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A new class of optimal four-point methods with convergence order 16 for solving nonlinear equations

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  • Sharifi, Somayeh
  • Salimi, Mehdi
  • Siegmund, Stefan
  • Lotfi, Taher

Abstract

We introduce a new class of optimal iterative methods without memory for approximating a simple root of a given nonlinear equation. The proposed class uses four function evaluations and one first derivative evaluation per iteration and it is therefore optimal in the sense of Kung and Traub’s conjecture. We present the construction, convergence analysis and numerical implementations, as well as comparisons of accuracy and basins of attraction between our method and existing optimal methods for several test problems.

Suggested Citation

  • Sharifi, Somayeh & Salimi, Mehdi & Siegmund, Stefan & Lotfi, Taher, 2016. "A new class of optimal four-point methods with convergence order 16 for solving nonlinear equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 119(C), pages 69-90.
    • Handle: RePEc:eee:matcom:v:119:y:2016:i:c:p:69-90
    DOI: 10.1016/j.matcom.2015.08.011
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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. 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.
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    Cited by:

    1. Vinay Kanwar & Puneet Sharma & Ioannis K. Argyros & Ramandeep Behl & Christopher Argyros & Ali Ahmadian & Mehdi Salimi, 2021. "Geometrically Constructed Family of the Simple Fixed Point Iteration Method," Mathematics, MDPI, vol. 9(6), pages 1-13, March.
    2. Ramandeep Behl & Munish Kansal & Mehdi Salimi, 2020. "Modified King’s Family for Multiple Zeros of Scalar Nonlinear Functions," Mathematics, MDPI, vol. 8(5), pages 1-17, May.
    3. Ramandeep Behl & Arwa Jeza Alsolami & Bruno Antonio Pansera & Waleed M. Al-Hamdan & Mehdi Salimi & Massimiliano Ferrara, 2019. "A New Optimal Family of Schröder’s Method for Multiple Zeros," Mathematics, MDPI, vol. 7(11), pages 1-14, November.
    4. Daniele Tommasini & David N. Olivieri, 2020. "Fast Switch and Spline Function Inversion Algorithm with Multistep Optimization and k-Vector Search for Solving Kepler’s Equation in Celestial Mechanics," Mathematics, MDPI, vol. 8(11), pages 1-18, November.
    5. Syahmi Afandi Sariman & Ishak Hashim & Faieza Samat & Mohammed Alshbool, 2021. "Modification of Newton-Househölder Method for Determining Multiple Roots of Unknown Multiplicity of Nonlinear Equations," Mathematics, MDPI, vol. 9(9), pages 1-12, April.
    6. Moin-ud-Din Junjua & Fiza Zafar & Nusrat Yasmin, 2019. "Optimal Derivative-Free Root Finding Methods Based on Inverse Interpolation," Mathematics, MDPI, vol. 7(2), pages 1-10, February.

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