1 SUBROUTINE ZGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
2 $ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
3 *
4 * -- LAPACK driver routine (version 3.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * November 2006
8 *
9 * .. Scalar Arguments ..
10 CHARACTER JOBVL, JOBVR
11 INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
12 * ..
13 * .. Array Arguments ..
14 DOUBLE PRECISION RWORK( * )
15 COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
16 $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
17 $ WORK( * )
18 * ..
19 *
20 * Purpose
21 * =======
22 *
23 * This routine is deprecated and has been replaced by routine ZGGEV.
24 *
25 * ZGEGV computes the eigenvalues and, optionally, the left and/or right
26 * eigenvectors of a complex matrix pair (A,B).
27 * Given two square matrices A and B,
28 * the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
29 * eigenvalues lambda and corresponding (non-zero) eigenvectors x such
30 * that
31 * A*x = lambda*B*x.
32 *
33 * An alternate form is to find the eigenvalues mu and corresponding
34 * eigenvectors y such that
35 * mu*A*y = B*y.
36 *
37 * These two forms are equivalent with mu = 1/lambda and x = y if
38 * neither lambda nor mu is zero. In order to deal with the case that
39 * lambda or mu is zero or small, two values alpha and beta are returned
40 * for each eigenvalue, such that lambda = alpha/beta and
41 * mu = beta/alpha.
42 *
43 * The vectors x and y in the above equations are right eigenvectors of
44 * the matrix pair (A,B). Vectors u and v satisfying
45 * u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
46 * are left eigenvectors of (A,B).
47 *
48 * Note: this routine performs "full balancing" on A and B -- see
49 * "Further Details", below.
50 *
51 * Arguments
52 * =========
53 *
54 * JOBVL (input) CHARACTER*1
55 * = 'N': do not compute the left generalized eigenvectors;
56 * = 'V': compute the left generalized eigenvectors (returned
57 * in VL).
58 *
59 * JOBVR (input) CHARACTER*1
60 * = 'N': do not compute the right generalized eigenvectors;
61 * = 'V': compute the right generalized eigenvectors (returned
62 * in VR).
63 *
64 * N (input) INTEGER
65 * The order of the matrices A, B, VL, and VR. N >= 0.
66 *
67 * A (input/output) COMPLEX*16 array, dimension (LDA, N)
68 * On entry, the matrix A.
69 * If JOBVL = 'V' or JOBVR = 'V', then on exit A
70 * contains the Schur form of A from the generalized Schur
71 * factorization of the pair (A,B) after balancing. If no
72 * eigenvectors were computed, then only the diagonal elements
73 * of the Schur form will be correct. See ZGGHRD and ZHGEQZ
74 * for details.
75 *
76 * LDA (input) INTEGER
77 * The leading dimension of A. LDA >= max(1,N).
78 *
79 * B (input/output) COMPLEX*16 array, dimension (LDB, N)
80 * On entry, the matrix B.
81 * If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
82 * upper triangular matrix obtained from B in the generalized
83 * Schur factorization of the pair (A,B) after balancing.
84 * If no eigenvectors were computed, then only the diagonal
85 * elements of B will be correct. See ZGGHRD and ZHGEQZ for
86 * details.
87 *
88 * LDB (input) INTEGER
89 * The leading dimension of B. LDB >= max(1,N).
90 *
91 * ALPHA (output) COMPLEX*16 array, dimension (N)
92 * The complex scalars alpha that define the eigenvalues of
93 * GNEP.
94 *
95 * BETA (output) COMPLEX*16 array, dimension (N)
96 * The complex scalars beta that define the eigenvalues of GNEP.
97 *
98 * Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
99 * represent the j-th eigenvalue of the matrix pair (A,B), in
100 * one of the forms lambda = alpha/beta or mu = beta/alpha.
101 * Since either lambda or mu may overflow, they should not,
102 * in general, be computed.
103 *
104 * VL (output) COMPLEX*16 array, dimension (LDVL,N)
105 * If JOBVL = 'V', the left eigenvectors u(j) are stored
106 * in the columns of VL, in the same order as their eigenvalues.
107 * Each eigenvector is scaled so that its largest component has
108 * abs(real part) + abs(imag. part) = 1, except for eigenvectors
109 * corresponding to an eigenvalue with alpha = beta = 0, which
110 * are set to zero.
111 * Not referenced if JOBVL = 'N'.
112 *
113 * LDVL (input) INTEGER
114 * The leading dimension of the matrix VL. LDVL >= 1, and
115 * if JOBVL = 'V', LDVL >= N.
116 *
117 * VR (output) COMPLEX*16 array, dimension (LDVR,N)
118 * If JOBVR = 'V', the right eigenvectors x(j) are stored
119 * in the columns of VR, in the same order as their eigenvalues.
120 * Each eigenvector is scaled so that its largest component has
121 * abs(real part) + abs(imag. part) = 1, except for eigenvectors
122 * corresponding to an eigenvalue with alpha = beta = 0, which
123 * are set to zero.
124 * Not referenced if JOBVR = 'N'.
125 *
126 * LDVR (input) INTEGER
127 * The leading dimension of the matrix VR. LDVR >= 1, and
128 * if JOBVR = 'V', LDVR >= N.
129 *
130 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
131 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
132 *
133 * LWORK (input) INTEGER
134 * The dimension of the array WORK. LWORK >= max(1,2*N).
135 * For good performance, LWORK must generally be larger.
136 * To compute the optimal value of LWORK, call ILAENV to get
137 * blocksizes (for ZGEQRF, ZUNMQR, and ZUNGQR.) Then compute:
138 * NB -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and ZUNGQR;
139 * The optimal LWORK is MAX( 2*N, N*(NB+1) ).
140 *
141 * If LWORK = -1, then a workspace query is assumed; the routine
142 * only calculates the optimal size of the WORK array, returns
143 * this value as the first entry of the WORK array, and no error
144 * message related to LWORK is issued by XERBLA.
145 *
146 * RWORK (workspace/output) DOUBLE PRECISION array, dimension (8*N)
147 *
148 * INFO (output) INTEGER
149 * = 0: successful exit
150 * < 0: if INFO = -i, the i-th argument had an illegal value.
151 * =1,...,N:
152 * The QZ iteration failed. No eigenvectors have been
153 * calculated, but ALPHA(j) and BETA(j) should be
154 * correct for j=INFO+1,...,N.
155 * > N: errors that usually indicate LAPACK problems:
156 * =N+1: error return from ZGGBAL
157 * =N+2: error return from ZGEQRF
158 * =N+3: error return from ZUNMQR
159 * =N+4: error return from ZUNGQR
160 * =N+5: error return from ZGGHRD
161 * =N+6: error return from ZHGEQZ (other than failed
162 * iteration)
163 * =N+7: error return from ZTGEVC
164 * =N+8: error return from ZGGBAK (computing VL)
165 * =N+9: error return from ZGGBAK (computing VR)
166 * =N+10: error return from ZLASCL (various calls)
167 *
168 * Further Details
169 * ===============
170 *
171 * Balancing
172 * ---------
173 *
174 * This driver calls ZGGBAL to both permute and scale rows and columns
175 * of A and B. The permutations PL and PR are chosen so that PL*A*PR
176 * and PL*B*R will be upper triangular except for the diagonal blocks
177 * A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
178 * possible. The diagonal scaling matrices DL and DR are chosen so
179 * that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
180 * one (except for the elements that start out zero.)
181 *
182 * After the eigenvalues and eigenvectors of the balanced matrices
183 * have been computed, ZGGBAK transforms the eigenvectors back to what
184 * they would have been (in perfect arithmetic) if they had not been
185 * balanced.
186 *
187 * Contents of A and B on Exit
188 * -------- -- - --- - -- ----
189 *
190 * If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
191 * both), then on exit the arrays A and B will contain the complex Schur
192 * form[*] of the "balanced" versions of A and B. If no eigenvectors
193 * are computed, then only the diagonal blocks will be correct.
194 *
195 * [*] In other words, upper triangular form.
196 *
197 * =====================================================================
198 *
199 * .. Parameters ..
200 DOUBLE PRECISION ZERO, ONE
201 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
202 COMPLEX*16 CZERO, CONE
203 PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ),
204 $ CONE = ( 1.0D0, 0.0D0 ) )
205 * ..
206 * .. Local Scalars ..
207 LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY
208 CHARACTER CHTEMP
209 INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
210 $ IN, IRIGHT, IROWS, IRWORK, ITAU, IWORK, JC, JR,
211 $ LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3
212 DOUBLE PRECISION ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
213 $ BNRM1, BNRM2, EPS, SAFMAX, SAFMIN, SALFAI,
214 $ SALFAR, SBETA, SCALE, TEMP
215 COMPLEX*16 X
216 * ..
217 * .. Local Arrays ..
218 LOGICAL LDUMMA( 1 )
219 * ..
220 * .. External Subroutines ..
221 EXTERNAL XERBLA, ZGEQRF, ZGGBAK, ZGGBAL, ZGGHRD, ZHGEQZ,
222 $ ZLACPY, ZLASCL, ZLASET, ZTGEVC, ZUNGQR, ZUNMQR
223 * ..
224 * .. External Functions ..
225 LOGICAL LSAME
226 INTEGER ILAENV
227 DOUBLE PRECISION DLAMCH, ZLANGE
228 EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE
229 * ..
230 * .. Intrinsic Functions ..
231 INTRINSIC ABS, DBLE, DCMPLX, DIMAG, INT, MAX
232 * ..
233 * .. Statement Functions ..
234 DOUBLE PRECISION ABS1
235 * ..
236 * .. Statement Function definitions ..
237 ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) )
238 * ..
239 * .. Executable Statements ..
240 *
241 * Decode the input arguments
242 *
243 IF( LSAME( JOBVL, 'N' ) ) THEN
244 IJOBVL = 1
245 ILVL = .FALSE.
246 ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
247 IJOBVL = 2
248 ILVL = .TRUE.
249 ELSE
250 IJOBVL = -1
251 ILVL = .FALSE.
252 END IF
253 *
254 IF( LSAME( JOBVR, 'N' ) ) THEN
255 IJOBVR = 1
256 ILVR = .FALSE.
257 ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
258 IJOBVR = 2
259 ILVR = .TRUE.
260 ELSE
261 IJOBVR = -1
262 ILVR = .FALSE.
263 END IF
264 ILV = ILVL .OR. ILVR
265 *
266 * Test the input arguments
267 *
268 LWKMIN = MAX( 2*N, 1 )
269 LWKOPT = LWKMIN
270 WORK( 1 ) = LWKOPT
271 LQUERY = ( LWORK.EQ.-1 )
272 INFO = 0
273 IF( IJOBVL.LE.0 ) THEN
274 INFO = -1
275 ELSE IF( IJOBVR.LE.0 ) THEN
276 INFO = -2
277 ELSE IF( N.LT.0 ) THEN
278 INFO = -3
279 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
280 INFO = -5
281 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
282 INFO = -7
283 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
284 INFO = -11
285 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
286 INFO = -13
287 ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
288 INFO = -15
289 END IF
290 *
291 IF( INFO.EQ.0 ) THEN
292 NB1 = ILAENV( 1, 'ZGEQRF', ' ', N, N, -1, -1 )
293 NB2 = ILAENV( 1, 'ZUNMQR', ' ', N, N, N, -1 )
294 NB3 = ILAENV( 1, 'ZUNGQR', ' ', N, N, N, -1 )
295 NB = MAX( NB1, NB2, NB3 )
296 LOPT = MAX( 2*N, N*( NB+1 ) )
297 WORK( 1 ) = LOPT
298 END IF
299 *
300 IF( INFO.NE.0 ) THEN
301 CALL XERBLA( 'ZGEGV ', -INFO )
302 RETURN
303 ELSE IF( LQUERY ) THEN
304 RETURN
305 END IF
306 *
307 * Quick return if possible
308 *
309 IF( N.EQ.0 )
310 $ RETURN
311 *
312 * Get machine constants
313 *
314 EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
315 SAFMIN = DLAMCH( 'S' )
316 SAFMIN = SAFMIN + SAFMIN
317 SAFMAX = ONE / SAFMIN
318 *
319 * Scale A
320 *
321 ANRM = ZLANGE( 'M', N, N, A, LDA, RWORK )
322 ANRM1 = ANRM
323 ANRM2 = ONE
324 IF( ANRM.LT.ONE ) THEN
325 IF( SAFMAX*ANRM.LT.ONE ) THEN
326 ANRM1 = SAFMIN
327 ANRM2 = SAFMAX*ANRM
328 END IF
329 END IF
330 *
331 IF( ANRM.GT.ZERO ) THEN
332 CALL ZLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
333 IF( IINFO.NE.0 ) THEN
334 INFO = N + 10
335 RETURN
336 END IF
337 END IF
338 *
339 * Scale B
340 *
341 BNRM = ZLANGE( 'M', N, N, B, LDB, RWORK )
342 BNRM1 = BNRM
343 BNRM2 = ONE
344 IF( BNRM.LT.ONE ) THEN
345 IF( SAFMAX*BNRM.LT.ONE ) THEN
346 BNRM1 = SAFMIN
347 BNRM2 = SAFMAX*BNRM
348 END IF
349 END IF
350 *
351 IF( BNRM.GT.ZERO ) THEN
352 CALL ZLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
353 IF( IINFO.NE.0 ) THEN
354 INFO = N + 10
355 RETURN
356 END IF
357 END IF
358 *
359 * Permute the matrix to make it more nearly triangular
360 * Also "balance" the matrix.
361 *
362 ILEFT = 1
363 IRIGHT = N + 1
364 IRWORK = IRIGHT + N
365 CALL ZGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ),
366 $ RWORK( IRIGHT ), RWORK( IRWORK ), IINFO )
367 IF( IINFO.NE.0 ) THEN
368 INFO = N + 1
369 GO TO 80
370 END IF
371 *
372 * Reduce B to triangular form, and initialize VL and/or VR
373 *
374 IROWS = IHI + 1 - ILO
375 IF( ILV ) THEN
376 ICOLS = N + 1 - ILO
377 ELSE
378 ICOLS = IROWS
379 END IF
380 ITAU = 1
381 IWORK = ITAU + IROWS
382 CALL ZGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
383 $ WORK( IWORK ), LWORK+1-IWORK, IINFO )
384 IF( IINFO.GE.0 )
385 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
386 IF( IINFO.NE.0 ) THEN
387 INFO = N + 2
388 GO TO 80
389 END IF
390 *
391 CALL ZUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
392 $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
393 $ LWORK+1-IWORK, IINFO )
394 IF( IINFO.GE.0 )
395 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
396 IF( IINFO.NE.0 ) THEN
397 INFO = N + 3
398 GO TO 80
399 END IF
400 *
401 IF( ILVL ) THEN
402 CALL ZLASET( 'Full', N, N, CZERO, CONE, VL, LDVL )
403 CALL ZLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
404 $ VL( ILO+1, ILO ), LDVL )
405 CALL ZUNGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
406 $ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
407 $ IINFO )
408 IF( IINFO.GE.0 )
409 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
410 IF( IINFO.NE.0 ) THEN
411 INFO = N + 4
412 GO TO 80
413 END IF
414 END IF
415 *
416 IF( ILVR )
417 $ CALL ZLASET( 'Full', N, N, CZERO, CONE, VR, LDVR )
418 *
419 * Reduce to generalized Hessenberg form
420 *
421 IF( ILV ) THEN
422 *
423 * Eigenvectors requested -- work on whole matrix.
424 *
425 CALL ZGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
426 $ LDVL, VR, LDVR, IINFO )
427 ELSE
428 CALL ZGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
429 $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
430 END IF
431 IF( IINFO.NE.0 ) THEN
432 INFO = N + 5
433 GO TO 80
434 END IF
435 *
436 * Perform QZ algorithm
437 *
438 IWORK = ITAU
439 IF( ILV ) THEN
440 CHTEMP = 'S'
441 ELSE
442 CHTEMP = 'E'
443 END IF
444 CALL ZHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
445 $ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK( IWORK ),
446 $ LWORK+1-IWORK, RWORK( IRWORK ), IINFO )
447 IF( IINFO.GE.0 )
448 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
449 IF( IINFO.NE.0 ) THEN
450 IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
451 INFO = IINFO
452 ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
453 INFO = IINFO - N
454 ELSE
455 INFO = N + 6
456 END IF
457 GO TO 80
458 END IF
459 *
460 IF( ILV ) THEN
461 *
462 * Compute Eigenvectors
463 *
464 IF( ILVL ) THEN
465 IF( ILVR ) THEN
466 CHTEMP = 'B'
467 ELSE
468 CHTEMP = 'L'
469 END IF
470 ELSE
471 CHTEMP = 'R'
472 END IF
473 *
474 CALL ZTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
475 $ VR, LDVR, N, IN, WORK( IWORK ), RWORK( IRWORK ),
476 $ IINFO )
477 IF( IINFO.NE.0 ) THEN
478 INFO = N + 7
479 GO TO 80
480 END IF
481 *
482 * Undo balancing on VL and VR, rescale
483 *
484 IF( ILVL ) THEN
485 CALL ZGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ),
486 $ RWORK( IRIGHT ), N, VL, LDVL, IINFO )
487 IF( IINFO.NE.0 ) THEN
488 INFO = N + 8
489 GO TO 80
490 END IF
491 DO 30 JC = 1, N
492 TEMP = ZERO
493 DO 10 JR = 1, N
494 TEMP = MAX( TEMP, ABS1( VL( JR, JC ) ) )
495 10 CONTINUE
496 IF( TEMP.LT.SAFMIN )
497 $ GO TO 30
498 TEMP = ONE / TEMP
499 DO 20 JR = 1, N
500 VL( JR, JC ) = VL( JR, JC )*TEMP
501 20 CONTINUE
502 30 CONTINUE
503 END IF
504 IF( ILVR ) THEN
505 CALL ZGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ),
506 $ RWORK( IRIGHT ), N, VR, LDVR, IINFO )
507 IF( IINFO.NE.0 ) THEN
508 INFO = N + 9
509 GO TO 80
510 END IF
511 DO 60 JC = 1, N
512 TEMP = ZERO
513 DO 40 JR = 1, N
514 TEMP = MAX( TEMP, ABS1( VR( JR, JC ) ) )
515 40 CONTINUE
516 IF( TEMP.LT.SAFMIN )
517 $ GO TO 60
518 TEMP = ONE / TEMP
519 DO 50 JR = 1, N
520 VR( JR, JC ) = VR( JR, JC )*TEMP
521 50 CONTINUE
522 60 CONTINUE
523 END IF
524 *
525 * End of eigenvector calculation
526 *
527 END IF
528 *
529 * Undo scaling in alpha, beta
530 *
531 * Note: this does not give the alpha and beta for the unscaled
532 * problem.
533 *
534 * Un-scaling is limited to avoid underflow in alpha and beta
535 * if they are significant.
536 *
537 DO 70 JC = 1, N
538 ABSAR = ABS( DBLE( ALPHA( JC ) ) )
539 ABSAI = ABS( DIMAG( ALPHA( JC ) ) )
540 ABSB = ABS( DBLE( BETA( JC ) ) )
541 SALFAR = ANRM*DBLE( ALPHA( JC ) )
542 SALFAI = ANRM*DIMAG( ALPHA( JC ) )
543 SBETA = BNRM*DBLE( BETA( JC ) )
544 ILIMIT = .FALSE.
545 SCALE = ONE
546 *
547 * Check for significant underflow in imaginary part of ALPHA
548 *
549 IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
550 $ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
551 ILIMIT = .TRUE.
552 SCALE = ( SAFMIN / ANRM1 ) / MAX( SAFMIN, ANRM2*ABSAI )
553 END IF
554 *
555 * Check for significant underflow in real part of ALPHA
556 *
557 IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
558 $ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
559 ILIMIT = .TRUE.
560 SCALE = MAX( SCALE, ( SAFMIN / ANRM1 ) /
561 $ MAX( SAFMIN, ANRM2*ABSAR ) )
562 END IF
563 *
564 * Check for significant underflow in BETA
565 *
566 IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
567 $ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
568 ILIMIT = .TRUE.
569 SCALE = MAX( SCALE, ( SAFMIN / BNRM1 ) /
570 $ MAX( SAFMIN, BNRM2*ABSB ) )
571 END IF
572 *
573 * Check for possible overflow when limiting scaling
574 *
575 IF( ILIMIT ) THEN
576 TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
577 $ ABS( SBETA ) )
578 IF( TEMP.GT.ONE )
579 $ SCALE = SCALE / TEMP
580 IF( SCALE.LT.ONE )
581 $ ILIMIT = .FALSE.
582 END IF
583 *
584 * Recompute un-scaled ALPHA, BETA if necessary.
585 *
586 IF( ILIMIT ) THEN
587 SALFAR = ( SCALE*DBLE( ALPHA( JC ) ) )*ANRM
588 SALFAI = ( SCALE*DIMAG( ALPHA( JC ) ) )*ANRM
589 SBETA = ( SCALE*BETA( JC ) )*BNRM
590 END IF
591 ALPHA( JC ) = DCMPLX( SALFAR, SALFAI )
592 BETA( JC ) = SBETA
593 70 CONTINUE
594 *
595 80 CONTINUE
596 WORK( 1 ) = LWKOPT
597 *
598 RETURN
599 *
600 * End of ZGEGV
601 *
602 END
2 $ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
3 *
4 * -- LAPACK driver routine (version 3.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * November 2006
8 *
9 * .. Scalar Arguments ..
10 CHARACTER JOBVL, JOBVR
11 INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
12 * ..
13 * .. Array Arguments ..
14 DOUBLE PRECISION RWORK( * )
15 COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
16 $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
17 $ WORK( * )
18 * ..
19 *
20 * Purpose
21 * =======
22 *
23 * This routine is deprecated and has been replaced by routine ZGGEV.
24 *
25 * ZGEGV computes the eigenvalues and, optionally, the left and/or right
26 * eigenvectors of a complex matrix pair (A,B).
27 * Given two square matrices A and B,
28 * the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
29 * eigenvalues lambda and corresponding (non-zero) eigenvectors x such
30 * that
31 * A*x = lambda*B*x.
32 *
33 * An alternate form is to find the eigenvalues mu and corresponding
34 * eigenvectors y such that
35 * mu*A*y = B*y.
36 *
37 * These two forms are equivalent with mu = 1/lambda and x = y if
38 * neither lambda nor mu is zero. In order to deal with the case that
39 * lambda or mu is zero or small, two values alpha and beta are returned
40 * for each eigenvalue, such that lambda = alpha/beta and
41 * mu = beta/alpha.
42 *
43 * The vectors x and y in the above equations are right eigenvectors of
44 * the matrix pair (A,B). Vectors u and v satisfying
45 * u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
46 * are left eigenvectors of (A,B).
47 *
48 * Note: this routine performs "full balancing" on A and B -- see
49 * "Further Details", below.
50 *
51 * Arguments
52 * =========
53 *
54 * JOBVL (input) CHARACTER*1
55 * = 'N': do not compute the left generalized eigenvectors;
56 * = 'V': compute the left generalized eigenvectors (returned
57 * in VL).
58 *
59 * JOBVR (input) CHARACTER*1
60 * = 'N': do not compute the right generalized eigenvectors;
61 * = 'V': compute the right generalized eigenvectors (returned
62 * in VR).
63 *
64 * N (input) INTEGER
65 * The order of the matrices A, B, VL, and VR. N >= 0.
66 *
67 * A (input/output) COMPLEX*16 array, dimension (LDA, N)
68 * On entry, the matrix A.
69 * If JOBVL = 'V' or JOBVR = 'V', then on exit A
70 * contains the Schur form of A from the generalized Schur
71 * factorization of the pair (A,B) after balancing. If no
72 * eigenvectors were computed, then only the diagonal elements
73 * of the Schur form will be correct. See ZGGHRD and ZHGEQZ
74 * for details.
75 *
76 * LDA (input) INTEGER
77 * The leading dimension of A. LDA >= max(1,N).
78 *
79 * B (input/output) COMPLEX*16 array, dimension (LDB, N)
80 * On entry, the matrix B.
81 * If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
82 * upper triangular matrix obtained from B in the generalized
83 * Schur factorization of the pair (A,B) after balancing.
84 * If no eigenvectors were computed, then only the diagonal
85 * elements of B will be correct. See ZGGHRD and ZHGEQZ for
86 * details.
87 *
88 * LDB (input) INTEGER
89 * The leading dimension of B. LDB >= max(1,N).
90 *
91 * ALPHA (output) COMPLEX*16 array, dimension (N)
92 * The complex scalars alpha that define the eigenvalues of
93 * GNEP.
94 *
95 * BETA (output) COMPLEX*16 array, dimension (N)
96 * The complex scalars beta that define the eigenvalues of GNEP.
97 *
98 * Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
99 * represent the j-th eigenvalue of the matrix pair (A,B), in
100 * one of the forms lambda = alpha/beta or mu = beta/alpha.
101 * Since either lambda or mu may overflow, they should not,
102 * in general, be computed.
103 *
104 * VL (output) COMPLEX*16 array, dimension (LDVL,N)
105 * If JOBVL = 'V', the left eigenvectors u(j) are stored
106 * in the columns of VL, in the same order as their eigenvalues.
107 * Each eigenvector is scaled so that its largest component has
108 * abs(real part) + abs(imag. part) = 1, except for eigenvectors
109 * corresponding to an eigenvalue with alpha = beta = 0, which
110 * are set to zero.
111 * Not referenced if JOBVL = 'N'.
112 *
113 * LDVL (input) INTEGER
114 * The leading dimension of the matrix VL. LDVL >= 1, and
115 * if JOBVL = 'V', LDVL >= N.
116 *
117 * VR (output) COMPLEX*16 array, dimension (LDVR,N)
118 * If JOBVR = 'V', the right eigenvectors x(j) are stored
119 * in the columns of VR, in the same order as their eigenvalues.
120 * Each eigenvector is scaled so that its largest component has
121 * abs(real part) + abs(imag. part) = 1, except for eigenvectors
122 * corresponding to an eigenvalue with alpha = beta = 0, which
123 * are set to zero.
124 * Not referenced if JOBVR = 'N'.
125 *
126 * LDVR (input) INTEGER
127 * The leading dimension of the matrix VR. LDVR >= 1, and
128 * if JOBVR = 'V', LDVR >= N.
129 *
130 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
131 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
132 *
133 * LWORK (input) INTEGER
134 * The dimension of the array WORK. LWORK >= max(1,2*N).
135 * For good performance, LWORK must generally be larger.
136 * To compute the optimal value of LWORK, call ILAENV to get
137 * blocksizes (for ZGEQRF, ZUNMQR, and ZUNGQR.) Then compute:
138 * NB -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and ZUNGQR;
139 * The optimal LWORK is MAX( 2*N, N*(NB+1) ).
140 *
141 * If LWORK = -1, then a workspace query is assumed; the routine
142 * only calculates the optimal size of the WORK array, returns
143 * this value as the first entry of the WORK array, and no error
144 * message related to LWORK is issued by XERBLA.
145 *
146 * RWORK (workspace/output) DOUBLE PRECISION array, dimension (8*N)
147 *
148 * INFO (output) INTEGER
149 * = 0: successful exit
150 * < 0: if INFO = -i, the i-th argument had an illegal value.
151 * =1,...,N:
152 * The QZ iteration failed. No eigenvectors have been
153 * calculated, but ALPHA(j) and BETA(j) should be
154 * correct for j=INFO+1,...,N.
155 * > N: errors that usually indicate LAPACK problems:
156 * =N+1: error return from ZGGBAL
157 * =N+2: error return from ZGEQRF
158 * =N+3: error return from ZUNMQR
159 * =N+4: error return from ZUNGQR
160 * =N+5: error return from ZGGHRD
161 * =N+6: error return from ZHGEQZ (other than failed
162 * iteration)
163 * =N+7: error return from ZTGEVC
164 * =N+8: error return from ZGGBAK (computing VL)
165 * =N+9: error return from ZGGBAK (computing VR)
166 * =N+10: error return from ZLASCL (various calls)
167 *
168 * Further Details
169 * ===============
170 *
171 * Balancing
172 * ---------
173 *
174 * This driver calls ZGGBAL to both permute and scale rows and columns
175 * of A and B. The permutations PL and PR are chosen so that PL*A*PR
176 * and PL*B*R will be upper triangular except for the diagonal blocks
177 * A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
178 * possible. The diagonal scaling matrices DL and DR are chosen so
179 * that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
180 * one (except for the elements that start out zero.)
181 *
182 * After the eigenvalues and eigenvectors of the balanced matrices
183 * have been computed, ZGGBAK transforms the eigenvectors back to what
184 * they would have been (in perfect arithmetic) if they had not been
185 * balanced.
186 *
187 * Contents of A and B on Exit
188 * -------- -- - --- - -- ----
189 *
190 * If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
191 * both), then on exit the arrays A and B will contain the complex Schur
192 * form[*] of the "balanced" versions of A and B. If no eigenvectors
193 * are computed, then only the diagonal blocks will be correct.
194 *
195 * [*] In other words, upper triangular form.
196 *
197 * =====================================================================
198 *
199 * .. Parameters ..
200 DOUBLE PRECISION ZERO, ONE
201 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
202 COMPLEX*16 CZERO, CONE
203 PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ),
204 $ CONE = ( 1.0D0, 0.0D0 ) )
205 * ..
206 * .. Local Scalars ..
207 LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY
208 CHARACTER CHTEMP
209 INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
210 $ IN, IRIGHT, IROWS, IRWORK, ITAU, IWORK, JC, JR,
211 $ LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3
212 DOUBLE PRECISION ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
213 $ BNRM1, BNRM2, EPS, SAFMAX, SAFMIN, SALFAI,
214 $ SALFAR, SBETA, SCALE, TEMP
215 COMPLEX*16 X
216 * ..
217 * .. Local Arrays ..
218 LOGICAL LDUMMA( 1 )
219 * ..
220 * .. External Subroutines ..
221 EXTERNAL XERBLA, ZGEQRF, ZGGBAK, ZGGBAL, ZGGHRD, ZHGEQZ,
222 $ ZLACPY, ZLASCL, ZLASET, ZTGEVC, ZUNGQR, ZUNMQR
223 * ..
224 * .. External Functions ..
225 LOGICAL LSAME
226 INTEGER ILAENV
227 DOUBLE PRECISION DLAMCH, ZLANGE
228 EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE
229 * ..
230 * .. Intrinsic Functions ..
231 INTRINSIC ABS, DBLE, DCMPLX, DIMAG, INT, MAX
232 * ..
233 * .. Statement Functions ..
234 DOUBLE PRECISION ABS1
235 * ..
236 * .. Statement Function definitions ..
237 ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) )
238 * ..
239 * .. Executable Statements ..
240 *
241 * Decode the input arguments
242 *
243 IF( LSAME( JOBVL, 'N' ) ) THEN
244 IJOBVL = 1
245 ILVL = .FALSE.
246 ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
247 IJOBVL = 2
248 ILVL = .TRUE.
249 ELSE
250 IJOBVL = -1
251 ILVL = .FALSE.
252 END IF
253 *
254 IF( LSAME( JOBVR, 'N' ) ) THEN
255 IJOBVR = 1
256 ILVR = .FALSE.
257 ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
258 IJOBVR = 2
259 ILVR = .TRUE.
260 ELSE
261 IJOBVR = -1
262 ILVR = .FALSE.
263 END IF
264 ILV = ILVL .OR. ILVR
265 *
266 * Test the input arguments
267 *
268 LWKMIN = MAX( 2*N, 1 )
269 LWKOPT = LWKMIN
270 WORK( 1 ) = LWKOPT
271 LQUERY = ( LWORK.EQ.-1 )
272 INFO = 0
273 IF( IJOBVL.LE.0 ) THEN
274 INFO = -1
275 ELSE IF( IJOBVR.LE.0 ) THEN
276 INFO = -2
277 ELSE IF( N.LT.0 ) THEN
278 INFO = -3
279 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
280 INFO = -5
281 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
282 INFO = -7
283 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
284 INFO = -11
285 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
286 INFO = -13
287 ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
288 INFO = -15
289 END IF
290 *
291 IF( INFO.EQ.0 ) THEN
292 NB1 = ILAENV( 1, 'ZGEQRF', ' ', N, N, -1, -1 )
293 NB2 = ILAENV( 1, 'ZUNMQR', ' ', N, N, N, -1 )
294 NB3 = ILAENV( 1, 'ZUNGQR', ' ', N, N, N, -1 )
295 NB = MAX( NB1, NB2, NB3 )
296 LOPT = MAX( 2*N, N*( NB+1 ) )
297 WORK( 1 ) = LOPT
298 END IF
299 *
300 IF( INFO.NE.0 ) THEN
301 CALL XERBLA( 'ZGEGV ', -INFO )
302 RETURN
303 ELSE IF( LQUERY ) THEN
304 RETURN
305 END IF
306 *
307 * Quick return if possible
308 *
309 IF( N.EQ.0 )
310 $ RETURN
311 *
312 * Get machine constants
313 *
314 EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
315 SAFMIN = DLAMCH( 'S' )
316 SAFMIN = SAFMIN + SAFMIN
317 SAFMAX = ONE / SAFMIN
318 *
319 * Scale A
320 *
321 ANRM = ZLANGE( 'M', N, N, A, LDA, RWORK )
322 ANRM1 = ANRM
323 ANRM2 = ONE
324 IF( ANRM.LT.ONE ) THEN
325 IF( SAFMAX*ANRM.LT.ONE ) THEN
326 ANRM1 = SAFMIN
327 ANRM2 = SAFMAX*ANRM
328 END IF
329 END IF
330 *
331 IF( ANRM.GT.ZERO ) THEN
332 CALL ZLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
333 IF( IINFO.NE.0 ) THEN
334 INFO = N + 10
335 RETURN
336 END IF
337 END IF
338 *
339 * Scale B
340 *
341 BNRM = ZLANGE( 'M', N, N, B, LDB, RWORK )
342 BNRM1 = BNRM
343 BNRM2 = ONE
344 IF( BNRM.LT.ONE ) THEN
345 IF( SAFMAX*BNRM.LT.ONE ) THEN
346 BNRM1 = SAFMIN
347 BNRM2 = SAFMAX*BNRM
348 END IF
349 END IF
350 *
351 IF( BNRM.GT.ZERO ) THEN
352 CALL ZLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
353 IF( IINFO.NE.0 ) THEN
354 INFO = N + 10
355 RETURN
356 END IF
357 END IF
358 *
359 * Permute the matrix to make it more nearly triangular
360 * Also "balance" the matrix.
361 *
362 ILEFT = 1
363 IRIGHT = N + 1
364 IRWORK = IRIGHT + N
365 CALL ZGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ),
366 $ RWORK( IRIGHT ), RWORK( IRWORK ), IINFO )
367 IF( IINFO.NE.0 ) THEN
368 INFO = N + 1
369 GO TO 80
370 END IF
371 *
372 * Reduce B to triangular form, and initialize VL and/or VR
373 *
374 IROWS = IHI + 1 - ILO
375 IF( ILV ) THEN
376 ICOLS = N + 1 - ILO
377 ELSE
378 ICOLS = IROWS
379 END IF
380 ITAU = 1
381 IWORK = ITAU + IROWS
382 CALL ZGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
383 $ WORK( IWORK ), LWORK+1-IWORK, IINFO )
384 IF( IINFO.GE.0 )
385 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
386 IF( IINFO.NE.0 ) THEN
387 INFO = N + 2
388 GO TO 80
389 END IF
390 *
391 CALL ZUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
392 $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
393 $ LWORK+1-IWORK, IINFO )
394 IF( IINFO.GE.0 )
395 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
396 IF( IINFO.NE.0 ) THEN
397 INFO = N + 3
398 GO TO 80
399 END IF
400 *
401 IF( ILVL ) THEN
402 CALL ZLASET( 'Full', N, N, CZERO, CONE, VL, LDVL )
403 CALL ZLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
404 $ VL( ILO+1, ILO ), LDVL )
405 CALL ZUNGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
406 $ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
407 $ IINFO )
408 IF( IINFO.GE.0 )
409 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
410 IF( IINFO.NE.0 ) THEN
411 INFO = N + 4
412 GO TO 80
413 END IF
414 END IF
415 *
416 IF( ILVR )
417 $ CALL ZLASET( 'Full', N, N, CZERO, CONE, VR, LDVR )
418 *
419 * Reduce to generalized Hessenberg form
420 *
421 IF( ILV ) THEN
422 *
423 * Eigenvectors requested -- work on whole matrix.
424 *
425 CALL ZGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
426 $ LDVL, VR, LDVR, IINFO )
427 ELSE
428 CALL ZGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
429 $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
430 END IF
431 IF( IINFO.NE.0 ) THEN
432 INFO = N + 5
433 GO TO 80
434 END IF
435 *
436 * Perform QZ algorithm
437 *
438 IWORK = ITAU
439 IF( ILV ) THEN
440 CHTEMP = 'S'
441 ELSE
442 CHTEMP = 'E'
443 END IF
444 CALL ZHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
445 $ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK( IWORK ),
446 $ LWORK+1-IWORK, RWORK( IRWORK ), IINFO )
447 IF( IINFO.GE.0 )
448 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
449 IF( IINFO.NE.0 ) THEN
450 IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
451 INFO = IINFO
452 ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
453 INFO = IINFO - N
454 ELSE
455 INFO = N + 6
456 END IF
457 GO TO 80
458 END IF
459 *
460 IF( ILV ) THEN
461 *
462 * Compute Eigenvectors
463 *
464 IF( ILVL ) THEN
465 IF( ILVR ) THEN
466 CHTEMP = 'B'
467 ELSE
468 CHTEMP = 'L'
469 END IF
470 ELSE
471 CHTEMP = 'R'
472 END IF
473 *
474 CALL ZTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
475 $ VR, LDVR, N, IN, WORK( IWORK ), RWORK( IRWORK ),
476 $ IINFO )
477 IF( IINFO.NE.0 ) THEN
478 INFO = N + 7
479 GO TO 80
480 END IF
481 *
482 * Undo balancing on VL and VR, rescale
483 *
484 IF( ILVL ) THEN
485 CALL ZGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ),
486 $ RWORK( IRIGHT ), N, VL, LDVL, IINFO )
487 IF( IINFO.NE.0 ) THEN
488 INFO = N + 8
489 GO TO 80
490 END IF
491 DO 30 JC = 1, N
492 TEMP = ZERO
493 DO 10 JR = 1, N
494 TEMP = MAX( TEMP, ABS1( VL( JR, JC ) ) )
495 10 CONTINUE
496 IF( TEMP.LT.SAFMIN )
497 $ GO TO 30
498 TEMP = ONE / TEMP
499 DO 20 JR = 1, N
500 VL( JR, JC ) = VL( JR, JC )*TEMP
501 20 CONTINUE
502 30 CONTINUE
503 END IF
504 IF( ILVR ) THEN
505 CALL ZGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ),
506 $ RWORK( IRIGHT ), N, VR, LDVR, IINFO )
507 IF( IINFO.NE.0 ) THEN
508 INFO = N + 9
509 GO TO 80
510 END IF
511 DO 60 JC = 1, N
512 TEMP = ZERO
513 DO 40 JR = 1, N
514 TEMP = MAX( TEMP, ABS1( VR( JR, JC ) ) )
515 40 CONTINUE
516 IF( TEMP.LT.SAFMIN )
517 $ GO TO 60
518 TEMP = ONE / TEMP
519 DO 50 JR = 1, N
520 VR( JR, JC ) = VR( JR, JC )*TEMP
521 50 CONTINUE
522 60 CONTINUE
523 END IF
524 *
525 * End of eigenvector calculation
526 *
527 END IF
528 *
529 * Undo scaling in alpha, beta
530 *
531 * Note: this does not give the alpha and beta for the unscaled
532 * problem.
533 *
534 * Un-scaling is limited to avoid underflow in alpha and beta
535 * if they are significant.
536 *
537 DO 70 JC = 1, N
538 ABSAR = ABS( DBLE( ALPHA( JC ) ) )
539 ABSAI = ABS( DIMAG( ALPHA( JC ) ) )
540 ABSB = ABS( DBLE( BETA( JC ) ) )
541 SALFAR = ANRM*DBLE( ALPHA( JC ) )
542 SALFAI = ANRM*DIMAG( ALPHA( JC ) )
543 SBETA = BNRM*DBLE( BETA( JC ) )
544 ILIMIT = .FALSE.
545 SCALE = ONE
546 *
547 * Check for significant underflow in imaginary part of ALPHA
548 *
549 IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
550 $ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
551 ILIMIT = .TRUE.
552 SCALE = ( SAFMIN / ANRM1 ) / MAX( SAFMIN, ANRM2*ABSAI )
553 END IF
554 *
555 * Check for significant underflow in real part of ALPHA
556 *
557 IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
558 $ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
559 ILIMIT = .TRUE.
560 SCALE = MAX( SCALE, ( SAFMIN / ANRM1 ) /
561 $ MAX( SAFMIN, ANRM2*ABSAR ) )
562 END IF
563 *
564 * Check for significant underflow in BETA
565 *
566 IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
567 $ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
568 ILIMIT = .TRUE.
569 SCALE = MAX( SCALE, ( SAFMIN / BNRM1 ) /
570 $ MAX( SAFMIN, BNRM2*ABSB ) )
571 END IF
572 *
573 * Check for possible overflow when limiting scaling
574 *
575 IF( ILIMIT ) THEN
576 TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
577 $ ABS( SBETA ) )
578 IF( TEMP.GT.ONE )
579 $ SCALE = SCALE / TEMP
580 IF( SCALE.LT.ONE )
581 $ ILIMIT = .FALSE.
582 END IF
583 *
584 * Recompute un-scaled ALPHA, BETA if necessary.
585 *
586 IF( ILIMIT ) THEN
587 SALFAR = ( SCALE*DBLE( ALPHA( JC ) ) )*ANRM
588 SALFAI = ( SCALE*DIMAG( ALPHA( JC ) ) )*ANRM
589 SBETA = ( SCALE*BETA( JC ) )*BNRM
590 END IF
591 ALPHA( JC ) = DCMPLX( SALFAR, SALFAI )
592 BETA( JC ) = SBETA
593 70 CONTINUE
594 *
595 80 CONTINUE
596 WORK( 1 ) = LWKOPT
597 *
598 RETURN
599 *
600 * End of ZGEGV
601 *
602 END