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

SUBROUTINE CGEEV(

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

CHARACTER JOBVL, JOBVR

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

REAL RWORK( * )

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

CGEEV 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.

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 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 array, dimension (N)

W contains the computed eigenvalues.

VL (output) COMPLEX 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 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 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 (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.