LAPACK  3.10.1
LAPACK: Linear Algebra PACKage

◆ 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 267 of file cgges.f.

270 *
271 * -- LAPACK driver routine --
272 * -- LAPACK is a software package provided by Univ. of Tennessee, --
273 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
274 *
275 * .. Scalar Arguments ..
276  CHARACTER JOBVSL, JOBVSR, SORT
277  INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
278 * ..
279 * .. Array Arguments ..
280  LOGICAL BWORK( * )
281  REAL RWORK( * )
282  COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
283  $ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
284  $ WORK( * )
285 * ..
286 * .. Function Arguments ..
287  LOGICAL SELCTG
288  EXTERNAL selctg
289 * ..
290 *
291 * =====================================================================
292 *
293 * .. Parameters ..
294  REAL ZERO, ONE
295  parameter( zero = 0.0e0, one = 1.0e0 )
296  COMPLEX CZERO, CONE
297  parameter( czero = ( 0.0e0, 0.0e0 ),
298  $ cone = ( 1.0e0, 0.0e0 ) )
299 * ..
300 * .. Local Scalars ..
301  LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
302  $ LQUERY, WANTST
303  INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
304  $ ILO, IRIGHT, IROWS, IRWRK, ITAU, IWRK, LWKMIN,
305  $ LWKOPT
306  REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
307  $ PVSR, SMLNUM
308 * ..
309 * .. Local Arrays ..
310  INTEGER IDUM( 1 )
311  REAL DIF( 2 )
312 * ..
313 * .. External Subroutines ..
314  EXTERNAL cgeqrf, cggbak, cggbal, cgghrd, chgeqz, clacpy,
316  $ xerbla
317 * ..
318 * .. External Functions ..
319  LOGICAL LSAME
320  INTEGER ILAENV
321  REAL CLANGE, SLAMCH
322  EXTERNAL lsame, ilaenv, clange, slamch
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. ( .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, 'CGEQRF', ' ', n, 1, n, 0 ) )
387  lwkopt = max( lwkopt, n +
388  $ n*ilaenv( 1, 'CUNMQR', ' ', n, 1, n, -1 ) )
389  IF( ilvsl ) THEN
390  lwkopt = max( lwkopt, n +
391  $ n*ilaenv( 1, 'CUNGQR', ' ', n, 1, n, -1 ) )
392  END IF
393  work( 1 ) = lwkopt
394 *
395  IF( lwork.LT.lwkmin .AND. .NOT.lquery )
396  $ info = -18
397  END IF
398 *
399  IF( info.NE.0 ) THEN
400  CALL xerbla( 'CGGES ', -info )
401  RETURN
402  ELSE IF( lquery ) THEN
403  RETURN
404  END IF
405 *
406 * Quick return if possible
407 *
408  IF( n.EQ.0 ) THEN
409  sdim = 0
410  RETURN
411  END IF
412 *
413 * Get machine constants
414 *
415  eps = slamch( 'P' )
416  smlnum = slamch( 'S' )
417  bignum = one / smlnum
418  CALL slabad( smlnum, bignum )
419  smlnum = sqrt( smlnum ) / eps
420  bignum = one / smlnum
421 *
422 * Scale A if max element outside range [SMLNUM,BIGNUM]
423 *
424  anrm = clange( '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 clascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
436 *
437 * Scale B if max element outside range [SMLNUM,BIGNUM]
438 *
439  bnrm = clange( '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 clascl( '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 cggbal( '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 cgeqrf( 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 cunmqr( '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 claset( 'Full', n, n, czero, cone, vsl, ldvsl )
483  IF( irows.GT.1 ) THEN
484  CALL clacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
485  $ vsl( ilo+1, ilo ), ldvsl )
486  END IF
487  CALL cungqr( 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 claset( 'Full', n, n, czero, cone, vsr, ldvsr )
495 *
496 * Reduce to generalized Hessenberg form
497 * (Workspace: none needed)
498 *
499  CALL cgghrd( 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 chgeqz( '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 clascl( 'G', 0, 0, anrm, anrmto, n, 1, alpha, n, ierr )
532  IF( ilbscl )
533  $ CALL clascl( 'G', 0, 0, bnrm, bnrmto, n, 1, beta, n, ierr )
534 *
535 * Select eigenvalues
536 *
537  DO 10 i = 1, n
538  bwork( i ) = selctg( alpha( i ), beta( i ) )
539  10 CONTINUE
540 *
541  CALL ctgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb, alpha,
542  $ beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl, pvsr,
543  $ dif, work( iwrk ), lwork-iwrk+1, idum, 1, ierr )
544  IF( ierr.EQ.1 )
545  $ info = n + 3
546 *
547  END IF
548 *
549 * Apply back-permutation to VSL and VSR
550 * (Workspace: none needed)
551 *
552  IF( ilvsl )
553  $ CALL cggbak( 'P', 'L', n, ilo, ihi, rwork( ileft ),
554  $ rwork( iright ), n, vsl, ldvsl, ierr )
555  IF( ilvsr )
556  $ CALL cggbak( 'P', 'R', n, ilo, ihi, rwork( ileft ),
557  $ rwork( iright ), n, vsr, ldvsr, ierr )
558 *
559 * Undo scaling
560 *
561  IF( ilascl ) THEN
562  CALL clascl( 'U', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
563  CALL clascl( 'G', 0, 0, anrmto, anrm, n, 1, alpha, n, ierr )
564  END IF
565 *
566  IF( ilbscl ) THEN
567  CALL clascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
568  CALL clascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
569  END IF
570 *
571  IF( wantst ) THEN
572 *
573 * Check if reordering is correct
574 *
575  lastsl = .true.
576  sdim = 0
577  DO 20 i = 1, n
578  cursl = selctg( alpha( i ), beta( i ) )
579  IF( cursl )
580  $ sdim = sdim + 1
581  IF( cursl .AND. .NOT.lastsl )
582  $ info = n + 2
583  lastsl = cursl
584  20 CONTINUE
585 *
586  END IF
587 *
588  30 CONTINUE
589 *
590  work( 1 ) = lwkopt
591 *
592  RETURN
593 *
594 * End of CGGES
595 *
subroutine slabad(SMALL, LARGE)
SLABAD
Definition: slabad.f:74
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 cggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
CGGBAL
Definition: cggbal.f:177
subroutine cggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
CGGBAK
Definition: cggbak.f:148
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:115
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:284
subroutine cgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
CGEQRF
Definition: cgeqrf.f:146
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:106
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:143
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
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:433
subroutine cunmqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMQR
Definition: cunmqr.f:168
subroutine cgghrd(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
CGGHRD
Definition: cgghrd.f:204
subroutine cungqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
CUNGQR
Definition: cungqr.f:128
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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