We present the results for a small example that can be easily repeated. We took the example from the collection of test matrices in [28].
We consider the bounded fineline dielectric waveguide generalized
eigenproblem BFW782 [28] of order 782. This problem stems from
a finite element discretization of the Maxwell equation for
propagating modes and magnetic field profiles of a rectangular
waveguide filled with dielectric and PEC structures. The resulting
matrix is non-symmetric and the matrix
is positive definite.
Of special interest are the generalized eigenvalues
with positive real part
(i.e.,
) and
their corresponding eigenvectors.
For this problem, the parameters were set to ,
, and
. In the first few
steps, until the size of the first residual was smaller than
, we replaced
in the correction equation
by
(as explained in note (36)).
The computed generalized eigenvalues, represented as ,
are given in Table 8.1. With Algorithm 8.1
we discovered all four positive generalized eigenvalues.
The convergence history is plotted in
Figure 8.1. We solved the correction
equation (1) by simply taking as
, denoted
by GMRES
;
(2) with
full GMRES [389] with a maximum of 10 steps, denoted
by GMRES
, and (3) with
Bi-CGSTAB(2) [409] with a maximum of 100 matrix
multiplications (Bi-CGSTAB refers to biconjugate gradient stabilized). We did not use preconditioning (
).
As stopping criterion for the iterative methods for
the correction equation, we used a residual reduction of
in the
th Jacobi-Davidson iteration
or on the maximum number of iterations permitted.
A summary of the results is given in
Table 8.2. We see that the Jacobi-Davidson QZ
method converges quite nicely for
GMRES
and Bi-CGSTAB(2). It should be noted that although it
seems that with Bi-CGSTAB(2) only
four generalized eigenvalues are computed,
in fact five
generalized eigenvalues are computed:
the two rightmost generalized eigenvalues, which are
relatively close, are found in the same Jacobi-Davidson iteration.