## 2 The Optimal Fourth-, Eighth- and \(2^{n}\) th Order Iterative Methods
** Let the varying parameter \(\lambda_{k}\) in the iterative Equation (17) be calculated by Equation (14). If an initial approximation \(x_{0}\) is sufficiently close to a simple zero \(a\) of \(f(x)\), then the \(R\)-order of convergence of the \(n\)-point Equation (17) with memory is at least \(2^{n}+2^{n-3}\) for \(n\geq 3\) and at least \((5+\sqrt{17})/2\approx 4.5616\) for \(n=2\). **Proof. ** First, let us consider the case \(n\geq 3\) and assume that the iterative sequences \(\{y_{k,n}\}\) and \(\{y_{k,n-1}\}\) have the \(R\)-order \(r\) and \(q\), respectively, we have  
\[\frac{H^{\prime\prime}_{m}(y_{k,0})}{2f^{\prime}(y_{k,0})}\sim\left(c_{2}-(-1) ^{m-1}c_{m+1}\prod_{j=1}^{m-1}e_{k-1,n-j}\right) \tag{19}\]0  
\[\frac{H^{\prime\prime}_{m}(y_{k,0})}{2f^{\prime}(y_{k,0})}\sim\left(c_{2}-(-1) ^{m-1}c_{m+1}\prod_{j=1}^{m-1}e_{k-1,n-j}\right) \tag{19}\]1  
where \(D_{k,j}\)\((j\in R)\) tends to the asymptotic error constant \(D_{j}\) when \(k\rightarrow\infty\). From Equation (13), we obtain the following error relations  
\[\frac{H^{\prime\prime}_{m}(y_{k,0})}{2f^{\prime}(y_{k,0})}\sim\left(c_{2}-(-1) ^{m-1}c_{m+1}\prod_{j=1}^{m-1}e_{k-1,n-j}\right) \tag{19}\]2  
\[\frac{H^{\prime\prime}_{m}(y_{k,0})}{2f^{\prime}(y_{k,0})}\sim\left(c_{2}-(-1) ^{m-1}c_{m+1}\prod_{j=1}^{m-1}e_{k-1,n-j}\right) \tag{19}\]3  
Using the Lemma 1 for \(m=2\), we obtain  
\[\frac{H^{\prime\prime}_{m}(y_{k,0})}{2f^{\prime}(y_{k,0})}\sim\left(c_{2}-(-1) ^{m-1}c_{m+1}\prod_{j=1}^{m-1}e_{k-1,n-j}\right) \tag{19}\]4  
Substituting Equation (32) into Equations (30) and (31) instead of \(\lambda\), we have  
\[\frac{H^{\prime\prime}_{m}(y_{k,0})}{2f^{\prime}(y_{k,0})}\sim\left(c_{2}-(-1) ^{m-1}c_{m+1}\prod_{j=1}^{m-1}e_{k-1,n-j}\right) \tag{19}\]5  
\[\frac{H^{\prime\prime}_{m}(y_{k,0})}{2f^{\prime}(y_{k,0})}\sim\left(c_{2}-(-1) ^{m-1}c_{m+1}\prod_{j=1}^{m-1}e_{k-1,n-j}\right) \tag{19}\]6  
\[\frac{H^{\prime\prime}_{m}(y_{k,0})}{2f^{\prime}(y_{k,0})}\sim\left(c_{2}-(-1) ^{m-1}c_{m+1}\prod_{j=1}^{m-1}e_{k-1,n-j}\right) \tag{19}\]7  
\[\frac{H^{\prime\prime}_{m}(y_{k,0})}{2f^{\prime}(y_{k,0})}\sim\left(c_{2}-(-1) ^{m-1}c_{m+1}\prod_{j=1}^{m-1}e_{k-1,n-j}\right) \tag{19}\]8By comparing exponents of \(e_{k-1}\) appearing in two pairs of relations Equations (29)-(33) and Equations (28)-(34), we get the following system of equations  
\[\left\{ \begin{array}{l}r2^{n-1}+q2^{n-3}=rq\\ r2^{n}+q2^{n-2}=r^{2}\end{array} \right. \tag{35}\]  
The solution of the system Equation (35) is given by \(r=2^{n}+2^{n-3}\) and \(q=2^{n-1}+2^{n-4}\). Therefore, the \(R\)-order of the methods with memory Equation (17) is at least \(r=2^{n}+2^{n-3}\) for \(n\geq 3\). For example, the R-order of the three-point family Equation (17) is at least 9, the four-point family has the R-order at least 18, assuming that \(\lambda_{k}\) is calculated by Equation (14). The case \(n=2\) differs from the previous analysis; [PERSON]'s interpolating polynomial is constructed at the nodes \(y_{k,0},y_{k-1,1}\). Substituting Equation (32) into Equation (2) and Equation (8) instead of \(\lambda\), we have  
\[e_{k,1}= y_{k,1}-a\sim(-c_{3}e_{k-1,1})d_{1}e_{k}^{2}\sim-c_{3}D_{k-1,q}e_{k-1 }^{q}d_{1}(D_{k-1,r}e_{k-1}^{r})^{2} \tag{36}\] \[\sim-c_{3}D_{k-1,q}d_{1}D_{k-1,r}^{2}e_{k-1}^{2r+q}\] \[e_{k+1}= e_{k,2}=y_{k,2}-a\sim(-c_{3}e_{k-1,1})d_{2}e_{k}^{4}\sim-c_{3}D_ {k-1,q}e_{k-1}^{q}d_{2}(D_{k-1,r}e_{k-1}^{r})^{4}\] (37) \[\sim-c_{3}D_{k-1,q}d_{1}D_{k-1,r}^{4}e_{k-1}^{4r+q}\]  
By comparing exponents of \(e_{k-1}\) appearing in two pairs of relations Equations (29)-(36) and Equations (28)-(37), we get the following system of equations  
\[\left\{ \begin{array}{l}2r+q=rq\\ 4r+q=r^{2}\end{array} \right. \tag{38}\]  
Positive solution of the system Equation (38) is given by \(r=(5+\sqrt{17})/2\) and \(q=(1+\sqrt{17})/2\). Therefore, the \(R\)-order of the methods with memory Equations (17) with (14) is at least \((5+\sqrt{17})/2\approx 4.5616\) for \(n=2\). **Theorem 4. ** Let the varying parameter \(\lambda_{k}\) in the iterative Equation (17) be calculated by Equation (15).