IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i9p2036-d1132461.html
   My bibliography  Save this article

Novel Parametric Families of with and without Memory Iterative Methods for Multiple Roots of Nonlinear Equations

Author

Listed:
  • G Thangkhenpau

    (Department of Mathematics, National Institute of Technology Manipur, Langol, Imphal 795004, Manipur, India)

  • Sunil Panday

    (Department of Mathematics, National Institute of Technology Manipur, Langol, Imphal 795004, Manipur, India)

  • Shubham Kumar Mittal

    (Department of Mathematics, National Institute of Technology Manipur, Langol, Imphal 795004, Manipur, India)

  • Lorentz Jäntschi

    (Department of Physics and Chemistry, Technical University of Cluj-Napoca, B.-dul Muncii nr. 103-105, 400641 Cluj-Napoca, Romania)

Abstract

The methods that use memory using accelerating parameters for computing multiple roots are almost non-existent in the literature. Furthermore, the only paper available in this direction showed an increase in the order of convergence of 0.5 from the without memory to the with memory extension. In this paper, we introduce a new fifth-order without memory method, which we subsequently extend to two higher-order with memory methods using a self-accelerating parameter. The proposed with memory methods extension demonstrate a significant improvement in the order of convergence from 5 to 7, making this the first paper to achieve at least a 2-order improvement. In addition to this improvement, our paper is also the first to use Hermite interpolating polynomials to approximate the accelerating parameter in the proposed with memory methods for multiple roots. We also provide rigorous theoretical proofs of convergence theorems to establish the order of the proposed methods. Finally, we demonstrate the potential impact of the proposed methods through numerical experimentation on a diverse range of problems. Overall, we believe that our proposed methods have significant potential for various applications in science and engineering.

Suggested Citation

  • G Thangkhenpau & Sunil Panday & Shubham Kumar Mittal & Lorentz Jäntschi, 2023. "Novel Parametric Families of with and without Memory Iterative Methods for Multiple Roots of Nonlinear Equations," Mathematics, MDPI, vol. 11(9), pages 1-18, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:9:p:2036-:d:1132461
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/9/2036/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/9/2036/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Geum, Young Hee & Kim, Young Ik & Neta, Beny, 2015. "A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 387-400.
    2. Ramandeep Behl, 2022. "A Derivative Free Fourth-Order Optimal Scheme for Applied Science Problems," Mathematics, MDPI, vol. 10(9), pages 1-17, April.
    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 & Chih-Wen Chang, 2024. "Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations," Mathematics, MDPI, vol. 12(7), pages 1-21, March.
    2. Ekta Sharma & Sunil Panday & Shubham Kumar Mittal & Dan-Marian Joița & Lavinia Lorena Pruteanu & Lorentz Jäntschi, 2023. "Derivative-Free Families of With- and Without-Memory Iterative Methods for Solving Nonlinear Equations and Their Engineering Applications," Mathematics, MDPI, vol. 11(21), pages 1-13, November.
    3. Chein-Shan Liu & Chih-Wen Chang, 2024. "New Memory-Updating Methods in Two-Step Newton’s Variants for Solving Nonlinear Equations with High Efficiency Index," Mathematics, MDPI, vol. 12(4), pages 1-22, February.

    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. Geum, Young Hee & Kim, Young Ik & Neta, Beny, 2016. "A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points," Applied Mathematics and Computation, Elsevier, vol. 283(C), pages 120-140.
    2. Fiza Zafar & Alicia Cordero & Juan R. Torregrosa, 2018. "An Efficient Family of Optimal Eighth-Order Multiple Root Finders," Mathematics, MDPI, vol. 6(12), pages 1-16, December.
    3. Janak Raj Sharma & Sunil Kumar & Lorentz Jäntschi, 2020. "On Derivative Free Multiple-Root Finders with Optimal Fourth Order Convergence," Mathematics, MDPI, vol. 8(7), pages 1-15, July.
    4. Saima Akram & Fiza Zafar & Nusrat Yasmin, 2019. "An Optimal Eighth-Order Family of Iterative Methods for Multiple Roots," Mathematics, MDPI, vol. 7(8), pages 1-14, July.
    5. Himani Arora & Alicia Cordero & Juan R. Torregrosa & Ramandeep Behl & Sattam Alharbi, 2022. "Derivative-Free Iterative Schemes for Multiple Roots of Nonlinear Functions," Mathematics, MDPI, vol. 10(9), pages 1-13, May.
    6. Min-Young Lee & Young Ik Kim & Beny Neta, 2019. "A Generic Family of Optimal Sixteenth-Order Multiple-Root Finders and Their Dynamics Underlying Purely Imaginary Extraneous Fixed Points," Mathematics, MDPI, vol. 7(6), pages 1-26, June.
    7. Ramandeep Behl & Ioannis K. Argyros & Michael Argyros & Mehdi Salimi & Arwa Jeza Alsolami, 2020. "An Iteration Function Having Optimal Eighth-Order of Convergence for Multiple Roots and Local Convergence," Mathematics, MDPI, vol. 8(9), pages 1-21, August.
    8. Young Hee Geum & Young Ik Kim & Beny Neta, 2018. "Developing an Optimal Class of Generic Sixteenth-Order Simple-Root Finders and Investigating Their Dynamics," Mathematics, MDPI, vol. 7(1), pages 1-32, December.
    9. Sharma, Janak Raj & Kumar, Sunil, 2021. "An excellent numerical technique for multiple roots," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 316-324.
    10. Deepak Kumar & Janak Raj Sharma & Clemente Cesarano, 2019. "One-Point Optimal Family of Multiple Root Solvers of Second-Order," Mathematics, MDPI, vol. 7(7), pages 1-11, July.
    11. Deepak Kumar & Janak Raj Sharma & Ioannis K. Argyros, 2020. "Optimal One-Point Iterative Function Free from Derivatives for Multiple Roots," Mathematics, MDPI, vol. 8(5), pages 1-14, May.
    12. Ramandeep Behl & Eulalia Martínez & Fabricio Cevallos & Diego Alarcón, 2019. "A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots," Mathematics, MDPI, vol. 7(4), pages 1-12, April.
    13. Munish Kansal & Ali Saleh Alshomrani & Sonia Bhalla & Ramandeep Behl & Mehdi Salimi, 2020. "One Parameter Optimal Derivative-Free Family to Find the Multiple Roots of Algebraic Nonlinear Equations," Mathematics, MDPI, vol. 8(12), pages 1-15, December.

    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:gam:jmathe:v:11:y:2023:i:9:p:2036-:d:1132461. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.