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High Order Schemes for Solving Nonlinear Systems of Equations

[11]A. Cordero, J.R. Torregrosa, On interpolation variants of Newton’s method for functions of several variables, Journal of Computational and Applied Mathematics 234 (2010) 34–43.

Proceedings of the 12th International Conference on Computational and Mathematical Methods

in Science and Engineering, CMMSE 2012 July, 2-5, 2012.

Cycles of period two in the family of Chebyshev-Halley type methods

Alicia Cordero1, Juan R. Torregrosa1 and Pura Vindel2

1 Instituto de Matem´atica Multidisciplinar, Universitat Polit`ecnica de Val`encia, Camino de Vera, s/n, 46022 Val`encia, Spain

2 Instituto de Matem´aticas y Aplicaciones de Castell´on, Universitat Jaume I, Av. de Vicent Sos Baynat s/n, 12071 Castell`o de la Plana, Spain

emails: acordero@mat.upv.es, jrtorre@mat.upv.es, vindel@uji.es

Abstract

In this paper, 2-cycles of the Chebyshev-Halley family are studied on quadratic polynomials. This analysis has been made on the basis of the parameter space, described as “cat set”. Some regions of this set are the loci of values of the parameter α that give rise to iterative methods of the family with problems of convergence.

Key words: Nonlinear equations, iterative methods, complex dynamics.

1Introduction

The application of iterative methods for solving nonlinear equations f (z) = 0, f : C → C, gives rise to rational functions whose dynamical behavior provide us important information about the stability and reliability of the corresponding iterative scheme. The best known iterative method, under the dynamical point of view, is Newton’s scheme (see, for example, [4]).

It is known that parameter space of Newton’s process applied on p(z) = z2 + c gives the Mandelbrot set (see [7]). This set has been widely studied (see, for example [6]), analyzing the dynamical behavior of the method for values of the parameter c in the di erent regions of the parameter space. For example, the bulbs rounding the main body of Mandelbrot set contain values of c for which Newton’s procedure has periodic orbits, of several periods.

This study has been extended by di erent authors to other point-to-point iterative methods for solving nonlinear equations (see, for example [1], [2] and, more recently, [8] and

Cycles of period two in the family of Chebyshev-Halley type methods

[10]). Some of the classical iterative schemes for solving nonlinear equations are included in the parametric family of Chebyshev-Halley, whose dynamical analysis has been started in [5].

The family of Chebyshev-Halley type methods can be written as the iterative scheme

In [5] the dynamics of this operator when it is applied on quadratic polynomial p (z) = z2 + c has been initially studied. For this polynomial, the operator (2) correspond to the rational function:

,

depending on parameters α and c.

The parameter c can be obviated by considering the conjugacy map

Now, let us recall some basic concepts on complex dynamics (see [3]). Given a rational

as:

z0, R (z0) , R2 (z0) , ..., Rn (z0) , ...

We are interested in the study of the asymptotic behavior of the orbits depending on the initial condition z0, that is, we are going to analyze the phase plane of the map R defined by the di erent iterative methods.

To obtain these phase spaces, the first of all is to classify the starting points from the asymptotic behavior of their orbits.

A. Cordero, J.R. Torregrosa, P. Vindel

p > 1 is a point such that Rp (z0) = z0 and Rk (z0) = z0, k < p. A pre-periodic point is a point z0 that is not periodic but there exists a k > 0 such that Rk (z0) is periodic. A critical point z0 is a point where the derivative of rational function vanishes, R (z0) = 0.

On the other hand, a fixed point z0 is called attractor if |R (z0)| < 1, superattractor if |R (z0)| = 0, repulsor if |R (z0)| > 1 and parabolic if |R (z0)| = 1. The stability of a periodic orbit is defined by the magnitude (lower than 1 or not) of |R (z1) . . . R (zp)|, where {z1, . . . , zp} are the points of the orbit of period p.

The basin of attraction of an attractor z¯ is defined as the set of pre-images of any order:

borhood U (z) , is the Fatou set, F (R) , that is, the Fatou set is composed by the set of points whose orbits tend to an attractor (fixed point, periodic orbit or infinity). Its comple-

ment in ˆ is the Julia set, J (R) ; therefore, the Julia set includes all repelling fixed points,

C

periodic orbits and their pre-images. That means that the basin of attraction of any fixed point belongs to the Fatou set. On the contrary, the boundaries of the basins of attraction belong to the Julia set.

The invariant Julia set for Newton’s method is the unit circle S1 and the Fatou set is defined by the two basins of atraction of the superattractor fixed points: 0 and ∞. On the other hand, the Julia set for Chebyshev’s method applied to quadratic polynomials is more complicated than for Newton’s method and it has been studied in [9]. These methods are two elements of the family (1).

2Previous results on Chebyshev-Halley family

Fixed points of the operator Op(z) are z = 0, z = ∞, which correspond to the roots of the

√

polynomial, and z = 1 and z = −3+2α± 5−12α+4α2 , denoted by s1 and s2, respectively.

2

Moreover, z = 0 and z = ∞ are superattractors and the stability of the other fixed points is established in the following results, which appear in [5].

Proposition 1 The fixed point z = 1 satisfies the following statements :

iii)If α − 136 > 13 , then z = 1 is a repulsive fixed point.

Cycles of period two in the family of Chebyshev-Halley type methods

Proposition 2 The fixed points z = si, i = 1, 2, satisfy the following statements:

On the other hand, the critical points of Op(z) are z = 0, z = ∞ and

z = 3 − 4α + 2α2 ± √−6α + 19α2 − 16α3 + 4α4 , 3 (α − 1)

which are denoted by c1 and c2, respectively.

It is known that there is at least one critical point associated with each invariant Fatou component. It is shown in [5] that the critical points ci, i = 1, 2, are inside the basin of attraction of z = 1 when it is attractive ( 116 < α < 52 ) and coincide with z = 1 for α = 2. Then, they move to the basins of attraction of s1 and s2 when these fixed points become attractive ( 52 < α < 72 ), critical and fixed points coincide for α = 3 and s1 and s2 become superattractors.

A powerful tool to analyze the dynamics of the rational function associated to an iterative method is the parameter space (see Figure 1): each point of the parameter plane is associated to a complex value of α, i.e., to an element of family (1). Every value of α belonging to the same connected component of the parameter space give rise to subsets of schemes of family (1) with similar dynamical behavior.

In this parameter space we observe a black figure (let us call it the cat set ), with a certain similarity with the Mandelbrot set: for values of α outside this cat set the Julia set is disconnected. The two disks in the main body of the cat set correspond to the α values for those the fixed points z = 1 (the head) and s1 and s2 became attractive (the body). We also observe a curve similar to a circle that passes through the cat’s neck, we call it the necklace. As we have proved in [5], the parameter space inside this curve is topologically equivalent to a disk.

The head of the cat corresponds to the values of the parameter for which the fixed point

The body of the cat set corresponds to values of the parameter |α − 3| < 2 . In this case, the fixed point z = 1 is a repulsor and s1,s2 are attractors and have their own basin of attraction, one critical point is in each basin.

Let us remark that the intersection point of the head and the body of the cat is in their common boundary and corresponds to α = 52 .

A. Cordero, J.R. Torregrosa, P. Vindel

Figure 1: Parameter plane

In fact, similarly to what happen in the Mandelbrot set, the boundary of the cat set is exactly the bifurcation locus of the family of Chebyshev-Halley type family acting on quadratic polynomial; that is, the set of parameters for which the dynamics changes abruptly under small changes of α. Let us observe that the head and the body are surrounded by bulbs, of di erent sizes, that yield to the appearance of attractive cycles of di erent periods. In this paper, we are interested in the study of the bulbs involving cycles of period 2.

3The bulb of period 2 of the head

Cycles of period two in the family of Chebyshev-Halley type methods

Therefore, for di erent θ values we find some bulbs with attractive cycles surrounding “the head of the cat”. In this section we study the bulb of period 2, whose intersection with the head of the cat corresponds to α = 116 .

As we have seen in Proposition 1, if α > 116 then Op (1) < 1, if α = 116 then Op (1) = 1, and when α < 116 then Op (1) > 1.

We are going to show that there is a doubling period bifurcation for α = 116 . For α < 116 the periodic point z = 1 became repulsive and one attractive cycle of period 2 appears.

This cycle has to satisfy the equation:

for α = come from the zeros of this function. In fact, we will focus our attention on

and we observe that f1 116 , z = (1 − z)2 and f2 116 , z = 13 3 − 4z + 3z2 . So, the cycle of period 2 that becomes attractive comes from f1 (α, z) = 0.

Cycles of period two in the family of Chebyshev-Halley type methods

1

0.8

0.6

0.4

0.2

0.064 where the 2-cycles are attractive. We can see it by evaluating |S(α)| in the points α = α0 + 0.064eitπ , where 0 ≤ t ≤ 2 and the step size is h = 0.1. The values are: 0.977843, 0.979721, 0.984619, 0.99065, 0.995595, 0.997797, 0.996763, 0.993236, 0.988793, 0.985214, 0.983854, 0.985214, 0.988793, 0.993236, 0.996763, 0.997797, 0.995595, 0.99065, 0.984619, 0.979721, 0.977843.

A similar study can be made on g(α, z), in order to obtain cycles in the bulb of period 2 surrounding the body of the cat set.

4Conclusions

The cat set as parameter space of the Chebyshev-Halley family on quadratic polynomials is dynamically very wealthy, as it happens with Mandelbrot set. The head and the body of the cat set are surrounded by bulbs of di erent sizes. Some of them have been analyzed in this work, obtaining cycles of period two for several values of the parameter, that is, for di erent members of the family of iterative methods.

Acknowledgments: This research was supported by Ministerio de Ciencia y Tecnolog´ıa MTM2011-28636-C02-02 and by Vicerrectorado de Investigaci´on, Universitat Poli- t`ecnica de Val`encia PAID-06-2010-2285.

A. Cordero, J.R. Torregrosa, P. Vindel

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