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

Author

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

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.

Suggested Citation

  • Chang, Chih-Wen & Qureshi, Sania & Argyros, Ioannis K. & Chicharro, Francisco I. & Soomro, Amanullah, 2025. "A modified two-step optimal iterative method for solving nonlinear models in one and higher dimensions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 229(C), pages 448-467.
    • Handle: RePEc:eee:matcom:v:229:y:2025:i:c:p:448-467
    DOI: 10.1016/j.matcom.2024.09.021
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    References listed on IDEAS

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    1. Ramandeep Behl & Ioannis K. Argyros & Hashim Alshehri & Samundra Regmi, 2024. "Generalized Convergence for Multi-Step Schemes under Weak Conditions," Mathematics, MDPI, vol. 12(2), pages 1-15, January.
    2. Sharma, Janak Raj & Sharma, Rajni & Kalra, Nitin, 2015. "A novel family of composite Newton–Traub methods for solving systems of nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 520-535.
    3. Ioannis K. Argyros, 2021. "Unified Convergence Criteria for Iterative Banach Space Valued Methods with Applications," Mathematics, MDPI, vol. 9(16), pages 1-15, August.
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