LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ cgges3()

 subroutine cgges3 ( 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 )

CGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

Purpose:
``` CGGES3 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

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 CGGES3. 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. 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 CLAQZ0 =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.```

Definition at line 266 of file cgges3.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, LWKOPT
304  REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
305  \$ PVSR, SMLNUM
306 * ..
307 * .. Local Arrays ..
308  INTEGER IDUM( 1 )
309  REAL DIF( 2 )
310 * ..
311 * .. External Subroutines ..
312  EXTERNAL cgeqrf, cggbak, cggbal, cgghd3, claqz0, clacpy,
314  \$ xerbla
315 * ..
316 * .. External Functions ..
317  LOGICAL LSAME
318  REAL CLANGE, SLAMCH
319  EXTERNAL lsame, clange, slamch
320 * ..
321 * .. Intrinsic Functions ..
322  INTRINSIC max, sqrt
323 * ..
324 * .. Executable Statements ..
325 *
326 * Decode the input arguments
327 *
328  IF( lsame( jobvsl, 'N' ) ) THEN
329  ijobvl = 1
330  ilvsl = .false.
331  ELSE IF( lsame( jobvsl, 'V' ) ) THEN
332  ijobvl = 2
333  ilvsl = .true.
334  ELSE
335  ijobvl = -1
336  ilvsl = .false.
337  END IF
338 *
339  IF( lsame( jobvsr, 'N' ) ) THEN
340  ijobvr = 1
341  ilvsr = .false.
342  ELSE IF( lsame( jobvsr, 'V' ) ) THEN
343  ijobvr = 2
344  ilvsr = .true.
345  ELSE
346  ijobvr = -1
347  ilvsr = .false.
348  END IF
349 *
350  wantst = lsame( sort, 'S' )
351 *
352 * Test the input arguments
353 *
354  info = 0
355  lquery = ( lwork.EQ.-1 )
356  IF( ijobvl.LE.0 ) THEN
357  info = -1
358  ELSE IF( ijobvr.LE.0 ) THEN
359  info = -2
360  ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
361  info = -3
362  ELSE IF( n.LT.0 ) THEN
363  info = -5
364  ELSE IF( lda.LT.max( 1, n ) ) THEN
365  info = -7
366  ELSE IF( ldb.LT.max( 1, n ) ) THEN
367  info = -9
368  ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
369  info = -14
370  ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
371  info = -16
372  ELSE IF( lwork.LT.max( 1, 2*n ) .AND. .NOT.lquery ) THEN
373  info = -18
374  END IF
375 *
376 * Compute workspace
377 *
378  IF( info.EQ.0 ) THEN
379  CALL cgeqrf( n, n, b, ldb, work, work, -1, ierr )
380  lwkopt = max( 1, n + int( work( 1 ) ) )
381  CALL cunmqr( 'L', 'C', n, n, n, b, ldb, work, a, lda, work,
382  \$ -1, ierr )
383  lwkopt = max( lwkopt, n + int( work( 1 ) ) )
384  IF( ilvsl ) THEN
385  CALL cungqr( n, n, n, vsl, ldvsl, work, work, -1,
386  \$ ierr )
387  lwkopt = max( lwkopt, n + int( work( 1 ) ) )
388  END IF
389  CALL cgghd3( jobvsl, jobvsr, n, 1, n, a, lda, b, ldb, vsl,
390  \$ ldvsl, vsr, ldvsr, work, -1, ierr )
391  lwkopt = max( lwkopt, n + int( work( 1 ) ) )
392  CALL claqz0( 'S', jobvsl, jobvsr, n, 1, n, a, lda, b, ldb,
393  \$ alpha, beta, vsl, ldvsl, vsr, ldvsr, work, -1,
394  \$ rwork, 0, ierr )
395  lwkopt = max( lwkopt, int( work( 1 ) ) )
396  IF( wantst ) THEN
397  CALL ctgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
398  \$ alpha, beta, vsl, ldvsl, vsr, ldvsr, sdim,
399  \$ pvsl, pvsr, dif, work, -1, idum, 1, ierr )
400  lwkopt = max( lwkopt, int( work( 1 ) ) )
401  END IF
402  work( 1 ) = cmplx( lwkopt )
403  END IF
404
405 *
406  IF( info.NE.0 ) THEN
407  CALL xerbla( 'CGGES3 ', -info )
408  RETURN
409  ELSE IF( lquery ) THEN
410  RETURN
411  END IF
412 *
413 * Quick return if possible
414 *
415  IF( n.EQ.0 ) THEN
416  sdim = 0
417  RETURN
418  END IF
419 *
420 * Get machine constants
421 *
422  eps = slamch( 'P' )
423  smlnum = slamch( 'S' )
424  bignum = one / smlnum
425  CALL slabad( smlnum, bignum )
426  smlnum = sqrt( smlnum ) / eps
427  bignum = one / smlnum
428 *
429 * Scale A if max element outside range [SMLNUM,BIGNUM]
430 *
431  anrm = clange( 'M', n, n, a, lda, rwork )
432  ilascl = .false.
433  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
434  anrmto = smlnum
435  ilascl = .true.
436  ELSE IF( anrm.GT.bignum ) THEN
437  anrmto = bignum
438  ilascl = .true.
439  END IF
440 *
441  IF( ilascl )
442  \$ CALL clascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
443 *
444 * Scale B if max element outside range [SMLNUM,BIGNUM]
445 *
446  bnrm = clange( 'M', n, n, b, ldb, rwork )
447  ilbscl = .false.
448  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
449  bnrmto = smlnum
450  ilbscl = .true.
451  ELSE IF( bnrm.GT.bignum ) THEN
452  bnrmto = bignum
453  ilbscl = .true.
454  END IF
455 *
456  IF( ilbscl )
457  \$ CALL clascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
458 *
459 * Permute the matrix to make it more nearly triangular
460 *
461  ileft = 1
462  iright = n + 1
463  irwrk = iright + n
464  CALL cggbal( 'P', n, a, lda, b, ldb, ilo, ihi, rwork( ileft ),
465  \$ rwork( iright ), rwork( irwrk ), ierr )
466 *
467 * Reduce B to triangular form (QR decomposition of B)
468 *
469  irows = ihi + 1 - ilo
470  icols = n + 1 - ilo
471  itau = 1
472  iwrk = itau + irows
473  CALL cgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
474  \$ work( iwrk ), lwork+1-iwrk, ierr )
475 *
476 * Apply the orthogonal transformation to matrix A
477 *
478  CALL cunmqr( 'L', 'C', irows, icols, irows, b( ilo, ilo ), ldb,
479  \$ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
480  \$ lwork+1-iwrk, ierr )
481 *
482 * Initialize VSL
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 *
501  CALL cgghd3( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
502  \$ ldvsl, vsr, ldvsr, work( iwrk ), lwork+1-iwrk, ierr )
503 *
504  sdim = 0
505 *
506 * Perform QZ algorithm, computing Schur vectors if desired
507 *
508  iwrk = itau
509  CALL claqz0( '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 ), 0, 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 *
525  IF( wantst ) THEN
526 *
527 * Undo scaling on eigenvalues before selecting
528 *
529  IF( ilascl )
530  \$ CALL clascl( 'G', 0, 0, anrm, anrmto, n, 1, alpha, n, ierr )
531  IF( ilbscl )
532  \$ CALL clascl( 'G', 0, 0, bnrm, bnrmto, n, 1, beta, n, ierr )
533 *
534 * Select eigenvalues
535 *
536  DO 10 i = 1, n
537  bwork( i ) = selctg( alpha( i ), beta( i ) )
538  10 CONTINUE
539 *
540  CALL ctgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb, alpha,
541  \$ beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl, pvsr,
542  \$ dif, work( iwrk ), lwork-iwrk+1, idum, 1, ierr )
543  IF( ierr.EQ.1 )
544  \$ info = n + 3
545 *
546  END IF
547 *
548 * Apply back-permutation to VSL and VSR
549 *
550  IF( ilvsl )
551  \$ CALL cggbak( 'P', 'L', n, ilo, ihi, rwork( ileft ),
552  \$ rwork( iright ), n, vsl, ldvsl, ierr )
553  IF( ilvsr )
554  \$ CALL cggbak( 'P', 'R', n, ilo, ihi, rwork( ileft ),
555  \$ rwork( iright ), n, vsr, ldvsr, ierr )
556 *
557 * Undo scaling
558 *
559  IF( ilascl ) THEN
560  CALL clascl( 'U', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
561  CALL clascl( 'G', 0, 0, anrmto, anrm, n, 1, alpha, n, ierr )
562  END IF
563 *
564  IF( ilbscl ) THEN
565  CALL clascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
566  CALL clascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
567  END IF
568 *
569  IF( wantst ) THEN
570 *
571 * Check if reordering is correct
572 *
573  lastsl = .true.
574  sdim = 0
575  DO 20 i = 1, n
576  cursl = selctg( alpha( i ), beta( i ) )
577  IF( cursl )
578  \$ sdim = sdim + 1
579  IF( cursl .AND. .NOT.lastsl )
580  \$ info = n + 2
581  lastsl = cursl
582  20 CONTINUE
583 *
584  END IF
585 *
586  30 CONTINUE
587 *
588  work( 1 ) = cmplx( lwkopt )
589 *
590  RETURN
591 *
592 * End of CGGES3
593 *
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 cgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
CGEQRF
Definition: cgeqrf.f:146
recursive subroutine claqz0(WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, REC, INFO)
CLAQZ0
Definition: claqz0.f:284
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 cgghd3(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
CGGHD3
Definition: cgghd3.f:231
subroutine cunmqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMQR
Definition: cunmqr.f:168
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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