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A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots

Author

Listed:
  • Ramandeep Behl

    (Department of Mathematics, King Abdualziz University, Jeddah 21589, Saudi Arabia)

  • Eulalia Martínez

    (Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain)

  • Fabricio Cevallos

    (Fac. de Ciencias Económicas, Universidad Laica “Eloy Alfaro de Manabí”, Manabí 130214, Ecuador)

  • Diego Alarcón

    (Departamento de Matemática Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain)

Abstract

The aim of this paper is to introduce new high order iterative methods for multiple roots of the nonlinear scalar equation; this is a demanding task in the area of computational mathematics and numerical analysis. Specifically, we present a new Chebyshev–Halley-type iteration function having at least sixth-order convergence and eighth-order convergence for a particular value in the case of multiple roots. With regard to computational cost, each member of our scheme needs four functional evaluations each step. Therefore, the maximum efficiency index of our scheme is 1.6818 for α = 2 , which corresponds to an optimal method in the sense of Kung and Traub’s conjecture. We obtain the theoretical convergence order by using Taylor developments. Finally, we consider some real-life situations for establishing some numerical experiments to corroborate the theoretical results.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:4:p:339-:d:221138
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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. Sharma, Janak Raj, 2015. "Improved Chebyshev–Halley methods with sixth and eighth order convergence," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 119-124.
    3. 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.
    4. Behl, Ramandeep & Cordero, Alicia & Motsa, S.S. & Torregrosa, Juan R., 2015. "On developing fourth-order optimal families of methods for multiple roots and their dynamics," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 520-532.
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