A family of Newton-Chebyshev type methods to find simple roots of nonlinear equations and their dynamics

In this work, a new family of Newton-Chebyshev type methods for solving nonlinear equations is presented. The dynamics of the Newton-Chebyshev family for the class of quadratic polynomials is analyzed and the convergence is established. We find the fixed and critical points. The stable and unstable behaviors are studied. The parameter space associated with the family is studied and finally, some dynamical planes that show different aspects of the dynamics of this family are presented.


Introduction
Iterative methods are usually necessary for solving scalar nonlinear equations.Several good methods exist in the literature: Newton, Halley and Chebyshev methods among others, see ( [1]- [3]).The study of the dynamics of various methods was also done, see [4] for example.In this paper, we give a new family of Newton-Halley type methods for solving scalar nonlinear equations.Here the author establishes the conjugacy class and when this family is applied to the class of quadratic polynomials, fixed and critical points of this family are obtained.Dynamical planes for different values of the parameter A selected from the parameter space are presented.To conclude this section, some preliminary basics are presented.Then, in section 2 the mentioned family and their convergence is presented.Subsequently, in section 3, results on the dynamics of the Newton-Halley family, with an emphasis on the stability of fixed points for then use the parameter space and thus represent the dynamic planes for different values of the parameter A. Finally, the concluding remarks are presented.

Basic preliminaries
We now recall some preliminaries of complex dynamics (see [4], [13] and [51]) that we use in this work.Given a rational function R : C → C, where C is the Riemann sphere Definition 1.1.For z ∈ C we define its orbit as the set orb(z then the fixed point is called superattractor.A superattractor fixed point is also a critical point. Definition 1.7.A fixed point z 0 that is not associated to the roots of the function f (z) is called strange fixed point.
Definition 1.8.The basin of attraction of a attractor α ∈ C is defined as the set of starting points whose orbits tend to α.
2 Newton-Halley type methods and their Convergence In this section we present the new family and its convergence.We recall that a sequence {x n } n≥0 converges to r with order of convergence p if there exists a K(r) > 0 such that and the error equation is e n+1 = K(r)e n + O(e n+1 n ) where e n = x n − r and K(r) is the asymptotic constant error (see).In 1993 Hernández and Salanova [52] developed a family of Chebyshev-Halley type methods.Here, we present an uni-parametric family that allows us to study the evolution of the dynamics of the Newton-Halley family given by and where parameter A is complex and L f is degree of logarithmic convexity of f (see [53]- [55]).This family includes Newton's method for A = 0 and Halley's method for A = 1 2 .To begin the study of this family, we present the convergence in the following theorem. 2 where e n = z n − α is the error in the nth iterate and B j = f ( j) (α) j! , j = 1, 2, • • • .Proof.By Taylor series expansion around the simple root α in the nth iteration, we have Furthermore, it can be easily found by substituting these terms in (2.1) that which gives (2.2).This proves the theorem So, the family has order of convergence two, except for Halley's method which has order three.

Dynamical behavior of the rational function associated with Newton-Halley family
Here the author establishes the conjugacy class and the analytical expressions for the fixed and critical points of the Newton-Halley family in terms of the parameter A. Then the study of the fixed points, critical points and parameter space are presented.To finish this section several dynamical planes for different values of A selected from the parameter space are shown.

Conjugacy classes
In what remains of this paper we study the dynamics of the rational map R arising from Newton-Halley family (2.1) Let us first remember the following definition.
Definition 3.1.[56].Let f and g be two maps from the Riemann sphere into itself.An analytic conjugacy between f and g is an analytic diffeomorphism h from the Riemann sphere onto itself such that h R f has the following property for an analytic function f Theorem 3.1.(The Scaling Theorem).Let f (z) be an analytical function on the Riemann sphere, and let Proof.With the iteration function R(z), we have We therefore have

. [10]. We say that a one-point iterative root-finding algorithm p → T p has a universal Julia set (for polynomials of degree d) if there exists a rational map S such that for every degree d polynomial p, J(T p ) is conjugate by a Möbius transformation to J(S)
The following theorem establishes a universal Julia set for quadratics for our method (2.1).
We then have We observe that parameters a and b do not appear in S(z), because the Newton-Halley family complies with theorem 3.

Study of the fixed points
The fixed points of S for S defined in (3.4) To study the stability of the fixed points, we calculate S ′ (z), so It is obvious from (3.5) that z = 0 and z = ∞ are superatractive fixed points.The study of stability of the other fixed points is now presented.The operator S ′ (z) in z = −1 gives If we analyze this function, we obtain an horizontal asymptote in |S ′ (−1)| = 1 when A → ±∞, and a vertical asymptote in A = 0 (Newton's method).
In the following result we present the stability of the fixed point z = −1.

If Re{A}
Proof.From (3.6), Let A = α + iβ be an arbitrary complex number.Then, If we analyze this function, we obtain an horizontal asymptote in |S ′ (1)| = 1 when A → ±∞, and a vertical asymptote in A = 1 (Newton's method for multiple roots).
In the following result we present the stability of the fixed point z = 1.
Theorem 3.4.The strange fixed point z = 1 satisfies the following statements: 1.If Re{A} > 3 2 , then z = 1 is an attractor and it is a superattractor for A = 2.
Proof.From (3.7), In Figure 1 the functions where are observed the regions of stability are graphed.These functions are given by Zones of stability are when S 1 (A) = 1.

Study of the critical points
Critical points of S(z) satisfy S ′ (z) = 0, that is, z = 0, z = ∞ and Observe that zc 2 = 1 zc 1 and zc 1 = zc 2 = 1 only when A = 2. zc 1 = zc 2 = −1 only when A = −1.When A = 0 or A = 1 the only one critical point is z = 0 and if A = 1  2 , z = 0 is a critical point with multiplicity two.In Figure 2, the author represent the behavior of the fixed points and critical points for real values of A between −4 and 4. Fixed points are represented by black solid lines and this is more thick when fixed points are attractors.Critical points zc 1 and zc 2 are represented by red solid line and blue dotted line respectively.

Study of parameter space
In this section the behavior of the iterative methods obtained for various values of parameter A when it is used in the calculation of the critical points that are used as initial iteration is analyzed graphically.In this way some members of the family of methods presented with good or bad behavior can be identified.In this study, we use a mesh of 1000 × 1000 points, a tolerance of 10 −2 and a maximum of 50 iterations.If the iteration begins with the critical point obtained by substituting the value of parameter A in the method for that parameter value and observing the convergence to z = 0 or to z = ∞ with the established tolerance, point A of the complex plane is represented in Figure 3 in red color.When the critical point generates iterates that do not converge, the point A is represented in blue; other colors indicate convergence to strange fixed points.The various tonalities are related to the speed of convergence; so, if the color is darker the method for that parameter value converges faster.Figure 3 on the right shows a zoom to observe in more detail the behavior of the method in non-convergence zones.

Dynamical Planes
In this section the dynamic planes are represented for various methods obtained by substituting some values of parameter A in the rational function S given in (3.4).These values of A were selected from different areas of the parameter space studied in the previous section.In these dynamical planes the convergence to 0 appear in light blue, in red appears the convergence to ∞, in dark blue the zones with no convergence to the roots and other colors show the convergence to strange fixed points.The various tonalities are related to the speed of convergence; so, if the color is darker the method converges more slowly.Now, in Figures 4-7 various stable dynamic planes for values of A selected in the parameter space are showed.In Figures 8-9, the dynamical planes of several members of the family with broad regions of no convergence is shown.

Periodic orbits
In this section we present different periodic orbits of period two and only one periodic orbit of period three for A = 1.35.First, S(S(z)) = z is resolved, where S is give in (3.4).In Figure 12  To calculate periodic orbits of period three is necessary that S(S(S(z))) = z.In this case, eight orbits can be obtained.In Figure 12

Result and discussion
In this paper we present a family of Newton-Halley type methods and then a study of the complex dynamics for this family for the second-degree polynomial class is made.For this, the scaling theorem and the conjugation mapping for that family were first established, then the fixed points and critical points of the obtained rational operator were studied.We also analyzed the parameter space, selecting different values of this parameter to make the respective dynamic planes.Thus dynamic planes of methods with stable, unstable behavior and with convergence to strange fixed points are presented.Finally, we show the existence of periodic orbits, representing graphically all orbits of period two and one of the eight orbits of period three.It is clear that more studies on the dynamics of this family are necessary.

Theorem 2 . 1 .
Let α ∈ I be a simple root of a sufficiently differentiable function f : B → R for an open interval B. If x 0 is sufficiently close to α, then the family Newton-Halley type methods defined by (2.1) has almost second-order convergence, and satisfies the error equation:

Theorem 3 .
1 allows the study of the dynamics of the iteration function of Newton-Halley family (2.1) for the polynomial P 2 (z) = a 2 (z − z 1 )(z − z 2 ) by means of the study of the polynomial p(z) = (z − a)(z − b) where a ̸ = b.Definition 3.2

Theorem 3 . 2 .
For a rational map R p (z) given by (2.1) applied to p(z) = (z − a)(z − b), a ̸ = b, R p (z) is conjugate via the Möbius transformation given by M(z) = z−a z−b to

Figure 2 :
Figure 2: Dynamical Behavior of strange fixed points and critical points for −4 < A < 4

Figure 3 :
Figure 3: Parameter plane associated to the critical point zc1 and zoom