## 2 The Optimal Fourth-, Eighth- and \(2^{n}\) th Order Iterative Methods
Then the \(n\)-point family Equation (4) converges with at least \(2^{n}\)th order and satisfies the error relation  
\[e_{k+1}=e_{k,n}=y_{k,n}-a=(c_{2}+\lambda)^{2^{n-2}}d_{n}e_{k}^{2^{n}}+O(e_{k}^ {2^{n}+1}) \tag{10}\]  
where \(e_{k}=e_{k,0}=y_{k,0}-a\) and \(d_{n}=d_{n-1}(c_{2}d_{n-1}+(-1)^{n-1}c_{n+1}d_{n-2}\cdots d_{1}d_{0}),\)\(n\geq 3\)  
**Proof. ** We prove the theorem by induction. For \(n=3\), the theorem is valid by Theorem 1. Let us assume that Equation (10) is true for the intermediate error relations, then the intermediate error relations are of the form  
\[e_{k,j}=y_{k,j}-a=(c_{2}+\lambda)^{2^{j-2}}d_{j}e_{k}^{2^{j}}+O(e_{k}^{2^{j}+ 1}) \tag{11}\]  
\[\left\{ \begin{aligned} y_{k,1}=& y_{k,0}-\frac{f(y_{k,0})}{ \lambda_{k}f(y_{k,0})+f^{\prime}(y_{k,0})}\\ y_{k,2}=& y_{k,1}-\frac{f(y_{k,1})}{f[y_{k,1},y_{k,0}]+f [y_{k,1},y_{k,0},y_{k,0}](y_{k,1}-y_{k,0})}\\ &\cdots\\ y_{k,n}=& y_{k,n-1}-\frac{f(y_{k,n-1})}{N(y_{k,n-1},y_{k,n-2 },\cdots,y_{k,1},y_{k,0})}\end{aligned} \right. \tag{17}\]  
where \(N(y_{k,n-1},y_{k,n-2},\cdots,y_{k,1},y_{k,0})=f[y_{k,n-1},y_{k,n-2}]+\cdots+f[y _{k,n-1},y_{k,n-2},\cdots,y_{k,1},y_{k,0},y_{k,0}]\)\(\times\)\((y_{k,n-1}-y_{k,n-2})\cdots(y_{k,n-1}-y_{k,0}),\) and the parameter\(\lambda_{k}\) is calculated by using one of the Equations (14)-(16) and depends on the data available from the current and the previous iterations. **Lemma 1. ** Let \(H_{m}\) be the Hermite interpolating polynomial of the degree \(m\) that interpolates a function \(f\) at \(m\) distinct interpolation nodes \(y_{k,0},y_{k-1,n-1},\cdots y_{k-1,n-m+1}\) contained in an interval \(I\)and the derivative \(f^{(m+1)}\) is continuous in \(I\) and the Hermite interpolating polynomial \(H_{m}(x)\) satisfied the condition \(H_{m}(y_{k,0})=f(y_{k,0}),H^{\prime}_{m}(y_{k,0})=f^{\prime}(y_{k,0}),H_{m}(y_ {k-1,n-j})=f(y_{k-1,n-j})(j=1,\cdots m-1).\) Define the errors \(e_{k-1,n-j}=y_{k-1,n-j}-a(i=1\cdots m-1)\) and assume that  
1. all nodes \(y_{k,0},y_{k-1,n-1},\cdots y_{k-1,n-m+1}\) are sufficiently close to the zero \(a\);
2. the condition \(e_{k}=O(e_{k-1,n-1}\cdots e_{k-1,n-m+1})\) holds. Then  
\[H^{\prime\prime}_{m}(y_{k,0})\sim 2f^{\prime}(a)\left(c_{2}-(-1)^{m-1}c_{m+1} \prod_{j=1}^{m-1}e_{k-1,n-j}\right) \tag{18}\]  
and  
\[\frac{H^{\prime\prime}_{m}(y_{k,0})}{2f^{\prime}(y_{k,0})}\sim\left(c_{2}-(-1) ^{m-1}c_{m+1}\prod_{j=1}^{m-1}e_{k-1,n-j}\right) \tag{19}\]  
**Proof. ** The error of the Hermite interpolation can be expressed as follows  
\[f(x)-H_{m}(x)=\frac{f^{(m+1)}(\xi)}{(m+1)!}{(x-y_{k,0})}^{2}\prod_{j=1}^{m-1} {(x-y_{k-1,n-j})}\;(\xi\in I) \tag{20}\]  
Differentiating Equation (20) at the point \(x=y_{k,0}\), we obtain  
\[f^{\prime\prime}(y_{k,0})-H^{\prime\prime}_{m}(y_{k,0})=2\frac{f^{(m+1)}(\xi) }{(m+1)!}\prod_{j=1}^{m-1}{(y_{k,0}-y_{k-1,n-j})}\;(\xi\in I) \tag{21}\]  
\[H^{\prime\prime}_{m}(y_{k,0})=f^{\prime\prime}(y_{k,0})-2\frac{f^{(m+1)}(\xi) }{(m+1)!}\prod_{j=1}^{m-1}{(y_{k,0}-y_{k-1,n-j})}\;(\xi\in I) \tag{22}\]  
[PERSON]'s series of derivatives of \(f\)at the point \(y_{k,0}\in I\) and \(\xi\in I\) about the zero \(a\) of \(f\)give  
\[f^{\prime}(y_{k,0})=f^{\prime}(a)(1+2c_{2}e_{k,0}+3c_{3}e_{k,0}^{2}+O(e_{k,0}^ {3})) \tag{23}\]\[f^{\prime\prime}(y_{k,0})=f^{\prime}(a)(2c_{2}+6c_{3}e_{k,0}+O(e_{k,0}^{2})) \tag{24}\]  
\[f^{(m+1)}(\xi)=f^{\prime}(a)((m+1)!c_{m+1}+(m+2)!c_{m+1}e_{\xi}+O(e_{\xi}^{2})) \tag{25}\]  
where \(e_{\xi}=\xi-a\). Substituting Equations (24) and (25) into Equation (22), we have  
\[H_{m}^{\prime\prime}(y_{k,0})=2f^{\prime}(a)(c_{2}-(-1)^{m-1}c_{m+1}\prod_{j=1 }^{m-1}e_{k-1,n-j}) \tag{26}\]  
and  
\[\frac{H_{m}^{\prime\prime}(y_{k,0})}{2f^{\prime}(y_{k,0})}\sim\left(c_{2}-(-1) ^{m-1}c_{m+1}\prod_{j=1}^{m-1}e_{k-1,n-j}\right) \tag{27}\]  
The concept of the \(R\)-order of convergence [1] and the following assertion (see [8]) will be applied to estimate the convergence order of the iterative method with memory Equation (17). Now we can state the following convergence theorem for iterative method with memory Equation (17). **Theorem 3.