Reducing Chaos and Bifurcations in Newton-Type Methods

Abstract

We study the dynamics of some Newton-type iterative methods when they are applied of polynomials degrees two and three. The methods are free of high-order derivatives which are the main limitation of the classical high-order iterative schemes. The iterative schemes consist of several steps of damped Newton's method with the same derivative. We introduce a damping factor in order to reduce the bad zones of convergence. The conclusion is that the damped schemes become real alternative to the classical Newton-type method since both chaos and bifurcations of the original schemes are reduced. Therefore, the new schemes can be utilized to obtain good starting points for the original schemes.

1. Introduction

One of the most important and usual problems in numerical analysis is finding the solutions of a nonlinear equation: 𝑓(𝑥)=0, (1) where 𝑓:ℝ→ℝ is a 𝐶^𝑟 function, with 𝑟≥1. To approximate the solutions of these equations, iterative methods can be used. There are many iterative methods with different properties but the most commonly used is Newton’s iterative method: 𝑁_𝑓(𝑥)𝑓(𝑥)=𝑥−𝑓^′(𝑥). (2)

This method and its variants have been studied by many authors since it was proposed by Newton in 1669. It is well known that under certain assumptions related to the starting point 𝑥₀ and to the function 𝑓 the method provides a sequence {𝑥_𝑛}_𝑛∈ℕ that converges quadratically to a solution of (1).

The classical third-order schemes, such as Halley or Chebyshev methods, evaluate second-order Fréchet derivatives. These evaluations are very time consuming for systems of equations. Indeed, let us observe that, for a nonlinear system of 𝑚 equations and 𝑚 unknowns, the second Fréchet derivative has 𝑚³ entries. In particular, these methods are hardly used in practice. For that reason, we are interested in the study of methods that increase the order but evaluating only first-order Fréchet derivatives. The methods consist of two (or more) steps of the Newton method having the same derivative, other main advantage of these methods relays on the fact that if we consider a system of equations only one 𝐿𝑈 decomposition is necessary in each iteration. Because of these two properties the schemes are considered a real alternative to the classical Newton method.

On the other hand, we can find a rigorous and useful procedure to find the best scheme in families of Newton-type methods with frozen derivatives in [1]. For most of the cases, the most efficient candidate of the family is the two-step iterative method. We will study all this family but with special attention to the two-step scheme.

In this paper, our main purpose is to study the dynamics of the discrete system defined by the methods but introducing a damping factor 𝜆∈(0,1]. We are interested to search chaos or sensitive dependence on initial conditions for this discrete dynamical system. In [2], Hurley and Martin showed that the classical Newton iterative method exhibits chaos for a large class of functions (see also [3]). The introduction of the damping factor not only reduces the order of convergence but also reduces the chaos and the bifurcations as we will see in Sections 3 and 4. In particular, we can utilize these schemes as starting points for the classical Newton schemes. We conclude this paper by comparing the damped method with the original one and the successive damped high-order methods that come from using more Newton’s steps.

The study of the dynamics of the iterative methods attracts the attention of many groups [4–8].

The paper is organized as follows. In Section 2 we introduce our new scheme and present the associated scaling theorem. The scaling theorem allows up to suitable change of coordinates, to reduce the study of the dynamics of iterations of general maps, to the study of specific families of iterations of simpler maps. The dynamics for polynomials of degree-two and three, having chaotic dynamics in some cases, are presented in Sections 3 and 4, respectively. In these sections, we observe that the inclusion of the damping factor reduces the bad zone of convergence. In Section 5 we give the main conclusions drawn from the study and we summarize the benefits to include the damping factor. Finally, we propose the use of damped methods in order to find good starting points for the original methods.

2. A Damped Newton-Type Iterative Method

Our first goal is to analyze the dynamics of a modification of the following third-order iterative root-finding method [9]: 𝑦_𝑛=𝑥_𝑛−𝑓𝑥_𝑛𝑓^′𝑥_𝑛,𝑥_𝑛+1=𝑦_𝑛−𝑓𝑦_𝑛𝑓^′𝑥_𝑛. (3)

This iterative root-finding algorithm is defined by the following iterative function: 𝑀_𝑓(𝑥)=𝑁_𝑓(𝑥)−𝑓𝑁_𝑓(𝑥)𝑓^′(𝑥), (4) where 𝑁_𝑓(𝑥)=𝑥−𝑢_𝑓(𝑥) and 𝑢_𝑓(𝑥)=𝑓(𝑥)/𝑓^′(𝑥). In other words, 𝑥_𝑛+1=𝑀_𝑓(𝑥_𝑛). The dynamics of this method were studied in [10]. Now, we introduce a damping factor 𝜆∈(0,1] and we analyze its influence on the dynamics in the iterative method (4). By introducing the damping factor the root-finding algorithm has the following form: 𝑦_𝑛=𝑥_𝑛𝑓𝑥−𝜆^𝑛𝑓^′𝑥_𝑛,𝑥_𝑛+1=𝑦_𝑛𝑓𝑦−𝜆^𝑛𝑓^′𝑥_𝑛. (5)

Thus the following iterative function describes the root-finding algorithm: 𝑀_𝜆,𝑓(𝑥)=𝑁_𝜆,𝑓(𝑥)𝑓𝑁−𝜆^𝜆,𝑓(𝑥)𝑓^′(𝑥),0<𝜆≤1, (6) where 𝑁_𝜆,𝑓(𝑥)=𝑥−𝜆𝑓(𝑥)/𝑓^′(𝑥).

Notice that it is easy to proof that the roots of the function 𝑓 are fixed points of 𝑀_𝜆,𝑓. Moreover, we have that 𝑀^′_𝜆,𝑓(𝑥)𝑓=1−𝜆^′(𝑥)²−𝑓(𝑥)𝑓^′′(𝑥)𝑓^′(𝑥)²−𝜆𝑓^′𝑓(𝑥)𝑥−𝜆𝑓^′(𝑥)×𝑓1−𝜆^′(𝑥)²−𝑓(𝑥)𝑓^′′(𝑥)𝑓^′(𝑥)²𝑓^′(𝑥)𝑓(𝑥)−𝑓𝑥−𝜆𝑓^′(𝑥)𝑓^′′(𝑥×𝑓_′)(𝑥)²⁻¹. (7)

In particular, if 𝛼 is a simple root of 𝑓, that is, 𝑓(𝛼)=0 and 𝑓^′(𝛼)≠0, then 𝑀^′_𝜆,𝑓(𝛼)=(1−𝜆)². Thus every simple root of 𝑓 is an attracting point of 𝑀_𝜆,𝑓 and for 𝜆=1 a superattracting fixed point. If 𝛽 is a multiple root of multiplicity 𝑚 of 𝑓, then it is an attracting fixed point for every value of 𝜆. There are fixed points of 𝑀_𝜆,𝑓(𝑥) different from the roots. These points are called extraneous fixed points. In [10], Amat et al. showed that this method for 𝜆=1 (without damping factor) presents chaos and bifurcations when it is applied to a one-parameter family of cubic polynomials. Chaos behavior are related to the existence of points 𝑥^∗ such that 𝑓^′(𝑥^∗)=0 and 𝑓(𝑥^∗)≠0.

When we apply the iterative method 𝑀_𝜆,𝑓 to a polynomial, we may have some problems, since we obtain a rational map, say 𝑀_𝜆,𝑝(𝑥)=𝑃(𝑥)/𝑄(𝑥), where 𝑃 and 𝑄, are polynomials, that we may suppose without common factors. The difficulty arises at those points where the evaluation of numerator is nonzero and the denominator is zero, one such points correspond to the poles of the iterative method.

In order to study the dynamics of 𝑀_𝜆,𝑓 by means of graphical analysis we consider it as a map 𝑀_𝜆,𝑓 on  [0,1[. For this, let 𝐺∶ℝ→]0,1[   be given by 𝐺(𝑥)=(1/𝜋)arctan(𝑥)+1/2. This map is an homeomorphism from ℝ into  ]0,1[. Define the maps 𝑀_𝜆,𝑓∶]0,1[→]0,1[  by 𝑀_𝜆,𝑓=(𝐺∘𝑀_𝜆,𝑓∘𝐺⁻¹)(𝑥), where 𝑀_𝜆,𝑓 is the iterative root-finding method introduced above for a map 𝑓∶ℝ→ℝ. We may extend 𝑀_𝜆,𝑓 to maps from [0,1] into itself. We use the same notation for this extension. Note that the extended function has fixed points at 𝑥=0 and at 𝑥=1 and that these fixed points are repelling.

Now we have the following useful result.

Theorem 1 (the scaling theorem). Let 𝑓(𝑥) be an analytic function, and let 𝑇(𝑥)=𝛼𝑥+𝛽, with 𝛼≠0, be an affine map. Let 𝑔(𝑥)=(𝑓∘𝑇)(𝑥). Then 𝑇∘𝑀_𝜆,𝑔∘𝑇⁻¹(𝑥)=𝑀_𝜆,𝑓(𝑥); that is, 𝑀_𝜆,𝑔 and 𝑀_𝜆,𝑓 are affine conjugated by 𝑇.

Proof. We have 𝑀_𝜆,𝑔𝑇⁻¹(𝑥)=𝑇⁻¹(𝑥)−𝜆𝑢_𝑔𝑇⁻¹(𝑥𝑔𝑇)−𝜆⁻¹(𝑥)−𝜆𝑢_𝑔𝑇⁻¹(𝑥)𝑔^′𝑇⁻¹(𝑥). (8) On the other hand, since 𝑔∘𝑇⁻¹(𝑥)=𝑓(𝑥) and (𝑔∘𝑇⁻¹)^′(𝑥)=(1/𝛼)𝑔^′(𝑇⁻¹(𝑥)), then 𝑔^′(𝑇⁻¹(𝑥))=𝛼(𝑔∘𝑇⁻¹)^′(𝑥)=𝛼𝑓^′(𝑥), and by an easy induction process it follows that 𝑔^(𝑘)(𝑇⁻¹(𝑥))=𝛼^𝑘𝑓^(𝑘)(𝑥). Hence, 𝑢_𝑔(𝑇⁻¹(𝑥))=(1/𝛼)𝑢_𝑓(𝑥).
Substituting these quantities in 𝑀_𝜆,𝑔(𝑇⁻¹(𝑥)) we obtain 𝑇∘𝑀_𝜆,𝑔∘𝑇⁻¹(𝑥)=𝑇𝑀_𝜆,𝑔𝑇⁻¹(𝑥)=𝛼𝑀_𝜆,𝑔𝑇⁻¹(𝑥)+𝛽=𝛼𝑇⁻¹(𝑥)−𝛼𝜆𝑢_𝑔𝑇⁻¹(𝑥)𝑔𝑇−𝛼𝜆⁻¹(𝑥)−𝜆𝑢_𝑔𝑇⁻¹(𝑥)𝑔^′𝑇⁻¹(𝑥)+𝛽=𝛼𝑇⁻¹(𝑥)−𝜆𝑢_𝑓(𝑥)𝑔𝑇−𝜆⁻¹(𝑥)−(𝜆/𝛼)𝑢_𝑓(𝑥)𝑓^′(𝑥)+𝛽=𝑥−𝜆𝑢_𝑓(𝑥)𝑔𝑇−𝜆⁻¹(𝑥)−(𝜆/𝛼)𝑢_𝑓(𝑥)𝑓^′(𝑥). (9)
Finally, by a comparison of Taylor series expansions of 𝑓 and 𝑔, we obtain 𝑔𝑇⁻¹(𝑥)−𝜆𝛼𝑢_𝑓(𝑥)=𝑔𝑇⁻¹(𝑥)−𝜆𝛼𝑔^′𝑇⁻¹(𝑥)𝑢_𝑓(𝑥)⋯=𝑓(𝑥)−𝜆𝛼𝑓^′(𝑥)1𝛼𝑢_𝑓(𝑥)(𝑥).+⋯=𝑓𝑥−𝜆𝑢𝑓 (10)
Therefore, 𝑇∘𝑀_𝜆,𝑔∘𝑇⁻¹(𝑥)=𝑀_𝜆,𝑓(𝑥).
This ends the proof.

The theorem remains valid for any nonzero constant 𝑐 such that 𝑔(𝑥)=𝑐(𝑓∘𝑇)(𝑥).

The Scaling Theorem allows up to suitable change of coordinates, to reduce the study of the dynamics of iterations 𝑀_𝜆,𝑓, to the study of specific families of iterations of simpler maps. For example, each quadratic polynomial 𝑓(𝑥)=𝑎𝑥²+𝑏𝑥+𝑐, with 𝑎≠0, that we suppose that is monic, by an affine change of variables reduces to one of the following polynomials: (i)𝑓₋(𝑥)=𝑥²−1, (ii)𝑓₊(𝑥)=𝑥²+1, (iii)𝑓₀(𝑥)=𝑥²,

depending on the number of real roots of 𝑓(𝑥)=0 (two, one (double), or not real roots). Therefore the study of the dynamics of 𝑀_𝜆,𝑓, when it is applied to a quadratic polynomial, reduces to the study of the dynamics of 𝑀_{𝜆,𝑓𝑠} where 𝑠=+,0,−, up to affine change of variables. On the other hand, the study of the dynamics of the iterative method 𝑀_𝜆,𝑓 reduces to the study of the dynamics of the interval map 𝑀_{𝜆,𝑓𝑠}. Similarly, any cubic polynomial reduces to one of the simplest cubic polynomials:(I)𝑓₋(𝑥)=𝑥³−𝑥,(II)𝑓₊(𝑥)=𝑥³+𝑥, (III)𝑓₀(𝑥)=𝑥³, (IV)𝑓_𝛾(𝑥)=𝑥³+𝛾𝑥+1.

The last one is an appropriate rescaling that puts 𝑀_𝜆,𝑓 inside the conjugacy class.

3. Quadratic Polynomials

Following the analysis of Section 2, we have to study three cases.

Case 1 (𝑓₀(𝑥)=𝑥²). In this case form (6) we have 𝑀_𝜆,𝑓0(𝑥)1=−8𝑥(−2+𝜆)(4+(−2+𝜆)𝜆). (11) This function for 𝜆∈(0,1] is a linear contraction. The unique fixed point are 𝑥=0, which is a global attractor, but not super-attracting. Therefore, its dynamics is trivial.

Case 2 (𝑓₋(𝑥)=𝑥²−1). For 𝑓₋(𝑥)=𝑥²−1 from (6) and (7) we have that 𝑀_𝜆,𝑓−(𝑥)=−𝜆³+𝑥⁴(−2+𝜆)(4+(−2+𝜆)𝜆)−2𝑥²𝜆(()𝜆)×4+−2+𝜆8𝑥³⁻¹,𝑀^′_𝜆,𝑓−(𝑥)=−−3𝜆³+𝑥⁴(−2+𝜆)(4+(−2+𝜆)𝜆)+2𝑥²𝜆(()𝜆)×4+−2+𝜆8𝑥⁴⁻¹. (12) Now the fixed points of 𝑀_𝜆,𝑓−(𝑥) are 𝑥_1,2=±1 and 𝑥_3,4=±𝜆/8−4𝜆+𝜆². The points 𝑥_1,2 are the roots of 𝑓₋, and we have that 𝑀^′_𝜆,𝑓−(𝑥_1,2)=(𝜆−1)²; thus 𝑥_1,2 are attracting fixed points for 𝜆∈(0,1), and superattracting for 𝜆=1. On the other hand, we have that 𝑀^′_𝜆,𝑓−(𝑥_3,4)=−15+16/𝜆+6𝜆−𝜆². Thus, the extraneous fixed points 𝑥_3,4 are repelling. Note that 𝑀_𝜆,𝑓−(𝑥) has a vertical asymptote at 𝑥=0.

Let 𝜆=1 and 𝐼=[𝑝₁,𝑝₂] be the interval determined by the repelling fixed points 𝑝₁=𝐺(𝑥₃)=0.36614… and 𝑝₂=𝐺(𝑥₄)=0.63386… of 𝑀_1.0,𝑓−. Figure 1 shows 𝑀_1.0,𝑓− and Figure 2 shows a zoom of the restriction of the function to the interval 𝐼.

Figure 1 

𝑀1.0,𝑓− as a circle map.

Figure 2 

𝑀1.0,𝑓− restricted to the interval 𝐼.

It is clear that 𝑀_1.0,𝑓− restricted to the interval 𝐼 has an invariant Cantor set Δ₋ of zero Lebesgue measure of nonescaping points; that is, Δ₋ is the set of points whose orbits remain in the interval 𝐼 under iteration by 𝑀_1.0,𝑓−. In other words, these points are not attracted to one of the superattracting fixed points of 𝑀_1.0,𝑓− (note that when 𝜆=1 the fixed points are superattracting).

From the above analysis, we have that the orbit of any point 𝑥∈[0,1]⧵(Δ₋∪𝑝₁,𝑝₂) is attracted to one of the superattracting fixed points of 𝑀_1.0,𝑓−. Consequently, the orbit of any point in ℝ⧵(𝐺⁻¹(Δ₋)∪{𝑥₃,𝑥₄}) is attracted to one of the fixed points of 𝑀_1.0,𝑓−.

Thus we can define the “bad zone” as BZ=Δ₋∪{𝑝₁,𝑝₂} since if 𝑥∈BZ, then it is not attracted by any of the attracting points of 𝑀_𝜆,𝑓−.

Lemma 2. The “bad zone” of the damping two-step Newton-like method defined in (6) applied to quadratic polynomials with two different roots decreases with the damping parameter 𝜆.

Proof. We begin by calculating 𝑝₁=𝐺𝑥₃=12−arctan𝜆/8+(−4+𝜆)𝜆𝜋,𝑝₂=𝐺𝑥₄=12+arctan𝜆/8+(−4+𝜆)𝜆𝜋. (13)
So we have that 𝐼_𝜆=12−arctan𝜆/8+(−4+𝜆)𝜆𝜋,12+arctan𝜆/8+(−4+𝜆)𝜆𝜋. (14)
Thus 𝐼 is an interval centered in 0.5 and width 2(arctan(𝜆/8+(−4+𝜆)𝜆)/𝜋)=2𝑎. It is easy to see that Δ₋ decreases when 𝑎 decreases. Taking into account that the functions (𝜆/8+(−4+𝜆)𝜆)(𝑥) and arctan(𝑥) are increasing functions, it follows that 𝑎 decreases as 𝜆 so with smaller 𝜆 we have smaller BZ.

Thus we have seen analytically and graphically, in Figures 3, 4, 5, and 6, that the size of (Δ₋∪{𝑝₁,𝑝₂}) decreases with the damping factor 𝜆 and as a consequence we can state that the chaotic zone decreases with 𝜆.

Figure 3 

Graphics of 𝑀𝜆,𝑓− with 𝜆=0.75, where the BZ interval is 𝐼0.75=[0.401996⋯,0.598004⋯].

Figure 4 

Graphics of 𝑀𝜆,𝑓− with 𝜆=0.5, where the BZ interval is 𝐼0.5=[0.437167⋯,0.562833⋯].

Figure 5 

Graphics of 𝑀𝜆,𝑓− with 𝜆=0.25, where the BZ interval is 𝐼0.25=[0.470144⋯,0.529856⋯].

Figure 6 

Graphics of 𝑀𝜆,𝑓− with 𝜆=0.01, where the BZ interval is 𝐼0.01=[0.498872⋯,0.501128⋯].

Case 3 (𝑓₊(𝑥)=𝑥²+1). In this case, the equation 𝑓₊ has no real roots, thus 𝑀_𝜆,𝑓+(𝑥) has not real fixed points, and its dynamics is chaotic. Figures 7 and 8 illustrate this behavior.

Figure 7 

Graphics of 𝑀1.0,𝑓+.

Figure 8 

Graphics of 𝑀1.0,𝑓+ as a circle map.

4. Cubic Polynomials

In this section we consider cubic polynomials. As we have seen in Section 2, by an affine change of coordinates, every cubic polynomial 𝑓 reduces to one of the simplest 𝑓₀,𝑓₋,𝑓₊ or to a member of the one-parameter family of cubic polynomials 𝑓_𝛾. Therefore, we analyze the dynamics of these simpler cubic polynomials.

Case 4 (𝑓₊(𝑥)=𝑥³+𝑥). In this case, 𝑓 has only one real root at 𝑥=0, and the method (6) has the following form: 𝑀_𝑓+,𝜆=𝑥−𝑥+𝑥³𝜆1+3𝑥²−𝜆𝑥−𝑥+𝑥³𝜆1+3𝑥²+𝑥−𝑥+𝑥³𝜆1+3𝑥²³×1+3𝑥²⁻¹. (15)

Since 𝑓^′₊(𝑥)=1+3𝑥²>0 for all 𝑥, it follows that 𝑓₊ has no critical points; hence, 𝑀_𝑓+,𝜆(𝑥) is a global homeomorphism from ℝ into itself and its dynamics is trivial.

Case 5 (𝑓₋(𝑥)=𝑥³−𝑥). In this case 𝑓 has 3 real roots, 𝑥=0, 𝑥=1 and 𝑥=−1. 𝑀_𝑓−,𝜆(𝑥) is given by (6) 𝑀_𝑓−,𝜆(𝑥)𝑥=𝑥−−1+𝑥²𝜆−1+3𝑥²−𝜆𝑥−𝑥+−1+𝑥²𝜆−1+3𝑥²+𝑥𝑥−−1+𝑥²𝜆−1+3𝑥²³−1+3𝑥²⁻¹. (16) Notice that 𝑀_𝑓−,𝜆(𝑥) has two asymptotes at the points 𝑎₁=√3/3 and 𝑎₂=−√3/3. We have that when 𝑥→𝑎₁, then 𝑀_𝑓−,𝜆(𝑥)→−∞, and when 𝑥→𝑎₂, then 𝑀_𝑓−,𝜆(𝑥)→∞. If we now calculate the fixed points of 𝑀_𝑓−,𝜆(𝑥), there exist 9 fixed points which are the three roots of 𝑓(𝑥₁=−1,𝑥₂=0,𝑥₃=1) and the following extraneous fixed points:

𝑥₄=−39−3𝜆+𝜆²−𝜆29−3𝜆+𝜆²+𝜆²29−3𝜆+𝜆²−𝜆9−2𝜆+𝜆²29−3𝜆+𝜆²,𝑥₅=39−3𝜆+𝜆²−𝜆29−3𝜆+𝜆²+𝜆²29−3𝜆+𝜆²−𝜆9−2𝜆+𝜆²29−3𝜆+𝜆²,𝑥₆=−39−3𝜆+𝜆²−𝜆29−3𝜆+𝜆²+𝜆²29−3𝜆+𝜆²+𝜆9−2𝜆+𝜆²29−3𝜆+𝜆²,𝑥₇=39−3𝜆+𝜆²−𝜆29−3𝜆+𝜆²+𝜆²29−3𝜆+𝜆²+𝜆9−2𝜆+𝜆²29−3𝜆+𝜆², (17)

where 𝑥₈=−(−2+𝜆)/(−6+𝜆) and 𝑥₉=(−2+𝜆)/(−6+𝜆). Evaluating all these points in 𝑀^′_𝑓−,𝜆 (7), we obtain that the first 3 ones (the roots of 𝑓) are attracting fixed points for every 𝜆≠1 and superattracting for 𝜆=1. The extraneous fixed points are repelling ones since 𝑀^′_𝑓−,𝜆(𝑥_1,2,3)=(−1+𝜆)²: 𝑀^′_𝑓−,𝜆𝑥_4,5=−9−3+9+(−2+𝜆)𝜆+𝜆47+𝜆−13+5𝜆+39+(−2+𝜆)𝜆(4𝜆)⁻¹,𝑀^′_𝑓−,𝜆𝑥_6,7=93+9+(−2+𝜆)𝜆+𝜆−47+𝜆13−5𝜆+39+(−2+𝜆)𝜆(4𝜆)⁻¹,𝑀^′_𝑓−,𝜆𝑥_8,9=13+(−8+𝜆)𝜆. (18) In particular, the dynamics is trivial.

Case 6 (𝑓₀(𝑥)=𝑥³). The function 𝑓₀ only has one triple root at 𝑥=0. We have that (6) has the following form: 𝑀_𝑓0,𝜆=1𝑥(()81−3+𝜆)−27+−3+𝜆²𝜆, (19) which is a linear contraction for every value of 𝜆∈(0,1]. The only fixed point is attracting, but not superattracting and the dynamics of 𝑀_𝑓0,𝜆 is trivial.

Case 7 (𝑓_𝛾(𝑥)=𝑥³+𝛾𝑥+1). We have from (6) that 𝑀_{𝑓𝛾,𝜆}(𝑥)=𝑥−1+𝑥³+𝑥𝛾𝜆3𝑥²−𝜆+𝛾1+𝛾𝑥−1+𝑥³+𝑥𝛾𝜆3𝑥²++𝛾𝑥−1+𝑥³+𝑥𝛾𝜆3𝑥²+𝛾³×3𝑥²+𝛾⁻¹. (20) We are going to analyze in this section the dynamics of 𝑀_{𝑓𝛾,𝜆}(𝑥) depending on the two parameters 𝛾 and 𝜆. Our main purpose is to compare the results by fixing the parameter 𝛾 for different values of the damping parameter 𝜆 and to study how it changes the dynamics.

We begin by studying the case 𝛾<0. In this case, our function 𝑓_𝛾 has two critical points: 𝛾_+,−=±−𝛾/3. Let be 𝛾_∗=−3/√³4≈−1.88988. Notice that 𝑓_𝛾∗ has a double root at the positive critical point 𝛾₊. Hence, this value of the parameter will be very significant in the study of the dynamics.

We have three different possibilities. (i)If 𝛾<𝛾_∗, the polynomial 𝑓_𝛾 has three real roots. (ii)If 𝛾=𝛾_∗, the polynomial 𝑓_𝛾 has one positive double real root and a simple one which is negative. (iii)If 𝛾>𝛾_∗, the polynomial 𝑓_𝛾 has only one real root.

For 𝛾<𝛾_∗ the iterative method 𝑀_{𝑓𝛾,𝜆}(𝑥) has two vertical asymptotes. 𝑀_{𝑓𝛾,𝜆}(𝑥) has 7 fixed points, the 3 real roots of the polynomial (attracting fixed points), and the extraneous fixed points (repelling). See Figure 9 as an example. As we can observe in Figures 10, and 11 if the damping factor decreases, then extraneous fixed points tend to be closer to the asymptotes.

Figure 9 

𝑀𝑓−3,𝜆 for 𝜆=0.9.

Figure 10 

𝑀𝑓−3,𝜆 for 𝜆=0.3.

Figure 11 

𝑀𝑓−3,𝜆 for 𝜆=0.1.

For 𝛾_∗≤𝛾<0 the iterative method 𝑀_{𝑓𝛾,𝜆}(𝑥) has two vertical asymptotes at 𝛾₋ and 𝛾₊, as we can see in Figure 12. Notice that lim_{𝑥→𝛾±−}𝑀_𝑓𝛾,0(𝑥)=∞ and lim_{𝑥→𝛾±+}𝑀_𝑓𝛾,0(𝑥)=∞. 𝑀_{𝑓𝛾,𝜆}(𝑥) has 3 fixed points, the one which corresponds to the real solution of the polynomial 𝑓_𝛾 is attracting and two repelling extraneous fixed points. Again we obtain that when the damping factor decreases the chaotic zone also decreases.

Figure 12 

𝑀𝑓−1,𝜆 for 𝜆=0.9.

Now we are going to consider the simplest case which corresponds to 𝛾=0, and we have 𝑀_𝑓0,𝜆(𝑥)=3𝑥³(−3+𝜆)𝜆³+𝜆⁴+3𝑥⁶𝜆−18+9𝜆−6𝜆²+𝜆³+𝑥⁹81−54𝜆+27𝜆²−9𝜆³+𝜆⁴×81𝑥⁸⁻¹. (21) This map has one attracting point at 𝑥₁=−1 which corresponds to the root of 𝑓₀. Moreover it has another two extraneous fixed points which are the real solutions of the equation: 𝜆³+−9𝜆²+2𝜆³𝑥³+−54+27𝜆−9𝜆²+𝜆³𝑥⁶=0. (22) As we see in Figure 13 both extraneous fixed points are repelling.

Figure 13 

𝑀𝑓0,𝜆 for 𝜆=0.9.

Now if we focus our attention on the damping factor, we obtain that when 𝜆 decreases, the extraneous fixed points tend to be closer to the asymptote 𝑥=0. This behavior can be observed in Figures 14 and 15. Notice that 𝑥=0 is the critical point of 𝑓₀.

Figure 14 

𝑀𝑓0,𝜆 for 𝜆=0.7.

Figure 15 

𝑀𝑓0,𝜆 for 𝜆=0.1.

The last study is related with 𝛾>0. Notice that for, every value of 𝛾≠0, 𝑀_{𝑓𝛾,𝜆} presents a maximum which increases when 𝛾 decreases. Moreover there exists a value 𝛾_𝜆 that depends on the value of the damping factor, such that, for each 𝛾>𝛾_𝜆, 𝑀_{𝑓𝛾,𝜆} has only one fixed point (the root of the function) and the dynamics is trivial. In Table 1, it shows the different approximations of 𝛾_𝜆 for some values of 𝜆. Figure 16 illustrates this case.

Table 1 

Approximations of 𝛾_𝜆 for some values of 𝜆.

Figure 16 

𝑀𝑓𝛾,𝜆 for 𝜆=0.5 and 𝛾=𝛾0.5=0.442.

When 𝛾=𝛾_𝜆, 𝑀_𝑓𝛾,1 presents another fixed point which is a saddle node (as Amat et al. showed in [10]) and when the parameter 𝛾 decreases, it appears bifurcations. If we now see Figures 17, 18, 19, and 20, it is clear that the problems decrease with 𝜆; even for small values (𝜆<0.2) they seem they disappear.

Figure 17 

Feigembaun diagram for 𝜆=1.0.

Figure 18 

Feigembaun diagram for 𝜆=0.8.

Figure 19 

Feigembaun diagram for 𝜆=0.5.

Figure 20 

Feigembaun diagram for 𝜆=0.3.

Finally, we conclude with the following conjecture: if 𝜆=𝑛/2^𝑘 with 𝑛,𝑘∈ℕ, then 𝑀_{𝑓𝛾,𝜆}  has only three fixed points.

5. Concluding Remarks

In this work we have analyzed the dynamics of a new damped third-order Newton-like method. We have stated the following conclusions drawn from the study (i)The connected component of the basin of attraction increases when the damping factor decreases. (ii)The sizes of the chaotic and the bifurcation zones decrease when the damping factor decreases. (iii)The nonconvergence zone decreases when the damping factor decreases.

In addition, we have performed a similar study including more Newton’s steps. The conclusions remain valid. Figures 21, 22, 23, and 24 show examples of the behavior with more Newton’s steps. Therefore, we can propose the use of damped methods in order to find good starting points for the original methods. We would like to analyze these hybrid methods in a forthcoming paper. We are also interested to analyze the situation of nonfixed damping factors.

Figure 21 

𝑀𝑓−,𝜆 for 𝜆=1.0 and 𝑓−(𝑥)=𝑥3−𝑥 with three Newton steps.

Figure 22 

𝑀𝑓−,𝜆 for 𝜆=0.25 and 𝑓−(𝑥)=𝑥3−𝑥 with three Newton steps.

Figure 23 

𝑀𝑓+,𝜆 for 𝜆=1.0 and 𝑓+(𝑥)=𝑥3+𝑥 with three Newton steps.

Figure 24 

𝑀𝑓+,𝜆 for 𝜆=0.25 and 𝑓+(𝑥)=𝑥3+𝑥 with three Newton steps.

Acknowledgments

The authors would like to thank the referees and the editor of AAA for their suggestions in the first draft of this paper. Research of the first two authors is supported in part by FEDER MTM 2010-17508 and 08662/PI/08. Research of the third author is supported in part by MTM2011-28636-C02-01.