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A New Family of Fourth-Order Optimal Iterative Schemes and Remark on Kung and Traub’s Conjecture

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  • Chein-Shan Liu
  • Tsung-Lin Lee
  • Xiaolong Qin

Abstract

Kung and Traub conjectured that a multipoint iterative scheme without memory based on m evaluations of functions has an optimal convergence order p=2m−1. In the paper, we first prove that the two-step fourth-order optimal iterative schemes of the same class have a common feature including a same term in the error equations, resorting on the conjecture of Kung and Traub. Based on the error equations, we derive a constantly weighting algorithm obtained from the combination of two iterative schemes, which converges faster than the departed ones. Then, a new family of fourth-order optimal iterative schemes is developed by using a new weight function technique, which needs three evaluations of functions and whose convergence order is proved to be p=23−1=4.

Suggested Citation

  • Chein-Shan Liu & Tsung-Lin Lee & Xiaolong Qin, 2021. "A New Family of Fourth-Order Optimal Iterative Schemes and Remark on Kung and Traub’s Conjecture," Journal of Mathematics, Hindawi, vol. 2021, pages 1-9, February.
    • Handle: RePEc:hin:jjmath:5516694
    DOI: 10.1155/2021/5516694
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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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