ZGGEV(1) LAPACK driver routine (version 3.2) ZGGEV(1)NAME
ZGGEV - computes for a pair of N-by-N complex nonsymmetric matrices
(A,B), the generalized eigenvalues, and optionally, the left and/or
right generalized eigenvectors
SYNOPSIS
SUBROUTINE ZGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL,
LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VL(
LDVL, * ), VR( LDVR, * ), WORK( * )
PURPOSE
ZGGEV computes for a pair of N-by-N complex nonsymmetric matrices
(A,B), the generalized eigenvalues, and optionally, the left and/or
right generalized 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 singular. 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 generalized eigenvector v(j) corresponding to the generalized
eigenvalue lambda(j) of (A,B) satisfies
A * v(j) = lambda(j) * B * v(j).
The left generalized eigenvector u(j) corresponding to the generalized
eigenvalues 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) COMPLEX*16 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) COMPLEX*16 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).
ALPHA (output) COMPLEX*16 array, dimension (N)
BETA (output) COMPLEX*16 array, dimension (N) On exit,
ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenval‐
ues. Note: the quotients ALPHA(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, ALPHA
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) COMPLEX*16 array, dimension (LDVL,N)
If JOBVL = 'V', the left generalized eigenvectors u(j) are
stored one after another in the columns of VL, in the same
order as their eigenvalues. Each eigenvector is scaled so the
largest component has 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) COMPLEX*16 array, dimension (LDVR,N)
If JOBVR = 'V', the right generalized eigenvectors v(j) are
stored one after another in the columns of VR, in the same
order as their eigenvalues. Each eigenvector is scaled so the
largest component has 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) COMPLEX*16 array, dimension (MAX(1,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 calcu‐
lates 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/output) DOUBLE PRECISION array, dimension (8*N)
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 ALPHA(j) and BETA(j) should be correct for
j=INFO+1,...,N. > N: =N+1: other then QZ iteration failed in
DHGEQZ,
=N+2: error return from DTGEVC.
LAPACK driver routine (version 3November 2008 ZGGEV(1)