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A biparametric extension of King’s fourth-order methods and their dynamics

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  • Geum, Young Hee
  • Kim, Young Ik
  • Magreñán, Á. Alberto

Abstract

A class of two-point quartic-order simple-zero finders and their dynamics are investigated in this paper by extending King’s fourth-order family of methods. With the introduction of an error corrector having a weight function dependent on a function-to-function ratio, higher-order convergence is obtained. Through a variety of test equations, numerical experiments strongly support the theory developed in this paper. In addition, relevant dynamics of the proposed methods is successfully explored for a prototype quadratic polynomial as well as parameter spaces and dynamical planes.

Suggested Citation

  • Geum, Young Hee & Kim, Young Ik & Magreñán, Á. Alberto, 2016. "A biparametric extension of King’s fourth-order methods and their dynamics," Applied Mathematics and Computation, Elsevier, vol. 282(C), pages 254-275.
    • Handle: RePEc:eee:apmaco:v:282:y:2016:i:c:p:254-275
    DOI: 10.1016/j.amc.2016.02.020
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    References listed on IDEAS

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    1. 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.
    2. Geum, Young Hee & Kim, Young Ik & Neta, Beny, 2015. "On developing a higher-order family of double-Newton methods with a bivariate weighting function," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 277-290.
    3. Cordero, Alicia & Lotfi, Taher & Mahdiani, Katayoun & Torregrosa, Juan R., 2015. "A stable family with high order of convergence for solving nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 240-251.
    4. 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.
    5. 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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    Cited by:

    1. Min-Young Lee & Young Ik Kim, 2020. "Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root Finders," Mathematics, MDPI, vol. 8(1), pages 1-13, January.
    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. Cordero, Alicia & Soleymani, Fazlollah & Torregrosa, Juan R. & Haghani, F. Khaksar, 2017. "A family of Kurchatov-type methods and its stability," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 264-279.
    5. Young Hee Geum & Young Ik Kim, 2019. "On Locating and Counting Satellite Components Born along the Stability Circle in the Parameter Space for a Family of Jarratt-Like Iterative Methods," Mathematics, MDPI, vol. 7(9), pages 1-16, September.

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