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The complex dynamical analysis of the parametric fourth-order Kim’s iterative family is made on quadratic polynomials, showing the MATLAB codes generated to draw the fractal images necessary to complete the study. The parameter spaces associated with the free critical points have been analyzed, showing the stable (and unstable) regions where the selection of the parameter will provide us the excellent schemes (or dreadful ones).

It is usual to find nonlinear equations in the modelization of many scientific and engineering problems, and a broadly extended tools to solve them are the iterative methods. In the last years, it has become an increasing and fruitful area of research. More recently, complex dynamics has been revealed as a very useful tool to deep in the understanding of the rational functions that rise when an iterative scheme is applied to solve the nonlinear equation

There is an extensive literature on the study of iteration of rational mappings of complex variables (see [

In the past decade Varona, in [

In order to study the dynamical behavior of an iterative method when it is applied to a polynomial

Let

The dynamical behavior of the orbit of a point on the complex plane can be classified depending on its asymptotic behavior. In this way, a point

If

The set of points whose orbits tends to an attracting fixed point

In this paper, Section

We will focus our attention on the dynamical analysis of a known parametric family of fourth-order methods for solving a nonlinear equation

In order to study the affine conjugacy classes of the iterative methods, the following scaling theorem can be easily checked.

Let

This result allows us to know the behavior of an iterative scheme on a family of polynomials with just the analysis of a few cases, from a suitable scaling.

In the following we will analyze the dynamical behavior of the fourth-order parametric family (

We apply the Möbius transformation

As we will see in the following, not only the number but also the stability of the fixed points depend on the parameter of the family. The expression of the differential operator, necessary for analyzing the stability of the fixed points and for obtaining the critical points, is

As they come from the roots of the polynomial, it is clear that the origin and

The character of the strange fixed point

If

When

If

It is easy to prove that

The critical points are

The relevance of the knowledge of the free critical points (critical points different from the associated with the roots) is the following known fact: each invariant Fatou component is associated with, at least, one critical point.

Analyzing the equation

If

If

If

In case of

In any other case,

Some of these properties determine the complexity of the operator, as we can see in the following results.

The only member of the family whose operator is always conjugated to the rational map

From (

In fact, the element of Kim’s class corresponding to

The element of the family corresponding to

From directly substituting

Then, in the particular case

From the previous analysis, it is clear that the dynamical behavior of the rational operator associated with each value of the parameter can be very different. Several parameter spaces associated with free critical points of this family are obtained. The process to obtain these parameter planes is the following: we associate each point of the parameter plane with a complex value of

As

Parameter plane

This figure has been generated for values of

Dynamical planes for

The generation of dynamical planes is very similar to the one of parameter spaces. In case of dynamical planes, the value of parameter

In Figure

Around the origin.

Detail of

Dynamical plane for

In fact, for

Around

Two periodic orbit for

Dynamical plane for

It is also interesting to note in Figure

A similar procedure can be carried out with the free critical points,

Parameter space

As in case of

Some dynamical planes from

Two periodic orbit

Two attracting strange fixed points

Four periodic orbit

Three periodic orbit

The bulbs on the top (see Figure

The main goal of drawing the dynamical and parameters planes is the comprehension of the family or method behavior at a glance. The procedure to generate a dynamical or a parameters plane is very similar. However, there are small differences, so both cases are developed below.

From a fixed point operator, that associates a polynomial with an iterative method, the dynamical plane illustrates the basins of attraction of the operator. The orbit of every point in the dynamical plane tends to a root (or to the infinity); this information and the speed that the points tend to the root can be displayed in the dynamical plane. In our pictures, each basin of attraction is drawn with a different color. Moreover, the brightness of the color points the number of iterations needed to reach the root of the polynomial.

Pseudocode

The code is divided into five different parts.

Values (lines 17-18): the bounds are renamed and the symbolic function introduced as

Fixed point operators (line 23).

Calculation of attractive fixed points (lines 26–36).

Image creation (lines 39–94): once the fixed point operator and the attracting points are set, the next step consists of the determination of the basins of attraction. The combination of the input parameters

Lines 58–87 are devoted to assign a color to each starting point. It depends on the basin of attraction and the number of iterations needed to reach the root. If the orbit tends to the attracting point set in the first index of line 35, the point is pictured in orange, as lines 67–69 show; for the second and third cases, the point is pictured in blue (lines 72–74) and green (lines 78–80), respectively. Otherwise, the point is not modified, so its color is black.

As the number of iterations needed to reach convergence increases, its corresponding color gets closer to white (black in the decreasing case). A coefficient in each case (lines 66, 71, and 77) is high if the number of iterations is low, and the RGB values are greater than in the slow orbit instance.

Image display (lines 90–94): the image display is based on the

Once the program is executed, the output values are the image

In order to apply the introduced code to different fixed point operators, the only part to be changed is the fixed point operators corresponding one. If the method can converge to more than three points, just add another

Pseudocode

Generation of the mesh of values of

Matrices startup (line 26-27).

Iterative process (lines 31–44). The value of the critical point depends on

Colors assignment (lines 43–51). If the critical point converges, it is drawn by a red

Image display (lines 55–61): the image display is based on the

Once the program is executed, the output values are the image

In order to apply the introduced code to different fixed point operators, the only part to be changed is the fixed point operators corresponding one. If the method can converge to more than three points, just add another

We have analyzed the dynamical properties of the parametric Kim’s family showing stability regions and elements of the family with interesting dynamical behavior but bad numerical features. The main tools used to get this aim are the parameter and dynamical planes implemented in Matlab, whose code is presented in the last section.

The authors thank the anonymous referees for their valuable comments and for the suggestions to improve the readability of the paper. This research was supported by Ministerio de Ciencia y Tecnología MTM2011-28636-C02-02.