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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 263893, 8 pages
http://dx.doi.org/10.1155/2012/263893
Research Article

Attracting Periodic Cycles for an Optimal Fourth-Order Nonlinear Solver

Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea

Received 28 November 2011; Accepted 14 February 2012

Academic Editor: Khalida Inayat Noor

Copyright © 2012 Mi Young Lee and Changbum Chun. 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

We consider an optimal fourth-order method for solving nonlinear equations and construct polynomials such that the rational map arising from the method applied to these polynomials has an attracting periodic orbit of any prescribed period.

1. Introduction

Recently, an optimal fourth-order iterative method to find a simple root , that is, and , of a nonlinear equation , which is given by where was proposed in [1] and its dynamics behavior was investigated and analyzed in detail. By an optimal method, we mean a multipoint method without memory which requires functional evaluations per iteration, but achieves the order of convergence [2].

The method (1.1) may be used to approximate both real and complex roots of the nonlinear equation with . If the initial guess is chosen sufficiently near a zero of , then an iterative method is expected to converge. This is, however, not true in general if there exists an attracting periodic orbit or cycle of period (whose definition will be introduced below). If the initial guess happens to be chosen from the basin of attraction of an attractive periodic cycle, the sequence converges to the attractive cycle, not to a zero of since any root of may be considered as a period orbit of period 1. Thus the existence of attracting periodic cycles of period greater than or equal to 2 could interfere with an iterative method searching for a root of the nonlinear equation. As a result, it has been an important concern from practical aspect in iteration theory to construct specific polynomials for a given method such that the map arising from the iterative method applied to the polynomials has an attractive periodic orbit. In this direction there was some result in [3] where attractive periodic orbits of any prescribed period were constructed for some classical third-order methods. Motivated by this, in this paper we extend the construction of attractive periodic cycles of any prescribed period to higher-order iterative methods. To this end, we will recall some preliminaries, see for example Milnor [4], Amat et al. [5], and Chun et al. [1]. Let be a rational map on the Riemann sphere.

Definition 1.1. For we define its orbit as the set

Definition 1.2. A point is a fixed point of if .

Definition 1.3. A periodic point of period is such that where is the smallest such integer. The set of the distinct points is called a periodic cycle.

Remark 1.4. If is periodic of period then it is a fixed point for .

Definition 1.5. If is a periodic point of period , then the derivative is called the eigenvalue of the periodic point .

Remark 1.6. By the chain rule, if is a periodic point of period , then its eigenvalue is the product of the derivatives of at each point on the orbit of , and we have that is, all the points of a cycle have the same eigenvalue.

We classify the fixed points of a map based on the magnitude of the derivative.

Definition 1.7. A point is called attracting if , repelling if , and neutral if . If the derivative is zero then the point is called superattracting.

Definition 1.8. The Julia set of a nonlinear map , denoted , is the closure of the set of its repelling periodic points. The complement of is the Fatou set .
By its definition, is a closed subset of . A point belongs to the Julia set if and only if dynamics in a neighborhood of displays sensitive dependence on the initial conditions, so that nearby initial conditions lead to wildly different behavior after a number of iterations. As a simple example, consider the map on . The entire open disk is contained in , since successive iterates on any compact subset converge uniformly to zero. Similarly the exterior is contained in . On the other hand if is on the unit circle than in any neighborhood of any limit of the iterates would necessarily have a jump discontinuity as we cross the unit circle. Therefore is the unit circle. Such smooth Julia sets are exceptional.

Lemma 1.9 (Invariance lemma (Milnor [4])). The Julia set of a holomorphic map is fully invariant under . That is, belongs to if and only if belongs to .

Lemma 1.10 (Iteration Lemma). For any , the Julia set of the -fold iterate coincides with .

Definition 1.11. If is an attracting periodic orbit of period , we define the basin of attraction to be the open set consisting of all points for which the successive iterates converge towards some point of .
The basin of attraction of a periodic orbit may have infinitely many components.

Definition 1.12. The immediate basin of attraction of a periodic orbit is the connected component containing the periodic orbit.

Lemma 1.13. Every attracting periodic orbit is contained in the Fatou set of . In fact the entire basin of attraction for an attracting periodic orbit is contained in the Fatou set. However, every repelling periodic orbit is contained in the Julia set.

2. Attractive Cycles Results

Let us consider the map associated to where for which the optimal method (1.1) is written as

Toward the aim to construct periodic orbits of any prescribed period for the method (1.1), we have the following characterization.

Proposition 2.1. Let be a set of distinct complex numbers, and let be a complex analytic function. Then is periodic orbit of period of iteration function if and only if where , .

Proof. Assume that is a periodic orbit of . Then for and . We have Therefore, we obtain Conversely, suppose that satisfies condition (2.4). Then we easily have

Proposition 2.2. For any positive integer , there exists a polynomial of degree less than or equal to for which has a periodic orbit of period .

Proof. Let be a given a set of distinct complex numbers, let be a set of nonzero complex numbers, and let be another set of nonzero complex numbers such that where .
Assume that there exists a polynomial such that Then by Proposition 2.1, is a periodic orbit of period of . Using (2.9) and condition (2.8) yields
We now show that such a polynomial exists. For this, we use the Hermite interpolation procedure. We begin the construction by writing as where the functions are polynomials defined as follows:
To determine the polynomial we must find suitable coefficients for . For this we must solve a linear system of equations with unknown of the form . The matrix associated with the linear system of equations is given by and the column vectors are given by and . The components of the upper triangle of the matrix are zero, and the components of the diagonal of are nonzero. Therefore, the linear system has a unique solution.

Example 2.3. Let us construct a polynomial for which the iterative method has a periodic orbit of period 2. For this, we let and . Then we have , , , and . We construct a polynomial , where , , , , , . Then matrix is given by By evaluating we obtain
Now we consider the system of linear equations , where and . Here, we can take . Hence we have the following system: whose solution is . Therefore, the polynomial is given by = . From (2.1), we obtain and . So, is a periodic orbit of period 2 for the method .

Proposition 2.4. Let be a polynomial for which has a periodic orbit of period , say . If and or and for some , then is an superattracting periodic orbit of period for .

Proof.. Without loss of generality, we may assume that and . By the chain rule, we have Note that on differentiating (2.1) we have We therefore obtain . By (2.17) we conclude that .

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011–0025877).

References

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