How to get high order without loosing efficiency for the resolution of systems of nonlinear equations: A short review of Shamanskii's m method

TL;DR

This paper reviews Shamanskii's m method, an efficient iterative approach for solving nonlinear systems that achieves high order convergence without sacrificing computational efficiency, emphasizing its relevance and encouraging its use.

Contribution

It provides a comprehensive review of Shamanskii's m method, highlighting implementation strategies and advocating for its broader adoption in nonlinear system solutions.

Findings

Shamanskii's m method converges with order m+1.

Efficient implementation via matrix factorization is discussed.

The method remains underutilized in current research.

Abstract

We present relations between some recently proposed methods for the solution of a nonlinear system of equations. In particular, we review the Shamanskii's m method, that is an iterative method derived from Newton's method that converge with order m+1. We discuss efficient implementation of this method via matrix factorization and some relevant properties. We believe that recent developments in the research of solutions of systems of equations did not take sufficiently into account this method. The hope, with this paper, is to encourage the entire community to remember this simple method and use it for comparison when new methods are introduced. This work is dedicated to Prof. Elvira Russo: a very special teacher.