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

Kim, Young Ik; Geum, Young Hee

doi:https://doi.org/10.1155/2016/8436759

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Real and Complex Dynamics of Iterative Methods

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Research Article | Open Access

Volume 2016 | Article ID 8436759 | https://doi.org/10.1155/2016/8436759

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

Young Ik Kim¹and Young Hee Geum¹

Academic Editor: Juan R. Torregrosa

Received04 Sept 2015

Revised26 Dec 2015

Accepted04 Jan 2016

Published07 Mar 2016

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 [1–7] 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 [11–17] 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 .

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 [28–30] 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 + . This completes the proof.

Theorem 14. Let in (34). Then the following hold:(a)If , then .(b)If , then .(c)If , then , where − .(d)Let . Let be a fixed point of satisfying . Then ; holds for .

Proof. The proofs of (a)–(c) immediately follow from the same argument as used in the proofs of Theorem 13. For the proof of (d), via direct computation with the aid of Mathematica symbolic capability, we find , where + . This completes the proof.

Table 2 summarizes the stability results for the strange fixed points of for special -values with .

We are ready to discuss the stability of the fixed points described in Theorems 10 and 11 in terms of parameter .

Theorem 15. Let and . Then the following hold:(a)The strange fixed point becomes an attractor, a parabolic (indifferent, neutral) point, and a repulser, respectively, when , , and .(b)The strange fixed point is a superattractor if .

Proof. (a) From the case of in (34), we find . Solving for , we obtain an ellipse in the cross-sectional -parameter plane for to be a parabolic point, where , . (b) Solving easily yields .

Theorem 16. Let and . Then the following hold:(a)The strange fixed point is a parabolic (neutral, indifferent) point, respectively, when , , and > .(b)The strange fixed point is a superattractor if .

Proof. From the case of in (34), we find ; = . Solving for , we obtain an ellipse in the cross-sectional -parameter plane for to be a parabolic point, where , . (b) Solving easily yields .

We now proceed to discuss the stability of the strange fixed points for conjugate map with using . As a consequence of Theorems 13 (g) and 14 (d) together with Corollary 12, the stability can be stated at most five strange fixed points including . Then the stability of these fixed points can be best described by illustrative conical surfaces shown in Figures 1-2. The top row of each figure refers to a stability surface for strange fixed point . The stability surfaces for the remaining fixed points are displayed in order from top to bottom and from left to right in each case of and . The underlying theory is clearly verified via cross-sectional views of the stability surfaces with -parameter domains.

4. Extraneous Fixed Points

In this section, we will consider different complex dynamics behind the extraneous fixed points to be defined now. The fixed points of are zeros of under consideration. The iteration function , however, might possess other fixed points that are not zeros of . Such fixed points different from zeros of are called the extraneous fixed points [31, 32] of the iteration function . Extraneous fixed points may form attractive, indifferent, repulsive cycles or periodic orbits to display chaotic dynamics behind the basin of attraction under investigation. The existence of such extraneous fixed points would affect the global iteration dynamics, which was demonstrated via König functions by Vrscay and Gilbert [32]. Particularly the presence of attractive cycles induced by the extraneous fixed points of may alter the basin of attractions due to the trapped sequence . Even in the case of repulsive or indifferent fixed points, an initial value chosen near a desired root may converge to another unwanted remote root. Indeed, these aspects of the Schröder functions [32, 33] were observed in an application to the family of functions for simple-root finders. By taking a particular member of the family into account for multiple-root finders, we are further interested in the dynamics applied to prototype quadratic polynomial raised to the power , being the multiplicity of zero under consideration. Such dynamical aspects motivate our investigation of the extraneous fixed points that may affect the basins of attraction for the proposed methods (5).

The structure of in (20) clearly characterizes a variety of iterative methods. The zero of is obviously a fixed point of . The points for which are extraneous fixed points of .

Let represent when is a finite-order rational function of . Then it would be of great interest for us to investigate the complex dynamics of the rational iterative map of the form [26]in connection with the basins of attraction for a variety of polynomials . Clearly, represents classical Newton’s method with weight function and may possess its fixed points as zeros of or extraneous fixed points associated with .

We now turn to complex dynamics [28–30] behind the basins of attraction of iterative map (36) applied to a prototype quadratic polynomial raised to the power of . We are interested also in the investigation of unified dynamics associated with these extraneous fixed points. To this end, we apply a simple quadratic polynomial raised to the power of multiplicity , that is, to , simple-root cases of which were introduced by Cayley [34] and Vrscay and Gilbert [32] in dynamical studies of the Schröder and König functions for a family of functions , , to minimize perturbations of the Julia set boundaries.

Hence in this section we will discuss the complex dynamics of (36) associated with its extraneous fixed points. To this end, we first write associated with applied to in the form ofwhere and with is a constant independent of ; are polynomials having no common factors withHence, the roots of may indeed express the desired extraneous fixed points of , provided that . In this paper, we limit ourselves to considering a simple form of by selecting , , and with and so that we have and . Consequently, and reduce to the following:

In order to compare the dynamics behavior of (36) behind the extraneous fixed points, let us now investigate the corresponding of the existing three optimal methods, Kan, Sol, and Li, introduced in Section 1. By similarly following the development procedure of as shown in (37), we find with where and

Since in (40) or (41) defines a high-order rational function as the multiplicity increases, it is convenient to study the typical cases of for locating the corresponding extraneous fixed points by solving for . In fact, Table 3 lists the extraneous fixed points and their stability from the value of applied to a prototype polynomial , respectively, for values of . As can be seen in the table, all extraneous fixed points of the listed methods are found to be repulsive. Observe that methods Sol and Li do not possess the extraneous fixed points when . In addition, critical points of the proposed methods applied to a polynomial are found and displayed in Table 4 for values of .

In the latter part of Section 6, complex dynamics behind the extraneous fixed points will be discussed along with chaotic behavior of rational iterative maps (36) when applied to various polynomials , based on visual description of their basins of attraction along with comparison of their dynamic properties and characteristics.

5. Numerical Experiment

We have conducted numerical experiments with a number of test functions using Mathematica Version 7 to confirm the optimal fourth-order convergence. We have assigned significant digits to the minimum number of precision digits and prescribed error bound of throughout the current experiment. The initial values are selected close to the sought zero for guaranteed convergence to the desired root. All computations have been performed by Mathematica Version 7 with AMD Kaveri 7850 CPU having 3700 Mhz of clock core speed under Windows 7 operating system.

Definition 17 (computational convergence order). Assume that theoretical asymptotic error constant and convergence order are known. Define as the computational convergence order. Note that .

Typical methods have been applied to the test functions F₁–F₆ below:As seen in Table 5, the order of convergence is four and the values of computational asymptotic error constant well approach theoretical value .

The following test functions listed below further confirm the convergence behavior of our proposed methods (5). ConsiderTable 6 shows the comparison of among listed methods Kan, Sol, and Li and Y1–Y6 described in Section 1. In Table 6, the least errors within the prescribed error bound are highlighted in bold face. Although we are limited to the selected current experiments, within three iterations, a strict comparison shows that method displays slightly better convergence for test functions and , and method Sol displays slightly better convergence for test function , while method Kan displays slightly better convergence for test functions and . In addition, both methods Sol and Li show similar convergence for test function . If we closely view the definition of the asymptotic error constant, we will find that the local convergence is dependent on the function , an initial guess , the multiplicity , and zero . Consequently, for a given set of test functions, one method is hardly expected to always show better performance than the others.

It is important to properly select initial values guaranteeing the convergence of iterative schemes. For ensured convergence of iterative map (5), it requires good initial values close to zero . It is not easy to determine how close the initial values are to zero , since initial values are generally dependent upon computational precision, error bound, and the given function under consideration. One efficient way of selecting stable initial guesses is to directly use visual basins of attraction [27, 35, 36]. Since the area of convergence can be seen on the basins of attraction, it would be a reasonable measure of convergence behavior. One would say that a method having a larger area of convergence implies a more stable method. Obviously a quantitative analysis becomes an essential tool for measuring the size of area of convergence. In the next section, we will illustrate the basins of attraction of the listed methods when applied to a variety of polynomials with multiple zeros and discuss underlying relevant dynamics.

6. Basins of Attraction

Throughout the current dynamics experiment, for effective constructing basins of attraction, we have employed a tolerance for convergence within a maximum of 40 iterations. To illustrate the desired basins of attraction, we first take a uniform grid point in a square region centered at the origin of the complex plane, which contains all roots of test functions selected. We then paint the initial points on the basins of attraction, with a diversity of colors ranging from bright ones to dark ones based on the iteration number for convergence. In Figures 3–8, the black points are regarded as the points for which the corresponding iteration scheme starting from an initial point does not converge to any root of the test polynomial under consideration. We have applied all the methods mentioned in Section 1 to a variety of polynomials having multiple roots with multiplicity of . In Tables 7–12, abbreviations CPU, TCON, AVG, and TDIV denote CPU time measured in units of seconds for convergence, the number of total convergent points, the number of average iteration for convergence, and the number of divergent points, respectively. At this point, we now begin by presenting various examples to display the desired basins of attraction.

(a)

(b)

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As a first example, we have taken the following polynomial: whose roots , are all real with multiplicity . Based on Table 7 and Figure 3, we find that has shown best AVG and TDIV, followed by Kan and . As can be seen in Figure 3, Sol has shown considerable amount of black points. These points causing divergence behavior were observed from the last column of Table 7. The best result for CPU is given by Zhou and the worst one is given by .

Our next sample has triple roots. The polynomial has three roots of multiplicity . The statistical results are listed in Table 8 and relevant basins of attraction are illustrated in Figure 4. The method has performed best AVG and TDIV. As can be seen in Figure 4, Sol has shown considerable amount of black points, while Zhou has shown a few black ones. The best result for CPU is given by and the worst one is given by .

As a third example, we have taken the following polynomial whose roots are all of multiplicity four: whose roots are . The statistical results are presented in Table 9 and relevant basins of attraction are illustrated in Figure 5. The method has shown best AVG and TDIV. As can be seen in Figure 5, Kan and have shown considerable amount of black points, while and have shown a few black ones. The best result for CPU is given by Zhou and the worst one is given by .

In the fourth example, we have taken the polynomial having two roots of unity The statistical results are presented in Table 10 and relevant basins of attraction are illustrated in Figure 6. The method has shown best AVG and TDIV. As can be seen in Figure 6, has shown considerable amount of black points, while Kan and Sol have shown a few black ones. We find that Zhou shows the best result for CPU, while shows the worst one.

In the fifth example, we have chosen the following polynomial: The statistical results are presented in Table 11 and relevant basins of attraction are displayed in Figure 7. The method has shown best AVG and TDIV. As can be seen in Figure 7, Sol has shown considerable amount of black points. We find that Zhou shows the best result for CPU, while shows the worst one.

In the last example, we have selected the polynomial whose roots are all of multiplicity five The experimented results are presented in Table 12 and relevant basins of attraction are shown in Figure 8. The method Sol has shown best AVG and TDIV. We find that Zhou shows the best result for CPU, while shows the worst one.

7. Conclusion

We have developed a triparametric family of optimal fourth-order methods and investigated their complex dynamics via Möbius conjugacy map applied to a polynomial of the form along with the stability analysis of strange fixed points. Besides, different complex dynamics have been also investigated from a viewpoint of extraneous fixed points of the proposed methods (5) when applied to a prototype quadratic polynomial raised to the power of known multiplicity . To extract information on better initial guesses of an iterative method, we need to carefully examine the basins of attraction. In this paper, we have illustrated a number of basins of attraction for the listed optimal fourth-order iterative methods when applied to various complex polynomials as well as statistical analysis featuring data for CPU time and other tabulated numbers for convergence behavior. From the area of convergence on the basins of attraction, more effective information can be obtained for selecting stable initial values of the iterative methods under consideration. As a future research work, we will devote ourselves to developing a higher-order family of iterative methods along with statistical data analysis as well as an illustrative investigation of the desired basins of attraction.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The corresponding author (Y. H. Geum) was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education under the research grant (Project no. 2015-R1D1A3A-01020808).

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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.

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