LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
Loading...
Searching...
No Matches
sgges.f
Go to the documentation of this file.
1*> \brief <b> SGGES 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 SGGES + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgges.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgges.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgges.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE SGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
22* SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR,
23* LDVSR, WORK, LWORK, BWORK, INFO )
24*
25* .. Scalar Arguments ..
26* CHARACTER JOBVSL, JOBVSR, SORT
27* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
28* ..
29* .. Array Arguments ..
30* LOGICAL BWORK( * )
31* REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
32* $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
33* $ VSR( LDVSR, * ), WORK( * )
34* ..
35* .. Function Arguments ..
36* LOGICAL SELCTG
37* EXTERNAL SELCTG
38* ..
39*
40*
41*> \par Purpose:
42* =============
43*>
44*> \verbatim
45*>
46*> SGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B),
47*> the generalized eigenvalues, the generalized real Schur form (S,T),
48*> optionally, the left and/or right matrices of Schur vectors (VSL and
49*> VSR). This gives the generalized Schur factorization
50*>
51*> (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
52*>
53*> Optionally, it also orders the eigenvalues so that a selected cluster
54*> of eigenvalues appears in the leading diagonal blocks of the upper
55*> quasi-triangular matrix S and the upper triangular matrix T.The
56*> leading columns of VSL and VSR then form an orthonormal basis for the
57*> corresponding left and right eigenspaces (deflating subspaces).
58*>
59*> (If only the generalized eigenvalues are needed, use the driver
60*> SGGEV instead, which is faster.)
61*>
62*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
63*> or a ratio alpha/beta = w, such that A - w*B is singular. It is
64*> usually represented as the pair (alpha,beta), as there is a
65*> reasonable interpretation for beta=0 or both being zero.
66*>
67*> A pair of matrices (S,T) is in generalized real Schur form if T is
68*> upper triangular with non-negative diagonal and S is block upper
69*> triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond
70*> to real generalized eigenvalues, while 2-by-2 blocks of S will be
71*> "standardized" by making the corresponding elements of T have the
72*> form:
73*> [ a 0 ]
74*> [ 0 b ]
75*>
76*> and the pair of corresponding 2-by-2 blocks in S and T will have a
77*> complex conjugate pair of generalized eigenvalues.
78*>
79*> \endverbatim
80*
81* Arguments:
82* ==========
83*
84*> \param[in] JOBVSL
85*> \verbatim
86*> JOBVSL is CHARACTER*1
87*> = 'N': do not compute the left Schur vectors;
88*> = 'V': compute the left Schur vectors.
89*> \endverbatim
90*>
91*> \param[in] JOBVSR
92*> \verbatim
93*> JOBVSR is CHARACTER*1
94*> = 'N': do not compute the right Schur vectors;
95*> = 'V': compute the right Schur vectors.
96*> \endverbatim
97*>
98*> \param[in] SORT
99*> \verbatim
100*> SORT is CHARACTER*1
101*> Specifies whether or not to order the eigenvalues on the
102*> diagonal of the generalized Schur form.
103*> = 'N': Eigenvalues are not ordered;
104*> = 'S': Eigenvalues are ordered (see SELCTG);
105*> \endverbatim
106*>
107*> \param[in] SELCTG
108*> \verbatim
109*> SELCTG is a LOGICAL FUNCTION of three REAL arguments
110*> SELCTG must be declared EXTERNAL in the calling subroutine.
111*> If SORT = 'N', SELCTG is not referenced.
112*> If SORT = 'S', SELCTG is used to select eigenvalues to sort
113*> to the top left of the Schur form.
114*> An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
115*> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
116*> one of a complex conjugate pair of eigenvalues is selected,
117*> then both complex eigenvalues are selected.
118*>
119*> Note that in the ill-conditioned case, a selected complex
120*> eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
121*> BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
122*> in this case.
123*> \endverbatim
124*>
125*> \param[in] N
126*> \verbatim
127*> N is INTEGER
128*> The order of the matrices A, B, VSL, and VSR. N >= 0.
129*> \endverbatim
130*>
131*> \param[in,out] A
132*> \verbatim
133*> A is REAL array, dimension (LDA, N)
134*> On entry, the first of the pair of matrices.
135*> On exit, A has been overwritten by its generalized Schur
136*> form S.
137*> \endverbatim
138*>
139*> \param[in] LDA
140*> \verbatim
141*> LDA is INTEGER
142*> The leading dimension of A. LDA >= max(1,N).
143*> \endverbatim
144*>
145*> \param[in,out] B
146*> \verbatim
147*> B is REAL array, dimension (LDB, N)
148*> On entry, the second of the pair of matrices.
149*> On exit, B has been overwritten by its generalized Schur
150*> form T.
151*> \endverbatim
152*>
153*> \param[in] LDB
154*> \verbatim
155*> LDB is INTEGER
156*> The leading dimension of B. LDB >= max(1,N).
157*> \endverbatim
158*>
159*> \param[out] SDIM
160*> \verbatim
161*> SDIM is INTEGER
162*> If SORT = 'N', SDIM = 0.
163*> If SORT = 'S', SDIM = number of eigenvalues (after sorting)
164*> for which SELCTG is true. (Complex conjugate pairs for which
165*> SELCTG is true for either eigenvalue count as 2.)
166*> \endverbatim
167*>
168*> \param[out] ALPHAR
169*> \verbatim
170*> ALPHAR is REAL array, dimension (N)
171*> \endverbatim
172*>
173*> \param[out] ALPHAI
174*> \verbatim
175*> ALPHAI is REAL array, dimension (N)
176*> \endverbatim
177*>
178*> \param[out] BETA
179*> \verbatim
180*> BETA is REAL array, dimension (N)
181*> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
182*> be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i,
183*> and BETA(j),j=1,...,N are the diagonals of the complex Schur
184*> form (S,T) that would result if the 2-by-2 diagonal blocks of
185*> the real Schur form of (A,B) were further reduced to
186*> triangular form using 2-by-2 complex unitary transformations.
187*> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
188*> positive, then the j-th and (j+1)-st eigenvalues are a
189*> complex conjugate pair, with ALPHAI(j+1) negative.
190*>
191*> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
192*> may easily over- or underflow, and BETA(j) may even be zero.
193*> Thus, the user should avoid naively computing the ratio.
194*> However, ALPHAR and ALPHAI will be always less than and
195*> usually comparable with norm(A) in magnitude, and BETA always
196*> less than and usually comparable with norm(B).
197*> \endverbatim
198*>
199*> \param[out] VSL
200*> \verbatim
201*> VSL is REAL array, dimension (LDVSL,N)
202*> If JOBVSL = 'V', VSL will contain the left Schur vectors.
203*> Not referenced if JOBVSL = 'N'.
204*> \endverbatim
205*>
206*> \param[in] LDVSL
207*> \verbatim
208*> LDVSL is INTEGER
209*> The leading dimension of the matrix VSL. LDVSL >=1, and
210*> if JOBVSL = 'V', LDVSL >= N.
211*> \endverbatim
212*>
213*> \param[out] VSR
214*> \verbatim
215*> VSR is REAL array, dimension (LDVSR,N)
216*> If JOBVSR = 'V', VSR will contain the right Schur vectors.
217*> Not referenced if JOBVSR = 'N'.
218*> \endverbatim
219*>
220*> \param[in] LDVSR
221*> \verbatim
222*> LDVSR is INTEGER
223*> The leading dimension of the matrix VSR. LDVSR >= 1, and
224*> if JOBVSR = 'V', LDVSR >= N.
225*> \endverbatim
226*>
227*> \param[out] WORK
228*> \verbatim
229*> WORK is REAL array, dimension (MAX(1,LWORK))
230*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
231*> \endverbatim
232*>
233*> \param[in] LWORK
234*> \verbatim
235*> LWORK is INTEGER
236*> The dimension of the array WORK.
237*> If N = 0, LWORK >= 1, else LWORK >= max(8*N,6*N+16).
238*> For good performance , LWORK must generally be larger.
239*>
240*> If LWORK = -1, then a workspace query is assumed; the routine
241*> only calculates the optimal size of the WORK array, returns
242*> this value as the first entry of the WORK array, and no error
243*> message related to LWORK is issued by XERBLA.
244*> \endverbatim
245*>
246*> \param[out] BWORK
247*> \verbatim
248*> BWORK is LOGICAL array, dimension (N)
249*> Not referenced if SORT = 'N'.
250*> \endverbatim
251*>
252*> \param[out] INFO
253*> \verbatim
254*> INFO is INTEGER
255*> = 0: successful exit
256*> < 0: if INFO = -i, the i-th argument had an illegal value.
257*> = 1,...,N:
258*> The QZ iteration failed. (A,B) are not in Schur
259*> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
260*> be correct for j=INFO+1,...,N.
261*> > N: =N+1: other than QZ iteration failed in SHGEQZ.
262*> =N+2: after reordering, roundoff changed values of
263*> some complex eigenvalues so that leading
264*> eigenvalues in the Generalized Schur form no
265*> longer satisfy SELCTG=.TRUE. This could also
266*> be caused due to scaling.
267*> =N+3: reordering failed in STGSEN.
268*> \endverbatim
269*
270* Authors:
271* ========
272*
273*> \author Univ. of Tennessee
274*> \author Univ. of California Berkeley
275*> \author Univ. of Colorado Denver
276*> \author NAG Ltd.
277*
278*> \ingroup realGEeigen
279*
280* =====================================================================
281 SUBROUTINE sgges( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
282 $ SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR,
283 $ LDVSR, WORK, LWORK, BWORK, INFO )
284*
285* -- LAPACK driver routine --
286* -- LAPACK is a software package provided by Univ. of Tennessee, --
287* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
288*
289* .. Scalar Arguments ..
290 CHARACTER JOBVSL, JOBVSR, SORT
291 INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
292* ..
293* .. Array Arguments ..
294 LOGICAL BWORK( * )
295 REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
296 $ b( ldb, * ), beta( * ), vsl( ldvsl, * ),
297 $ vsr( ldvsr, * ), work( * )
298* ..
299* .. Function Arguments ..
300 LOGICAL SELCTG
301 EXTERNAL SELCTG
302* ..
303*
304* =====================================================================
305*
306* .. Parameters ..
307 REAL ZERO, ONE
308 PARAMETER ( ZERO = 0.0e+0, one = 1.0e+0 )
309* ..
310* .. Local Scalars ..
311 LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
312 $ LQUERY, LST2SL, WANTST
313 INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
314 $ ILO, IP, IRIGHT, IROWS, ITAU, IWRK, MAXWRK,
315 $ minwrk
316 REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
317 $ PVSR, SAFMAX, SAFMIN, SMLNUM
318* ..
319* .. Local Arrays ..
320 INTEGER IDUM( 1 )
321 REAL DIF( 2 )
322* ..
323* .. External Subroutines ..
324 EXTERNAL sgeqrf, sggbak, sggbal, sgghrd, shgeqz, slabad,
326 $ xerbla
327* ..
328* .. External Functions ..
329 LOGICAL LSAME
330 INTEGER ILAENV
331 REAL SLAMCH, SLANGE
332 EXTERNAL lsame, ilaenv, slamch, slange
333* ..
334* .. Intrinsic Functions ..
335 INTRINSIC abs, max, sqrt
336* ..
337* .. Executable Statements ..
338*
339* Decode the input arguments
340*
341 IF( lsame( jobvsl, 'N' ) ) THEN
342 ijobvl = 1
343 ilvsl = .false.
344 ELSE IF( lsame( jobvsl, 'V' ) ) THEN
345 ijobvl = 2
346 ilvsl = .true.
347 ELSE
348 ijobvl = -1
349 ilvsl = .false.
350 END IF
351*
352 IF( lsame( jobvsr, 'N' ) ) THEN
353 ijobvr = 1
354 ilvsr = .false.
355 ELSE IF( lsame( jobvsr, 'V' ) ) THEN
356 ijobvr = 2
357 ilvsr = .true.
358 ELSE
359 ijobvr = -1
360 ilvsr = .false.
361 END IF
362*
363 wantst = lsame( sort, 'S' )
364*
365* Test the input arguments
366*
367 info = 0
368 lquery = ( lwork.EQ.-1 )
369 IF( ijobvl.LE.0 ) THEN
370 info = -1
371 ELSE IF( ijobvr.LE.0 ) THEN
372 info = -2
373 ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
374 info = -3
375 ELSE IF( n.LT.0 ) THEN
376 info = -5
377 ELSE IF( lda.LT.max( 1, n ) ) THEN
378 info = -7
379 ELSE IF( ldb.LT.max( 1, n ) ) THEN
380 info = -9
381 ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
382 info = -15
383 ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
384 info = -17
385 END IF
386*
387* Compute workspace
388* (Note: Comments in the code beginning "Workspace:" describe the
389* minimal amount of workspace needed at that point in the code,
390* as well as the preferred amount for good performance.
391* NB refers to the optimal block size for the immediately
392* following subroutine, as returned by ILAENV.)
393*
394 IF( info.EQ.0 ) THEN
395 IF( n.GT.0 )THEN
396 minwrk = max( 8*n, 6*n + 16 )
397 maxwrk = minwrk - n +
398 $ n*ilaenv( 1, 'SGEQRF', ' ', n, 1, n, 0 )
399 maxwrk = max( maxwrk, minwrk - n +
400 $ n*ilaenv( 1, 'SORMQR', ' ', n, 1, n, -1 ) )
401 IF( ilvsl ) THEN
402 maxwrk = max( maxwrk, minwrk - n +
403 $ n*ilaenv( 1, 'SORGQR', ' ', n, 1, n, -1 ) )
404 END IF
405 ELSE
406 minwrk = 1
407 maxwrk = 1
408 END IF
409 work( 1 ) = maxwrk
410*
411 IF( lwork.LT.minwrk .AND. .NOT.lquery )
412 $ info = -19
413 END IF
414*
415 IF( info.NE.0 ) THEN
416 CALL xerbla( 'SGGES ', -info )
417 RETURN
418 ELSE IF( lquery ) THEN
419 RETURN
420 END IF
421*
422* Quick return if possible
423*
424 IF( n.EQ.0 ) THEN
425 sdim = 0
426 RETURN
427 END IF
428*
429* Get machine constants
430*
431 eps = slamch( 'P' )
432 safmin = slamch( 'S' )
433 safmax = one / safmin
434 CALL slabad( safmin, safmax )
435 smlnum = sqrt( safmin ) / eps
436 bignum = one / smlnum
437*
438* Scale A if max element outside range [SMLNUM,BIGNUM]
439*
440 anrm = slange( 'M', n, n, a, lda, work )
441 ilascl = .false.
442 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
443 anrmto = smlnum
444 ilascl = .true.
445 ELSE IF( anrm.GT.bignum ) THEN
446 anrmto = bignum
447 ilascl = .true.
448 END IF
449 IF( ilascl )
450 $ CALL slascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
451*
452* Scale B if max element outside range [SMLNUM,BIGNUM]
453*
454 bnrm = slange( 'M', n, n, b, ldb, work )
455 ilbscl = .false.
456 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
457 bnrmto = smlnum
458 ilbscl = .true.
459 ELSE IF( bnrm.GT.bignum ) THEN
460 bnrmto = bignum
461 ilbscl = .true.
462 END IF
463 IF( ilbscl )
464 $ CALL slascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
465*
466* Permute the matrix to make it more nearly triangular
467* (Workspace: need 6*N + 2*N space for storing balancing factors)
468*
469 ileft = 1
470 iright = n + 1
471 iwrk = iright + n
472 CALL sggbal( 'P', n, a, lda, b, ldb, ilo, ihi, work( ileft ),
473 $ work( iright ), work( iwrk ), ierr )
474*
475* Reduce B to triangular form (QR decomposition of B)
476* (Workspace: need N, prefer N*NB)
477*
478 irows = ihi + 1 - ilo
479 icols = n + 1 - ilo
480 itau = iwrk
481 iwrk = itau + irows
482 CALL sgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
483 $ work( iwrk ), lwork+1-iwrk, ierr )
484*
485* Apply the orthogonal transformation to matrix A
486* (Workspace: need N, prefer N*NB)
487*
488 CALL sormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
489 $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
490 $ lwork+1-iwrk, ierr )
491*
492* Initialize VSL
493* (Workspace: need N, prefer N*NB)
494*
495 IF( ilvsl ) THEN
496 CALL slaset( 'Full', n, n, zero, one, vsl, ldvsl )
497 IF( irows.GT.1 ) THEN
498 CALL slacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
499 $ vsl( ilo+1, ilo ), ldvsl )
500 END IF
501 CALL sorgqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
502 $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
503 END IF
504*
505* Initialize VSR
506*
507 IF( ilvsr )
508 $ CALL slaset( 'Full', n, n, zero, one, vsr, ldvsr )
509*
510* Reduce to generalized Hessenberg form
511* (Workspace: none needed)
512*
513 CALL sgghrd( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
514 $ ldvsl, vsr, ldvsr, ierr )
515*
516* Perform QZ algorithm, computing Schur vectors if desired
517* (Workspace: need N)
518*
519 iwrk = itau
520 CALL shgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
521 $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
522 $ work( iwrk ), lwork+1-iwrk, ierr )
523 IF( ierr.NE.0 ) THEN
524 IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
525 info = ierr
526 ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
527 info = ierr - n
528 ELSE
529 info = n + 1
530 END IF
531 GO TO 40
532 END IF
533*
534* Sort eigenvalues ALPHA/BETA if desired
535* (Workspace: need 4*N+16 )
536*
537 sdim = 0
538 IF( wantst ) THEN
539*
540* Undo scaling on eigenvalues before SELCTGing
541*
542 IF( ilascl ) THEN
543 CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n,
544 $ ierr )
545 CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n,
546 $ ierr )
547 END IF
548 IF( ilbscl )
549 $ CALL slascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
550*
551* Select eigenvalues
552*
553 DO 10 i = 1, n
554 bwork( i ) = selctg( alphar( i ), alphai( i ), beta( i ) )
555 10 CONTINUE
556*
557 CALL stgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb, alphar,
558 $ alphai, beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl,
559 $ pvsr, dif, work( iwrk ), lwork-iwrk+1, idum, 1,
560 $ ierr )
561 IF( ierr.EQ.1 )
562 $ info = n + 3
563*
564 END IF
565*
566* Apply back-permutation to VSL and VSR
567* (Workspace: none needed)
568*
569 IF( ilvsl )
570 $ CALL sggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
571 $ work( iright ), n, vsl, ldvsl, ierr )
572*
573 IF( ilvsr )
574 $ CALL sggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
575 $ work( iright ), n, vsr, ldvsr, ierr )
576*
577* Check if unscaling would cause over/underflow, if so, rescale
578* (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
579* B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
580*
581 IF( ilascl )THEN
582 DO 50 i = 1, n
583 IF( alphai( i ).NE.zero ) THEN
584 IF( ( alphar( i )/safmax ).GT.( anrmto/anrm ) .OR.
585 $ ( safmin/alphar( i ) ).GT.( anrm/anrmto ) ) THEN
586 work( 1 ) = abs( a( i, i )/alphar( i ) )
587 beta( i ) = beta( i )*work( 1 )
588 alphar( i ) = alphar( i )*work( 1 )
589 alphai( i ) = alphai( i )*work( 1 )
590 ELSE IF( ( alphai( i )/safmax ).GT.( anrmto/anrm ) .OR.
591 $ ( safmin/alphai( i ) ).GT.( anrm/anrmto ) ) THEN
592 work( 1 ) = abs( a( i, i+1 )/alphai( i ) )
593 beta( i ) = beta( i )*work( 1 )
594 alphar( i ) = alphar( i )*work( 1 )
595 alphai( i ) = alphai( i )*work( 1 )
596 END IF
597 END IF
598 50 CONTINUE
599 END IF
600*
601 IF( ilbscl )THEN
602 DO 60 i = 1, n
603 IF( alphai( i ).NE.zero ) THEN
604 IF( ( beta( i )/safmax ).GT.( bnrmto/bnrm ) .OR.
605 $ ( safmin/beta( i ) ).GT.( bnrm/bnrmto ) ) THEN
606 work( 1 ) = abs(b( i, i )/beta( i ))
607 beta( i ) = beta( i )*work( 1 )
608 alphar( i ) = alphar( i )*work( 1 )
609 alphai( i ) = alphai( i )*work( 1 )
610 END IF
611 END IF
612 60 CONTINUE
613 END IF
614*
615* Undo scaling
616*
617 IF( ilascl ) THEN
618 CALL slascl( 'H', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
619 CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
620 CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
621 END IF
622*
623 IF( ilbscl ) THEN
624 CALL slascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
625 CALL slascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
626 END IF
627*
628 IF( wantst ) THEN
629*
630* Check if reordering is correct
631*
632 lastsl = .true.
633 lst2sl = .true.
634 sdim = 0
635 ip = 0
636 DO 30 i = 1, n
637 cursl = selctg( alphar( i ), alphai( i ), beta( i ) )
638 IF( alphai( i ).EQ.zero ) THEN
639 IF( cursl )
640 $ sdim = sdim + 1
641 ip = 0
642 IF( cursl .AND. .NOT.lastsl )
643 $ info = n + 2
644 ELSE
645 IF( ip.EQ.1 ) THEN
646*
647* Last eigenvalue of conjugate pair
648*
649 cursl = cursl .OR. lastsl
650 lastsl = cursl
651 IF( cursl )
652 $ sdim = sdim + 2
653 ip = -1
654 IF( cursl .AND. .NOT.lst2sl )
655 $ info = n + 2
656 ELSE
657*
658* First eigenvalue of conjugate pair
659*
660 ip = 1
661 END IF
662 END IF
663 lst2sl = lastsl
664 lastsl = cursl
665 30 CONTINUE
666*
667 END IF
668*
669 40 CONTINUE
670*
671 work( 1 ) = maxwrk
672*
673 RETURN
674*
675* End of SGGES
676*
677 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 sgges(JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, BWORK, INFO)
SGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE m...
Definition: sgges.f:284
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