## 3 Numerical Algorithms with Non-Constant Step Size  
As we can see, the application of the numerical algorithms considered in [3] to the continuous [PERSON]'s method Equation (2) generates iterative root finding methods that does not present a high numerical efficiency. The situation does not improve if we consider higher order numerical methods for the initial value problem Equation (2). In fact, if we consider the second order [PERSON]'s method  
\[z_{n+1}=z_{n}+hf(t_{n},z_{n})+\frac{h^{2}}{2}\left(\frac{\partial f}{\partial t }(t_{n},z_{n})+\frac{\partial f}{\partial z}(t_{n},z_{n})\right)f(t_{n},z_{n})\]  
we obtain the following iterative method  
\[z_{n+1}=F_{6}(z_{n})=z_{n}-h\frac{p(z_{n})}{p^{\prime}(z_{n})}+\frac{h^{2}}{2} (1-L_{p}(z_{n}))\frac{p(z_{n})}{p^{\prime}(z_{n})}\] \begin{table}
\begin{tabular}{l c c} \hline \hline
**Method** & **A.E.C.** & **I.C.** & \(\mathbf{h^{*}}\) \\ \hline Euler Equation (5) & \(1-h\) & \((0,2)\) & \(1\) \\ Refined Euler Equation (6) & \((1+(1-h)^{2})/2\) & \((0,2)\) & \(1\) \\ Heun Equation (7) & \((1+(1-h)^{2})/2\) & \((0,2)\) & \(1\) \\ Runge-Kutta 2 Equation (8) & \((1+(1-h)^{2})/2\) & \((0,2)\) & \(1\) \\ Runge-Kutta 4 Equation (9) & \((h^{4}-4h^{3}+12h^{2}-24h+24)/24\) & \((0,2.7853)\) & \(1.5961\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Some numerical properties of the root-finding methods obtained after applying methods Equations (5)–(9) to continuous [PERSON]’s method Equation (2). where \(L_{p}(z)\) has been defined in Equation (13). But, once again,  
\[F_{6}(z^{*})=z^{*},\quad F_{6}^{\prime}(z^{*})=\frac{1+(1-h)^{2}}{2}\]  
so that the convergence of the method is just linear although the number of functional evaluations have been increased. To avoid these difficulties, we can consider iterative methods with a non-constant step size. For instance, instead of considering method Equation (12) derived from [PERSON]'s method, we consider methods where the step \(h\) is substituted by a non-constant function \(H(z_{n})\):  
\[z_{n+1}=F(z_{n})=z_{n}-H(z_{n})\frac{p(z_{n})}{p^{\prime}(z_{n})},\quad n\geq 1 \tag{15}\]  
As in the previous section, if we consider a simple root \(z^{*}\) of \(p(z)=0\), we have \(F(z^{*})=z^{*}\). In addition,  
\[F^{\prime}(z^{*}) =0\text{ if }H(z^{*})=1 \tag{16}\] \[F^{\prime\prime}(z^{*}) =0\text{ if }H^{\prime}(z^{*})=L_{p}^{\prime}(z^{*})/2 \tag{17}\]  
So, if we consider a function \(H\) that satisfies conditions Equations (16) and (17), we can obtain iterative methods with at least cubic convergence. One choice (not the only one) for function \(H\) that fulfills Equations (16) and (17) is  
\[H(z)=1+\frac{1}{2}L_{p}(z)\]  
that leads us to the well-known [PERSON]'s method ([4,5]):  
\[z_{n+1}=F_{7}(z_{n})=z_{n}-\left(1+\frac{1}{2}L_{p}(z_{n})\right)\frac{p(z_{n })}{p^{\prime}(z_{n})} \tag{18}\]  
Even more, if we choose  
\[H(z)=1+\frac{L_{p}(z)}{2(1-\alpha L_{p}(z))}\]  
we deduce the well-known Chebyshev-Halley family of methods ([4,5]):  
\[z_{n+1}=F_{\alpha}(z_{n})=z_{n}-\left(1+\frac{L_{p}(z_{n})}{2(1-\alpha L_{p}( z_{n}))}\right)\frac{p(z_{n})}{p^{\prime}(z_{n})},\quad\alpha\in\mathbb{C} \tag{19}\]  
Note that [PERSON]'s iterative method defined in Equation (18) is one of the methods defined in Equation (19), actually for \(\alpha=0\). Other methods belonging to this family are [PERSON]'s method (\(\alpha=1/2\)) or super-Halley's method (\(\alpha=1\)). The dynamical behavior of methods in the family Equation (19) have been studied in detail in the work of [PERSON] _et al._[6]. The structure and dynamical properties of the basins of attraction of the roots of a given function changes depending on the choice of the iterative method considered for solving the equation. As a visual sample we show in Figure 2 the basins of attraction of three methods in family Equation (19) when they applied to the polynomial \(p(z)=z^{3}-1\)We can generalize family Equation (19) to obtain the general form of methods with cubic convergence given by [PERSON] in [7].