## 2 Numerical Algorithms Applied to Continuous Newton's Method  
In [3] the dynamical properties of six numerical methods for solving differential equations are considered when they are applied to continuous [PERSON]'s method Equation (2). In particular, they compare the basins of attraction of these methods applied to Equation (2) for \(p(z)=z^{3}-1\). They estimate the corresponding fractal dimension using a box-counting algorithm. The main conclusion is that higher-order algorithms are not necessarily related with basin boundaries with smaller fractal dimension. In this section we are concerned with some numerical properties of root-finding methods derived from the application of numerical methods for solving differential equations  
\[\left\{ \begin{array}{rcl}\frac{dz}{dt}&=&f(t,z(t))\\ \\ z(t_{0})&=&z_{0}\end{array} \right. \tag{4}\]  
as continuous [PERSON]'s method Equation (2). So, as it has been done in [3], we use \(y_{n}\) for the approximations of \(z(t_{n})\), where \(t_{n+1}=t_{n}+h\) and \(h\) for the step size. The six methods considered by [PERSON] _et al._ in [3] are:  
1. [PERSON]'s method: \[y_{n+1}=y_{n}+hf(t_{n},y_{n})\] (5)
2. Refined [PERSON]'s method: \[\left\{ \begin{array}{rcl}y^{*}&=&y_{n}+h/2f(t_{n},y_{n})\\ \\ y_{n+1}&=&y_{n}+hf(t_{n}+h/2,y^{*})\end{array} \right.\] (6)
3. [PERSON]'s method: \[\left\{ \begin{array}{rcl}y^{*}&=&y_{n}+hf(t_{n},y_{n})\\ \\ y_{n+1}&=&y_{n}+h/2\left(f(t_{n},y_{n})+f(t_{n}+h,y^{*})\right)\end{array} \right.\] (7)
4. Runge-Kutta method of order 2: \[\left\{ \begin{array}{rcl}y^{*}&=&y_{n}+2h/3f(t_{n},y_{n})\\ \\ y_{n+1}&=&y_{n}+h/4\left(f(t_{n},y_{n})+3f(t_{n}+2h/3,y^{*})\right)\end{array} \right.\] (8)
5. Runge-Kutta method of order 4: \[\left\{ \begin{array}{rcl}k_{1}&=&hf(t_{n},y_{n})\\ \\ k_{2}&=&hf(t_{n}+h/2,y_{n}+k_{1}/2)\\ \\ k_{3}&=&hf(t_{n}+h/2,y_{n}+k_{2}/2)\\ \\ k_{4}&=&hf(t_{n}+h,y_{n}+k_{3})\\ \\ y_{n+1}&=&y_{n}+1/6\left(k_{1}+2k_{2}+2k_{3}+k_{4}\right)\end{array} \right.\] (9)6. Adams-Bashforth method of order 2: \[\left\{ \begin{array}{rcl}&y^{*}&=&y_{0}+2h/3f(t_{n},y_{0})\\ \\ &y_{1}&=&y_{0}+h/4\left(f(t_{n},y_{0})+3f(t_{n}+2h/3,y^{*})\right)\\ \\ &y_{n+1}&=&y_{n}+y_{n}+h\left(3/2f(t_{n},y_{n})-1/2f(t_{n-1},y_{n-1})\right) \end{array} \right.\] (10)  
Note that, in the case of continuous [PERSON]'s method, the function \(f(t,z(t))\) that appears in differential equation Equation (4) is  
\[f(t,z(t))=-\frac{p(z(t))}{p^{\prime}(z(t))}\]  
So, the application of [PERSON]'s method Equation (5) gives rise to the root-finding algorithm  
\[z_{n+1}=F_{1}(z_{n})=z_{n}-h\frac{p(z_{n})}{p^{\prime}(z_{n})},\quad n\geq 0 \tag{11}\]  
that is the wel-known relaxed [PERSON]'s method. Note that the role of the step size \(h\) in the methods for solving differential equations is moved to a relaxing parameter in Equation (12). As it was stated in [3] and we have previously mentioned, \(h\) has a clear influence in the dynamical properties of an iterative method. We analyze now the influence of \(h\) in the numerical properties of an iterative method. Let \(F_{1}\) be the iteration map related to the relaxed [PERSON]'s method Equation (12) and let \(z^{*}\) be a simple root of \(p(z)=0\). It is a straightforward calculation to show that  
\[F_{1}(z^{*})=z^{*},\quad F_{1}^{\prime}(z^{*})=1-h\]  
So, we have that method Equation (12) is consistent (the roots of \(p\) are attracting fixed points of the iteration map \(F_{1}\)) only for \(h\in(0,2)\). In addition, we obtain iterative methods with just linear convergence, except for the case \(h=1\), that is the classical [PERSON]'s method with quadratic convergence. Now, we see what happens with the refined [PERSON]'s method defined in Equation (6).