## 2 Optimal Derivative-Free Methods  
The well-known [PERSON]'s method (see [9]) replaces the derivative of Equation (1) by the forward difference, so its iterative expression is  
\[z_{n+1}=z_{n}-\frac{[f(z_{n})]^{2}}{f(v_{n})-f(z_{n})} \tag{2}\]  
where \(v_{n}=z_{n}+f(z_{n})\). The composition technique (described, for instance, in [10]) allows up the implementation of high-order methods. From two methods with orders of convergence \(p_{1}\) and \(p_{2}\) is possible to perform a method of \(p_{1}\cdot p_{2}\) order. In [11], the authors describe a technique based on composition and Pade-like approximants to obtain optimal derivative-free methods. We first compose [PERSON]'s method with [PERSON]'s scheme obtaining the fourth-order scheme  
\[ \begin{split} y_{n}&=z_{n}-\frac{f(z_{n})^{2}}{f(v_ {n})-f(z_{n})}\\ z_{n+1}&=y_{n}-\frac{f(y_{n})}{f^{\prime}(y_{n})} \end{split}  \tag{3}\]where \(v_{n}=z_{n}+f(z_{n})\). Now, in order to avoid the evaluation of \(f^{\prime}(y_{n})\), we approximate it by the derivative \(m^{\prime}(y_{n})\) of the following first degree Pade-like approximant  
\[m(t)=\frac{a_{1}+a_{2}(t-y_{n})}{1+a_{3}(t-y_{n})}\]  
where \(a_{1}\), \(a_{2}\) and \(a_{3}\) are real parameters to be determined satisfying the following conditions:  
\[m(z_{n})=f(z_{n}) \tag{4}\]  
\[m(y_{n})=f(y_{n}) \tag{5}\]  
and  
\[m(v_{n})=f(v_{n}) \tag{6}\]  
From Equation (5), it is immediately obtained  
\[a_{1}=f(y_{n})\]  
Conditions (4) and (6), give, respectively,  
\[a_{2}-f(z_{n})a_{3}=f[z_{n},y_{n}]\]  
and  
\[a_{2}-f(v_{n})a_{3}=f[v_{n},y_{n}]\]  
where \(f[z_{n},y_{n}]\) denotes the divided difference \(\dfrac{f(z_{n})-f(y_{n})}{z_{n}-y_{n}}\). After some algebraic manipulations, the following values are obtained for the parameters:  
\[a_{2}=f[y_{n},v_{n}]-\dfrac{f(v_{n})f[z_{n},y_{n},v_{n}]}{f[v_{n},z_{n}]}\]  
and  
\[a_{3}=-\dfrac{f[z_{n},y_{n},v_{n}]}{f[z_{n},v_{n}]}\]  
where \(f[z_{n},y_{n},v_{n}]=\dfrac{f[z_{n},y_{n}]-f[y_{n},v_{n}]}{z_{n}-v_{n}}\) is a second order divided difference. The derivative of the Pade approximant evaluated in \(y_{n}\) can be expressed as  
\[m^{\prime}(y_{n})=\dfrac{f[z_{n},y_{n}]f[y_{n},v_{n}]}{f[z_{n},v_{n}]} \tag{7}\]  
Substituting \(f^{\prime}(y_{n})\) from Equation (3) by its approximation \(m^{\prime}(y_{n})\), [PERSON] _et al._ in [11] obtained an optimal fourth-order iterative method denoted by M4, whose expression is:  
\[m(z_{n})=f(z_{n}) \tag{4}\]0 \[m(z_{n})=f(z_{n}) \tag{4}\]1By composing again the resulting scheme with [PERSON]'s method, and estimating the last derivative by a Pade-like approximant of order two, the obtained optimal eighth-order method, denoted by M8, has the iterative scheme  
\[\left\{ \begin{array}{l}y_{n}=z_{n}-\frac{[f(z_{n})]^{2}}{f(v_{n})-f(z_{n})}\\ u_{n}=y_{n}-\frac{f(y_{n})f[z_{n},v_{n}]}{f[z_{n},y_{n}]f[y_{n},v_{n}]}\\ z_{n+1}=u_{n}-\frac{f(u_{n})f[z_{n},y_{n},v_{n}]}{f[y_{n},v_{n},u_{n}]f[u_{n},z _{n},y_{n}](u_{n}-z_{n})+f[z_{n},y_{n},v_{n}]f[y_{n},u_{n}]}\end{array} \right. \tag{9}\]  
In a similar way, we can obtain a new optimal 16 th-order method, denoted by M16, by composing the scheme M8 with [PERSON]'s method and estimating the last derivative by a Pade-like approximant of third degree  
\[m(t)=\frac{c_{1}+c_{2}(t-z_{n})+c_{3}(t-z_{n})^{2}+c_{4}(t-z_{n})^{3}}{1+c_{5}( t-z_{n})}\]  
obtaining the iterative expression  
\[\left\{ \begin{array}{l}y_{n}=z_{n}-\frac{[f(z_{n})]^{2}}{f(v_{n})-f(z_{n})} \\ u_{n}=y_{n}-\frac{f(y_{n})f[z_{n},v_{n}]}{f[z_{n},y_{n}]f[y_{n},u_{n}]}\\ w_{n}=u_{n}-\frac{f(u_{n})f[z_{n},y_{n},v_{n}]}{f[y_{n},u_{n}]f[u_{n},z_{n},y_{ n}](u_{n}-z_{n})+f[z_{n},y_{n},v_{n}]f[y_{n},u_{n}]}\\ z_{n+1}=w_{n}-\frac{f(w_{n})}{f[z_{n},w_{n}]+(z_{n}-w_{n})\{c_{5}f[z_{n},w_{n}]- c_{3}-c_{4}(z_{n}-w_{n})\}}\end{array} \right.