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A new class of methods with higher order of convergence for solving systems of nonlinear equations

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  • Xiao, Xiaoyong
  • Yin, Hongwei

Abstract

By studying the commonness of some fifth order methods modified from third order ones for solving systems of nonlinear equations, we propose a new class of three-step methods of convergence order five by modifying a class of two-step methods with cubic convergence. Next, for a given method of order p ≥ 2 which uses the extended Newton iteration yk = xk − aF′(xk)−1F(xk) as a predictor, a new method of order p + 2 is proposed. For example, we construct a class of m + 2-step methods of convergence order 2m + 3 by introducing only one evaluation of the function to each of the last m steps for any positive integer m. In this paper, we mainly focus on the class of fifth order methods when m = 1. Computational efficiency in the general form is considered. Several examples for numerical tests are given to show the asymptotic behavior and the computational efficiency of these higher order methods.

Suggested Citation

  • Xiao, Xiaoyong & Yin, Hongwei, 2015. "A new class of methods with higher order of convergence for solving systems of nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 264(C), pages 300-309.
    • Handle: RePEc:eee:apmaco:v:264:y:2015:i:c:p:300-309
    DOI: 10.1016/j.amc.2015.04.094
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    References listed on IDEAS

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    1. J. A. Ezquerro & M. Grau-Sánchez & A. Grau & M. A. Hernández & M. Noguera & N. Romero, 2011. "On Iterative Methods with Accelerated Convergence for Solving Systems of Nonlinear Equations," Journal of Optimization Theory and Applications, Springer, vol. 151(1), pages 163-174, October.
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    Cited by:

    1. Bahl, Ashu & Cordero, Alicia & Sharma, Rajni & R. Torregrosa, Juan, 2019. "A novel bi-parametric sixth order iterative scheme for solving nonlinear systems and its dynamics," Applied Mathematics and Computation, Elsevier, vol. 357(C), pages 147-166.
    2. Ramandeep Behl & Ioannis K. Argyros & Sattam Alharbi, 2024. "Accelerating the Speed of Convergence for High-Order Methods to Solve Equations," Mathematics, MDPI, vol. 12(17), pages 1-22, September.
    3. Sharma, Janak Raj & Sharma, Rajni & Bahl, Ashu, 2016. "An improved Newton–Traub composition for solving systems of nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 290(C), pages 98-110.
    4. Xiao, Xiao-Yong & Yin, Hong-Wei, 2018. "Accelerating the convergence speed of iterative methods for solving nonlinear systems," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 8-19.
    5. Ramandeep Behl & Ioannis K. Argyros & Sattam Alharbi, 2024. "A One-Parameter Family of Methods with a Higher Order of Convergence for Equations in a Banach Space," Mathematics, MDPI, vol. 12(9), pages 1-18, April.
    6. Zhanlav, T. & Otgondorj, Kh., 2021. "Higher order Jarratt-like iterations for solving systems of nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 395(C).

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