A New Family of Optimal Fourth-Order Iterative Methods for Solving Nonlinear Equations With Applications

Abstract

A new family of fourth-order iterative methods for solving nonlinear equations is proposed using the weight function procedure. This family is optimal in the sense of the Kung–Traub conjecture, as it requires three function evaluations per iteration. Due to its flexible structure, the new family offers a variety of options, demonstrating that it includes several well-known and recent methods as special cases. In particular, three new specific methods are designed to achieve better results compared to existing methods within the same family. Various nonlinear functions and engineering problems are used to illustrate the performance of these new specific methods, comparing them with existing ones. Furthermore, the analysis of complex dynamics and basins of attraction shows that the newly proposed methods yield the best results, with wider sets of initial points that lead to convergence.