Journal of Nonlinear Dynamics

Volume 2015, Article ID 162818, 9 pages

http://dx.doi.org/10.1155/2015/162818

## On the Complex Dynamics of Continued and Discrete Cauchy’s Method

Department of Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria

Received 2 May 2015; Accepted 17 September 2015

Academic Editor: Ivo Petras

Copyright © 2015 Mohamed Lamine Sahari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Let be a complex polynomial of fixed degree . In this paper we show that Cauchy’s method may fail to find all zeros of for any initial guess lying in the complex plane and we propose several ways to find all zeros of a given polynomial using scaled Cauchy’s methods.

#### 1. Introduction

For a large class of computational problems a zero of a polynomial in the complex plane has to be found. Cauchy’s method [1] is an interesting candidate for the numerical solution. Let be a starting point for a zero of . We setIf is a simple root of and is selected in such way that the following criterion is satisfied,the Cauchy method has the following properties:(a)is an attractive fixed point of a map ().(b)For sufficiently close to , the sequence of iterates,will converge to the root and that is, the sequence (3) converges linearly.

However, someone may wonder about the basin of attractionof a root of and the relationship with the vector field and trajectories of the differential system:known as the continuous Cauchy’s method. It is assumed that discrete Cauchy’s method (1) may be interpreted as Euler step for the differential equation (6).

#### 2. Examples and Graphics

Let us consider a famous example given by Cayley (1879, [2]). Letfor , , and , the three roots of , and color a point blue if converges to , color it green if converges to , and color it red if converges to Any remaining point gets coloured white (e.g., if ). Figure 1 shows basins of attraction for (7) with a value of , respectively, taken and for the region in Figure 1(a) and in Figure 1(b). Figure 2 indicates corresponding basins of attraction for the gradient method with and for the region in Figure 2(a) and in Figure 2(b). Note that the boundary of the basins of attraction for each root of exhibits a fractal structure (see Figures 1–5). These figures show also the role played by the parameter ; this parameter is used on one hand to enlarge the domains in the basins of attraction which contain the roots of the polynomial and on the other hand to ensure the convergence of the method. The question which arises naturally is of knowing what happens when in (1). The answer can be given by differential system (6). Denote , where To see this, let Rearranging yields Finally, letting we have