Some Classes of third and Fourth-order iterative methods for solving nonlinear equations

TL;DR

This paper introduces new classes of third and fourth-order iterative methods for solving nonlinear equations, improving convergence efficiency and unifying existing methods through novel weighting strategies.

Contribution

The paper presents a unified framework for third and fourth-order methods, using a single weight function to enhance convergence and include existing methods as special cases.

Findings

The new methods demonstrate faster convergence in examples.

They encompass existing methods as special cases.

Performance comparisons show improved efficiency.

Abstract

The object of the present work is to present the new classes of third-order and fourth-order iterative methods for solving nonlinear equations. Our third-order method includes methods of Weerakoon \cite{Weerakoon}, Homeier \cite{Homeier2}, Chun \cite{Chun} e.t.c. as particular cases. After that we make this third-order method to fourth-order (optimal) by using a single weight function rather than using two different weight functions in \cite{Soleymani}. Finally some examples are given to illustrate the performance of the our method by comparing with new existing third and fourth-order methods.