How to get high order without loosing efficiency for the resolution of systems of nonlinear equations: A short review of Shamanskii's m method
For researchers developing new iterative solvers, this paper serves as a reminder of an existing efficient method that should be used for comparison, though it is a review without new results.
This review highlights Shamanskii's m method for solving nonlinear systems, which achieves order m+1 convergence via Newton's method, and advocates for its use as a baseline in future research.
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.