LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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zggevx.f
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1*> \brief <b> ZGGEVX 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 ZGGEVX + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggevx.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zggevx.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zggevx.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
22* ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI,
23* LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV,
24* WORK, LWORK, RWORK, IWORK, BWORK, INFO )
25*
26* .. Scalar Arguments ..
27* CHARACTER BALANC, JOBVL, JOBVR, SENSE
28* INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
29* DOUBLE PRECISION ABNRM, BBNRM
30* ..
31* .. Array Arguments ..
32* LOGICAL BWORK( * )
33* INTEGER IWORK( * )
34* DOUBLE PRECISION LSCALE( * ), RCONDE( * ), RCONDV( * ),
35* $ RSCALE( * ), RWORK( * )
36* COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
37* $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
38* $ WORK( * )
39* ..
40*
41*
42*> \par Purpose:
43* =============
44*>
45*> \verbatim
46*>
47*> ZGGEVX computes for a pair of N-by-N complex nonsymmetric matrices
48*> (A,B) the generalized eigenvalues, and optionally, the left and/or
49*> right generalized eigenvectors.
50*>
51*> Optionally, it also computes a balancing transformation to improve
52*> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
53*> LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
54*> the eigenvalues (RCONDE), and reciprocal condition numbers for the
55*> right eigenvectors (RCONDV).
56*>
57*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
58*> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
59*> singular. It is usually represented as the pair (alpha,beta), as
60*> there is a reasonable interpretation for beta=0, and even for both
61*> being zero.
62*>
63*> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
64*> of (A,B) satisfies
65*> A * v(j) = lambda(j) * B * v(j) .
66*> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
67*> of (A,B) satisfies
68*> u(j)**H * A = lambda(j) * u(j)**H * B.
69*> where u(j)**H is the conjugate-transpose of u(j).
70*>
71*> \endverbatim
72*
73* Arguments:
74* ==========
75*
76*> \param[in] BALANC
77*> \verbatim
78*> BALANC is CHARACTER*1
79*> Specifies the balance option to be performed:
80*> = 'N': do not diagonally scale or permute;
81*> = 'P': permute only;
82*> = 'S': scale only;
83*> = 'B': both permute and scale.
84*> Computed reciprocal condition numbers will be for the
85*> matrices after permuting and/or balancing. Permuting does
86*> not change condition numbers (in exact arithmetic), but
87*> balancing does.
88*> \endverbatim
89*>
90*> \param[in] JOBVL
91*> \verbatim
92*> JOBVL is CHARACTER*1
93*> = 'N': do not compute the left generalized eigenvectors;
94*> = 'V': compute the left generalized eigenvectors.
95*> \endverbatim
96*>
97*> \param[in] JOBVR
98*> \verbatim
99*> JOBVR is CHARACTER*1
100*> = 'N': do not compute the right generalized eigenvectors;
101*> = 'V': compute the right generalized eigenvectors.
102*> \endverbatim
103*>
104*> \param[in] SENSE
105*> \verbatim
106*> SENSE is CHARACTER*1
107*> Determines which reciprocal condition numbers are computed.
108*> = 'N': none are computed;
109*> = 'E': computed for eigenvalues only;
110*> = 'V': computed for eigenvectors only;
111*> = 'B': computed for eigenvalues and eigenvectors.
112*> \endverbatim
113*>
114*> \param[in] N
115*> \verbatim
116*> N is INTEGER
117*> The order of the matrices A, B, VL, and VR. N >= 0.
118*> \endverbatim
119*>
120*> \param[in,out] A
121*> \verbatim
122*> A is COMPLEX*16 array, dimension (LDA, N)
123*> On entry, the matrix A in the pair (A,B).
124*> On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
125*> or both, then A contains the first part of the complex Schur
126*> form of the "balanced" versions of the input A and B.
127*> \endverbatim
128*>
129*> \param[in] LDA
130*> \verbatim
131*> LDA is INTEGER
132*> The leading dimension of A. LDA >= max(1,N).
133*> \endverbatim
134*>
135*> \param[in,out] B
136*> \verbatim
137*> B is COMPLEX*16 array, dimension (LDB, N)
138*> On entry, the matrix B in the pair (A,B).
139*> On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
140*> or both, then B contains the second part of the complex
141*> Schur form of the "balanced" versions of the input A and B.
142*> \endverbatim
143*>
144*> \param[in] LDB
145*> \verbatim
146*> LDB is INTEGER
147*> The leading dimension of B. LDB >= max(1,N).
148*> \endverbatim
149*>
150*> \param[out] ALPHA
151*> \verbatim
152*> ALPHA is COMPLEX*16 array, dimension (N)
153*> \endverbatim
154*>
155*> \param[out] BETA
156*> \verbatim
157*> BETA is COMPLEX*16 array, dimension (N)
158*> On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized
159*> eigenvalues.
160*>
161*> Note: the quotient ALPHA(j)/BETA(j) ) may easily over- or
162*> underflow, and BETA(j) may even be zero. Thus, the user
163*> should avoid naively computing the ratio ALPHA/BETA.
164*> However, ALPHA will be always less than and usually
165*> comparable with norm(A) in magnitude, and BETA always less
166*> than and usually comparable with norm(B).
167*> \endverbatim
168*>
169*> \param[out] VL
170*> \verbatim
171*> VL is COMPLEX*16 array, dimension (LDVL,N)
172*> If JOBVL = 'V', the left generalized eigenvectors u(j) are
173*> stored one after another in the columns of VL, in the same
174*> order as their eigenvalues.
175*> Each eigenvector will be scaled so the largest component
176*> will have abs(real part) + abs(imag. part) = 1.
177*> Not referenced if JOBVL = 'N'.
178*> \endverbatim
179*>
180*> \param[in] LDVL
181*> \verbatim
182*> LDVL is INTEGER
183*> The leading dimension of the matrix VL. LDVL >= 1, and
184*> if JOBVL = 'V', LDVL >= N.
185*> \endverbatim
186*>
187*> \param[out] VR
188*> \verbatim
189*> VR is COMPLEX*16 array, dimension (LDVR,N)
190*> If JOBVR = 'V', the right generalized eigenvectors v(j) are
191*> stored one after another in the columns of VR, in the same
192*> order as their eigenvalues.
193*> Each eigenvector will be scaled so the largest component
194*> will have abs(real part) + abs(imag. part) = 1.
195*> Not referenced if JOBVR = 'N'.
196*> \endverbatim
197*>
198*> \param[in] LDVR
199*> \verbatim
200*> LDVR is INTEGER
201*> The leading dimension of the matrix VR. LDVR >= 1, and
202*> if JOBVR = 'V', LDVR >= N.
203*> \endverbatim
204*>
205*> \param[out] ILO
206*> \verbatim
207*> ILO is INTEGER
208*> \endverbatim
209*>
210*> \param[out] IHI
211*> \verbatim
212*> IHI is INTEGER
213*> ILO and IHI are integer values such that on exit
214*> A(i,j) = 0 and B(i,j) = 0 if i > j and
215*> j = 1,...,ILO-1 or i = IHI+1,...,N.
216*> If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
217*> \endverbatim
218*>
219*> \param[out] LSCALE
220*> \verbatim
221*> LSCALE is DOUBLE PRECISION array, dimension (N)
222*> Details of the permutations and scaling factors applied
223*> to the left side of A and B. If PL(j) is the index of the
224*> row interchanged with row j, and DL(j) is the scaling
225*> factor applied to row j, then
226*> LSCALE(j) = PL(j) for j = 1,...,ILO-1
227*> = DL(j) for j = ILO,...,IHI
228*> = PL(j) for j = IHI+1,...,N.
229*> The order in which the interchanges are made is N to IHI+1,
230*> then 1 to ILO-1.
231*> \endverbatim
232*>
233*> \param[out] RSCALE
234*> \verbatim
235*> RSCALE is DOUBLE PRECISION array, dimension (N)
236*> Details of the permutations and scaling factors applied
237*> to the right side of A and B. If PR(j) is the index of the
238*> column interchanged with column j, and DR(j) is the scaling
239*> factor applied to column j, then
240*> RSCALE(j) = PR(j) for j = 1,...,ILO-1
241*> = DR(j) for j = ILO,...,IHI
242*> = PR(j) for j = IHI+1,...,N
243*> The order in which the interchanges are made is N to IHI+1,
244*> then 1 to ILO-1.
245*> \endverbatim
246*>
247*> \param[out] ABNRM
248*> \verbatim
249*> ABNRM is DOUBLE PRECISION
250*> The one-norm of the balanced matrix A.
251*> \endverbatim
252*>
253*> \param[out] BBNRM
254*> \verbatim
255*> BBNRM is DOUBLE PRECISION
256*> The one-norm of the balanced matrix B.
257*> \endverbatim
258*>
259*> \param[out] RCONDE
260*> \verbatim
261*> RCONDE is DOUBLE PRECISION array, dimension (N)
262*> If SENSE = 'E' or 'B', the reciprocal condition numbers of
263*> the eigenvalues, stored in consecutive elements of the array.
264*> If SENSE = 'N' or 'V', RCONDE is not referenced.
265*> \endverbatim
266*>
267*> \param[out] RCONDV
268*> \verbatim
269*> RCONDV is DOUBLE PRECISION array, dimension (N)
270*> If JOB = 'V' or 'B', the estimated reciprocal condition
271*> numbers of the eigenvectors, stored in consecutive elements
272*> of the array. If the eigenvalues cannot be reordered to
273*> compute RCONDV(j), RCONDV(j) is set to 0; this can only occur
274*> when the true value would be very small anyway.
275*> If SENSE = 'N' or 'E', RCONDV is not referenced.
276*> \endverbatim
277*>
278*> \param[out] WORK
279*> \verbatim
280*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
281*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
282*> \endverbatim
283*>
284*> \param[in] LWORK
285*> \verbatim
286*> LWORK is INTEGER
287*> The dimension of the array WORK. LWORK >= max(1,2*N).
288*> If SENSE = 'E', LWORK >= max(1,4*N).
289*> If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*N).
290*>
291*> If LWORK = -1, then a workspace query is assumed; the routine
292*> only calculates the optimal size of the WORK array, returns
293*> this value as the first entry of the WORK array, and no error
294*> message related to LWORK is issued by XERBLA.
295*> \endverbatim
296*>
297*> \param[out] RWORK
298*> \verbatim
299*> RWORK is DOUBLE PRECISION array, dimension (lrwork)
300*> lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B',
301*> and at least max(1,2*N) otherwise.
302*> Real workspace.
303*> \endverbatim
304*>
305*> \param[out] IWORK
306*> \verbatim
307*> IWORK is INTEGER array, dimension (N+2)
308*> If SENSE = 'E', IWORK is not referenced.
309*> \endverbatim
310*>
311*> \param[out] BWORK
312*> \verbatim
313*> BWORK is LOGICAL array, dimension (N)
314*> If SENSE = 'N', BWORK is not referenced.
315*> \endverbatim
316*>
317*> \param[out] INFO
318*> \verbatim
319*> INFO is INTEGER
320*> = 0: successful exit
321*> < 0: if INFO = -i, the i-th argument had an illegal value.
322*> = 1,...,N:
323*> The QZ iteration failed. No eigenvectors have been
324*> calculated, but ALPHA(j) and BETA(j) should be correct
325*> for j=INFO+1,...,N.
326*> > N: =N+1: other than QZ iteration failed in ZHGEQZ.
327*> =N+2: error return from ZTGEVC.
328*> \endverbatim
329*
330* Authors:
331* ========
332*
333*> \author Univ. of Tennessee
334*> \author Univ. of California Berkeley
335*> \author Univ. of Colorado Denver
336*> \author NAG Ltd.
337*
338*> \ingroup complex16GEeigen
339*
340*> \par Further Details:
341* =====================
342*>
343*> \verbatim
344*>
345*> Balancing a matrix pair (A,B) includes, first, permuting rows and
346*> columns to isolate eigenvalues, second, applying diagonal similarity
347*> transformation to the rows and columns to make the rows and columns
348*> as close in norm as possible. The computed reciprocal condition
349*> numbers correspond to the balanced matrix. Permuting rows and columns
350*> will not change the condition numbers (in exact arithmetic) but
351*> diagonal scaling will. For further explanation of balancing, see
352*> section 4.11.1.2 of LAPACK Users' Guide.
353*>
354*> An approximate error bound on the chordal distance between the i-th
355*> computed generalized eigenvalue w and the corresponding exact
356*> eigenvalue lambda is
357*>
358*> chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
359*>
360*> An approximate error bound for the angle between the i-th computed
361*> eigenvector VL(i) or VR(i) is given by
362*>
363*> EPS * norm(ABNRM, BBNRM) / DIF(i).
364*>
365*> For further explanation of the reciprocal condition numbers RCONDE
366*> and RCONDV, see section 4.11 of LAPACK User's Guide.
367*> \endverbatim
368*>
369* =====================================================================
370 SUBROUTINE zggevx( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
371 $ ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI,
372 $ LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV,
373 $ WORK, LWORK, RWORK, IWORK, BWORK, INFO )
374*
375* -- LAPACK driver routine --
376* -- LAPACK is a software package provided by Univ. of Tennessee, --
377* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
378*
379* .. Scalar Arguments ..
380 CHARACTER BALANC, JOBVL, JOBVR, SENSE
381 INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
382 DOUBLE PRECISION ABNRM, BBNRM
383* ..
384* .. Array Arguments ..
385 LOGICAL BWORK( * )
386 INTEGER IWORK( * )
387 DOUBLE PRECISION LSCALE( * ), RCONDE( * ), RCONDV( * ),
388 $ rscale( * ), rwork( * )
389 COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
390 $ beta( * ), vl( ldvl, * ), vr( ldvr, * ),
391 $ work( * )
392* ..
393*
394* =====================================================================
395*
396* .. Parameters ..
397 DOUBLE PRECISION ZERO, ONE
398 PARAMETER ( ZERO = 0.0d+0, one = 1.0d+0 )
399 COMPLEX*16 CZERO, CONE
400 parameter( czero = ( 0.0d+0, 0.0d+0 ),
401 $ cone = ( 1.0d+0, 0.0d+0 ) )
402* ..
403* .. Local Scalars ..
404 LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, NOSCL,
405 $ WANTSB, WANTSE, WANTSN, WANTSV
406 CHARACTER CHTEMP
407 INTEGER I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS,
408 $ ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK, MINWRK
409 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
410 $ smlnum, temp
411 COMPLEX*16 X
412* ..
413* .. Local Arrays ..
414 LOGICAL LDUMMA( 1 )
415* ..
416* .. External Subroutines ..
417 EXTERNAL dlabad, dlascl, xerbla, zgeqrf, zggbak, zggbal,
420* ..
421* .. External Functions ..
422 LOGICAL LSAME
423 INTEGER ILAENV
424 DOUBLE PRECISION DLAMCH, ZLANGE
425 EXTERNAL lsame, ilaenv, dlamch, zlange
426* ..
427* .. Intrinsic Functions ..
428 INTRINSIC abs, dble, dimag, max, sqrt
429* ..
430* .. Statement Functions ..
431 DOUBLE PRECISION ABS1
432* ..
433* .. Statement Function definitions ..
434 abs1( x ) = abs( dble( x ) ) + abs( dimag( x ) )
435* ..
436* .. Executable Statements ..
437*
438* Decode the input arguments
439*
440 IF( lsame( jobvl, 'N' ) ) THEN
441 ijobvl = 1
442 ilvl = .false.
443 ELSE IF( lsame( jobvl, 'V' ) ) THEN
444 ijobvl = 2
445 ilvl = .true.
446 ELSE
447 ijobvl = -1
448 ilvl = .false.
449 END IF
450*
451 IF( lsame( jobvr, 'N' ) ) THEN
452 ijobvr = 1
453 ilvr = .false.
454 ELSE IF( lsame( jobvr, 'V' ) ) THEN
455 ijobvr = 2
456 ilvr = .true.
457 ELSE
458 ijobvr = -1
459 ilvr = .false.
460 END IF
461 ilv = ilvl .OR. ilvr
462*
463 noscl = lsame( balanc, 'N' ) .OR. lsame( balanc, 'P' )
464 wantsn = lsame( sense, 'N' )
465 wantse = lsame( sense, 'E' )
466 wantsv = lsame( sense, 'V' )
467 wantsb = lsame( sense, 'B' )
468*
469* Test the input arguments
470*
471 info = 0
472 lquery = ( lwork.EQ.-1 )
473 IF( .NOT.( noscl .OR. lsame( balanc,'S' ) .OR.
474 $ lsame( balanc, 'B' ) ) ) THEN
475 info = -1
476 ELSE IF( ijobvl.LE.0 ) THEN
477 info = -2
478 ELSE IF( ijobvr.LE.0 ) THEN
479 info = -3
480 ELSE IF( .NOT.( wantsn .OR. wantse .OR. wantsb .OR. wantsv ) )
481 $ THEN
482 info = -4
483 ELSE IF( n.LT.0 ) THEN
484 info = -5
485 ELSE IF( lda.LT.max( 1, n ) ) THEN
486 info = -7
487 ELSE IF( ldb.LT.max( 1, n ) ) THEN
488 info = -9
489 ELSE IF( ldvl.LT.1 .OR. ( ilvl .AND. ldvl.LT.n ) ) THEN
490 info = -13
491 ELSE IF( ldvr.LT.1 .OR. ( ilvr .AND. ldvr.LT.n ) ) THEN
492 info = -15
493 END IF
494*
495* Compute workspace
496* (Note: Comments in the code beginning "Workspace:" describe the
497* minimal amount of workspace needed at that point in the code,
498* as well as the preferred amount for good performance.
499* NB refers to the optimal block size for the immediately
500* following subroutine, as returned by ILAENV. The workspace is
501* computed assuming ILO = 1 and IHI = N, the worst case.)
502*
503 IF( info.EQ.0 ) THEN
504 IF( n.EQ.0 ) THEN
505 minwrk = 1
506 maxwrk = 1
507 ELSE
508 minwrk = 2*n
509 IF( wantse ) THEN
510 minwrk = 4*n
511 ELSE IF( wantsv .OR. wantsb ) THEN
512 minwrk = 2*n*( n + 1)
513 END IF
514 maxwrk = minwrk
515 maxwrk = max( maxwrk,
516 $ n + n*ilaenv( 1, 'ZGEQRF', ' ', n, 1, n, 0 ) )
517 maxwrk = max( maxwrk,
518 $ n + n*ilaenv( 1, 'ZUNMQR', ' ', n, 1, n, 0 ) )
519 IF( ilvl ) THEN
520 maxwrk = max( maxwrk, n +
521 $ n*ilaenv( 1, 'ZUNGQR', ' ', n, 1, n, 0 ) )
522 END IF
523 END IF
524 work( 1 ) = maxwrk
525*
526 IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
527 info = -25
528 END IF
529 END IF
530*
531 IF( info.NE.0 ) THEN
532 CALL xerbla( 'ZGGEVX', -info )
533 RETURN
534 ELSE IF( lquery ) THEN
535 RETURN
536 END IF
537*
538* Quick return if possible
539*
540 IF( n.EQ.0 )
541 $ RETURN
542*
543* Get machine constants
544*
545 eps = dlamch( 'P' )
546 smlnum = dlamch( 'S' )
547 bignum = one / smlnum
548 CALL dlabad( smlnum, bignum )
549 smlnum = sqrt( smlnum ) / eps
550 bignum = one / smlnum
551*
552* Scale A if max element outside range [SMLNUM,BIGNUM]
553*
554 anrm = zlange( 'M', n, n, a, lda, rwork )
555 ilascl = .false.
556 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
557 anrmto = smlnum
558 ilascl = .true.
559 ELSE IF( anrm.GT.bignum ) THEN
560 anrmto = bignum
561 ilascl = .true.
562 END IF
563 IF( ilascl )
564 $ CALL zlascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
565*
566* Scale B if max element outside range [SMLNUM,BIGNUM]
567*
568 bnrm = zlange( 'M', n, n, b, ldb, rwork )
569 ilbscl = .false.
570 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
571 bnrmto = smlnum
572 ilbscl = .true.
573 ELSE IF( bnrm.GT.bignum ) THEN
574 bnrmto = bignum
575 ilbscl = .true.
576 END IF
577 IF( ilbscl )
578 $ CALL zlascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
579*
580* Permute and/or balance the matrix pair (A,B)
581* (Real Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise)
582*
583 CALL zggbal( balanc, n, a, lda, b, ldb, ilo, ihi, lscale, rscale,
584 $ rwork, ierr )
585*
586* Compute ABNRM and BBNRM
587*
588 abnrm = zlange( '1', n, n, a, lda, rwork( 1 ) )
589 IF( ilascl ) THEN
590 rwork( 1 ) = abnrm
591 CALL dlascl( 'G', 0, 0, anrmto, anrm, 1, 1, rwork( 1 ), 1,
592 $ ierr )
593 abnrm = rwork( 1 )
594 END IF
595*
596 bbnrm = zlange( '1', n, n, b, ldb, rwork( 1 ) )
597 IF( ilbscl ) THEN
598 rwork( 1 ) = bbnrm
599 CALL dlascl( 'G', 0, 0, bnrmto, bnrm, 1, 1, rwork( 1 ), 1,
600 $ ierr )
601 bbnrm = rwork( 1 )
602 END IF
603*
604* Reduce B to triangular form (QR decomposition of B)
605* (Complex Workspace: need N, prefer N*NB )
606*
607 irows = ihi + 1 - ilo
608 IF( ilv .OR. .NOT.wantsn ) THEN
609 icols = n + 1 - ilo
610 ELSE
611 icols = irows
612 END IF
613 itau = 1
614 iwrk = itau + irows
615 CALL zgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
616 $ work( iwrk ), lwork+1-iwrk, ierr )
617*
618* Apply the unitary transformation to A
619* (Complex Workspace: need N, prefer N*NB)
620*
621 CALL zunmqr( 'L', 'C', irows, icols, irows, b( ilo, ilo ), ldb,
622 $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
623 $ lwork+1-iwrk, ierr )
624*
625* Initialize VL and/or VR
626* (Workspace: need N, prefer N*NB)
627*
628 IF( ilvl ) THEN
629 CALL zlaset( 'Full', n, n, czero, cone, vl, ldvl )
630 IF( irows.GT.1 ) THEN
631 CALL zlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
632 $ vl( ilo+1, ilo ), ldvl )
633 END IF
634 CALL zungqr( irows, irows, irows, vl( ilo, ilo ), ldvl,
635 $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
636 END IF
637*
638 IF( ilvr )
639 $ CALL zlaset( 'Full', n, n, czero, cone, vr, ldvr )
640*
641* Reduce to generalized Hessenberg form
642* (Workspace: none needed)
643*
644 IF( ilv .OR. .NOT.wantsn ) THEN
645*
646* Eigenvectors requested -- work on whole matrix.
647*
648 CALL zgghrd( jobvl, jobvr, n, ilo, ihi, a, lda, b, ldb, vl,
649 $ ldvl, vr, ldvr, ierr )
650 ELSE
651 CALL zgghrd( 'N', 'N', irows, 1, irows, a( ilo, ilo ), lda,
652 $ b( ilo, ilo ), ldb, vl, ldvl, vr, ldvr, ierr )
653 END IF
654*
655* Perform QZ algorithm (Compute eigenvalues, and optionally, the
656* Schur forms and Schur vectors)
657* (Complex Workspace: need N)
658* (Real Workspace: need N)
659*
660 iwrk = itau
661 IF( ilv .OR. .NOT.wantsn ) THEN
662 chtemp = 'S'
663 ELSE
664 chtemp = 'E'
665 END IF
666*
667 CALL zhgeqz( chtemp, jobvl, jobvr, n, ilo, ihi, a, lda, b, ldb,
668 $ alpha, beta, vl, ldvl, vr, ldvr, work( iwrk ),
669 $ lwork+1-iwrk, rwork, ierr )
670 IF( ierr.NE.0 ) THEN
671 IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
672 info = ierr
673 ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
674 info = ierr - n
675 ELSE
676 info = n + 1
677 END IF
678 GO TO 90
679 END IF
680*
681* Compute Eigenvectors and estimate condition numbers if desired
682* ZTGEVC: (Complex Workspace: need 2*N )
683* (Real Workspace: need 2*N )
684* ZTGSNA: (Complex Workspace: need 2*N*N if SENSE='V' or 'B')
685* (Integer Workspace: need N+2 )
686*
687 IF( ilv .OR. .NOT.wantsn ) THEN
688 IF( ilv ) THEN
689 IF( ilvl ) THEN
690 IF( ilvr ) THEN
691 chtemp = 'B'
692 ELSE
693 chtemp = 'L'
694 END IF
695 ELSE
696 chtemp = 'R'
697 END IF
698*
699 CALL ztgevc( chtemp, 'B', ldumma, n, a, lda, b, ldb, vl,
700 $ ldvl, vr, ldvr, n, in, work( iwrk ), rwork,
701 $ ierr )
702 IF( ierr.NE.0 ) THEN
703 info = n + 2
704 GO TO 90
705 END IF
706 END IF
707*
708 IF( .NOT.wantsn ) THEN
709*
710* compute eigenvectors (ZTGEVC) and estimate condition
711* numbers (ZTGSNA). Note that the definition of the condition
712* number is not invariant under transformation (u,v) to
713* (Q*u, Z*v), where (u,v) are eigenvectors of the generalized
714* Schur form (S,T), Q and Z are orthogonal matrices. In order
715* to avoid using extra 2*N*N workspace, we have to
716* re-calculate eigenvectors and estimate the condition numbers
717* one at a time.
718*
719 DO 20 i = 1, n
720*
721 DO 10 j = 1, n
722 bwork( j ) = .false.
723 10 CONTINUE
724 bwork( i ) = .true.
725*
726 iwrk = n + 1
727 iwrk1 = iwrk + n
728*
729 IF( wantse .OR. wantsb ) THEN
730 CALL ztgevc( 'B', 'S', bwork, n, a, lda, b, ldb,
731 $ work( 1 ), n, work( iwrk ), n, 1, m,
732 $ work( iwrk1 ), rwork, ierr )
733 IF( ierr.NE.0 ) THEN
734 info = n + 2
735 GO TO 90
736 END IF
737 END IF
738*
739 CALL ztgsna( sense, 'S', bwork, n, a, lda, b, ldb,
740 $ work( 1 ), n, work( iwrk ), n, rconde( i ),
741 $ rcondv( i ), 1, m, work( iwrk1 ),
742 $ lwork-iwrk1+1, iwork, ierr )
743*
744 20 CONTINUE
745 END IF
746 END IF
747*
748* Undo balancing on VL and VR and normalization
749* (Workspace: none needed)
750*
751 IF( ilvl ) THEN
752 CALL zggbak( balanc, 'L', n, ilo, ihi, lscale, rscale, n, vl,
753 $ ldvl, ierr )
754*
755 DO 50 jc = 1, n
756 temp = zero
757 DO 30 jr = 1, n
758 temp = max( temp, abs1( vl( jr, jc ) ) )
759 30 CONTINUE
760 IF( temp.LT.smlnum )
761 $ GO TO 50
762 temp = one / temp
763 DO 40 jr = 1, n
764 vl( jr, jc ) = vl( jr, jc )*temp
765 40 CONTINUE
766 50 CONTINUE
767 END IF
768*
769 IF( ilvr ) THEN
770 CALL zggbak( balanc, 'R', n, ilo, ihi, lscale, rscale, n, vr,
771 $ ldvr, ierr )
772 DO 80 jc = 1, n
773 temp = zero
774 DO 60 jr = 1, n
775 temp = max( temp, abs1( vr( jr, jc ) ) )
776 60 CONTINUE
777 IF( temp.LT.smlnum )
778 $ GO TO 80
779 temp = one / temp
780 DO 70 jr = 1, n
781 vr( jr, jc ) = vr( jr, jc )*temp
782 70 CONTINUE
783 80 CONTINUE
784 END IF
785*
786* Undo scaling if necessary
787*
788 90 CONTINUE
789*
790 IF( ilascl )
791 $ CALL zlascl( 'G', 0, 0, anrmto, anrm, n, 1, alpha, n, ierr )
792*
793 IF( ilbscl )
794 $ CALL zlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
795*
796 work( 1 ) = maxwrk
797 RETURN
798*
799* End of ZGGEVX
800*
801 END
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:74
subroutine dlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: dlascl.f:143
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
ZGGBAL
Definition: zggbal.f:177
subroutine zggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
ZGGBAK
Definition: zggbak.f:148
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 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 zggevx(BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK, LWORK, RWORK, IWORK, BWORK, INFO)
ZGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
Definition: zggevx.f:374
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 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 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 zungqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
ZUNGQR
Definition: zungqr.f:128
subroutine ztgsna(JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO)
ZTGSNA
Definition: ztgsna.f:311
subroutine zgghrd(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
ZGGHRD
Definition: zgghrd.f:204
subroutine zunmqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
ZUNMQR
Definition: zunmqr.f:167
subroutine zgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
ZGEQRF VARIANT: left-looking Level 3 BLAS of the algorithm.
Definition: zgeqrf.f:152