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

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.