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DGGEV(l)			       )			      DGGEV(l)

NAME
       DGGEV - compute for a pair of N-by-N real nonsymmetric matrices (A,B)

SYNOPSIS
       SUBROUTINE DGGEV( JOBVL,	 JOBVR,	 N,  A,	 LDA,  B, LDB, ALPHAR, ALPHAI,
			 BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )

	   CHARACTER	 JOBVL, JOBVR

	   INTEGER	 INFO, LDA, LDB, LDVL, LDVR, LWORK, N

	   DOUBLE	 PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( *	),  B(
			 LDB,  *  ),  BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
			 WORK( * )

PURPOSE
       DGGEV computes for a pair of N-by-N real	 nonsymmetric  matrices	 (A,B)
       the generalized eigenvalues, and optionally, the left and/or right gen‐
       eralized eigenvectors.

       A generalized eigenvalue for a pair  of	matrices  (A,B)	 is  a	scalar
       lambda or a ratio alpha/beta = lambda, such that A - lambda*B is singu‐
       lar. It is usually represented as the pair (alpha,beta), as there is  a
       reasonable interpretation for beta=0, and even for both being zero.

       The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of
       (A,B) satisfies

			A * v(j) = lambda(j) * B * v(j).

       The left eigenvector u(j) corresponding to the eigenvalue lambda(j)  of
       (A,B) satisfies

			u(j)**H * A  = lambda(j) * u(j)**H * B .

       where u(j)**H is the conjugate-transpose of u(j).

ARGUMENTS
       JOBVL   (input) CHARACTER*1
	       = 'N':  do not compute the left generalized eigenvectors;
	       = 'V':  compute the left generalized eigenvectors.

       JOBVR   (input) CHARACTER*1
	       = 'N':  do not compute the right generalized eigenvectors;
	       = 'V':  compute the right generalized eigenvectors.

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

       A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
	       On  entry, the matrix A in the pair (A,B).  On exit, A has been
	       overwritten.

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

       B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
	       On entry, the matrix B in the pair (A,B).  On exit, B has  been
	       overwritten.

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

       ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
	       ALPHAI	(output)  DOUBLE  PRECISION  array, dimension (N) BETA
	       (output)	 DOUBLE	 PRECISION  array,  dimension  (N)  On	 exit,
	       (ALPHAR(j)  + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the gen‐
	       eralized eigenvalues.  If ALPHAI(j) is zero, then the j-th  ei‐
	       genvalue	 is  real; if positive, then the j-th and (j+1)-st ei‐
	       genvalues are a complex conjugate pair, with ALPHAI(j+1)	 nega‐
	       tive.

	       Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI will be always less than and usually
	       comparable with norm(A) in magnitude, and BETA always less than
	       and usually comparable with norm(B).

       VL      (output) DOUBLE PRECISION array, dimension (LDVL,N)
	       If JOBVL = 'V', the left eigenvectors u(j) are stored one after
	       another in the columns of VL, in the same order as their eigen‐
	       values. If the j-th eigenvalue is real, then  u(j)  =  VL(:,j),
	       the  j-th  column  of  VL. If the j-th and (j+1)-th eigenvalues
	       form a complex conjugate pair, then u(j) =  VL(:,j)+i*VL(:,j+1)
	       and  u(j+1)  =  VL(:,j)-i*VL(:,j+1).   Each eigenvector will be
	       scaled so the largest component have  abs(real  part)+abs(imag.
	       part)=1.	 Not referenced if JOBVL = 'N'.

       LDVL    (input) INTEGER
	       The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL
	       = 'V', LDVL >= N.

       VR      (output) DOUBLE PRECISION array, dimension (LDVR,N)
	       If JOBVR = 'V', the right  eigenvectors	v(j)  are  stored  one
	       after  another in the columns of VR, in the same order as their
	       eigenvalues. If the  j-th  eigenvalue  is  real,	 then  v(j)  =
	       VR(:,j), the j-th column of VR. If the j-th and (j+1)-th eigen‐
	       values  form  a	 complex   conjugate   pair,   then   v(j)   =
	       VR(:,j)+i*VR(:,j+1)  and	 v(j+1)	 =  VR(:,j)-i*VR(:,j+1).  Each
	       eigenvector will	 be  scaled  so	 the  largest  component  have
	       abs(real	 part)+abs(imag.  part)=1.   Not referenced if JOBVR =
	       'N'.

       LDVR    (input) INTEGER
	       The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR
	       = 'V', LDVR >= N.

       WORK    (workspace/output) DOUBLE PRECISION 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(1,8*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.

       INFO    (output) INTEGER
	       = 0:  successful exit
	       < 0:  if INFO = -i, the i-th argument had an illegal value.
	       =  1,...,N: The QZ iteration failed.  No eigenvectors have been
	       calculated, but ALPHAR(j), ALPHAI(j),  and  BETA(j)  should  be
	       correct	for  j=INFO+1,...,N.  > N:  =N+1: other than QZ itera‐
	       tion failed in DHGEQZ.
	       =N+2: error return from DTGEVC.

LAPACK version 3.0		 15 June 2000			      DGGEV(l)
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