New Family of Two-Step Iterative Methods Based on Heron Mean for Solving Nonlinear Equations

Noori Yasir  Abdul-Hassan; Ali Hasan Ali; Firas  Ghanim

doi:10.24996/ijs.2026.67.8.%g

Authors

Noori Yasir Abdul-Hassan Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah, Iraq
Ali Hasan Ali Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah, Iraq https://orcid.org/0000-0003-2959-4212
Firas Ghanim Department of Mathematics, College of Sciences, University of Sharjah, Sharjah, United Arab Emirates

DOI:

https://doi.org/10.24996/ijs.2026.67.8.%25g

Keywords:

Nonlinear equations, Newton-type method, Heron mean, Order of convergence, Efficiency index

Abstract

In this paper, we propose and analyze a new one-parameter family of two-step iterative methods without requiring higher derivatives for solving nonlinear equations. This family is constructed based on a Newton-type method utilizing the Heron mean, a predictor-corrector technique, and an appropriate choice of derivative approximations. Each iteration of the new methods requires two function evaluations and one first derivative evaluation. Convergence analysis shows that the proposed methods achieve third- and fourth-order convergence, with corresponding efficiency indices of 1.44225 and 1.58740, respectively. Numerical examples are provided to demonstrate the performance and efficiency of the proposed methods.