LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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zgges3.f
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1*> \brief <b> ZGGES3 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
9*> Download ZGGES3 + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgges3.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgges3.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgges3.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZGGES3( 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* DOUBLE PRECISION RWORK( * )
32* COMPLEX*16 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*> ZGGES3 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*> ZGGEV 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*16 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*16 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*16 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*16 array, dimension (N)
163*> \endverbatim
164*>
165*> \param[out] BETA
166*> \verbatim
167*> BETA is COMPLEX*16 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 ZGGES3. 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*16 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*16 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*16 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 DOUBLE PRECISION 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 ZLAQZ0
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 ZTGSEN.
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 zgges3( 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 DOUBLE PRECISION RWORK( * )
281 COMPLEX*16 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 DOUBLE PRECISION ZERO, ONE
294 PARAMETER ( ZERO = 0.0d0, one = 1.0d0 )
295 COMPLEX*16 CZERO, CONE
296 parameter( czero = ( 0.0d0, 0.0d0 ),
297 $ cone = ( 1.0d0, 0.0d0 ) )
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 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
305 $ PVSR, SMLNUM
306* ..
307* .. Local Arrays ..
308 INTEGER IDUM( 1 )
309 DOUBLE PRECISION DIF( 2 )
310* ..
311* .. External Subroutines ..
312 EXTERNAL xerbla, zgeqrf, zggbak, zggbal, zgghd3, zlaqz0,
314* ..
315* .. External Functions ..
316 LOGICAL LSAME
317 DOUBLE PRECISION DLAMCH, ZLANGE
318 EXTERNAL lsame, dlamch, zlange
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 zgeqrf( n, n, b, ldb, work, work, -1, ierr )
379 lwkopt = max( 1, n + int( work( 1 ) ) )
380 CALL zunmqr( '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 zungqr( n, n, n, vsl, ldvsl, work, work, -1, ierr )
385 lwkopt = max( lwkopt, n + int( work( 1 ) ) )
386 END IF
387 CALL zgghd3( jobvsl, jobvsr, n, 1, n, a, lda, b, ldb, vsl,
388 $ ldvsl, vsr, ldvsr, work, -1, ierr )
389 lwkopt = max( lwkopt, n + int( work( 1 ) ) )
390 CALL zlaqz0( 'S', jobvsl, jobvsr, n, 1, n, a, lda, b, ldb,
391 $ alpha, beta, vsl, ldvsl, vsr, ldvsr, work, -1,
392 $ rwork, 0, ierr )
393 lwkopt = max( lwkopt, int( work( 1 ) ) )
394 IF( wantst ) THEN
395 CALL ztgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
396 $ alpha, beta, vsl, ldvsl, vsr, ldvsr, sdim,
397 $ pvsl, pvsr, dif, work, -1, idum, 1, ierr )
398 lwkopt = max( lwkopt, int( work( 1 ) ) )
399 END IF
400 work( 1 ) = dcmplx( lwkopt )
401 END IF
402*
403 IF( info.NE.0 ) THEN
404 CALL xerbla( 'ZGGES3 ', -info )
405 RETURN
406 ELSE IF( lquery ) THEN
407 RETURN
408 END IF
409*
410* Quick return if possible
411*
412 IF( n.EQ.0 ) THEN
413 sdim = 0
414 RETURN
415 END IF
416*
417* Get machine constants
418*
419 eps = dlamch( 'P' )
420 smlnum = dlamch( 'S' )
421 bignum = one / smlnum
422 smlnum = sqrt( smlnum ) / eps
423 bignum = one / smlnum
424*
425* Scale A if max element outside range [SMLNUM,BIGNUM]
426*
427 anrm = zlange( 'M', n, n, a, lda, rwork )
428 ilascl = .false.
429 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
430 anrmto = smlnum
431 ilascl = .true.
432 ELSE IF( anrm.GT.bignum ) THEN
433 anrmto = bignum
434 ilascl = .true.
435 END IF
436*
437 IF( ilascl )
438 $ CALL zlascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
439*
440* Scale B if max element outside range [SMLNUM,BIGNUM]
441*
442 bnrm = zlange( 'M', n, n, b, ldb, rwork )
443 ilbscl = .false.
444 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
445 bnrmto = smlnum
446 ilbscl = .true.
447 ELSE IF( bnrm.GT.bignum ) THEN
448 bnrmto = bignum
449 ilbscl = .true.
450 END IF
451*
452 IF( ilbscl )
453 $ CALL zlascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
454*
455* Permute the matrix to make it more nearly triangular
456*
457 ileft = 1
458 iright = n + 1
459 irwrk = iright + n
460 CALL zggbal( 'P', n, a, lda, b, ldb, ilo, ihi, rwork( ileft ),
461 $ rwork( iright ), rwork( irwrk ), ierr )
462*
463* Reduce B to triangular form (QR decomposition of B)
464*
465 irows = ihi + 1 - ilo
466 icols = n + 1 - ilo
467 itau = 1
468 iwrk = itau + irows
469 CALL zgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
470 $ work( iwrk ), lwork+1-iwrk, ierr )
471*
472* Apply the orthogonal transformation to matrix A
473*
474 CALL zunmqr( 'L', 'C', irows, icols, irows, b( ilo, ilo ), ldb,
475 $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
476 $ lwork+1-iwrk, ierr )
477*
478* Initialize VSL
479*
480 IF( ilvsl ) THEN
481 CALL zlaset( 'Full', n, n, czero, cone, vsl, ldvsl )
482 IF( irows.GT.1 ) THEN
483 CALL zlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
484 $ vsl( ilo+1, ilo ), ldvsl )
485 END IF
486 CALL zungqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
487 $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
488 END IF
489*
490* Initialize VSR
491*
492 IF( ilvsr )
493 $ CALL zlaset( 'Full', n, n, czero, cone, vsr, ldvsr )
494*
495* Reduce to generalized Hessenberg form
496*
497 CALL zgghd3( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
498 $ ldvsl, vsr, ldvsr, work( iwrk ), lwork+1-iwrk, ierr )
499*
500 sdim = 0
501*
502* Perform QZ algorithm, computing Schur vectors if desired
503*
504 iwrk = itau
505 CALL zlaqz0( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
506 $ alpha, beta, vsl, ldvsl, vsr, ldvsr, work( iwrk ),
507 $ lwork+1-iwrk, rwork( irwrk ), 0, ierr )
508 IF( ierr.NE.0 ) THEN
509 IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
510 info = ierr
511 ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
512 info = ierr - n
513 ELSE
514 info = n + 1
515 END IF
516 GO TO 30
517 END IF
518*
519* Sort eigenvalues ALPHA/BETA if desired
520*
521 IF( wantst ) THEN
522*
523* Undo scaling on eigenvalues before selecting
524*
525 IF( ilascl )
526 $ CALL zlascl( 'G', 0, 0, anrm, anrmto, n, 1, alpha, n, ierr )
527 IF( ilbscl )
528 $ CALL zlascl( 'G', 0, 0, bnrm, bnrmto, n, 1, beta, n, ierr )
529*
530* Select eigenvalues
531*
532 DO 10 i = 1, n
533 bwork( i ) = selctg( alpha( i ), beta( i ) )
534 10 CONTINUE
535*
536 CALL ztgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb, alpha,
537 $ beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl, pvsr,
538 $ dif, work( iwrk ), lwork-iwrk+1, idum, 1, ierr )
539 IF( ierr.EQ.1 )
540 $ info = n + 3
541*
542 END IF
543*
544* Apply back-permutation to VSL and VSR
545*
546 IF( ilvsl )
547 $ CALL zggbak( 'P', 'L', n, ilo, ihi, rwork( ileft ),
548 $ rwork( iright ), n, vsl, ldvsl, ierr )
549 IF( ilvsr )
550 $ CALL zggbak( 'P', 'R', n, ilo, ihi, rwork( ileft ),
551 $ rwork( iright ), n, vsr, ldvsr, ierr )
552*
553* Undo scaling
554*
555 IF( ilascl ) THEN
556 CALL zlascl( 'U', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
557 CALL zlascl( 'G', 0, 0, anrmto, anrm, n, 1, alpha, n, ierr )
558 END IF
559*
560 IF( ilbscl ) THEN
561 CALL zlascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
562 CALL zlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
563 END IF
564*
565 IF( wantst ) THEN
566*
567* Check if reordering is correct
568*
569 lastsl = .true.
570 sdim = 0
571 DO 20 i = 1, n
572 cursl = selctg( alpha( i ), beta( i ) )
573 IF( cursl )
574 $ sdim = sdim + 1
575 IF( cursl .AND. .NOT.lastsl )
576 $ info = n + 2
577 lastsl = cursl
578 20 CONTINUE
579*
580 END IF
581*
582 30 CONTINUE
583*
584 work( 1 ) = dcmplx( lwkopt )
585*
586 RETURN
587*
588* End of ZGGES3
589*
590 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 zgges3(jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, rwork, bwork, info)
ZGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE ...
Definition zgges3.f:269
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 ztgsen(ijob, wantq, wantz, select, n, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork, info)
ZTGSEN
Definition ztgsen.f:433
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