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A new time-efficient and convergent nonlinear solver

Author

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  • Abro, Hameer Akhtar
  • Shaikh, Muhammad Mujtaba

Abstract

Nonlinear equations arise in various fields of science and engineering. The present era of computational science – where one needs maximum achievement in minimum time – demands proposal of new and efficient iterative methods for solving nonlinear equations and systems. While the new methods are expected to be higher order convergent, the time efficiency and lesser computational information used are the top priorities. In this paper, we propose a new three-step iterative nonlinear solver for nonlinear equations and systems. The proposed method requires three evaluations of function and two evaluations of the first-order derivative per iteration. The proposed method is sixth order convergent, which is also proved theoretically. The performance of the proposed method is tested against other existing methods on the basis of error distributions, computational efficiency and CPU times. The numerical results on the application of the discussed methods on various nonlinear equations and systems, including an application problem related to combustion for a temperature of 3000 °C, show that the proposed method is comparable with existing methods with the main feature of the proposed method being its time-effectiveness.

Suggested Citation

  • Abro, Hameer Akhtar & Shaikh, Muhammad Mujtaba, 2019. "A new time-efficient and convergent nonlinear solver," Applied Mathematics and Computation, Elsevier, vol. 355(C), pages 516-536.
    • Handle: RePEc:eee:apmaco:v:355:y:2019:i:c:p:516-536
    DOI: 10.1016/j.amc.2019.03.012
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    References listed on IDEAS

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    1. Alicia Cordero & Esther Gómez & Juan R. Torregrosa, 2017. "Efficient High-Order Iterative Methods for Solving Nonlinear Systems and Their Application on Heat Conduction Problems," Complexity, Hindawi, vol. 2017, pages 1-11, January.
    2. Herceg, Djordje & Herceg, Dragoslav, 2015. "A family of methods for solving nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 882-895.
    3. Sharma, Janak Raj & Sharma, Rajni & Kalra, Nitin, 2015. "A novel family of composite Newton–Traub methods for solving systems of nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 520-535.
    4. Waseem, Muhammad & Noor, Muhammad Aslam & Noor, Khalida Inayat, 2016. "Efficient method for solving a system of nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 134-146.
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    Cited by:

    1. Sania Qureshi & Higinio Ramos & Abdul Karim Soomro, 2021. "A New Nonlinear Ninth-Order Root-Finding Method with Error Analysis and Basins of Attraction," Mathematics, MDPI, vol. 9(16), pages 1-19, August.

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