A note on a new cubically convergent one-parameter root solver

TL;DR

This paper introduces a new family of one-parameter iterative methods with cubic convergence for solving nonlinear equations, including variants for simple and multiple zeros, and demonstrates their effectiveness through numerical examples.

Contribution

It develops a novel one-parameter family of third-order methods, encompassing existing methods like Halley's as special cases, with proven convergence properties.

Findings

Methods achieve cubic convergence for simple and multiple zeros.

Numerical examples confirm the convergence and effectiveness of the proposed methods.

The family includes a variety of third-order methods, broadening existing solution techniques.

Abstract

A new one-parameter family of iterative method for solving nonlinear equations is constructed and studied. Two variants, both with cubic convergence, are developed, one for finding simple zeros and other for multiple zeros of known multiplicities. This family generates a variety of different third order methods, including Halley-like method as a special case. Four numerical examples are given to demonstrate convergence properties of the proposed methods for multiple zeros and various values of the parameter.