NAME

ZGEEV - compute for an N-by-N complex nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors

SYNOPSIS

SUBROUTINE ZGEEV(
JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )

CHARACTER JOBVL, JOBVR

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

DOUBLE PRECISION RWORK( ∗ )

COMPLEX∗16 A( LDA, ∗ ), VL( LDVL, ∗ ), VR( LDVR, ∗ ), W( ∗ ), WORK( ∗ )

PURPOSE

ZGEEV computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues 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 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) COMPLEX∗16 array, dimension (LDA,N)
On entry, the N-by-N matrix A. On exit, A has been overwritten.

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

W       (output) COMPLEX∗16 array, dimension (N)
W contains the computed eigenvalues.

VL      (output) COMPLEX∗16 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 eigenvalues. If JOBVL = ’N’, VL is not referenced. u(j) = VL(:,j), the j-th column of VL.

LDVL    (input) INTEGER
The leading dimension of the array VL.  LDVL >= 1; if JOBVL = ’V’, LDVL >= N.

VR      (output) COMPLEX∗16 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. v(j) = VR(:,j), the j-th column of VR.

LDVR    (input) INTEGER
The leading dimension of the array VR.  LDVR >= 1; if JOBVR = ’V’, LDVR >= 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.

RWORK   (workspace) DOUBLE PRECISION array, dimension (2∗N)

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 and i+1:N of W contain eigenvalues which have converged.

  —  LAPACK version 2.0  —  08 October 1994