LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
Searching...
No Matches
cgges3.f
Go to the documentation of this file.
1*> \brief <b> CGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors 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/cgges3.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgges3.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgges3.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CGGES3( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B,
22* \$ LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR,
23* \$ WORK, LWORK, RWORK, BWORK, INFO )
24*
25* .. Scalar Arguments ..
26* CHARACTER JOBVSL, JOBVSR, SORT
27* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
28* ..
29* .. Array Arguments ..
30* LOGICAL BWORK( * )
31* REAL RWORK( * )
32* COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
33* \$ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
34* \$ WORK( * )
35* ..
36* .. Function Arguments ..
37* LOGICAL SELCTG
38* EXTERNAL SELCTG
39* ..
40*
41*
42*> \par Purpose:
43* =============
44*>
45*> \verbatim
46*>
47*> CGGES3 computes for a pair of N-by-N complex nonsymmetric matrices
48*> (A,B), the generalized eigenvalues, the generalized complex Schur
49*> form (S, T), and optionally left and/or right Schur vectors (VSL
50*> and VSR). This gives the generalized Schur factorization
51*>
52*> (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )
53*>
54*> where (VSR)**H is the conjugate-transpose of VSR.
55*>
56*> Optionally, it also orders the eigenvalues so that a selected cluster
57*> of eigenvalues appears in the leading diagonal blocks of the upper
58*> triangular matrix S and the upper triangular matrix T. The leading
59*> columns of VSL and VSR then form an unitary basis for the
60*> corresponding left and right eigenspaces (deflating subspaces).
61*>
62*> (If only the generalized eigenvalues are needed, use the driver
63*> CGGEV instead, which is faster.)
64*>
65*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
66*> or a ratio alpha/beta = w, such that A - w*B is singular. It is
67*> usually represented as the pair (alpha,beta), as there is a
68*> reasonable interpretation for beta=0, and even for both being zero.
69*>
70*> A pair of matrices (S,T) is in generalized complex Schur form if S
71*> and T are upper triangular and, in addition, the diagonal elements
72*> of T are non-negative real numbers.
73*> \endverbatim
74*
75* Arguments:
76* ==========
77*
78*> \param[in] JOBVSL
79*> \verbatim
80*> JOBVSL is CHARACTER*1
81*> = 'N': do not compute the left Schur vectors;
82*> = 'V': compute the left Schur vectors.
83*> \endverbatim
84*>
85*> \param[in] JOBVSR
86*> \verbatim
87*> JOBVSR is CHARACTER*1
88*> = 'N': do not compute the right Schur vectors;
89*> = 'V': compute the right Schur vectors.
90*> \endverbatim
91*>
92*> \param[in] SORT
93*> \verbatim
94*> SORT is CHARACTER*1
95*> Specifies whether or not to order the eigenvalues on the
96*> diagonal of the generalized Schur form.
97*> = 'N': Eigenvalues are not ordered;
98*> = 'S': Eigenvalues are ordered (see SELCTG).
99*> \endverbatim
100*>
101*> \param[in] SELCTG
102*> \verbatim
103*> SELCTG is a LOGICAL FUNCTION of two COMPLEX arguments
104*> SELCTG must be declared EXTERNAL in the calling subroutine.
105*> If SORT = 'N', SELCTG is not referenced.
106*> If SORT = 'S', SELCTG is used to select eigenvalues to sort
107*> to the top left of the Schur form.
108*> An eigenvalue ALPHA(j)/BETA(j) is selected if
109*> SELCTG(ALPHA(j),BETA(j)) is true.
110*>
111*> Note that a selected complex eigenvalue may no longer satisfy
112*> SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
113*> ordering may change the value of complex eigenvalues
114*> (especially if the eigenvalue is ill-conditioned), in this
115*> case INFO is set to N+2 (See INFO below).
116*> \endverbatim
117*>
118*> \param[in] N
119*> \verbatim
120*> N is INTEGER
121*> The order of the matrices A, B, VSL, and VSR. N >= 0.
122*> \endverbatim
123*>
124*> \param[in,out] A
125*> \verbatim
126*> A is COMPLEX array, dimension (LDA, N)
127*> On entry, the first of the pair of matrices.
128*> On exit, A has been overwritten by its generalized Schur
129*> form S.
130*> \endverbatim
131*>
132*> \param[in] LDA
133*> \verbatim
134*> LDA is INTEGER
135*> The leading dimension of A. LDA >= max(1,N).
136*> \endverbatim
137*>
138*> \param[in,out] B
139*> \verbatim
140*> B is COMPLEX array, dimension (LDB, N)
141*> On entry, the second of the pair of matrices.
142*> On exit, B has been overwritten by its generalized Schur
143*> form T.
144*> \endverbatim
145*>
146*> \param[in] LDB
147*> \verbatim
148*> LDB is INTEGER
149*> The leading dimension of B. LDB >= max(1,N).
150*> \endverbatim
151*>
152*> \param[out] SDIM
153*> \verbatim
154*> SDIM is INTEGER
155*> If SORT = 'N', SDIM = 0.
156*> If SORT = 'S', SDIM = number of eigenvalues (after sorting)
157*> for which SELCTG is true.
158*> \endverbatim
159*>
160*> \param[out] ALPHA
161*> \verbatim
162*> ALPHA is COMPLEX array, dimension (N)
163*> \endverbatim
164*>
165*> \param[out] BETA
166*> \verbatim
167*> BETA is COMPLEX array, dimension (N)
168*> On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
169*> generalized eigenvalues. ALPHA(j), j=1,...,N and BETA(j),
170*> j=1,...,N are the diagonals of the complex Schur form (A,B)
171*> output by CGGES3. The BETA(j) will be non-negative real.
172*>
173*> Note: the quotients ALPHA(j)/BETA(j) may easily over- or
174*> underflow, and BETA(j) may even be zero. Thus, the user
175*> should avoid naively computing the ratio alpha/beta.
176*> However, ALPHA will be always less than and usually
177*> comparable with norm(A) in magnitude, and BETA always less
178*> than and usually comparable with norm(B).
179*> \endverbatim
180*>
181*> \param[out] VSL
182*> \verbatim
183*> VSL is COMPLEX array, dimension (LDVSL,N)
184*> If JOBVSL = 'V', VSL will contain the left Schur vectors.
185*> Not referenced if JOBVSL = 'N'.
186*> \endverbatim
187*>
188*> \param[in] LDVSL
189*> \verbatim
190*> LDVSL is INTEGER
191*> The leading dimension of the matrix VSL. LDVSL >= 1, and
192*> if JOBVSL = 'V', LDVSL >= N.
193*> \endverbatim
194*>
195*> \param[out] VSR
196*> \verbatim
197*> VSR is COMPLEX array, dimension (LDVSR,N)
198*> If JOBVSR = 'V', VSR will contain the right Schur vectors.
199*> Not referenced if JOBVSR = 'N'.
200*> \endverbatim
201*>
202*> \param[in] LDVSR
203*> \verbatim
204*> LDVSR is INTEGER
205*> The leading dimension of the matrix VSR. LDVSR >= 1, and
206*> if JOBVSR = 'V', LDVSR >= N.
207*> \endverbatim
208*>
209*> \param[out] WORK
210*> \verbatim
211*> WORK is COMPLEX array, dimension (MAX(1,LWORK))
212*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
213*> \endverbatim
214*>
215*> \param[in] LWORK
216*> \verbatim
217*> LWORK is INTEGER
218*> The dimension of the array WORK.
219*>
220*> If LWORK = -1, then a workspace query is assumed; the routine
221*> only calculates the optimal size of the WORK array, returns
222*> this value as the first entry of the WORK array, and no error
223*> message related to LWORK is issued by XERBLA.
224*> \endverbatim
225*>
226*> \param[out] RWORK
227*> \verbatim
228*> RWORK is REAL array, dimension (8*N)
229*> \endverbatim
230*>
231*> \param[out] BWORK
232*> \verbatim
233*> BWORK is LOGICAL array, dimension (N)
234*> Not referenced if SORT = 'N'.
235*> \endverbatim
236*>
237*> \param[out] INFO
238*> \verbatim
239*> INFO is INTEGER
240*> = 0: successful exit
241*> < 0: if INFO = -i, the i-th argument had an illegal value.
242*> =1,...,N:
243*> The QZ iteration failed. (A,B) are not in Schur
244*> form, but ALPHA(j) and BETA(j) should be correct for
245*> j=INFO+1,...,N.
246*> > N: =N+1: other than QZ iteration failed in CLAQZ0
247*> =N+2: after reordering, roundoff changed values of
248*> some complex eigenvalues so that leading
249*> eigenvalues in the Generalized Schur form no
250*> longer satisfy SELCTG=.TRUE. This could also
251*> be caused due to scaling.
252*> =N+3: reordering failed in CTGSEN.
253*> \endverbatim
254*
255* Authors:
256* ========
257*
258*> \author Univ. of Tennessee
259*> \author Univ. of California Berkeley
260*> \author Univ. of Colorado Denver
261*> \author NAG Ltd.
262*
263*> \ingroup gges3
264*
265* =====================================================================
266 SUBROUTINE cgges3( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B,
267 \$ LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR,
268 \$ WORK, LWORK, RWORK, BWORK, INFO )
269*
270* -- LAPACK driver routine --
271* -- LAPACK is a software package provided by Univ. of Tennessee, --
272* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
273*
274* .. Scalar Arguments ..
275 CHARACTER JOBVSL, JOBVSR, SORT
276 INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
277* ..
278* .. Array Arguments ..
279 LOGICAL BWORK( * )
280 REAL RWORK( * )
281 COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
282 \$ beta( * ), vsl( ldvsl, * ), vsr( ldvsr, * ),
283 \$ work( * )
284* ..
285* .. Function Arguments ..
286 LOGICAL SELCTG
287 EXTERNAL SELCTG
288* ..
289*
290* =====================================================================
291*
292* .. Parameters ..
293 REAL ZERO, ONE
294 PARAMETER ( ZERO = 0.0e0, one = 1.0e0 )
295 COMPLEX CZERO, CONE
296 parameter( czero = ( 0.0e0, 0.0e0 ),
297 \$ cone = ( 1.0e0, 0.0e0 ) )
298* ..
299* .. Local Scalars ..
300 LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
301 \$ LQUERY, WANTST
302 INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
303 \$ ILO, IRIGHT, IROWS, IRWRK, ITAU, IWRK, LWKOPT
304 REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
305 \$ PVSR, SMLNUM
306* ..
307* .. Local Arrays ..
308 INTEGER IDUM( 1 )
309 REAL DIF( 2 )
310* ..
311* .. External Subroutines ..
312 EXTERNAL cgeqrf, cggbak, cggbal, cgghd3, claqz0, clacpy,
314* ..
315* .. External Functions ..
316 LOGICAL LSAME
317 REAL CLANGE, SLAMCH
318 EXTERNAL lsame, clange, slamch
319* ..
320* .. Intrinsic Functions ..
321 INTRINSIC max, sqrt
322* ..
323* .. Executable Statements ..
324*
325* Decode the input arguments
326*
327 IF( lsame( jobvsl, 'N' ) ) THEN
328 ijobvl = 1
329 ilvsl = .false.
330 ELSE IF( lsame( jobvsl, 'V' ) ) THEN
331 ijobvl = 2
332 ilvsl = .true.
333 ELSE
334 ijobvl = -1
335 ilvsl = .false.
336 END IF
337*
338 IF( lsame( jobvsr, 'N' ) ) THEN
339 ijobvr = 1
340 ilvsr = .false.
341 ELSE IF( lsame( jobvsr, 'V' ) ) THEN
342 ijobvr = 2
343 ilvsr = .true.
344 ELSE
345 ijobvr = -1
346 ilvsr = .false.
347 END IF
348*
349 wantst = lsame( sort, 'S' )
350*
351* Test the input arguments
352*
353 info = 0
354 lquery = ( lwork.EQ.-1 )
355 IF( ijobvl.LE.0 ) THEN
356 info = -1
357 ELSE IF( ijobvr.LE.0 ) THEN
358 info = -2
359 ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
360 info = -3
361 ELSE IF( n.LT.0 ) THEN
362 info = -5
363 ELSE IF( lda.LT.max( 1, n ) ) THEN
364 info = -7
365 ELSE IF( ldb.LT.max( 1, n ) ) THEN
366 info = -9
367 ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
368 info = -14
369 ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
370 info = -16
371 ELSE IF( lwork.LT.max( 1, 2*n ) .AND. .NOT.lquery ) THEN
372 info = -18
373 END IF
374*
375* Compute workspace
376*
377 IF( info.EQ.0 ) THEN
378 CALL cgeqrf( n, n, b, ldb, work, work, -1, ierr )
379 lwkopt = max( 1, n + int( work( 1 ) ) )
380 CALL cunmqr( 'L', 'C', n, n, n, b, ldb, work, a, lda, work,
381 \$ -1, ierr )
382 lwkopt = max( lwkopt, n + int( work( 1 ) ) )
383 IF( ilvsl ) THEN
384 CALL cungqr( n, n, n, vsl, ldvsl, work, work, -1,
385 \$ ierr )
386 lwkopt = max( lwkopt, n + int( work( 1 ) ) )
387 END IF
388 CALL cgghd3( jobvsl, jobvsr, n, 1, n, a, lda, b, ldb, vsl,
389 \$ ldvsl, vsr, ldvsr, work, -1, ierr )
390 lwkopt = max( lwkopt, n + int( work( 1 ) ) )
391 CALL claqz0( 'S', jobvsl, jobvsr, n, 1, n, a, lda, b, ldb,
392 \$ alpha, beta, vsl, ldvsl, vsr, ldvsr, work, -1,
393 \$ rwork, 0, ierr )
394 lwkopt = max( lwkopt, int( work( 1 ) ) )
395 IF( wantst ) THEN
396 CALL ctgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
397 \$ alpha, beta, vsl, ldvsl, vsr, ldvsr, sdim,
398 \$ pvsl, pvsr, dif, work, -1, idum, 1, ierr )
399 lwkopt = max( lwkopt, int( work( 1 ) ) )
400 END IF
401 work( 1 ) = cmplx( lwkopt )
402 END IF
403
404*
405 IF( info.NE.0 ) THEN
406 CALL xerbla( 'CGGES3 ', -info )
407 RETURN
408 ELSE IF( lquery ) THEN
409 RETURN
410 END IF
411*
412* Quick return if possible
413*
414 IF( n.EQ.0 ) THEN
415 sdim = 0
416 RETURN
417 END IF
418*
419* Get machine constants
420*
421 eps = slamch( 'P' )
422 smlnum = slamch( 'S' )
423 bignum = one / smlnum
424 smlnum = sqrt( smlnum ) / eps
425 bignum = one / smlnum
426*
427* Scale A if max element outside range [SMLNUM,BIGNUM]
428*
429 anrm = clange( 'M', n, n, a, lda, rwork )
430 ilascl = .false.
431 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
432 anrmto = smlnum
433 ilascl = .true.
434 ELSE IF( anrm.GT.bignum ) THEN
435 anrmto = bignum
436 ilascl = .true.
437 END IF
438*
439 IF( ilascl )
440 \$ CALL clascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
441*
442* Scale B if max element outside range [SMLNUM,BIGNUM]
443*
444 bnrm = clange( 'M', n, n, b, ldb, rwork )
445 ilbscl = .false.
446 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
447 bnrmto = smlnum
448 ilbscl = .true.
449 ELSE IF( bnrm.GT.bignum ) THEN
450 bnrmto = bignum
451 ilbscl = .true.
452 END IF
453*
454 IF( ilbscl )
455 \$ CALL clascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
456*
457* Permute the matrix to make it more nearly triangular
458*
459 ileft = 1
460 iright = n + 1
461 irwrk = iright + n
462 CALL cggbal( 'P', n, a, lda, b, ldb, ilo, ihi, rwork( ileft ),
463 \$ rwork( iright ), rwork( irwrk ), ierr )
464*
465* Reduce B to triangular form (QR decomposition of B)
466*
467 irows = ihi + 1 - ilo
468 icols = n + 1 - ilo
469 itau = 1
470 iwrk = itau + irows
471 CALL cgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
472 \$ work( iwrk ), lwork+1-iwrk, ierr )
473*
474* Apply the orthogonal transformation to matrix A
475*
476 CALL cunmqr( 'L', 'C', irows, icols, irows, b( ilo, ilo ), ldb,
477 \$ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
478 \$ lwork+1-iwrk, ierr )
479*
480* Initialize VSL
481*
482 IF( ilvsl ) THEN
483 CALL claset( 'Full', n, n, czero, cone, vsl, ldvsl )
484 IF( irows.GT.1 ) THEN
485 CALL clacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
486 \$ vsl( ilo+1, ilo ), ldvsl )
487 END IF
488 CALL cungqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
489 \$ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
490 END IF
491*
492* Initialize VSR
493*
494 IF( ilvsr )
495 \$ CALL claset( 'Full', n, n, czero, cone, vsr, ldvsr )
496*
497* Reduce to generalized Hessenberg form
498*
499 CALL cgghd3( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
500 \$ ldvsl, vsr, ldvsr, work( iwrk ), lwork+1-iwrk, ierr )
501*
502 sdim = 0
503*
504* Perform QZ algorithm, computing Schur vectors if desired
505*
506 iwrk = itau
507 CALL claqz0( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
508 \$ alpha, beta, vsl, ldvsl, vsr, ldvsr, work( iwrk ),
509 \$ lwork+1-iwrk, rwork( irwrk ), 0, ierr )
510 IF( ierr.NE.0 ) THEN
511 IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
512 info = ierr
513 ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
514 info = ierr - n
515 ELSE
516 info = n + 1
517 END IF
518 GO TO 30
519 END IF
520*
521* Sort eigenvalues ALPHA/BETA if desired
522*
523 IF( wantst ) THEN
524*
525* Undo scaling on eigenvalues before selecting
526*
527 IF( ilascl )
528 \$ CALL clascl( 'G', 0, 0, anrm, anrmto, n, 1, alpha, n, ierr )
529 IF( ilbscl )
530 \$ CALL clascl( 'G', 0, 0, bnrm, bnrmto, n, 1, beta, n, ierr )
531*
532* Select eigenvalues
533*
534 DO 10 i = 1, n
535 bwork( i ) = selctg( alpha( i ), beta( i ) )
536 10 CONTINUE
537*
538 CALL ctgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb, alpha,
539 \$ beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl, pvsr,
540 \$ dif, work( iwrk ), lwork-iwrk+1, idum, 1, ierr )
541 IF( ierr.EQ.1 )
542 \$ info = n + 3
543*
544 END IF
545*
546* Apply back-permutation to VSL and VSR
547*
548 IF( ilvsl )
549 \$ CALL cggbak( 'P', 'L', n, ilo, ihi, rwork( ileft ),
550 \$ rwork( iright ), n, vsl, ldvsl, ierr )
551 IF( ilvsr )
552 \$ CALL cggbak( 'P', 'R', n, ilo, ihi, rwork( ileft ),
553 \$ rwork( iright ), n, vsr, ldvsr, ierr )
554*
555* Undo scaling
556*
557 IF( ilascl ) THEN
558 CALL clascl( 'U', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
559 CALL clascl( 'G', 0, 0, anrmto, anrm, n, 1, alpha, n, ierr )
560 END IF
561*
562 IF( ilbscl ) THEN
563 CALL clascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
564 CALL clascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
565 END IF
566*
567 IF( wantst ) THEN
568*
569* Check if reordering is correct
570*
571 lastsl = .true.
572 sdim = 0
573 DO 20 i = 1, n
574 cursl = selctg( alpha( i ), beta( i ) )
575 IF( cursl )
576 \$ sdim = sdim + 1
577 IF( cursl .AND. .NOT.lastsl )
578 \$ info = n + 2
579 lastsl = cursl
580 20 CONTINUE
581*
582 END IF
583*
584 30 CONTINUE
585*
586 work( 1 ) = cmplx( lwkopt )
587*
588 RETURN
589*
590* End of CGGES3
591*
592 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine cgeqrf(m, n, a, lda, tau, work, lwork, info)
CGEQRF
Definition cgeqrf.f:146
subroutine cggbak(job, side, n, ilo, ihi, lscale, rscale, m, v, ldv, info)
CGGBAK
Definition cggbak.f:148
subroutine cggbal(job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)
CGGBAL
Definition cggbal.f:177
subroutine cgges3(jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, rwork, bwork, info)
CGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE ...
Definition cgges3.f:269
subroutine cgghd3(compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, work, lwork, info)
CGGHD3
Definition cgghd3.f:231
subroutine clacpy(uplo, m, n, a, lda, b, ldb)
CLACPY copies all or part of one two-dimensional array to another.
Definition clacpy.f:103
recursive subroutine claqz0(wants, wantq, wantz, n, ilo, ihi, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, rec, info)
CLAQZ0
Definition claqz0.f:284
subroutine clascl(type, kl, ku, cfrom, cto, m, n, a, lda, info)
CLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition clascl.f:143
subroutine claset(uplo, m, n, alpha, beta, a, lda)
CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition claset.f:106
subroutine ctgsen(ijob, wantq, wantz, select, n, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork, info)
CTGSEN
Definition ctgsen.f:433
subroutine cungqr(m, n, k, a, lda, tau, work, lwork, info)
CUNGQR
Definition cungqr.f:128
subroutine cunmqr(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
CUNMQR
Definition cunmqr.f:168