IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v193y2022icp317-330.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475421003839
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2021.10.019?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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. 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.
    3. 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.
    4. 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.
    5. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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.
    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. Yanlin Tao & Kalyanasundaram Madhu, 2019. "Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniques and Their Basins of Attraction and Its Application," Mathematics, MDPI, vol. 7(4), pages 1-22, March.
    4. Prem B. Chand & Francisco I. Chicharro & Neus Garrido & Pankaj Jain, 2019. "Design and Complex Dynamics of Potra–Pták-Type Optimal Methods for Solving Nonlinear Equations and Its Applications," Mathematics, MDPI, vol. 7(10), pages 1-21, October.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:matcom:v:193:y:2022:i:c:p:317-330. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.