LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ cgeesx()

subroutine cgeesx ( character  jobvs,
character  sort,
external  select,
character  sense,
integer  n,
complex, dimension( lda, * )  a,
integer  lda,
integer  sdim,
complex, dimension( * )  w,
complex, dimension( ldvs, * )  vs,
integer  ldvs,
real  rconde,
real  rcondv,
complex, dimension( * )  work,
integer  lwork,
real, dimension( * )  rwork,
logical, dimension( * )  bwork,
integer  info 
)

CGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Download CGEESX + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 CGEESX computes for an N-by-N complex nonsymmetric matrix A, the
 eigenvalues, the Schur form T, and, optionally, the matrix of Schur
 vectors Z.  This gives the Schur factorization A = Z*T*(Z**H).

 Optionally, it also orders the eigenvalues on the diagonal of the
 Schur form so that selected eigenvalues are at the top left;
 computes a reciprocal condition number for the average of the
 selected eigenvalues (RCONDE); and computes a reciprocal condition
 number for the right invariant subspace corresponding to the
 selected eigenvalues (RCONDV).  The leading columns of Z form an
 orthonormal basis for this invariant subspace.

 For further explanation of the reciprocal condition numbers RCONDE
 and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
 these quantities are called s and sep respectively).

 A complex matrix is in Schur form if it is upper triangular.
Parameters
[in]JOBVS
          JOBVS is CHARACTER*1
          = 'N': Schur vectors are not computed;
          = 'V': Schur vectors are computed.
[in]SORT
          SORT is CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the Schur form.
          = 'N': Eigenvalues are not ordered;
          = 'S': Eigenvalues are ordered (see SELECT).
[in]SELECT
          SELECT is a LOGICAL FUNCTION of one COMPLEX argument
          SELECT must be declared EXTERNAL in the calling subroutine.
          If SORT = 'S', SELECT is used to select eigenvalues to order
          to the top left of the Schur form.
          If SORT = 'N', SELECT is not referenced.
          An eigenvalue W(j) is selected if SELECT(W(j)) is true.
[in]SENSE
          SENSE is CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N': None are computed;
          = 'E': Computed for average of selected eigenvalues only;
          = 'V': Computed for selected right invariant subspace only;
          = 'B': Computed for both.
          If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
[in]N
          N is INTEGER
          The order of the matrix A. N >= 0.
[in,out]A
          A is COMPLEX array, dimension (LDA, N)
          On entry, the N-by-N matrix A.
          On exit, A is overwritten by its Schur form T.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
[out]SDIM
          SDIM is INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues for which
                         SELECT is true.
[out]W
          W is COMPLEX array, dimension (N)
          W contains the computed eigenvalues, in the same order
          that they appear on the diagonal of the output Schur form T.
[out]VS
          VS is COMPLEX array, dimension (LDVS,N)
          If JOBVS = 'V', VS contains the unitary matrix Z of Schur
          vectors.
          If JOBVS = 'N', VS is not referenced.
[in]LDVS
          LDVS is INTEGER
          The leading dimension of the array VS.  LDVS >= 1, and if
          JOBVS = 'V', LDVS >= N.
[out]RCONDE
          RCONDE is REAL
          If SENSE = 'E' or 'B', RCONDE contains the reciprocal
          condition number for the average of the selected eigenvalues.
          Not referenced if SENSE = 'N' or 'V'.
[out]RCONDV
          RCONDV is REAL
          If SENSE = 'V' or 'B', RCONDV contains the reciprocal
          condition number for the selected right invariant subspace.
          Not referenced if SENSE = 'N' or 'E'.
[out]WORK
          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
[in]LWORK
          LWORK is INTEGER
          The dimension of the array WORK.  LWORK >= max(1,2*N).
          Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),
          where SDIM is the number of selected eigenvalues computed by
          this routine.  Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also
          that an error is only returned if LWORK < max(1,2*N), but if
          SENSE = 'E' or 'V' or 'B' this may not be large enough.
          For good performance, LWORK must generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates upper bound on the optimal size of the
          array WORK, returns this value as the first entry of the WORK
          array, and no error message related to LWORK is issued by
          XERBLA.
[out]RWORK
          RWORK is REAL array, dimension (N)
[out]BWORK
          BWORK is LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.
[out]INFO
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value.
          > 0: if INFO = i, and i is
             <= N: the QR algorithm failed to compute all the
                   eigenvalues; elements 1:ILO-1 and i+1:N of W
                   contain those eigenvalues which have converged; if
                   JOBVS = 'V', VS contains the transformation which
                   reduces A to its partially converged Schur form.
             = N+1: the eigenvalues could not be reordered because some
                   eigenvalues were too close to separate (the problem
                   is very ill-conditioned);
             = N+2: after reordering, roundoff changed values of some
                   complex eigenvalues so that leading eigenvalues in
                   the Schur form no longer satisfy SELECT=.TRUE.  This
                   could also be caused by underflow due to scaling.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 236 of file cgeesx.f.

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, SROUNDUP_LWORK
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 ) = sroundup_lwork(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 smlnum = sqrt( smlnum ) / eps
380 bignum = one / smlnum
381*
382* Scale A if max element outside range [SMLNUM,BIGNUM]
383*
384 anrm = clange( 'M', n, n, a, lda, dum )
385 scalea = .false.
386 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
387 scalea = .true.
388 cscale = smlnum
389 ELSE IF( anrm.GT.bignum ) THEN
390 scalea = .true.
391 cscale = bignum
392 END IF
393 IF( scalea )
394 $ CALL clascl( 'G', 0, 0, anrm, cscale, n, n, a, lda, ierr )
395*
396*
397* Permute the matrix to make it more nearly triangular
398* (CWorkspace: none)
399* (RWorkspace: need N)
400*
401 ibal = 1
402 CALL cgebal( 'P', n, a, lda, ilo, ihi, rwork( ibal ), ierr )
403*
404* Reduce to upper Hessenberg form
405* (CWorkspace: need 2*N, prefer N+N*NB)
406* (RWorkspace: none)
407*
408 itau = 1
409 iwrk = n + itau
410 CALL cgehrd( n, ilo, ihi, a, lda, work( itau ), work( iwrk ),
411 $ lwork-iwrk+1, ierr )
412*
413 IF( wantvs ) THEN
414*
415* Copy Householder vectors to VS
416*
417 CALL clacpy( 'L', n, n, a, lda, vs, ldvs )
418*
419* Generate unitary matrix in VS
420* (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
421* (RWorkspace: none)
422*
423 CALL cunghr( n, ilo, ihi, vs, ldvs, work( itau ), work( iwrk ),
424 $ lwork-iwrk+1, ierr )
425 END IF
426*
427 sdim = 0
428*
429* Perform QR iteration, accumulating Schur vectors in VS if desired
430* (CWorkspace: need 1, prefer HSWORK (see comments) )
431* (RWorkspace: none)
432*
433 iwrk = itau
434 CALL chseqr( 'S', jobvs, n, ilo, ihi, a, lda, w, vs, ldvs,
435 $ work( iwrk ), lwork-iwrk+1, ieval )
436 IF( ieval.GT.0 )
437 $ info = ieval
438*
439* Sort eigenvalues if desired
440*
441 IF( wantst .AND. info.EQ.0 ) THEN
442 IF( scalea )
443 $ CALL clascl( 'G', 0, 0, cscale, anrm, n, 1, w, n, ierr )
444 DO 10 i = 1, n
445 bwork( i ) = SELECT( w( i ) )
446 10 CONTINUE
447*
448* Reorder eigenvalues, transform Schur vectors, and compute
449* reciprocal condition numbers
450* (CWorkspace: if SENSE is not 'N', need 2*SDIM*(N-SDIM)
451* otherwise, need none )
452* (RWorkspace: none)
453*
454 CALL ctrsen( sense, jobvs, bwork, n, a, lda, vs, ldvs, w, sdim,
455 $ rconde, rcondv, work( iwrk ), lwork-iwrk+1,
456 $ icond )
457 IF( .NOT.wantsn )
458 $ maxwrk = max( maxwrk, 2*sdim*( n-sdim ) )
459 IF( icond.EQ.-14 ) THEN
460*
461* Not enough complex workspace
462*
463 info = -15
464 END IF
465 END IF
466*
467 IF( wantvs ) THEN
468*
469* Undo balancing
470* (CWorkspace: none)
471* (RWorkspace: need N)
472*
473 CALL cgebak( 'P', 'R', n, ilo, ihi, rwork( ibal ), n, vs, ldvs,
474 $ ierr )
475 END IF
476*
477 IF( scalea ) THEN
478*
479* Undo scaling for the Schur form of A
480*
481 CALL clascl( 'U', 0, 0, cscale, anrm, n, n, a, lda, ierr )
482 CALL ccopy( n, a, lda+1, w, 1 )
483 IF( ( wantsv .OR. wantsb ) .AND. info.EQ.0 ) THEN
484 dum( 1 ) = rcondv
485 CALL slascl( 'G', 0, 0, cscale, anrm, 1, 1, dum, 1, ierr )
486 rcondv = dum( 1 )
487 END IF
488 END IF
489*
490 work( 1 ) = sroundup_lwork(maxwrk)
491 RETURN
492*
493* End of CGEESX
494*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine ccopy(n, cx, incx, cy, incy)
CCOPY
Definition ccopy.f:81
subroutine cgebak(job, side, n, ilo, ihi, scale, m, v, ldv, info)
CGEBAK
Definition cgebak.f:131
subroutine cgebal(job, n, a, lda, ilo, ihi, scale, info)
CGEBAL
Definition cgebal.f:165
subroutine cgehrd(n, ilo, ihi, a, lda, tau, work, lwork, info)
CGEHRD
Definition cgehrd.f:167
subroutine chseqr(job, compz, n, ilo, ihi, h, ldh, w, z, ldz, work, lwork, info)
CHSEQR
Definition chseqr.f:299
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
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
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
real function clange(norm, m, n, a, lda, work)
CLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition clange.f:115
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 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
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
real function sroundup_lwork(lwork)
SROUNDUP_LWORK
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
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