ZGGEV - compute for a pair of N-by-N complex nonsymmetric
matrices (A,B), the generalized eigenvalues, and option
ally, 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 option
ally, 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 eigen
vectors;
= 'V': compute the left generalized eigenvectors.
JOBVR (input) CHARACTER*1
= 'N': do not compute the right generalized
eigenvectors;
= 'V': compute the right generalized eigenvec
tors.
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 eigenvalues.
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 will be scaled so the largest
component will have 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 will be scaled so the largest
component will have abs(real part) + abs(imag.
part) = 1. Not referenced if JOBVR = 'N'.
The leading dimension of the matrix VR. LDVR >= 1,
and 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 gen
erally 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/output) DOUBLE PRECISION array,
dimension (8*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille
gal value.
=1,...,N: The QZ iteration failed. No eigenvec
tors 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.
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