Methods for large sparse eigenvalue problems from waveguide analysis

We discuss several techniques for finding leading eigenvalues and eigenvectors for large sparse matrices. The techniques are demonstrated on a scalar Helmholtz equation derived from a model semiconductor rib waveguide problem. We compare the simple inverse iteration approach with more sophisticated methods, including minimum degree reordering, Arnoldi and Lanczos methods. We then propose a new Arnoldi method designed particularly for the constrained generalized eigenvalue problem, a formulation arising naturally from the scalar waveguide problem.