 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ zgegs()

 subroutine zgegs ( character JOBVSL, character JOBVSR, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( ldvsl, * ) VSL, integer LDVSL, complex*16, dimension( ldvsr, * ) VSR, integer LDVSR, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer INFO )

ZGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:
``` This routine is deprecated and has been replaced by routine ZGGES.

ZGEGS computes the eigenvalues, Schur form, and, optionally, the
left and or/right Schur vectors of a complex matrix pair (A,B).
Given two square matrices A and B, the generalized Schur
factorization has the form

A = Q*S*Z**H,  B = Q*T*Z**H

where Q and Z are unitary matrices and S and T are upper triangular.
The columns of Q are the left Schur vectors
and the columns of Z are the right Schur vectors.

If only the eigenvalues of (A,B) are needed, the driver routine
ZGEGV should be used instead.  See ZGEGV for a description of the
eigenvalues of the generalized nonsymmetric eigenvalue problem
(GNEP).```
Parameters
 [in] JOBVSL ``` JOBVSL is CHARACTER*1 = 'N': do not compute the left Schur vectors; = 'V': compute the left Schur vectors (returned in VSL).``` [in] JOBVSR ``` JOBVSR is CHARACTER*1 = 'N': do not compute the right Schur vectors; = 'V': compute the right Schur vectors (returned in VSR).``` [in] N ``` N is INTEGER The order of the matrices A, B, VSL, and VSR. N >= 0.``` [in,out] A ``` A is COMPLEX*16 array, dimension (LDA, N) On entry, the matrix A. On exit, the upper triangular matrix S from the generalized Schur factorization.``` [in] LDA ``` LDA is INTEGER The leading dimension of A. LDA >= max(1,N).``` [in,out] B ``` B is COMPLEX*16 array, dimension (LDB, N) On entry, the matrix B. On exit, the upper triangular matrix T from the generalized Schur factorization.``` [in] LDB ``` LDB is INTEGER The leading dimension of B. LDB >= max(1,N).``` [out] ALPHA ``` ALPHA is COMPLEX*16 array, dimension (N) The complex scalars alpha that define the eigenvalues of GNEP. ALPHA(j) = S(j,j), the diagonal element of the Schur form of A.``` [out] BETA ``` BETA is COMPLEX*16 array, dimension (N) The non-negative real scalars beta that define the eigenvalues of GNEP. BETA(j) = T(j,j), the diagonal element of the triangular factor T. Together, the quantities alpha = ALPHA(j) and beta = BETA(j) represent the j-th eigenvalue of the matrix pair (A,B), in one of the forms lambda = alpha/beta or mu = beta/alpha. Since either lambda or mu may overflow, they should not, in general, be computed.``` [out] VSL ``` VSL is COMPLEX*16 array, dimension (LDVSL,N) If JOBVSL = 'V', the matrix of left Schur vectors Q. Not referenced if JOBVSL = 'N'.``` [in] LDVSL ``` LDVSL is INTEGER The leading dimension of the matrix VSL. LDVSL >= 1, and if JOBVSL = 'V', LDVSL >= N.``` [out] VSR ``` VSR is COMPLEX*16 array, dimension (LDVSR,N) If JOBVSR = 'V', the matrix of right Schur vectors Z. Not referenced if JOBVSR = 'N'.``` [in] LDVSR ``` LDVSR is INTEGER The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N.``` [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). For good performance, LWORK must generally be larger. To compute the optimal value of LWORK, call ILAENV to get blocksizes (for ZGEQRF, ZUNMQR, and CUNGQR.) Then compute: NB -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and CUNGQR; the optimal LWORK is N*(NB+1). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, 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 (3*N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. =1,...,N: The QZ iteration failed. (A,B) are not in Schur form, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N. > N: errors that usually indicate LAPACK problems: =N+1: error return from ZGGBAL =N+2: error return from ZGEQRF =N+3: error return from ZUNMQR =N+4: error return from ZUNGQR =N+5: error return from ZGGHRD =N+6: error return from ZHGEQZ (other than failed iteration) =N+7: error return from ZGGBAK (computing VSL) =N+8: error return from ZGGBAK (computing VSR) =N+9: error return from ZLASCL (various places)```
Date
December 2016

Definition at line 227 of file zgegs.f.

227 *
228 * -- LAPACK driver routine (version 3.7.0) --
229 * -- LAPACK is a software package provided by Univ. of Tennessee, --
230 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
231 * December 2016
232 *
233 * .. Scalar Arguments ..
234  CHARACTER jobvsl, jobvsr
235  INTEGER info, lda, ldb, ldvsl, ldvsr, lwork, n
236 * ..
237 * .. Array Arguments ..
238  DOUBLE PRECISION rwork( * )
239  COMPLEX*16 a( lda, * ), alpha( * ), b( ldb, * ),
240  \$ beta( * ), vsl( ldvsl, * ), vsr( ldvsr, * ),
241  \$ work( * )
242 * ..
243 *
244 * =====================================================================
245 *
246 * .. Parameters ..
247  DOUBLE PRECISION zero, one
248  parameter( zero = 0.0d0, one = 1.0d0 )
249  COMPLEX*16 czero, cone
250  parameter( czero = ( 0.0d0, 0.0d0 ),
251  \$ cone = ( 1.0d0, 0.0d0 ) )
252 * ..
253 * .. Local Scalars ..
254  LOGICAL ilascl, ilbscl, ilvsl, ilvsr, lquery
255  INTEGER icols, ihi, iinfo, ijobvl, ijobvr, ileft, ilo,
256  \$ iright, irows, irwork, itau, iwork, lopt,
257  \$ lwkmin, lwkopt, nb, nb1, nb2, nb3
258  DOUBLE PRECISION anrm, anrmto, bignum, bnrm, bnrmto, eps,
259  \$ safmin, smlnum
260 * ..
261 * .. External Subroutines ..
262  EXTERNAL xerbla, zgeqrf, zggbak, zggbal, zgghrd, zhgeqz,
264 * ..
265 * .. External Functions ..
266  LOGICAL lsame
267  INTEGER ilaenv
268  DOUBLE PRECISION dlamch, zlange
269  EXTERNAL lsame, ilaenv, dlamch, zlange
270 * ..
271 * .. Intrinsic Functions ..
272  INTRINSIC int, max
273 * ..
274 * .. Executable Statements ..
275 *
276 * Decode the input arguments
277 *
278  IF( lsame( jobvsl, 'N' ) ) THEN
279  ijobvl = 1
280  ilvsl = .false.
281  ELSE IF( lsame( jobvsl, 'V' ) ) THEN
282  ijobvl = 2
283  ilvsl = .true.
284  ELSE
285  ijobvl = -1
286  ilvsl = .false.
287  END IF
288 *
289  IF( lsame( jobvsr, 'N' ) ) THEN
290  ijobvr = 1
291  ilvsr = .false.
292  ELSE IF( lsame( jobvsr, 'V' ) ) THEN
293  ijobvr = 2
294  ilvsr = .true.
295  ELSE
296  ijobvr = -1
297  ilvsr = .false.
298  END IF
299 *
300 * Test the input arguments
301 *
302  lwkmin = max( 2*n, 1 )
303  lwkopt = lwkmin
304  work( 1 ) = lwkopt
305  lquery = ( lwork.EQ.-1 )
306  info = 0
307  IF( ijobvl.LE.0 ) THEN
308  info = -1
309  ELSE IF( ijobvr.LE.0 ) THEN
310  info = -2
311  ELSE IF( n.LT.0 ) THEN
312  info = -3
313  ELSE IF( lda.LT.max( 1, n ) ) THEN
314  info = -5
315  ELSE IF( ldb.LT.max( 1, n ) ) THEN
316  info = -7
317  ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
318  info = -11
319  ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
320  info = -13
321  ELSE IF( lwork.LT.lwkmin .AND. .NOT.lquery ) THEN
322  info = -15
323  END IF
324 *
325  IF( info.EQ.0 ) THEN
326  nb1 = ilaenv( 1, 'ZGEQRF', ' ', n, n, -1, -1 )
327  nb2 = ilaenv( 1, 'ZUNMQR', ' ', n, n, n, -1 )
328  nb3 = ilaenv( 1, 'ZUNGQR', ' ', n, n, n, -1 )
329  nb = max( nb1, nb2, nb3 )
330  lopt = n*( nb+1 )
331  work( 1 ) = lopt
332  END IF
333 *
334  IF( info.NE.0 ) THEN
335  CALL xerbla( 'ZGEGS ', -info )
336  RETURN
337  ELSE IF( lquery ) THEN
338  RETURN
339  END IF
340 *
341 * Quick return if possible
342 *
343  IF( n.EQ.0 )
344  \$ RETURN
345 *
346 * Get machine constants
347 *
348  eps = dlamch( 'E' )*dlamch( 'B' )
349  safmin = dlamch( 'S' )
350  smlnum = n*safmin / eps
351  bignum = one / smlnum
352 *
353 * Scale A if max element outside range [SMLNUM,BIGNUM]
354 *
355  anrm = zlange( 'M', n, n, a, lda, rwork )
356  ilascl = .false.
357  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
358  anrmto = smlnum
359  ilascl = .true.
360  ELSE IF( anrm.GT.bignum ) THEN
361  anrmto = bignum
362  ilascl = .true.
363  END IF
364 *
365  IF( ilascl ) THEN
366  CALL zlascl( 'G', -1, -1, anrm, anrmto, n, n, a, lda, iinfo )
367  IF( iinfo.NE.0 ) THEN
368  info = n + 9
369  RETURN
370  END IF
371  END IF
372 *
373 * Scale B if max element outside range [SMLNUM,BIGNUM]
374 *
375  bnrm = zlange( 'M', n, n, b, ldb, rwork )
376  ilbscl = .false.
377  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
378  bnrmto = smlnum
379  ilbscl = .true.
380  ELSE IF( bnrm.GT.bignum ) THEN
381  bnrmto = bignum
382  ilbscl = .true.
383  END IF
384 *
385  IF( ilbscl ) THEN
386  CALL zlascl( 'G', -1, -1, bnrm, bnrmto, n, n, b, ldb, iinfo )
387  IF( iinfo.NE.0 ) THEN
388  info = n + 9
389  RETURN
390  END IF
391  END IF
392 *
393 * Permute the matrix to make it more nearly triangular
394 *
395  ileft = 1
396  iright = n + 1
397  irwork = iright + n
398  iwork = 1
399  CALL zggbal( 'P', n, a, lda, b, ldb, ilo, ihi, rwork( ileft ),
400  \$ rwork( iright ), rwork( irwork ), iinfo )
401  IF( iinfo.NE.0 ) THEN
402  info = n + 1
403  GO TO 10
404  END IF
405 *
406 * Reduce B to triangular form, and initialize VSL and/or VSR
407 *
408  irows = ihi + 1 - ilo
409  icols = n + 1 - ilo
410  itau = iwork
411  iwork = itau + irows
412  CALL zgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
413  \$ work( iwork ), lwork+1-iwork, iinfo )
414  IF( iinfo.GE.0 )
415  \$ lwkopt = max( lwkopt, int( work( iwork ) )+iwork-1 )
416  IF( iinfo.NE.0 ) THEN
417  info = n + 2
418  GO TO 10
419  END IF
420 *
421  CALL zunmqr( 'L', 'C', irows, icols, irows, b( ilo, ilo ), ldb,
422  \$ work( itau ), a( ilo, ilo ), lda, work( iwork ),
423  \$ lwork+1-iwork, iinfo )
424  IF( iinfo.GE.0 )
425  \$ lwkopt = max( lwkopt, int( work( iwork ) )+iwork-1 )
426  IF( iinfo.NE.0 ) THEN
427  info = n + 3
428  GO TO 10
429  END IF
430 *
431  IF( ilvsl ) THEN
432  CALL zlaset( 'Full', n, n, czero, cone, vsl, ldvsl )
433  CALL zlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
434  \$ vsl( ilo+1, ilo ), ldvsl )
435  CALL zungqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
436  \$ work( itau ), work( iwork ), lwork+1-iwork,
437  \$ iinfo )
438  IF( iinfo.GE.0 )
439  \$ lwkopt = max( lwkopt, int( work( iwork ) )+iwork-1 )
440  IF( iinfo.NE.0 ) THEN
441  info = n + 4
442  GO TO 10
443  END IF
444  END IF
445 *
446  IF( ilvsr )
447  \$ CALL zlaset( 'Full', n, n, czero, cone, vsr, ldvsr )
448 *
449 * Reduce to generalized Hessenberg form
450 *
451  CALL zgghrd( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
452  \$ ldvsl, vsr, ldvsr, iinfo )
453  IF( iinfo.NE.0 ) THEN
454  info = n + 5
455  GO TO 10
456  END IF
457 *
458 * Perform QZ algorithm, computing Schur vectors if desired
459 *
460  iwork = itau
461  CALL zhgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
462  \$ alpha, beta, vsl, ldvsl, vsr, ldvsr, work( iwork ),
463  \$ lwork+1-iwork, rwork( irwork ), iinfo )
464  IF( iinfo.GE.0 )
465  \$ lwkopt = max( lwkopt, int( work( iwork ) )+iwork-1 )
466  IF( iinfo.NE.0 ) THEN
467  IF( iinfo.GT.0 .AND. iinfo.LE.n ) THEN
468  info = iinfo
469  ELSE IF( iinfo.GT.n .AND. iinfo.LE.2*n ) THEN
470  info = iinfo - n
471  ELSE
472  info = n + 6
473  END IF
474  GO TO 10
475  END IF
476 *
477 * Apply permutation to VSL and VSR
478 *
479  IF( ilvsl ) THEN
480  CALL zggbak( 'P', 'L', n, ilo, ihi, rwork( ileft ),
481  \$ rwork( iright ), n, vsl, ldvsl, iinfo )
482  IF( iinfo.NE.0 ) THEN
483  info = n + 7
484  GO TO 10
485  END IF
486  END IF
487  IF( ilvsr ) THEN
488  CALL zggbak( 'P', 'R', n, ilo, ihi, rwork( ileft ),
489  \$ rwork( iright ), n, vsr, ldvsr, iinfo )
490  IF( iinfo.NE.0 ) THEN
491  info = n + 8
492  GO TO 10
493  END IF
494  END IF
495 *
496 * Undo scaling
497 *
498  IF( ilascl ) THEN
499  CALL zlascl( 'U', -1, -1, anrmto, anrm, n, n, a, lda, iinfo )
500  IF( iinfo.NE.0 ) THEN
501  info = n + 9
502  RETURN
503  END IF
504  CALL zlascl( 'G', -1, -1, anrmto, anrm, n, 1, alpha, n, iinfo )
505  IF( iinfo.NE.0 ) THEN
506  info = n + 9
507  RETURN
508  END IF
509  END IF
510 *
511  IF( ilbscl ) THEN
512  CALL zlascl( 'U', -1, -1, bnrmto, bnrm, n, n, b, ldb, iinfo )
513  IF( iinfo.NE.0 ) THEN
514  info = n + 9
515  RETURN
516  END IF
517  CALL zlascl( 'G', -1, -1, bnrmto, bnrm, n, 1, beta, n, iinfo )
518  IF( iinfo.NE.0 ) THEN
519  info = n + 9
520  RETURN
521  END IF
522  END IF
523 *
524  10 CONTINUE
525  work( 1 ) = lwkopt
526 *
527  RETURN
528 *
529 * End of ZGEGS
530 *
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 zgghrd(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
ZGGHRD
Definition: zgghrd.f:206
subroutine zggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
ZGGBAK
Definition: zggbak.f:150
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine zlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values...
Definition: zlaset.f:108
subroutine zggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
ZGGBAL
Definition: zggbal.f:179
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zhgeqz(JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO)
ZHGEQZ
Definition: zhgeqz.f:286
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: tstiee.f:83
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
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 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
subroutine zungqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
ZUNGQR
Definition: zungqr.f:130
subroutine zunmqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
ZUNMQR
Definition: zunmqr.f:169
subroutine zgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
ZGEQRF VARIANT: left-looking Level 3 BLAS of the algorithm.
Definition: zgeqrf.f:151
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