summaryrefslogtreecommitdiff
path: root/SRC/zgges.f
blob: 555abcd86e8da0d9eb32a4886c6d596d36f5e87e (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
*> \brief <b> ZGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZGGES + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgges.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgges.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgges.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE ZGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
*                         SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
*                         LWORK, RWORK, BWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBVSL, JOBVSR, SORT
*       INTEGER            INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
*       ..
*       .. Array Arguments ..
*       LOGICAL            BWORK( * )
*       DOUBLE PRECISION   RWORK( * )
*       COMPLEX*16         A( LDA, * ), ALPHA( * ), B( LDB, * ),
*      $                   BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
*      $                   WORK( * )
*       ..
*       .. Function Arguments ..
*       LOGICAL            SELCTG
*       EXTERNAL           SELCTG
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZGGES computes for a pair of N-by-N complex nonsymmetric matrices
*> (A,B), the generalized eigenvalues, the generalized complex Schur
*> form (S, T), and optionally left and/or right Schur vectors (VSL
*> and VSR). This gives the generalized Schur factorization
*>
*>         (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )
*>
*> where (VSR)**H is the conjugate-transpose of VSR.
*>
*> Optionally, it also orders the eigenvalues so that a selected cluster
*> of eigenvalues appears in the leading diagonal blocks of the upper
*> triangular matrix S and the upper triangular matrix T. The leading
*> columns of VSL and VSR then form an unitary basis for the
*> corresponding left and right eigenspaces (deflating subspaces).
*>
*> (If only the generalized eigenvalues are needed, use the driver
*> ZGGEV instead, which is faster.)
*>
*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
*> or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
*> usually represented as the pair (alpha,beta), as there is a
*> reasonable interpretation for beta=0, and even for both being zero.
*>
*> A pair of matrices (S,T) is in generalized complex Schur form if S
*> and T are upper triangular and, in addition, the diagonal elements
*> of T are non-negative real numbers.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBVSL
*> \verbatim
*>          JOBVSL is CHARACTER*1
*>          = 'N':  do not compute the left Schur vectors;
*>          = 'V':  compute the left Schur vectors.
*> \endverbatim
*>
*> \param[in] JOBVSR
*> \verbatim
*>          JOBVSR is CHARACTER*1
*>          = 'N':  do not compute the right Schur vectors;
*>          = 'V':  compute the right Schur vectors.
*> \endverbatim
*>
*> \param[in] SORT
*> \verbatim
*>          SORT is CHARACTER*1
*>          Specifies whether or not to order the eigenvalues on the
*>          diagonal of the generalized Schur form.
*>          = 'N':  Eigenvalues are not ordered;
*>          = 'S':  Eigenvalues are ordered (see SELCTG).
*> \endverbatim
*>
*> \param[in] SELCTG
*> \verbatim
*>          SELCTG is a LOGICAL FUNCTION of two COMPLEX*16 arguments
*>          SELCTG must be declared EXTERNAL in the calling subroutine.
*>          If SORT = 'N', SELCTG is not referenced.
*>          If SORT = 'S', SELCTG is used to select eigenvalues to sort
*>          to the top left of the Schur form.
*>          An eigenvalue ALPHA(j)/BETA(j) is selected if
*>          SELCTG(ALPHA(j),BETA(j)) is true.
*>
*>          Note that a selected complex eigenvalue may no longer satisfy
*>          SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
*>          ordering may change the value of complex eigenvalues
*>          (especially if the eigenvalue is ill-conditioned), in this
*>          case INFO is set to N+2 (See INFO below).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A, B, VSL, and VSR.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is COMPLEX*16 array, dimension (LDA, N)
*>          On entry, the first of the pair of matrices.
*>          On exit, A has been overwritten by its generalized Schur
*>          form S.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is COMPLEX*16 array, dimension (LDB, N)
*>          On entry, the second of the pair of matrices.
*>          On exit, B has been overwritten by its generalized Schur
*>          form T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] SDIM
*> \verbatim
*>          SDIM is INTEGER
*>          If SORT = 'N', SDIM = 0.
*>          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
*>          for which SELCTG is true.
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*>          ALPHA is COMPLEX*16 array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is COMPLEX*16 array, dimension (N)
*>          On exit,  ALPHA(j)/BETA(j), j=1,...,N, will be the
*>          generalized eigenvalues.  ALPHA(j), j=1,...,N  and  BETA(j),
*>          j=1,...,N  are the diagonals of the complex Schur form (A,B)
*>          output by ZGGES. The  BETA(j) will be non-negative real.
*>
*>          Note: the quotients ALPHA(j)/BETA(j) may easily over- or
*>          underflow, and BETA(j) may even be zero.  Thus, the user
*>          should avoid naively computing the ratio alpha/beta.
*>          However, ALPHA will be always less than and usually
*>          comparable with norm(A) in magnitude, and BETA always less
*>          than and usually comparable with norm(B).
*> \endverbatim
*>
*> \param[out] VSL
*> \verbatim
*>          VSL is COMPLEX*16 array, dimension (LDVSL,N)
*>          If JOBVSL = 'V', VSL will contain the left Schur vectors.
*>          Not referenced if JOBVSL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSL
*> \verbatim
*>          LDVSL is INTEGER
*>          The leading dimension of the matrix VSL. LDVSL >= 1, and
*>          if JOBVSL = 'V', LDVSL >= N.
*> \endverbatim
*>
*> \param[out] VSR
*> \verbatim
*>          VSR is COMPLEX*16 array, dimension (LDVSR,N)
*>          If JOBVSR = 'V', VSR will contain the right Schur vectors.
*>          Not referenced if JOBVSR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSR
*> \verbatim
*>          LDVSR is INTEGER
*>          The leading dimension of the matrix VSR. LDVSR >= 1, and
*>          if JOBVSR = 'V', LDVSR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.  LWORK >= max(1,2*N).
*>          For good performance, LWORK must generally be larger.
*>
*>          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] RWORK
*> \verbatim
*>          RWORK is DOUBLE PRECISION array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] BWORK
*> \verbatim
*>          BWORK is LOGICAL array, dimension (N)
*>          Not referenced if SORT = 'N'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          =1,...,N:
*>                The QZ iteration failed.  (A,B) are not in Schur
*>                form, but ALPHA(j) and BETA(j) should be correct for
*>                j=INFO+1,...,N.
*>          > N:  =N+1: other than QZ iteration failed in ZHGEQZ
*>                =N+2: after reordering, roundoff changed values of
*>                      some complex eigenvalues so that leading
*>                      eigenvalues in the Generalized Schur form no
*>                      longer satisfy SELCTG=.TRUE.  This could also
*>                      be caused due to scaling.
*>                =N+3: reordering failed in ZTGSEN.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup complex16GEeigen
*
*  =====================================================================
      SUBROUTINE ZGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
     $                  SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
     $                  LWORK, RWORK, BWORK, INFO )
*
*  -- LAPACK driver routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
*     .. Scalar Arguments ..
      CHARACTER          JOBVSL, JOBVSR, SORT
      INTEGER            INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
*     ..
*     .. Array Arguments ..
      LOGICAL            BWORK( * )
      DOUBLE PRECISION   RWORK( * )
      COMPLEX*16         A( LDA, * ), ALPHA( * ), B( LDB, * ),
     $                   BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
     $                   WORK( * )
*     ..
*     .. Function Arguments ..
      LOGICAL            SELCTG
      EXTERNAL           SELCTG
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
      COMPLEX*16         CZERO, CONE
      PARAMETER          ( CZERO = ( 0.0D0, 0.0D0 ),
     $                   CONE = ( 1.0D0, 0.0D0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
     $                   LQUERY, WANTST
      INTEGER            I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
     $                   ILO, IRIGHT, IROWS, IRWRK, ITAU, IWRK, LWKMIN,
     $                   LWKOPT
      DOUBLE PRECISION   ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
     $                   PVSR, SMLNUM
*     ..
*     .. Local Arrays ..
      INTEGER            IDUM( 1 )
      DOUBLE PRECISION   DIF( 2 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DLABAD, XERBLA, ZGEQRF, ZGGBAK, ZGGBAL, ZGGHRD,
     $                   ZHGEQZ, ZLACPY, ZLASCL, ZLASET, ZTGSEN, ZUNGQR,
     $                   ZUNMQR
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, ZLANGE
      EXTERNAL           LSAME, ILAENV, DLAMCH, ZLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Decode the input arguments
*
      IF( LSAME( JOBVSL, 'N' ) ) THEN
         IJOBVL = 1
         ILVSL = .FALSE.
      ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
         IJOBVL = 2
         ILVSL = .TRUE.
      ELSE
         IJOBVL = -1
         ILVSL = .FALSE.
      END IF
*
      IF( LSAME( JOBVSR, 'N' ) ) THEN
         IJOBVR = 1
         ILVSR = .FALSE.
      ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
         IJOBVR = 2
         ILVSR = .TRUE.
      ELSE
         IJOBVR = -1
         ILVSR = .FALSE.
      END IF
*
      WANTST = LSAME( SORT, 'S' )
*
*     Test the input arguments
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      IF( IJOBVL.LE.0 ) THEN
         INFO = -1
      ELSE IF( IJOBVR.LE.0 ) THEN
         INFO = -2
      ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
         INFO = -14
      ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
         INFO = -16
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace needed at that point in the code,
*       as well as the preferred amount for good performance.
*       NB refers to the optimal block size for the immediately
*       following subroutine, as returned by ILAENV.)
*
      IF( INFO.EQ.0 ) THEN
         LWKMIN = MAX( 1, 2*N )
         LWKOPT = MAX( 1, N + N*ILAENV( 1, 'ZGEQRF', ' ', N, 1, N, 0 ) )
         LWKOPT = MAX( LWKOPT, N +
     $                 N*ILAENV( 1, 'ZUNMQR', ' ', N, 1, N, -1 ) )
         IF( ILVSL ) THEN
            LWKOPT = MAX( LWKOPT, N +
     $                    N*ILAENV( 1, 'ZUNGQR', ' ', N, 1, N, -1 ) )
         END IF
         WORK( 1 ) = LWKOPT
*
         IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
     $      INFO = -18
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'ZGGES ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 ) THEN
         SDIM = 0
         RETURN
      END IF
*
*     Get machine constants
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' )
      BIGNUM = ONE / SMLNUM
      CALL DLABAD( SMLNUM, BIGNUM )
      SMLNUM = SQRT( SMLNUM ) / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = ZLANGE( 'M', N, N, A, LDA, RWORK )
      ILASCL = .FALSE.
      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
         ANRMTO = SMLNUM
         ILASCL = .TRUE.
      ELSE IF( ANRM.GT.BIGNUM ) THEN
         ANRMTO = BIGNUM
         ILASCL = .TRUE.
      END IF
*
      IF( ILASCL )
     $   CALL ZLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
*
*     Scale B if max element outside range [SMLNUM,BIGNUM]
*
      BNRM = ZLANGE( 'M', N, N, B, LDB, RWORK )
      ILBSCL = .FALSE.
      IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
         BNRMTO = SMLNUM
         ILBSCL = .TRUE.
      ELSE IF( BNRM.GT.BIGNUM ) THEN
         BNRMTO = BIGNUM
         ILBSCL = .TRUE.
      END IF
*
      IF( ILBSCL )
     $   CALL ZLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
*
*     Permute the matrix to make it more nearly triangular
*     (Real Workspace: need 6*N)
*
      ILEFT = 1
      IRIGHT = N + 1
      IRWRK = IRIGHT + N
      CALL ZGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ),
     $             RWORK( IRIGHT ), RWORK( IRWRK ), IERR )
*
*     Reduce B to triangular form (QR decomposition of B)
*     (Complex Workspace: need N, prefer N*NB)
*
      IROWS = IHI + 1 - ILO
      ICOLS = N + 1 - ILO
      ITAU = 1
      IWRK = ITAU + IROWS
      CALL ZGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
     $             WORK( IWRK ), LWORK+1-IWRK, IERR )
*
*     Apply the orthogonal transformation to matrix A
*     (Complex Workspace: need N, prefer N*NB)
*
      CALL ZUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
     $             WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
     $             LWORK+1-IWRK, IERR )
*
*     Initialize VSL
*     (Complex Workspace: need N, prefer N*NB)
*
      IF( ILVSL ) THEN
         CALL ZLASET( 'Full', N, N, CZERO, CONE, VSL, LDVSL )
         IF( IROWS.GT.1 ) THEN
            CALL ZLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
     $                   VSL( ILO+1, ILO ), LDVSL )
         END IF
         CALL ZUNGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
     $                WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
      END IF
*
*     Initialize VSR
*
      IF( ILVSR )
     $   CALL ZLASET( 'Full', N, N, CZERO, CONE, VSR, LDVSR )
*
*     Reduce to generalized Hessenberg form
*     (Workspace: none needed)
*
      CALL ZGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
     $             LDVSL, VSR, LDVSR, IERR )
*
      SDIM = 0
*
*     Perform QZ algorithm, computing Schur vectors if desired
*     (Complex Workspace: need N)
*     (Real Workspace: need N)
*
      IWRK = ITAU
      CALL ZHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
     $             ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK( IWRK ),
     $             LWORK+1-IWRK, RWORK( IRWRK ), IERR )
      IF( IERR.NE.0 ) THEN
         IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
            INFO = IERR
         ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
            INFO = IERR - N
         ELSE
            INFO = N + 1
         END IF
         GO TO 30
      END IF
*
*     Sort eigenvalues ALPHA/BETA if desired
*     (Workspace: none needed)
*
      IF( WANTST ) THEN
*
*        Undo scaling on eigenvalues before selecting
*
         IF( ILASCL )
     $      CALL ZLASCL( 'G', 0, 0, ANRM, ANRMTO, N, 1, ALPHA, N, IERR )
         IF( ILBSCL )
     $      CALL ZLASCL( 'G', 0, 0, BNRM, BNRMTO, N, 1, BETA, N, IERR )
*
*        Select eigenvalues
*
         DO 10 I = 1, N
            BWORK( I ) = SELCTG( ALPHA( I ), BETA( I ) )
   10    CONTINUE
*
         CALL ZTGSEN( 0, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB, ALPHA,
     $                BETA, VSL, LDVSL, VSR, LDVSR, SDIM, PVSL, PVSR,
     $                DIF, WORK( IWRK ), LWORK-IWRK+1, IDUM, 1, IERR )
         IF( IERR.EQ.1 )
     $      INFO = N + 3
*
      END IF
*
*     Apply back-permutation to VSL and VSR
*     (Workspace: none needed)
*
      IF( ILVSL )
     $   CALL ZGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ),
     $                RWORK( IRIGHT ), N, VSL, LDVSL, IERR )
      IF( ILVSR )
     $   CALL ZGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ),
     $                RWORK( IRIGHT ), N, VSR, LDVSR, IERR )
*
*     Undo scaling
*
      IF( ILASCL ) THEN
         CALL ZLASCL( 'U', 0, 0, ANRMTO, ANRM, N, N, A, LDA, IERR )
         CALL ZLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHA, N, IERR )
      END IF
*
      IF( ILBSCL ) THEN
         CALL ZLASCL( 'U', 0, 0, BNRMTO, BNRM, N, N, B, LDB, IERR )
         CALL ZLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
      END IF
*
      IF( WANTST ) THEN
*
*        Check if reordering is correct
*
         LASTSL = .TRUE.
         SDIM = 0
         DO 20 I = 1, N
            CURSL = SELCTG( ALPHA( I ), BETA( I ) )
            IF( CURSL )
     $         SDIM = SDIM + 1
            IF( CURSL .AND. .NOT.LASTSL )
     $         INFO = N + 2
            LASTSL = CURSL
   20    CONTINUE
*
      END IF
*
   30 CONTINUE
*
      WORK( 1 ) = LWKOPT
*
      RETURN
*
*     End of ZGGES
*
      END