Damped Traub’s method: Convergence and stability

Accepted Manuscript Damped Traub’s method: Convergence and stability Alicia Cordero, Alfredo Ferrero, Juan R. Torregrosa PII: DOI: Reference:

S0378-4754(15)00177-9 http://dx.doi.org/10.1016/j.matcom.2015.08.012 MATCOM 4221

To appear in:

Mathematics and Computers in Simulation

Received date: 24 November 2014 Revised date: 18 August 2015 Accepted date: 18 August 2015 Please cite this article as: A. Cordero, A. Ferrero, J.R. Torregrosa, Damped Traub’s method: Convergence and stability, Math. Comput. Simulation (2015), http://dx.doi.org/10.1016/j.matcom.2015.08.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Damped Traub’s method: convergence and stability I Alicia Corderoa,∗, Alfredo Ferrerob , Juan R. Torregrosaa a Instituto

Universitario de Matem´atica Multidisciplinar de Matem´atica Aplicada Universitat Polit`ecnica de Val`encia Camino de Vera s/n, 46022 Val`encia, Spain b Departamento

Abstract In this paper, a parametric family including Newton’ and Traub’s iterative schemes is presented. Its local convergence and dynamical behavior on quadratic polynomials is studied. The analysis of fixed and critical points and the associated parameter plane show the dynamical richness of the family and allows us to find members of it with good numerical properties, as well as other ones with very unstable behavior. Keywords: Nonlinear equations; iterative methods; dynamical behavior; quadratic polynomials; Fatou and Julia sets; Traub’s family; convergence regions.

1. Introduction Real world phenomena are often modeled by means of nonlinear equations f (z) = 0; for example, the numerical solution of nonlinear equations are needed in the study of dynamical models of chemical reactors [11], in radioactive transfer [17], preliminary orbit determination of satellites [4] or even the approximation of the eigenvalues of square matrices, which is known have many applications in areas as image processing, dynamical systems, control theory, etc (see, for example, [12]). For solving these equations, iterative schemes must be used. The best known iterative approach is Newton’s method. In last decades, many researchers have proposed different iterative methods to improve Newton’s scheme (see, for example, the review [25], and the references therein). These variants of Newton’s method have been designed by means of different techniques, providing in the most of cases multistep schemes. Some of them come from Adomian decomposition (see [1] and [6], for example). Another procedure to develop iterative methods is the replacement of the second derivative in Chebyshev-type methods by some approximation: in [27], Traub presented a family of multi-point methods based on approximating the second derivative that appears in the iterative formula of Chebyshev’s scheme and Babajee et al. in [7] designed two Chebyshev-like methods free from second derivatives. A common way to generate new schemes is the direct composition of known methods with a later treatment to reduce the number of functional evaluations. For example, by composing Newton’s method with itself, holding the derivative ”frozen” in the second step, third-order Traub’s method [27] is obtained. Recently, the weight-function procedure has been used to increase the order of convergence of known methods ([25, 26]), allowing to get optimal methods, under the point of view of Kung-Traub’s conjecture [21]. These authors conjectured that an iterative methods without memory which uses d functional evaluations per iteration can reach, at most, order of convergence 2d−1 . When this bound is reached, the scheme is called optimal. Although the aim of many researches in this area is to design optimal high-order methods (see, for example [14, 18, 20]), it is also known that the higher the order is, the more sensitive the scheme to initial estimations will be [22]. On the other hand, recent studies on damped Newton’s procedure show (see, for example [23]) that small damping parameters widen the set of initial guesses that make the method convergent, although the speed of convergence decreases. In this paper, we present a damped Traub’s-type family of iterative methods. We will prove that it is an uniparametric set of second and third-order iterative schemes to estimate simple roots of nonlinear equations, whose iterative expression is zn+1 = yn − γ

f (yn ) , n = 0, 1, . . . , f ′ (zn )

(1)

f (zn ) and γ is the damping parameter. Let us note that, if γ = 1 we get Traub’s scheme and γ = 0 corresponds f ′ (zn ) to Newton’s method.

where yn = zn −

I This research was supported by Ministerio de Econom´ıa y Competitividad MTM2014-52016-C2-02 and FONDOCYT 2014-1C1- 088 Rep´ ublica Dominicana. ∗ Corresponding author Email addresses: [email protected] (Alicia Cordero ), [email protected] (Alfredo Ferrero), [email protected] (Juan R. Torregrosa)

Preprint submitted to Elsevier

August 18, 2015

The application of iterative methods on polynomials gives rise to rational functions whose complex dynamics is not wellknown. However, Amat et al. in [3] studied real dynamics of Traub’s scheme on quadratic and cubic polynomials. From the numerical point of view, the dynamical behavior of the rational function associated with an iterative method gives us important information about its stability and reliability. In these terms, Amat et al. in [2] described the dynamical behavior of several well-known families of iterative methods. More recently, in [15, 16, 19, 5, 24], the authors analyze the qualitative behavior of different known iterative families. The most of these studies show different pathological numerical behavior, such as periodic orbits, attracting fixed points different from the solution of the problem, etc. Indeed, parameter planes associated to a family of methods allow us to understand the behavior of the different members of the family of methods, helping us in the election of a particular one. The rest of the paper is organized as follows: in Section 2 we introduce the basic concepts on complex dynamics, in Section 3 we study the local convergence of family (1) of iterative methods. Section 4 is devoted to the analysis of the fixed and critical points of the operator Oγ (z) and the study the stability of the fixed points is showed in Section 5. These regions of stability also appear in the associated parameter spaces (Section 6), as well as the stability region of the attractive 2-periodic orbits, whose analytical expression is found, depending on the damping parameter. We finish the work with some remarks and conclusions. 2. Basic concepts Under the point of view of complex dynamics, we will study the general convergence of family (1) on quadratic polynomials. It is known that the roots of a polynomial can be transformed by an affine map with no qualitative changes on the dynamics of the family. So, we can use the quadratic polynomial p (z) = (z − a)(z − b). For p(z), the operator of the family is the rational function: Tp,γ,a,b (z) =

1 (a3 b + 2a2 b2 + ab3 − 4a2 bz − 4ab2 z − a2 z 2 + 2abz 2 − b2 z 2 + 4az 3 + 4bz 3 − (a + b − 2z)3

4z 4 + a2 b2 γ − 2a2 bzγ − 2ab2 zγ + a2 z 2 γ + 4abz 2γ + b2 z 2 γ − 2az 3 γ − 2bz 3γ + z 4 γ), depending on the parameters γ, a and b. Blanchard in [9] considered the conjugacy map h (z) = i) h (∞) = 1,

z−a , (a M¨obius transformation) with the following properties: z−b

ii) h (a) = 0,

iii) h (b) = ∞,

and proved that, for quadratic polynomials, Newton’s operator is conjugate to the rational map z 2 . In an analogous way, operator Tp,γ,a,b(z) on quadratic polynomials is conjugated to operator Oγ (z),
1 + 2z + z 2 − γ . Oγ (z) = h ◦ Tp,γ,a,b ◦ h−1 (z) = z 2 1 + 2z + z 2 (1 − γ)

(2)

We observe that the parameters a and b have been obviated in Oγ (z). Now, we are going to recall some dynamical concepts of complex dynamics (see [10]) that we use in this work. Given a ˆ → C, ˆ where C ˆ is the Riemann sphere, the orbit of a point z0 ∈ C ˆ is defined as: rational function R : C {z0 , R (z0 ) , R2 (z0 ) , ..., Rn (z0 ) , ...}. ˆ We analyze the phase plane of the map R by classifying the starting points from the asymptotic behavior of their orbits. A z0 ∈ C p k is called a fixed point if R (z0 ) = z0 . A periodic point z0 of period p > 1 is a point such that R (z0 ) = z0 and R (z0 ) 6= z0 , for 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 the rational function vanishes, R′ (z0 ) = 0. Moreover, 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 basin of attraction of an attractor α is defined as: ˆ : Rn (z0 ) →α, n→∞}. A (α) = {z0 ∈ C ˆ whose orbits tend to an attractor (fixed point, The Fatou set of the rational function R, F (R) , is the set of points z ∈ C ˆ periodic orbit or infinity). Its complement in C is the Julia set, J (R). That means that the basin of attraction of any fixed point belongs to the Fatou set and the boundaries of these basins of attraction belong to the Julia set. 3. Convergence analysis The following result show the local convergence of family (1) of iterative methods.

2

Theorem 1. Let f : D ⊂ R → R be a sufficiently differentiable function in an open interval D. If α ∈ D is a simple root of f (z) = 0 and z0 is close enough to α, then the members of family (1) converge to α, being their error equation: en+1 = (1 − γ)c2 e2n + ((4γ − 2)c22 + (2 − 2γ)c3 )e3n + O(e4n ), where en = zn − α, and cj = (1/j!)f (j) (α)/f ′ (α), j = 2, 3, . . . Moreover if γ = 1, Traub’s method is obtained and it has third order of convergence. Proof. By using Taylor expansions around α, 1 (2) 1 f (α)e2n + f (3) (α)e3n + O(e4n ) 2! 3! = f ′ (α)[en + c2 e2n + c3 e3n + O(e4n )]

f (zn ) = f (α + en ) = f (α) + f ′ (α)en +

(3)

and f ′ (zn ) = f ′ (α + en ) = f ′ (α) + f (2) (α)en +

1 (3) f (α)e2n + O(e3n ) 2!

= f ′ (α)[1 + 2c2 en + 3c3 e2n + O(e3n )].

(4)

By direct division of (3) and (4), f (zn ) =[en + c2 e2n + c3 e3n + O(e4n )][1 + 2c2 en + 3c3 e2n + O(e3n )]−1 f ′ (zn )

(5)

=en − c2 e2n + (2c22 − 2c3 )e3n + O(e4n ). Then, by (5) yn = zn −

f (zn ) = α + c2 e2n + (2c3 − 2c22 )e3n + O(e4n ). f ′ (zn )

(6)

Again by (6) and Taylor’s expression, f (yn ) = f ′ (α)[c2 e2n + (2c3 − 2c22 )e3n + O(e4n )].

(7)

f (yn ) =[c2 e2n + (2c3 − 2c22 )e3n + O(e4n )][1 + 2c2 en + 3c3 e2n + O(e3n )]−1 f ′ (zn )

(8)

Dividing (7) by (4),

=c2 e2n + (2c3 − 4c22 )e3n + O(e4n ). Finally, from (6) and (8), en+1 =zn+1 − α = yn − α −

γf (yn ) f ′ (zn )

=c2 e2n + (2c3 − 2c22 )e3n − γ[c2 e2n + (2c3 − 4c22 )e3n ] + O(e4n ) =(1 − γ)c2 e2n + ((4γ − 2)c22 + (2 − 2γ)c3 )e3n + O(e4n ),

and the proof is finished. In the next sections we are going to analyze, under the dynamical point of view, the stability and reliability of the members of the proposed family. Firstly, we will study the fixed points of the rational function Oγ (z) that are not related with the original roots of the polynomial p(z) (called strange fixed points), and the free critical points, that is, the critical points of Oγ (z) different from 0 and ∞. 4. Analysis of the fixed and critical points Fixed points of Oγ (z) are the roots of the equation Oγ (z) = z, that is, z = 0, z = ∞ and, for γ 6= 0, the strange fixed points • ex1 (γ) = 1 for γ 6= 4 and γ 6= 0, √ √ • ex2 (γ) = 12 (−2 − γ − γ 4 + γ), 3

• ex3 (γ) = 21 (−2 − γ +

√ √ γ 4 + γ).

Some relations between the strange fixed points are described in the following result. Lemma 1. The number of simple strange fixed points of operator Oγ (z) is three, except in cases: i) If γ = 0, then the operator’s expression is O0 (z) = z 2 , so there are no strange fixed points. ii) If γ = 4, there are two simple strange fixed points, ex2 (4) and ex3 (4), being the associated rational function O4 (z) = z 2

−3 + 2z + z 2 . 1 + 2z − 3z 2 2

5+3z+z iii) If γ = −4, then ex2 (−4) = ex3 (−4) = ex1 (−4) = 1, as O−4 (z) = z 2 1+2z+5z 2.

In order to determine the critical points, we calculate the first derivative of Oγ (z), Oγ′ (z) = 2z(1 + z)2

1 + z 2 (1 − γ) − γ + z(2 + γ) . (1 + 2z − z 2 (γ − 1))2

A classical result establishes that there is at least one critical point associated with each invariant Fatou component. It is clear that z = 0 and z = ∞ (related to the roots of the polynomial by means of M¨obius map) are critical points and give rise to their respective Fatou components, but there exist in the family some free critical points, some of them depending on the value of the parameter. Lemma 2. Analyzing the equation Oγ′ (z) = 0, we obtain that a) If γ = 0, there is no free critical points of operator Oγ (z). b) If γ = 4 or γ = 1, then z = −1 is the only free critical point. c) In any other case, cr1 (γ) = −1,

cr2 (γ) =

√ √ 2+γ− 3 4γ−γ 2 2(γ−1)

and cr3 (γ) =

√ √ 2+γ+ 3 4γ−γ 2 2(γ−1)

=

1 cr2

are free critical points. Moreover, the following result can be stated. From it, the only member of family (1) that satisfies Cayley’s test (see [8]) is Newton’s method. Theorem 2. The only member of Traub’s family whose operator is always conjugated to the rational map z 2 corresponds to γ = 0 that is, Newton’s scheme. Proof. From (2), we denote r(z) = z 2 + 2z + (1 − γ) and q(z) = (1 − γ)z 2 + 2z + 1. By factorizing both polynomials, we can observe that the unique value of γ verifying r(z) = q(z) is γ = 0. From the previous results, let us summarize: • At most, there are two independent free critical points. So, we will consider only cr1 (γ) and cr2 (γ). • When γ = 4, cr2 (4) = cr3 (4) = −1 that is a pre-image of z = 1, that in this case is not a fixed point, and the associated 2 ) operator is O4 (z) = z 2 (−3+2z+z 1+2z−3z 2 . It can be noticed that z = 1 is a pole of O4 (z). 2

) • If γ = −4, then the associated operator is O−4 (z) = z 2 (5+3z+z 1+2z+5z 2 and there is only one strange fixed point, z = 1.

• When γ = 0, the associated operator is O0 (z) = z 2 and there are not strange fixed points, nor free critical points. As we will see in the following section, not only the number but also the stability of the fixed points depend on the parameter of the family. The relevance of this study yields in the fact that the existence of attracting strange fixed points can make the iterative scheme converge to a ”false” solution.

4

5. Stability of the fixed points As the order of convergence of the family is at least two, it is clear that the origin and ∞ are always superattractive fixed points, but the stability of the other fixed points gives us interesting numerical information. In the following results we show the stability of the strange fixed points. Theorem 3. The character of the strange fixed point ex1 (γ) = 1, γ 6= 4, is as follows: i) If |γ − 4| > 8 , then ex1 (γ) = 1 is an attractor, but it can not be a superattractor. ii) When |γ − 4| = 8, ex1 (γ) = 1 is a parabolic point. iii) If |γ − 4| < 8, then ex1 (γ) = 1 is a repulsor. Proof. It is easy to proof that Oγ′ (1) = − So,

−

and

8 ≤1 −4 + γ

Therefore,

−

8 . −4 + γ

is equivalent to

8 6= 0 −4 + γ

8 ≤ |−4 + γ| ,

for any γ in R.

′ Oγ (1) ≤ 1 if and only if |γ − 4| ≥ 8 Moreover, if γ satisfies 0 < |γ − 4| < 8, then Oγ′ (1) > 1 and ex1 (γ) = 1 is a repulsive point. Finally, if |γ − 4| = 0 then γ = 4 and ex1 (γ) = 1 is not a fixed point. Similar results can be proved for the rest of strange fixed points.

Theorem 4. The analysis of the stability of strange points ex2 (γ) and ex3 (γ) shows that: i) If γ + 38 < 43 , then both points are attractors and they are superattractors when γ = −2. ii) If γ + 38 = 43 , then ex2 (γ) and ex3 (γ) are parabolic. iii) In any other case, both are repulsors.

The proof of this theorem is analogous to that of Theorem 3, by using the stability function of ex2 (γ) and ex3 (γ),     1 1 4 √ p √ p ′ ′ Oγ (−2 − γ − γ 4 + γ) = Oγ (−2 − γ + γ 4 + γ) = + 2. 2 2 γ

In Figure 1, we represent the stability regions of exi (γ), i = 1, 2, 3, that we get from Theorem 3 and Theorem 4.

Figure 1: Stability regions of ex1 (γ) (left) and exi (γ), i = 2, 3 (right), respectively.

5

−8

−6

−6

−4

−4

−2

−2

IIm{α}

IIm{α}

−8

0

0

2

2

4

4

6

6

8 −4

−2

0

2

4 IRe{α}

6

8

10

8 −4

12

Figure 2: Parameter plane P1 associated to cr1 (γ) = −1

−2

0

2

4 IRe{α}

6

8

10

12

Figure 3: Parameter plane P2 associated to cri (γ), i = 2, 3

6. The parameter space As we have seen, the dynamical behavior of operator Oγ (z) depends on the values of the parameter γ. The parameter space associated with a free critical point of operator (2) is obtained by associating each point of the parameter plane with a complex value of γ, i.e., with an element of family (1). Every value of γ belonging to the same connected component of the parameter space give rise to subsets of schemes of family (1) with similar dynamical behavior. So, it is interesting to find regions of the parameter plane as much stable as possible, because these values of γ will give us the best members of the family in terms of numerical stability. As cr2 (γ) = cr31(γ) , we have at most two free independent critical points, so we can obtain different parameter planes, with complementary information. When we consider the free critical point cr1 (γ) = −1 as a starting point of the iterative scheme of the family associated to each complex value of γ, we paint this point of the complex plane in red if the method converges to any of the roots (zero and infinity) and they are white in other cases. The color used is brighter when the number of iterations is lower. Then, the parameter plane P1 is obtained; it is showed in Figure 2. Each parameter plane has been generated by slightly modifying the routines described in [13]. A mesh of 2000 × 2000 points has been used, 1000 has been the maximum number of iterations involved and 10−3 the tolerance used as a stopping criterium. A similar procedure can be carried out with the free critical points, z = cri (γ), i = 2, 3, obtaining in both cases the same parameter space P2 , showed in Figure 3. Let us remark that both parameter planes have common regions: the disk that defines the values of γ where ex1 (γ) is attractive (defined in Theorem 3) and a white disk on the right of both figures, which we denote by D1 . In case of P1 , let us remark that Oγ (−1) = 1, and 1 is a repulsive strange fixed point inside the disk centered at 4 and with radius 8. Then, the rounding error in the calculations will determine its orbit. In the following, we will focus our attention on parameter plane P2 , due to its dynamical richness. In it, joint with the previously mentioned disks, we can observe a smaller white disk on the left (which we denote by D2 ): it is the region where strange fixed points ex2 (γ) and ex3 (γ) are attractive or superattractive (see Theorem 4). Now we are going to show that D1 corresponds to values of γ for which there are attractive orbits of period 2. 6.1. Orbits of period two In order to obtain the analytical expression of the elements of 2-periodic orbits, depending on γ, we calculate Oγ (Oγ (z)), that will be denoted by Oγ2 (z) Oγ2 (z)

=


2 

−γ 3 z 4 − γ(z + 1)2 z 4z 3 + 5z + 2 + 1 + 2γ 2 (z(3z + 2) + 1)z 2 + (z + 1)4 z 2 + 1
. − γ 3 z 4 − 2γ 2 (z(z + 2) + 3)z 4 + γ(z + 1)2 ((z(z + 2) + 5)z 2 + 4) z 2 − (z + 1)4 (z 2 + 1)2 (z(−γz + z + 2) + 1)2 z 4 (z + 1)2 − γ


2 

The periodic points of Oγ (z) with period two are the roots of the equation Oγ2 (z) = z, that is, the fixed points, the 2-periodic points q q √ √ √ √ + γ − 32 − 43 , pe{3,4} (γ) = 41 4γ + 1 ∓ 21 12(4−γ)−51 + γ − 23 − 43 , pe{1,2} (γ) = − 41 4γ + 1 ± 12 − 12(4−γ)−51 2 4γ+1 2 4γ+1

and also the roots of polynomial 1 + (4 − γ)z + (8 − 7γ + γ 2 )z 2 + (12 − 11γ + 2γ 2 )z 3 + (14 − 10γ + 6γ 2 − γ 3 )z 4 + (12 − 11γ + 2γ 2 )z 5 + (8 − 7γ + γ 2 )z 6 + (4 − γ)z 7 + z 8 whose analytical expression has not been obtained. In Figure 4, we represent the stability regions of pei (γ), i = 1, 2, 3, 4, of which the first one correspond to D1 , that is, the 2 complex area where |Oγ′ (z)| < 1. In Figure 5 we show all stability regions including those without analytic expression (they correspond to the biggest bulbs of disks D1 and D2 ). 6

Figure 4: Stability regions of 2-periodic orbits pei (γ), i = 1, 2 (left) and pei (γ), i = 3, 4 (right)

Figure 5: Stability regions of strange fixed points and 2-periodic points

Moreover, it can be checked that there exist two √ values of parameter γ that make the 2-periodic orbits superattracting, that is, 2 satisfy Oγ′ (z) = 0. These values are γ = 4 ± 3 2. 6.2. Dynamical Planes In this section we will show, by means of dynamical planes, the qualitative behavior of the different elements of family (1). We will select this elements by using the conclusions obtained by analyzing the parameter planes of the family. −4

−4 −3

−3 −2

−2

−1 IIm{z}

IIm{z}

−1

0

0

1

1

2

2

3

3

4 −4

−3

−2

−1

0 IRe{z}

1

2

3

4 −4

4

−3

−2

(a) γ = 0

−1

0 IRe{z}

1

2

3

4

(b) γ = 1 −4

−4

−3

−3

−2

−2

−1

IIm{z}

IIm{z}

−1

0

0

1

1

2

2

3

4 −5

3

−4

−3

−2

−1 IRe{z}

0

1

2

4 −4

3

(c) γ = 2.12 − 1.76i

−3

−2

−1

0 IRe{z}

1

2

3

4

(d) γ = 5.9 − 4.75i

Figure 6: Some dynamical planes with stable behavior

As in case of parameter planes, these dynamical planes has been generated by using the routines appearing in [13]. The dynamical plane associated to a value of the parameter γ, that is, obtained by iterating an element of family (1), is generated by 7

using each point of the complex plane as initial estimation (we have used a mesh of 400 × 400 points). We paint in blue the points whose orbit converges to infinity, in orange the points converging to zero (with a tolerance of 10−3 ), in green those points whose orbit converges to one of the strange fixed points (all fixed points appear marked as a white star in the figures) and in black if it reaches the maximum number of 40 iterations without converging to any of the fixed points. There are some regions in parameter space P2 whose corresponding iterative methods have good numerical behavior, in terms of stability and efficiency. They correspond to values of γ painted in red (Figure 3) within the disk |γ − 4| < 8. In Figure 6 we show different stable behavior corresponding to several values of γ selected in this red region; in particular, we use γ = 0 (Newton’s method), γ = 1 (Traub’s method), γ = 2.12 − 1.76i and γ = 5.9 − 4.75i; being the last one a case whose basins of attraction are unconnected. On the other hand, unstable behavior is found when we choose values of γ in the white region of parameter plane P2 . In Figure 7, the dynamical plane of the iterative method corresponding to γ = −2 ∈ D2 is presented, showing the existence of four different basins of attraction, two of them of the superattractors 0 and ∞ and the other two corresponding to the superattractors ex2 (−2) and ex3 (−2). z=0.0060126+i1 −4

−3

−3

−2

−2

−1

−1

IIm{z}

IIm{z}

z=0.0057324+i−1.0004 −4

0

0

1

1

2

2

3

3

4 −4

−3

−2

−1

0 IRe{z}

1

2

3

4 −4

4

−3

(a) ex2 (−2)

−2

−1

0 IRe{z}

1

2

3

4

(b) ex3 (−2)

Figure 7: Dynamical planes associated to γ = −2 with trajectories (in yellow) converging to ex2 (−2) and ex3 (−2)

In Figure 8, we represent the dynamical plane corresponding to γ = 10.3 + 0.7920i, which is in D1 , showing the existence of three different basins of attraction, two of them of the superattractors 0 and ∞ and the other one corresponding to the atractive orbit of period 2. z=0.00018766+i−0.00088601 −10

−8

−8

−6

−6

−4

−4

−2

−2 IIm{z}

IIm{z}

z=0.62459+i0.40399 −10

0

0

2

2

4

4

6

6

8

8

10

10 −10

−5

0

5

10

−10

−5

0

IRe{z}

IRe{z}

(a) 2-periodic orbit

(b) z = 0

5

10

Figure 8: Dynamical planes associated to γ = 10.3 + 0.7920i with trajectories (in yellow) converging to a two-periodic orbit and to zero

Other regions of the parameter plane P2 where there are attractive orbits of period 2 is the left central antennas. A detail of these region is presented in Figures 9a and 9b. A particular value of γ in this region is γ = −0.2. It is placed in the Mandelbrot set. The corresponding dynamical plane is showed in Figures 9c and 9d. We can observe the existence of three basin of attraction, two of them associated to the roots and the other to an attracting periodic orbit of period 2. 7. Conclusions A parametric two-point family is presented, showing the second- and third-order of convergence for any non-zero value of the parameter. This class contains the well-known Traub’s scheme. The dynamical study of the designed family on quadratic polynomials gives us important information about its stability, depending on the parameter. From parameter planes, it has been proved that there are many values of parameter γ, that is, elements of the family, with no convergence to the roots of the polynomial, and the existence of periodic orbits of period two has been showed and its analytical expression has been obtained in terms of parameter γ. These members of the family show a clear bad numerical behavior. Nevertheless, there are some regions in 8

0.1 1 0.08

0.8

0.06

0.4

0.04

0.2

0.02 Im{γ}

Im{γ}

0.6

0

0

−0.2

−0.02

−0.4

−0.04

−0.6

−0.06

−0.8

−1 −2

−0.08

−1.8

−1.6

−1.4

−1.2

−1 −0.8 Re{γ}

−0.6

−0.4

−0.2

−0.1 −0.3

0

(a) Details of the antennas of P2

−0.28

−0.26

−0.24

−3

−3

−2

−2

−1

−1

0

1

2

2

3

3

−1

0 IRe{z}

1

−0.16

−0.14

−0.12

−0.1

0

1

−2

−0.18

z=−0.59391+i0.81186 −4

IIm{z}

IIm{z}

z=−0.6876+i−0.74352

−3

−0.2 Re{γ}

(b) Details of the antennas of P2

−4

4 −4

−0.22

2

3

4 −4

4

(c) Periodic orbit for γ = −0.2

−3

−2

−1

0 IRe{z}

1

2

3

4

(d) Periodic orbit for γ = −0.2

Figure 9: Details of parameter plane of P2 ((a) and (b)) and two 2-periodic orbits corresponding to a γ in the Mandelbrot set of (b)

parameter spaces whose corresponding iterative methods have good numerical behavior, mainly in the disk of the complex plane centered in the origin and with radius two. This region includes Traub’s method, but also many other new stable schemes. Acknowledgments: The authors thank to the anonymous referee and to the editor for their valuable comments and for the suggestions that have improved the final version of the paper. 8. References References [1] G. Adomian, Solving Frontier Problem of Physics: The Decomposition Method, Kluwer Academic Publishers, Dordrecht, 1994. [2] S. Amat, S. Busquier, S. Plaza, Review of some iterative root-finding methods from a dynamical point of view, Sci. Ser. A: Math. Sci. 10 (2004) 3–35. [3] S. Amat, S. Busquier, S. Plaza, Chaotic dynamics of a third-order Newton-type method, J. Math. Anal. Appl. 366 (2010) 24–32. [4] V. Arroyo, A. Cordero, J.R. Torregrosa, Approximation of artificial satellites preliminary orbits: The efficiency challenge, Math. Comput. Mod. 54 (2011) 1802-1807. [5] S. Artidiello, F. Chicharro, A. Cordero, J.R. Torregrosa, Local convergence and dynamical analysis of a new family of optimal fourth-order iterative methods, Int. J. Comput. Math. 90 (10) (2013) 2049–2060. [6] D.K.R. Babajee, M.Z. Dauhoo, M.T. Darvishi, A. Barati, A note on the local convergence of iterative methods based on Adomian decomposition method and 3-node quadrature rule, Appl. Math. Comput. 200(1) (2008) 452–458. [7] D.K.R. Babajee, M.Z. Dauhoo, M.T. Darvishi, A. Karami, A. Barati, Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations, J. Comput. Appl. Math. 233(8) (2010) 2002–2012. [8] D.K.R. Babajee, A. Cordero, J.R. Torregrosa, Study of iterative methods through the Cayley Quadratic Test, J. Comput. Appl. Math. doi:10.1016/j.cam.2014.09.020. [9] P. Blanchard, The Dynamics of Newton’s Method, Proc. Symp. Appl. Math. 49 (1994) 139–154. 9

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