 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ 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.```
Date
June 2016

Definition at line 241 of file zgeesx.f.

241 *
242 * -- LAPACK driver routine (version 3.7.0) --
243 * -- LAPACK is a software package provided by Univ. of Tennessee, --
244 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
245 * June 2016
246 *
247 * .. Scalar Arguments ..
248  CHARACTER jobvs, sense, sort
249  INTEGER info, lda, ldvs, lwork, n, sdim
250  DOUBLE PRECISION rconde, rcondv
251 * ..
252 * .. Array Arguments ..
253  LOGICAL bwork( * )
254  DOUBLE PRECISION rwork( * )
255  COMPLEX*16 a( lda, * ), vs( ldvs, * ), w( * ), work( * )
256 * ..
257 * .. Function Arguments ..
258  LOGICAL select
259  EXTERNAL SELECT
260 * ..
261 *
262 * =====================================================================
263 *
264 * .. Parameters ..
265  DOUBLE PRECISION zero, one
266  parameter( zero = 0.0d0, one = 1.0d0 )
267 * ..
268 * .. Local Scalars ..
269  LOGICAL lquery, scalea, wantsb, wantse, wantsn, wantst,
270  \$ wantsv, wantvs
271  INTEGER hswork, i, ibal, icond, ierr, ieval, ihi, ilo,
272  \$ itau, iwrk, lwrk, maxwrk, minwrk
273  DOUBLE PRECISION anrm, bignum, cscale, eps, smlnum
274 * ..
275 * .. Local Arrays ..
276  DOUBLE PRECISION dum( 1 )
277 * ..
278 * .. External Subroutines ..
279  EXTERNAL dlabad, dlascl, xerbla, zcopy, zgebak, zgebal,
281 * ..
282 * .. External Functions ..
283  LOGICAL lsame
284  INTEGER ilaenv
285  DOUBLE PRECISION dlamch, zlange
286  EXTERNAL lsame, ilaenv, dlamch, zlange
287 * ..
288 * .. Intrinsic Functions ..
289  INTRINSIC max, sqrt
290 * ..
291 * .. Executable Statements ..
292 *
293 * Test the input arguments
294 *
295  info = 0
296  wantvs = lsame( jobvs, 'V' )
297  wantst = lsame( sort, 'S' )
298  wantsn = lsame( sense, 'N' )
299  wantse = lsame( sense, 'E' )
300  wantsv = lsame( sense, 'V' )
301  wantsb = lsame( sense, 'B' )
302  lquery = ( lwork.EQ.-1 )
303 *
304  IF( ( .NOT.wantvs ) .AND. ( .NOT.lsame( jobvs, 'N' ) ) ) THEN
305  info = -1
306  ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
307  info = -2
308  ELSE IF( .NOT.( wantsn .OR. wantse .OR. wantsv .OR. wantsb ) .OR.
309  \$ ( .NOT.wantst .AND. .NOT.wantsn ) ) THEN
310  info = -4
311  ELSE IF( n.LT.0 ) THEN
312  info = -5
313  ELSE IF( lda.LT.max( 1, n ) ) THEN
314  info = -7
315  ELSE IF( ldvs.LT.1 .OR. ( wantvs .AND. ldvs.LT.n ) ) THEN
316  info = -11
317  END IF
318 *
319 * Compute workspace
320 * (Note: Comments in the code beginning "Workspace:" describe the
321 * minimal amount of real workspace needed at that point in the
322 * code, as well as the preferred amount for good performance.
323 * CWorkspace refers to complex workspace, and RWorkspace to real
324 * workspace. NB refers to the optimal block size for the
325 * immediately following subroutine, as returned by ILAENV.
326 * HSWORK refers to the workspace preferred by ZHSEQR, as
327 * calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
328 * the worst case.
329 * If SENSE = 'E', 'V' or 'B', then the amount of workspace needed
330 * depends on SDIM, which is computed by the routine ZTRSEN later
331 * in the code.)
332 *
333  IF( info.EQ.0 ) THEN
334  IF( n.EQ.0 ) THEN
335  minwrk = 1
336  lwrk = 1
337  ELSE
338  maxwrk = n + n*ilaenv( 1, 'ZGEHRD', ' ', n, 1, n, 0 )
339  minwrk = 2*n
340 *
341  CALL zhseqr( 'S', jobvs, n, 1, n, a, lda, w, vs, ldvs,
342  \$ work, -1, ieval )
343  hswork = work( 1 )
344 *
345  IF( .NOT.wantvs ) THEN
346  maxwrk = max( maxwrk, hswork )
347  ELSE
348  maxwrk = max( maxwrk, n + ( n - 1 )*ilaenv( 1, 'ZUNGHR',
349  \$ ' ', n, 1, n, -1 ) )
350  maxwrk = max( maxwrk, hswork )
351  END IF
352  lwrk = maxwrk
353  IF( .NOT.wantsn )
354  \$ lwrk = max( lwrk, ( n*n )/2 )
355  END IF
356  work( 1 ) = lwrk
357 *
358  IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
359  info = -15
360  END IF
361  END IF
362 *
363  IF( info.NE.0 ) THEN
364  CALL xerbla( 'ZGEESX', -info )
365  RETURN
366  ELSE IF( lquery ) THEN
367  RETURN
368  END IF
369 *
370 * Quick return if possible
371 *
372  IF( n.EQ.0 ) THEN
373  sdim = 0
374  RETURN
375  END IF
376 *
377 * Get machine constants
378 *
379  eps = dlamch( 'P' )
380  smlnum = dlamch( 'S' )
381  bignum = one / smlnum
382  CALL dlabad( smlnum, bignum )
383  smlnum = sqrt( smlnum ) / eps
384  bignum = one / smlnum
385 *
386 * Scale A if max element outside range [SMLNUM,BIGNUM]
387 *
388  anrm = zlange( 'M', n, n, a, lda, dum )
389  scalea = .false.
390  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
391  scalea = .true.
392  cscale = smlnum
393  ELSE IF( anrm.GT.bignum ) THEN
394  scalea = .true.
395  cscale = bignum
396  END IF
397  IF( scalea )
398  \$ CALL zlascl( 'G', 0, 0, anrm, cscale, n, n, a, lda, ierr )
399 *
400 *
401 * Permute the matrix to make it more nearly triangular
402 * (CWorkspace: none)
403 * (RWorkspace: need N)
404 *
405  ibal = 1
406  CALL zgebal( 'P', n, a, lda, ilo, ihi, rwork( ibal ), ierr )
407 *
408 * Reduce to upper Hessenberg form
409 * (CWorkspace: need 2*N, prefer N+N*NB)
410 * (RWorkspace: none)
411 *
412  itau = 1
413  iwrk = n + itau
414  CALL zgehrd( n, ilo, ihi, a, lda, work( itau ), work( iwrk ),
415  \$ lwork-iwrk+1, ierr )
416 *
417  IF( wantvs ) THEN
418 *
419 * Copy Householder vectors to VS
420 *
421  CALL zlacpy( 'L', n, n, a, lda, vs, ldvs )
422 *
423 * Generate unitary matrix in VS
424 * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
425 * (RWorkspace: none)
426 *
427  CALL zunghr( n, ilo, ihi, vs, ldvs, work( itau ), work( iwrk ),
428  \$ lwork-iwrk+1, ierr )
429  END IF
430 *
431  sdim = 0
432 *
433 * Perform QR iteration, accumulating Schur vectors in VS if desired
434 * (CWorkspace: need 1, prefer HSWORK (see comments) )
435 * (RWorkspace: none)
436 *
437  iwrk = itau
438  CALL zhseqr( 'S', jobvs, n, ilo, ihi, a, lda, w, vs, ldvs,
439  \$ work( iwrk ), lwork-iwrk+1, ieval )
440  IF( ieval.GT.0 )
441  \$ info = ieval
442 *
443 * Sort eigenvalues if desired
444 *
445  IF( wantst .AND. info.EQ.0 ) THEN
446  IF( scalea )
447  \$ CALL zlascl( 'G', 0, 0, cscale, anrm, n, 1, w, n, ierr )
448  DO 10 i = 1, n
449  bwork( i ) = SELECT( w( i ) )
450  10 CONTINUE
451 *
452 * Reorder eigenvalues, transform Schur vectors, and compute
453 * reciprocal condition numbers
454 * (CWorkspace: if SENSE is not 'N', need 2*SDIM*(N-SDIM)
455 * otherwise, need none )
456 * (RWorkspace: none)
457 *
458  CALL ztrsen( sense, jobvs, bwork, n, a, lda, vs, ldvs, w, sdim,
459  \$ rconde, rcondv, work( iwrk ), lwork-iwrk+1,
460  \$ icond )
461  IF( .NOT.wantsn )
462  \$ maxwrk = max( maxwrk, 2*sdim*( n-sdim ) )
463  IF( icond.EQ.-14 ) THEN
464 *
465 * Not enough complex workspace
466 *
467  info = -15
468  END IF
469  END IF
470 *
471  IF( wantvs ) THEN
472 *
473 * Undo balancing
474 * (CWorkspace: none)
475 * (RWorkspace: need N)
476 *
477  CALL zgebak( 'P', 'R', n, ilo, ihi, rwork( ibal ), n, vs, ldvs,
478  \$ ierr )
479  END IF
480 *
481  IF( scalea ) THEN
482 *
483 * Undo scaling for the Schur form of A
484 *
485  CALL zlascl( 'U', 0, 0, cscale, anrm, n, n, a, lda, ierr )
486  CALL zcopy( n, a, lda+1, w, 1 )
487  IF( ( wantsv .OR. wantsb ) .AND. info.EQ.0 ) THEN
488  dum( 1 ) = rcondv
489  CALL dlascl( 'G', 0, 0, cscale, anrm, 1, 1, dum, 1, ierr )
490  rcondv = dum( 1 )
491  END IF
492  END IF
493 *
494  work( 1 ) = maxwrk
495  RETURN
496 *
497 * End of ZGEESX
498 *
subroutine zunghr(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
ZUNGHR
Definition: zunghr.f:128
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:117
subroutine zgehrd(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
ZGEHRD
Definition: zgehrd.f:169
subroutine zgebak(JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO)
ZGEBAK
Definition: zgebak.f:133
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:83
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine zhseqr(JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK, INFO)
ZHSEQR
Definition: zhseqr.f:301
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: tstiee.f:83
subroutine ztrsen(JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S, SEP, WORK, LWORK, INFO)
ZTRSEN
Definition: ztrsen.f:266
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
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:145
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:105
subroutine zgebal(JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
ZGEBAL
Definition: zgebal.f:164
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:145
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