A modified two-step optimal iterative method for solving nonlinear models in one and higher dimensions

Chih-Wen Chang, Sania Qureshi, Ioannis K. Argyros, Francisco I. Chicharro and Amanullah Soomro

Mathematics and Computers in Simulation (MATCOM), 2025, vol. 229, issue C, 448-467

Abstract: Iterative methods are essential tools in computational science, particularly for addressing nonlinear models. This study introduces a novel two-step optimal iterative root-finding method designed to solve nonlinear equations and systems of nonlinear equations. The proposed method exhibits the optimal convergence, adhering to the Kung-Traub conjecture, and necessitates only three function evaluations per iteration to achieve a fourth-order optimal iterative process. The development of this method involves the amalgamation of two well-established third-order iterative techniques. Comprehensive local and semilocal convergence analyses are conducted, accompanied by a stability investigation of the proposed approach. This method marks a substantial enhancement over existing optimal iterative methods, as evidenced by its performance in various nonlinear models. Extensive testing demonstrates that the proposed method consistently yields accurate and efficient results, surpassing existing algorithms in both speed and accuracy. Numerical simulations, including real-world models such as boundary value problems and integral equations, indicate that the proposed optimal method outperforms several contemporary optimal iterative techniques.

Keywords: Root-finding methods; Nonlinear models; Iterative algorithms; Efficiency index; Computational order of convergence; Computational cost (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:229:y:2025:i:c:p:448-467

DOI: 10.1016/j.matcom.2024.09.021

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