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Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction

Author

Listed:
  • Jian Li

    (Inner Mongolia Vocational College of Chemical Engineering, Hohhot 010070, China)

  • Xiaomeng Wang

    (Inner Mongolia Vocational College of Chemical Engineering, Hohhot 010070, China)

  • Kalyanasundaram Madhu

    (Department of Mathematics, Saveetha Engineering College, Chennai 602105, India)

Abstract

Based on the Steffensen-type method, we develop fourth-, eighth-, and sixteenth-order algorithms for solving one-variable equations. The new methods are fourth-, eighth-, and sixteenth-order converging and require at each iteration three, four, and five function evaluations, respectively. Therefore, all these algorithms are optimal in the sense of Kung–Traub conjecture; the new schemes have an efficiency index of 1.587, 1.682, and 1.741, respectively. We have given convergence analyses of the proposed methods and also given comparisons with already established known schemes having the same convergence order, demonstrating the efficiency of the present techniques numerically. We also studied basins of attraction to demonstrate their dynamical behavior in the complex plane.

Suggested Citation

  • Jian Li & Xiaomeng Wang & Kalyanasundaram Madhu, 2019. "Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction," Mathematics, MDPI, vol. 7(11), pages 1-15, November.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:11:p:1052-:d:283397
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    References listed on IDEAS

    as
    1. F. Soleymani & S. Karimi Vanani & M. Jamali Paghaleh, 2012. "A Class of Three-Step Derivative-Free Root Solvers with Optimal Convergence Order," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-15, February.
    2. Farahnaz Soleimani & Fazlollah Soleymani & Stanford Shateyi, 2013. "Some Iterative Methods Free from Derivatives and Their Basins of Attraction for Nonlinear Equations," Discrete Dynamics in Nature and Society, Hindawi, vol. 2013, pages 1-10, April.
    3. 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.
    Full references (including those not matched with items on IDEAS)

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