How to increase convergence order of Newton's method to $2\times m$?

TL;DR

This paper introduces a straightforward technique to enhance Newton's method, enabling the creation of iterative algorithms with higher convergence orders up to ten, with easy implementation and demonstrated effectiveness on example problems.

Contribution

A simple modification technique for Newton's method that achieves higher convergence orders, up to ten, and can be integrated into existing software.

Findings

Methods of convergence orders 4, 6, 8, and 10 are constructed.

The technique is easy to implement in C++.

Proven effectiveness on example problems.

Abstract

We present a simple yet powerful technique for forming iterative methods of various convergence orders. Methods of various convergence orders (four, six, eight and ten) are formed through a modest modification of the classical Newton method. The technique can be easily implemented in existing software packages as suggested by the presented C$^{++}$ algorithm. Finally some problems are solved through the proposed algorithm.