LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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cgeesx.f
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1*> \brief <b> CGEESX 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 CGEESX + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgeesx.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgeesx.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgeesx.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W,
22* VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK,
23* BWORK, INFO )
24*
25* .. Scalar Arguments ..
26* CHARACTER JOBVS, SENSE, SORT
27* INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
28* REAL RCONDE, RCONDV
29* ..
30* .. Array Arguments ..
31* LOGICAL BWORK( * )
32* REAL RWORK( * )
33* COMPLEX A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )
34* ..
35* .. Function Arguments ..
36* LOGICAL SELECT
37* EXTERNAL SELECT
38* ..
39*
40*
41*> \par Purpose:
42* =============
43*>
44*> \verbatim
45*>
46*> CGEESX computes for an N-by-N complex nonsymmetric matrix A, the
47*> eigenvalues, the Schur form T, and, optionally, the matrix of Schur
48*> vectors Z. This gives the Schur factorization A = Z*T*(Z**H).
49*>
50*> Optionally, it also orders the eigenvalues on the diagonal of the
51*> Schur form so that selected eigenvalues are at the top left;
52*> computes a reciprocal condition number for the average of the
53*> selected eigenvalues (RCONDE); and computes a reciprocal condition
54*> number for the right invariant subspace corresponding to the
55*> selected eigenvalues (RCONDV). The leading columns of Z form an
56*> orthonormal basis for this invariant subspace.
57*>
58*> For further explanation of the reciprocal condition numbers RCONDE
59*> and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
60*> these quantities are called s and sep respectively).
61*>
62*> A complex matrix is in Schur form if it is upper triangular.
63*> \endverbatim
64*
65* Arguments:
66* ==========
67*
68*> \param[in] JOBVS
69*> \verbatim
70*> JOBVS is CHARACTER*1
71*> = 'N': Schur vectors are not computed;
72*> = 'V': Schur vectors are computed.
73*> \endverbatim
74*>
75*> \param[in] SORT
76*> \verbatim
77*> SORT is CHARACTER*1
78*> Specifies whether or not to order the eigenvalues on the
79*> diagonal of the Schur form.
80*> = 'N': Eigenvalues are not ordered;
81*> = 'S': Eigenvalues are ordered (see SELECT).
82*> \endverbatim
83*>
84*> \param[in] SELECT
85*> \verbatim
86*> SELECT is a LOGICAL FUNCTION of one COMPLEX argument
87*> SELECT must be declared EXTERNAL in the calling subroutine.
88*> If SORT = 'S', SELECT is used to select eigenvalues to order
89*> to the top left of the Schur form.
90*> If SORT = 'N', SELECT is not referenced.
91*> An eigenvalue W(j) is selected if SELECT(W(j)) is true.
92*> \endverbatim
93*>
94*> \param[in] SENSE
95*> \verbatim
96*> SENSE is CHARACTER*1
97*> Determines which reciprocal condition numbers are computed.
98*> = 'N': None are computed;
99*> = 'E': Computed for average of selected eigenvalues only;
100*> = 'V': Computed for selected right invariant subspace only;
101*> = 'B': Computed for both.
102*> If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
103*> \endverbatim
104*>
105*> \param[in] N
106*> \verbatim
107*> N is INTEGER
108*> The order of the matrix A. N >= 0.
109*> \endverbatim
110*>
111*> \param[in,out] A
112*> \verbatim
113*> A is COMPLEX array, dimension (LDA, N)
114*> On entry, the N-by-N matrix A.
115*> On exit, A is overwritten by its Schur form T.
116*> \endverbatim
117*>
118*> \param[in] LDA
119*> \verbatim
120*> LDA is INTEGER
121*> The leading dimension of the array A. LDA >= max(1,N).
122*> \endverbatim
123*>
124*> \param[out] SDIM
125*> \verbatim
126*> SDIM is INTEGER
127*> If SORT = 'N', SDIM = 0.
128*> If SORT = 'S', SDIM = number of eigenvalues for which
129*> SELECT is true.
130*> \endverbatim
131*>
132*> \param[out] W
133*> \verbatim
134*> W is COMPLEX array, dimension (N)
135*> W contains the computed eigenvalues, in the same order
136*> that they appear on the diagonal of the output Schur form T.
137*> \endverbatim
138*>
139*> \param[out] VS
140*> \verbatim
141*> VS is COMPLEX array, dimension (LDVS,N)
142*> If JOBVS = 'V', VS contains the unitary matrix Z of Schur
143*> vectors.
144*> If JOBVS = 'N', VS is not referenced.
145*> \endverbatim
146*>
147*> \param[in] LDVS
148*> \verbatim
149*> LDVS is INTEGER
150*> The leading dimension of the array VS. LDVS >= 1, and if
151*> JOBVS = 'V', LDVS >= N.
152*> \endverbatim
153*>
154*> \param[out] RCONDE
155*> \verbatim
156*> RCONDE is REAL
157*> If SENSE = 'E' or 'B', RCONDE contains the reciprocal
158*> condition number for the average of the selected eigenvalues.
159*> Not referenced if SENSE = 'N' or 'V'.
160*> \endverbatim
161*>
162*> \param[out] RCONDV
163*> \verbatim
164*> RCONDV is REAL
165*> If SENSE = 'V' or 'B', RCONDV contains the reciprocal
166*> condition number for the selected right invariant subspace.
167*> Not referenced if SENSE = 'N' or 'E'.
168*> \endverbatim
169*>
170*> \param[out] WORK
171*> \verbatim
172*> WORK is COMPLEX array, dimension (MAX(1,LWORK))
173*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
174*> \endverbatim
175*>
176*> \param[in] LWORK
177*> \verbatim
178*> LWORK is INTEGER
179*> The dimension of the array WORK. LWORK >= max(1,2*N).
180*> Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),
181*> where SDIM is the number of selected eigenvalues computed by
182*> this routine. Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also
183*> that an error is only returned if LWORK < max(1,2*N), but if
184*> SENSE = 'E' or 'V' or 'B' this may not be large enough.
185*> For good performance, LWORK must generally be larger.
186*>
187*> If LWORK = -1, then a workspace query is assumed; the routine
188*> only calculates upper bound on the optimal size of the
189*> array WORK, returns this value as the first entry of the WORK
190*> array, and no error message related to LWORK is issued by
191*> XERBLA.
192*> \endverbatim
193*>
194*> \param[out] RWORK
195*> \verbatim
196*> RWORK is REAL array, dimension (N)
197*> \endverbatim
198*>
199*> \param[out] BWORK
200*> \verbatim
201*> BWORK is LOGICAL array, dimension (N)
202*> Not referenced if SORT = 'N'.
203*> \endverbatim
204*>
205*> \param[out] INFO
206*> \verbatim
207*> INFO is INTEGER
208*> = 0: successful exit
209*> < 0: if INFO = -i, the i-th argument had an illegal value.
210*> > 0: if INFO = i, and i is
211*> <= N: the QR algorithm failed to compute all the
212*> eigenvalues; elements 1:ILO-1 and i+1:N of W
213*> contain those eigenvalues which have converged; if
214*> JOBVS = 'V', VS contains the transformation which
215*> reduces A to its partially converged Schur form.
216*> = N+1: the eigenvalues could not be reordered because some
217*> eigenvalues were too close to separate (the problem
218*> is very ill-conditioned);
219*> = N+2: after reordering, roundoff changed values of some
220*> complex eigenvalues so that leading eigenvalues in
221*> the Schur form no longer satisfy SELECT=.TRUE. This
222*> could also be caused by underflow due to scaling.
223*> \endverbatim
224*
225* Authors:
226* ========
227*
228*> \author Univ. of Tennessee
229*> \author Univ. of California Berkeley
230*> \author Univ. of Colorado Denver
231*> \author NAG Ltd.
232*
233*> \ingroup complexGEeigen
234*
235* =====================================================================
236 SUBROUTINE cgeesx( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W,
237 $ VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK,
238 $ BWORK, INFO )
239*
240* -- LAPACK driver routine --
241* -- LAPACK is a software package provided by Univ. of Tennessee, --
242* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
243*
244* .. Scalar Arguments ..
245 CHARACTER JOBVS, SENSE, SORT
246 INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
247 REAL RCONDE, RCONDV
248* ..
249* .. Array Arguments ..
250 LOGICAL BWORK( * )
251 REAL RWORK( * )
252 COMPLEX A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )
253* ..
254* .. Function Arguments ..
255 LOGICAL SELECT
256 EXTERNAL SELECT
257* ..
258*
259* =====================================================================
260*
261* .. Parameters ..
262 REAL ZERO, ONE
263 PARAMETER ( ZERO = 0.0e0, one = 1.0e0 )
264* ..
265* .. Local Scalars ..
266 LOGICAL LQUERY, SCALEA, WANTSB, WANTSE, WANTSN, WANTST,
267 $ WANTSV, WANTVS
268 INTEGER HSWORK, I, IBAL, ICOND, IERR, IEVAL, IHI, ILO,
269 $ ITAU, IWRK, LWRK, MAXWRK, MINWRK
270 REAL ANRM, BIGNUM, CSCALE, EPS, SMLNUM
271* ..
272* .. Local Arrays ..
273 REAL DUM( 1 )
274* ..
275* .. External Subroutines ..
276 EXTERNAL ccopy, cgebak, cgebal, cgehrd, chseqr, clacpy,
278* ..
279* .. External Functions ..
280 LOGICAL LSAME
281 INTEGER ILAENV
282 REAL CLANGE, SLAMCH
283 EXTERNAL lsame, ilaenv, clange, slamch
284* ..
285* .. Intrinsic Functions ..
286 INTRINSIC max, sqrt
287* ..
288* .. Executable Statements ..
289*
290* Test the input arguments
291*
292 info = 0
293 wantvs = lsame( jobvs, 'V' )
294 wantst = lsame( sort, 'S' )
295 wantsn = lsame( sense, 'N' )
296 wantse = lsame( sense, 'E' )
297 wantsv = lsame( sense, 'V' )
298 wantsb = lsame( sense, 'B' )
299 lquery = ( lwork.EQ.-1 )
300*
301 IF( ( .NOT.wantvs ) .AND. ( .NOT.lsame( jobvs, 'N' ) ) ) THEN
302 info = -1
303 ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
304 info = -2
305 ELSE IF( .NOT.( wantsn .OR. wantse .OR. wantsv .OR. wantsb ) .OR.
306 $ ( .NOT.wantst .AND. .NOT.wantsn ) ) THEN
307 info = -4
308 ELSE IF( n.LT.0 ) THEN
309 info = -5
310 ELSE IF( lda.LT.max( 1, n ) ) THEN
311 info = -7
312 ELSE IF( ldvs.LT.1 .OR. ( wantvs .AND. ldvs.LT.n ) ) THEN
313 info = -11
314 END IF
315*
316* Compute workspace
317* (Note: Comments in the code beginning "Workspace:" describe the
318* minimal amount of real workspace needed at that point in the
319* code, as well as the preferred amount for good performance.
320* CWorkspace refers to complex workspace, and RWorkspace to real
321* workspace. NB refers to the optimal block size for the
322* immediately following subroutine, as returned by ILAENV.
323* HSWORK refers to the workspace preferred by CHSEQR, as
324* calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
325* the worst case.
326* If SENSE = 'E', 'V' or 'B', then the amount of workspace needed
327* depends on SDIM, which is computed by the routine CTRSEN later
328* in the code.)
329*
330 IF( info.EQ.0 ) THEN
331 IF( n.EQ.0 ) THEN
332 minwrk = 1
333 lwrk = 1
334 ELSE
335 maxwrk = n + n*ilaenv( 1, 'CGEHRD', ' ', n, 1, n, 0 )
336 minwrk = 2*n
337*
338 CALL chseqr( 'S', jobvs, n, 1, n, a, lda, w, vs, ldvs,
339 $ work, -1, ieval )
340 hswork = int( work( 1 ) )
341*
342 IF( .NOT.wantvs ) THEN
343 maxwrk = max( maxwrk, hswork )
344 ELSE
345 maxwrk = max( maxwrk, n + ( n - 1 )*ilaenv( 1, 'CUNGHR',
346 $ ' ', n, 1, n, -1 ) )
347 maxwrk = max( maxwrk, hswork )
348 END IF
349 lwrk = maxwrk
350 IF( .NOT.wantsn )
351 $ lwrk = max( lwrk, ( n*n )/2 )
352 END IF
353 work( 1 ) = lwrk
354*
355 IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
356 info = -15
357 END IF
358 END IF
359*
360 IF( info.NE.0 ) THEN
361 CALL xerbla( 'CGEESX', -info )
362 RETURN
363 ELSE IF( lquery ) THEN
364 RETURN
365 END IF
366*
367* Quick return if possible
368*
369 IF( n.EQ.0 ) THEN
370 sdim = 0
371 RETURN
372 END IF
373*
374* Get machine constants
375*
376 eps = slamch( 'P' )
377 smlnum = slamch( 'S' )
378 bignum = one / smlnum
379 CALL slabad( smlnum, bignum )
380 smlnum = sqrt( smlnum ) / eps
381 bignum = one / smlnum
382*
383* Scale A if max element outside range [SMLNUM,BIGNUM]
384*
385 anrm = clange( 'M', n, n, a, lda, dum )
386 scalea = .false.
387 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
388 scalea = .true.
389 cscale = smlnum
390 ELSE IF( anrm.GT.bignum ) THEN
391 scalea = .true.
392 cscale = bignum
393 END IF
394 IF( scalea )
395 $ CALL clascl( 'G', 0, 0, anrm, cscale, n, n, a, lda, ierr )
396*
397*
398* Permute the matrix to make it more nearly triangular
399* (CWorkspace: none)
400* (RWorkspace: need N)
401*
402 ibal = 1
403 CALL cgebal( 'P', n, a, lda, ilo, ihi, rwork( ibal ), ierr )
404*
405* Reduce to upper Hessenberg form
406* (CWorkspace: need 2*N, prefer N+N*NB)
407* (RWorkspace: none)
408*
409 itau = 1
410 iwrk = n + itau
411 CALL cgehrd( n, ilo, ihi, a, lda, work( itau ), work( iwrk ),
412 $ lwork-iwrk+1, ierr )
413*
414 IF( wantvs ) THEN
415*
416* Copy Householder vectors to VS
417*
418 CALL clacpy( 'L', n, n, a, lda, vs, ldvs )
419*
420* Generate unitary matrix in VS
421* (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
422* (RWorkspace: none)
423*
424 CALL cunghr( n, ilo, ihi, vs, ldvs, work( itau ), work( iwrk ),
425 $ lwork-iwrk+1, ierr )
426 END IF
427*
428 sdim = 0
429*
430* Perform QR iteration, accumulating Schur vectors in VS if desired
431* (CWorkspace: need 1, prefer HSWORK (see comments) )
432* (RWorkspace: none)
433*
434 iwrk = itau
435 CALL chseqr( 'S', jobvs, n, ilo, ihi, a, lda, w, vs, ldvs,
436 $ work( iwrk ), lwork-iwrk+1, ieval )
437 IF( ieval.GT.0 )
438 $ info = ieval
439*
440* Sort eigenvalues if desired
441*
442 IF( wantst .AND. info.EQ.0 ) THEN
443 IF( scalea )
444 $ CALL clascl( 'G', 0, 0, cscale, anrm, n, 1, w, n, ierr )
445 DO 10 i = 1, n
446 bwork( i ) = SELECT( w( i ) )
447 10 CONTINUE
448*
449* Reorder eigenvalues, transform Schur vectors, and compute
450* reciprocal condition numbers
451* (CWorkspace: if SENSE is not 'N', need 2*SDIM*(N-SDIM)
452* otherwise, need none )
453* (RWorkspace: none)
454*
455 CALL ctrsen( sense, jobvs, bwork, n, a, lda, vs, ldvs, w, sdim,
456 $ rconde, rcondv, work( iwrk ), lwork-iwrk+1,
457 $ icond )
458 IF( .NOT.wantsn )
459 $ maxwrk = max( maxwrk, 2*sdim*( n-sdim ) )
460 IF( icond.EQ.-14 ) THEN
461*
462* Not enough complex workspace
463*
464 info = -15
465 END IF
466 END IF
467*
468 IF( wantvs ) THEN
469*
470* Undo balancing
471* (CWorkspace: none)
472* (RWorkspace: need N)
473*
474 CALL cgebak( 'P', 'R', n, ilo, ihi, rwork( ibal ), n, vs, ldvs,
475 $ ierr )
476 END IF
477*
478 IF( scalea ) THEN
479*
480* Undo scaling for the Schur form of A
481*
482 CALL clascl( 'U', 0, 0, cscale, anrm, n, n, a, lda, ierr )
483 CALL ccopy( n, a, lda+1, w, 1 )
484 IF( ( wantsv .OR. wantsb ) .AND. info.EQ.0 ) THEN
485 dum( 1 ) = rcondv
486 CALL slascl( 'G', 0, 0, cscale, anrm, 1, 1, dum, 1, ierr )
487 rcondv = dum( 1 )
488 END IF
489 END IF
490*
491 work( 1 ) = maxwrk
492 RETURN
493*
494* End of CGEESX
495*
496 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 xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine cgehrd(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
CGEHRD
Definition: cgehrd.f:167
subroutine cgebal(JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
CGEBAL
Definition: cgebal.f:161
subroutine cgebak(JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO)
CGEBAK
Definition: cgebak.f:131
subroutine cgeesx(JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W, VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK, BWORK, INFO)
CGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE ...
Definition: cgeesx.f:239
subroutine clascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
CLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: clascl.f:143
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
subroutine ctrsen(JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S, SEP, WORK, LWORK, INFO)
CTRSEN
Definition: ctrsen.f:264
subroutine cunghr(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
CUNGHR
Definition: cunghr.f:126
subroutine chseqr(JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK, INFO)
CHSEQR
Definition: chseqr.f:299