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Two-point generalized Hermite interpolation: Double-weight function and functional recursion methods for solving nonlinear equations

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  • Liu, Dongjie
  • Liu, Chein-Shan

Abstract

Based on the two-point Hermite interpolation technique, the paper proposes a two-point generalized Hermite interpolation and its inversion in terms of weight functions. We prove that upon combining fourth-order optimal iterative scheme to the double Newton’s method (DNM), we can yield a generalized Hermite interpolation formula to relate the first-order derivatives at two points, and the converse is also true. Resorted on the DNM and the derived formula for the generalized inverse Hermite interpolation, some new third-order iterative schemes of quadrature type are constructed. Then, the fourth-order optimal iterative schemes are devised by using a double-weight function. A functional recursion formula is developed which can generate a sequence of two-point generalized Hermite interpolations for any two given weight functions with certain constraints; hence, a more general class of fourth-order optimal iterative schemes is developed from the functional recursion formula. The constructions of fourth-order optimal iterative schemes by using the techniques of double-weight function and the recursion formula obtained from a single weight function are appeared in the literature at the first time. The novelties involve deriving a two-point generalized Hermite interpolation and its inversion in terms of weight functions subjected to two conditions and through the recursion formula, relating the DNM to the third-order iterative schemes by the inverse Hermite interpolation, formulating a functional recursion formula, deriving a broad class fourth-order optimal iterative schemes through double-weight functions rather than the previous technique with a single-weight function, and finding that the new double-weight function and the newly developed fourth-order optimal iterative schemes are inclusive being convergent faster and competitive to other iterative schemes.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:193:y:2022:i:c:p:317-330
    DOI: 10.1016/j.matcom.2021.10.019
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    References listed on IDEAS

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    1. Liu, Chein-Shan & El-Zahar, Essam R. & Chang, Chih-Wen, 2021. "Three novel fifth-order iterative schemes for solving nonlinear equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 282-293.
    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. Chein-Shan Liu & Tsung-Lin Lee & Xiaolong Qin, 2021. "A New Family of Fourth-Order Optimal Iterative Schemes and Remark on Kung and Traub’s Conjecture," Journal of Mathematics, Hindawi, vol. 2021, pages 1-9, February.
    4. 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.
    5. Rajni Sharma & Ashu Bahl, 2015. "An Optimal Fourth Order Iterative Method for Solving Nonlinear Equations and Its Dynamics," Journal of Complex Analysis, Hindawi, vol. 2015, pages 1-9, November.
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    Cited by:

    1. 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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