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Three novel fifth-order iterative schemes for solving nonlinear equations

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  • Liu, Chein-Shan
  • El-Zahar, Essam R.
  • Chang, Chih-Wen

Abstract

Kung and Traub’s conjecture indicates that a multipoint iterative scheme without memory and based on m evaluations of functions has an optimal convergence order p=2m−1. Consequently, a fifth-order iterative scheme requires at least four evaluations of functions. Herein, we derive three novel iterative schemes that have fifth-order convergence and involve four evaluations of functions, such that the efficiency index is E.I.=1.49535. On the basis of the analysis of error equations, we obtain our first iterative scheme from the constant weight combinations of three first- and second-class fourth-order iterative schemes. For the second iterative scheme, we devise a new weight function to derive another fifth-order iterative scheme. Finally, we derive our third iterative scheme from a combination of two second-class fourth-order iterative schemes. For testing the practical application of our schemes, we apply them to solve the van der Waals equation of state.

Suggested Citation

  • Liu, Chein-Shan & El-Zahar, Essam R. & Chang, Chih-Wen, 2021. "Three novel fifth-order iterative schemes for solving nonlinear equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 282-293.
    • Handle: RePEc:eee:matcom:v:187:y:2021:i:c:p:282-293
    DOI: 10.1016/j.matcom.2021.03.002
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    Cited by:

    1. Liu, Dongjie & Liu, Chein-Shan, 2022. "Two-point generalized Hermite interpolation: Double-weight function and functional recursion methods for solving nonlinear equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 317-330.

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