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
|
*> \brief \b ZTGEVC
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition
* ==========
*
* SUBROUTINE ZTGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL,
* LDVL, VR, LDVR, MM, M, WORK, RWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER HOWMNY, SIDE
* INTEGER INFO, LDP, LDS, LDVL, LDVR, M, MM, N
* ..
* .. Array Arguments ..
* LOGICAL SELECT( * )
* DOUBLE PRECISION RWORK( * )
* COMPLEX*16 P( LDP, * ), S( LDS, * ), VL( LDVL, * ),
* $ VR( LDVR, * ), WORK( * )
* ..
*
*
* Purpose
* =======
*
*>\details \b Purpose:
*>\verbatim
*>
*> ZTGEVC computes some or all of the right and/or left eigenvectors of
*> a pair of complex matrices (S,P), where S and P are upper triangular.
*> Matrix pairs of this type are produced by the generalized Schur
*> factorization of a complex matrix pair (A,B):
*>
*> A = Q*S*Z**H, B = Q*P*Z**H
*>
*> as computed by ZGGHRD + ZHGEQZ.
*>
*> The right eigenvector x and the left eigenvector y of (S,P)
*> corresponding to an eigenvalue w are defined by:
*>
*> S*x = w*P*x, (y**H)*S = w*(y**H)*P,
*>
*> where y**H denotes the conjugate tranpose of y.
*> The eigenvalues are not input to this routine, but are computed
*> directly from the diagonal elements of S and P.
*>
*> This routine returns the matrices X and/or Y of right and left
*> eigenvectors of (S,P), or the products Z*X and/or Q*Y,
*> where Z and Q are input matrices.
*> If Q and Z are the unitary factors from the generalized Schur
*> factorization of a matrix pair (A,B), then Z*X and Q*Y
*> are the matrices of right and left eigenvectors of (A,B).
*>
*>\endverbatim
*
* Arguments
* =========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> = 'R': compute right eigenvectors only;
*> = 'L': compute left eigenvectors only;
*> = 'B': compute both right and left eigenvectors.
*> \endverbatim
*>
*> \param[in] HOWMNY
*> \verbatim
*> HOWMNY is CHARACTER*1
*> = 'A': compute all right and/or left eigenvectors;
*> = 'B': compute all right and/or left eigenvectors,
*> backtransformed by the matrices in VR and/or VL;
*> = 'S': compute selected right and/or left eigenvectors,
*> specified by the logical array SELECT.
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*> SELECT is LOGICAL array, dimension (N)
*> If HOWMNY='S', SELECT specifies the eigenvectors to be
*> computed. The eigenvector corresponding to the j-th
*> eigenvalue is computed if SELECT(j) = .TRUE..
*> Not referenced if HOWMNY = 'A' or 'B'.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices S and P. N >= 0.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is COMPLEX*16 array, dimension (LDS,N)
*> The upper triangular matrix S from a generalized Schur
*> factorization, as computed by ZHGEQZ.
*> \endverbatim
*>
*> \param[in] LDS
*> \verbatim
*> LDS is INTEGER
*> The leading dimension of array S. LDS >= max(1,N).
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is COMPLEX*16 array, dimension (LDP,N)
*> The upper triangular matrix P from a generalized Schur
*> factorization, as computed by ZHGEQZ. P must have real
*> diagonal elements.
*> \endverbatim
*>
*> \param[in] LDP
*> \verbatim
*> LDP is INTEGER
*> The leading dimension of array P. LDP >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*> VL is COMPLEX*16 array, dimension (LDVL,MM)
*> On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
*> contain an N-by-N matrix Q (usually the unitary matrix Q
*> of left Schur vectors returned by ZHGEQZ).
*> On exit, if SIDE = 'L' or 'B', VL contains:
*> if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
*> if HOWMNY = 'B', the matrix Q*Y;
*> if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
*> SELECT, stored consecutively in the columns of
*> VL, in the same order as their eigenvalues.
*> Not referenced if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of array VL. LDVL >= 1, and if
*> SIDE = 'L' or 'l' or 'B' or 'b', LDVL >= N.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*> VR is COMPLEX*16 array, dimension (LDVR,MM)
*> On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
*> contain an N-by-N matrix Q (usually the unitary matrix Z
*> of right Schur vectors returned by ZHGEQZ).
*> On exit, if SIDE = 'R' or 'B', VR contains:
*> if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
*> if HOWMNY = 'B', the matrix Z*X;
*> if HOWMNY = 'S', the right eigenvectors of (S,P) specified by
*> SELECT, stored consecutively in the columns of
*> VR, in the same order as their eigenvalues.
*> Not referenced if SIDE = 'L'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the array VR. LDVR >= 1, and if
*> SIDE = 'R' or 'B', LDVR >= N.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*> MM is INTEGER
*> The number of columns in the arrays VL and/or VR. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The number of columns in the arrays VL and/or VR actually
*> used to store the eigenvectors. If HOWMNY = 'A' or 'B', M
*> is set to N. Each selected eigenvector occupies one column.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX*16 array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is DOUBLE PRECISION array, dimension (2*N)
*> \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 complex16GEcomputational
*
* =====================================================================
SUBROUTINE ZTGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL,
$ LDVL, VR, LDVR, MM, M, WORK, RWORK, INFO )
*
* -- LAPACK computational routine (version 3.2) --
* -- 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 HOWMNY, SIDE
INTEGER INFO, LDP, LDS, LDVL, LDVR, M, MM, N
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
DOUBLE PRECISION RWORK( * )
COMPLEX*16 P( LDP, * ), S( LDS, * ), VL( LDVL, * ),
$ VR( LDVR, * ), WORK( * )
* ..
*
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
COMPLEX*16 CZERO, CONE
PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
$ CONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL COMPL, COMPR, ILALL, ILBACK, ILBBAD, ILCOMP,
$ LSA, LSB
INTEGER I, IBEG, IEIG, IEND, IHWMNY, IM, ISIDE, ISRC,
$ J, JE, JR
DOUBLE PRECISION ACOEFA, ACOEFF, ANORM, ASCALE, BCOEFA, BIG,
$ BIGNUM, BNORM, BSCALE, DMIN, SAFMIN, SBETA,
$ SCALE, SMALL, TEMP, ULP, XMAX
COMPLEX*16 BCOEFF, CA, CB, D, SALPHA, SUM, SUMA, SUMB, X
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
COMPLEX*16 ZLADIV
EXTERNAL LSAME, DLAMCH, ZLADIV
* ..
* .. External Subroutines ..
EXTERNAL DLABAD, XERBLA, ZGEMV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN
* ..
* .. Statement Functions ..
DOUBLE PRECISION ABS1
* ..
* .. Statement Function definitions ..
ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) )
* ..
* .. Executable Statements ..
*
* Decode and Test the input parameters
*
IF( LSAME( HOWMNY, 'A' ) ) THEN
IHWMNY = 1
ILALL = .TRUE.
ILBACK = .FALSE.
ELSE IF( LSAME( HOWMNY, 'S' ) ) THEN
IHWMNY = 2
ILALL = .FALSE.
ILBACK = .FALSE.
ELSE IF( LSAME( HOWMNY, 'B' ) ) THEN
IHWMNY = 3
ILALL = .TRUE.
ILBACK = .TRUE.
ELSE
IHWMNY = -1
END IF
*
IF( LSAME( SIDE, 'R' ) ) THEN
ISIDE = 1
COMPL = .FALSE.
COMPR = .TRUE.
ELSE IF( LSAME( SIDE, 'L' ) ) THEN
ISIDE = 2
COMPL = .TRUE.
COMPR = .FALSE.
ELSE IF( LSAME( SIDE, 'B' ) ) THEN
ISIDE = 3
COMPL = .TRUE.
COMPR = .TRUE.
ELSE
ISIDE = -1
END IF
*
INFO = 0
IF( ISIDE.LT.0 ) THEN
INFO = -1
ELSE IF( IHWMNY.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDS.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDP.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZTGEVC', -INFO )
RETURN
END IF
*
* Count the number of eigenvectors
*
IF( .NOT.ILALL ) THEN
IM = 0
DO 10 J = 1, N
IF( SELECT( J ) )
$ IM = IM + 1
10 CONTINUE
ELSE
IM = N
END IF
*
* Check diagonal of B
*
ILBBAD = .FALSE.
DO 20 J = 1, N
IF( DIMAG( P( J, J ) ).NE.ZERO )
$ ILBBAD = .TRUE.
20 CONTINUE
*
IF( ILBBAD ) THEN
INFO = -7
ELSE IF( COMPL .AND. LDVL.LT.N .OR. LDVL.LT.1 ) THEN
INFO = -10
ELSE IF( COMPR .AND. LDVR.LT.N .OR. LDVR.LT.1 ) THEN
INFO = -12
ELSE IF( MM.LT.IM ) THEN
INFO = -13
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZTGEVC', -INFO )
RETURN
END IF
*
* Quick return if possible
*
M = IM
IF( N.EQ.0 )
$ RETURN
*
* Machine Constants
*
SAFMIN = DLAMCH( 'Safe minimum' )
BIG = ONE / SAFMIN
CALL DLABAD( SAFMIN, BIG )
ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
SMALL = SAFMIN*N / ULP
BIG = ONE / SMALL
BIGNUM = ONE / ( SAFMIN*N )
*
* Compute the 1-norm of each column of the strictly upper triangular
* part of A and B to check for possible overflow in the triangular
* solver.
*
ANORM = ABS1( S( 1, 1 ) )
BNORM = ABS1( P( 1, 1 ) )
RWORK( 1 ) = ZERO
RWORK( N+1 ) = ZERO
DO 40 J = 2, N
RWORK( J ) = ZERO
RWORK( N+J ) = ZERO
DO 30 I = 1, J - 1
RWORK( J ) = RWORK( J ) + ABS1( S( I, J ) )
RWORK( N+J ) = RWORK( N+J ) + ABS1( P( I, J ) )
30 CONTINUE
ANORM = MAX( ANORM, RWORK( J )+ABS1( S( J, J ) ) )
BNORM = MAX( BNORM, RWORK( N+J )+ABS1( P( J, J ) ) )
40 CONTINUE
*
ASCALE = ONE / MAX( ANORM, SAFMIN )
BSCALE = ONE / MAX( BNORM, SAFMIN )
*
* Left eigenvectors
*
IF( COMPL ) THEN
IEIG = 0
*
* Main loop over eigenvalues
*
DO 140 JE = 1, N
IF( ILALL ) THEN
ILCOMP = .TRUE.
ELSE
ILCOMP = SELECT( JE )
END IF
IF( ILCOMP ) THEN
IEIG = IEIG + 1
*
IF( ABS1( S( JE, JE ) ).LE.SAFMIN .AND.
$ ABS( DBLE( P( JE, JE ) ) ).LE.SAFMIN ) THEN
*
* Singular matrix pencil -- return unit eigenvector
*
DO 50 JR = 1, N
VL( JR, IEIG ) = CZERO
50 CONTINUE
VL( IEIG, IEIG ) = CONE
GO TO 140
END IF
*
* Non-singular eigenvalue:
* Compute coefficients a and b in
* H
* y ( a A - b B ) = 0
*
TEMP = ONE / MAX( ABS1( S( JE, JE ) )*ASCALE,
$ ABS( DBLE( P( JE, JE ) ) )*BSCALE, SAFMIN )
SALPHA = ( TEMP*S( JE, JE ) )*ASCALE
SBETA = ( TEMP*DBLE( P( JE, JE ) ) )*BSCALE
ACOEFF = SBETA*ASCALE
BCOEFF = SALPHA*BSCALE
*
* Scale to avoid underflow
*
LSA = ABS( SBETA ).GE.SAFMIN .AND. ABS( ACOEFF ).LT.SMALL
LSB = ABS1( SALPHA ).GE.SAFMIN .AND. ABS1( BCOEFF ).LT.
$ SMALL
*
SCALE = ONE
IF( LSA )
$ SCALE = ( SMALL / ABS( SBETA ) )*MIN( ANORM, BIG )
IF( LSB )
$ SCALE = MAX( SCALE, ( SMALL / ABS1( SALPHA ) )*
$ MIN( BNORM, BIG ) )
IF( LSA .OR. LSB ) THEN
SCALE = MIN( SCALE, ONE /
$ ( SAFMIN*MAX( ONE, ABS( ACOEFF ),
$ ABS1( BCOEFF ) ) ) )
IF( LSA ) THEN
ACOEFF = ASCALE*( SCALE*SBETA )
ELSE
ACOEFF = SCALE*ACOEFF
END IF
IF( LSB ) THEN
BCOEFF = BSCALE*( SCALE*SALPHA )
ELSE
BCOEFF = SCALE*BCOEFF
END IF
END IF
*
ACOEFA = ABS( ACOEFF )
BCOEFA = ABS1( BCOEFF )
XMAX = ONE
DO 60 JR = 1, N
WORK( JR ) = CZERO
60 CONTINUE
WORK( JE ) = CONE
DMIN = MAX( ULP*ACOEFA*ANORM, ULP*BCOEFA*BNORM, SAFMIN )
*
* H
* Triangular solve of (a A - b B) y = 0
*
* H
* (rowwise in (a A - b B) , or columnwise in a A - b B)
*
DO 100 J = JE + 1, N
*
* Compute
* j-1
* SUM = sum conjg( a*S(k,j) - b*P(k,j) )*x(k)
* k=je
* (Scale if necessary)
*
TEMP = ONE / XMAX
IF( ACOEFA*RWORK( J )+BCOEFA*RWORK( N+J ).GT.BIGNUM*
$ TEMP ) THEN
DO 70 JR = JE, J - 1
WORK( JR ) = TEMP*WORK( JR )
70 CONTINUE
XMAX = ONE
END IF
SUMA = CZERO
SUMB = CZERO
*
DO 80 JR = JE, J - 1
SUMA = SUMA + DCONJG( S( JR, J ) )*WORK( JR )
SUMB = SUMB + DCONJG( P( JR, J ) )*WORK( JR )
80 CONTINUE
SUM = ACOEFF*SUMA - DCONJG( BCOEFF )*SUMB
*
* Form x(j) = - SUM / conjg( a*S(j,j) - b*P(j,j) )
*
* with scaling and perturbation of the denominator
*
D = DCONJG( ACOEFF*S( J, J )-BCOEFF*P( J, J ) )
IF( ABS1( D ).LE.DMIN )
$ D = DCMPLX( DMIN )
*
IF( ABS1( D ).LT.ONE ) THEN
IF( ABS1( SUM ).GE.BIGNUM*ABS1( D ) ) THEN
TEMP = ONE / ABS1( SUM )
DO 90 JR = JE, J - 1
WORK( JR ) = TEMP*WORK( JR )
90 CONTINUE
XMAX = TEMP*XMAX
SUM = TEMP*SUM
END IF
END IF
WORK( J ) = ZLADIV( -SUM, D )
XMAX = MAX( XMAX, ABS1( WORK( J ) ) )
100 CONTINUE
*
* Back transform eigenvector if HOWMNY='B'.
*
IF( ILBACK ) THEN
CALL ZGEMV( 'N', N, N+1-JE, CONE, VL( 1, JE ), LDVL,
$ WORK( JE ), 1, CZERO, WORK( N+1 ), 1 )
ISRC = 2
IBEG = 1
ELSE
ISRC = 1
IBEG = JE
END IF
*
* Copy and scale eigenvector into column of VL
*
XMAX = ZERO
DO 110 JR = IBEG, N
XMAX = MAX( XMAX, ABS1( WORK( ( ISRC-1 )*N+JR ) ) )
110 CONTINUE
*
IF( XMAX.GT.SAFMIN ) THEN
TEMP = ONE / XMAX
DO 120 JR = IBEG, N
VL( JR, IEIG ) = TEMP*WORK( ( ISRC-1 )*N+JR )
120 CONTINUE
ELSE
IBEG = N + 1
END IF
*
DO 130 JR = 1, IBEG - 1
VL( JR, IEIG ) = CZERO
130 CONTINUE
*
END IF
140 CONTINUE
END IF
*
* Right eigenvectors
*
IF( COMPR ) THEN
IEIG = IM + 1
*
* Main loop over eigenvalues
*
DO 250 JE = N, 1, -1
IF( ILALL ) THEN
ILCOMP = .TRUE.
ELSE
ILCOMP = SELECT( JE )
END IF
IF( ILCOMP ) THEN
IEIG = IEIG - 1
*
IF( ABS1( S( JE, JE ) ).LE.SAFMIN .AND.
$ ABS( DBLE( P( JE, JE ) ) ).LE.SAFMIN ) THEN
*
* Singular matrix pencil -- return unit eigenvector
*
DO 150 JR = 1, N
VR( JR, IEIG ) = CZERO
150 CONTINUE
VR( IEIG, IEIG ) = CONE
GO TO 250
END IF
*
* Non-singular eigenvalue:
* Compute coefficients a and b in
*
* ( a A - b B ) x = 0
*
TEMP = ONE / MAX( ABS1( S( JE, JE ) )*ASCALE,
$ ABS( DBLE( P( JE, JE ) ) )*BSCALE, SAFMIN )
SALPHA = ( TEMP*S( JE, JE ) )*ASCALE
SBETA = ( TEMP*DBLE( P( JE, JE ) ) )*BSCALE
ACOEFF = SBETA*ASCALE
BCOEFF = SALPHA*BSCALE
*
* Scale to avoid underflow
*
LSA = ABS( SBETA ).GE.SAFMIN .AND. ABS( ACOEFF ).LT.SMALL
LSB = ABS1( SALPHA ).GE.SAFMIN .AND. ABS1( BCOEFF ).LT.
$ SMALL
*
SCALE = ONE
IF( LSA )
$ SCALE = ( SMALL / ABS( SBETA ) )*MIN( ANORM, BIG )
IF( LSB )
$ SCALE = MAX( SCALE, ( SMALL / ABS1( SALPHA ) )*
$ MIN( BNORM, BIG ) )
IF( LSA .OR. LSB ) THEN
SCALE = MIN( SCALE, ONE /
$ ( SAFMIN*MAX( ONE, ABS( ACOEFF ),
$ ABS1( BCOEFF ) ) ) )
IF( LSA ) THEN
ACOEFF = ASCALE*( SCALE*SBETA )
ELSE
ACOEFF = SCALE*ACOEFF
END IF
IF( LSB ) THEN
BCOEFF = BSCALE*( SCALE*SALPHA )
ELSE
BCOEFF = SCALE*BCOEFF
END IF
END IF
*
ACOEFA = ABS( ACOEFF )
BCOEFA = ABS1( BCOEFF )
XMAX = ONE
DO 160 JR = 1, N
WORK( JR ) = CZERO
160 CONTINUE
WORK( JE ) = CONE
DMIN = MAX( ULP*ACOEFA*ANORM, ULP*BCOEFA*BNORM, SAFMIN )
*
* Triangular solve of (a A - b B) x = 0 (columnwise)
*
* WORK(1:j-1) contains sums w,
* WORK(j+1:JE) contains x
*
DO 170 JR = 1, JE - 1
WORK( JR ) = ACOEFF*S( JR, JE ) - BCOEFF*P( JR, JE )
170 CONTINUE
WORK( JE ) = CONE
*
DO 210 J = JE - 1, 1, -1
*
* Form x(j) := - w(j) / d
* with scaling and perturbation of the denominator
*
D = ACOEFF*S( J, J ) - BCOEFF*P( J, J )
IF( ABS1( D ).LE.DMIN )
$ D = DCMPLX( DMIN )
*
IF( ABS1( D ).LT.ONE ) THEN
IF( ABS1( WORK( J ) ).GE.BIGNUM*ABS1( D ) ) THEN
TEMP = ONE / ABS1( WORK( J ) )
DO 180 JR = 1, JE
WORK( JR ) = TEMP*WORK( JR )
180 CONTINUE
END IF
END IF
*
WORK( J ) = ZLADIV( -WORK( J ), D )
*
IF( J.GT.1 ) THEN
*
* w = w + x(j)*(a S(*,j) - b P(*,j) ) with scaling
*
IF( ABS1( WORK( J ) ).GT.ONE ) THEN
TEMP = ONE / ABS1( WORK( J ) )
IF( ACOEFA*RWORK( J )+BCOEFA*RWORK( N+J ).GE.
$ BIGNUM*TEMP ) THEN
DO 190 JR = 1, JE
WORK( JR ) = TEMP*WORK( JR )
190 CONTINUE
END IF
END IF
*
CA = ACOEFF*WORK( J )
CB = BCOEFF*WORK( J )
DO 200 JR = 1, J - 1
WORK( JR ) = WORK( JR ) + CA*S( JR, J ) -
$ CB*P( JR, J )
200 CONTINUE
END IF
210 CONTINUE
*
* Back transform eigenvector if HOWMNY='B'.
*
IF( ILBACK ) THEN
CALL ZGEMV( 'N', N, JE, CONE, VR, LDVR, WORK, 1,
$ CZERO, WORK( N+1 ), 1 )
ISRC = 2
IEND = N
ELSE
ISRC = 1
IEND = JE
END IF
*
* Copy and scale eigenvector into column of VR
*
XMAX = ZERO
DO 220 JR = 1, IEND
XMAX = MAX( XMAX, ABS1( WORK( ( ISRC-1 )*N+JR ) ) )
220 CONTINUE
*
IF( XMAX.GT.SAFMIN ) THEN
TEMP = ONE / XMAX
DO 230 JR = 1, IEND
VR( JR, IEIG ) = TEMP*WORK( ( ISRC-1 )*N+JR )
230 CONTINUE
ELSE
IEND = 0
END IF
*
DO 240 JR = IEND + 1, N
VR( JR, IEIG ) = CZERO
240 CONTINUE
*
END IF
250 CONTINUE
END IF
*
RETURN
*
* End of ZTGEVC
*
END
|