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

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 Dzˇunic´’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.

Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations

“…For the Newton method, there exists some weak points as pointed out in [11]. In addition, for an odd function with f (−x) = − f (x), there exist two-cycle points of the Newton method.…”

“…By resulting in x 0 and γ 0 and incorporating the memory of x n , Traub's iterative scheme can proceed to find the solution of f (x) = 0 upon convergence. For a recent report on the progress of the memory method with accelerating parameters in the one-step iterative method, one can refer to [2], while for a recent progress report on the memory method with accelerating parameters in the two-step iterative method, one can refer to [11]. One major goal of the paper is the development of multi-step iterative schemes with a new memory method to determine the accelerating parameters by updating the technique with information at the current step.…”

A New Adaptive Eleventh-Order Memory Algorithm for Solving Nonlinear Equations

“…In recent years, the development of with-memory iterative methods has garnered considerable interest among researchers. For a deeper understanding, one can refer to [23][24][25][26][27][28][29][30][31][32][33][34] and the references cited therein.…”