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Optimal Derivative-Free Root Finding Methods Based on Inverse Interpolation

Author

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  • Moin-ud-Din Junjua

    (Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan)

  • Fiza Zafar

    (Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan)

  • Nusrat Yasmin

    (Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan)

Abstract

Finding a simple root for a nonlinear equation f ( x ) = 0 , f : I ⊆ R → R has always been of much interest due to its wide applications in many fields of science and engineering. Newton’s method is usually applied to solve this kind of problems. In this paper, for such problems, we present a family of optimal derivative-free root finding methods of arbitrary high order based on inverse interpolation and modify it by using a transformation of first order derivative. Convergence analysis of the modified methods confirms that the optimal order of convergence is preserved according to the Kung-Traub conjecture. To examine the effectiveness and significance of the newly developed methods numerically, several nonlinear equations including the van der Waals equation are tested.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:2:p:164-:d:205163
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    References listed on IDEAS

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    1. Sharifi, Somayeh & Salimi, Mehdi & Siegmund, Stefan & Lotfi, Taher, 2016. "A new class of optimal four-point methods with convergence order 16 for solving nonlinear equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 119(C), pages 69-90.
    2. Behl, Ramandeep & Argyros, Ioannis K. & Motsa, S.S., 2016. "A new highly efficient and optimal family of eighth-order methods for solving nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 282(C), pages 175-186.
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    Cited by:

    1. 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.
    2. Syahmi Afandi Sariman & Ishak Hashim & Faieza Samat & Mohammed Alshbool, 2021. "Modification of Newton-Househölder Method for Determining Multiple Roots of Unknown Multiplicity of Nonlinear Equations," Mathematics, MDPI, vol. 9(9), pages 1-12, April.
    3. 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.

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