LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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cggevx.f
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1*> \brief <b> CGGEVX 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 CGGEVX + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cggevx.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cggevx.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cggevx.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CGGEVX( 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* REAL ABNRM, BBNRM
30* ..
31* .. Array Arguments ..
32* LOGICAL BWORK( * )
33* INTEGER IWORK( * )
34* REAL LSCALE( * ), RCONDE( * ), RCONDV( * ),
35* $ RSCALE( * ), RWORK( * )
36* COMPLEX 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*> CGGEVX 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 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 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 array, dimension (N)
153*> \endverbatim
154*>
155*> \param[out] BETA
156*> \verbatim
157*> BETA is COMPLEX 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 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 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 REAL 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 REAL 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 REAL
250*> The one-norm of the balanced matrix A.
251*> \endverbatim
252*>
253*> \param[out] BBNRM
254*> \verbatim
255*> BBNRM is REAL
256*> The one-norm of the balanced matrix B.
257*> \endverbatim
258*>
259*> \param[out] RCONDE
260*> \verbatim
261*> RCONDE is REAL 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 REAL array, dimension (N)
270*> If SENSE = '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 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 REAL 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 CHGEQZ.
327*> =N+2: error return from CTGEVC.
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 ggevx
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 cggevx( 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 REAL ABNRM, BBNRM
383* ..
384* .. Array Arguments ..
385 LOGICAL BWORK( * )
386 INTEGER IWORK( * )
387 REAL LSCALE( * ), RCONDE( * ), RCONDV( * ),
388 $ rscale( * ), rwork( * )
389 COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
390 $ beta( * ), vl( ldvl, * ), vr( ldvr, * ),
391 $ work( * )
392* ..
393*
394* =====================================================================
395*
396* .. Parameters ..
397 REAL ZERO, ONE
398 PARAMETER ( ZERO = 0.0e+0, one = 1.0e+0 )
399 COMPLEX CZERO, CONE
400 parameter( czero = ( 0.0e+0, 0.0e+0 ),
401 $ cone = ( 1.0e+0, 0.0e+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 REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
410 $ smlnum, temp
411 COMPLEX X
412* ..
413* .. Local Arrays ..
414 LOGICAL LDUMMA( 1 )
415* ..
416* .. External Subroutines ..
417 EXTERNAL cgeqrf, cggbak, cggbal, cgghrd, chgeqz, clacpy,
419 $ slascl, xerbla
420* ..
421* .. External Functions ..
422 LOGICAL LSAME
423 INTEGER ILAENV
424 REAL CLANGE, SLAMCH, SROUNDUP_LWORK
425 EXTERNAL lsame, ilaenv, clange, slamch, sroundup_lwork
426* ..
427* .. Intrinsic Functions ..
428 INTRINSIC abs, aimag, max, real, sqrt
429* ..
430* .. Statement Functions ..
431 REAL ABS1
432* ..
433* .. Statement Function definitions ..
434 abs1( x ) = abs( real( x ) ) + abs( aimag( 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, 'CGEQRF', ' ', n, 1, n, 0 ) )
517 maxwrk = max( maxwrk,
518 $ n + n*ilaenv( 1, 'CUNMQR', ' ', n, 1, n, 0 ) )
519 IF( ilvl ) THEN
520 maxwrk = max( maxwrk, n +
521 $ n*ilaenv( 1, 'CUNGQR', ' ', n, 1, n, 0 ) )
522 END IF
523 END IF
524 work( 1 ) = sroundup_lwork(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( 'CGGEVX', -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 = slamch( 'P' )
546 smlnum = slamch( 'S' )
547 bignum = one / smlnum
548 smlnum = sqrt( smlnum ) / eps
549 bignum = one / smlnum
550*
551* Scale A if max element outside range [SMLNUM,BIGNUM]
552*
553 anrm = clange( 'M', n, n, a, lda, rwork )
554 ilascl = .false.
555 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
556 anrmto = smlnum
557 ilascl = .true.
558 ELSE IF( anrm.GT.bignum ) THEN
559 anrmto = bignum
560 ilascl = .true.
561 END IF
562 IF( ilascl )
563 $ CALL clascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
564*
565* Scale B if max element outside range [SMLNUM,BIGNUM]
566*
567 bnrm = clange( 'M', n, n, b, ldb, rwork )
568 ilbscl = .false.
569 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
570 bnrmto = smlnum
571 ilbscl = .true.
572 ELSE IF( bnrm.GT.bignum ) THEN
573 bnrmto = bignum
574 ilbscl = .true.
575 END IF
576 IF( ilbscl )
577 $ CALL clascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
578*
579* Permute and/or balance the matrix pair (A,B)
580* (Real Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise)
581*
582 CALL cggbal( balanc, n, a, lda, b, ldb, ilo, ihi, lscale, rscale,
583 $ rwork, ierr )
584*
585* Compute ABNRM and BBNRM
586*
587 abnrm = clange( '1', n, n, a, lda, rwork( 1 ) )
588 IF( ilascl ) THEN
589 rwork( 1 ) = abnrm
590 CALL slascl( 'G', 0, 0, anrmto, anrm, 1, 1, rwork( 1 ), 1,
591 $ ierr )
592 abnrm = rwork( 1 )
593 END IF
594*
595 bbnrm = clange( '1', n, n, b, ldb, rwork( 1 ) )
596 IF( ilbscl ) THEN
597 rwork( 1 ) = bbnrm
598 CALL slascl( 'G', 0, 0, bnrmto, bnrm, 1, 1, rwork( 1 ), 1,
599 $ ierr )
600 bbnrm = rwork( 1 )
601 END IF
602*
603* Reduce B to triangular form (QR decomposition of B)
604* (Complex Workspace: need N, prefer N*NB )
605*
606 irows = ihi + 1 - ilo
607 IF( ilv .OR. .NOT.wantsn ) THEN
608 icols = n + 1 - ilo
609 ELSE
610 icols = irows
611 END IF
612 itau = 1
613 iwrk = itau + irows
614 CALL cgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
615 $ work( iwrk ), lwork+1-iwrk, ierr )
616*
617* Apply the unitary transformation to A
618* (Complex Workspace: need N, prefer N*NB)
619*
620 CALL cunmqr( 'L', 'C', irows, icols, irows, b( ilo, ilo ), ldb,
621 $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
622 $ lwork+1-iwrk, ierr )
623*
624* Initialize VL and/or VR
625* (Workspace: need N, prefer N*NB)
626*
627 IF( ilvl ) THEN
628 CALL claset( 'Full', n, n, czero, cone, vl, ldvl )
629 IF( irows.GT.1 ) THEN
630 CALL clacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
631 $ vl( ilo+1, ilo ), ldvl )
632 END IF
633 CALL cungqr( irows, irows, irows, vl( ilo, ilo ), ldvl,
634 $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
635 END IF
636*
637 IF( ilvr )
638 $ CALL claset( 'Full', n, n, czero, cone, vr, ldvr )
639*
640* Reduce to generalized Hessenberg form
641* (Workspace: none needed)
642*
643 IF( ilv .OR. .NOT.wantsn ) THEN
644*
645* Eigenvectors requested -- work on whole matrix.
646*
647 CALL cgghrd( jobvl, jobvr, n, ilo, ihi, a, lda, b, ldb, vl,
648 $ ldvl, vr, ldvr, ierr )
649 ELSE
650 CALL cgghrd( 'N', 'N', irows, 1, irows, a( ilo, ilo ), lda,
651 $ b( ilo, ilo ), ldb, vl, ldvl, vr, ldvr, ierr )
652 END IF
653*
654* Perform QZ algorithm (Compute eigenvalues, and optionally, the
655* Schur forms and Schur vectors)
656* (Complex Workspace: need N)
657* (Real Workspace: need N)
658*
659 iwrk = itau
660 IF( ilv .OR. .NOT.wantsn ) THEN
661 chtemp = 'S'
662 ELSE
663 chtemp = 'E'
664 END IF
665*
666 CALL chgeqz( chtemp, jobvl, jobvr, n, ilo, ihi, a, lda, b, ldb,
667 $ alpha, beta, vl, ldvl, vr, ldvr, work( iwrk ),
668 $ lwork+1-iwrk, rwork, ierr )
669 IF( ierr.NE.0 ) THEN
670 IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
671 info = ierr
672 ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
673 info = ierr - n
674 ELSE
675 info = n + 1
676 END IF
677 GO TO 90
678 END IF
679*
680* Compute Eigenvectors and estimate condition numbers if desired
681* CTGEVC: (Complex Workspace: need 2*N )
682* (Real Workspace: need 2*N )
683* CTGSNA: (Complex Workspace: need 2*N*N if SENSE='V' or 'B')
684* (Integer Workspace: need N+2 )
685*
686 IF( ilv .OR. .NOT.wantsn ) THEN
687 IF( ilv ) THEN
688 IF( ilvl ) THEN
689 IF( ilvr ) THEN
690 chtemp = 'B'
691 ELSE
692 chtemp = 'L'
693 END IF
694 ELSE
695 chtemp = 'R'
696 END IF
697*
698 CALL ctgevc( chtemp, 'B', ldumma, n, a, lda, b, ldb, vl,
699 $ ldvl, vr, ldvr, n, in, work( iwrk ), rwork,
700 $ ierr )
701 IF( ierr.NE.0 ) THEN
702 info = n + 2
703 GO TO 90
704 END IF
705 END IF
706*
707 IF( .NOT.wantsn ) THEN
708*
709* compute eigenvectors (CTGEVC) and estimate condition
710* numbers (CTGSNA). Note that the definition of the condition
711* number is not invariant under transformation (u,v) to
712* (Q*u, Z*v), where (u,v) are eigenvectors of the generalized
713* Schur form (S,T), Q and Z are orthogonal matrices. In order
714* to avoid using extra 2*N*N workspace, we have to
715* re-calculate eigenvectors and estimate the condition numbers
716* one at a time.
717*
718 DO 20 i = 1, n
719*
720 DO 10 j = 1, n
721 bwork( j ) = .false.
722 10 CONTINUE
723 bwork( i ) = .true.
724*
725 iwrk = n + 1
726 iwrk1 = iwrk + n
727*
728 IF( wantse .OR. wantsb ) THEN
729 CALL ctgevc( 'B', 'S', bwork, n, a, lda, b, ldb,
730 $ work( 1 ), n, work( iwrk ), n, 1, m,
731 $ work( iwrk1 ), rwork, ierr )
732 IF( ierr.NE.0 ) THEN
733 info = n + 2
734 GO TO 90
735 END IF
736 END IF
737*
738 CALL ctgsna( sense, 'S', bwork, n, a, lda, b, ldb,
739 $ work( 1 ), n, work( iwrk ), n, rconde( i ),
740 $ rcondv( i ), 1, m, work( iwrk1 ),
741 $ lwork-iwrk1+1, iwork, ierr )
742*
743 20 CONTINUE
744 END IF
745 END IF
746*
747* Undo balancing on VL and VR and normalization
748* (Workspace: none needed)
749*
750 IF( ilvl ) THEN
751 CALL cggbak( balanc, 'L', n, ilo, ihi, lscale, rscale, n, vl,
752 $ ldvl, ierr )
753*
754 DO 50 jc = 1, n
755 temp = zero
756 DO 30 jr = 1, n
757 temp = max( temp, abs1( vl( jr, jc ) ) )
758 30 CONTINUE
759 IF( temp.LT.smlnum )
760 $ GO TO 50
761 temp = one / temp
762 DO 40 jr = 1, n
763 vl( jr, jc ) = vl( jr, jc )*temp
764 40 CONTINUE
765 50 CONTINUE
766 END IF
767*
768 IF( ilvr ) THEN
769 CALL cggbak( balanc, 'R', n, ilo, ihi, lscale, rscale, n, vr,
770 $ ldvr, ierr )
771 DO 80 jc = 1, n
772 temp = zero
773 DO 60 jr = 1, n
774 temp = max( temp, abs1( vr( jr, jc ) ) )
775 60 CONTINUE
776 IF( temp.LT.smlnum )
777 $ GO TO 80
778 temp = one / temp
779 DO 70 jr = 1, n
780 vr( jr, jc ) = vr( jr, jc )*temp
781 70 CONTINUE
782 80 CONTINUE
783 END IF
784*
785* Undo scaling if necessary
786*
787 90 CONTINUE
788*
789 IF( ilascl )
790 $ CALL clascl( 'G', 0, 0, anrmto, anrm, n, 1, alpha, n, ierr )
791*
792 IF( ilbscl )
793 $ CALL clascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
794*
795 work( 1 ) = sroundup_lwork(maxwrk)
796 RETURN
797*
798* End of CGGEVX
799*
800 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 cggevx(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)
CGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
Definition cggevx.f:374
subroutine cgghrd(compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)
CGGHRD
Definition cgghrd.f:204
subroutine chgeqz(job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, info)
CHGEQZ
Definition chgeqz.f:284
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
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 slascl(type, kl, ku, cfrom, cto, m, n, a, lda, info)
SLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition slascl.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 ctgevc(side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, rwork, info)
CTGEVC
Definition ctgevc.f:219
subroutine ctgsna(job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr, ldvr, s, dif, mm, m, work, lwork, iwork, info)
CTGSNA
Definition ctgsna.f:311
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