An empirical study of the convergence area and convergence speed of Agarwal et al. fixed point iteration procedure

Argyros

2023

Axioms

In the present paper, we introduced a quadratically convergent Newton-like normal S-iteration method free from the second derivative for the solution of nonlinear equations permitting f′(x)=0 at some points in the neighborhood of the root. Our proposed method works well when the Newton method fails and performs even better than some higher-order converging methods. Numerical results verified that the Newton-like normal S-iteration method converges faster than Fang et al.’s method. We studied different aspects of the normal S-iteration method regarding the faster convergence to the root. Lastly, the dynamic results support the numerical results and explain the convergence, divergence, and stability of the proposed method.

Section: Average Number Of Iterations In Normal-s Iteration Methodsmentioning

confidence: 99%

“…Although Newton's method is the most important known and the most basic used method to solve Equation (1), some weaknesses of Newton's method [6][7][8][9][10][11][12] are as follows:…”

Section: Introductionmentioning

confidence: 99%

Newton-like Normal S-iteration under Weak Conditions

Argyros

2023

Axioms

Computational Science – ICCS 2021

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

Gdawiec

Kotarski²,

Lisowska

2021

In this paper, we propose an improvement of the Robust Newton's Method (RNM). The RNM is a generalisation of the known Newton's root finding method restricted to polynomials. Unfortunately, the RNM is slow. Thus, in this paper, we propose the acceleration of this method by replacing the standard Picard iteration in the RNM by the S-iteration. This leads to an essential acceleration of the modified method. We present the advantages of the proposed algorithm over the RNM using polynomiagraphs and some numerical measures. Moreover, we present its possible application to the generation of artistic patterns.

Higher Order Methods of the Basic Family of Iterations via S-Iteration Scheme with s-Convexity

2020

There are many methods for solving a polynomial equation and many different modifications of those methods have been proposed in the literature. One of such modifications is the use of various iteration processes taken from the fixed point theory. In this paper, we propose a modification of the iteration processes used in the Basic Family of iterations by replacing the convex combination with an s-convex one. In our study, we concentrate only on the S-iteration with s-convexity. We present some graphical examples, the so-called polynomiographs, and numerical experiments showing the dependency of polynomiograph's generation time on the value of the s parameter in the s-convex combination.