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

Chein-Shan Liu and Chih-Wen Chang ()
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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

Mathematics, 2024, vol. 12, issue 4, 1-22

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.

Keywords: nonlinear equation; two-step iterative schemes; new memory updating method; relaxation factor; supplementary variable (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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