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A Two-Dimensional Variant of Newton’s Method and a Three-Point Hermite Interpolation: Fourth- and Eighth-Order Optimal Iterative Schemes

Author

Listed:
  • Chein-Shan Liu

    (Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan)

  • Essam R. El-Zahar

    (Department of Mathematics, College of Sciences and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Alkharj 11942, Saudi Arabia
    Department of Basic Engineering Science, Faculty of Engineering, Menofia University, Shebin El-Kom 32511, Egypt)

  • Chih-Wen Chang

    (Department of Mechanical Engineering, National United University, Miaoli 36063, Taiwan)

Abstract

A nonlinear equation f ( x ) = 0 is mathematically transformed to a coupled system of quasi-linear equations in the two-dimensional space. Then, a linearized approximation renders a fractional iterative scheme x n + 1 = x n − f ( x n ) / [ a + b f ( x n ) ] , which requires one evaluation of the given function per iteration. A local convergence analysis is adopted to determine the optimal values of a and b . Moreover, upon combining the fractional iterative scheme to the generalized quadrature methods, the fourth-order optimal iterative schemes are derived. The finite differences based on three data are used to estimate the optimal values of a and b . We recast the Newton iterative method to two types of derivative-free iterative schemes by using the finite difference technique. A three-point generalized Hermite interpolation technique is developed, which includes the weight functions with certain constraints. Inserting the derived interpolation formulas into the triple Newton method, the eighth-order optimal iterative schemes are constructed, of which four evaluations of functions per iteration are required.

Suggested Citation

  • Chein-Shan Liu & Essam R. El-Zahar & Chih-Wen Chang, 2023. "A Two-Dimensional Variant of Newton’s Method and a Three-Point Hermite Interpolation: Fourth- and Eighth-Order Optimal Iterative Schemes," Mathematics, MDPI, vol. 11(21), pages 1-21, November.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:21:p:4529-:d:1273502
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    References listed on IDEAS

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    1. F. Soleymani & S. Shateyi & H. Salmani, 2012. "Computing Simple Roots by an Optimal Sixteenth-Order Class," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-13, November.
    2. Moin-ud-Din Junjua & Fiza Zafar & Nusrat Yasmin, 2019. "Optimal Derivative-Free Root Finding Methods Based on Inverse Interpolation," Mathematics, MDPI, vol. 7(2), pages 1-10, February.
    3. Faisal Ali & Waqas Aslam & Kashif Ali & Muhammad Adnan Anwar & Akbar Nadeem, 2018. "New Family of Iterative Methods for Solving Nonlinear Models," Discrete Dynamics in Nature and Society, Hindawi, vol. 2018, pages 1-12, April.
    4. Liu, Dongjie & Liu, Chein-Shan, 2022. "Two-point generalized Hermite interpolation: Double-weight function and functional recursion methods for solving nonlinear equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 317-330.
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