Naseem, AmirGdawiec, KrzysztofQureshi, SaniaArgyros, Ioannis K.Rehman, Muhammad Aziz UrSoomro, AmanullahPadder, Ausifur Rehman, Muhammad Aziz2025-10-152025-10-1520260885-064X1090-270810.1016/j.jco.2025.1019942-s2.0-105016993225https://doi.org/10.1016/j.jco.2025.101994This study introduces an optimal fourth-order iterative method derived by combining two established methods, resulting in enhanced convergence when solving nonlinear equations. Through rigorous convergence analysis using both Taylor expansion and the Banach space framework, the fourth-order optimality condition is verified. We demonstrate the superior efficiency and stability of this new method compared to traditional alternatives. Numerical experiments confirm its effectiveness, showing a reduction in the average number of iterations and computational time. Visual analysis with polynomiographs confirms the method's robustness, focusing on convergence area index, iteration count, computational time, fractal dimension, and Wada measure of basins. These findings underscore the potential of this optimal method for tackling complex nonlinear problems in various scientific and engineering fields. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC license (http:// creativecommons.org/licenses/by-nc/4.0/).eninfo:eu-repo/semantics/closedAccessNonlinear EquationsLocal and Semi-Local Convergence AnalysisEfficiency IndexKung-Traub ConjectureNumerical MeasuresPolynomiographyArticle