Numerical Example

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 $A$ is non-symmetric and the matrix $B$ is positive definite. Of special interest are the generalized eigenvalues $(\alpha,\beta)$ with positive real part (i.e., $\re(\alpha/\beta)\geq 0$) and their corresponding eigenvectors.

For this problem, the parameters were set to $\tau=2750.0$, $k_{\max}=5$, and $\epsilon=10^{-9}$. In the first few steps, until the size of the first residual was smaller than $10^{-6}$, we replaced $(\zeta,\eta)$ in the correction equation by $(1,\tau)$ (as explained in note (36)).

The computed generalized eigenvalues, represented as $\alpha/\beta$, are given in Table 8.1. With Algorithm 8.1 we discovered all four positive generalized eigenvalues.

Table 8.1: Five generalized eigenvalues of BFW$782$, computed by Jacobi-Davidson QZ

$-1.1373e+03$

$5.6467e+02$

$1.2634e+03$

$2.4843e+03$

$2.5233e+03$

Figure 8.1: Convergence history for BFW$782$.

Table 8.2: Summary of results for BFW$782$.

Method for the correction equation

JD iterations

MVs

$\mbox{flops}\times 10^6$

GMRES$_{1}$

143

143

$67$

GMRES$_{10}$

37

233

$31.7$

Bi-CGSTAB(2)

32

429

$38.8$

The convergence history is plotted in Figure 8.1. We solved the correction equation (1) by simply taking $t$ as $-\hat r$, denoted by GMRES$_{1}$; (2) with full GMRES [389] with a maximum of 10 steps, denoted by GMRES$_{10}$, 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 ($K=I$). As stopping criterion for the iterative methods for the correction equation, we used a residual reduction of $2^{-\ell}$ in the $\ell$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$_{10}$ 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.