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

Abstract

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.

Some Escape Time Results for General Complex Polynomials and Biomorphs Generation by a New Iteration Process

“…The Mann, Ishikawa and Noor iterations have been employed for generating Julia and Mandelbrot sets [23], superior Julia and Mandelbrot sets [19,20], relatively superior Julia and Mandelbrot sets [33][34][35], superfractals [36], generalized Julia sets [17] and polynomiographs [37][38][39]. Recently, Gdwaiec et al [29] also considered the Mann and Ishikawa iterations with the Pickover algorithm for generating biomorphs.…”

Visual Analysis of Mixed Algorithms with Newton and Abbasbandy Methods Using Periodic Parameters

“…Instead of using the standard Picard iteration, they used several different iterations, such as the Mann, Ishikawa, Noor, S and SP iterations. Many other authors have proposed several polynomiographs obtained using generalized versions of Newton's method with different iteration processes, for instance, [10,11] and references therein. Moreover, a survey of stability analysis for solving systems of nonlinear equations can be found in [12,13].…”

Improved Newton Iterative Algorithm for Fractal Art Graphic Design

“…Fractal geometry is often used to describe irregular things in nature. e well-known Euclidean geometry describes objects composed of points, straight lines, common polygons and curves in two dimensions, and boxes and surfaces in three dimensions [1,2]. Most common man-made objects can be described by Euclidean geometry, such as books, desks, lighthouses, houses, and other buildings.…”