LAPACK  3.8.0
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.
Date
December 2016

Definition at line 272 of file cgges.f.

272 *
273 * -- LAPACK driver routine (version 3.7.0) --
274 * -- LAPACK is a software package provided by Univ. of Tennessee, --
275 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
276 * December 2016
277 *
278 * .. Scalar Arguments ..
279  CHARACTER jobvsl, jobvsr, sort
280  INTEGER info, lda, ldb, ldvsl, ldvsr, lwork, n, sdim
281 * ..
282 * .. Array Arguments ..
283  LOGICAL bwork( * )
284  REAL rwork( * )
285  COMPLEX a( lda, * ), alpha( * ), b( ldb, * ),
286  $ beta( * ), vsl( ldvsl, * ), vsr( ldvsr, * ),
287  $ work( * )
288 * ..
289 * .. Function Arguments ..
290  LOGICAL selctg
291  EXTERNAL selctg
292 * ..
293 *
294 * =====================================================================
295 *
296 * .. Parameters ..
297  REAL zero, one
298  parameter( zero = 0.0e0, one = 1.0e0 )
299  COMPLEX czero, cone
300  parameter( czero = ( 0.0e0, 0.0e0 ),
301  $ cone = ( 1.0e0, 0.0e0 ) )
302 * ..
303 * .. Local Scalars ..
304  LOGICAL cursl, ilascl, ilbscl, ilvsl, ilvsr, lastsl,
305  $ lquery, wantst
306  INTEGER i, icols, ierr, ihi, ijobvl, ijobvr, ileft,
307  $ ilo, iright, irows, irwrk, itau, iwrk, lwkmin,
308  $ lwkopt
309  REAL anrm, anrmto, bignum, bnrm, bnrmto, eps, pvsl,
310  $ pvsr, smlnum
311 * ..
312 * .. Local Arrays ..
313  INTEGER idum( 1 )
314  REAL dif( 2 )
315 * ..
316 * .. External Subroutines ..
317  EXTERNAL cgeqrf, cggbak, cggbal, cgghrd, chgeqz, clacpy,
319  $ xerbla
320 * ..
321 * .. External Functions ..
322  LOGICAL lsame
323  INTEGER ilaenv
324  REAL clange, slamch
325  EXTERNAL lsame, ilaenv, clange, slamch
326 * ..
327 * .. Intrinsic Functions ..
328  INTRINSIC max, sqrt
329 * ..
330 * .. Executable Statements ..
331 *
332 * Decode the input arguments
333 *
334  IF( lsame( jobvsl, 'N' ) ) THEN
335  ijobvl = 1
336  ilvsl = .false.
337  ELSE IF( lsame( jobvsl, 'V' ) ) THEN
338  ijobvl = 2
339  ilvsl = .true.
340  ELSE
341  ijobvl = -1
342  ilvsl = .false.
343  END IF
344 *
345  IF( lsame( jobvsr, 'N' ) ) THEN
346  ijobvr = 1
347  ilvsr = .false.
348  ELSE IF( lsame( jobvsr, 'V' ) ) THEN
349  ijobvr = 2
350  ilvsr = .true.
351  ELSE
352  ijobvr = -1
353  ilvsr = .false.
354  END IF
355 *
356  wantst = lsame( sort, 'S' )
357 *
358 * Test the input arguments
359 *
360  info = 0
361  lquery = ( lwork.EQ.-1 )
362  IF( ijobvl.LE.0 ) THEN
363  info = -1
364  ELSE IF( ijobvr.LE.0 ) THEN
365  info = -2
366  ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
367  info = -3
368  ELSE IF( n.LT.0 ) THEN
369  info = -5
370  ELSE IF( lda.LT.max( 1, n ) ) THEN
371  info = -7
372  ELSE IF( ldb.LT.max( 1, n ) ) THEN
373  info = -9
374  ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
375  info = -14
376  ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
377  info = -16
378  END IF
379 *
380 * Compute workspace
381 * (Note: Comments in the code beginning "Workspace:" describe the
382 * minimal amount of workspace needed at that point in the code,
383 * as well as the preferred amount for good performance.
384 * NB refers to the optimal block size for the immediately
385 * following subroutine, as returned by ILAENV.)
386 *
387  IF( info.EQ.0 ) THEN
388  lwkmin = max( 1, 2*n )
389  lwkopt = max( 1, n + n*ilaenv( 1, 'CGEQRF', ' ', n, 1, n, 0 ) )
390  lwkopt = max( lwkopt, n +
391  $ n*ilaenv( 1, 'CUNMQR', ' ', n, 1, n, -1 ) )
392  IF( ilvsl ) THEN
393  lwkopt = max( lwkopt, n +
394  $ n*ilaenv( 1, 'CUNGQR', ' ', n, 1, n, -1 ) )
395  END IF
396  work( 1 ) = lwkopt
397 *
398  IF( lwork.LT.lwkmin .AND. .NOT.lquery )
399  $ info = -18
400  END IF
401 *
402  IF( info.NE.0 ) THEN
403  CALL xerbla( 'CGGES ', -info )
404  RETURN
405  ELSE IF( lquery ) THEN
406  RETURN
407  END IF
408 *
409 * Quick return if possible
410 *
411  IF( n.EQ.0 ) THEN
412  sdim = 0
413  RETURN
414  END IF
415 *
416 * Get machine constants
417 *
418  eps = slamch( 'P' )
419  smlnum = slamch( 'S' )
420  bignum = one / smlnum
421  CALL slabad( smlnum, bignum )
422  smlnum = sqrt( smlnum ) / eps
423  bignum = one / smlnum
424 *
425 * Scale A if max element outside range [SMLNUM,BIGNUM]
426 *
427  anrm = clange( 'M', n, n, a, lda, rwork )
428  ilascl = .false.
429  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
430  anrmto = smlnum
431  ilascl = .true.
432  ELSE IF( anrm.GT.bignum ) THEN
433  anrmto = bignum
434  ilascl = .true.
435  END IF
436 *
437  IF( ilascl )
438  $ CALL clascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
439 *
440 * Scale B if max element outside range [SMLNUM,BIGNUM]
441 *
442  bnrm = clange( 'M', n, n, b, ldb, rwork )
443  ilbscl = .false.
444  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
445  bnrmto = smlnum
446  ilbscl = .true.
447  ELSE IF( bnrm.GT.bignum ) THEN
448  bnrmto = bignum
449  ilbscl = .true.
450  END IF
451 *
452  IF( ilbscl )
453  $ CALL clascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
454 *
455 * Permute the matrix to make it more nearly triangular
456 * (Real Workspace: need 6*N)
457 *
458  ileft = 1
459  iright = n + 1
460  irwrk = iright + n
461  CALL cggbal( 'P', n, a, lda, b, ldb, ilo, ihi, rwork( ileft ),
462  $ rwork( iright ), rwork( irwrk ), ierr )
463 *
464 * Reduce B to triangular form (QR decomposition of B)
465 * (Complex Workspace: need N, prefer N*NB)
466 *
467  irows = ihi + 1 - ilo
468  icols = n + 1 - ilo
469  itau = 1
470  iwrk = itau + irows
471  CALL cgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
472  $ work( iwrk ), lwork+1-iwrk, ierr )
473 *
474 * Apply the orthogonal transformation to matrix A
475 * (Complex Workspace: need N, prefer N*NB)
476 *
477  CALL cunmqr( 'L', 'C', irows, icols, irows, b( ilo, ilo ), ldb,
478  $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
479  $ lwork+1-iwrk, ierr )
480 *
481 * Initialize VSL
482 * (Complex Workspace: need N, prefer N*NB)
483 *
484  IF( ilvsl ) THEN
485  CALL claset( 'Full', n, n, czero, cone, vsl, ldvsl )
486  IF( irows.GT.1 ) THEN
487  CALL clacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
488  $ vsl( ilo+1, ilo ), ldvsl )
489  END IF
490  CALL cungqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
491  $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
492  END IF
493 *
494 * Initialize VSR
495 *
496  IF( ilvsr )
497  $ CALL claset( 'Full', n, n, czero, cone, vsr, ldvsr )
498 *
499 * Reduce to generalized Hessenberg form
500 * (Workspace: none needed)
501 *
502  CALL cgghrd( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
503  $ ldvsl, vsr, ldvsr, ierr )
504 *
505  sdim = 0
506 *
507 * Perform QZ algorithm, computing Schur vectors if desired
508 * (Complex Workspace: need N)
509 * (Real Workspace: need N)
510 *
511  iwrk = itau
512  CALL chgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
513  $ alpha, beta, vsl, ldvsl, vsr, ldvsr, work( iwrk ),
514  $ lwork+1-iwrk, rwork( irwrk ), ierr )
515  IF( ierr.NE.0 ) THEN
516  IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
517  info = ierr
518  ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
519  info = ierr - n
520  ELSE
521  info = n + 1
522  END IF
523  GO TO 30
524  END IF
525 *
526 * Sort eigenvalues ALPHA/BETA if desired
527 * (Workspace: none needed)
528 *
529  IF( wantst ) THEN
530 *
531 * Undo scaling on eigenvalues before selecting
532 *
533  IF( ilascl )
534  $ CALL clascl( 'G', 0, 0, anrm, anrmto, n, 1, alpha, n, ierr )
535  IF( ilbscl )
536  $ CALL clascl( 'G', 0, 0, bnrm, bnrmto, n, 1, beta, n, 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, 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 cggbak( 'P', 'L', n, ilo, ihi, rwork( ileft ),
557  $ rwork( iright ), n, vsl, ldvsl, ierr )
558  IF( ilvsr )
559  $ CALL cggbak( '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 clascl( 'U', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
566  CALL clascl( 'G', 0, 0, anrmto, anrm, n, 1, alpha, n, ierr )
567  END IF
568 *
569  IF( ilbscl ) THEN
570  CALL clascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
571  CALL clascl( '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 CGGES
598 *
subroutine cggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
CGGBAL
Definition: cggbal.f:179
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:108
subroutine cungqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
CUNGQR
Definition: cungqr.f:130
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:145
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:286
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: tstiee.f:83
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:117
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine cgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
CGEQRF
Definition: cgeqrf.f:138
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:435
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:105
subroutine slabad(SMALL, LARGE)
SLABAD
Definition: slabad.f:76
subroutine cgghrd(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
CGGHRD
Definition: cgghrd.f:206
subroutine cggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
CGGBAK
Definition: cggbak.f:150
subroutine cunmqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMQR
Definition: cunmqr.f:170
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