A novel Newton-Raphson style root finding algorithm

TL;DR

This paper introduces a new family of high-order root finding algorithms inspired by Newton-Raphson, designed to improve efficiency by reducing the number of function evaluations needed per iteration.

Contribution

The paper proposes a novel Newton-Raphson style method family that achieves high order convergence with fewer function evaluations, enhancing efficiency in root finding.

Findings

Achieves high order convergence with minimal function evaluations

Reduces computational cost compared to traditional methods

Improves efficiency in solving applied mathematics problems

Abstract

Many problems in applied mathematics require root finding algorithms. Unfortunately, root finding methods have limitations. Firstly, regarding the convergence, there is a trade-off between the size of it's domain and it's rate. Secondly the numerous evaluations of the function and its derivatives penalize the efficiency of high order methods. In this article, we present a family of high order methods, that require few functional evaluations ( One for each step plus one for each considered derivative at the start of the method), thus increasing the efficiency of the methods.