A Methodology for Obtaining the Different Convergence Orders of Numerical Method under Weaker Conditions

A process for solving an algebraic equation was presented by Newton in 1669 and later by Raphson in 1690. This technique is called Newton's method or Newton-Raphson method and is even today a popular technique for solving nonlinear equations in abstract spaces. The objective of this article is to update developments in the convergence of this method. In particular, it is shown that the Kantorovich theory for solving nonlinear equations using Newton's method can be replaced by a finer one with no additional and even weaker conditions. Moreover, the convergence order two is proven under these conditions. Furthermore, the new ratio of convergence is at least as small. The same methodology can be used to extend the applicability of other numerical methods. Numerical experiments complement this study.

Automated summary

This article updates the convergence developments of Newton's method for solving nonlinear equations. It introduces a finer theory to replace the Kantorovich theory, requiring weaker conditions. The article also proves the convergence order is two under these conditions, and the new convergence ratio is at least as small. Numerical experiments are conducted to complement the study.