ = 'N', then Z is not
* accessed.
*
* = -7777: PDLAQR0 failed to converge and PDLAQR1 was called
* instead. This could happen. Mostly due to a bug.
* Please, send a bug report to the authors.
* = -8888: PDLAQR1 failed to converge and PDLAQR0 was called
* instead. This should not happen.
*
* ================================================================
* Based on contributions by
* Robert Granat, Department of Computing Science and HPC2N,
* Umea University, Sweden.
* ================================================================
*
* Restrictions: The block size in H and Z must be square and larger
* than or equal to six (6) due to restrictions in PDLAQR1, PDLAQR5
* and DLAQR6. Moreover, H and Z need to be distributed identically
* with the same context.
*
* ================================================================
* References:
* K. Braman, R. Byers, and R. Mathias,
* The Multi-Shift QR Algorithm Part I: Maintaining Well Focused
* Shifts, and Level 3 Performance.
* SIAM J. Matrix Anal. Appl., 23(4):929--947, 2002.
*
* K. Braman, R. Byers, and R. Mathias,
* The Multi-Shift QR Algorithm Part II: Aggressive Early
* Deflation.
* SIAM J. Matrix Anal. Appl., 23(4):948--973, 2002.
*
* R. Granat, B. Kagstrom, and D. Kressner,
* A Novel Parallel QR Algorithm for Hybrid Distributed Momory HPC
* Systems.
* SIAM J. Sci. Comput., 32(4):2345--2378, 2010.
*
* ================================================================
* .. Parameters ..
INTEGER BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_,
$ LLD_, MB_, M_, NB_, N_, RSRC_
LOGICAL CRSOVER
PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1,
$ CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6,
$ RSRC_ = 7, CSRC_ = 8, LLD_ = 9,
$ CRSOVER = .TRUE. )
INTEGER NTINY
PARAMETER ( NTINY = 11 )
INTEGER NL
PARAMETER ( NL = 49 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
INTEGER I, KBOT, NMIN, LLDH, LLDZ, ICTXT, NPROW, NPCOL,
$ MYROW, MYCOL, HROWS, HCOLS, IPW, NH, NB,
$ II, JJ, HRSRC, HCSRC, NPROCS, ILOC1, JLOC1,
$ HRSRC1, HCSRC1, K, ILOC2, JLOC2, ILOC3, JLOC3,
$ ILOC4, JLOC4, HRSRC2, HCSRC2, HRSRC3, HCSRC3,
$ HRSRC4, HCSRC4, LIWKOPT
LOGICAL INITZ, LQUERY, WANTT, WANTZ, PAIR, BORDER
DOUBLE PRECISION TMP1, TMP2, TMP3, TMP4, DUM1, DUM2, DUM3,
$ DUM4, ELEM1, ELEM2, ELEM3, ELEM4,
$ CS, SN, ELEM5, TMP, LWKOPT
* ..
* .. Local Arrays ..
INTEGER DESCH2( DLEN_ )
* ..
* .. External Functions ..
INTEGER PILAENVX, NUMROC, ICEIL
LOGICAL LSAME
EXTERNAL PILAENVX, LSAME, NUMROC, ICEIL
* ..
* .. External Subroutines ..
EXTERNAL PDLACPY, PDLAQR1, PDLAQR0, PDLASET, PXERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, MIN
* ..
* .. Executable Statements ..
*
* Decode and check the input parameters.
*
INFO = 0
ICTXT = DESCH( CTXT_ )
CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL )
NPROCS = NPROW*NPCOL
IF( NPROW.EQ.-1 ) INFO = -(600+CTXT_)
IF( INFO.EQ.0 ) THEN
WANTT = LSAME( JOB, 'S' )
INITZ = LSAME( COMPZ, 'I' )
WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
LLDH = DESCH( LLD_ )
LLDZ = DESCZ( LLD_ )
NB = DESCH( MB_ )
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -5
ELSEIF( DESCZ( CTXT_ ).NE.DESCH( CTXT_ ) ) THEN
INFO = -( 1000+CTXT_ )
ELSEIF( DESCH( MB_ ).NE.DESCH( NB_ ) ) THEN
INFO = -( 700+NB_ )
ELSEIF( DESCZ( MB_ ).NE.DESCZ( NB_ ) ) THEN
INFO = -( 1000+NB_ )
ELSEIF( DESCH( MB_ ).NE.DESCZ( MB_ ) ) THEN
INFO = -( 1000+MB_ )
ELSEIF( DESCH( MB_ ).LT.6 ) THEN
INFO = -( 700+NB_ )
ELSEIF( DESCZ( MB_ ).LT.6 ) THEN
INFO = -( 1000+MB_ )
ELSE
CALL CHK1MAT( N, 3, N, 3, 1, 1, DESCH, 7, INFO )
IF( INFO.EQ.0 )
$ CALL CHK1MAT( N, 3, N, 3, 1, 1, DESCZ, 11, INFO )
IF( INFO.EQ.0 )
$ CALL PCHK2MAT( N, 3, N, 3, 1, 1, DESCH, 7, N, 3, N, 3,
$ 1, 1, DESCZ, 11, 0, IWORK, IWORK, INFO )
END IF
END IF
*
* Compute required workspace.
*
CALL PDLAQR1( WANTT, WANTZ, N, ILO, IHI, H, DESCH, WR, WI,
$ ILO, IHI, Z, DESCZ, WORK, -1, IWORK, -1, INFO )
LWKOPT = WORK(1)
LIWKOPT = IWORK(1)
CALL PDLAQR0( WANTT, WANTZ, N, ILO, IHI, H, DESCH, WR, WI,
$ ILO, IHI, Z, DESCZ, WORK, -1, IWORK, -1, INFO, 0 )
IF( N.LT.NL ) THEN
HROWS = NUMROC( NL, NB, MYROW, DESCH(RSRC_), NPROW )
HCOLS = NUMROC( NL, NB, MYCOL, DESCH(CSRC_), NPCOL )
WORK(1) = WORK(1) + DBLE(2*HROWS*HCOLS)
END IF
LWKOPT = MAX( LWKOPT, WORK(1) )
LIWKOPT = MAX( LIWKOPT, IWORK(1) )
WORK(1) = LWKOPT
IWORK(1) = LIWKOPT
*
IF( .NOT.LQUERY .AND. LWORK.LT.INT(LWKOPT) ) THEN
INFO = -13
ELSEIF( .NOT.LQUERY .AND. LIWORK.LT.LIWKOPT ) THEN
INFO = -15
END IF
*
IF( INFO.NE.0 ) THEN
*
* Quick return in case of invalid argument.
*
CALL PXERBLA( ICTXT, 'PDHSEQR', -INFO )
RETURN
*
ELSE IF( N.EQ.0 ) THEN
*
* Quick return in case N = 0; nothing to do.
*
RETURN
*
ELSE IF( LQUERY ) THEN
*
* Quick return in case of a workspace query.
*
RETURN
*
ELSE
*
* Copy eigenvalues isolated by PDGEBAL.
*
DO 10 I = 1, ILO - 1
CALL INFOG2L( I, I, DESCH, NPROW, NPCOL, MYROW, MYCOL, II,
$ JJ, HRSRC, HCSRC )
IF( MYROW.EQ.HRSRC .AND. MYCOL.EQ.HCSRC ) THEN
WR( I ) = H( (JJ-1)*LLDH + II )
ELSE
WR( I ) = ZERO
END IF
WI( I ) = ZERO
10 CONTINUE
IF( ILO.GT.1 )
$ CALL DGSUM2D( ICTXT, 'All', '1-Tree', ILO-1, 1, WR, N, -1,
$ -1 )
DO 20 I = IHI + 1, N
CALL INFOG2L( I, I, DESCH, NPROW, NPCOL, MYROW, MYCOL, II,
$ JJ, HRSRC, HCSRC )
IF( MYROW.EQ.HRSRC .AND. MYCOL.EQ.HCSRC ) THEN
WR( I ) = H( (JJ-1)*LLDH + II )
ELSE
WR( I ) = ZERO
END IF
WI( I ) = ZERO
20 CONTINUE
IF( IHI.LT.N )
$ CALL DGSUM2D( ICTXT, 'All', '1-Tree', N-IHI, 1, WR(IHI+1),
$ N, -1, -1 )
*
* Initialize Z, if requested.
*
IF( INITZ )
$ CALL PDLASET( 'A', N, N, ZERO, ONE, Z, 1, 1, DESCZ )
*
* Quick return if possible.
*
NPROCS = NPROW*NPCOL
IF( ILO.EQ.IHI ) THEN
CALL INFOG2L( ILO, ILO, DESCH, NPROW, NPCOL, MYROW,
$ MYCOL, II, JJ, HRSRC, HCSRC )
IF( MYROW.EQ.HRSRC .AND. MYCOL.EQ.HCSRC ) THEN
WR( ILO ) = H( (JJ-1)*LLDH + II )
IF( NPROCS.GT.1 )
$ CALL DGEBS2D( ICTXT, 'All', '1-Tree', 1, 1, WR(ILO),
$ 1 )
ELSE
CALL DGEBR2D( ICTXT, 'All', '1-Tree', 1, 1, WR(ILO),
$ 1, HRSRC, HCSRC )
END IF
WI( ILO ) = ZERO
RETURN
END IF
*
* PDLAQR1/PDLAQR0 crossover point.
*
NH = IHI-ILO+1
NMIN = PILAENVX( ICTXT, 12, 'PDHSEQR',
$ JOB( : 1 ) // COMPZ( : 1 ), N, ILO, IHI, LWORK )
NMIN = MAX( NTINY, NMIN )
*
* PDLAQR0 for big matrices; PDLAQR1 for small ones.
*
IF( (.NOT. CRSOVER .AND. NH.GT.NTINY) .OR. NH.GT.NMIN .OR.
$ DESCH(RSRC_).NE.0 .OR. DESCH(CSRC_).NE.0 ) THEN
CALL PDLAQR0( WANTT, WANTZ, N, ILO, IHI, H, DESCH, WR, WI,
$ ILO, IHI, Z, DESCZ, WORK, LWORK, IWORK, LIWORK, INFO,
$ 0 )
IF( INFO.GT.0 .AND. ( DESCH(RSRC_).NE.0 .OR.
$ DESCH(CSRC_).NE.0 ) ) THEN
*
* A rare PDLAQR0 failure! PDLAQR1 sometimes succeeds
* when PDLAQR0 fails.
*
KBOT = INFO
CALL PDLAQR1( WANTT, WANTZ, N, ILO, IHI, H, DESCH, WR,
$ WI, ILO, IHI, Z, DESCZ, WORK, LWORK, IWORK,
$ LIWORK, INFO )
INFO = -7777
END IF
ELSE
*
* Small matrix.
*
CALL PDLAQR1( WANTT, WANTZ, N, ILO, IHI, H, DESCH, WR, WI,
$ ILO, IHI, Z, DESCZ, WORK, LWORK, IWORK, LIWORK, INFO )
*
IF( INFO.GT.0 ) THEN
*
* A rare PDLAQR1 failure! PDLAQR0 sometimes succeeds
* when PDLAQR1 fails.
*
KBOT = INFO
*
IF( N.GE.NL ) THEN
*
* Larger matrices have enough subdiagonal scratch
* space to call PDLAQR0 directly.
*
CALL PDLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, DESCH,
$ WR, WI, ILO, IHI, Z, DESCZ, WORK, LWORK,
$ IWORK, LIWORK, INFO, 0 )
ELSE
*
* Tiny matrices don't have enough subdiagonal
* scratch space to benefit from PDLAQR0. Hence,
* tiny matrices must be copied into a larger
* array before calling PDLAQR0.
*
HROWS = NUMROC( NL, NB, MYROW, DESCH(RSRC_), NPROW )
HCOLS = NUMROC( NL, NB, MYCOL, DESCH(CSRC_), NPCOL )
CALL DESCINIT( DESCH2, NL, NL, NB, NB, DESCH(RSRC_),
$ DESCH(CSRC_), ICTXT, MAX(1, HROWS), INFO )
CALL PDLACPY( 'All', N, N, H, 1, 1, DESCH, WORK, 1,
$ 1, DESCH2 )
CALL PDELSET( WORK, N+1, N, DESCH2, ZERO )
CALL PDLASET( 'All', NL, NL-N, ZERO, ZERO, WORK, 1,
$ N+1, DESCH2 )
IPW = 1 + DESCH2(LLD_)*HCOLS
CALL PDLAQR0( WANTT, WANTZ, NL, ILO, KBOT, WORK,
$ DESCH2, WR, WI, ILO, IHI, Z, DESCZ,
$ WORK(IPW), LWORK-IPW+1, IWORK,
$ LIWORK, INFO, 0 )
IF( WANTT .OR. INFO.NE.0 )
$ CALL PDLACPY( 'All', N, N, WORK, 1, 1, DESCH2,
$ H, 1, 1, DESCH )
END IF
INFO = -8888
END IF
END IF
*
* Clear out the trash, if necessary.
*
IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 )
$ CALL PDLASET( 'L', N-2, N-2, ZERO, ZERO, H, 3, 1, DESCH )
*
* Force any 2-by-2 blocks to be complex conjugate pairs of
* eigenvalues by removing false such blocks.
*
DO 30 I = ILO, IHI-1
CALL PDELGET( 'All', ' ', TMP3, H, I+1, I, DESCH )
IF( TMP3.NE.0.0D+00 ) THEN
CALL PDELGET( 'All', ' ', TMP1, H, I, I, DESCH )
CALL PDELGET( 'All', ' ', TMP2, H, I, I+1, DESCH )
CALL PDELGET( 'All', ' ', TMP4, H, I+1, I+1, DESCH )
CALL DLANV2( TMP1, TMP2, TMP3, TMP4, DUM1, DUM2, DUM3,
$ DUM4, CS, SN )
IF( TMP3.EQ.0.0D+00 ) THEN
IF( WANTT ) THEN
IF( I+2.LE.N )
$ CALL PDROT( N-I-1, H, I, I+2, DESCH,
$ DESCH(M_), H, I+1, I+2, DESCH, DESCH(M_),
$ CS, SN, WORK, LWORK, INFO )
CALL PDROT( I-1, H, 1, I, DESCH, 1, H, 1, I+1,
$ DESCH, 1, CS, SN, WORK, LWORK, INFO )
END IF
IF( WANTZ ) THEN
CALL PDROT( N, Z, 1, I, DESCZ, 1, Z, 1, I+1, DESCZ,
$ 1, CS, SN, WORK, LWORK, INFO )
END IF
CALL PDELSET( H, I, I, DESCH, TMP1 )
CALL PDELSET( H, I, I+1, DESCH, TMP2 )
CALL PDELSET( H, I+1, I, DESCH, TMP3 )
CALL PDELSET( H, I+1, I+1, DESCH, TMP4 )
END IF
END IF
30 CONTINUE
*
* Read out eigenvalues: first let all the processes compute the
* eigenvalue inside their diagonal blocks in parallel, except for
* the eigenvalue located next to a block border. After that,
* compute all eigenvalues located next to the block borders.
* Finally, do a global summation over WR and WI so that all
* processors receive the result.
*
DO 40 K = ILO, IHI
WR( K ) = ZERO
WI( K ) = ZERO
40 CONTINUE
NB = DESCH( MB_ )
*
* Loop 50: extract eigenvalues from the blocks which are not laid
* out across a border of the processor mesh, except for those 1x1
* blocks on the border.
*
PAIR = .FALSE.
DO 50 K = ILO, IHI
IF( .NOT. PAIR ) THEN
BORDER = MOD( K, NB ).EQ.0 .OR. ( K.NE.1 .AND.
$ MOD( K, NB ).EQ.1 )
IF( .NOT. BORDER ) THEN
CALL INFOG2L( K, K, DESCH, NPROW, NPCOL, MYROW,
$ MYCOL, ILOC1, JLOC1, HRSRC1, HCSRC1 )
IF( MYROW.EQ.HRSRC1 .AND. MYCOL.EQ.HCSRC1 ) THEN
ELEM1 = H((JLOC1-1)*LLDH+ILOC1)
IF( K.LT.N ) THEN
ELEM3 = H((JLOC1-1)*LLDH+ILOC1+1)
ELSE
ELEM3 = ZERO
END IF
IF( ELEM3.NE.ZERO ) THEN
ELEM2 = H((JLOC1)*LLDH+ILOC1)
ELEM4 = H((JLOC1)*LLDH+ILOC1+1)
CALL DLANV2( ELEM1, ELEM2, ELEM3, ELEM4,
$ WR( K ), WI( K ), WR( K+1 ), WI( K+1 ),
$ SN, CS )
PAIR = .TRUE.
ELSE
IF( K.GT.1 ) THEN
TMP = H((JLOC1-2)*LLDH+ILOC1)
IF( TMP.NE.ZERO ) THEN
ELEM1 = H((JLOC1-2)*LLDH+ILOC1-1)
ELEM2 = H((JLOC1-1)*LLDH+ILOC1-1)
ELEM3 = H((JLOC1-2)*LLDH+ILOC1)
ELEM4 = H((JLOC1-1)*LLDH+ILOC1)
CALL DLANV2( ELEM1, ELEM2, ELEM3,
$ ELEM4, WR( K-1 ), WI( K-1 ),
$ WR( K ), WI( K ), SN, CS )
ELSE
WR( K ) = ELEM1
END IF
ELSE
WR( K ) = ELEM1
END IF
END IF
END IF
END IF
ELSE
PAIR = .FALSE.
END IF
50 CONTINUE
*
* Loop 60: extract eigenvalues from the blocks which are laid
* out across a border of the processor mesh. The processors are
* numbered as below:
*
* 1 | 2
* --+--
* 3 | 4
*
DO 60 K = ICEIL(ILO,NB)*NB, IHI-1, NB
CALL INFOG2L( K, K, DESCH, NPROW, NPCOL, MYROW, MYCOL,
$ ILOC1, JLOC1, HRSRC1, HCSRC1 )
CALL INFOG2L( K, K+1, DESCH, NPROW, NPCOL, MYROW, MYCOL,
$ ILOC2, JLOC2, HRSRC2, HCSRC2 )
CALL INFOG2L( K+1, K, DESCH, NPROW, NPCOL, MYROW, MYCOL,
$ ILOC3, JLOC3, HRSRC3, HCSRC3 )
CALL INFOG2L( K+1, K+1, DESCH, NPROW, NPCOL, MYROW, MYCOL,
$ ILOC4, JLOC4, HRSRC4, HCSRC4 )
IF( MYROW.EQ.HRSRC2 .AND. MYCOL.EQ.HCSRC2 ) THEN
ELEM2 = H((JLOC2-1)*LLDH+ILOC2)
IF( HRSRC1.NE.HRSRC2 .OR. HCSRC1.NE.HCSRC2 )
$ CALL DGESD2D( ICTXT, 1, 1, ELEM2, 1, HRSRC1, HCSRC1)
END IF
IF( MYROW.EQ.HRSRC3 .AND. MYCOL.EQ.HCSRC3 ) THEN
ELEM3 = H((JLOC3-1)*LLDH+ILOC3)
IF( HRSRC1.NE.HRSRC3 .OR. HCSRC1.NE.HCSRC3 )
$ CALL DGESD2D( ICTXT, 1, 1, ELEM3, 1, HRSRC1, HCSRC1)
END IF
IF( MYROW.EQ.HRSRC4 .AND. MYCOL.EQ.HCSRC4 ) THEN
WORK(1) = H((JLOC4-1)*LLDH+ILOC4)
IF( K+1.LT.N ) THEN
WORK(2) = H((JLOC4-1)*LLDH+ILOC4+1)
ELSE
WORK(2) = ZERO
END IF
IF( HRSRC1.NE.HRSRC4 .OR. HCSRC1.NE.HCSRC4 )
$ CALL DGESD2D( ICTXT, 2, 1, WORK, 2, HRSRC1, HCSRC1 )
END IF
IF( MYROW.EQ.HRSRC1 .AND. MYCOL.EQ.HCSRC1 ) THEN
ELEM1 = H((JLOC1-1)*LLDH+ILOC1)
IF( HRSRC1.NE.HRSRC2 .OR. HCSRC1.NE.HCSRC2 )
$ CALL DGERV2D( ICTXT, 1, 1, ELEM2, 1, HRSRC2, HCSRC2)
IF( HRSRC1.NE.HRSRC3 .OR. HCSRC1.NE.HCSRC3 )
$ CALL DGERV2D( ICTXT, 1, 1, ELEM3, 1, HRSRC3, HCSRC3)
IF( HRSRC1.NE.HRSRC4 .OR. HCSRC1.NE.HCSRC4 )
$ CALL DGERV2D( ICTXT, 2, 1, WORK, 2, HRSRC4, HCSRC4 )
ELEM4 = WORK(1)
ELEM5 = WORK(2)
IF( ELEM5.EQ.ZERO ) THEN
IF( WR( K ).EQ.ZERO .AND. WI( K ).EQ.ZERO ) THEN
CALL DLANV2( ELEM1, ELEM2, ELEM3, ELEM4, WR( K ),
$ WI( K ), WR( K+1 ), WI( K+1 ), SN, CS )
ELSEIF( WR( K+1 ).EQ.ZERO .AND. WI( K+1 ).EQ.ZERO )
$ THEN
WR( K+1 ) = ELEM4
END IF
ELSEIF( WR( K ).EQ.ZERO .AND. WI( K ).EQ.ZERO )
$ THEN
WR( K ) = ELEM1
END IF
END IF
60 CONTINUE
*
IF( NPROCS.GT.1 ) THEN
CALL DGSUM2D( ICTXT, 'All', ' ', IHI-ILO+1, 1, WR(ILO), N,
$ -1, -1 )
CALL DGSUM2D( ICTXT, 'All', ' ', IHI-ILO+1, 1, WI(ILO), N,
$ -1, -1 )
END IF
*
END IF
*
WORK(1) = LWKOPT
IWORK(1) = LIWKOPT
RETURN
*
* End of PDHSEQR
*
END
|
