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On the convergence of an optimal fourth-order family of methods and its dynamics

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  • Argyros, Ioannis K.
  • Magreñán, Á. Alberto

Abstract

In this paper, we present the study of the semilocal and local convergence of an optimal fourth-order family of methods. Moreover, the dynamical behavior of this family of iterative methods applied to quadratic polynomials is studied. Some anomalies are found in this family be means of studying the dynamical behavior. Parameter spaces are shown and the study of the stability of all the fixed points is presented.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:252:y:2015:i:c:p:336-346
    DOI: 10.1016/j.amc.2014.11.074
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    Citations

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    Cited by:

    1. 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.
    2. Ioannis K. Argyros & Ángel Alberto Magreñán & Lara Orcos & Íñigo Sarría, 2019. "Unified Local Convergence for Newton’s Method and Uniqueness of the Solution of Equations under Generalized Conditions in a Banach Space," Mathematics, MDPI, vol. 7(5), pages 1-13, May.
    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. 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.
    5. 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.
    6. 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.
    7. Hueso, José L. & Martínez, Eulalia & Gupta, D.K. & Cevallos, Fabricio, 2018. "A note on “Convergence radius of Osada’s method under Hölder continuous condition”," Applied Mathematics and Computation, Elsevier, vol. 321(C), pages 689-699.
    8. Campos, Beatriz & Cordero, Alicia & Torregrosa, Juan R. & Vindel, Pura, 2016. "Dynamics of a multipoint variant of Chebyshev–Halley’s family," Applied Mathematics and Computation, Elsevier, vol. 284(C), pages 195-208.
    9. Chun, Changbum & Neta, Beny, 2015. "Basins of attraction for several third order methods to find multiple roots of nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 129-137.
    10. Lee, Min-Young & Ik Kim, Young & Alberto Magreñán, Á., 2017. "On the dynamics of a triparametric family of optimal fourth-order multiple-zero finders with a weight function of the principal mth root of a function-to function ratio," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 564-590.
    11. Behl, Ramandeep & Cordero, Alicia & Motsa, Sandile S. & Torregrosa, Juan R., 2015. "Construction of fourth-order optimal families of iterative methods and their dynamics," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 89-101.
    12. Janak Raj Sharma & Deepak Kumar & Ioannis K. Argyros & Ángel Alberto Magreñán, 2019. "On a Bi-Parametric Family of Fourth Order Composite Newton–Jarratt Methods for Nonlinear Systems," Mathematics, MDPI, vol. 7(6), pages 1-27, May.
    13. Behl, Ramandeep & Cordero, Alicia & Motsa, Sandile S. & Torregrosa, Juan R., 2017. "Stable high-order iterative methods for solving nonlinear models," Applied Mathematics and Computation, Elsevier, vol. 303(C), pages 70-88.
    14. Petković, I. & Herceg, Ð., 2017. "Symbolic computation and computer graphics as tools for developing and studying new root-finding methods," Applied Mathematics and Computation, Elsevier, vol. 295(C), pages 95-113.
    15. 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.
    16. 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.
    17. García Calcines, José M. & Gutiérrez, José M. & Hernández Paricio, Luis J. & Rivas Rodríguez, M. Teresa, 2015. "Graphical representations for the homogeneous bivariate Newton’s method," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 988-1006.
    18. Argyros, Ioannis K. & Kansal, Munish & Kanwar, Vinay & Bajaj, Sugandha, 2017. "Higher-order derivative-free families of Chebyshev–Halley type methods with or without memory for solving nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 224-245.
    19. Ramandeep Behl & Ioannis K. Argyros, 2020. "Local Convergence for Multi-Step High Order Solvers under Weak Conditions," Mathematics, MDPI, vol. 8(2), pages 1-14, February.
    20. Young Hee Geum & Young Ik Kim, 2020. "Computational Bifurcations Occurring on Red Fixed Components in the λ -Parameter Plane for a Family of Optimal Fourth-Order Multiple-Root Finders under the Möbius Conjugacy Map," Mathematics, MDPI, vol. 8(5), pages 1-17, May.
    21. Chun, Changbum & Neta, Beny, 2016. "Comparison of several families of optimal eighth order methods," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 762-773.

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