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Discrete Dynamics in Nature and Society
Volume 2016, Article ID 8436759, 23 pages
http://dx.doi.org/10.1155/2016/8436759
Research Article

A Triparametric Family of Optimal Fourth-Order Multiple-Root Finders and Their Dynamics

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

Received 4 September 2015; Revised 26 December 2015; Accepted 4 January 2016

Academic Editor: Juan R. Torregrosa

Copyright © 2016 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

We investigate the complex dynamics of a triparametric family of optimal fourth-order multiple-root solvers by analyzing their basins of attraction along with extensive study of Möbius conjugacy maps and extraneous fixed points applied to a prototype quadratic polynomial raised to the power of the known integer multiplicity . A uniform grid centered at the origin covering square region is chosen to display the initial points on each basin of attraction according to a coloring scheme based on their orbit behavior. With illustrative basins of attractions applied to various test polynomials and the corresponding statistical data for convergence as well as a number of comparisons made among the listed methods, we confirm our investigation and analysis developed in this paper.

1. Introduction

Many researchers [17] have shown their interest in the dynamics of iterative methods locating the multiple roots [8, 9] of a nonlinear equation. To ensure the convergence of an iterative method in a root-finding problem [10], it is very important to take a good initial value [1117] close to the desired zero of the given nonlinear equation under consideration. In connection with such a choice of a good initial value, we pay a special attention to the complex dynamics for a number of optimal fourth-order multiple-root finders by investigating their basins of attraction.

Definition 1. Let be a sequence converging to and let be the th iterate error. If there exist real numbers and a nonzero constant such that the following error equation holdsthen or is called the asymptotic error constant and is called the order of convergence [18].

Definition 2. Let be the number of distinct functional or derivative evaluations per iteration. The efficiency index [19] is defined by , where is the order of convergence.

In our study, all the listed methods have the same agreeing with Kung-Traub optimality [19] conjecture. We investigate the basins of attraction of a number of iterative methods with various test polynomials. Typical fourth-order multiple-root finders developed by Kanwar et al. [20], Soleymani and Babajee [21], and Shengguo et al. [6] are conveniently denoted by Kan, Sol, and Li for later use. Besides, by extending the work of Geum and Kim [22, 23], we propose a triparametric family of optimal fourth-order methods Yk’s whose developments will be described in Section 2. They are listed below in their respective order.

Kan:

Sol:

Li:

Yk:where ,  ,  ,  ,  , and are parameters to be chosen for fourth order of optimal convergence [19, 24]. Typical cases of methods Yk’s are presented in Table 1 for with selected parameters , , and .

Table 1: Typical methods with selected parameters and constants .

2. Convergence Analysis

We describe the main theorem regarding the convergence behavior of proposed family of methods (5) and select parameters ,  , and for the quartic convergence with the aid of Taylor expansion and symbolic computation of Mathematica [25].

Theorem 3. Let have a zero with integer multiplicity and be analytic in a small neighborhood of . Let be an initial guess chosen in a sufficiently small neighborhood of . Let , and be free constant parameters. Let + , , + , and . Then iterative method (5) is of order four and defines a triparametric family of iterative methods with the following error equation: where + , for , and .

Proof. Using Taylor’s series about , we have where ,  ,  , and for .
Dividing (7) by (8) gives uswhere ,  , and + .
Letting with the above relation (9), we haveSubstituting (7)–(10) into (5), we get the error equation:where   and coefficients depend on the parameters ,  ,  ,  ,  ,  , and and the function .
Solving and for and , we haveAfter substituting and into , we solve . Due to the fact that is independent of and , solving for and , we havewhere and .
Applying into (12)–(14) with and , we have the following relations:Thanks to symbolic computation of Mathematica [25], we reach the error equation below:where withcompleting the proof.

Remark 4. If ,  , and   are selected, then we find relations: In this case, proposed method (5) reduces to method Li given by (4).

3. Conjugacy Maps and Dynamics

Multipoint iterative methods [19] solving a nonlinear equation of the form can be generally written as a discrete dynamical systemwhere is the iteration function. We begin by writing (5) in the form of a complex discrete dynamical system:where , , , , and ; ,  , and   are given, respectively, by (12), (13), and (14); are free parameters.

Definition 5. Let and be two functions (dynamical systems). One says that and are conjugate if there is a function such that . Then the map is called a conjugacy [26].

Remark 6. Note that a conjugacy indeed preserves the dynamical behavior between the two dynamical systems; for example, if is conjugate to via and is a fixed point of , then is a fixed point of .
Furthermore, if is a homeomorphism, that is, if is topologically conjugate to via , and is a fixed point of , then is a fixed point of . Also, we find and . If and are invertible, then the topological conjugacy maps an orbit of , onto an orbit of , where and the order of points is preserved. Hence, the orbits of the two maps behave similarly under homeomorphism or .

Via Möbius conjugacy map ,  ,  , considered by Blanchard [27], in (20) is conjugated to satisfyingwhen applied to a quadratic polynomial raised to the power of , where and are polynomials with no common factors whose coefficients are generally dependent upon parameters ,  ,  ,  , and  . The following theorem favorably indicates that is dependent only on but independent of parameters ,  ,  , and .

Theorem 7. Let with and ,  ,  . Then is conjugate to satisfying where  ,  ,  ,  ,  ,   + ,  ,  ,  ,  ,  , and  .

Proof. Since the inverse of is easily found to be , we find after a lengthy computation with the aid of Mathematica [25] symbolic capability: where and are polynomials of degree at most in with a single free parameter . This gives the desired result, completing the proof.

The result of Theorem 7 enables us to discover that (corresponding to fixed point of or root of ) and (corresponding to fixed point of or root of are clearly two of their fixed points of the conjugate map , regardless of -values. Besides, by direct computation, we find that is a strange fixed point [2830] of (that is not a root of ) due to the fact that , regardless of -values.

We now seek further strange fixed points including (corresponding to the original convergence to infinity in view of the fact that or ). To do so, we first investigate some properties of stated in the following theorem.

Theorem 8. Let and be given by (25). Then the following hold: (a)The leading highest-order term of is given by , provided that .(b) has a factor , provided that .(c)=, and , provided that + , with = and .(d) approaches as tends to , provided that .

Proof. After a lengthy computation and careful algebraic treatments with the aid of Mathematica, and   follow without difficulty. For the proof of , we directly compute the values of and . The proof of follows from the fact that , by using along with a highest-order term of having degree at most .

We now will begin with locating the fixed points of the iteration function . Let , whose zeros are the desired fixed points of . The result of Theorem 8 shows that and are the roots of . Hence the expression of will take the following form: where + and = + are polynomials in with ,  ,  ,   + ,  ,  , and   + and with  ,  ,  ,  ,  , and   given in Theorem 7.

As a result, ,  , and are clearly the fixed points of . Among these fixed points, is a strange fixed point that is not the root or . Further strange fixed points are possible from the roots of . The following theorem describes some properties of .

Theorem 9. Let be given by (26). Then the following hold: (a) for , regardless of -values.(b) has double roots at and ; that is, it has a factor , provided that .(c) for , regardless of -values, where ,  .(d) has also double roots at and ; that is, it has a common factor as shown in , provided that .(e), and , for , provided that .

Proof. Via careful algebraic treatments and symbolic computation with the aid of Mathematica, ,  , and   follow without difficulty. For the proof of , we directly compute the values of and , for . In view of the relations, ,  , and  , we also find and . The proof of follows from the fact that and , for any . We also find and , for . We also find and , for .

With the use of properties of , we now consider some strange fixed points along with their stability for selected values of and .

To continue our investigation of dynamics behind iterative map (20) applied to a quadratic polynomial raised to the power of , , we will describe the fixed points of in (25) and their stability. In view of the fact that is a fixed point of for a fixed point of with , we are interested in the explicit form of for below:where we conveniently denote

This enables us to discover that (corresponding to fixed point of or root of ) and (corresponding to fixed point of or root of are clearly two of their fixed points regardless of . To find further strange fixed points, we solve equations in (27) for with typical values of .

We now investigate further strange fixed points including (corresponding to the original convergence to infinity in view of the fact that or ). By direct computation, we will describe the roots of for . To this end, we first check the existence of -values for common factors (divisors) of and . Besides, will be checked if it has a divisor or . The following theorem best describes relevant properties of such existence as well as explicit strange fixed points.

Theorem 10. Let in (27). Then the following hold:(a)If , then and the strange fixed points are given by and .(b)If , then and the strange fixed points are given by and .(c)If , then and the strange fixed points are given by ,  .(d)If , then and the strange fixed points are given by and .(e)Let . Then holds for . Hence, if is a root of , then so is .

Proof. (a)–(c) Suppose that and for some values of . Observe that parameter exists in a linear fashion in all coefficients of both polynomials. By eliminating from the two polynomials, we obtain the relation . Hence, any root of is a candidate for a common divisor of and . Substituting all the roots of into and , we find required relations for and solving them for , we find . The remaining part of the proof is straightforward. (d) If is a divisor of , then yielding , which is already handled in (b). If is a divisor of , then , yielding . Then remaining proof is trivial. (e) By direct substitution, we find without difficulty. Hence if and only if for . This completes the proof.

Theorem 11. Let in (27). Then the following hold:(a)If , then and the strange fixed points are given by , , and .(b)If , then and the strange fixed points are given by and (triple).(c)If , then + and the strange fixed points are given by .(d)Let . Then holds for . Hence, if is a root of , then so is .

Proof. The proofs immediately follow from the same argument as used in the proofs of Theorem 10.

As a result of Theorem 9 (a), we find the fixed points of , that is, the roots of explicitly as stated in the following corollary.

Corollary 12. Let be a root of , that is, a root of   for in (27). Suppose that and have no common factors for some suitable -values. Then the roots of for are explicitly given by the following:(a)The four roots of are explicitly found to be where and .(b)The eight roots of are explicitly found to be where ,  ,  ,  ,  ,  ,  , and + .

Proof. Since is a root of for , so is from the result of Theorem 9 (a). For the proof of (a), can be written as a product of two factors: By expanding the right side of the above equation and comparing the coefficients of the first- and second-order terms, we find two relations: which gives the desired values of and  . Then the four roots can be found explicitly from or . Similarly for the proof of (b), can be written as a product of two factors each of which is further decomposed into two factors:By the same argument as used in the proof of (a), the desired result follows. This completes the proof.

We are now ready to determine the stability of the fixed points. To do so, it is necessary to compute the derivative of from (24): whereWe first check the existence of -values for common factors (divisors) of and . Besides, will be checked if it has divisors ,  ,  . The following theorem best describes relevant properties of such existence as well as explicit strange fixed points.

Theorem 13. Let in (34). Then the following hold:(a)If , then .(b)If , then .(c)If , then .(d)If , then .(e)If , then .(f)If , then .(g)Let . Let be a fixed point of satisfying . Then holds for .

Proof. The proofs of (a)–(f) immediately follow from the same argument as used in the proofs of Theorem 10. Eliminating from the two polynomials and plays a key role in obtaining the relation , whose roots enable us to deduce some desired -values. Additional requirement that are candidates for common divisors of and gives only . For the proof of (g), via direct computation with the aid of Mathematica symbolic capability, we find , where +