LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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sggesx.f
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1*> \brief <b> SGGESX 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 SGGESX + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sggesx.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sggesx.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sggesx.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE SGGESX( 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* REAL 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*> SGGESX 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 REAL 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 REAL 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 REAL 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 REAL array, dimension (N)
187*> \endverbatim
188*>
189*> \param[out] ALPHAI
190*> \verbatim
191*> ALPHAI is REAL array, dimension (N)
192*> \endverbatim
193*>
194*> \param[out] BETA
195*> \verbatim
196*> BETA is REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 SHGEQZ
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 STGSEN.
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 realGEeigen
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 sggesx( 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 REAL 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 REAL ZERO, ONE
392 PARAMETER ( ZERO = 0.0e+0, one = 1.0e+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 REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PL,
402 $ PR, SAFMAX, SAFMIN, SMLNUM
403* ..
404* .. Local Arrays ..
405 REAL DIF( 2 )
406* ..
407* .. External Subroutines ..
408 EXTERNAL sgeqrf, sggbak, sggbal, sgghrd, shgeqz, slabad,
410 $ xerbla
411* ..
412* .. External Functions ..
413 LOGICAL LSAME
414 INTEGER ILAENV
415 REAL SLAMCH, SLANGE
416 EXTERNAL lsame, ilaenv, slamch, slange
417* ..
418* .. Intrinsic Functions ..
419 INTRINSIC abs, max, sqrt
420* ..
421* .. Executable Statements ..
422*
423* Decode the input arguments
424*
425 IF( lsame( jobvsl, 'N' ) ) THEN
426 ijobvl = 1
427 ilvsl = .false.
428 ELSE IF( lsame( jobvsl, 'V' ) ) THEN
429 ijobvl = 2
430 ilvsl = .true.
431 ELSE
432 ijobvl = -1
433 ilvsl = .false.
434 END IF
435*
436 IF( lsame( jobvsr, 'N' ) ) THEN
437 ijobvr = 1
438 ilvsr = .false.
439 ELSE IF( lsame( jobvsr, 'V' ) ) THEN
440 ijobvr = 2
441 ilvsr = .true.
442 ELSE
443 ijobvr = -1
444 ilvsr = .false.
445 END IF
446*
447 wantst = lsame( sort, 'S' )
448 wantsn = lsame( sense, 'N' )
449 wantse = lsame( sense, 'E' )
450 wantsv = lsame( sense, 'V' )
451 wantsb = lsame( sense, 'B' )
452 lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
453 IF( wantsn ) THEN
454 ijob = 0
455 ELSE IF( wantse ) THEN
456 ijob = 1
457 ELSE IF( wantsv ) THEN
458 ijob = 2
459 ELSE IF( wantsb ) THEN
460 ijob = 4
461 END IF
462*
463* Test the input arguments
464*
465 info = 0
466 IF( ijobvl.LE.0 ) THEN
467 info = -1
468 ELSE IF( ijobvr.LE.0 ) THEN
469 info = -2
470 ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
471 info = -3
472 ELSE IF( .NOT.( wantsn .OR. wantse .OR. wantsv .OR. wantsb ) .OR.
473 $ ( .NOT.wantst .AND. .NOT.wantsn ) ) THEN
474 info = -5
475 ELSE IF( n.LT.0 ) THEN
476 info = -6
477 ELSE IF( lda.LT.max( 1, n ) ) THEN
478 info = -8
479 ELSE IF( ldb.LT.max( 1, n ) ) THEN
480 info = -10
481 ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
482 info = -16
483 ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
484 info = -18
485 END IF
486*
487* Compute workspace
488* (Note: Comments in the code beginning "Workspace:" describe the
489* minimal amount of workspace needed at that point in the code,
490* as well as the preferred amount for good performance.
491* NB refers to the optimal block size for the immediately
492* following subroutine, as returned by ILAENV.)
493*
494 IF( info.EQ.0 ) THEN
495 IF( n.GT.0) THEN
496 minwrk = max( 8*n, 6*n + 16 )
497 maxwrk = minwrk - n +
498 $ n*ilaenv( 1, 'SGEQRF', ' ', n, 1, n, 0 )
499 maxwrk = max( maxwrk, minwrk - n +
500 $ n*ilaenv( 1, 'SORMQR', ' ', n, 1, n, -1 ) )
501 IF( ilvsl ) THEN
502 maxwrk = max( maxwrk, minwrk - n +
503 $ n*ilaenv( 1, 'SORGQR', ' ', n, 1, n, -1 ) )
504 END IF
505 lwrk = maxwrk
506 IF( ijob.GE.1 )
507 $ lwrk = max( lwrk, n*n/2 )
508 ELSE
509 minwrk = 1
510 maxwrk = 1
511 lwrk = 1
512 END IF
513 work( 1 ) = lwrk
514 IF( wantsn .OR. n.EQ.0 ) THEN
515 liwmin = 1
516 ELSE
517 liwmin = n + 6
518 END IF
519 iwork( 1 ) = liwmin
520*
521 IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
522 info = -22
523 ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
524 info = -24
525 END IF
526 END IF
527*
528 IF( info.NE.0 ) THEN
529 CALL xerbla( 'SGGESX', -info )
530 RETURN
531 ELSE IF (lquery) THEN
532 RETURN
533 END IF
534*
535* Quick return if possible
536*
537 IF( n.EQ.0 ) THEN
538 sdim = 0
539 RETURN
540 END IF
541*
542* Get machine constants
543*
544 eps = slamch( 'P' )
545 safmin = slamch( 'S' )
546 safmax = one / safmin
547 CALL slabad( safmin, safmax )
548 smlnum = sqrt( safmin ) / eps
549 bignum = one / smlnum
550*
551* Scale A if max element outside range [SMLNUM,BIGNUM]
552*
553 anrm = slange( 'M', n, n, a, lda, work )
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 slascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
564*
565* Scale B if max element outside range [SMLNUM,BIGNUM]
566*
567 bnrm = slange( 'M', n, n, b, ldb, work )
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 slascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
578*
579* Permute the matrix to make it more nearly triangular
580* (Workspace: need 6*N + 2*N for permutation parameters)
581*
582 ileft = 1
583 iright = n + 1
584 iwrk = iright + n
585 CALL sggbal( 'P', n, a, lda, b, ldb, ilo, ihi, work( ileft ),
586 $ work( iright ), work( iwrk ), ierr )
587*
588* Reduce B to triangular form (QR decomposition of B)
589* (Workspace: need N, prefer N*NB)
590*
591 irows = ihi + 1 - ilo
592 icols = n + 1 - ilo
593 itau = iwrk
594 iwrk = itau + irows
595 CALL sgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
596 $ work( iwrk ), lwork+1-iwrk, ierr )
597*
598* Apply the orthogonal transformation to matrix A
599* (Workspace: need N, prefer N*NB)
600*
601 CALL sormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
602 $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
603 $ lwork+1-iwrk, ierr )
604*
605* Initialize VSL
606* (Workspace: need N, prefer N*NB)
607*
608 IF( ilvsl ) THEN
609 CALL slaset( 'Full', n, n, zero, one, vsl, ldvsl )
610 IF( irows.GT.1 ) THEN
611 CALL slacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
612 $ vsl( ilo+1, ilo ), ldvsl )
613 END IF
614 CALL sorgqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
615 $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
616 END IF
617*
618* Initialize VSR
619*
620 IF( ilvsr )
621 $ CALL slaset( 'Full', n, n, zero, one, vsr, ldvsr )
622*
623* Reduce to generalized Hessenberg form
624* (Workspace: none needed)
625*
626 CALL sgghrd( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
627 $ ldvsl, vsr, ldvsr, ierr )
628*
629 sdim = 0
630*
631* Perform QZ algorithm, computing Schur vectors if desired
632* (Workspace: need N)
633*
634 iwrk = itau
635 CALL shgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
636 $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
637 $ work( iwrk ), lwork+1-iwrk, ierr )
638 IF( ierr.NE.0 ) THEN
639 IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
640 info = ierr
641 ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
642 info = ierr - n
643 ELSE
644 info = n + 1
645 END IF
646 GO TO 50
647 END IF
648*
649* Sort eigenvalues ALPHA/BETA and compute the reciprocal of
650* condition number(s)
651* (Workspace: If IJOB >= 1, need MAX( 8*(N+1), 2*SDIM*(N-SDIM) )
652* otherwise, need 8*(N+1) )
653*
654 IF( wantst ) THEN
655*
656* Undo scaling on eigenvalues before SELCTGing
657*
658 IF( ilascl ) THEN
659 CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n,
660 $ ierr )
661 CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n,
662 $ ierr )
663 END IF
664 IF( ilbscl )
665 $ CALL slascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
666*
667* Select eigenvalues
668*
669 DO 10 i = 1, n
670 bwork( i ) = selctg( alphar( i ), alphai( i ), beta( i ) )
671 10 CONTINUE
672*
673* Reorder eigenvalues, transform Generalized Schur vectors, and
674* compute reciprocal condition numbers
675*
676 CALL stgsen( ijob, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
677 $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
678 $ sdim, pl, pr, dif, work( iwrk ), lwork-iwrk+1,
679 $ iwork, liwork, ierr )
680*
681 IF( ijob.GE.1 )
682 $ maxwrk = max( maxwrk, 2*sdim*( n-sdim ) )
683 IF( ierr.EQ.-22 ) THEN
684*
685* not enough real workspace
686*
687 info = -22
688 ELSE
689 IF( ijob.EQ.1 .OR. ijob.EQ.4 ) THEN
690 rconde( 1 ) = pl
691 rconde( 2 ) = pr
692 END IF
693 IF( ijob.EQ.2 .OR. ijob.EQ.4 ) THEN
694 rcondv( 1 ) = dif( 1 )
695 rcondv( 2 ) = dif( 2 )
696 END IF
697 IF( ierr.EQ.1 )
698 $ info = n + 3
699 END IF
700*
701 END IF
702*
703* Apply permutation to VSL and VSR
704* (Workspace: none needed)
705*
706 IF( ilvsl )
707 $ CALL sggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
708 $ work( iright ), n, vsl, ldvsl, ierr )
709*
710 IF( ilvsr )
711 $ CALL sggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
712 $ work( iright ), n, vsr, ldvsr, ierr )
713*
714* Check if unscaling would cause over/underflow, if so, rescale
715* (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
716* B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
717*
718 IF( ilascl ) THEN
719 DO 20 i = 1, n
720 IF( alphai( i ).NE.zero ) THEN
721 IF( ( alphar( i ) / safmax ).GT.( anrmto / anrm ) .OR.
722 $ ( safmin / alphar( i ) ).GT.( anrm / anrmto ) )
723 $ THEN
724 work( 1 ) = abs( a( i, i ) / alphar( i ) )
725 beta( i ) = beta( i )*work( 1 )
726 alphar( i ) = alphar( i )*work( 1 )
727 alphai( i ) = alphai( i )*work( 1 )
728 ELSE IF( ( alphai( i ) / safmax ).GT.( anrmto / anrm )
729 $ .OR. ( safmin / alphai( i ) ).GT.( anrm / anrmto ) )
730 $ THEN
731 work( 1 ) = abs( a( i, i+1 ) / alphai( i ) )
732 beta( i ) = beta( i )*work( 1 )
733 alphar( i ) = alphar( i )*work( 1 )
734 alphai( i ) = alphai( i )*work( 1 )
735 END IF
736 END IF
737 20 CONTINUE
738 END IF
739*
740 IF( ilbscl ) THEN
741 DO 25 i = 1, n
742 IF( alphai( i ).NE.zero ) THEN
743 IF( ( beta( i ) / safmax ).GT.( bnrmto / bnrm ) .OR.
744 $ ( safmin / beta( i ) ).GT.( bnrm / bnrmto ) ) THEN
745 work( 1 ) = abs( b( i, i ) / beta( i ) )
746 beta( i ) = beta( i )*work( 1 )
747 alphar( i ) = alphar( i )*work( 1 )
748 alphai( i ) = alphai( i )*work( 1 )
749 END IF
750 END IF
751 25 CONTINUE
752 END IF
753*
754* Undo scaling
755*
756 IF( ilascl ) THEN
757 CALL slascl( 'H', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
758 CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
759 CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
760 END IF
761*
762 IF( ilbscl ) THEN
763 CALL slascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
764 CALL slascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
765 END IF
766*
767 IF( wantst ) THEN
768*
769* Check if reordering is correct
770*
771 lastsl = .true.
772 lst2sl = .true.
773 sdim = 0
774 ip = 0
775 DO 40 i = 1, n
776 cursl = selctg( alphar( i ), alphai( i ), beta( i ) )
777 IF( alphai( i ).EQ.zero ) THEN
778 IF( cursl )
779 $ sdim = sdim + 1
780 ip = 0
781 IF( cursl .AND. .NOT.lastsl )
782 $ info = n + 2
783 ELSE
784 IF( ip.EQ.1 ) THEN
785*
786* Last eigenvalue of conjugate pair
787*
788 cursl = cursl .OR. lastsl
789 lastsl = cursl
790 IF( cursl )
791 $ sdim = sdim + 2
792 ip = -1
793 IF( cursl .AND. .NOT.lst2sl )
794 $ info = n + 2
795 ELSE
796*
797* First eigenvalue of conjugate pair
798*
799 ip = 1
800 END IF
801 END IF
802 lst2sl = lastsl
803 lastsl = cursl
804 40 CONTINUE
805*
806 END IF
807*
808 50 CONTINUE
809*
810 work( 1 ) = maxwrk
811 iwork( 1 ) = liwmin
812*
813 RETURN
814*
815* End of SGGESX
816*
817 END
subroutine slabad(SMALL, LARGE)
SLABAD
Definition: slabad.f:74
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 slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: slaset.f:110
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine sggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
SGGBAK
Definition: sggbak.f:147
subroutine sggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
SGGBAL
Definition: sggbal.f:177
subroutine sgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRF
Definition: sgeqrf.f:146
subroutine shgeqz(JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
SHGEQZ
Definition: shgeqz.f:304
subroutine sggesx(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)
SGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE ...
Definition: sggesx.f:365
subroutine stgsen(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)
STGSEN
Definition: stgsen.f:451
subroutine sorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGQR
Definition: sorgqr.f:128
subroutine sormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMQR
Definition: sormqr.f:168
subroutine sgghrd(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
SGGHRD
Definition: sgghrd.f:207