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RedHat 9 (Linux i386) - man page for zgges (redhat section l)

ZGGES(l)					)					 ZGGES(l)

NAME
       ZGGES  - 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 ZGGES( JOBVSL, JOBVSR, SORT, DELCTG, 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( * )

	   DOUBLE	 PRECISION RWORK( * )

	   COMPLEX*16	 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VSL( LDVSL, *  ),  VSR(
			 LDVSR, * ), WORK( * )

	   LOGICAL	 DELCTG

	   EXTERNAL	 DELCTG

PURPOSE
       ZGGES  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 tri-
       angular	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 ZGGEV  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 trian-
       gular 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 diagonal of the  general-
	       ized Schur form.  = 'N':  Eigenvalues are not ordered;
	       = 'S':  Eigenvalues are ordered (see DELZTG).

       DELZTG  (input) LOGICAL FUNCTION of two COMPLEX*16 arguments
	       DELZTG must be declared EXTERNAL in the calling subroutine.  If SORT = 'N', DELZTG
	       is not referenced.  If SORT = 'S', DELZTG is used to select eigenvalues to sort to
	       the  top  left  of  the Schur form.  An eigenvalue ALPHA(j)/BETA(j) is selected if
	       DELZTG(ALPHA(j),BETA(j)) is true.

	       Note   that   a	 selected   complex   eigenvalue   may	  no	longer	  satisfy
	       DELZTG(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).

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

       A       (input/output) COMPLEX*16 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*16 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 eigenvalues (after sort-
	       ing) for which DELZTG is true.

       ALPHA   (output) COMPLEX*16 array, dimension (N)
	       BETA    (output) COMPLEX*16  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
	       ZGGES. 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).

       VSL     (output) COMPLEX*16 array, dimension (LDVSL,N)
	       If  JOBVSL = 'V', VSL will contain the left Schur vectors.  Not referenced if JOB-
	       VSL = 'N'.

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

       VSR     (output) COMPLEX*16 array, dimension (LDVSR,N)
	       If JOBVSR = 'V', VSR will contain the right Schur vectors.  Not referenced if JOB-
	       VSR = 'N'.

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

       WORK    (workspace/output) COMPLEX*16 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) DOUBLE PRECISION 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 ZHGEQZ
	       =N+2: after reordering, roundoff changed values of  some  complex  eigenvalues  so
	       that  leading  eigenvalues  in  the  Generalized  Schur	form  no  longer  satisfy
	       DELZTG=.TRUE.  This could also be caused due to scaling.  =N+3: reordering  falied
	       in ZTGSEN.

LAPACK version 3.0			   15 June 2000 				 ZGGES(l)


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