## 3 Complex Dynamics of Iterative Methods  
The dynamical planes provides an at-a-glance status of the convergence and stability of a method. In this section, the dynamical planes of the introduced methods are shown. In order to get a better understanding, some dynamical concepts are recalled. For a more extensive and comprehensive review of these concepts, see for example [7,8]. Let \(R:\hat{\mathbf{C}}\rightarrow\hat{\mathbf{C}}\) be a rational function, where \(\hat{\mathbf{C}}\) is the Riemann sphere. The orbit of a point \(z_{0}\in\hat{\mathbf{C}}\) is defined as \(\{z_{0},R(z_{0}),\ldots,R^{n}(z_{0}),\ldots\}\). The dynamical behavior of the orbit of a point on the complex plane can be classified depending on its asymptotic behavior. In this way, a point \(z_{0}\in\mathbf{C}\) is a fixed point of \(R\) if \(R(z_{0})=z_{0}\). A fixed point is attracting, repelling or neutral if \(|R^{\prime}(z_{0})|\) is less than, greater than or equal to 1, respectively. Moreover, if \(|R^{\prime}(z_{0})|=0\), the fixed point is superattracting. Let \(z_{f}^{*}\) be an attracting fixed point of the rational function \(R\). Its basin of attraction \(\mathcal{A}(z_{f}^{*})\) is defined as the set of pre-images of any order such that \(\mathcal{A}(z_{f}^{*})=\left\{z_{0}\in\hat{\mathbf{C}}:R^{n}(z_{0})\to z _{f}^{*},n\rightarrow\infty\right\}\). In our calculations, we usually consider the region \([-5,5]\times[-5,5]\) of the complex plane, with \(600\times 600\)points and we apply the corresponding iterative method starting in every \(z_{0}\) in this area. If the sequence generated by iterative method reaches a zero \(z^{*}\) of the polynomial with a tolerance \(|z_{k}-z^{*}|<10^{-3}\) and a maximum of 40 iterations, we decide that \(z_{0}\) is in the basin of attraction of these zero and we paint this point in a color previously selected for this root. In the same basin of attraction, the number of iterations needed to achieve the solution is showed in darker or brighter colors (the less iterations, the brighter color). Black color denotes lack of convergence to any of the roots (with the maximum of iterations established) or convergence to the infinity. The set of points whose orbits tends to an attracting fixed point (or an attracting periodic orbit) \(z_{f}^{*}\) is defined as the Fatou set, \(\mathcal{F}(R)\). The complementary set, the Julia set \(\mathcal{J}(R)\), is the closure of the set consisting of its repelling fixed points, and establishes the boundaries between the basins of attraction. If the rational function \(R\) is associated to the fixed point operator of the developed methods in Section 2 applied on a polynomial \(f(z)\), denoted generically as \(O_{f}(z)\), it is possible to find its fixed and critical points. The fixed points \(z_{f}\) verify \(O_{f}(z)=z\), and the critical points \(z_{c}\) validate \(O_{f}^{\prime}(z)=0\). The expressions  
\[S_{f}(z) = z-\frac{f^{2}(z)}{f(v)-f(z)} \tag{11}\] \[F_{f}(z) = y-\frac{f(y)f[z,v]}{f[z,y]f[y,v]}\] (12) \[E_{f}(z) = u-\frac{f(u)f[z,y,v]}{f[y,v,u]f[u,z,y](u-z)+f[z,y,v]f[y,u]}\] (13) \[X_{f}(z) = w-\frac{f(w)}{f[z,w]+(z-w)\{c_{5}f[z,w]-c_{3}-c_{4}(z-w)\}} \tag{14}\]  
where \(v=z+f(z)\), \(y=S_{f}(z)\), \(u=F_{f}(z)\) and \(w=E_{f}(z)\), are the fixed point operators of [PERSON]'s, M4, M8 and M16 methods, respectively. From the dynamical point of view, conjugacy classes play an important role in the understanding of the behavior of classes of maps in the following sense. Let us consider a map \(z\to M_{f}(z)\) whose \(M_{f}\) is any iterative root-finding map. Since a conjugacy preserves fixed and periodic points, as well as their basins of attraction, the dynamical information concerning \(f\) is carried by the character of the fixed points of \(M_{f}\). So, for polynomials \(p\) of degree greater than or equal to \(k\), it is interesting to build up parametrized families of polynomial \(p_{\mu}\), as simple as possible, so that there exist a conjugacy between \(M_{p_{\mu}}\) and \(M_{p}\).