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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
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    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.
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    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. Sunil Panday & Shubham Kumar Mittal & Carmen Elena Stoenoiu & Lorentz Jäntschi, 2024. "A New Adaptive Eleventh-Order Memory Algorithm for Solving Nonlinear Equations," Mathematics, MDPI, vol. 12(12), pages 1-14, June.
    4. 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.

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