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CGGES(l)			       )			      CGGES(l)

NAME
       CGGES  -	 compute  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)

SYNOPSIS
       SUBROUTINE CGGES( JOBVSL,  JOBVSR,  SORT,  SELCTG,  N,  A, LDA, B, LDB,
			 SDIM, ALPHA, BETA,  VSL,  LDVSL,  VSR,	 LDVSR,	 WORK,
			 LWORK, RWORK, BWORK, INFO )

	   CHARACTER	 JOBVSL, JOBVSR, SORT

	   INTEGER	 INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM

	   LOGICAL	 BWORK( * )

	   REAL		 RWORK( * )

	   COMPLEX	 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VSL(
			 LDVSL, * ), VSR( LDVSR, * ), WORK( * )

	   LOGICAL	 SELCTG

	   EXTERNAL	 SELCTG

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 tri‐
       angular 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	inter‐
       pretation 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.

ARGUMENTS
       JOBVSL  (input) CHARACTER*1
	       = 'N':  do not compute the left Schur vectors;
	       = 'V':  compute the left Schur vectors.

       JOBVSR  (input) CHARACTER*1
	       = 'N':  do not compute the right Schur vectors;
	       = 'V':  compute the right Schur vectors.

       SORT    (input) CHARACTER*1
	       Specifies whether or not to order the eigenvalues on the diago‐
	       nal of the generalized Schur form.  = 'N':  Eigenvalues are not
	       ordered;
	       = 'S':  Eigenvalues are ordered (see SELCTG).

       SELCTG  (input) 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	order‐
	       ing  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).

       N       (input) INTEGER
	       The order of the matrices A, B, VSL, and VSR.  N >= 0.

       A       (input/output) 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.

       LDA     (input) INTEGER
	       The leading dimension of A.  LDA >= max(1,N).

       B       (input/output) 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.

       LDB     (input) INTEGER
	       The leading dimension of B.  LDB >= max(1,N).

       SDIM    (output) INTEGER
	       If  SORT	 = 'N', SDIM = 0.  If SORT = 'S', SDIM = number of ei‐
	       genvalues (after sorting) for which SELCTG is true.

       ALPHA   (output) COMPLEX array, dimension (N)
	       BETA	(output)  COMPLEX  array,  dimension  (N)   On	 exit,
	       ALPHA(j)/BETA(j),  j=1,...,N, will be the generalized eigenval‐
	       ues.  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 under‐
	       flow,  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).

       VSL     (output) COMPLEX array, dimension (LDVSL,N)
	       If  JOBVSL = 'V', VSL will contain the left Schur vectors.  Not
	       referenced if JOBVSL = 'N'.

       LDVSL   (input) INTEGER
	       The leading dimension of the matrix VSL. LDVSL  >=  1,  and  if
	       JOBVSL = 'V', LDVSL >= N.

       VSR     (output) COMPLEX array, dimension (LDVSR,N)
	       If JOBVSR = 'V', VSR will contain the right Schur vectors.  Not
	       referenced if JOBVSR = 'N'.

       LDVSR   (input) INTEGER
	       The leading dimension of the matrix VSR. LDVSR  >=  1,  and  if
	       JOBVSR = 'V', LDVSR >= N.

       WORK    (workspace/output) COMPLEX array, dimension (LWORK)
	       On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK   (input) 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.

       RWORK   (workspace) REAL array, dimension (8*N)

       BWORK   (workspace) LOGICAL array, dimension (N)
	       Not referenced if SORT = 'N'.

       INFO    (output) 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 falied in CTGSEN.

LAPACK version 3.0		 15 June 2000			      CGGES(l)
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