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DGEEV(l)			       )			      DGEEV(l)

NAME
       DGEEV - compute for an N-by-N real nonsymmetric matrix A, the eigenval‐
       ues and, optionally, the left and/or right eigenvectors

SYNOPSIS
       SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR,	 LDVR,
			 WORK, LWORK, INFO )

	   CHARACTER	 JOBVL, JOBVR

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

	   DOUBLE	 PRECISION  A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
			 WI( * ), WORK( * ), WR( * )

PURPOSE
       DGEEV computes for an N-by-N real nonsymmetric matrix A, the  eigenval‐
       ues  and,  optionally,  the  left and/or right eigenvectors.  The right
       eigenvector v(j) of A satisfies
			A * v(j) = lambda(j) * v(j)
       where lambda(j) is its eigenvalue.
       The left eigenvector u(j) of A satisfies
		     u(j)**H * A = lambda(j) * u(j)**H
       where u(j)**H denotes the conjugate transpose of u(j).

       The computed eigenvectors are normalized to have Euclidean  norm	 equal
       to 1 and largest component real.

ARGUMENTS
       JOBVL   (input) CHARACTER*1
	       = 'N': left eigenvectors of A are not computed;
	       = 'V': left eigenvectors of A are computed.

       JOBVR   (input) CHARACTER*1
	       = 'N': right eigenvectors of A are not computed;
	       = 'V': right eigenvectors of A are computed.

       N       (input) INTEGER
	       The order of the matrix A. N >= 0.

       A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
	       On  entry,  the N-by-N matrix A.	 On exit, A has been overwrit‐
	       ten.

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

       WR      (output) DOUBLE PRECISION array, dimension (N)
	       WI      (output) DOUBLE PRECISION array, dimension (N)  WR  and
	       WI  contain  the real and imaginary parts, respectively, of the
	       computed eigenvalues.  Complex conjugate pairs  of  eigenvalues
	       appear  consecutively  with  the eigenvalue having the positive
	       imaginary part first.

       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 JOBVL = 'N', VL is not referenced.  If the j-th ei‐
	       genvalue	 is  real, then u(j) = VL(:,j), the j-th column of VL.
	       If the j-th and (j+1)-st 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).

       LDVL    (input) INTEGER
	       The  leading  dimension of the array VL.	 LDVL >= 1; 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 JOBVR = 'N', VR is not referenced.  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)-st eigenvalues form a complex conju‐
	       gate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
	       v(j+1) = VR(:,j) - i*VR(:,j+1).

       LDVR    (input) INTEGER
	       The  leading  dimension of the array VR.	 LDVR >= 1; 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(1,3*N),  and  if
	       JOBVL  =	 'V'  or  JOBVR = 'V', LWORK >= 4*N.  For good perfor‐
	       mance, 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.
	       >  0:   if INFO = i, the QR algorithm failed to compute all the
	       eigenvalues, and no eigenvectors have been  computed;  elements
	       i+1:N of WR and WI contain eigenvalues which have converged.

LAPACK version 3.0		 15 June 2000			      DGEEV(l)
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