DGEEV - compute for an N-by-N real nonsymmetric matrix A,
the eigenvalues 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 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 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 overwritten.
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
parts, respectively, of the computed eigenvalues.
Complex conjugate pairs of eigenvalues appear con
secutively 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 eigenvalues. If JOBVL =
'N', VL is not referenced. 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)-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 conjugate 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 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.
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille
gal value.
> 0: if INFO = i, the QR algorithm failed to com
pute all the eigenvalues, and no eigenvectors have
been computed; elements i+1:N of WR and WI contain
eigenvalues which have converged.
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