Quadrature based optimal iterative methods

TL;DR

This paper introduces a quadrature-based scheme for constructing optimal iterative methods that achieve high convergence orders, demonstrating efficiency through computational results.

Contribution

It develops a new quadrature-based scheme to create optimal iterative methods of orders four and eight, advancing the design of high-order root-finding algorithms.

Findings

Developed methods of orders four and eight

Methods outperform many well-known algorithms

Quadrature scheme can be extended to higher orders

Abstract

We present a simple yet powerful and applicable quadrature based scheme for constructing optimal iterative methods. According to the, still unproved, Kung-Traub conjecture an optimal iterative method based on $n+1$ evaluations could achieve a maximum convergence order of $2^n$. Through quadrature, we develop optimal iterative methods of orders four and eight. The scheme can be further applied to develop iterative methods of even higher order. Computational results demonstrate that the developed methods are efficient as compared with many well known methods.