Numerically Computing Zeros of the Evans Function

This paper presents a method of numerically computing zeros of an analytic function for the specific application of computing eigenvalues of the Sturm-Liouville problem. The Sturm-Liouville problem is an infinite dimensional eigenvalue problem that often arises in solving partial differential equations, including the heat and wave equations. To compute eigenvalues of the Sturm-Liouville problem, we construct the Evans function, whose zeros correspond to eigenvalues of the Sturm-Liouville problem. Our method requires defining a contour integral based on an rough approximation of the zero. To apply this method to find zeros of the Evans function, we make rough approximates of zeros by a finite difference calculation for eigenvalues of the Sturm-Liouville problem. For cases where the exact zeros are known, we do a comparison to find that the numerical method in this paper has an error as small as O(10−16).