Discrete Dynamics in Nature and Society

Volume 2015, Article ID 378517, 12 pages

http://dx.doi.org/10.1155/2015/378517

## On Constructing Two-Point Optimal Fourth-Order Multiple-Root Finders with a Generic Error Corrector and Illustrating Their Dynamics

Department of Applied Mathematics, Dankook University, Cheonan 330-714, Republic of Korea

Received 2 September 2015; Accepted 15 October 2015

Academic Editor: Alicia Cordero

Copyright © 2015 Young Ik Kim and Young Hee Geum. 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

With an error corrector via principal branch of the *m*th root of a function-to-function ratio, we propose optimal quartic-order multiple-root finders for nonlinear equations. The relevant optimal order satisfies Kung-Traub conjecture made in 1974. Numerical experiments performed for various test equations demonstrate convergence behavior agreeing with theory and the basins of attractions for several examples are presented.

#### 1. Introduction

Iterative root-finding methods have been constantly developed by numerous researchers [1–3] to solve more accurately the root of a nonlinear equation that arises frequently in a scientific world. Classical Newtons method below, is widely used for a simple root under normal circumstances, provided that an initial guess is chosen close enough to . Likewise, modified Newtons method [4] of the formis most popular for a multiple root with its integer multiplicity .

*Definition 1. *Let be a sequence converging to and be the th iterate error. If there exist real numbers and such that the following error equation holds:then or is called the asymptotic error constant and is called the order of convergence [5].

Note that both methods (1) and (2) are quadratically convergent one-point optimal [3] methods. Two-point higher-order methods for multiple roots can be found in papers [6–11]. Among these, we introduce, here, respectively, in (4), (5), and (6), some interesting works of Soleymani and Babajee [12], Kanwar et al. [13], and Zhou et al. [14] who have recently developed fourth-order multiple-root finders:

By a close inspection of iterative methods (1), (2), and (4)–(6), we find that an iterative method can be constructed in the following form: where is implicitly dependent upon , for example, with in (1) and in (6). We may regard as an error-correcting function. Consequently, it would be natural to call the function as the error corrector. Usually takes the form of , with as a weighting function that is widely known among many researchers. A more generic form of the error corrector will be investigated during the course of developing new quartic-order multiple-root finders.

The main aim of this paper is to design new two-step two-point optimal quartic-order multiple-root finders with multiplicity of . The first step is to compute using usually with a Newton-like method or other. The second step is to update in the first step by introducing an error corrector formed by and a principal branch of . We will check the optimality based on the Kung-Traub conjecture [3] that a multipoint method [15] without memory can achieve its convergence order at most of for functional evaluations.

This paper is comprised of four sections as follows. Following this introductory section, Section 2 describes main results with convergence analysis for newly proposed two-point optimal fourth-order multiple-root finders. Principal branch of a logarithmic function plays a crucial role in developing new methods in view of the relation . The convergence analysis includes the derivation of the error equation for the proposed methods. In Section 3, special cases of error-correcting functions are treated with tabulated results and labeled case numbers. Two types of error-correcting functions are constructed based on bivariate polynomials and rational functions. In the first part of Section 4, with error-correcting functions properly chosen from Section 3, a variety of numerical examples are presented for a wide selection of test functions. A comparison for the convergence behavior is made among the proposed methods and the listed existing methods (4)–(6). The second part of Section 4 discusses related dynamics of maps (8) behind the basins of attraction. Dynamical properties of the proposed methods along with their illustrative basins of attraction are displayed with detailed analyses and comments. Section 5 describes overall conclusion as well as possible future work.

#### 2. Main Results

We first assume that a function has a multiple root with integer multiplicity and is analytic [16] in a small neighborhood of . Then with the concept of error corrector introduced in (7) we propose new two-step iterative multiple-root finders below, given an initial guess sufficiently close to : for ,where is a parameter and is holomorphic [17] in a neighborhood of , where is to be determined later for optimal quartic-order convergence. Since is a one-to- multiple-valued function, we choose as a principal analytic branch given by , with for ; this convention of arg() for agrees with that of command of Mathematica to be adopted in numerical experiments of Section 4. By means of further inspection of , we find that is characterized in such a way that , as shown by (15).

These suggested methods require one derivative and two functions in order to achieve optimal order of four. In this section, we establish a main theorem describing the convergence analysis regarding proposed methods (8) and find out how to select the parameter and the error-correcting function for optimal fourth-order convergence.

Theorem 2. *Let have a zero with integer multiplicity and be analytic in a small neighborhood of . Let and for . Let be an initial guess chosen in a sufficiently small neighborhood of . Let be holomorphic in a neighborhood of . Let for , . Suppose that relations , , and hold. Then iterative methods (8) are optimal and of order four and possess the following error equation: *

*Proof. *Three functional evaluations evidently are eligible for optimal convergence order in the sense of Kung-Traub. Hence, it suffices to determine the constant parameter and some properties of the error-correcting function for fourth-order convergence. Applying the Taylors series expansion of about , we get the following relations:where , , and for .

Dividing (10) by (11), we havewhere , , and .

Letting for convenience with the above relation (12), we obtainExpanding about the multiple root leads us to relationHence, we getHence, mentioned in the description of (8) is selected to be as desired. It must be emphasized that denotes a principal analytic branch as mentioned earlier from (8). Noting that , Taylor expansion of about up to fourth-order terms in both variables yields after retaining up to fourth-order terms in with :Hence by substituting (10)–(16) into the proposed method (8), we obtain the error equation as where and the coefficients , , generally depend on the parameters , , and and the function .

Solving for , we obtain Substituting (18) into and simplifying, we have from which and must be satisfied independently of . Hence, Substituting (18) and (20) into , we get Solving the above equation independently of for , and , we obtain Substituting (18), (20), and (22) into and simplifying, we have To make independently of and , we obtain With the aid of symbolic computation of Mathematica [18], we substitute (18)–(24) into to arrive at the following relation: This completes the proof.

#### 3. Special Cases of Error-Correcting Functions

Using relations (18)–(24), the bivariate Taylor polynomial of is easily given by Here notaions and are introduced for simplicity. Special cases of are considered here. In each case, relevant coefficients are determined based on relations (18)–(24). If , then (26) yields

*Remark 3. *If we take the Taylor polynomial of being in the form of , then and hold. Hence, method (6) is a subcase of this case in which is satisfied under the restriction of to .

Although a variety of forms of error-correcting functions are available in view of (26), we will limit ourselves to considering two cases of error correctors comprising low-order bivariate polynomials or simple rational functions.

*Case 1 ( with a bivariate polynomial). *In this case, , and can be regarded as parameters to be chosen to satisfy We list typical subcases with selected parameters and in Table 1, where SN stands for the corresponding subcase identification number.