On the robust Newton’s method with the Mann iteration and the artistic patterns from its dynamics

Abstract

There are two main aims of this paper. The first one is to show some improvement of the robust Newton’s method (RNM) introduced recently by Kalantari. The RNM is a generalisation of the well-known Newton’s root finding method. Since the base method is undefined at critical points, the RNM allows working also at such points. In this paper, we improve the RNM method by applying the Mann iteration instead of the standard Picard iteration. This leads to an essential decrease in the number of root finding steps without visible destroying the sharp boundaries among the basins of attractions presented in polynomiographs. Furthermore, we investigate visually the dynamics of the RNM with the Mann iteration together with the basins of attraction for varying Mann’s iteration parameter with the help of polynomiographs for several polynomials. The second aim of this paper is to present the intriguing polynomiographs obtained from the dynamics of the RNM with the Mann iteration under various sequences used in this iteration. The obtained polynomiographs differ considerably from the ones obtained with the RNM and are interesting from the artistic perspective. Moreover, they can easily find applications in wallpaper or fabric design.

Acceleration of the Robust Newton Method by the Use of the S-iteration

“…It means that in this case, the S-RNM needs nearly 17 times iterations less in comparison to the RNM to achieve the same goal in considerably lower computation time. Dynamics of a discrete dynamical system can give rise to fascinating patterns, see for instance [5,7,12]. The introduction of the S-iteration in the RNM can give birth to very complicated patterns of possible artistic applications.…”

From Halley to Secant: Redefining root finding with memory‐based methods including convergence and stability

“…Polynomiographs can also be used to obtain several numerical measures that can allow the analysis to be broadened [28]. The three most popular measures are average number of iterations (ANI) [28], convergence area index (CAI, i.e., the ratio of the number of starting points that converged to a root to the number of all points in the considered area) [29], and generation time [5]. We will use these measures in our analysis performed in this section.…”

“…They are also employed in the analysis of nonlinear dynamics and the research of phase changes. Root finding has several applications in computer science, ranging from graphical rendering in games, generation of artistic patterns [5] to the improvement of machine learning methods [6]. Reinforcement learning and neural networks rely heavily on it, two areas of artificial intelligence (AI) where it plays a crucial role.…”

A General Optimal Iterative Scheme with Arbitrary Order of Convergence

“…The dynamical study of a family of iterative methods allows classifying them, based on their behavior with respect to the initial approximations taken, into stable methods and chaotic ones. This analysis also provides important information on the reliability of the methods [15][16][17][18][19].…”