## 2 A Steffensen-Type Family of Optimal Order without Memory  
Let \(x_{n}\) be an approximation of the simple root of a nonlinear equation \(f(x)=0\) and \(z_{n}=x_{n}+\gamma_{n}f(x_{n})\). By direct Newtonian interpolatory polynomial of degree one, such that \(N_{1}(x_{n})=f(x_{n})\) and \(N_{1}(z_{n})=f(z_{n})\), we have  
\[N_{1}(x)=f(x_{n})+f[x_{n},z_{n}](x-x_{n})\]and \(f(x)\approx N_{1}(x)\), where \(R_{1}(x)=f(x)-N_{1}(x)=f[x_{n},z_{n},x](x-x_{n})(x-z_{n})\). So, for some \(\mu_{n}\approx f[x_{n},z_{n},x]\), we have  
\[\widetilde{N}_{2}(x)=f(x_{n})+f[x_{n},z_{n}](x-x_{n})+\mu_{n}(x-x_{n})(x-z_{n})\]  
and \(f(x)\approx\widetilde{N}_{2}(x)\), which is a polynomial of degree two based still on \(f(x_{n})\) and \(f(z_{n})\), but \(f(x)\approx\widetilde{N}_{2}(x)\) could be better than \(f(x)\approx N_{1}(x)\) by adding a higher-order term. We suggest that the next approximation \(x_{n+1}\) of the root of \(f(x)\) be obtained from [PERSON]'s iteration for \(\widetilde{N}_{2}(x)\) as \(x_{n+1}=x_{n}-\frac{\widetilde{N}_{2}(x_{n})}{\widetilde{N}_{2}^{\prime}(x_{ n})}=x_{n}-\frac{f(x_{n})}{f[x_{n},z_{n}]+\mu_{n}(x_{n}-z_{n})}\). Then, we have an optimal second-order Steffensen-type method:  
\[x_{n+1}=x_{n}-\frac{f(x_{n})}{f[x_{n},z_{n}]+\mu_{n}(x_{n}-z_{n})},\qquad\qquad n =0,1,2,\ldots \tag{4}\]  
where \(z_{n}=x_{n}+\gamma_{n}f(x_{n})\), \(\{\gamma_{n}\}\) and \(\{\mu_{n}\}\) are bounded constant sequences. This method gives SM when \(\gamma_{n}\equiv 1\) and \(\mu_{n}\equiv 0\). Similarly, an optimal fourth-order Steffensen-type method is obtained as follows:  
\[\left\{ \begin{array}{l}y_{n}=x_{n}-\frac{f(x_{n})}{f[x_{n},z_{n}]},\\ x_{n+1}=y_{n}-\frac{f(y_{n})}{f[y_{n},x_{n}]+f[y_{n},x_{n},z_{n}](y_{n}-x_{n})+ \alpha_{n}(y_{n}-x_{n})(y_{n}-z_{n})},\quad n=0,1,2,\ldots\end{array} \right. \tag{5}\]  
where \(z_{n}=x_{n}+\gamma_{n}f(x_{n})\), \(\{\gamma_{n}\}\) and \(\{\alpha_{n}\}\) are bounded constant sequences. This method gives RWBM when \(\gamma_{n}\equiv 1\) and \(\alpha_{n}\equiv\alpha\). **Theorem 1**. : _Let \(f:D\rightarrow\Re\) be a sufficiently differentiable function with a simple root \(a\in D\), \(D\subset\Re\) be an open set, \(x_{0}\) be close enough to \(a\), then the method Equations (4) and (5) are at least of second-order and fourth-order, respectively, and satisfy the error equations:_  
\[e_{n+1}=[(1+\gamma_{n}f^{\prime}(a))c_{2}-\mu_{n}\gamma_{n}]e_{n}^{2}+O(e_{n} ^{3}) \tag{6}\]  
\[e_{n+1}=(1+\gamma_{n}f^{\prime}(a))^{2}c_{2}[c_{2}^{2}-c_{3}+\frac{\alpha_{n}} {f^{\prime}(a)}]e_{n}^{4}+O(e_{n}^{5}) \tag{7}\]  
_respectively, where \(c_{k}=\frac{f^{(k)}(a)}{k!f^{\prime}(a)}\), \(e_{n}=x_{n}-a,n=0,1,2,\cdots\)._  
**Proof. ** The theorem can be proved by the definition of divided difference and Taylor formula, see [9] or the proof of Theorem 2. By successive Newtonian interpolatory polynomials up to \(m+1\) points, we can derive the optimal \(2^{m}\)th-order Steffensen-type family, moreover we are able to write it in a preferable explicit form as follows: for any \(m>0\), \(x_{n+1}=y_{m}\) is obtained for \(n=0,1,\cdots\), by  
\[\left\{ \begin{array}{l}y_{1}=y_{0}-\frac{f(y_{0})}{f[y_{0},y_{-1}]},\\ y_{2}=y_{1}-\frac{f(y_{1})}{f[y_{1},y_{0}]+f[y_{1},y_{0},y_{-1}](y_{1}-y_{0})}, \\ \cdots\\ y_{m}=y_{m-1}-\frac{f(y_{m-1})}{f[y_{m-1},y_{m-2}]+\cdots+f[y_{m-1},\cdots,y_{- 1}](y_{m-1}-y_{m-2})\cdots(y_{m-1}-y_{0})+\
u_{n}(y_{m-1}-y_{m-2})\cdots(y_{m-1 }-y_{-1})}\end{array} \right. \tag{8}\]  
where \(y_{-1}=x_{n}+\gamma_{n}f(x_{n})\), \(y_{0}=x_{n}\), \(\{\gamma_{n}\}\) and \(\{\
u_{n}\}\) are bounded constant sequences. When \(\
u_{n}\equiv 0\), it gives the general optimal Steffensen-type family in [9]. **Theorem 2**.