LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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◆ zgges()

subroutine zgges ( character jobvsl,
character jobvsr,
character sort,
external selctg,
integer n,
complex*16, dimension( lda, * ) a,
integer lda,
complex*16, dimension( ldb, * ) b,
integer ldb,
integer sdim,
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,
logical, dimension( * ) bwork,
integer info )

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

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

Purpose:
!>
!> ZGGES computes for a pair of N-by-N complex nonsymmetric matrices
!> (A,B), the generalized eigenvalues, the generalized complex Schur
!> form (S, T), and optionally left and/or right Schur vectors (VSL
!> and VSR). This gives the generalized Schur factorization
!>
!>         (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )
!>
!> where (VSR)**H is the conjugate-transpose of VSR.
!>
!> Optionally, it also orders the eigenvalues so that a selected cluster
!> of eigenvalues appears in the leading diagonal blocks of the upper
!> triangular matrix S and the upper triangular matrix T. The leading
!> columns of VSL and VSR then form an unitary basis for the
!> corresponding left and right eigenspaces (deflating subspaces).
!>
!> (If only the generalized eigenvalues are needed, use the driver
!> ZGGEV instead, which is faster.)
!>
!> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
!> or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
!> usually represented as the pair (alpha,beta), as there is a
!> reasonable interpretation for beta=0, and even for both being zero.
!>
!> A pair of matrices (S,T) is in generalized complex Schur form if S
!> and T are upper triangular and, in addition, the diagonal elements
!> of T are non-negative real numbers.
!>
Parameters
[in]JOBVSL
!>          JOBVSL is CHARACTER*1
!>          = 'N':  do not compute the left Schur vectors;
!>          = 'V':  compute the left Schur vectors.
!>
[in]JOBVSR
!>          JOBVSR is CHARACTER*1
!>          = 'N':  do not compute the right Schur vectors;
!>          = 'V':  compute the right Schur vectors.
!>
[in]SORT
!>          SORT is CHARACTER*1
!>          Specifies whether or not to order the eigenvalues on the
!>          diagonal of the generalized Schur form.
!>          = 'N':  Eigenvalues are not ordered;
!>          = 'S':  Eigenvalues are ordered (see SELCTG).
!>
[in]SELCTG
!>          SELCTG is a LOGICAL FUNCTION of two COMPLEX*16 arguments
!>          SELCTG must be declared EXTERNAL in the calling subroutine.
!>          If SORT = 'N', SELCTG is not referenced.
!>          If SORT = 'S', SELCTG is used to select eigenvalues to sort
!>          to the top left of the Schur form.
!>          An eigenvalue ALPHA(j)/BETA(j) is selected if
!>          SELCTG(ALPHA(j),BETA(j)) is true.
!>
!>          Note that a selected complex eigenvalue may no longer satisfy
!>          SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
!>          ordering may change the value of complex eigenvalues
!>          (especially if the eigenvalue is ill-conditioned), in this
!>          case INFO is set to N+2 (See INFO below).
!>
[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 first of the pair of matrices.
!>          On exit, A has been overwritten by its generalized Schur
!>          form S.
!>
[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 second of the pair of matrices.
!>          On exit, B has been overwritten by its generalized Schur
!>          form T.
!>
[in]LDB
!>          LDB is INTEGER
!>          The leading dimension of B.  LDB >= max(1,N).
!>
[out]SDIM
!>          SDIM is INTEGER
!>          If SORT = 'N', SDIM = 0.
!>          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
!>          for which SELCTG is true.
!>
[out]ALPHA
!>          ALPHA is COMPLEX*16 array, dimension (N)
!>
[out]BETA
!>          BETA is COMPLEX*16 array, dimension (N)
!>          On exit,  ALPHA(j)/BETA(j), j=1,...,N, will be the
!>          generalized eigenvalues.  ALPHA(j), j=1,...,N  and  BETA(j),
!>          j=1,...,N  are the diagonals of the complex Schur form (A,B)
!>          output by ZGGES. The  BETA(j) will be non-negative real.
!>
!>          Note: the quotients ALPHA(j)/BETA(j) may easily over- or
!>          underflow, and BETA(j) may even be zero.  Thus, the user
!>          should avoid naively computing the ratio alpha/beta.
!>          However, ALPHA will be always less than and usually
!>          comparable with norm(A) in magnitude, and BETA always less
!>          than and usually comparable with norm(B).
!>
[out]VSL
!>          VSL is COMPLEX*16 array, dimension (LDVSL,N)
!>          If JOBVSL = 'V', VSL will contain the left Schur vectors.
!>          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', VSR will contain the right Schur vectors.
!>          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.
!>
!>          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 (8*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.
!>          =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:  =N+1: other than QZ iteration failed in ZHGEQZ
!>                =N+2: after reordering, roundoff changed values of
!>                      some complex eigenvalues so that leading
!>                      eigenvalues in the Generalized Schur form no
!>                      longer satisfy SELCTG=.TRUE.  This could also
!>                      be caused due to scaling.
!>                =N+3: reordering failed in ZTGSEN.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 265 of file zgges.f.

269*
270* -- LAPACK driver routine --
271* -- LAPACK is a software package provided by Univ. of Tennessee, --
272* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
273*
274* .. Scalar Arguments ..
275 CHARACTER JOBVSL, JOBVSR, SORT
276 INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
277* ..
278* .. Array Arguments ..
279 LOGICAL BWORK( * )
280 DOUBLE PRECISION RWORK( * )
281 COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
282 $ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
283 $ WORK( * )
284* ..
285* .. Function Arguments ..
286 LOGICAL SELCTG
287 EXTERNAL selctg
288* ..
289*
290* =====================================================================
291*
292* .. Parameters ..
293 DOUBLE PRECISION ZERO, ONE
294 parameter( zero = 0.0d0, one = 1.0d0 )
295 COMPLEX*16 CZERO, CONE
296 parameter( czero = ( 0.0d0, 0.0d0 ),
297 $ cone = ( 1.0d0, 0.0d0 ) )
298* ..
299* .. Local Scalars ..
300 LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
301 $ LQUERY, WANTST
302 INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
303 $ ILO, IRIGHT, IROWS, IRWRK, ITAU, IWRK, LWKMIN,
304 $ LWKOPT
305 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
306 $ PVSR, SMLNUM
307* ..
308* .. Local Arrays ..
309 INTEGER IDUM( 1 )
310 DOUBLE PRECISION DIF( 2 )
311* ..
312* .. External Subroutines ..
313 EXTERNAL xerbla, zgeqrf, zggbak, zggbal, zgghrd,
314 $ zhgeqz,
315 $ zlacpy, zlascl, zlaset, ztgsen, zungqr, zunmqr
316* ..
317* .. External Functions ..
318 LOGICAL LSAME
319 INTEGER ILAENV
320 DOUBLE PRECISION DLAMCH, ZLANGE
321 EXTERNAL lsame, ilaenv, dlamch, zlange
322* ..
323* .. Intrinsic Functions ..
324 INTRINSIC max, sqrt
325* ..
326* .. Executable Statements ..
327*
328* Decode the input arguments
329*
330 IF( lsame( jobvsl, 'N' ) ) THEN
331 ijobvl = 1
332 ilvsl = .false.
333 ELSE IF( lsame( jobvsl, 'V' ) ) THEN
334 ijobvl = 2
335 ilvsl = .true.
336 ELSE
337 ijobvl = -1
338 ilvsl = .false.
339 END IF
340*
341 IF( lsame( jobvsr, 'N' ) ) THEN
342 ijobvr = 1
343 ilvsr = .false.
344 ELSE IF( lsame( jobvsr, 'V' ) ) THEN
345 ijobvr = 2
346 ilvsr = .true.
347 ELSE
348 ijobvr = -1
349 ilvsr = .false.
350 END IF
351*
352 wantst = lsame( sort, 'S' )
353*
354* Test the input arguments
355*
356 info = 0
357 lquery = ( lwork.EQ.-1 )
358 IF( ijobvl.LE.0 ) THEN
359 info = -1
360 ELSE IF( ijobvr.LE.0 ) THEN
361 info = -2
362 ELSE IF( ( .NOT.wantst ) .AND.
363 $ ( .NOT.lsame( sort, 'N' ) ) ) THEN
364 info = -3
365 ELSE IF( n.LT.0 ) THEN
366 info = -5
367 ELSE IF( lda.LT.max( 1, n ) ) THEN
368 info = -7
369 ELSE IF( ldb.LT.max( 1, n ) ) THEN
370 info = -9
371 ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
372 info = -14
373 ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
374 info = -16
375 END IF
376*
377* Compute workspace
378* (Note: Comments in the code beginning "Workspace:" describe the
379* minimal amount of workspace needed at that point in the code,
380* as well as the preferred amount for good performance.
381* NB refers to the optimal block size for the immediately
382* following subroutine, as returned by ILAENV.)
383*
384 IF( info.EQ.0 ) THEN
385 lwkmin = max( 1, 2*n )
386 lwkopt = max( 1, n + n*ilaenv( 1, 'ZGEQRF', ' ', n, 1, n,
387 $ 0 ) )
388 lwkopt = max( lwkopt, n +
389 $ n*ilaenv( 1, 'ZUNMQR', ' ', n, 1, n, -1 ) )
390 IF( ilvsl ) THEN
391 lwkopt = max( lwkopt, n +
392 $ n*ilaenv( 1, 'ZUNGQR', ' ', n, 1, n, -1 ) )
393 END IF
394 work( 1 ) = lwkopt
395*
396 IF( lwork.LT.lwkmin .AND. .NOT.lquery )
397 $ info = -18
398 END IF
399*
400 IF( info.NE.0 ) THEN
401 CALL xerbla( 'ZGGES ', -info )
402 RETURN
403 ELSE IF( lquery ) THEN
404 RETURN
405 END IF
406*
407* Quick return if possible
408*
409 IF( n.EQ.0 ) THEN
410 sdim = 0
411 RETURN
412 END IF
413*
414* Get machine constants
415*
416 eps = dlamch( 'P' )
417 smlnum = dlamch( 'S' )
418 bignum = one / smlnum
419 smlnum = sqrt( smlnum ) / eps
420 bignum = one / smlnum
421*
422* Scale A if max element outside range [SMLNUM,BIGNUM]
423*
424 anrm = zlange( 'M', n, n, a, lda, rwork )
425 ilascl = .false.
426 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
427 anrmto = smlnum
428 ilascl = .true.
429 ELSE IF( anrm.GT.bignum ) THEN
430 anrmto = bignum
431 ilascl = .true.
432 END IF
433*
434 IF( ilascl )
435 $ CALL zlascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
436*
437* Scale B if max element outside range [SMLNUM,BIGNUM]
438*
439 bnrm = zlange( 'M', n, n, b, ldb, rwork )
440 ilbscl = .false.
441 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
442 bnrmto = smlnum
443 ilbscl = .true.
444 ELSE IF( bnrm.GT.bignum ) THEN
445 bnrmto = bignum
446 ilbscl = .true.
447 END IF
448*
449 IF( ilbscl )
450 $ CALL zlascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
451*
452* Permute the matrix to make it more nearly triangular
453* (Real Workspace: need 6*N)
454*
455 ileft = 1
456 iright = n + 1
457 irwrk = iright + n
458 CALL zggbal( 'P', n, a, lda, b, ldb, ilo, ihi, rwork( ileft ),
459 $ rwork( iright ), rwork( irwrk ), ierr )
460*
461* Reduce B to triangular form (QR decomposition of B)
462* (Complex Workspace: need N, prefer N*NB)
463*
464 irows = ihi + 1 - ilo
465 icols = n + 1 - ilo
466 itau = 1
467 iwrk = itau + irows
468 CALL zgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
469 $ work( iwrk ), lwork+1-iwrk, ierr )
470*
471* Apply the orthogonal transformation to matrix A
472* (Complex Workspace: need N, prefer N*NB)
473*
474 CALL zunmqr( 'L', 'C', irows, icols, irows, b( ilo, ilo ), ldb,
475 $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
476 $ lwork+1-iwrk, ierr )
477*
478* Initialize VSL
479* (Complex Workspace: need N, prefer N*NB)
480*
481 IF( ilvsl ) THEN
482 CALL zlaset( 'Full', n, n, czero, cone, vsl, ldvsl )
483 IF( irows.GT.1 ) THEN
484 CALL zlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
485 $ vsl( ilo+1, ilo ), ldvsl )
486 END IF
487 CALL zungqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
488 $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
489 END IF
490*
491* Initialize VSR
492*
493 IF( ilvsr )
494 $ CALL zlaset( 'Full', n, n, czero, cone, vsr, ldvsr )
495*
496* Reduce to generalized Hessenberg form
497* (Workspace: none needed)
498*
499 CALL zgghrd( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
500 $ ldvsl, vsr, ldvsr, ierr )
501*
502 sdim = 0
503*
504* Perform QZ algorithm, computing Schur vectors if desired
505* (Complex Workspace: need N)
506* (Real Workspace: need N)
507*
508 iwrk = itau
509 CALL zhgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
510 $ alpha, beta, vsl, ldvsl, vsr, ldvsr, work( iwrk ),
511 $ lwork+1-iwrk, rwork( irwrk ), ierr )
512 IF( ierr.NE.0 ) THEN
513 IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
514 info = ierr
515 ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
516 info = ierr - n
517 ELSE
518 info = n + 1
519 END IF
520 GO TO 30
521 END IF
522*
523* Sort eigenvalues ALPHA/BETA if desired
524* (Workspace: none needed)
525*
526 IF( wantst ) THEN
527*
528* Undo scaling on eigenvalues before selecting
529*
530 IF( ilascl )
531 $ CALL zlascl( 'G', 0, 0, anrm, anrmto, n, 1, alpha, n,
532 $ ierr )
533 IF( ilbscl )
534 $ CALL zlascl( 'G', 0, 0, bnrm, bnrmto, n, 1, beta, n,
535 $ ierr )
536*
537* Select eigenvalues
538*
539 DO 10 i = 1, n
540 bwork( i ) = selctg( alpha( i ), beta( i ) )
541 10 CONTINUE
542*
543 CALL ztgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
544 $ alpha,
545 $ beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl, pvsr,
546 $ dif, work( iwrk ), lwork-iwrk+1, idum, 1, ierr )
547 IF( ierr.EQ.1 )
548 $ info = n + 3
549*
550 END IF
551*
552* Apply back-permutation to VSL and VSR
553* (Workspace: none needed)
554*
555 IF( ilvsl )
556 $ CALL zggbak( 'P', 'L', n, ilo, ihi, rwork( ileft ),
557 $ rwork( iright ), n, vsl, ldvsl, ierr )
558 IF( ilvsr )
559 $ CALL zggbak( 'P', 'R', n, ilo, ihi, rwork( ileft ),
560 $ rwork( iright ), n, vsr, ldvsr, ierr )
561*
562* Undo scaling
563*
564 IF( ilascl ) THEN
565 CALL zlascl( 'U', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
566 CALL zlascl( 'G', 0, 0, anrmto, anrm, n, 1, alpha, n, ierr )
567 END IF
568*
569 IF( ilbscl ) THEN
570 CALL zlascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
571 CALL zlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
572 END IF
573*
574 IF( wantst ) THEN
575*
576* Check if reordering is correct
577*
578 lastsl = .true.
579 sdim = 0
580 DO 20 i = 1, n
581 cursl = selctg( alpha( i ), beta( i ) )
582 IF( cursl )
583 $ sdim = sdim + 1
584 IF( cursl .AND. .NOT.lastsl )
585 $ info = n + 2
586 lastsl = cursl
587 20 CONTINUE
588*
589 END IF
590*
591 30 CONTINUE
592*
593 work( 1 ) = lwkopt
594*
595 RETURN
596*
597* End of ZGGES
598*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
zgeqrf
subroutine zgeqrf(m, n, a, lda, tau, work, lwork, info)
ZGEQRF
Definition zgeqrf.f:144
zggbak
subroutine zggbak(job, side, n, ilo, ihi, lscale, rscale, m, v, ldv, info)
ZGGBAK
Definition zggbak.f:147
zggbal
subroutine zggbal(job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)
ZGGBAL
Definition zggbal.f:175
zgghrd
subroutine zgghrd(compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)
ZGGHRD
Definition zgghrd.f:203
zhgeqz
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:283
ilaenv
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:160
zlacpy
subroutine zlacpy(uplo, m, n, a, lda, b, ldb)
ZLACPY copies all or part of one two-dimensional array to another.
Definition zlacpy.f:101
dlamch
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
zlange
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:113
zlascl
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:142
zlaset
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:104
lsame
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
ztgsen
subroutine ztgsen(ijob, wantq, wantz, select, n, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork, info)
ZTGSEN
Definition ztgsen.f:432
zungqr
subroutine zungqr(m, n, k, a, lda, tau, work, lwork, info)
ZUNGQR
Definition zungqr.f:126
zunmqr
subroutine zunmqr(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
ZUNMQR
Definition zunmqr.f:165
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