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Developing an Optimal Class of Generic Sixteenth-Order Simple-Root Finders and Investigating Their Dynamics

Author

Listed:
  • Young Hee Geum

    (Department of Applied Mathematics, Dankook University, Cheonan 330-714, Korea)

  • Young Ik Kim

    (Department of Applied Mathematics, Dankook University, Cheonan 330-714, Korea)

  • Beny Neta

    (Naval Postgraduate School, Department of Applied Mathematics, Monterey, CA 93943, USA)

Abstract

Developed here are sixteenth-order simple-root-finding optimal methods with generic weight functions. Their numerical and dynamical aspects are investigated with the establishment of a main theorem describing the desired optimal convergence. Special cases with polynomial and rational weight functions have been extensively studied for applications to real-world problems. A number of computational experiments clearly support the underlying theory on the local convergence of the proposed methods. In addition, to investigate the relevant global convergence, we focus on the dynamics of the developed methods, as well as other known methods through the visual description of attraction basins. Finally, we summarized the results, discussion, conclusion, and future work.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:7:y:2018:i:1:p:8-:d:192399
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    References listed on IDEAS

    as
    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. 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.
    3. 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.
    4. Sharma, Janak Raj & Arora, Himani, 2016. "A new family of optimal eighth order methods with dynamics for nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 924-933.
    5. 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.
    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. 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.
    8. 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.
    9. 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.
    Full references (including those not matched with items on IDEAS)

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