LAPACK  3.6.0
LAPACK: Linear Algebra PACKage
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
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11 *> [TGZ]</a>
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13 *> [ZIP]</a>
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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 procedure) 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 *> \date November 2011
341 *
342 *> \ingroup doubleGEeigen
343 *
344 *> \par Further Details:
345 * =====================
346 *>
347 *> \verbatim
348 *>
349 *> An approximate (asymptotic) bound on the average absolute error of
350 *> the selected eigenvalues is
351 *>
352 *> EPS * norm((A, B)) / RCONDE( 1 ).
353 *>
354 *> An approximate (asymptotic) bound on the maximum angular error in
355 *> the computed deflating subspaces is
356 *>
357 *> EPS * norm((A, B)) / RCONDV( 2 ).
358 *>
359 *> See LAPACK User's Guide, section 4.11 for more information.
360 *> \endverbatim
361 *>
362 * =====================================================================
363  SUBROUTINE dggesx( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA,
364  $ b, ldb, sdim, alphar, alphai, beta, vsl, ldvsl,
365  $ vsr, ldvsr, rconde, rcondv, work, lwork, iwork,
366  $ liwork, bwork, info )
367 *
368 * -- LAPACK driver routine (version 3.4.0) --
369 * -- LAPACK is a software package provided by Univ. of Tennessee, --
370 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
371 * November 2011
372 *
373 * .. Scalar Arguments ..
374  CHARACTER JOBVSL, JOBVSR, SENSE, SORT
375  INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N,
376  $ sdim
377 * ..
378 * .. Array Arguments ..
379  LOGICAL BWORK( * )
380  INTEGER IWORK( * )
381  DOUBLE PRECISION A( lda, * ), ALPHAI( * ), ALPHAR( * ),
382  $ b( ldb, * ), beta( * ), rconde( 2 ),
383  $ rcondv( 2 ), vsl( ldvsl, * ), vsr( ldvsr, * ),
384  $ work( * )
385 * ..
386 * .. Function Arguments ..
387  LOGICAL SELCTG
388  EXTERNAL selctg
389 * ..
390 *
391 * =====================================================================
392 *
393 * .. Parameters ..
394  DOUBLE PRECISION ZERO, ONE
395  parameter( zero = 0.0d+0, one = 1.0d+0 )
396 * ..
397 * .. Local Scalars ..
398  LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
399  $ lquery, lst2sl, wantsb, wantse, wantsn, wantst,
400  $ wantsv
401  INTEGER I, ICOLS, IERR, IHI, IJOB, IJOBVL, IJOBVR,
402  $ ileft, ilo, ip, iright, irows, itau, iwrk,
403  $ liwmin, lwrk, maxwrk, minwrk
404  DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PL,
405  $ pr, safmax, safmin, smlnum
406 * ..
407 * .. Local Arrays ..
408  DOUBLE PRECISION DIF( 2 )
409 * ..
410 * .. External Subroutines ..
411  EXTERNAL dgeqrf, dggbak, dggbal, dgghrd, dhgeqz, dlabad,
413  $ xerbla
414 * ..
415 * .. External Functions ..
416  LOGICAL LSAME
417  INTEGER ILAENV
418  DOUBLE PRECISION DLAMCH, DLANGE
419  EXTERNAL lsame, ilaenv, dlamch, dlange
420 * ..
421 * .. Intrinsic Functions ..
422  INTRINSIC abs, max, sqrt
423 * ..
424 * .. Executable Statements ..
425 *
426 * Decode the input arguments
427 *
428  IF( lsame( jobvsl, 'N' ) ) THEN
429  ijobvl = 1
430  ilvsl = .false.
431  ELSE IF( lsame( jobvsl, 'V' ) ) THEN
432  ijobvl = 2
433  ilvsl = .true.
434  ELSE
435  ijobvl = -1
436  ilvsl = .false.
437  END IF
438 *
439  IF( lsame( jobvsr, 'N' ) ) THEN
440  ijobvr = 1
441  ilvsr = .false.
442  ELSE IF( lsame( jobvsr, 'V' ) ) THEN
443  ijobvr = 2
444  ilvsr = .true.
445  ELSE
446  ijobvr = -1
447  ilvsr = .false.
448  END IF
449 *
450  wantst = lsame( sort, 'S' )
451  wantsn = lsame( sense, 'N' )
452  wantse = lsame( sense, 'E' )
453  wantsv = lsame( sense, 'V' )
454  wantsb = lsame( sense, 'B' )
455  lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
456  IF( wantsn ) THEN
457  ijob = 0
458  ELSE IF( wantse ) THEN
459  ijob = 1
460  ELSE IF( wantsv ) THEN
461  ijob = 2
462  ELSE IF( wantsb ) THEN
463  ijob = 4
464  END IF
465 *
466 * Test the input arguments
467 *
468  info = 0
469  IF( ijobvl.LE.0 ) THEN
470  info = -1
471  ELSE IF( ijobvr.LE.0 ) THEN
472  info = -2
473  ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
474  info = -3
475  ELSE IF( .NOT.( wantsn .OR. wantse .OR. wantsv .OR. wantsb ) .OR.
476  $ ( .NOT.wantst .AND. .NOT.wantsn ) ) THEN
477  info = -5
478  ELSE IF( n.LT.0 ) THEN
479  info = -6
480  ELSE IF( lda.LT.max( 1, n ) ) THEN
481  info = -8
482  ELSE IF( ldb.LT.max( 1, n ) ) THEN
483  info = -10
484  ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
485  info = -16
486  ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
487  info = -18
488  END IF
489 *
490 * Compute workspace
491 * (Note: Comments in the code beginning "Workspace:" describe the
492 * minimal amount of workspace needed at that point in the code,
493 * as well as the preferred amount for good performance.
494 * NB refers to the optimal block size for the immediately
495 * following subroutine, as returned by ILAENV.)
496 *
497  IF( info.EQ.0 ) THEN
498  IF( n.GT.0) THEN
499  minwrk = max( 8*n, 6*n + 16 )
500  maxwrk = minwrk - n +
501  $ n*ilaenv( 1, 'DGEQRF', ' ', n, 1, n, 0 )
502  maxwrk = max( maxwrk, minwrk - n +
503  $ n*ilaenv( 1, 'DORMQR', ' ', n, 1, n, -1 ) )
504  IF( ilvsl ) THEN
505  maxwrk = max( maxwrk, minwrk - n +
506  $ n*ilaenv( 1, 'DORGQR', ' ', n, 1, n, -1 ) )
507  END IF
508  lwrk = maxwrk
509  IF( ijob.GE.1 )
510  $ lwrk = max( lwrk, n*n/2 )
511  ELSE
512  minwrk = 1
513  maxwrk = 1
514  lwrk = 1
515  END IF
516  work( 1 ) = lwrk
517  IF( wantsn .OR. n.EQ.0 ) THEN
518  liwmin = 1
519  ELSE
520  liwmin = n + 6
521  END IF
522  iwork( 1 ) = liwmin
523 *
524  IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
525  info = -22
526  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
527  info = -24
528  END IF
529  END IF
530 *
531  IF( info.NE.0 ) THEN
532  CALL xerbla( 'DGGESX', -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 ) THEN
541  sdim = 0
542  RETURN
543  END IF
544 *
545 * Get machine constants
546 *
547  eps = dlamch( 'P' )
548  safmin = dlamch( 'S' )
549  safmax = one / safmin
550  CALL dlabad( safmin, safmax )
551  smlnum = sqrt( safmin ) / eps
552  bignum = one / smlnum
553 *
554 * Scale A if max element outside range [SMLNUM,BIGNUM]
555 *
556  anrm = dlange( 'M', n, n, a, lda, work )
557  ilascl = .false.
558  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
559  anrmto = smlnum
560  ilascl = .true.
561  ELSE IF( anrm.GT.bignum ) THEN
562  anrmto = bignum
563  ilascl = .true.
564  END IF
565  IF( ilascl )
566  $ CALL dlascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
567 *
568 * Scale B if max element outside range [SMLNUM,BIGNUM]
569 *
570  bnrm = dlange( 'M', n, n, b, ldb, work )
571  ilbscl = .false.
572  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
573  bnrmto = smlnum
574  ilbscl = .true.
575  ELSE IF( bnrm.GT.bignum ) THEN
576  bnrmto = bignum
577  ilbscl = .true.
578  END IF
579  IF( ilbscl )
580  $ CALL dlascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
581 *
582 * Permute the matrix to make it more nearly triangular
583 * (Workspace: need 6*N + 2*N for permutation parameters)
584 *
585  ileft = 1
586  iright = n + 1
587  iwrk = iright + n
588  CALL dggbal( 'P', n, a, lda, b, ldb, ilo, ihi, work( ileft ),
589  $ work( iright ), work( iwrk ), ierr )
590 *
591 * Reduce B to triangular form (QR decomposition of B)
592 * (Workspace: need N, prefer N*NB)
593 *
594  irows = ihi + 1 - ilo
595  icols = n + 1 - ilo
596  itau = iwrk
597  iwrk = itau + irows
598  CALL dgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
599  $ work( iwrk ), lwork+1-iwrk, ierr )
600 *
601 * Apply the orthogonal transformation to matrix A
602 * (Workspace: need N, prefer N*NB)
603 *
604  CALL dormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
605  $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
606  $ lwork+1-iwrk, ierr )
607 *
608 * Initialize VSL
609 * (Workspace: need N, prefer N*NB)
610 *
611  IF( ilvsl ) THEN
612  CALL dlaset( 'Full', n, n, zero, one, vsl, ldvsl )
613  IF( irows.GT.1 ) THEN
614  CALL dlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
615  $ vsl( ilo+1, ilo ), ldvsl )
616  END IF
617  CALL dorgqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
618  $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
619  END IF
620 *
621 * Initialize VSR
622 *
623  IF( ilvsr )
624  $ CALL dlaset( 'Full', n, n, zero, one, vsr, ldvsr )
625 *
626 * Reduce to generalized Hessenberg form
627 * (Workspace: none needed)
628 *
629  CALL dgghrd( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
630  $ ldvsl, vsr, ldvsr, ierr )
631 *
632  sdim = 0
633 *
634 * Perform QZ algorithm, computing Schur vectors if desired
635 * (Workspace: need N)
636 *
637  iwrk = itau
638  CALL dhgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
639  $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
640  $ work( iwrk ), lwork+1-iwrk, ierr )
641  IF( ierr.NE.0 ) THEN
642  IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
643  info = ierr
644  ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
645  info = ierr - n
646  ELSE
647  info = n + 1
648  END IF
649  GO TO 60
650  END IF
651 *
652 * Sort eigenvalues ALPHA/BETA and compute the reciprocal of
653 * condition number(s)
654 * (Workspace: If IJOB >= 1, need MAX( 8*(N+1), 2*SDIM*(N-SDIM) )
655 * otherwise, need 8*(N+1) )
656 *
657  IF( wantst ) THEN
658 *
659 * Undo scaling on eigenvalues before SELCTGing
660 *
661  IF( ilascl ) THEN
662  CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n,
663  $ ierr )
664  CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n,
665  $ ierr )
666  END IF
667  IF( ilbscl )
668  $ CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
669 *
670 * Select eigenvalues
671 *
672  DO 10 i = 1, n
673  bwork( i ) = selctg( alphar( i ), alphai( i ), beta( i ) )
674  10 CONTINUE
675 *
676 * Reorder eigenvalues, transform Generalized Schur vectors, and
677 * compute reciprocal condition numbers
678 *
679  CALL dtgsen( ijob, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
680  $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
681  $ sdim, pl, pr, dif, work( iwrk ), lwork-iwrk+1,
682  $ iwork, liwork, ierr )
683 *
684  IF( ijob.GE.1 )
685  $ maxwrk = max( maxwrk, 2*sdim*( n-sdim ) )
686  IF( ierr.EQ.-22 ) THEN
687 *
688 * not enough real workspace
689 *
690  info = -22
691  ELSE
692  IF( ijob.EQ.1 .OR. ijob.EQ.4 ) THEN
693  rconde( 1 ) = pl
694  rconde( 2 ) = pr
695  END IF
696  IF( ijob.EQ.2 .OR. ijob.EQ.4 ) THEN
697  rcondv( 1 ) = dif( 1 )
698  rcondv( 2 ) = dif( 2 )
699  END IF
700  IF( ierr.EQ.1 )
701  $ info = n + 3
702  END IF
703 *
704  END IF
705 *
706 * Apply permutation to VSL and VSR
707 * (Workspace: none needed)
708 *
709  IF( ilvsl )
710  $ CALL dggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
711  $ work( iright ), n, vsl, ldvsl, ierr )
712 *
713  IF( ilvsr )
714  $ CALL dggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
715  $ work( iright ), n, vsr, ldvsr, ierr )
716 *
717 * Check if unscaling would cause over/underflow, if so, rescale
718 * (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
719 * B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
720 *
721  IF( ilascl ) THEN
722  DO 20 i = 1, n
723  IF( alphai( i ).NE.zero ) THEN
724  IF( ( alphar( i ) / safmax ).GT.( anrmto / anrm ) .OR.
725  $ ( safmin / alphar( i ) ).GT.( anrm / anrmto ) ) THEN
726  work( 1 ) = abs( a( i, i ) / alphar( i ) )
727  beta( i ) = beta( i )*work( 1 )
728  alphar( i ) = alphar( i )*work( 1 )
729  alphai( i ) = alphai( i )*work( 1 )
730  ELSE IF( ( alphai( i ) / safmax ).GT.
731  $ ( anrmto / anrm ) .OR.
732  $ ( safmin / alphai( i ) ).GT.( anrm / anrmto ) )
733  $ THEN
734  work( 1 ) = abs( a( i, i+1 ) / alphai( i ) )
735  beta( i ) = beta( i )*work( 1 )
736  alphar( i ) = alphar( i )*work( 1 )
737  alphai( i ) = alphai( i )*work( 1 )
738  END IF
739  END IF
740  20 CONTINUE
741  END IF
742 *
743  IF( ilbscl ) THEN
744  DO 30 i = 1, n
745  IF( alphai( i ).NE.zero ) THEN
746  IF( ( beta( i ) / safmax ).GT.( bnrmto / bnrm ) .OR.
747  $ ( safmin / beta( i ) ).GT.( bnrm / bnrmto ) ) THEN
748  work( 1 ) = abs( b( i, i ) / beta( i ) )
749  beta( i ) = beta( i )*work( 1 )
750  alphar( i ) = alphar( i )*work( 1 )
751  alphai( i ) = alphai( i )*work( 1 )
752  END IF
753  END IF
754  30 CONTINUE
755  END IF
756 *
757 * Undo scaling
758 *
759  IF( ilascl ) THEN
760  CALL dlascl( 'H', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
761  CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
762  CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
763  END IF
764 *
765  IF( ilbscl ) THEN
766  CALL dlascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
767  CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
768  END IF
769 *
770  IF( wantst ) THEN
771 *
772 * Check if reordering is correct
773 *
774  lastsl = .true.
775  lst2sl = .true.
776  sdim = 0
777  ip = 0
778  DO 50 i = 1, n
779  cursl = selctg( alphar( i ), alphai( i ), beta( i ) )
780  IF( alphai( i ).EQ.zero ) THEN
781  IF( cursl )
782  $ sdim = sdim + 1
783  ip = 0
784  IF( cursl .AND. .NOT.lastsl )
785  $ info = n + 2
786  ELSE
787  IF( ip.EQ.1 ) THEN
788 *
789 * Last eigenvalue of conjugate pair
790 *
791  cursl = cursl .OR. lastsl
792  lastsl = cursl
793  IF( cursl )
794  $ sdim = sdim + 2
795  ip = -1
796  IF( cursl .AND. .NOT.lst2sl )
797  $ info = n + 2
798  ELSE
799 *
800 * First eigenvalue of conjugate pair
801 *
802  ip = 1
803  END IF
804  END IF
805  lst2sl = lastsl
806  lastsl = cursl
807  50 CONTINUE
808 *
809  END IF
810 *
811  60 CONTINUE
812 *
813  work( 1 ) = maxwrk
814  iwork( 1 ) = liwmin
815 *
816  RETURN
817 *
818 * End of DGGESX
819 *
820  END
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:112
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:105
subroutine dgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGEQRF
Definition: dgeqrf.f:138
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:76
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dgghrd(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
DGGHRD
Definition: dgghrd.f:209
subroutine dormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMQR
Definition: dormqr.f:169
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:306
subroutine dorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
DORGQR
Definition: dorgqr.f:130
subroutine dggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
DGGBAL
Definition: dggbal.f:179
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:367
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:141
subroutine dggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
DGGBAK
Definition: dggbak.f:149
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:454