Real and Complex Dynamics of Iterative MethodsView this Special Issue
Research Article | Open Access
A Triparametric Family of Optimal Fourth-Order Multiple-Root Finders and Their Dynamics
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.
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 , 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 .
Definition 2. Let be the number of distinct functional or derivative evaluations per iteration. The efficiency index  is defined by , where is the order of convergence.
In our study, all the listed methods have the same agreeing with Kung-Traub optimality  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. , Soleymani and Babajee , and Shengguo et al.  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.
|MT: method, , .|
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 .
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 , we reach the error equation below:where withcompleting the proof.
3. Conjugacy Maps and Dynamics
Multipoint iterative methods  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 .
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 , 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  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