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.