Analysis and application of the nonlinear power method

Abstract

The power method is a standard technique in numerical linear algebra for approximating the dominant eigenpair of a square matrix. Algorithms for implementing the method are simple and efficient, since only knowledge of the matrix action is required. In this thesis, we develop a nonlinear power method (NLPM) for computing the dominant eigenpair of the linearization of a nonlinear map. We implement the method in a matrix-free way by utilizing function differences, and also by reconstructing the entire matrix when matrix-free methods are unavailable. We apply the NLPM to the problems of determining the stability of a system of differential equations, and also to examining the issue of linearization error for computational error estimates for nonlinear elliptic two-point boundary value problems. We examine properties of the map between the forward solution and the linearized dual solution and use the NLPM in various forms to determine the norm of the derivative of this map, hence bounding the effect of linearization. Several numerical examples are presented in Chapter 5.