Fast solution of large complex non-Hermitian sparse eigenvalue problems

Dielectric waveguide formulations usually result in generalized eigenvalue problems which are non-Hermitian in nature. We solve this class of problems by a two stage process. In the first stage, a standard lumped eigenvalue problem is solved for a few eigenpairs of interest (usually the guided modes) by Arnoldi´s method accelerated by Chebyshev polynomials. The solution is then refined for the actual unlumped generalised eigenvalue problem by using inflated inverse iteration. After the eigenvalues and eigenvectors have been calculated at the first point (preferably the highest frequency), the eigenpairs for the rest of the frequencies of interest are calculated for progressively smaller frequencies by inflated inverse iteration alone, using the eigenvectors of the last step as the initial guess. To demonstrate the efficacy of this method, we use a variational formulation for anisotropic, dielectric waveguides based only on the E/sub s/ components or only on the H/sub s/ components of the electromagnetic fields that was presented by Chew and Nasir (1989). In this formulation it was shown that due of the imposition of the divergence condition the spurious waveguide modes were eliminated.