LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ zgeesx()

 subroutine zgeesx ( character JOBVS, character SORT, external SELECT, character SENSE, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer SDIM, complex*16, dimension( * ) W, complex*16, dimension( ldvs, * ) VS, integer LDVS, double precision RCONDE, double precision RCONDV, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, logical, dimension( * ) BWORK, integer INFO )

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

Purpose:
``` ZGEESX 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*16 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*16 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*16 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*16 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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*16 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 DOUBLE PRECISION 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.```

Definition at line 236 of file zgeesx.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 DOUBLE PRECISION RCONDE, RCONDV
248* ..
249* .. Array Arguments ..
250 LOGICAL BWORK( * )
251 DOUBLE PRECISION RWORK( * )
252 COMPLEX*16 A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )
253* ..
254* .. Function Arguments ..
255 LOGICAL SELECT
256 EXTERNAL SELECT
257* ..
258*
259* =====================================================================
260*
261* .. Parameters ..
262 DOUBLE PRECISION ZERO, ONE
263 parameter( zero = 0.0d0, one = 1.0d0 )
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 DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, SMLNUM
271* ..
272* .. Local Arrays ..
273 DOUBLE PRECISION DUM( 1 )
274* ..
275* .. External Subroutines ..
276 EXTERNAL dlabad, dlascl, xerbla, zcopy, zgebak, zgebal,
278* ..
279* .. External Functions ..
280 LOGICAL LSAME
281 INTEGER ILAENV
282 DOUBLE PRECISION DLAMCH, ZLANGE
283 EXTERNAL lsame, ilaenv, dlamch, zlange
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 ZHSEQR, 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 ZTRSEN 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, 'ZGEHRD', ' ', n, 1, n, 0 )
336 minwrk = 2*n
337*
338 CALL zhseqr( '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, 'ZUNGHR',
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( 'ZGEESX', -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 = dlamch( 'P' )
377 smlnum = dlamch( 'S' )
378 bignum = one / smlnum
379 CALL dlabad( 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 = zlange( '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 zlascl( '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 zgebal( '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 zgehrd( 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 zlacpy( '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 zunghr( 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 zhseqr( '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 zlascl( '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 ztrsen( 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 zgebak( '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 zlascl( 'U', 0, 0, cscale, anrm, n, n, a, lda, ierr )
483 CALL zcopy( n, a, lda+1, w, 1 )
484 IF( ( wantsv .OR. wantsb ) .AND. info.EQ.0 ) THEN
485 dum( 1 ) = rcondv
486 CALL dlascl( '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 ZGEESX
495*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
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:143
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
double precision function zlange(NORM, M, N, A, LDA, WORK)
ZLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: zlange.f:115
subroutine zgebal(JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
ZGEBAL
Definition: zgebal.f:162
subroutine zgehrd(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
ZGEHRD
Definition: zgehrd.f:167
subroutine zgebak(JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO)
ZGEBAK
Definition: zgebak.f:131
subroutine zlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
ZLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: zlascl.f:143
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine zhseqr(JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK, INFO)
ZHSEQR
Definition: zhseqr.f:299
subroutine ztrsen(JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S, SEP, WORK, LWORK, INFO)
ZTRSEN
Definition: ztrsen.f:264
subroutine zunghr(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
ZUNGHR
Definition: zunghr.f:126
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