Root Finding by High Order Iterative Methods Based on Quadratures

TL;DR

This paper introduces a recursive family of high-order iterative methods for root finding, based on Newton-Cotes quadrature rules, achieving convergence orders of at least n+2 for quadrature rules with n+1 nodes.

Contribution

The paper presents a novel recursive framework for root-finding methods that leverage quadrature rules to attain higher convergence orders than traditional methods.

Findings

Methods achieve convergence order at least n+2

For n=0, the method reduces to Newton's method

Higher-order methods improve root approximation efficiency

Abstract

We discuss a recursive family of iterative methods for the numerical approximation of roots of nonlinear functions in one variable. These methods are based on Newton-Cotes closed quadrature rules. We prove that when a quadrature rule with $n+1$ nodes is used the resulting iterative method has convergence order at least $n+2$, starting with the case $n=0$ (which corresponds to the Newton's method).