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A Triparametric Family of Optimal Fourth‐Order Multiple‐Root Finders and Their Dynamics

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  • Young Ik Kim
  • Young Hee Geum

Abstract

We investigate the complex dynamics of a triparametric family of optimal fourth‐order multiple‐root solvers by analyzing their basins of attraction along with extensive study of Möbius conjugacy maps and extraneous fixed points applied to a prototype quadratic polynomial raised to the power of the known integer multiplicity m. A 600 × 600 uniform grid centered at the origin covering 6 × 6 square region is chosen to display the initial points on each basin of attraction according to a coloring scheme based on their orbit behavior. With illustrative basins of attractions applied to various test polynomials and the corresponding statistical data for convergence as well as a number of comparisons made among the listed methods, we confirm our investigation and analysis developed in this paper.

Suggested Citation

  • Young Ik Kim & Young Hee Geum, 2016. "A Triparametric Family of Optimal Fourth‐Order Multiple‐Root Finders and Their Dynamics," Discrete Dynamics in Nature and Society, John Wiley & Sons, vol. 2016(1).
    • Handle: RePEc:wly:jnddns:v:2016:y:2016:i:1:n:8436759
    DOI: 10.1155/2016/8436759
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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. Young Ik Kim & Young Hee Geum, 2014. "A New Biparametric Family of Two-Point Optimal Fourth-Order Multiple-Root Finders," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-7, September.
    3. 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.
    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.
    5. Young Ik Kim & Young Hee Geum, 2013. "A Two-Parameter Family of Fourth-Order Iterative Methods with Optimal Convergence for Multiple Zeros," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-7, February.
    6. Young Ik Kim & Young Hee Geum, 2013. "A Two‐Parameter Family of Fourth‐Order Iterative Methods with Optimal Convergence for Multiple Zeros," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    7. Young Ik Kim & Young Hee Geum, 2014. "A New Biparametric Family of Two‐Point Optimal Fourth‐Order Multiple‐Root Finders," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).
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