Observations on greedy composite Newton methods

Abstract

The only robust general-purpose numerical methods for approximating the solution to systems of nonlinear algebraic equations (NAEs) are based on Newton's method. Many variants of Newton's method exist in order to take advantage of problem structure; it is often computationally infeasible to solve a given problem without taking some advantage of this structure. It is generally impossible to know a priori which variant of Newton's method will be optimal for a given problem. In this paper, we describe an algorithm for automatically selecting a composite Newton method, i.e., a sequential combination of Newton variants, for solving NAEs. The algorithm is based on a greedy principle that updates the current state at regular intervals according to the best performing Newton variant. Preliminary results show that it is possible for composite Newton methods to outperform optimal classical implementations of Newton's method, i.e., ones that only use one Newton variant on a given problem.