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New Memory-Updating Methods in Two-Step Newton’s Variants for Solving Nonlinear Equations with High Efficiency Index

Author

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  • Chein-Shan Liu

    (Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan)

  • Chih-Wen Chang

    (Department of Mechanical Engineering, National United University, Miaoli 360302, Taiwan)

Abstract

In the paper, we iteratively solve a scalar nonlinear equation f ( x ) = 0 , where f ∈ C ( I , R ) , x ∈ I ⊂ R , and I includes at least one real root r . Three novel two-step iterative schemes equipped with memory updating methods are developed; they are variants of the fixed-point Newton method. A triple data interpolation is carried out by the two-degree Newton polynomial, which is used to update the values of f ′ ( r ) and f ″ ( r ) . The relaxation factor in the supplementary variable is accelerated by imposing an extra condition on the interpolant. The new memory method (NMM) can raise the efficiency index (E.I.) significantly. We apply the NMM to five existing fourth-order iterative methods, and the computed order of convergence (COC) and E.I. are evaluated by numerical tests. When the relaxation factor acceleration technique is combined with the modified D z ˇ uni c ´ ’s memory method, the value of E.I. is much larger than that predicted by the paper [Kung, H.T.; Traub, J.F. J. Assoc. Comput. Machinery 1974 , 21 ]. for the iterative method without memory.

Suggested Citation

  • Chein-Shan Liu & Chih-Wen Chang, 2024. "New Memory-Updating Methods in Two-Step Newton’s Variants for Solving Nonlinear Equations with High Efficiency Index," Mathematics, MDPI, vol. 12(4), pages 1-22, February.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:4:p:581-:d:1339078
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    References listed on IDEAS

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    1. G Thangkhenpau & Sunil Panday & Shubham Kumar Mittal & Lorentz Jäntschi, 2023. "Novel Parametric Families of with and without Memory Iterative Methods for Multiple Roots of Nonlinear Equations," Mathematics, MDPI, vol. 11(9), pages 1-18, April.
    2. Manoj K. Singh & Ioannis K. Argyros, 2022. "The Dynamics of a Continuous Newton-like Method," Mathematics, MDPI, vol. 10(19), pages 1-14, October.
    3. Lee, Min-Young & Ik Kim, Young & Alberto Magreñán, Á., 2017. "On the dynamics of a triparametric family of optimal fourth-order multiple-zero finders with a weight function of the principal mth root of a function-to function ratio," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 564-590.
    4. Ioannis K. Argyros & Stepan Shakhno, 2019. "Extended Local Convergence for the Combined Newton-Kurchatov Method Under the Generalized Lipschitz Conditions," Mathematics, MDPI, vol. 7(2), pages 1-12, February.
    5. T. Lotfi & F. Soleymani & Z. Noori & A. Kılıçman & F. Khaksar Haghani, 2014. "Efficient Iterative Methods with and without Memory Possessing High Efficiency Indices," Discrete Dynamics in Nature and Society, Hindawi, vol. 2014, pages 1-9, September.
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