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A new family of optimal eighth order methods with dynamics for nonlinear equations

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  • Sharma, Janak Raj
  • Arora, Himani

Abstract

We propose a simple yet efficient family of three-point iterative methods for solving nonlinear equations. Each method of the family requires four evaluations, namely three functions and one derivative, per full iteration and possesses eighth order of convergence. Thus, the family is optimal in the sense of Kung–Traub conjecture and has the efficiency index 1.682 which is better than that of Newton’s and many other higher order methods. Various numerical examples are considered to check the performance and to verify the theoretical results. Computational results including the elapsed CPU-time, confirm the efficient and robust character of proposed technique. Moreover, the presented basins of attraction also confirm better performance of the methods as compared to other established methods in literature.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:273:y:2016:i:c:p:924-933
    DOI: 10.1016/j.amc.2015.10.049
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    Citations

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    Cited by:

    1. Zhanlav, T. & Chuluunbaatar, O. & Ulziibayar, V., 2017. "Generating function method for constructing new iterations," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 414-423.
    2. 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.
    3. Yanlin Tao & Kalyanasundaram Madhu, 2019. "Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniques and Their Basins of Attraction and Its Application," Mathematics, MDPI, vol. 7(4), pages 1-22, March.
    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. Janak Raj Sharma & Ioannis K. Argyros & Sunil Kumar, 2019. "Convergence Analysis of Weighted-Newton Methods of Optimal Eighth Order in Banach Spaces," Mathematics, MDPI, vol. 7(2), pages 1-14, February.

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