1 SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
2 $ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, 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 A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
15 $ B( LDB, * ), BETA( * ), VL( LDVL, * ),
16 $ VR( LDVR, * ), WORK( * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * This routine is deprecated and has been replaced by routine DGGEV.
23 *
24 * DGEGV computes the eigenvalues and, optionally, the left and/or right
25 * eigenvectors of a real matrix pair (A,B).
26 * Given two square matrices A and B,
27 * the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
28 * eigenvalues lambda and corresponding (non-zero) eigenvectors x such
29 * that
30 *
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 *
36 * mu*A*y = B*y.
37 *
38 * These two forms are equivalent with mu = 1/lambda and x = y if
39 * neither lambda nor mu is zero. In order to deal with the case that
40 * lambda or mu is zero or small, two values alpha and beta are returned
41 * for each eigenvalue, such that lambda = alpha/beta and
42 * mu = beta/alpha.
43 *
44 * The vectors x and y in the above equations are right eigenvectors of
45 * the matrix pair (A,B). Vectors u and v satisfying
46 *
47 * u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
48 *
49 * are left eigenvectors of (A,B).
50 *
51 * Note: this routine performs "full balancing" on A and B -- see
52 * "Further Details", below.
53 *
54 * Arguments
55 * =========
56 *
57 * JOBVL (input) CHARACTER*1
58 * = 'N': do not compute the left generalized eigenvectors;
59 * = 'V': compute the left generalized eigenvectors (returned
60 * in VL).
61 *
62 * JOBVR (input) CHARACTER*1
63 * = 'N': do not compute the right generalized eigenvectors;
64 * = 'V': compute the right generalized eigenvectors (returned
65 * in VR).
66 *
67 * N (input) INTEGER
68 * The order of the matrices A, B, VL, and VR. N >= 0.
69 *
70 * A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
71 * On entry, the matrix A.
72 * If JOBVL = 'V' or JOBVR = 'V', then on exit A
73 * contains the real Schur form of A from the generalized Schur
74 * factorization of the pair (A,B) after balancing.
75 * If no eigenvectors were computed, then only the diagonal
76 * blocks from the Schur form will be correct. See DGGHRD and
77 * DHGEQZ for details.
78 *
79 * LDA (input) INTEGER
80 * The leading dimension of A. LDA >= max(1,N).
81 *
82 * B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
83 * On entry, the matrix B.
84 * If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
85 * upper triangular matrix obtained from B in the generalized
86 * Schur factorization of the pair (A,B) after balancing.
87 * If no eigenvectors were computed, then only those elements of
88 * B corresponding to the diagonal blocks from the Schur form of
89 * A will be correct. See DGGHRD and DHGEQZ for details.
90 *
91 * LDB (input) INTEGER
92 * The leading dimension of B. LDB >= max(1,N).
93 *
94 * ALPHAR (output) DOUBLE PRECISION array, dimension (N)
95 * The real parts of each scalar alpha defining an eigenvalue of
96 * GNEP.
97 *
98 * ALPHAI (output) DOUBLE PRECISION array, dimension (N)
99 * The imaginary parts of each scalar alpha defining an
100 * eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
101 * eigenvalue is real; if positive, then the j-th and
102 * (j+1)-st eigenvalues are a complex conjugate pair, with
103 * ALPHAI(j+1) = -ALPHAI(j).
104 *
105 * BETA (output) DOUBLE PRECISION array, dimension (N)
106 * The scalars beta that define the eigenvalues of GNEP.
107 *
108 * Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
109 * beta = BETA(j) represent the j-th eigenvalue of the matrix
110 * pair (A,B), in one of the forms lambda = alpha/beta or
111 * mu = beta/alpha. Since either lambda or mu may overflow,
112 * they should not, in general, be computed.
113 *
114 * VL (output) DOUBLE PRECISION array, dimension (LDVL,N)
115 * If JOBVL = 'V', the left eigenvectors u(j) are stored
116 * in the columns of VL, in the same order as their eigenvalues.
117 * If the j-th eigenvalue is real, then u(j) = VL(:,j).
118 * If the j-th and (j+1)-st eigenvalues form a complex conjugate
119 * pair, then
120 * u(j) = VL(:,j) + i*VL(:,j+1)
121 * and
122 * u(j+1) = VL(:,j) - i*VL(:,j+1).
123 *
124 * Each eigenvector is scaled so that its largest component has
125 * abs(real part) + abs(imag. part) = 1, except for eigenvectors
126 * corresponding to an eigenvalue with alpha = beta = 0, which
127 * are set to zero.
128 * Not referenced if JOBVL = 'N'.
129 *
130 * LDVL (input) INTEGER
131 * The leading dimension of the matrix VL. LDVL >= 1, and
132 * if JOBVL = 'V', LDVL >= N.
133 *
134 * VR (output) DOUBLE PRECISION array, dimension (LDVR,N)
135 * If JOBVR = 'V', the right eigenvectors x(j) are stored
136 * in the columns of VR, in the same order as their eigenvalues.
137 * If the j-th eigenvalue is real, then x(j) = VR(:,j).
138 * If the j-th and (j+1)-st eigenvalues form a complex conjugate
139 * pair, then
140 * x(j) = VR(:,j) + i*VR(:,j+1)
141 * and
142 * x(j+1) = VR(:,j) - i*VR(:,j+1).
143 *
144 * Each eigenvector is scaled so that its largest component has
145 * abs(real part) + abs(imag. part) = 1, except for eigenvalues
146 * corresponding to an eigenvalue with alpha = beta = 0, which
147 * are set to zero.
148 * Not referenced if JOBVR = 'N'.
149 *
150 * LDVR (input) INTEGER
151 * The leading dimension of the matrix VR. LDVR >= 1, and
152 * if JOBVR = 'V', LDVR >= N.
153 *
154 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
155 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
156 *
157 * LWORK (input) INTEGER
158 * The dimension of the array WORK. LWORK >= max(1,8*N).
159 * For good performance, LWORK must generally be larger.
160 * To compute the optimal value of LWORK, call ILAENV to get
161 * blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute:
162 * NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR;
163 * The optimal LWORK is:
164 * 2*N + MAX( 6*N, N*(NB+1) ).
165 *
166 * If LWORK = -1, then a workspace query is assumed; the routine
167 * only calculates the optimal size of the WORK array, returns
168 * this value as the first entry of the WORK array, and no error
169 * message related to LWORK is issued by XERBLA.
170 *
171 * INFO (output) INTEGER
172 * = 0: successful exit
173 * < 0: if INFO = -i, the i-th argument had an illegal value.
174 * = 1,...,N:
175 * The QZ iteration failed. No eigenvectors have been
176 * calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
177 * should be correct for j=INFO+1,...,N.
178 * > N: errors that usually indicate LAPACK problems:
179 * =N+1: error return from DGGBAL
180 * =N+2: error return from DGEQRF
181 * =N+3: error return from DORMQR
182 * =N+4: error return from DORGQR
183 * =N+5: error return from DGGHRD
184 * =N+6: error return from DHGEQZ (other than failed
185 * iteration)
186 * =N+7: error return from DTGEVC
187 * =N+8: error return from DGGBAK (computing VL)
188 * =N+9: error return from DGGBAK (computing VR)
189 * =N+10: error return from DLASCL (various calls)
190 *
191 * Further Details
192 * ===============
193 *
194 * Balancing
195 * ---------
196 *
197 * This driver calls DGGBAL to both permute and scale rows and columns
198 * of A and B. The permutations PL and PR are chosen so that PL*A*PR
199 * and PL*B*R will be upper triangular except for the diagonal blocks
200 * A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
201 * possible. The diagonal scaling matrices DL and DR are chosen so
202 * that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
203 * one (except for the elements that start out zero.)
204 *
205 * After the eigenvalues and eigenvectors of the balanced matrices
206 * have been computed, DGGBAK transforms the eigenvectors back to what
207 * they would have been (in perfect arithmetic) if they had not been
208 * balanced.
209 *
210 * Contents of A and B on Exit
211 * -------- -- - --- - -- ----
212 *
213 * If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
214 * both), then on exit the arrays A and B will contain the real Schur
215 * form[*] of the "balanced" versions of A and B. If no eigenvectors
216 * are computed, then only the diagonal blocks will be correct.
217 *
218 * [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations",
219 * by Golub & van Loan, pub. by Johns Hopkins U. Press.
220 *
221 * =====================================================================
222 *
223 * .. Parameters ..
224 DOUBLE PRECISION ZERO, ONE
225 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
226 * ..
227 * .. Local Scalars ..
228 LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY
229 CHARACTER CHTEMP
230 INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
231 $ IN, IRIGHT, IROWS, ITAU, IWORK, JC, JR, LOPT,
232 $ LWKMIN, LWKOPT, NB, NB1, NB2, NB3
233 DOUBLE PRECISION ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
234 $ BNRM1, BNRM2, EPS, ONEPLS, SAFMAX, SAFMIN,
235 $ SALFAI, SALFAR, SBETA, SCALE, TEMP
236 * ..
237 * .. Local Arrays ..
238 LOGICAL LDUMMA( 1 )
239 * ..
240 * .. External Subroutines ..
241 EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY,
242 $ DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, XERBLA
243 * ..
244 * .. External Functions ..
245 LOGICAL LSAME
246 INTEGER ILAENV
247 DOUBLE PRECISION DLAMCH, DLANGE
248 EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
249 * ..
250 * .. Intrinsic Functions ..
251 INTRINSIC ABS, INT, MAX
252 * ..
253 * .. Executable Statements ..
254 *
255 * Decode the input arguments
256 *
257 IF( LSAME( JOBVL, 'N' ) ) THEN
258 IJOBVL = 1
259 ILVL = .FALSE.
260 ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
261 IJOBVL = 2
262 ILVL = .TRUE.
263 ELSE
264 IJOBVL = -1
265 ILVL = .FALSE.
266 END IF
267 *
268 IF( LSAME( JOBVR, 'N' ) ) THEN
269 IJOBVR = 1
270 ILVR = .FALSE.
271 ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
272 IJOBVR = 2
273 ILVR = .TRUE.
274 ELSE
275 IJOBVR = -1
276 ILVR = .FALSE.
277 END IF
278 ILV = ILVL .OR. ILVR
279 *
280 * Test the input arguments
281 *
282 LWKMIN = MAX( 8*N, 1 )
283 LWKOPT = LWKMIN
284 WORK( 1 ) = LWKOPT
285 LQUERY = ( LWORK.EQ.-1 )
286 INFO = 0
287 IF( IJOBVL.LE.0 ) THEN
288 INFO = -1
289 ELSE IF( IJOBVR.LE.0 ) THEN
290 INFO = -2
291 ELSE IF( N.LT.0 ) THEN
292 INFO = -3
293 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
294 INFO = -5
295 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
296 INFO = -7
297 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
298 INFO = -12
299 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
300 INFO = -14
301 ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
302 INFO = -16
303 END IF
304 *
305 IF( INFO.EQ.0 ) THEN
306 NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 )
307 NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 )
308 NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 )
309 NB = MAX( NB1, NB2, NB3 )
310 LOPT = 2*N + MAX( 6*N, N*( NB+1 ) )
311 WORK( 1 ) = LOPT
312 END IF
313 *
314 IF( INFO.NE.0 ) THEN
315 CALL XERBLA( 'DGEGV ', -INFO )
316 RETURN
317 ELSE IF( LQUERY ) THEN
318 RETURN
319 END IF
320 *
321 * Quick return if possible
322 *
323 IF( N.EQ.0 )
324 $ RETURN
325 *
326 * Get machine constants
327 *
328 EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
329 SAFMIN = DLAMCH( 'S' )
330 SAFMIN = SAFMIN + SAFMIN
331 SAFMAX = ONE / SAFMIN
332 ONEPLS = ONE + ( 4*EPS )
333 *
334 * Scale A
335 *
336 ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
337 ANRM1 = ANRM
338 ANRM2 = ONE
339 IF( ANRM.LT.ONE ) THEN
340 IF( SAFMAX*ANRM.LT.ONE ) THEN
341 ANRM1 = SAFMIN
342 ANRM2 = SAFMAX*ANRM
343 END IF
344 END IF
345 *
346 IF( ANRM.GT.ZERO ) THEN
347 CALL DLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
348 IF( IINFO.NE.0 ) THEN
349 INFO = N + 10
350 RETURN
351 END IF
352 END IF
353 *
354 * Scale B
355 *
356 BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
357 BNRM1 = BNRM
358 BNRM2 = ONE
359 IF( BNRM.LT.ONE ) THEN
360 IF( SAFMAX*BNRM.LT.ONE ) THEN
361 BNRM1 = SAFMIN
362 BNRM2 = SAFMAX*BNRM
363 END IF
364 END IF
365 *
366 IF( BNRM.GT.ZERO ) THEN
367 CALL DLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
368 IF( IINFO.NE.0 ) THEN
369 INFO = N + 10
370 RETURN
371 END IF
372 END IF
373 *
374 * Permute the matrix to make it more nearly triangular
375 * Workspace layout: (8*N words -- "work" requires 6*N words)
376 * left_permutation, right_permutation, work...
377 *
378 ILEFT = 1
379 IRIGHT = N + 1
380 IWORK = IRIGHT + N
381 CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
382 $ WORK( IRIGHT ), WORK( IWORK ), IINFO )
383 IF( IINFO.NE.0 ) THEN
384 INFO = N + 1
385 GO TO 120
386 END IF
387 *
388 * Reduce B to triangular form, and initialize VL and/or VR
389 * Workspace layout: ("work..." must have at least N words)
390 * left_permutation, right_permutation, tau, work...
391 *
392 IROWS = IHI + 1 - ILO
393 IF( ILV ) THEN
394 ICOLS = N + 1 - ILO
395 ELSE
396 ICOLS = IROWS
397 END IF
398 ITAU = IWORK
399 IWORK = ITAU + IROWS
400 CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
401 $ WORK( IWORK ), LWORK+1-IWORK, IINFO )
402 IF( IINFO.GE.0 )
403 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
404 IF( IINFO.NE.0 ) THEN
405 INFO = N + 2
406 GO TO 120
407 END IF
408 *
409 CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
410 $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
411 $ LWORK+1-IWORK, IINFO )
412 IF( IINFO.GE.0 )
413 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
414 IF( IINFO.NE.0 ) THEN
415 INFO = N + 3
416 GO TO 120
417 END IF
418 *
419 IF( ILVL ) THEN
420 CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
421 CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
422 $ VL( ILO+1, ILO ), LDVL )
423 CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
424 $ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
425 $ IINFO )
426 IF( IINFO.GE.0 )
427 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
428 IF( IINFO.NE.0 ) THEN
429 INFO = N + 4
430 GO TO 120
431 END IF
432 END IF
433 *
434 IF( ILVR )
435 $ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
436 *
437 * Reduce to generalized Hessenberg form
438 *
439 IF( ILV ) THEN
440 *
441 * Eigenvectors requested -- work on whole matrix.
442 *
443 CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
444 $ LDVL, VR, LDVR, IINFO )
445 ELSE
446 CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
447 $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
448 END IF
449 IF( IINFO.NE.0 ) THEN
450 INFO = N + 5
451 GO TO 120
452 END IF
453 *
454 * Perform QZ algorithm
455 * Workspace layout: ("work..." must have at least 1 word)
456 * left_permutation, right_permutation, work...
457 *
458 IWORK = ITAU
459 IF( ILV ) THEN
460 CHTEMP = 'S'
461 ELSE
462 CHTEMP = 'E'
463 END IF
464 CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
465 $ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
466 $ WORK( IWORK ), LWORK+1-IWORK, IINFO )
467 IF( IINFO.GE.0 )
468 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
469 IF( IINFO.NE.0 ) THEN
470 IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
471 INFO = IINFO
472 ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
473 INFO = IINFO - N
474 ELSE
475 INFO = N + 6
476 END IF
477 GO TO 120
478 END IF
479 *
480 IF( ILV ) THEN
481 *
482 * Compute Eigenvectors (DTGEVC requires 6*N words of workspace)
483 *
484 IF( ILVL ) THEN
485 IF( ILVR ) THEN
486 CHTEMP = 'B'
487 ELSE
488 CHTEMP = 'L'
489 END IF
490 ELSE
491 CHTEMP = 'R'
492 END IF
493 *
494 CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
495 $ VR, LDVR, N, IN, WORK( IWORK ), IINFO )
496 IF( IINFO.NE.0 ) THEN
497 INFO = N + 7
498 GO TO 120
499 END IF
500 *
501 * Undo balancing on VL and VR, rescale
502 *
503 IF( ILVL ) THEN
504 CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
505 $ WORK( IRIGHT ), N, VL, LDVL, IINFO )
506 IF( IINFO.NE.0 ) THEN
507 INFO = N + 8
508 GO TO 120
509 END IF
510 DO 50 JC = 1, N
511 IF( ALPHAI( JC ).LT.ZERO )
512 $ GO TO 50
513 TEMP = ZERO
514 IF( ALPHAI( JC ).EQ.ZERO ) THEN
515 DO 10 JR = 1, N
516 TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
517 10 CONTINUE
518 ELSE
519 DO 20 JR = 1, N
520 TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
521 $ ABS( VL( JR, JC+1 ) ) )
522 20 CONTINUE
523 END IF
524 IF( TEMP.LT.SAFMIN )
525 $ GO TO 50
526 TEMP = ONE / TEMP
527 IF( ALPHAI( JC ).EQ.ZERO ) THEN
528 DO 30 JR = 1, N
529 VL( JR, JC ) = VL( JR, JC )*TEMP
530 30 CONTINUE
531 ELSE
532 DO 40 JR = 1, N
533 VL( JR, JC ) = VL( JR, JC )*TEMP
534 VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
535 40 CONTINUE
536 END IF
537 50 CONTINUE
538 END IF
539 IF( ILVR ) THEN
540 CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
541 $ WORK( IRIGHT ), N, VR, LDVR, IINFO )
542 IF( IINFO.NE.0 ) THEN
543 INFO = N + 9
544 GO TO 120
545 END IF
546 DO 100 JC = 1, N
547 IF( ALPHAI( JC ).LT.ZERO )
548 $ GO TO 100
549 TEMP = ZERO
550 IF( ALPHAI( JC ).EQ.ZERO ) THEN
551 DO 60 JR = 1, N
552 TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
553 60 CONTINUE
554 ELSE
555 DO 70 JR = 1, N
556 TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
557 $ ABS( VR( JR, JC+1 ) ) )
558 70 CONTINUE
559 END IF
560 IF( TEMP.LT.SAFMIN )
561 $ GO TO 100
562 TEMP = ONE / TEMP
563 IF( ALPHAI( JC ).EQ.ZERO ) THEN
564 DO 80 JR = 1, N
565 VR( JR, JC ) = VR( JR, JC )*TEMP
566 80 CONTINUE
567 ELSE
568 DO 90 JR = 1, N
569 VR( JR, JC ) = VR( JR, JC )*TEMP
570 VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
571 90 CONTINUE
572 END IF
573 100 CONTINUE
574 END IF
575 *
576 * End of eigenvector calculation
577 *
578 END IF
579 *
580 * Undo scaling in alpha, beta
581 *
582 * Note: this does not give the alpha and beta for the unscaled
583 * problem.
584 *
585 * Un-scaling is limited to avoid underflow in alpha and beta
586 * if they are significant.
587 *
588 DO 110 JC = 1, N
589 ABSAR = ABS( ALPHAR( JC ) )
590 ABSAI = ABS( ALPHAI( JC ) )
591 ABSB = ABS( BETA( JC ) )
592 SALFAR = ANRM*ALPHAR( JC )
593 SALFAI = ANRM*ALPHAI( JC )
594 SBETA = BNRM*BETA( JC )
595 ILIMIT = .FALSE.
596 SCALE = ONE
597 *
598 * Check for significant underflow in ALPHAI
599 *
600 IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
601 $ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
602 ILIMIT = .TRUE.
603 SCALE = ( ONEPLS*SAFMIN / ANRM1 ) /
604 $ MAX( ONEPLS*SAFMIN, ANRM2*ABSAI )
605 *
606 ELSE IF( SALFAI.EQ.ZERO ) THEN
607 *
608 * If insignificant underflow in ALPHAI, then make the
609 * conjugate eigenvalue real.
610 *
611 IF( ALPHAI( JC ).LT.ZERO .AND. JC.GT.1 ) THEN
612 ALPHAI( JC-1 ) = ZERO
613 ELSE IF( ALPHAI( JC ).GT.ZERO .AND. JC.LT.N ) THEN
614 ALPHAI( JC+1 ) = ZERO
615 END IF
616 END IF
617 *
618 * Check for significant underflow in ALPHAR
619 *
620 IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
621 $ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
622 ILIMIT = .TRUE.
623 SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / ANRM1 ) /
624 $ MAX( ONEPLS*SAFMIN, ANRM2*ABSAR ) )
625 END IF
626 *
627 * Check for significant underflow in BETA
628 *
629 IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
630 $ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
631 ILIMIT = .TRUE.
632 SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / BNRM1 ) /
633 $ MAX( ONEPLS*SAFMIN, BNRM2*ABSB ) )
634 END IF
635 *
636 * Check for possible overflow when limiting scaling
637 *
638 IF( ILIMIT ) THEN
639 TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
640 $ ABS( SBETA ) )
641 IF( TEMP.GT.ONE )
642 $ SCALE = SCALE / TEMP
643 IF( SCALE.LT.ONE )
644 $ ILIMIT = .FALSE.
645 END IF
646 *
647 * Recompute un-scaled ALPHAR, ALPHAI, BETA if necessary.
648 *
649 IF( ILIMIT ) THEN
650 SALFAR = ( SCALE*ALPHAR( JC ) )*ANRM
651 SALFAI = ( SCALE*ALPHAI( JC ) )*ANRM
652 SBETA = ( SCALE*BETA( JC ) )*BNRM
653 END IF
654 ALPHAR( JC ) = SALFAR
655 ALPHAI( JC ) = SALFAI
656 BETA( JC ) = SBETA
657 110 CONTINUE
658 *
659 120 CONTINUE
660 WORK( 1 ) = LWKOPT
661 *
662 RETURN
663 *
664 * End of DGEGV
665 *
666 END
2 $ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, 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 A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
15 $ B( LDB, * ), BETA( * ), VL( LDVL, * ),
16 $ VR( LDVR, * ), WORK( * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * This routine is deprecated and has been replaced by routine DGGEV.
23 *
24 * DGEGV computes the eigenvalues and, optionally, the left and/or right
25 * eigenvectors of a real matrix pair (A,B).
26 * Given two square matrices A and B,
27 * the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
28 * eigenvalues lambda and corresponding (non-zero) eigenvectors x such
29 * that
30 *
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 *
36 * mu*A*y = B*y.
37 *
38 * These two forms are equivalent with mu = 1/lambda and x = y if
39 * neither lambda nor mu is zero. In order to deal with the case that
40 * lambda or mu is zero or small, two values alpha and beta are returned
41 * for each eigenvalue, such that lambda = alpha/beta and
42 * mu = beta/alpha.
43 *
44 * The vectors x and y in the above equations are right eigenvectors of
45 * the matrix pair (A,B). Vectors u and v satisfying
46 *
47 * u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
48 *
49 * are left eigenvectors of (A,B).
50 *
51 * Note: this routine performs "full balancing" on A and B -- see
52 * "Further Details", below.
53 *
54 * Arguments
55 * =========
56 *
57 * JOBVL (input) CHARACTER*1
58 * = 'N': do not compute the left generalized eigenvectors;
59 * = 'V': compute the left generalized eigenvectors (returned
60 * in VL).
61 *
62 * JOBVR (input) CHARACTER*1
63 * = 'N': do not compute the right generalized eigenvectors;
64 * = 'V': compute the right generalized eigenvectors (returned
65 * in VR).
66 *
67 * N (input) INTEGER
68 * The order of the matrices A, B, VL, and VR. N >= 0.
69 *
70 * A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
71 * On entry, the matrix A.
72 * If JOBVL = 'V' or JOBVR = 'V', then on exit A
73 * contains the real Schur form of A from the generalized Schur
74 * factorization of the pair (A,B) after balancing.
75 * If no eigenvectors were computed, then only the diagonal
76 * blocks from the Schur form will be correct. See DGGHRD and
77 * DHGEQZ for details.
78 *
79 * LDA (input) INTEGER
80 * The leading dimension of A. LDA >= max(1,N).
81 *
82 * B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
83 * On entry, the matrix B.
84 * If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
85 * upper triangular matrix obtained from B in the generalized
86 * Schur factorization of the pair (A,B) after balancing.
87 * If no eigenvectors were computed, then only those elements of
88 * B corresponding to the diagonal blocks from the Schur form of
89 * A will be correct. See DGGHRD and DHGEQZ for details.
90 *
91 * LDB (input) INTEGER
92 * The leading dimension of B. LDB >= max(1,N).
93 *
94 * ALPHAR (output) DOUBLE PRECISION array, dimension (N)
95 * The real parts of each scalar alpha defining an eigenvalue of
96 * GNEP.
97 *
98 * ALPHAI (output) DOUBLE PRECISION array, dimension (N)
99 * The imaginary parts of each scalar alpha defining an
100 * eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
101 * eigenvalue is real; if positive, then the j-th and
102 * (j+1)-st eigenvalues are a complex conjugate pair, with
103 * ALPHAI(j+1) = -ALPHAI(j).
104 *
105 * BETA (output) DOUBLE PRECISION array, dimension (N)
106 * The scalars beta that define the eigenvalues of GNEP.
107 *
108 * Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
109 * beta = BETA(j) represent the j-th eigenvalue of the matrix
110 * pair (A,B), in one of the forms lambda = alpha/beta or
111 * mu = beta/alpha. Since either lambda or mu may overflow,
112 * they should not, in general, be computed.
113 *
114 * VL (output) DOUBLE PRECISION array, dimension (LDVL,N)
115 * If JOBVL = 'V', the left eigenvectors u(j) are stored
116 * in the columns of VL, in the same order as their eigenvalues.
117 * If the j-th eigenvalue is real, then u(j) = VL(:,j).
118 * If the j-th and (j+1)-st eigenvalues form a complex conjugate
119 * pair, then
120 * u(j) = VL(:,j) + i*VL(:,j+1)
121 * and
122 * u(j+1) = VL(:,j) - i*VL(:,j+1).
123 *
124 * Each eigenvector is scaled so that its largest component has
125 * abs(real part) + abs(imag. part) = 1, except for eigenvectors
126 * corresponding to an eigenvalue with alpha = beta = 0, which
127 * are set to zero.
128 * Not referenced if JOBVL = 'N'.
129 *
130 * LDVL (input) INTEGER
131 * The leading dimension of the matrix VL. LDVL >= 1, and
132 * if JOBVL = 'V', LDVL >= N.
133 *
134 * VR (output) DOUBLE PRECISION array, dimension (LDVR,N)
135 * If JOBVR = 'V', the right eigenvectors x(j) are stored
136 * in the columns of VR, in the same order as their eigenvalues.
137 * If the j-th eigenvalue is real, then x(j) = VR(:,j).
138 * If the j-th and (j+1)-st eigenvalues form a complex conjugate
139 * pair, then
140 * x(j) = VR(:,j) + i*VR(:,j+1)
141 * and
142 * x(j+1) = VR(:,j) - i*VR(:,j+1).
143 *
144 * Each eigenvector is scaled so that its largest component has
145 * abs(real part) + abs(imag. part) = 1, except for eigenvalues
146 * corresponding to an eigenvalue with alpha = beta = 0, which
147 * are set to zero.
148 * Not referenced if JOBVR = 'N'.
149 *
150 * LDVR (input) INTEGER
151 * The leading dimension of the matrix VR. LDVR >= 1, and
152 * if JOBVR = 'V', LDVR >= N.
153 *
154 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
155 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
156 *
157 * LWORK (input) INTEGER
158 * The dimension of the array WORK. LWORK >= max(1,8*N).
159 * For good performance, LWORK must generally be larger.
160 * To compute the optimal value of LWORK, call ILAENV to get
161 * blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute:
162 * NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR;
163 * The optimal LWORK is:
164 * 2*N + MAX( 6*N, N*(NB+1) ).
165 *
166 * If LWORK = -1, then a workspace query is assumed; the routine
167 * only calculates the optimal size of the WORK array, returns
168 * this value as the first entry of the WORK array, and no error
169 * message related to LWORK is issued by XERBLA.
170 *
171 * INFO (output) INTEGER
172 * = 0: successful exit
173 * < 0: if INFO = -i, the i-th argument had an illegal value.
174 * = 1,...,N:
175 * The QZ iteration failed. No eigenvectors have been
176 * calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
177 * should be correct for j=INFO+1,...,N.
178 * > N: errors that usually indicate LAPACK problems:
179 * =N+1: error return from DGGBAL
180 * =N+2: error return from DGEQRF
181 * =N+3: error return from DORMQR
182 * =N+4: error return from DORGQR
183 * =N+5: error return from DGGHRD
184 * =N+6: error return from DHGEQZ (other than failed
185 * iteration)
186 * =N+7: error return from DTGEVC
187 * =N+8: error return from DGGBAK (computing VL)
188 * =N+9: error return from DGGBAK (computing VR)
189 * =N+10: error return from DLASCL (various calls)
190 *
191 * Further Details
192 * ===============
193 *
194 * Balancing
195 * ---------
196 *
197 * This driver calls DGGBAL to both permute and scale rows and columns
198 * of A and B. The permutations PL and PR are chosen so that PL*A*PR
199 * and PL*B*R will be upper triangular except for the diagonal blocks
200 * A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
201 * possible. The diagonal scaling matrices DL and DR are chosen so
202 * that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
203 * one (except for the elements that start out zero.)
204 *
205 * After the eigenvalues and eigenvectors of the balanced matrices
206 * have been computed, DGGBAK transforms the eigenvectors back to what
207 * they would have been (in perfect arithmetic) if they had not been
208 * balanced.
209 *
210 * Contents of A and B on Exit
211 * -------- -- - --- - -- ----
212 *
213 * If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
214 * both), then on exit the arrays A and B will contain the real Schur
215 * form[*] of the "balanced" versions of A and B. If no eigenvectors
216 * are computed, then only the diagonal blocks will be correct.
217 *
218 * [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations",
219 * by Golub & van Loan, pub. by Johns Hopkins U. Press.
220 *
221 * =====================================================================
222 *
223 * .. Parameters ..
224 DOUBLE PRECISION ZERO, ONE
225 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
226 * ..
227 * .. Local Scalars ..
228 LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY
229 CHARACTER CHTEMP
230 INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
231 $ IN, IRIGHT, IROWS, ITAU, IWORK, JC, JR, LOPT,
232 $ LWKMIN, LWKOPT, NB, NB1, NB2, NB3
233 DOUBLE PRECISION ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
234 $ BNRM1, BNRM2, EPS, ONEPLS, SAFMAX, SAFMIN,
235 $ SALFAI, SALFAR, SBETA, SCALE, TEMP
236 * ..
237 * .. Local Arrays ..
238 LOGICAL LDUMMA( 1 )
239 * ..
240 * .. External Subroutines ..
241 EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY,
242 $ DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, XERBLA
243 * ..
244 * .. External Functions ..
245 LOGICAL LSAME
246 INTEGER ILAENV
247 DOUBLE PRECISION DLAMCH, DLANGE
248 EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
249 * ..
250 * .. Intrinsic Functions ..
251 INTRINSIC ABS, INT, MAX
252 * ..
253 * .. Executable Statements ..
254 *
255 * Decode the input arguments
256 *
257 IF( LSAME( JOBVL, 'N' ) ) THEN
258 IJOBVL = 1
259 ILVL = .FALSE.
260 ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
261 IJOBVL = 2
262 ILVL = .TRUE.
263 ELSE
264 IJOBVL = -1
265 ILVL = .FALSE.
266 END IF
267 *
268 IF( LSAME( JOBVR, 'N' ) ) THEN
269 IJOBVR = 1
270 ILVR = .FALSE.
271 ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
272 IJOBVR = 2
273 ILVR = .TRUE.
274 ELSE
275 IJOBVR = -1
276 ILVR = .FALSE.
277 END IF
278 ILV = ILVL .OR. ILVR
279 *
280 * Test the input arguments
281 *
282 LWKMIN = MAX( 8*N, 1 )
283 LWKOPT = LWKMIN
284 WORK( 1 ) = LWKOPT
285 LQUERY = ( LWORK.EQ.-1 )
286 INFO = 0
287 IF( IJOBVL.LE.0 ) THEN
288 INFO = -1
289 ELSE IF( IJOBVR.LE.0 ) THEN
290 INFO = -2
291 ELSE IF( N.LT.0 ) THEN
292 INFO = -3
293 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
294 INFO = -5
295 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
296 INFO = -7
297 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
298 INFO = -12
299 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
300 INFO = -14
301 ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
302 INFO = -16
303 END IF
304 *
305 IF( INFO.EQ.0 ) THEN
306 NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 )
307 NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 )
308 NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 )
309 NB = MAX( NB1, NB2, NB3 )
310 LOPT = 2*N + MAX( 6*N, N*( NB+1 ) )
311 WORK( 1 ) = LOPT
312 END IF
313 *
314 IF( INFO.NE.0 ) THEN
315 CALL XERBLA( 'DGEGV ', -INFO )
316 RETURN
317 ELSE IF( LQUERY ) THEN
318 RETURN
319 END IF
320 *
321 * Quick return if possible
322 *
323 IF( N.EQ.0 )
324 $ RETURN
325 *
326 * Get machine constants
327 *
328 EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
329 SAFMIN = DLAMCH( 'S' )
330 SAFMIN = SAFMIN + SAFMIN
331 SAFMAX = ONE / SAFMIN
332 ONEPLS = ONE + ( 4*EPS )
333 *
334 * Scale A
335 *
336 ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
337 ANRM1 = ANRM
338 ANRM2 = ONE
339 IF( ANRM.LT.ONE ) THEN
340 IF( SAFMAX*ANRM.LT.ONE ) THEN
341 ANRM1 = SAFMIN
342 ANRM2 = SAFMAX*ANRM
343 END IF
344 END IF
345 *
346 IF( ANRM.GT.ZERO ) THEN
347 CALL DLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
348 IF( IINFO.NE.0 ) THEN
349 INFO = N + 10
350 RETURN
351 END IF
352 END IF
353 *
354 * Scale B
355 *
356 BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
357 BNRM1 = BNRM
358 BNRM2 = ONE
359 IF( BNRM.LT.ONE ) THEN
360 IF( SAFMAX*BNRM.LT.ONE ) THEN
361 BNRM1 = SAFMIN
362 BNRM2 = SAFMAX*BNRM
363 END IF
364 END IF
365 *
366 IF( BNRM.GT.ZERO ) THEN
367 CALL DLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
368 IF( IINFO.NE.0 ) THEN
369 INFO = N + 10
370 RETURN
371 END IF
372 END IF
373 *
374 * Permute the matrix to make it more nearly triangular
375 * Workspace layout: (8*N words -- "work" requires 6*N words)
376 * left_permutation, right_permutation, work...
377 *
378 ILEFT = 1
379 IRIGHT = N + 1
380 IWORK = IRIGHT + N
381 CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
382 $ WORK( IRIGHT ), WORK( IWORK ), IINFO )
383 IF( IINFO.NE.0 ) THEN
384 INFO = N + 1
385 GO TO 120
386 END IF
387 *
388 * Reduce B to triangular form, and initialize VL and/or VR
389 * Workspace layout: ("work..." must have at least N words)
390 * left_permutation, right_permutation, tau, work...
391 *
392 IROWS = IHI + 1 - ILO
393 IF( ILV ) THEN
394 ICOLS = N + 1 - ILO
395 ELSE
396 ICOLS = IROWS
397 END IF
398 ITAU = IWORK
399 IWORK = ITAU + IROWS
400 CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
401 $ WORK( IWORK ), LWORK+1-IWORK, IINFO )
402 IF( IINFO.GE.0 )
403 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
404 IF( IINFO.NE.0 ) THEN
405 INFO = N + 2
406 GO TO 120
407 END IF
408 *
409 CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
410 $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
411 $ LWORK+1-IWORK, IINFO )
412 IF( IINFO.GE.0 )
413 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
414 IF( IINFO.NE.0 ) THEN
415 INFO = N + 3
416 GO TO 120
417 END IF
418 *
419 IF( ILVL ) THEN
420 CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
421 CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
422 $ VL( ILO+1, ILO ), LDVL )
423 CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
424 $ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
425 $ IINFO )
426 IF( IINFO.GE.0 )
427 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
428 IF( IINFO.NE.0 ) THEN
429 INFO = N + 4
430 GO TO 120
431 END IF
432 END IF
433 *
434 IF( ILVR )
435 $ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
436 *
437 * Reduce to generalized Hessenberg form
438 *
439 IF( ILV ) THEN
440 *
441 * Eigenvectors requested -- work on whole matrix.
442 *
443 CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
444 $ LDVL, VR, LDVR, IINFO )
445 ELSE
446 CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
447 $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
448 END IF
449 IF( IINFO.NE.0 ) THEN
450 INFO = N + 5
451 GO TO 120
452 END IF
453 *
454 * Perform QZ algorithm
455 * Workspace layout: ("work..." must have at least 1 word)
456 * left_permutation, right_permutation, work...
457 *
458 IWORK = ITAU
459 IF( ILV ) THEN
460 CHTEMP = 'S'
461 ELSE
462 CHTEMP = 'E'
463 END IF
464 CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
465 $ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
466 $ WORK( IWORK ), LWORK+1-IWORK, IINFO )
467 IF( IINFO.GE.0 )
468 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
469 IF( IINFO.NE.0 ) THEN
470 IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
471 INFO = IINFO
472 ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
473 INFO = IINFO - N
474 ELSE
475 INFO = N + 6
476 END IF
477 GO TO 120
478 END IF
479 *
480 IF( ILV ) THEN
481 *
482 * Compute Eigenvectors (DTGEVC requires 6*N words of workspace)
483 *
484 IF( ILVL ) THEN
485 IF( ILVR ) THEN
486 CHTEMP = 'B'
487 ELSE
488 CHTEMP = 'L'
489 END IF
490 ELSE
491 CHTEMP = 'R'
492 END IF
493 *
494 CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
495 $ VR, LDVR, N, IN, WORK( IWORK ), IINFO )
496 IF( IINFO.NE.0 ) THEN
497 INFO = N + 7
498 GO TO 120
499 END IF
500 *
501 * Undo balancing on VL and VR, rescale
502 *
503 IF( ILVL ) THEN
504 CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
505 $ WORK( IRIGHT ), N, VL, LDVL, IINFO )
506 IF( IINFO.NE.0 ) THEN
507 INFO = N + 8
508 GO TO 120
509 END IF
510 DO 50 JC = 1, N
511 IF( ALPHAI( JC ).LT.ZERO )
512 $ GO TO 50
513 TEMP = ZERO
514 IF( ALPHAI( JC ).EQ.ZERO ) THEN
515 DO 10 JR = 1, N
516 TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
517 10 CONTINUE
518 ELSE
519 DO 20 JR = 1, N
520 TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
521 $ ABS( VL( JR, JC+1 ) ) )
522 20 CONTINUE
523 END IF
524 IF( TEMP.LT.SAFMIN )
525 $ GO TO 50
526 TEMP = ONE / TEMP
527 IF( ALPHAI( JC ).EQ.ZERO ) THEN
528 DO 30 JR = 1, N
529 VL( JR, JC ) = VL( JR, JC )*TEMP
530 30 CONTINUE
531 ELSE
532 DO 40 JR = 1, N
533 VL( JR, JC ) = VL( JR, JC )*TEMP
534 VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
535 40 CONTINUE
536 END IF
537 50 CONTINUE
538 END IF
539 IF( ILVR ) THEN
540 CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
541 $ WORK( IRIGHT ), N, VR, LDVR, IINFO )
542 IF( IINFO.NE.0 ) THEN
543 INFO = N + 9
544 GO TO 120
545 END IF
546 DO 100 JC = 1, N
547 IF( ALPHAI( JC ).LT.ZERO )
548 $ GO TO 100
549 TEMP = ZERO
550 IF( ALPHAI( JC ).EQ.ZERO ) THEN
551 DO 60 JR = 1, N
552 TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
553 60 CONTINUE
554 ELSE
555 DO 70 JR = 1, N
556 TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
557 $ ABS( VR( JR, JC+1 ) ) )
558 70 CONTINUE
559 END IF
560 IF( TEMP.LT.SAFMIN )
561 $ GO TO 100
562 TEMP = ONE / TEMP
563 IF( ALPHAI( JC ).EQ.ZERO ) THEN
564 DO 80 JR = 1, N
565 VR( JR, JC ) = VR( JR, JC )*TEMP
566 80 CONTINUE
567 ELSE
568 DO 90 JR = 1, N
569 VR( JR, JC ) = VR( JR, JC )*TEMP
570 VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
571 90 CONTINUE
572 END IF
573 100 CONTINUE
574 END IF
575 *
576 * End of eigenvector calculation
577 *
578 END IF
579 *
580 * Undo scaling in alpha, beta
581 *
582 * Note: this does not give the alpha and beta for the unscaled
583 * problem.
584 *
585 * Un-scaling is limited to avoid underflow in alpha and beta
586 * if they are significant.
587 *
588 DO 110 JC = 1, N
589 ABSAR = ABS( ALPHAR( JC ) )
590 ABSAI = ABS( ALPHAI( JC ) )
591 ABSB = ABS( BETA( JC ) )
592 SALFAR = ANRM*ALPHAR( JC )
593 SALFAI = ANRM*ALPHAI( JC )
594 SBETA = BNRM*BETA( JC )
595 ILIMIT = .FALSE.
596 SCALE = ONE
597 *
598 * Check for significant underflow in ALPHAI
599 *
600 IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
601 $ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
602 ILIMIT = .TRUE.
603 SCALE = ( ONEPLS*SAFMIN / ANRM1 ) /
604 $ MAX( ONEPLS*SAFMIN, ANRM2*ABSAI )
605 *
606 ELSE IF( SALFAI.EQ.ZERO ) THEN
607 *
608 * If insignificant underflow in ALPHAI, then make the
609 * conjugate eigenvalue real.
610 *
611 IF( ALPHAI( JC ).LT.ZERO .AND. JC.GT.1 ) THEN
612 ALPHAI( JC-1 ) = ZERO
613 ELSE IF( ALPHAI( JC ).GT.ZERO .AND. JC.LT.N ) THEN
614 ALPHAI( JC+1 ) = ZERO
615 END IF
616 END IF
617 *
618 * Check for significant underflow in ALPHAR
619 *
620 IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
621 $ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
622 ILIMIT = .TRUE.
623 SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / ANRM1 ) /
624 $ MAX( ONEPLS*SAFMIN, ANRM2*ABSAR ) )
625 END IF
626 *
627 * Check for significant underflow in BETA
628 *
629 IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
630 $ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
631 ILIMIT = .TRUE.
632 SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / BNRM1 ) /
633 $ MAX( ONEPLS*SAFMIN, BNRM2*ABSB ) )
634 END IF
635 *
636 * Check for possible overflow when limiting scaling
637 *
638 IF( ILIMIT ) THEN
639 TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
640 $ ABS( SBETA ) )
641 IF( TEMP.GT.ONE )
642 $ SCALE = SCALE / TEMP
643 IF( SCALE.LT.ONE )
644 $ ILIMIT = .FALSE.
645 END IF
646 *
647 * Recompute un-scaled ALPHAR, ALPHAI, BETA if necessary.
648 *
649 IF( ILIMIT ) THEN
650 SALFAR = ( SCALE*ALPHAR( JC ) )*ANRM
651 SALFAI = ( SCALE*ALPHAI( JC ) )*ANRM
652 SBETA = ( SCALE*BETA( JC ) )*BNRM
653 END IF
654 ALPHAR( JC ) = SALFAR
655 ALPHAI( JC ) = SALFAI
656 BETA( JC ) = SBETA
657 110 CONTINUE
658 *
659 120 CONTINUE
660 WORK( 1 ) = LWKOPT
661 *
662 RETURN
663 *
664 * End of DGEGV
665 *
666 END