LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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dggesx.f
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1*> \brief <b> DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors 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 DGGESX + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggesx.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggesx.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggesx.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE DGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA,
22* B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL,
23* VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK,
24* LIWORK, BWORK, INFO )
25*
26* .. Scalar Arguments ..
27* CHARACTER JOBVSL, JOBVSR, SENSE, SORT
28* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N,
29* $ SDIM
30* ..
31* .. Array Arguments ..
32* LOGICAL BWORK( * )
33* INTEGER IWORK( * )
34* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
35* $ B( LDB, * ), BETA( * ), RCONDE( 2 ),
36* $ RCONDV( 2 ), VSL( LDVSL, * ), VSR( LDVSR, * ),
37* $ WORK( * )
38* ..
39* .. Function Arguments ..
40* LOGICAL SELCTG
41* EXTERNAL SELCTG
42* ..
43*
44*
45*> \par Purpose:
46* =============
47*>
48*> \verbatim
49*>
50*> DGGESX computes for a pair of N-by-N real nonsymmetric matrices
51*> (A,B), the generalized eigenvalues, the real Schur form (S,T), and,
52*> optionally, the left and/or right matrices of Schur vectors (VSL and
53*> VSR). This gives the generalized Schur factorization
54*>
55*> (A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T )
56*>
57*> Optionally, it also orders the eigenvalues so that a selected cluster
58*> of eigenvalues appears in the leading diagonal blocks of the upper
59*> quasi-triangular matrix S and the upper triangular matrix T; computes
60*> a reciprocal condition number for the average of the selected
61*> eigenvalues (RCONDE); and computes a reciprocal condition number for
62*> the right and left deflating subspaces corresponding to the selected
63*> eigenvalues (RCONDV). The leading columns of VSL and VSR then form
64*> an orthonormal basis for the corresponding left and right eigenspaces
65*> (deflating subspaces).
66*>
67*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
68*> or a ratio alpha/beta = w, such that A - w*B is singular. It is
69*> usually represented as the pair (alpha,beta), as there is a
70*> reasonable interpretation for beta=0 or for both being zero.
71*>
72*> A pair of matrices (S,T) is in generalized real Schur form if T is
73*> upper triangular with non-negative diagonal and S is block upper
74*> triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond
75*> to real generalized eigenvalues, while 2-by-2 blocks of S will be
76*> "standardized" by making the corresponding elements of T have the
77*> form:
78*> [ a 0 ]
79*> [ 0 b ]
80*>
81*> and the pair of corresponding 2-by-2 blocks in S and T will have a
82*> complex conjugate pair of generalized eigenvalues.
83*>
84*> \endverbatim
85*
86* Arguments:
87* ==========
88*
89*> \param[in] JOBVSL
90*> \verbatim
91*> JOBVSL is CHARACTER*1
92*> = 'N': do not compute the left Schur vectors;
93*> = 'V': compute the left Schur vectors.
94*> \endverbatim
95*>
96*> \param[in] JOBVSR
97*> \verbatim
98*> JOBVSR is CHARACTER*1
99*> = 'N': do not compute the right Schur vectors;
100*> = 'V': compute the right Schur vectors.
101*> \endverbatim
102*>
103*> \param[in] SORT
104*> \verbatim
105*> SORT is CHARACTER*1
106*> Specifies whether or not to order the eigenvalues on the
107*> diagonal of the generalized Schur form.
108*> = 'N': Eigenvalues are not ordered;
109*> = 'S': Eigenvalues are ordered (see SELCTG).
110*> \endverbatim
111*>
112*> \param[in] SELCTG
113*> \verbatim
114*> SELCTG is a LOGICAL FUNCTION of three DOUBLE PRECISION arguments
115*> SELCTG must be declared EXTERNAL in the calling subroutine.
116*> If SORT = 'N', SELCTG is not referenced.
117*> If SORT = 'S', SELCTG is used to select eigenvalues to sort
118*> to the top left of the Schur form.
119*> An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
120*> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
121*> one of a complex conjugate pair of eigenvalues is selected,
122*> then both complex eigenvalues are selected.
123*> Note that a selected complex eigenvalue may no longer satisfy
124*> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering,
125*> since ordering may change the value of complex eigenvalues
126*> (especially if the eigenvalue is ill-conditioned), in this
127*> case INFO is set to N+3.
128*> \endverbatim
129*>
130*> \param[in] SENSE
131*> \verbatim
132*> SENSE is CHARACTER*1
133*> Determines which reciprocal condition numbers are computed.
134*> = 'N': None are computed;
135*> = 'E': Computed for average of selected eigenvalues only;
136*> = 'V': Computed for selected deflating subspaces only;
137*> = 'B': Computed for both.
138*> If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.
139*> \endverbatim
140*>
141*> \param[in] N
142*> \verbatim
143*> N is INTEGER
144*> The order of the matrices A, B, VSL, and VSR. N >= 0.
145*> \endverbatim
146*>
147*> \param[in,out] A
148*> \verbatim
149*> A is DOUBLE PRECISION array, dimension (LDA, N)
150*> On entry, the first of the pair of matrices.
151*> On exit, A has been overwritten by its generalized Schur
152*> form S.
153*> \endverbatim
154*>
155*> \param[in] LDA
156*> \verbatim
157*> LDA is INTEGER
158*> The leading dimension of A. LDA >= max(1,N).
159*> \endverbatim
160*>
161*> \param[in,out] B
162*> \verbatim
163*> B is DOUBLE PRECISION array, dimension (LDB, N)
164*> On entry, the second of the pair of matrices.
165*> On exit, B has been overwritten by its generalized Schur
166*> form T.
167*> \endverbatim
168*>
169*> \param[in] LDB
170*> \verbatim
171*> LDB is INTEGER
172*> The leading dimension of B. LDB >= max(1,N).
173*> \endverbatim
174*>
175*> \param[out] SDIM
176*> \verbatim
177*> SDIM is INTEGER
178*> If SORT = 'N', SDIM = 0.
179*> If SORT = 'S', SDIM = number of eigenvalues (after sorting)
180*> for which SELCTG is true. (Complex conjugate pairs for which
181*> SELCTG is true for either eigenvalue count as 2.)
182*> \endverbatim
183*>
184*> \param[out] ALPHAR
185*> \verbatim
186*> ALPHAR is DOUBLE PRECISION array, dimension (N)
187*> \endverbatim
188*>
189*> \param[out] ALPHAI
190*> \verbatim
191*> ALPHAI is DOUBLE PRECISION array, dimension (N)
192*> \endverbatim
193*>
194*> \param[out] BETA
195*> \verbatim
196*> BETA is DOUBLE PRECISION array, dimension (N)
197*> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
198*> be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i
199*> and BETA(j),j=1,...,N are the diagonals of the complex Schur
200*> form (S,T) that would result if the 2-by-2 diagonal blocks of
201*> the real Schur form of (A,B) were further reduced to
202*> triangular form using 2-by-2 complex unitary transformations.
203*> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
204*> positive, then the j-th and (j+1)-st eigenvalues are a
205*> complex conjugate pair, with ALPHAI(j+1) negative.
206*>
207*> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
208*> may easily over- or underflow, and BETA(j) may even be zero.
209*> Thus, the user should avoid naively computing the ratio.
210*> However, ALPHAR and ALPHAI will be always less than and
211*> usually comparable with norm(A) in magnitude, and BETA always
212*> less than and usually comparable with norm(B).
213*> \endverbatim
214*>
215*> \param[out] VSL
216*> \verbatim
217*> VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
218*> If JOBVSL = 'V', VSL will contain the left Schur vectors.
219*> Not referenced if JOBVSL = 'N'.
220*> \endverbatim
221*>
222*> \param[in] LDVSL
223*> \verbatim
224*> LDVSL is INTEGER
225*> The leading dimension of the matrix VSL. LDVSL >=1, and
226*> if JOBVSL = 'V', LDVSL >= N.
227*> \endverbatim
228*>
229*> \param[out] VSR
230*> \verbatim
231*> VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
232*> If JOBVSR = 'V', VSR will contain the right Schur vectors.
233*> Not referenced if JOBVSR = 'N'.
234*> \endverbatim
235*>
236*> \param[in] LDVSR
237*> \verbatim
238*> LDVSR is INTEGER
239*> The leading dimension of the matrix VSR. LDVSR >= 1, and
240*> if JOBVSR = 'V', LDVSR >= N.
241*> \endverbatim
242*>
243*> \param[out] RCONDE
244*> \verbatim
245*> RCONDE is DOUBLE PRECISION array, dimension ( 2 )
246*> If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
247*> reciprocal condition numbers for the average of the selected
248*> eigenvalues.
249*> Not referenced if SENSE = 'N' or 'V'.
250*> \endverbatim
251*>
252*> \param[out] RCONDV
253*> \verbatim
254*> RCONDV is DOUBLE PRECISION array, dimension ( 2 )
255*> If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
256*> reciprocal condition numbers for the selected deflating
257*> subspaces.
258*> Not referenced if SENSE = 'N' or 'E'.
259*> \endverbatim
260*>
261*> \param[out] WORK
262*> \verbatim
263*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
264*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
265*> \endverbatim
266*>
267*> \param[in] LWORK
268*> \verbatim
269*> LWORK is INTEGER
270*> The dimension of the array WORK.
271*> If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
272*> LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else
273*> LWORK >= max( 8*N, 6*N+16 ).
274*> Note that 2*SDIM*(N-SDIM) <= N*N/2.
275*> Note also that an error is only returned if
276*> LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B'
277*> this may not be large enough.
278*>
279*> If LWORK = -1, then a workspace query is assumed; the routine
280*> only calculates the bound on the optimal size of the WORK
281*> array and the minimum size of the IWORK array, returns these
282*> values as the first entries of the WORK and IWORK arrays, and
283*> no error message related to LWORK or LIWORK is issued by
284*> XERBLA.
285*> \endverbatim
286*>
287*> \param[out] IWORK
288*> \verbatim
289*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
290*> On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
291*> \endverbatim
292*>
293*> \param[in] LIWORK
294*> \verbatim
295*> LIWORK is INTEGER
296*> The dimension of the array IWORK.
297*> If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
298*> LIWORK >= N+6.
299*>
300*> If LIWORK = -1, then a workspace query is assumed; the
301*> routine only calculates the bound on the optimal size of the
302*> WORK array and the minimum size of the IWORK array, returns
303*> these values as the first entries of the WORK and IWORK
304*> arrays, and no error message related to LWORK or LIWORK is
305*> issued by XERBLA.
306*> \endverbatim
307*>
308*> \param[out] BWORK
309*> \verbatim
310*> BWORK is LOGICAL array, dimension (N)
311*> Not referenced if SORT = 'N'.
312*> \endverbatim
313*>
314*> \param[out] INFO
315*> \verbatim
316*> INFO is INTEGER
317*> = 0: successful exit
318*> < 0: if INFO = -i, the i-th argument had an illegal value.
319*> = 1,...,N:
320*> The QZ iteration failed. (A,B) are not in Schur
321*> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
322*> be correct for j=INFO+1,...,N.
323*> > N: =N+1: other than QZ iteration failed in DHGEQZ
324*> =N+2: after reordering, roundoff changed values of
325*> some complex eigenvalues so that leading
326*> eigenvalues in the Generalized Schur form no
327*> longer satisfy SELCTG=.TRUE. This could also
328*> be caused due to scaling.
329*> =N+3: reordering failed in DTGSEN.
330*> \endverbatim
331*
332* Authors:
333* ========
334*
335*> \author Univ. of Tennessee
336*> \author Univ. of California Berkeley
337*> \author Univ. of Colorado Denver
338*> \author NAG Ltd.
339*
340*> \ingroup ggesx
341*
342*> \par Further Details:
343* =====================
344*>
345*> \verbatim
346*>
347*> An approximate (asymptotic) bound on the average absolute error of
348*> the selected eigenvalues is
349*>
350*> EPS * norm((A, B)) / RCONDE( 1 ).
351*>
352*> An approximate (asymptotic) bound on the maximum angular error in
353*> the computed deflating subspaces is
354*>
355*> EPS * norm((A, B)) / RCONDV( 2 ).
356*>
357*> See LAPACK User's Guide, section 4.11 for more information.
358*> \endverbatim
359*>
360* =====================================================================
361 SUBROUTINE dggesx( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA,
362 $ B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL,
363 $ VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK,
364 $ LIWORK, BWORK, INFO )
365*
366* -- LAPACK driver routine --
367* -- LAPACK is a software package provided by Univ. of Tennessee, --
368* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
369*
370* .. Scalar Arguments ..
371 CHARACTER JOBVSL, JOBVSR, SENSE, SORT
372 INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N,
373 $ SDIM
374* ..
375* .. Array Arguments ..
376 LOGICAL BWORK( * )
377 INTEGER IWORK( * )
378 DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
379 $ b( ldb, * ), beta( * ), rconde( 2 ),
380 $ rcondv( 2 ), vsl( ldvsl, * ), vsr( ldvsr, * ),
381 $ work( * )
382* ..
383* .. Function Arguments ..
384 LOGICAL SELCTG
385 EXTERNAL SELCTG
386* ..
387*
388* =====================================================================
389*
390* .. Parameters ..
391 DOUBLE PRECISION ZERO, ONE
392 PARAMETER ( ZERO = 0.0d+0, one = 1.0d+0 )
393* ..
394* .. Local Scalars ..
395 LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
396 $ LQUERY, LST2SL, WANTSB, WANTSE, WANTSN, WANTST,
397 $ WANTSV
398 INTEGER I, ICOLS, IERR, IHI, IJOB, IJOBVL, IJOBVR,
399 $ ILEFT, ILO, IP, IRIGHT, IROWS, ITAU, IWRK,
400 $ LIWMIN, LWRK, MAXWRK, MINWRK
401 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PL,
402 $ PR, SAFMAX, SAFMIN, SMLNUM
403* ..
404* .. Local Arrays ..
405 DOUBLE PRECISION DIF( 2 )
406* ..
407* .. External Subroutines ..
408 EXTERNAL dgeqrf, dggbak, dggbal, dgghrd, dhgeqz, dlacpy,
410* ..
411* .. External Functions ..
412 LOGICAL LSAME
413 INTEGER ILAENV
414 DOUBLE PRECISION DLAMCH, DLANGE
415 EXTERNAL lsame, ilaenv, dlamch, dlange
416* ..
417* .. Intrinsic Functions ..
418 INTRINSIC abs, max, sqrt
419* ..
420* .. Executable Statements ..
421*
422* Decode the input arguments
423*
424 IF( lsame( jobvsl, 'N' ) ) THEN
425 ijobvl = 1
426 ilvsl = .false.
427 ELSE IF( lsame( jobvsl, 'V' ) ) THEN
428 ijobvl = 2
429 ilvsl = .true.
430 ELSE
431 ijobvl = -1
432 ilvsl = .false.
433 END IF
434*
435 IF( lsame( jobvsr, 'N' ) ) THEN
436 ijobvr = 1
437 ilvsr = .false.
438 ELSE IF( lsame( jobvsr, 'V' ) ) THEN
439 ijobvr = 2
440 ilvsr = .true.
441 ELSE
442 ijobvr = -1
443 ilvsr = .false.
444 END IF
445*
446 wantst = lsame( sort, 'S' )
447 wantsn = lsame( sense, 'N' )
448 wantse = lsame( sense, 'E' )
449 wantsv = lsame( sense, 'V' )
450 wantsb = lsame( sense, 'B' )
451 lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
452 IF( wantsn ) THEN
453 ijob = 0
454 ELSE IF( wantse ) THEN
455 ijob = 1
456 ELSE IF( wantsv ) THEN
457 ijob = 2
458 ELSE IF( wantsb ) THEN
459 ijob = 4
460 END IF
461*
462* Test the input arguments
463*
464 info = 0
465 IF( ijobvl.LE.0 ) THEN
466 info = -1
467 ELSE IF( ijobvr.LE.0 ) THEN
468 info = -2
469 ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
470 info = -3
471 ELSE IF( .NOT.( wantsn .OR. wantse .OR. wantsv .OR. wantsb ) .OR.
472 $ ( .NOT.wantst .AND. .NOT.wantsn ) ) THEN
473 info = -5
474 ELSE IF( n.LT.0 ) THEN
475 info = -6
476 ELSE IF( lda.LT.max( 1, n ) ) THEN
477 info = -8
478 ELSE IF( ldb.LT.max( 1, n ) ) THEN
479 info = -10
480 ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
481 info = -16
482 ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
483 info = -18
484 END IF
485*
486* Compute workspace
487* (Note: Comments in the code beginning "Workspace:" describe the
488* minimal amount of workspace needed at that point in the code,
489* as well as the preferred amount for good performance.
490* NB refers to the optimal block size for the immediately
491* following subroutine, as returned by ILAENV.)
492*
493 IF( info.EQ.0 ) THEN
494 IF( n.GT.0) THEN
495 minwrk = max( 8*n, 6*n + 16 )
496 maxwrk = minwrk - n +
497 $ n*ilaenv( 1, 'DGEQRF', ' ', n, 1, n, 0 )
498 maxwrk = max( maxwrk, minwrk - n +
499 $ n*ilaenv( 1, 'DORMQR', ' ', n, 1, n, -1 ) )
500 IF( ilvsl ) THEN
501 maxwrk = max( maxwrk, minwrk - n +
502 $ n*ilaenv( 1, 'DORGQR', ' ', n, 1, n, -1 ) )
503 END IF
504 lwrk = maxwrk
505 IF( ijob.GE.1 )
506 $ lwrk = max( lwrk, n*n/2 )
507 ELSE
508 minwrk = 1
509 maxwrk = 1
510 lwrk = 1
511 END IF
512 work( 1 ) = lwrk
513 IF( wantsn .OR. n.EQ.0 ) THEN
514 liwmin = 1
515 ELSE
516 liwmin = n + 6
517 END IF
518 iwork( 1 ) = liwmin
519*
520 IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
521 info = -22
522 ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
523 info = -24
524 END IF
525 END IF
526*
527 IF( info.NE.0 ) THEN
528 CALL xerbla( 'DGGESX', -info )
529 RETURN
530 ELSE IF (lquery) THEN
531 RETURN
532 END IF
533*
534* Quick return if possible
535*
536 IF( n.EQ.0 ) THEN
537 sdim = 0
538 RETURN
539 END IF
540*
541* Get machine constants
542*
543 eps = dlamch( 'P' )
544 safmin = dlamch( 'S' )
545 safmax = one / safmin
546 smlnum = sqrt( safmin ) / eps
547 bignum = one / smlnum
548*
549* Scale A if max element outside range [SMLNUM,BIGNUM]
550*
551 anrm = dlange( 'M', n, n, a, lda, work )
552 ilascl = .false.
553 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
554 anrmto = smlnum
555 ilascl = .true.
556 ELSE IF( anrm.GT.bignum ) THEN
557 anrmto = bignum
558 ilascl = .true.
559 END IF
560 IF( ilascl )
561 $ CALL dlascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
562*
563* Scale B if max element outside range [SMLNUM,BIGNUM]
564*
565 bnrm = dlange( 'M', n, n, b, ldb, work )
566 ilbscl = .false.
567 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
568 bnrmto = smlnum
569 ilbscl = .true.
570 ELSE IF( bnrm.GT.bignum ) THEN
571 bnrmto = bignum
572 ilbscl = .true.
573 END IF
574 IF( ilbscl )
575 $ CALL dlascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
576*
577* Permute the matrix to make it more nearly triangular
578* (Workspace: need 6*N + 2*N for permutation parameters)
579*
580 ileft = 1
581 iright = n + 1
582 iwrk = iright + n
583 CALL dggbal( 'P', n, a, lda, b, ldb, ilo, ihi, work( ileft ),
584 $ work( iright ), work( iwrk ), ierr )
585*
586* Reduce B to triangular form (QR decomposition of B)
587* (Workspace: need N, prefer N*NB)
588*
589 irows = ihi + 1 - ilo
590 icols = n + 1 - ilo
591 itau = iwrk
592 iwrk = itau + irows
593 CALL dgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
594 $ work( iwrk ), lwork+1-iwrk, ierr )
595*
596* Apply the orthogonal transformation to matrix A
597* (Workspace: need N, prefer N*NB)
598*
599 CALL dormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
600 $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
601 $ lwork+1-iwrk, ierr )
602*
603* Initialize VSL
604* (Workspace: need N, prefer N*NB)
605*
606 IF( ilvsl ) THEN
607 CALL dlaset( 'Full', n, n, zero, one, vsl, ldvsl )
608 IF( irows.GT.1 ) THEN
609 CALL dlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
610 $ vsl( ilo+1, ilo ), ldvsl )
611 END IF
612 CALL dorgqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
613 $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
614 END IF
615*
616* Initialize VSR
617*
618 IF( ilvsr )
619 $ CALL dlaset( 'Full', n, n, zero, one, vsr, ldvsr )
620*
621* Reduce to generalized Hessenberg form
622* (Workspace: none needed)
623*
624 CALL dgghrd( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
625 $ ldvsl, vsr, ldvsr, ierr )
626*
627 sdim = 0
628*
629* Perform QZ algorithm, computing Schur vectors if desired
630* (Workspace: need N)
631*
632 iwrk = itau
633 CALL dhgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
634 $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
635 $ work( iwrk ), lwork+1-iwrk, ierr )
636 IF( ierr.NE.0 ) THEN
637 IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
638 info = ierr
639 ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
640 info = ierr - n
641 ELSE
642 info = n + 1
643 END IF
644 GO TO 60
645 END IF
646*
647* Sort eigenvalues ALPHA/BETA and compute the reciprocal of
648* condition number(s)
649* (Workspace: If IJOB >= 1, need MAX( 8*(N+1), 2*SDIM*(N-SDIM) )
650* otherwise, need 8*(N+1) )
651*
652 IF( wantst ) THEN
653*
654* Undo scaling on eigenvalues before SELCTGing
655*
656 IF( ilascl ) THEN
657 CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n,
658 $ ierr )
659 CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n,
660 $ ierr )
661 END IF
662 IF( ilbscl )
663 $ CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
664*
665* Select eigenvalues
666*
667 DO 10 i = 1, n
668 bwork( i ) = selctg( alphar( i ), alphai( i ), beta( i ) )
669 10 CONTINUE
670*
671* Reorder eigenvalues, transform Generalized Schur vectors, and
672* compute reciprocal condition numbers
673*
674 CALL dtgsen( ijob, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
675 $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
676 $ sdim, pl, pr, dif, work( iwrk ), lwork-iwrk+1,
677 $ iwork, liwork, ierr )
678*
679 IF( ijob.GE.1 )
680 $ maxwrk = max( maxwrk, 2*sdim*( n-sdim ) )
681 IF( ierr.EQ.-22 ) THEN
682*
683* not enough real workspace
684*
685 info = -22
686 ELSE
687 IF( ijob.EQ.1 .OR. ijob.EQ.4 ) THEN
688 rconde( 1 ) = pl
689 rconde( 2 ) = pr
690 END IF
691 IF( ijob.EQ.2 .OR. ijob.EQ.4 ) THEN
692 rcondv( 1 ) = dif( 1 )
693 rcondv( 2 ) = dif( 2 )
694 END IF
695 IF( ierr.EQ.1 )
696 $ info = n + 3
697 END IF
698*
699 END IF
700*
701* Apply permutation to VSL and VSR
702* (Workspace: none needed)
703*
704 IF( ilvsl )
705 $ CALL dggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
706 $ work( iright ), n, vsl, ldvsl, ierr )
707*
708 IF( ilvsr )
709 $ CALL dggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
710 $ work( iright ), n, vsr, ldvsr, ierr )
711*
712* Check if unscaling would cause over/underflow, if so, rescale
713* (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
714* B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
715*
716 IF( ilascl ) THEN
717 DO 20 i = 1, n
718 IF( alphai( i ).NE.zero ) THEN
719 IF( ( alphar( i ) / safmax ).GT.( anrmto / anrm ) .OR.
720 $ ( safmin / alphar( i ) ).GT.( anrm / anrmto ) ) THEN
721 work( 1 ) = abs( a( i, i ) / alphar( i ) )
722 beta( i ) = beta( i )*work( 1 )
723 alphar( i ) = alphar( i )*work( 1 )
724 alphai( i ) = alphai( i )*work( 1 )
725 ELSE IF( ( alphai( i ) / safmax ).GT.
726 $ ( anrmto / anrm ) .OR.
727 $ ( safmin / alphai( i ) ).GT.( anrm / anrmto ) )
728 $ THEN
729 work( 1 ) = abs( a( i, i+1 ) / alphai( i ) )
730 beta( i ) = beta( i )*work( 1 )
731 alphar( i ) = alphar( i )*work( 1 )
732 alphai( i ) = alphai( i )*work( 1 )
733 END IF
734 END IF
735 20 CONTINUE
736 END IF
737*
738 IF( ilbscl ) THEN
739 DO 30 i = 1, n
740 IF( alphai( i ).NE.zero ) THEN
741 IF( ( beta( i ) / safmax ).GT.( bnrmto / bnrm ) .OR.
742 $ ( safmin / beta( i ) ).GT.( bnrm / bnrmto ) ) THEN
743 work( 1 ) = abs( b( i, i ) / beta( i ) )
744 beta( i ) = beta( i )*work( 1 )
745 alphar( i ) = alphar( i )*work( 1 )
746 alphai( i ) = alphai( i )*work( 1 )
747 END IF
748 END IF
749 30 CONTINUE
750 END IF
751*
752* Undo scaling
753*
754 IF( ilascl ) THEN
755 CALL dlascl( 'H', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
756 CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
757 CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
758 END IF
759*
760 IF( ilbscl ) THEN
761 CALL dlascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
762 CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
763 END IF
764*
765 IF( wantst ) THEN
766*
767* Check if reordering is correct
768*
769 lastsl = .true.
770 lst2sl = .true.
771 sdim = 0
772 ip = 0
773 DO 50 i = 1, n
774 cursl = selctg( alphar( i ), alphai( i ), beta( i ) )
775 IF( alphai( i ).EQ.zero ) THEN
776 IF( cursl )
777 $ sdim = sdim + 1
778 ip = 0
779 IF( cursl .AND. .NOT.lastsl )
780 $ info = n + 2
781 ELSE
782 IF( ip.EQ.1 ) THEN
783*
784* Last eigenvalue of conjugate pair
785*
786 cursl = cursl .OR. lastsl
787 lastsl = cursl
788 IF( cursl )
789 $ sdim = sdim + 2
790 ip = -1
791 IF( cursl .AND. .NOT.lst2sl )
792 $ info = n + 2
793 ELSE
794*
795* First eigenvalue of conjugate pair
796*
797 ip = 1
798 END IF
799 END IF
800 lst2sl = lastsl
801 lastsl = cursl
802 50 CONTINUE
803*
804 END IF
805*
806 60 CONTINUE
807*
808 work( 1 ) = maxwrk
809 iwork( 1 ) = liwmin
810*
811 RETURN
812*
813* End of DGGESX
814*
815 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine dgeqrf(m, n, a, lda, tau, work, lwork, info)
DGEQRF
Definition dgeqrf.f:146
subroutine dggbak(job, side, n, ilo, ihi, lscale, rscale, m, v, ldv, info)
DGGBAK
Definition dggbak.f:147
subroutine dggbal(job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)
DGGBAL
Definition dggbal.f:177
subroutine dggesx(jobvsl, jobvsr, sort, selctg, sense, n, a, lda, b, ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv, work, lwork, iwork, liwork, bwork, info)
DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE ...
Definition dggesx.f:365
subroutine dgghrd(compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)
DGGHRD
Definition dgghrd.f:207
subroutine dhgeqz(job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, info)
DHGEQZ
Definition dhgeqz.f:304
subroutine dlacpy(uplo, m, n, a, lda, b, ldb)
DLACPY copies all or part of one two-dimensional array to another.
Definition dlacpy.f:103
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 dlaset(uplo, m, n, alpha, beta, a, lda)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition dlaset.f:110
subroutine dtgsen(ijob, wantq, wantz, select, n, a, lda, b, ldb, alphar, alphai, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork, info)
DTGSEN
Definition dtgsen.f:451
subroutine dorgqr(m, n, k, a, lda, tau, work, lwork, info)
DORGQR
Definition dorgqr.f:128
subroutine dormqr(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
DORMQR
Definition dormqr.f:167