LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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◆ cgges()

subroutine cgges ( character jobvsl,
character jobvsr,
character sort,
external selctg,
integer n,
complex, dimension( lda, * ) a,
integer lda,
complex, dimension( ldb, * ) b,
integer ldb,
integer sdim,
complex, dimension( * ) alpha,
complex, dimension( * ) beta,
complex, dimension( ldvsl, * ) vsl,
integer ldvsl,
complex, dimension( ldvsr, * ) vsr,
integer ldvsr,
complex, dimension( * ) work,
integer lwork,
real, dimension( * ) rwork,
logical, dimension( * ) bwork,
integer info )

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

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

Purpose:
!>
!> CGGES 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
!> CGGEV 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 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 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 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 array, dimension (N)
!>
[out]BETA
!>          BETA is COMPLEX 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 CGGES. 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 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 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 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 REAL 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 CHGEQZ
!>                =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 CTGSEN.
!>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 265 of file cgges.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 REAL RWORK( * )
281 COMPLEX 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 REAL ZERO, ONE
294 parameter( zero = 0.0e0, one = 1.0e0 )
295 COMPLEX CZERO, CONE
296 parameter( czero = ( 0.0e0, 0.0e0 ),
297 $ cone = ( 1.0e0, 0.0e0 ) )
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 REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
306 $ PVSR, SMLNUM
307* ..
308* .. Local Arrays ..
309 INTEGER IDUM( 1 )
310 REAL DIF( 2 )
311* ..
312* .. External Subroutines ..
313 EXTERNAL cgeqrf, cggbak, cggbal, cgghrd, chgeqz,
314 $ clacpy,
315 $ clascl, claset, ctgsen, cungqr, cunmqr, xerbla
316* ..
317* .. External Functions ..
318 LOGICAL LSAME
319 INTEGER ILAENV
320 REAL CLANGE, SLAMCH, SROUNDUP_LWORK
321 EXTERNAL lsame, ilaenv, clange, slamch,
322 $ sroundup_lwork
323* ..
324* .. Intrinsic Functions ..
325 INTRINSIC max, sqrt
326* ..
327* .. Executable Statements ..
328*
329* Decode the input arguments
330*
331 IF( lsame( jobvsl, 'N' ) ) THEN
332 ijobvl = 1
333 ilvsl = .false.
334 ELSE IF( lsame( jobvsl, 'V' ) ) THEN
335 ijobvl = 2
336 ilvsl = .true.
337 ELSE
338 ijobvl = -1
339 ilvsl = .false.
340 END IF
341*
342 IF( lsame( jobvsr, 'N' ) ) THEN
343 ijobvr = 1
344 ilvsr = .false.
345 ELSE IF( lsame( jobvsr, 'V' ) ) THEN
346 ijobvr = 2
347 ilvsr = .true.
348 ELSE
349 ijobvr = -1
350 ilvsr = .false.
351 END IF
352*
353 wantst = lsame( sort, 'S' )
354*
355* Test the input arguments
356*
357 info = 0
358 lquery = ( lwork.EQ.-1 )
359 IF( ijobvl.LE.0 ) THEN
360 info = -1
361 ELSE IF( ijobvr.LE.0 ) THEN
362 info = -2
363 ELSE IF( ( .NOT.wantst ) .AND.
364 $ ( .NOT.lsame( sort, 'N' ) ) ) THEN
365 info = -3
366 ELSE IF( n.LT.0 ) THEN
367 info = -5
368 ELSE IF( lda.LT.max( 1, n ) ) THEN
369 info = -7
370 ELSE IF( ldb.LT.max( 1, n ) ) THEN
371 info = -9
372 ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
373 info = -14
374 ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
375 info = -16
376 END IF
377*
378* Compute workspace
379* (Note: Comments in the code beginning "Workspace:" describe the
380* minimal amount of workspace needed at that point in the code,
381* as well as the preferred amount for good performance.
382* NB refers to the optimal block size for the immediately
383* following subroutine, as returned by ILAENV.)
384*
385 IF( info.EQ.0 ) THEN
386 lwkmin = max( 1, 2*n )
387 lwkopt = max( 1, n + n*ilaenv( 1, 'CGEQRF', ' ', n, 1, n,
388 $ 0 ) )
389 lwkopt = max( lwkopt, n +
390 $ n*ilaenv( 1, 'CUNMQR', ' ', n, 1, n, -1 ) )
391 IF( ilvsl ) THEN
392 lwkopt = max( lwkopt, n +
393 $ n*ilaenv( 1, 'CUNGQR', ' ', n, 1, n, -1 ) )
394 END IF
395 work( 1 ) = sroundup_lwork(lwkopt)
396*
397 IF( lwork.LT.lwkmin .AND. .NOT.lquery )
398 $ info = -18
399 END IF
400*
401 IF( info.NE.0 ) THEN
402 CALL xerbla( 'CGGES ', -info )
403 RETURN
404 ELSE IF( lquery ) THEN
405 RETURN
406 END IF
407*
408* Quick return if possible
409*
410 IF( n.EQ.0 ) THEN
411 sdim = 0
412 RETURN
413 END IF
414*
415* Get machine constants
416*
417 eps = slamch( 'P' )
418 smlnum = slamch( 'S' )
419 bignum = one / smlnum
420 smlnum = sqrt( smlnum ) / eps
421 bignum = one / smlnum
422*
423* Scale A if max element outside range [SMLNUM,BIGNUM]
424*
425 anrm = clange( 'M', n, n, a, lda, rwork )
426 ilascl = .false.
427 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
428 anrmto = smlnum
429 ilascl = .true.
430 ELSE IF( anrm.GT.bignum ) THEN
431 anrmto = bignum
432 ilascl = .true.
433 END IF
434*
435 IF( ilascl )
436 $ CALL clascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
437*
438* Scale B if max element outside range [SMLNUM,BIGNUM]
439*
440 bnrm = clange( 'M', n, n, b, ldb, rwork )
441 ilbscl = .false.
442 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
443 bnrmto = smlnum
444 ilbscl = .true.
445 ELSE IF( bnrm.GT.bignum ) THEN
446 bnrmto = bignum
447 ilbscl = .true.
448 END IF
449*
450 IF( ilbscl )
451 $ CALL clascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
452*
453* Permute the matrix to make it more nearly triangular
454* (Real Workspace: need 6*N)
455*
456 ileft = 1
457 iright = n + 1
458 irwrk = iright + n
459 CALL cggbal( 'P', n, a, lda, b, ldb, ilo, ihi, rwork( ileft ),
460 $ rwork( iright ), rwork( irwrk ), ierr )
461*
462* Reduce B to triangular form (QR decomposition of B)
463* (Complex Workspace: need N, prefer N*NB)
464*
465 irows = ihi + 1 - ilo
466 icols = n + 1 - ilo
467 itau = 1
468 iwrk = itau + irows
469 CALL cgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
470 $ work( iwrk ), lwork+1-iwrk, ierr )
471*
472* Apply the orthogonal transformation to matrix A
473* (Complex Workspace: need N, prefer N*NB)
474*
475 CALL cunmqr( 'L', 'C', irows, icols, irows, b( ilo, ilo ), ldb,
476 $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
477 $ lwork+1-iwrk, ierr )
478*
479* Initialize VSL
480* (Complex Workspace: need N, prefer N*NB)
481*
482 IF( ilvsl ) THEN
483 CALL claset( 'Full', n, n, czero, cone, vsl, ldvsl )
484 IF( irows.GT.1 ) THEN
485 CALL clacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
486 $ vsl( ilo+1, ilo ), ldvsl )
487 END IF
488 CALL cungqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
489 $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
490 END IF
491*
492* Initialize VSR
493*
494 IF( ilvsr )
495 $ CALL claset( 'Full', n, n, czero, cone, vsr, ldvsr )
496*
497* Reduce to generalized Hessenberg form
498* (Workspace: none needed)
499*
500 CALL cgghrd( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
501 $ ldvsl, vsr, ldvsr, ierr )
502*
503 sdim = 0
504*
505* Perform QZ algorithm, computing Schur vectors if desired
506* (Complex Workspace: need N)
507* (Real Workspace: need N)
508*
509 iwrk = itau
510 CALL chgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
511 $ alpha, beta, vsl, ldvsl, vsr, ldvsr, work( iwrk ),
512 $ lwork+1-iwrk, rwork( irwrk ), ierr )
513 IF( ierr.NE.0 ) THEN
514 IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
515 info = ierr
516 ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
517 info = ierr - n
518 ELSE
519 info = n + 1
520 END IF
521 GO TO 30
522 END IF
523*
524* Sort eigenvalues ALPHA/BETA if desired
525* (Workspace: none needed)
526*
527 IF( wantst ) THEN
528*
529* Undo scaling on eigenvalues before selecting
530*
531 IF( ilascl )
532 $ CALL clascl( 'G', 0, 0, anrm, anrmto, n, 1, alpha, n,
533 $ ierr )
534 IF( ilbscl )
535 $ CALL clascl( 'G', 0, 0, bnrm, bnrmto, n, 1, beta, n,
536 $ ierr )
537*
538* Select eigenvalues
539*
540 DO 10 i = 1, n
541 bwork( i ) = selctg( alpha( i ), beta( i ) )
542 10 CONTINUE
543*
544 CALL ctgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
545 $ alpha,
546 $ beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl, pvsr,
547 $ dif, work( iwrk ), lwork-iwrk+1, idum, 1, ierr )
548 IF( ierr.EQ.1 )
549 $ info = n + 3
550*
551 END IF
552*
553* Apply back-permutation to VSL and VSR
554* (Workspace: none needed)
555*
556 IF( ilvsl )
557 $ CALL cggbak( 'P', 'L', n, ilo, ihi, rwork( ileft ),
558 $ rwork( iright ), n, vsl, ldvsl, ierr )
559 IF( ilvsr )
560 $ CALL cggbak( 'P', 'R', n, ilo, ihi, rwork( ileft ),
561 $ rwork( iright ), n, vsr, ldvsr, ierr )
562*
563* Undo scaling
564*
565 IF( ilascl ) THEN
566 CALL clascl( 'U', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
567 CALL clascl( 'G', 0, 0, anrmto, anrm, n, 1, alpha, n, ierr )
568 END IF
569*
570 IF( ilbscl ) THEN
571 CALL clascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
572 CALL clascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
573 END IF
574*
575 IF( wantst ) THEN
576*
577* Check if reordering is correct
578*
579 lastsl = .true.
580 sdim = 0
581 DO 20 i = 1, n
582 cursl = selctg( alpha( i ), beta( i ) )
583 IF( cursl )
584 $ sdim = sdim + 1
585 IF( cursl .AND. .NOT.lastsl )
586 $ info = n + 2
587 lastsl = cursl
588 20 CONTINUE
589*
590 END IF
591*
592 30 CONTINUE
593*
594 work( 1 ) = sroundup_lwork(lwkopt)
595*
596 RETURN
597*
598* End of CGGES
599*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
cgeqrf
subroutine cgeqrf(m, n, a, lda, tau, work, lwork, info)
CGEQRF
Definition cgeqrf.f:144
cggbak
subroutine cggbak(job, side, n, ilo, ihi, lscale, rscale, m, v, ldv, info)
CGGBAK
Definition cggbak.f:147
cggbal
subroutine cggbal(job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)
CGGBAL
Definition cggbal.f:175
cgghrd
subroutine cgghrd(compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)
CGGHRD
Definition cgghrd.f:203
chgeqz
subroutine chgeqz(job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, info)
CHGEQZ
Definition chgeqz.f:283
ilaenv
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:160
clacpy
subroutine clacpy(uplo, m, n, a, lda, b, ldb)
CLACPY copies all or part of one two-dimensional array to another.
Definition clacpy.f:101
slamch
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
clange
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:113
clascl
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:142
claset
subroutine claset(uplo, m, n, alpha, beta, a, lda)
CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition claset.f:104
lsame
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
sroundup_lwork
real function sroundup_lwork(lwork)
SROUNDUP_LWORK
Definition sroundup_lwork.f:59
ctgsen
subroutine ctgsen(ijob, wantq, wantz, select, n, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork, info)
CTGSEN
Definition ctgsen.f:432
cungqr
subroutine cungqr(m, n, k, a, lda, tau, work, lwork, info)
CUNGQR
Definition cungqr.f:126
cunmqr
subroutine cunmqr(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
CUNMQR
Definition cunmqr.f:166
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