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A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points

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  • Geum, Young Hee
  • Kim, Young Ik
  • Neta, Beny

Abstract

A class of three-point sixth-order multiple-root finders and the dynamics behind their extraneous fixed points are investigated by extending modified Newton-like methods with the introduction of the multivariate weight functions in the intermediate steps. The multivariate weight functions dependent on function-to-function ratios play a key role in constructing higher-order iterative methods. Extensive investigation of extraneous fixed points of the proposed iterative methods is carried out for the study of the dynamics associated with corresponding basins of attraction. Numerical experiments applied to a number of test equations strongly support the underlying theory pursued in this paper. Relevant dynamics of the proposed methods is well presented with a variety of illustrative basins of attraction applied to various test polynomials.

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  • 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.
    • Handle: RePEc:eee:apmaco:v:283:y:2016:i:c:p:120-140
    DOI: 10.1016/j.amc.2016.02.029
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    References listed on IDEAS

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    1. Neta, Beny & Chun, Changbum, 2014. "Basins of attraction for several optimal fourth order methods for multiple roots," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 103(C), pages 39-59.
    2. Magreñán, Á. Alberto & Cordero, Alicia & Gutiérrez, José M. & Torregrosa, Juan R., 2014. "Real qualitative behavior of a fourth-order family of iterative methods by using the convergence plane," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 105(C), pages 49-61.
    3. 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.
    4. Chun, Changbum & Neta, Beny, 2015. "Basins of attraction for Zhou–Chen–Song fourth order family of methods for multiple roots," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 109(C), pages 74-91.
    5. Andreu, Carlos & Cambil, Noelia & Cordero, Alicia & Torregrosa, Juan R., 2014. "A class of optimal eighth-order derivative-free methods for solving the Danchick–Gauss problem," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 237-246.
    6. Argyros, Ioannis K. & Magreñán, Á. Alberto, 2015. "On the convergence of an optimal fourth-order family of methods and its dynamics," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 336-346.
    7. Chun, Changbum & Neta, Beny, 2015. "Comparing the basins of attraction for Kanwar–Bhatia–Kansal family to the best fourth order method," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 277-292.
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    Cited by:

    1. Amiri, Abdolreza & Cordero, Alicia & Taghi Darvishi, M. & Torregrosa, Juan R., 2018. "Stability analysis of a parametric family of seventh-order iterative methods for solving nonlinear systems," Applied Mathematics and Computation, Elsevier, vol. 323(C), pages 43-57.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. Ramandeep Behl & Ioannis K. Argyros & Fouad Othman Mallawi, 2021. "Some High-Order Convergent Iterative Procedures for Nonlinear Systems with Local Convergence," Mathematics, MDPI, vol. 9(12), pages 1-13, June.
    8. Vázquez-Lozano, J. Enrique & Cordero, Alicia & Torregrosa, Juan R., 2018. "Dynamical analysis on cubic polynomials of Damped Traub’s method for approximating multiple roots," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 82-99.
    9. Alicia Cordero & Beny Neta & Juan R. Torregrosa, 2021. "Memorizing Schröder’s Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity," Mathematics, MDPI, vol. 9(20), pages 1-13, October.
    10. 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.
    11. 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.

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