Design and analysis of a new iterative family for solving nonlinear equations

Abstract

This article addresses the problem of improving the convergence order and stability of iterative methods for solving nonlinear equations. The main objective is to design a new multipoint iterative family with sixth-order convergence and to analyze both its convergence behavior and complex dynamics. The methodology combines the theoretical analysis of the convergence order, the derivation of the associated rational operator, and the use of complex dynamics tools such as stability surfaces, parameter planes, and dynamical planes. Numerical experiments conducted on nonlinear test equations confirm the results obtained from the convergence and stability analysis. The proposed method achieves high accuracy in few iterations, maintaining the Approximate Computational Order of Convergence (ACOC) around six and exhibiting competitive efficiency compared to classical methods such as Newton, Ostrowski, Jarratt, and CMT. The conclusions highlight the robustness of the family with respect to initial conditions. The findings have theoretical implications for the design of high-order iterative methods and practical implications for solving scientific and engineering problems more efficiently.