Novel Parametric Families of with and without Memory Iterative Methods for Multiple Roots of Nonlinear Equations

Abstract

The methods that use memory using accelerating parameters for computing multiple roots are almost non-existent in the literature. Furthermore, the only paper available in this direction showed an increase in the order of convergence of 0.5 from the without memory to the with memory extension. In this paper, we introduce a new fifth-order without memory method, which we subsequently extend to two higher-order with memory methods using a self-accelerating parameter. The proposed with memory methods extension demonstrate a significant improvement in the order of convergence from 5 to 7, making this the first paper to achieve at least a 2-order improvement. In addition to this improvement, our paper is also the first to use Hermite interpolating polynomials to approximate the accelerating parameter in the proposed with memory methods for multiple roots. We also provide rigorous theoretical proofs of convergence theorems to establish the order of the proposed methods. Finally, we demonstrate the potential impact of the proposed methods through numerical experimentation on a diverse range of problems. Overall, we believe that our proposed methods have significant potential for various applications in science and engineering.

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

“…The iterative scheme (10) was developed to be a one-step continuation Newton-like method [2], and it was used as the first step in the multistep iterative schemes in [3][4][5]. Some dynamical analysis of Equation ( 10) can be seen in [6,7].…”

“…Traub's technique is a typical method with memory, in which the data that appeared in the previous iteration were adopted in the iteration. For recent progress of the memory method with accelerating parameters in the multistep iterative methods, one can refer to [5,27,[34][35][36][37][38][39][40]. One major goal of this paper is to develop the two-step iterative schemes with a new memory method to determine the accelerating parameters by updating technique with the information at the current step.…”

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

“…The parameters a 0 and b 0 can be updated to speed up the convergence by the memory-dependent method [2]. One can refer to [3][4][5][6][7][8][9][10] for more memory-dependent iterative methods.…”

“…For f 3 (x) = 0, we obtain the root r = 0.1906879703188765. There are some self-accelerating iterative methods for simple roots [3,52,53], which are then extended to the self-accelerating technique for the iterative methods for multiple roots [8,54]. In Equations ( 86)-( 88), the self-accelerating technique for A n and B n is quite simple as compared to that in the literature.…”

Derivative-Free Families of With- and Without-Memory Iterative Methods for Solving Nonlinear Equations and Their Engineering Applications

“…Furthermore, Chanu et al [33] proposed optimal memoryless techniques of fourth and eighth orders, extending them to incorporate memory. In the pursuit of resolving nonlinear equations with multiple roots, Thangkhenpau et al [34] introduced a novel scheme that offers both with-and without-memory-based variants.…”