With the advancement of computer algebra, finding higher-order multi-point methods, not requiring the computation of second-order derivative for multiple roots, becomes very important and is an interesting task from the practical point of view. These multi-point methods are of great practical importance since they overcome theoretical limits of one-point methods concerning the order and computational efficiency. Further, these multi-point iterative methods are also capable to generate root approximations of high accuracy.
Here, discuss the dynamical analysis of iterative methods for finding a zero of a continuously differentiable function
This is quadratically convergent and is optimal in the sense of Kung and Traub [5]. Existing third and fourth-order methods of finding simple roots to multiple roots have been extended (see, for example, [7]-[9]). However, the number of families of optimal iterative methods for finding multiple roots of nonlinear equations available in the literature, such as [1]- [10], is very much reduced.
Recently, Hueso et al. [4] considered the following fourth-order iterative method
where
and
Here, our main concern is to discuss the dynamical analysis of the rational map associated with the above mentioned scheme for multiple roots. First, we are going to recall some dynamical concepts of complex dynamics (see [3]) that we use in this work. Given a rational function
We analysed the phase plane of the map
On the other hand, the basin of attraction of an attractor
In the Fatou set of the rational function
By using these tools of complex dynamics, we study the general convergence of family (1) on polynomials with multiple roots of multiplicity 2 and 3. 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 polynomials
The rest of the paper is organised as follows. In Section 2 the complex dynamics of the family is studied on low-degree polynomials with multiplicity
In order to study the stability of the family on polynomials with double roots, the operator of the family on
Blanchard [3] considered the conjugacy map (a Möbius transformation)
with the following properties:
and proved that, for quadratic polynomials, Newton’s operator is conjugate to the rational map
Next, we are going to analyze, under the dynamical point of view, the stability and reliability of the members of the proposed family. First, we will study the fixed points of the rational function that are not related to the original roots of the polynomial and then the free critical points, that is, the critical points of the associated rational function different from 0 and ¥.
For
where
Let us observe that the parameters
As we have seen, the fourth-order family of iterative methods (1), applied on the polynomial
that are denoted by
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
where
and
As it comes from the double root of the polynomial, it is clear that
The rest of the proof is straightforward as the stability function of the infinity is
So,
Let us consider
that is,
Therefore,
In addition, if
The analysis of the stability function of the strange fixed points,
Stability functions of the strange fixed pointsFig. 1
Regarding the critical points, they are calculated by solving
In this section, we show, by means of dynamical planes, the qualitative behaviour of the different elements of the proposed family by using the conclusions obtained in the analysis of the stability of strange fixed points.
The dynamical plane associated with a value of the parameter
Some values of the parameter whose associated iterative method shows stable behavior with convergence only to the "roots" (multiple
Dynamical planes with stable behaviorFig. 2
On the other hand, unstable behaviour is found when we choose values of the parameter in the stability region of attracting strange fixed points (Figures 3(a), 3(b), 3(e), 3(f) and 3(i)) or attracting periodic orbits (Figures 3(c), 3(d), 3(g) and 3(h)). The periodic orbits are marked with yellow lines, with yellow circles at the elements of the orbit (Figures 3(d) and 3(h)).
Dynamical planes with unstable behaviorFig. 3
For
where
and
Let us observe that the parameters
As we have seen, the fourth-order family of iterative methods (1), applied on the polynomial
to the roots of
Nevertheless, the complexity of the operator can be lower depending on the value of the parameter, as we can see in the following result.
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
where
and
Regarding the stability of the fixed points, it is clear that 0 is a superattractive fixed point. However, as in case
Let us remark that area of the complex plane where
Stability function of the fixed points Fig. 4
Regarding the rest of strange fixed points, we remark some interesting aspects that have been stated both numerical and graphically:
Most of them are repulsive, for any complex value of parameter Four of the strange fixed points are superattracting if In addition, one of these points is superattracting if
Stability functions of the strange fixed points Fig. 5
For the global behaviour of the strange fixed points, Figure 5(a) can be observed. In it, it is clear that, except for values of parameter close to 2
For better understanding the behaviour of the elements of the family applied on polynomials with cubic multiplicity, it is necessary to analyse the number of critical points of the associated operator, as a lower number decreases the number of attracting areas different from the roots. In the following result these items are discussed, from the analysis of the equation
Let us also remark that the first critical points
In this section, some of the values of
Dynamical planes with stable behavior for Fig. 6
There have been also appeared some values of
Dynamical planes with unstable behaviorFig. 7
In this article, investigation has been made on the complex plane for class (1) to reveal its dynamical behaviour on polynomials with double and triple roots. The dynamical study of family (1) of iterative methods allows us to select iterative schemes with good stability and reliability properties and detect iterative methods with dangerous numerical behaviour. Indeed, wide regions for parameter