LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
Searching...
No Matches
zggev3.f
Go to the documentation of this file.
1*> \brief <b> ZGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)</b>
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggev3.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zggev3.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zggev3.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZGGEV3( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
22* VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
23*
24* .. Scalar Arguments ..
25* CHARACTER JOBVL, JOBVR
26* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
27* ..
28* .. Array Arguments ..
29* DOUBLE PRECISION RWORK( * )
30* COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
31* \$ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
32* \$ WORK( * )
33* ..
34*
35*
36*> \par Purpose:
37* =============
38*>
39*> \verbatim
40*>
41*> ZGGEV3 computes for a pair of N-by-N complex nonsymmetric matrices
42*> (A,B), the generalized eigenvalues, and optionally, the left and/or
43*> right generalized eigenvectors.
44*>
45*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
46*> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
47*> singular. It is usually represented as the pair (alpha,beta), as
48*> there is a reasonable interpretation for beta=0, and even for both
49*> being zero.
50*>
51*> The right generalized eigenvector v(j) corresponding to the
52*> generalized eigenvalue lambda(j) of (A,B) satisfies
53*>
54*> A * v(j) = lambda(j) * B * v(j).
55*>
56*> The left generalized eigenvector u(j) corresponding to the
57*> generalized eigenvalues lambda(j) of (A,B) satisfies
58*>
59*> u(j)**H * A = lambda(j) * u(j)**H * B
60*>
61*> where u(j)**H is the conjugate-transpose of u(j).
62*> \endverbatim
63*
64* Arguments:
65* ==========
66*
67*> \param[in] JOBVL
68*> \verbatim
69*> JOBVL is CHARACTER*1
70*> = 'N': do not compute the left generalized eigenvectors;
71*> = 'V': compute the left generalized eigenvectors.
72*> \endverbatim
73*>
74*> \param[in] JOBVR
75*> \verbatim
76*> JOBVR is CHARACTER*1
77*> = 'N': do not compute the right generalized eigenvectors;
78*> = 'V': compute the right generalized eigenvectors.
79*> \endverbatim
80*>
81*> \param[in] N
82*> \verbatim
83*> N is INTEGER
84*> The order of the matrices A, B, VL, and VR. N >= 0.
85*> \endverbatim
86*>
87*> \param[in,out] A
88*> \verbatim
89*> A is COMPLEX*16 array, dimension (LDA, N)
90*> On entry, the matrix A in the pair (A,B).
91*> On exit, A has been overwritten.
92*> \endverbatim
93*>
94*> \param[in] LDA
95*> \verbatim
96*> LDA is INTEGER
97*> The leading dimension of A. LDA >= max(1,N).
98*> \endverbatim
99*>
100*> \param[in,out] B
101*> \verbatim
102*> B is COMPLEX*16 array, dimension (LDB, N)
103*> On entry, the matrix B in the pair (A,B).
104*> On exit, B has been overwritten.
105*> \endverbatim
106*>
107*> \param[in] LDB
108*> \verbatim
109*> LDB is INTEGER
110*> The leading dimension of B. LDB >= max(1,N).
111*> \endverbatim
112*>
113*> \param[out] ALPHA
114*> \verbatim
115*> ALPHA is COMPLEX*16 array, dimension (N)
116*> \endverbatim
117*>
118*> \param[out] BETA
119*> \verbatim
120*> BETA is COMPLEX*16 array, dimension (N)
121*> On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
122*> generalized eigenvalues.
123*>
124*> Note: the quotients ALPHA(j)/BETA(j) may easily over- or
125*> underflow, and BETA(j) may even be zero. Thus, the user
126*> should avoid naively computing the ratio alpha/beta.
127*> However, ALPHA will be always less than and usually
128*> comparable with norm(A) in magnitude, and BETA always less
129*> than and usually comparable with norm(B).
130*> \endverbatim
131*>
132*> \param[out] VL
133*> \verbatim
134*> VL is COMPLEX*16 array, dimension (LDVL,N)
135*> If JOBVL = 'V', the left generalized eigenvectors u(j) are
136*> stored one after another in the columns of VL, in the same
137*> order as their eigenvalues.
138*> Each eigenvector is scaled so the largest component has
139*> abs(real part) + abs(imag. part) = 1.
140*> Not referenced if JOBVL = 'N'.
141*> \endverbatim
142*>
143*> \param[in] LDVL
144*> \verbatim
145*> LDVL is INTEGER
146*> The leading dimension of the matrix VL. LDVL >= 1, and
147*> if JOBVL = 'V', LDVL >= N.
148*> \endverbatim
149*>
150*> \param[out] VR
151*> \verbatim
152*> VR is COMPLEX*16 array, dimension (LDVR,N)
153*> If JOBVR = 'V', the right generalized eigenvectors v(j) are
154*> stored one after another in the columns of VR, in the same
155*> order as their eigenvalues.
156*> Each eigenvector is scaled so the largest component has
157*> abs(real part) + abs(imag. part) = 1.
158*> Not referenced if JOBVR = 'N'.
159*> \endverbatim
160*>
161*> \param[in] LDVR
162*> \verbatim
163*> LDVR is INTEGER
164*> The leading dimension of the matrix VR. LDVR >= 1, and
165*> if JOBVR = 'V', LDVR >= N.
166*> \endverbatim
167*>
168*> \param[out] WORK
169*> \verbatim
170*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
171*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
172*> \endverbatim
173*>
174*> \param[in] LWORK
175*> \verbatim
176*> LWORK is INTEGER
177*> The dimension of the array WORK.
178*>
179*> If LWORK = -1, then a workspace query is assumed; the routine
180*> only calculates the optimal size of the WORK array, returns
181*> this value as the first entry of the WORK array, and no error
182*> message related to LWORK is issued by XERBLA.
183*> \endverbatim
184*>
185*> \param[out] RWORK
186*> \verbatim
187*> RWORK is DOUBLE PRECISION array, dimension (8*N)
188*> \endverbatim
189*>
190*> \param[out] INFO
191*> \verbatim
192*> INFO is INTEGER
193*> = 0: successful exit
194*> < 0: if INFO = -i, the i-th argument had an illegal value.
195*> =1,...,N:
196*> The QZ iteration failed. No eigenvectors have been
197*> calculated, but ALPHA(j) and BETA(j) should be
198*> correct for j=INFO+1,...,N.
199*> > N: =N+1: other then QZ iteration failed in ZHGEQZ,
200*> =N+2: error return from ZTGEVC.
201*> \endverbatim
202*
203* Authors:
204* ========
205*
206*> \author Univ. of Tennessee
207*> \author Univ. of California Berkeley
208*> \author Univ. of Colorado Denver
209*> \author NAG Ltd.
210*
211*> \ingroup ggev3
212*
213* =====================================================================
214 SUBROUTINE zggev3( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
215 \$ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
216*
217* -- LAPACK driver routine --
218* -- LAPACK is a software package provided by Univ. of Tennessee, --
219* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
220*
221* .. Scalar Arguments ..
222 CHARACTER JOBVL, JOBVR
223 INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
224* ..
225* .. Array Arguments ..
226 DOUBLE PRECISION RWORK( * )
227 COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
228 \$ beta( * ), vl( ldvl, * ), vr( ldvr, * ),
229 \$ work( * )
230* ..
231*
232* =====================================================================
233*
234* .. Parameters ..
235 DOUBLE PRECISION ZERO, ONE
236 parameter( zero = 0.0d0, one = 1.0d0 )
237 COMPLEX*16 CZERO, CONE
238 parameter( czero = ( 0.0d0, 0.0d0 ),
239 \$ cone = ( 1.0d0, 0.0d0 ) )
240* ..
241* .. Local Scalars ..
242 LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY
243 CHARACTER CHTEMP
244 INTEGER ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO,
245 \$ in, iright, irows, irwrk, itau, iwrk, jc, jr,
246 \$ lwkopt
247 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
248 \$ smlnum, temp
249 COMPLEX*16 X
250* ..
251* .. Local Arrays ..
252 LOGICAL LDUMMA( 1 )
253* ..
254* .. External Subroutines ..
255 EXTERNAL xerbla, zgeqrf, zggbak, zggbal, zgghd3, zlaqz0,
257* ..
258* .. External Functions ..
259 LOGICAL LSAME
260 DOUBLE PRECISION DLAMCH, ZLANGE
261 EXTERNAL lsame, dlamch, zlange
262* ..
263* .. Intrinsic Functions ..
264 INTRINSIC abs, dble, dimag, max, sqrt
265* ..
266* .. Statement Functions ..
267 DOUBLE PRECISION ABS1
268* ..
269* .. Statement Function definitions ..
270 abs1( x ) = abs( dble( x ) ) + abs( dimag( x ) )
271* ..
272* .. Executable Statements ..
273*
274* Decode the input arguments
275*
276 IF( lsame( jobvl, 'N' ) ) THEN
277 ijobvl = 1
278 ilvl = .false.
279 ELSE IF( lsame( jobvl, 'V' ) ) THEN
280 ijobvl = 2
281 ilvl = .true.
282 ELSE
283 ijobvl = -1
284 ilvl = .false.
285 END IF
286*
287 IF( lsame( jobvr, 'N' ) ) THEN
288 ijobvr = 1
289 ilvr = .false.
290 ELSE IF( lsame( jobvr, 'V' ) ) THEN
291 ijobvr = 2
292 ilvr = .true.
293 ELSE
294 ijobvr = -1
295 ilvr = .false.
296 END IF
297 ilv = ilvl .OR. ilvr
298*
299* Test the input arguments
300*
301 info = 0
302 lquery = ( lwork.EQ.-1 )
303 IF( ijobvl.LE.0 ) THEN
304 info = -1
305 ELSE IF( ijobvr.LE.0 ) THEN
306 info = -2
307 ELSE IF( n.LT.0 ) THEN
308 info = -3
309 ELSE IF( lda.LT.max( 1, n ) ) THEN
310 info = -5
311 ELSE IF( ldb.LT.max( 1, n ) ) THEN
312 info = -7
313 ELSE IF( ldvl.LT.1 .OR. ( ilvl .AND. ldvl.LT.n ) ) THEN
314 info = -11
315 ELSE IF( ldvr.LT.1 .OR. ( ilvr .AND. ldvr.LT.n ) ) THEN
316 info = -13
317 ELSE IF( lwork.LT.max( 1, 2*n ) .AND. .NOT.lquery ) THEN
318 info = -15
319 END IF
320*
321* Compute workspace
322*
323 IF( info.EQ.0 ) THEN
324 CALL zgeqrf( n, n, b, ldb, work, work, -1, ierr )
325 lwkopt = max( 1, n+int( work( 1 ) ) )
326 CALL zunmqr( 'L', 'C', n, n, n, b, ldb, work, a, lda, work,
327 \$ -1, ierr )
328 lwkopt = max( lwkopt, n+int( work( 1 ) ) )
329 IF( ilvl ) THEN
330 CALL zungqr( n, n, n, vl, ldvl, work, work, -1, ierr )
331 lwkopt = max( lwkopt, n+int( work( 1 ) ) )
332 END IF
333 IF( ilv ) THEN
334 CALL zgghd3( jobvl, jobvr, n, 1, n, a, lda, b, ldb, vl,
335 \$ ldvl, vr, ldvr, work, -1, ierr )
336 lwkopt = max( lwkopt, n+int( work( 1 ) ) )
337 CALL zlaqz0( 'S', jobvl, jobvr, n, 1, n, a, lda, b, ldb,
338 \$ alpha, beta, vl, ldvl, vr, ldvr, work, -1,
339 \$ rwork, 0, ierr )
340 lwkopt = max( lwkopt, n+int( work( 1 ) ) )
341 ELSE
342 CALL zgghd3( jobvl, jobvr, n, 1, n, a, lda, b, ldb, vl,
343 \$ ldvl, vr, ldvr, work, -1, ierr )
344 lwkopt = max( lwkopt, n+int( work( 1 ) ) )
345 CALL zlaqz0( 'E', jobvl, jobvr, n, 1, n, a, lda, b, ldb,
346 \$ alpha, beta, vl, ldvl, vr, ldvr, work, -1,
347 \$ rwork, 0, ierr )
348 lwkopt = max( lwkopt, n+int( work( 1 ) ) )
349 END IF
350 work( 1 ) = dcmplx( lwkopt )
351 END IF
352*
353 IF( info.NE.0 ) THEN
354 CALL xerbla( 'ZGGEV3 ', -info )
355 RETURN
356 ELSE IF( lquery ) THEN
357 RETURN
358 END IF
359*
360* Quick return if possible
361*
362 IF( n.EQ.0 )
363 \$ RETURN
364*
365* Get machine constants
366*
367 eps = dlamch( 'E' )*dlamch( 'B' )
368 smlnum = dlamch( 'S' )
369 bignum = one / smlnum
370 smlnum = sqrt( smlnum ) / eps
371 bignum = one / smlnum
372*
373* Scale A if max element outside range [SMLNUM,BIGNUM]
374*
375 anrm = zlange( 'M', n, n, a, lda, rwork )
376 ilascl = .false.
377 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
378 anrmto = smlnum
379 ilascl = .true.
380 ELSE IF( anrm.GT.bignum ) THEN
381 anrmto = bignum
382 ilascl = .true.
383 END IF
384 IF( ilascl )
385 \$ CALL zlascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
386*
387* Scale B if max element outside range [SMLNUM,BIGNUM]
388*
389 bnrm = zlange( 'M', n, n, b, ldb, rwork )
390 ilbscl = .false.
391 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
392 bnrmto = smlnum
393 ilbscl = .true.
394 ELSE IF( bnrm.GT.bignum ) THEN
395 bnrmto = bignum
396 ilbscl = .true.
397 END IF
398 IF( ilbscl )
399 \$ CALL zlascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
400*
401* Permute the matrices A, B to isolate eigenvalues if possible
402*
403 ileft = 1
404 iright = n + 1
405 irwrk = iright + n
406 CALL zggbal( 'P', n, a, lda, b, ldb, ilo, ihi, rwork( ileft ),
407 \$ rwork( iright ), rwork( irwrk ), ierr )
408*
409* Reduce B to triangular form (QR decomposition of B)
410*
411 irows = ihi + 1 - ilo
412 IF( ilv ) THEN
413 icols = n + 1 - ilo
414 ELSE
415 icols = irows
416 END IF
417 itau = 1
418 iwrk = itau + irows
419 CALL zgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
420 \$ work( iwrk ), lwork+1-iwrk, ierr )
421*
422* Apply the orthogonal transformation to matrix A
423*
424 CALL zunmqr( 'L', 'C', irows, icols, irows, b( ilo, ilo ), ldb,
425 \$ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
426 \$ lwork+1-iwrk, ierr )
427*
428* Initialize VL
429*
430 IF( ilvl ) THEN
431 CALL zlaset( 'Full', n, n, czero, cone, vl, ldvl )
432 IF( irows.GT.1 ) THEN
433 CALL zlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
434 \$ vl( ilo+1, ilo ), ldvl )
435 END IF
436 CALL zungqr( irows, irows, irows, vl( ilo, ilo ), ldvl,
437 \$ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
438 END IF
439*
440* Initialize VR
441*
442 IF( ilvr )
443 \$ CALL zlaset( 'Full', n, n, czero, cone, vr, ldvr )
444*
445* Reduce to generalized Hessenberg form
446*
447 IF( ilv ) THEN
448*
449* Eigenvectors requested -- work on whole matrix.
450*
451 CALL zgghd3( jobvl, jobvr, n, ilo, ihi, a, lda, b, ldb, vl,
452 \$ ldvl, vr, ldvr, work( iwrk ), lwork+1-iwrk, ierr )
453 ELSE
454 CALL zgghd3( 'N', 'N', irows, 1, irows, a( ilo, ilo ), lda,
455 \$ b( ilo, ilo ), ldb, vl, ldvl, vr, ldvr,
456 \$ work( iwrk ), lwork+1-iwrk, ierr )
457 END IF
458*
459* Perform QZ algorithm (Compute eigenvalues, and optionally, the
460* Schur form and Schur vectors)
461*
462 iwrk = itau
463 IF( ilv ) THEN
464 chtemp = 'S'
465 ELSE
466 chtemp = 'E'
467 END IF
468 CALL zlaqz0( chtemp, jobvl, jobvr, n, ilo, ihi, a, lda, b, ldb,
469 \$ alpha, beta, vl, ldvl, vr, ldvr, work( iwrk ),
470 \$ lwork+1-iwrk, rwork( irwrk ), 0, ierr )
471 IF( ierr.NE.0 ) THEN
472 IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
473 info = ierr
474 ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
475 info = ierr - n
476 ELSE
477 info = n + 1
478 END IF
479 GO TO 70
480 END IF
481*
482* Compute Eigenvectors
483*
484 IF( ilv ) THEN
485 IF( ilvl ) THEN
486 IF( ilvr ) THEN
487 chtemp = 'B'
488 ELSE
489 chtemp = 'L'
490 END IF
491 ELSE
492 chtemp = 'R'
493 END IF
494*
495 CALL ztgevc( chtemp, 'B', ldumma, n, a, lda, b, ldb, vl, ldvl,
496 \$ vr, ldvr, n, in, work( iwrk ), rwork( irwrk ),
497 \$ ierr )
498 IF( ierr.NE.0 ) THEN
499 info = n + 2
500 GO TO 70
501 END IF
502*
503* Undo balancing on VL and VR and normalization
504*
505 IF( ilvl ) THEN
506 CALL zggbak( 'P', 'L', n, ilo, ihi, rwork( ileft ),
507 \$ rwork( iright ), n, vl, ldvl, ierr )
508 DO 30 jc = 1, n
509 temp = zero
510 DO 10 jr = 1, n
511 temp = max( temp, abs1( vl( jr, jc ) ) )
512 10 CONTINUE
513 IF( temp.LT.smlnum )
514 \$ GO TO 30
515 temp = one / temp
516 DO 20 jr = 1, n
517 vl( jr, jc ) = vl( jr, jc )*temp
518 20 CONTINUE
519 30 CONTINUE
520 END IF
521 IF( ilvr ) THEN
522 CALL zggbak( 'P', 'R', n, ilo, ihi, rwork( ileft ),
523 \$ rwork( iright ), n, vr, ldvr, ierr )
524 DO 60 jc = 1, n
525 temp = zero
526 DO 40 jr = 1, n
527 temp = max( temp, abs1( vr( jr, jc ) ) )
528 40 CONTINUE
529 IF( temp.LT.smlnum )
530 \$ GO TO 60
531 temp = one / temp
532 DO 50 jr = 1, n
533 vr( jr, jc ) = vr( jr, jc )*temp
534 50 CONTINUE
535 60 CONTINUE
536 END IF
537 END IF
538*
539* Undo scaling if necessary
540*
541 70 CONTINUE
542*
543 IF( ilascl )
544 \$ CALL zlascl( 'G', 0, 0, anrmto, anrm, n, 1, alpha, n, ierr )
545*
546 IF( ilbscl )
547 \$ CALL zlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
548*
549 work( 1 ) = dcmplx( lwkopt )
550 RETURN
551*
552* End of ZGGEV3
553*
554 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zgeqrf(m, n, a, lda, tau, work, lwork, info)
ZGEQRF
Definition zgeqrf.f:146
subroutine zggbak(job, side, n, ilo, ihi, lscale, rscale, m, v, ldv, info)
ZGGBAK
Definition zggbak.f:148
subroutine zggbal(job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)
ZGGBAL
Definition zggbal.f:177
subroutine zggev3(jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info)
ZGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (...
Definition zggev3.f:216
subroutine zgghd3(compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, work, lwork, info)
ZGGHD3
Definition zgghd3.f:227
subroutine zlacpy(uplo, m, n, a, lda, b, ldb)
ZLACPY copies all or part of one two-dimensional array to another.
Definition zlacpy.f:103
recursive subroutine zlaqz0(wants, wantq, wantz, n, ilo, ihi, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, rec, info)
ZLAQZ0
Definition zlaqz0.f:284
subroutine zlascl(type, kl, ku, cfrom, cto, m, n, a, lda, info)
ZLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition zlascl.f:143
subroutine zlaset(uplo, m, n, alpha, beta, a, lda)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition zlaset.f:106
subroutine ztgevc(side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, rwork, info)
ZTGEVC
Definition ztgevc.f:219
subroutine zungqr(m, n, k, a, lda, tau, work, lwork, info)
ZUNGQR
Definition zungqr.f:128
subroutine zunmqr(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
ZUNMQR
Definition zunmqr.f:167