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Real and Complex Dynamics of Iterative Methods

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Research Article | Open Access

Volume 2015 |Article ID 378517 | https://doi.org/10.1155/2015/378517

Young Ik Kim, Young Hee Geum, "On Constructing Two-Point Optimal Fourth-Order Multiple-Root Finders with a Generic Error Corrector and Illustrating Their Dynamics", Discrete Dynamics in Nature and Society, vol. 2015, Article ID 378517, 12 pages, 2015. https://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

Accepted15 Oct 2015
Published16 Dec 2015

Abstract

With an error corrector via principal branch of the mth 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  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  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 .

Note that both methods (1) and (2) are quadratically convergent one-point optimal  methods. Two-point higher-order methods for multiple roots can be found in papers . Among these, we introduce, here, respectively, in (4), (5), and (6), some interesting works of Soleymani and Babajee , Kanwar et al. , and Zhou et al.  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  that a multipoint method  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  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  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 , 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.

 SN 1A 0 0 0 0 0 1B 1 0 0 0 0 1C 0 0 0 0 1D 0 1 0 0 0 1E 0 1 0 0 1F 0 0 0 1 0 1G 0 0 0 1

Case 2 ( with a bivariate polynomial and rational functions). Consider where , and and are determined using (18)–(24) with two of them as free parameters. We find that , and and the remaining parameters , and are free to be chosen.
In Table 2, we list typical subcases with interesting choices of parameters and .

 SN 2A 0 −2 0 0 0 0 0 0 2B 0 0 −1 −1 0 0 0 0 0 2C 0 −1 0 0 0 0 0 0 2D 0 −2 0 1 0 0 0 0 2E 0 0 −1 −1 0 1 0 0 0 2F 0 −1 0 0 0 1 0 1 2G 0 −2 0 0 0 1 1 0 2H 0 0 −1 −1 0 0 0 0 1

4. Numerical Experiments and Dynamics

In this section, we will first describe the computational experiments of proposed methods (8) and then illustrate the complex dynamics  related to the basins of attraction  of iterative maps (26) along with comparisons among existing methods.

Throughout the experiments, we have moderately assigned 112 significant digits as the minimum number of precision digits, via Mathematica  command , to achieve the desired accuracy ensuring convergence of the proposed methods. It is necessary to compute with high accuracy for desired numerical results. In case that is not exact, it is replaced by a more accurate value which has a larger number of significant digits than the assigned .

Definition 4 (computational convergence order). Assume that theoretical asymptotic error constant and convergence order are known. Define as the computational convergence order. Note that .
If and both mutually have the same accuracy of , then would be nearly zero as becomes large and thus computing would unfavorably cause a numeric overflow. Computed values of are accurate up to 112 significant digits. To observe the reliable convergence behavior, we desire with enough accuracy of 16 digits higher than , which has 128 significant digits. To supply such , a set of following Mathematica commands are used:Although the number of significant digits of and is 112 and 128, respectively, we list the two values at most up to 15 significant digits due to the limited paper space. We set the error bound to satisfying

Iterative methods (26) with all subcases of both Cases 1 and 2 were, respectively, identified by Y1A, Y1B, Y1C, Y1D, Y1E, Y1F, and Y1G and Y2A, Y2B, Y2C, Y2D, Y2E, Y2F, Y2G, and Y2H, being Y-prefixed. Among them, typical three methods have been successfully applied to three test functions shown below:

Methods Y1D, Y2B, and Y2E in Table 3 have clearly confirmed quartic-order convergence. Table 3 lists iteration indexes , approximate zeros , residual errors , errors , and computational asymptotic error constants as well as the theoretical asymptotic error constant and computational asymptotic convergence order . The values of agree up to 10 significant digits with . Undoubtedly, the computed asymptotic order of convergence well approaches 4. The computational asymptotic error constant reveals a good agreement with the theory developed in Section 3.

 0 −3.3 5.68515 0.3 1 −3.00038943156575 0.04807797108 0.06814101693 4.28967 2 −3.00000000000000 0.06811440433 4.00005 3 −3.00000000000000 0.06814101693 4.00000 4 −3.00000000000000 0 0.000338602 0.0650381 1 55.20587710 0.02458677729 1.17628 2 0.02459092337 3.99998 3 0.02458677729 4.00000 4 0 1.6 0.400529 0.0900869 1 1.70503564060025 0.0149488 226.9649843 0.3269289253 1.28174 2 1.69008689870921 0.3158415015 4.00821 3 1.69008688293711 0.3269289253 4.00000 4 1.69008688293711
MT means method, , and .

Additional functions below are tested to further confirm the convergence of methods (8):