LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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zggev.f
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1*> \brief <b> ZGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download ZGGEV + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggev.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zggev.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zggev.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZGGEV( 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*> ZGGEV 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. LWORK >= max(1,2*N).
178*> For good performance, LWORK must generally be larger.
179*>
180*> If LWORK = -1, then a workspace query is assumed; the routine
181*> only calculates the optimal size of the WORK array, returns
182*> this value as the first entry of the WORK array, and no error
183*> message related to LWORK is issued by XERBLA.
184*> \endverbatim
185*>
186*> \param[out] RWORK
187*> \verbatim
188*> RWORK is DOUBLE PRECISION array, dimension (8*N)
189*> \endverbatim
190*>
191*> \param[out] INFO
192*> \verbatim
193*> INFO is INTEGER
194*> = 0: successful exit
195*> < 0: if INFO = -i, the i-th argument had an illegal value.
196*> =1,...,N:
197*> The QZ iteration failed. No eigenvectors have been
198*> calculated, but ALPHA(j) and BETA(j) should be
199*> correct for j=INFO+1,...,N.
200*> > N: =N+1: other then QZ iteration failed in ZHGEQZ,
201*> =N+2: error return from ZTGEVC.
202*> \endverbatim
203*
204* Authors:
205* ========
206*
207*> \author Univ. of Tennessee
208*> \author Univ. of California Berkeley
209*> \author Univ. of Colorado Denver
210*> \author NAG Ltd.
211*
212*> \ingroup ggev
213*
214* =====================================================================
215 SUBROUTINE zggev( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
216 $ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
217*
218* -- LAPACK driver routine --
219* -- LAPACK is a software package provided by Univ. of Tennessee, --
220* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
221*
222* .. Scalar Arguments ..
223 CHARACTER JOBVL, JOBVR
224 INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
225* ..
226* .. Array Arguments ..
227 DOUBLE PRECISION RWORK( * )
228 COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
229 $ beta( * ), vl( ldvl, * ), vr( ldvr, * ),
230 $ work( * )
231* ..
232*
233* =====================================================================
234*
235* .. Parameters ..
236 DOUBLE PRECISION ZERO, ONE
237 parameter( zero = 0.0d0, one = 1.0d0 )
238 COMPLEX*16 CZERO, CONE
239 parameter( czero = ( 0.0d0, 0.0d0 ),
240 $ cone = ( 1.0d0, 0.0d0 ) )
241* ..
242* .. Local Scalars ..
243 LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY
244 CHARACTER CHTEMP
245 INTEGER ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO,
246 $ in, iright, irows, irwrk, itau, iwrk, jc, jr,
247 $ lwkmin, lwkopt
248 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
249 $ smlnum, temp
250 COMPLEX*16 X
251* ..
252* .. Local Arrays ..
253 LOGICAL LDUMMA( 1 )
254* ..
255* .. External Subroutines ..
256 EXTERNAL xerbla, zgeqrf, zggbak, zggbal, zgghrd, zhgeqz,
258* ..
259* .. External Functions ..
260 LOGICAL LSAME
261 INTEGER ILAENV
262 DOUBLE PRECISION DLAMCH, ZLANGE
263 EXTERNAL lsame, ilaenv, dlamch, zlange
264* ..
265* .. Intrinsic Functions ..
266 INTRINSIC abs, dble, dimag, max, sqrt
267* ..
268* .. Statement Functions ..
269 DOUBLE PRECISION ABS1
270* ..
271* .. Statement Function definitions ..
272 abs1( x ) = abs( dble( x ) ) + abs( dimag( x ) )
273* ..
274* .. Executable Statements ..
275*
276* Decode the input arguments
277*
278 IF( lsame( jobvl, 'N' ) ) THEN
279 ijobvl = 1
280 ilvl = .false.
281 ELSE IF( lsame( jobvl, 'V' ) ) THEN
282 ijobvl = 2
283 ilvl = .true.
284 ELSE
285 ijobvl = -1
286 ilvl = .false.
287 END IF
288*
289 IF( lsame( jobvr, 'N' ) ) THEN
290 ijobvr = 1
291 ilvr = .false.
292 ELSE IF( lsame( jobvr, 'V' ) ) THEN
293 ijobvr = 2
294 ilvr = .true.
295 ELSE
296 ijobvr = -1
297 ilvr = .false.
298 END IF
299 ilv = ilvl .OR. ilvr
300*
301* Test the input arguments
302*
303 info = 0
304 lquery = ( lwork.EQ.-1 )
305 IF( ijobvl.LE.0 ) THEN
306 info = -1
307 ELSE IF( ijobvr.LE.0 ) THEN
308 info = -2
309 ELSE IF( n.LT.0 ) THEN
310 info = -3
311 ELSE IF( lda.LT.max( 1, n ) ) THEN
312 info = -5
313 ELSE IF( ldb.LT.max( 1, n ) ) THEN
314 info = -7
315 ELSE IF( ldvl.LT.1 .OR. ( ilvl .AND. ldvl.LT.n ) ) THEN
316 info = -11
317 ELSE IF( ldvr.LT.1 .OR. ( ilvr .AND. ldvr.LT.n ) ) THEN
318 info = -13
319 END IF
320*
321* Compute workspace
322* (Note: Comments in the code beginning "Workspace:" describe the
323* minimal amount of workspace needed at that point in the code,
324* as well as the preferred amount for good performance.
325* NB refers to the optimal block size for the immediately
326* following subroutine, as returned by ILAENV. The workspace is
327* computed assuming ILO = 1 and IHI = N, the worst case.)
328*
329 IF( info.EQ.0 ) THEN
330 lwkmin = max( 1, 2*n )
331 lwkopt = max( 1, n + n*ilaenv( 1, 'ZGEQRF', ' ', n, 1, n, 0 ) )
332 lwkopt = max( lwkopt, n +
333 $ n*ilaenv( 1, 'ZUNMQR', ' ', n, 1, n, 0 ) )
334 IF( ilvl ) THEN
335 lwkopt = max( lwkopt, n +
336 $ n*ilaenv( 1, 'ZUNGQR', ' ', n, 1, n, -1 ) )
337 END IF
338 work( 1 ) = lwkopt
339*
340 IF( lwork.LT.lwkmin .AND. .NOT.lquery )
341 $ info = -15
342 END IF
343*
344 IF( info.NE.0 ) THEN
345 CALL xerbla( 'ZGGEV ', -info )
346 RETURN
347 ELSE IF( lquery ) THEN
348 RETURN
349 END IF
350*
351* Quick return if possible
352*
353 IF( n.EQ.0 )
354 $ RETURN
355*
356* Get machine constants
357*
358 eps = dlamch( 'E' )*dlamch( 'B' )
359 smlnum = dlamch( 'S' )
360 bignum = one / smlnum
361 smlnum = sqrt( smlnum ) / eps
362 bignum = one / smlnum
363*
364* Scale A if max element outside range [SMLNUM,BIGNUM]
365*
366 anrm = zlange( 'M', n, n, a, lda, rwork )
367 ilascl = .false.
368 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
369 anrmto = smlnum
370 ilascl = .true.
371 ELSE IF( anrm.GT.bignum ) THEN
372 anrmto = bignum
373 ilascl = .true.
374 END IF
375 IF( ilascl )
376 $ CALL zlascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
377*
378* Scale B if max element outside range [SMLNUM,BIGNUM]
379*
380 bnrm = zlange( 'M', n, n, b, ldb, rwork )
381 ilbscl = .false.
382 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
383 bnrmto = smlnum
384 ilbscl = .true.
385 ELSE IF( bnrm.GT.bignum ) THEN
386 bnrmto = bignum
387 ilbscl = .true.
388 END IF
389 IF( ilbscl )
390 $ CALL zlascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
391*
392* Permute the matrices A, B to isolate eigenvalues if possible
393* (Real Workspace: need 6*N)
394*
395 ileft = 1
396 iright = n + 1
397 irwrk = iright + n
398 CALL zggbal( 'P', n, a, lda, b, ldb, ilo, ihi, rwork( ileft ),
399 $ rwork( iright ), rwork( irwrk ), ierr )
400*
401* Reduce B to triangular form (QR decomposition of B)
402* (Complex Workspace: need N, prefer N*NB)
403*
404 irows = ihi + 1 - ilo
405 IF( ilv ) THEN
406 icols = n + 1 - ilo
407 ELSE
408 icols = irows
409 END IF
410 itau = 1
411 iwrk = itau + irows
412 CALL zgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
413 $ work( iwrk ), lwork+1-iwrk, ierr )
414*
415* Apply the orthogonal transformation to matrix A
416* (Complex Workspace: need N, prefer N*NB)
417*
418 CALL zunmqr( 'L', 'C', irows, icols, irows, b( ilo, ilo ), ldb,
419 $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
420 $ lwork+1-iwrk, ierr )
421*
422* Initialize VL
423* (Complex Workspace: need N, prefer N*NB)
424*
425 IF( ilvl ) THEN
426 CALL zlaset( 'Full', n, n, czero, cone, vl, ldvl )
427 IF( irows.GT.1 ) THEN
428 CALL zlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
429 $ vl( ilo+1, ilo ), ldvl )
430 END IF
431 CALL zungqr( irows, irows, irows, vl( ilo, ilo ), ldvl,
432 $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
433 END IF
434*
435* Initialize VR
436*
437 IF( ilvr )
438 $ CALL zlaset( 'Full', n, n, czero, cone, vr, ldvr )
439*
440* Reduce to generalized Hessenberg form
441*
442 IF( ilv ) THEN
443*
444* Eigenvectors requested -- work on whole matrix.
445*
446 CALL zgghrd( jobvl, jobvr, n, ilo, ihi, a, lda, b, ldb, vl,
447 $ ldvl, vr, ldvr, ierr )
448 ELSE
449 CALL zgghrd( 'N', 'N', irows, 1, irows, a( ilo, ilo ), lda,
450 $ b( ilo, ilo ), ldb, vl, ldvl, vr, ldvr, ierr )
451 END IF
452*
453* Perform QZ algorithm (Compute eigenvalues, and optionally, the
454* Schur form and Schur vectors)
455* (Complex Workspace: need N)
456* (Real Workspace: need N)
457*
458 iwrk = itau
459 IF( ilv ) THEN
460 chtemp = 'S'
461 ELSE
462 chtemp = 'E'
463 END IF
464 CALL zhgeqz( chtemp, jobvl, jobvr, n, ilo, ihi, a, lda, b, ldb,
465 $ alpha, beta, vl, ldvl, vr, ldvr, work( iwrk ),
466 $ lwork+1-iwrk, rwork( irwrk ), ierr )
467 IF( ierr.NE.0 ) THEN
468 IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
469 info = ierr
470 ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
471 info = ierr - n
472 ELSE
473 info = n + 1
474 END IF
475 GO TO 70
476 END IF
477*
478* Compute Eigenvectors
479* (Real Workspace: need 2*N)
480* (Complex Workspace: need 2*N)
481*
482 IF( ilv ) THEN
483 IF( ilvl ) THEN
484 IF( ilvr ) THEN
485 chtemp = 'B'
486 ELSE
487 chtemp = 'L'
488 END IF
489 ELSE
490 chtemp = 'R'
491 END IF
492*
493 CALL ztgevc( chtemp, 'B', ldumma, n, a, lda, b, ldb, vl, ldvl,
494 $ vr, ldvr, n, in, work( iwrk ), rwork( irwrk ),
495 $ ierr )
496 IF( ierr.NE.0 ) THEN
497 info = n + 2
498 GO TO 70
499 END IF
500*
501* Undo balancing on VL and VR and normalization
502* (Workspace: none needed)
503*
504 IF( ilvl ) THEN
505 CALL zggbak( 'P', 'L', n, ilo, ihi, rwork( ileft ),
506 $ rwork( iright ), n, vl, ldvl, ierr )
507 DO 30 jc = 1, n
508 temp = zero
509 DO 10 jr = 1, n
510 temp = max( temp, abs1( vl( jr, jc ) ) )
511 10 CONTINUE
512 IF( temp.LT.smlnum )
513 $ GO TO 30
514 temp = one / temp
515 DO 20 jr = 1, n
516 vl( jr, jc ) = vl( jr, jc )*temp
517 20 CONTINUE
518 30 CONTINUE
519 END IF
520 IF( ilvr ) THEN
521 CALL zggbak( 'P', 'R', n, ilo, ihi, rwork( ileft ),
522 $ rwork( iright ), n, vr, ldvr, ierr )
523 DO 60 jc = 1, n
524 temp = zero
525 DO 40 jr = 1, n
526 temp = max( temp, abs1( vr( jr, jc ) ) )
527 40 CONTINUE
528 IF( temp.LT.smlnum )
529 $ GO TO 60
530 temp = one / temp
531 DO 50 jr = 1, n
532 vr( jr, jc ) = vr( jr, jc )*temp
533 50 CONTINUE
534 60 CONTINUE
535 END IF
536 END IF
537*
538* Undo scaling if necessary
539*
540 70 CONTINUE
541*
542 IF( ilascl )
543 $ CALL zlascl( 'G', 0, 0, anrmto, anrm, n, 1, alpha, n, ierr )
544*
545 IF( ilbscl )
546 $ CALL zlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
547*
548 work( 1 ) = lwkopt
549 RETURN
550*
551* End of ZGGEV
552*
553 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 zggev(jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info)
ZGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
Definition zggev.f:217
subroutine zgghrd(compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)
ZGGHRD
Definition zgghrd.f:204
subroutine zhgeqz(job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, info)
ZHGEQZ
Definition zhgeqz.f:284
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
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