Journal of Complex Analysis

Volume 2015 (2015), Article ID 259167, 9 pages

http://dx.doi.org/10.1155/2015/259167

## An Optimal Fourth Order Iterative Method for Solving Nonlinear Equations and Its Dynamics

^{1}Department of Applied Sciences, DAV Institute of Engineering and Technology, Kabir Nagar, Jalandhar 144008, India^{2}Department of Mathematics, DAV College, Jalandhar 144008, India^{3}I.K. Gujral Punjab Technical University, Kapurthala 144601, India

Received 12 July 2015; Revised 7 October 2015; Accepted 12 October 2015

Academic Editor: Ying Hu

Copyright © 2015 Rajni Sharma and Ashu Bahl. 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 present a new fourth order method for finding simple roots of a nonlinear equation . In terms of computational cost, per iteration the method uses one evaluation of the function and two evaluations of its first derivative. Therefore, the method has optimal order with efficiency index 1.587 which is better than efficiency index 1.414 of Newton method and the same with Jarratt method and King’s family. Numerical examples are given to support that the method thus obtained is competitive with other similar robust methods. The conjugacy maps and extraneous fixed points of the presented method and other existing fourth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane.

#### 1. Introduction

Solving nonlinear equations is a common and important problem in science and engineering [1, 2]. Analytic methods for solving such equations are almost nonexistent and therefore it is only possible to obtain approximate solutions by relying on numerical methods based on iterative procedures. With the advancement of computers, the problem of solving nonlinear equations by numerical methods has gained more importance than before.

In this paper, we consider the problem of finding simple root of a nonlinear equation , where is a continuously differentiable function. Newton method is probably the most widely used algorithm for finding simple roots, which starts with an initial approximation closer to the root (say, ) and generates a sequence of successive iterates converging quadratically to simple roots (see [3]). It is given by

In order to improve the order of convergence of Newton method, many higher order multistep methods [4] have been proposed and analyzed by various researchers at the expense of additional evaluations of functions, derivatives, and changes in the points of iterations. An extensive survey of the literature dealing with these methods of improved order is found in [3, 5, 6] and references therein. Euler method and Chebyshev method (see Traub [3]) Weerakoon and Fernando [7], Ostrowski’s square root method [5], Halley [8], Hansen and Patrick [9], and so forth are well-known third order methods requiring the evaluation of , , and per step. The famous Ostrowski’s method [5] is an important example of fourth order multipoint methods without memory. The method requires two and one evaluations per step and is seen to be efficient compared to classical Newton method. Another well-known example of fourth order multipoint methods with same number of evaluations is King’s family of methods [10], which contains Ostrowski’s method as a special case. Chun et al. [11–13], Cordero et al. [14], and Kou et al. [15, 16] have also proposed fourth order methods requiring two evaluations and one evaluation per iteration. Jarratt [17] proposed fourth order methods requiring one evaluation and two evaluations per iteration. All of these methods are classified as multistep methods in which a Newton or weighted-Newton step is followed by a faster Newton-like step.

Through this work, we contribute a little more in the theory of iterative methods by developing the formula of optimal order four for computing simple roots of a nonlinear equation. The algorithm is based on the composition of two weighted-Newton steps and uses three function evaluations, namely, one evaluation and two evaluations.

On the other hand, we analyze the behavior of this method in the complex plane using some tools from complex dynamics. Several authors have used these techniques on different iterative methods. In this sense, Curry et al. [18] and Vrscay and Gilbert [19, 20] described the dynamical behavior of some well-known iterative methods. The complex dynamics of various other known iterative methods, such as King’s and Chebyshev-Halley’s families, Jarratt method, has also been analyzed by various researchers; for example, see [13, 21–26].

The paper is organized as follows. Some basic definitions relevant to the present work are presented in Section 2. In Section 3, the method is developed and its convergence behavior is analyzed. In Section 4, we compare the presented method with some existing methods of fourth order in a series of numerical examples. In Section 5, we obtain the conjugacy maps and possible extraneous fixed points of these methods to make a comparison from dynamical point of view. In Section 6, the methods are compared in the complex plane using basins of attraction. Concluding remarks are given in Section 7.

#### 2. Basic Definitions

*Definition 1. *Let be a simple root and let , , be a sequence of real or complex numbers that converges towards . Then, one says that the order of convergence of the sequence is if there exists such thatfor some , is known as the asymptotic error constant.

*Definition 2. *Let be the error in the th iteration; one calls the relationthe error equation. If one can obtain error equation for any iterative method, then the value of is the order of convergence.

*Definition 3. *Let be the number of new pieces of information required by a method. A “piece of information” typically is any evaluation of a function or one of its derivatives. The efficiency of the method is measured by the concept of efficiency index [27] and is defined bywhere is the order of the method.

*Definition 4. *Suppose that , , and are three successive iterations closer to the root . Then, the computational order of convergence (see [7]) is approximated by using (3) as

#### 3. Development of the Method

Let us consider the two-step weighted-Newton iteration scheme of the typewhere , , , and are some constants which are to be determined. A natural question arises: is it possible to find , , , and such that the iteration method (6) has maximum order of convergence? The answer to this question is affirmative and is proved in the following theorem.

Theorem 5. *Let be a real or complex function. Assuming that is sufficiently differentiable in an interval , if has a simple root and is sufficiently close to , then (6) has fourth order convergence if , , , and .*

*Proof. *Let be the error at th iteration, then . Expanding and about and using the fact that , , we havewhere , .

Using (7) and (8) and then simplifying, we obtainSubstituting (9) in first substep of (6), we get Expanding about and using (10), we have From (8) and (11), we have Using (9) and (12) in second substep of (6), we obtainwhereIn order to achieve fourth order convergence, the coefficients , , and must vanish. Therefore, , , and yield , , , and . With these values, error equation (13) turns out to beThus, (15) establishes the fourth order convergence for iterative scheme (6). This completes proof of the theorem.

Hence, the proposed scheme is given bywhere .

We denote this method by SBM.

Thus, we have derived fourth order method (16) for finding simple roots of a nonlinear equation. It is clear that this method requires three evaluations per iteration and therefore it is of optimal order.

#### 4. Numerical Results

In this section, we present the numerical results obtained by employing the presented method SBM equation (16) to solve some nonlinear equations. We compare the presented method with quadratically convergent Newton method denoted by NM defined by (1) and some existing fourth order iterative methods, namely, the method proposed by Chun et al. [13], Cordero et al. method [14], King’s family of methods [10], and Kou et al. method [16]. These above-mentioned methods are given as follows.

Chun et al. method (CLM) isCordero et al. method (CM) is King’s family of methods (KM) iswhere .

Kou et al. method (KLM) is

Test functions along with root correcting up to 28 decimal places are displayed in Table 1. Table 2 shows the values of initial approximation () chosen from both sides to the root, values of the error calculated by costing the same total number of function evaluations (NFE) for each method. Table 3 displays the computational order of convergence () defined by (5). The NFE is counted as sum of the number of evaluations of the function and the number of evaluations of the derivatives. In calculations, the NFE used for all the methods is 12. That means, for NM, the error is calculated at the sixth iteration, whereas for the remaining methods this is calculated at the fourth iteration.