Methods for Large Sparse Eigenvalue Problems from Waveguide Analysis

We discuss several techniques for nding 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. In the waveguide analysis (see e.g. 9]), the analysis of the propagation of the electric and magnetic elds in waveguides, based on the use of Maxwell's equations, often lead to scalar Helmholtz equations of the general form r 2 T E x + k 2 E x = 2 E x , where E x denotes the x component of the electric eld, k is the dielectric constant, is the unknown propagation constant, and r 2 T denotes the Laplacian operator in the transverse (x; y) coordinates (z being the longitudinal coordinate). When this Helmholtz equation is solved for the x coordinate only, one must impose internal continuity conditions across the boundaries between materials of diierent dielectric constants. The combined equations are then discretized by nite diierences yielding a large sparse ordinary matrix eigenvalue problem Ax = x 9]. Discretizing the vector elds equations using nite element formulations leads to a large sparse generalized eigenvalue problem 2]: Ax = Bx (1) where the matrix A nn is real and non-singular, B nn is real, symmetric, and positive deenite. Solving problem (1) constitutes the largest part of the computational eeort. In this paper we are discussing several iterative algorithms that can eeciently deal with problem (1) of several thousands in size on workstations. When B is banded, as is the case in many problem formulations from waveguide analysis, the Cholesky decomposition B = LL T problem (1) can be used to reduce (1) to an ordinary eigenvalue problem: (2) where L is lower triangular and y = L T x. In an iterative method, L ?1 AL ?T dose not need to be explicitly formed. This is possible because most iterative methods only need to form matrix-vector products. When B is not banded but its Cholesky decomposition gives a sparse L, the reduction is still useful. But if L is not sparse, then solving problems such as (1) becomes signiicantly more diicult and expensive. Since waveguide analysis …