## 2 The Optimal Fourth-, Eighth- and \(2^{n}\) th Order Iterative Methods  
Based on [PERSON]'s Equation [2] and [PERSON]'s \(n\)-point methods [6], we derive a general optimal \(2^{n}\)th order family and write it in the following form:  
\[\left\{ \begin{aligned} y_{k,1}=& y_{k,0}- \frac{f(y_{k,0})}{\lambda f(y_{k,0})+f^{\prime}(y_{k,0})}\\ y_{k,2}=& y_{k,1}-\frac{f(y_{k,1})}{f[y_{k,1},y_{k,0}]+f[y_{k,1},y_{k,0},y_{k,0}](y_{k,1}-y_{k,0})}\\ &\cdots\\ y_{k,n}=& y_{k,n-1}-\frac{f(y_{k,n-1})}{N(y_{k,n-1},y_{k,n-2}\cdots,y_{k,1},y_{k,0})}\end{aligned} \right. \tag{4}\]  
where \(N(y_{k,n-1},y_{k,n-2},\cdots,y_{k,1},y_{k,0})=f[y_{k,n-1},y_{k,n-2}]+\cdots+f[y _{k,n-1},y_{k,n-2},\cdots,y_{k,1},y_{k,0},y_{k,0}]\)\((y_{k,n-1}-y_{k,n-2})\cdots(y_{k,n-1}-y_{k,0}),y_{k,0}=x_{k},\lambda\in R\) is a constant and \(k\) being the iteration index. The entries \(y_{k,0},\cdots y_{k,n}\) are approximations with the associated error \(e_{k,j}=y_{k,j}-a\)\((j=0,1,\cdots,n).\)  
Using the Taylor series and symbolic computation in the programming package Mathematica, we can find the order of convergence and the asymptotic error constant (AEC) of the \(n\)-point methods Equation (4) for \(n=1,n=2\) and \(n=3\), respectively. For simplicity, we sometimes omit the iteration index \(n\) and write \(e\) instead of \(e_{k}\). The approximation \(x_{k+1}\) to the root \(a\) will be denoted by \(\hat{x}\). Regarding Equation (4), let us define \(x=y_{k,0},y=y_{k,1},z=y_{k,2},e=x-a,d=y-a,p=z-a,e1=\hat{x}-a.\)  
The following abbreviations are used in the program. \[\text{ck}=f^{(k)}(a)/(k!f^{\prime}(a)),\;\text{e}=x-a,\;\text{d} =y-a,\;\text{p}=z-a,\;\text{e1}=\hat{x}-a,\;\text{fx}=f(y_{k,0}),\;\text{fy}= f(y_{k,1})\] \[\text{dfx}=f^{\prime}(y_{k,0}),\;\text{fxxy}=f[y_{k,0},y_{k,0},y _{k,1}],\;\text{fla}=f^{\prime}(a),\;\;\text{fyz}=f[y_{k,1},y_{k,2}],\text{fxz }=f[y_{k,0},y_{k,2}],\] \[\text{fz}=f(y_{k,2}),\;\text{L}=\lambda,\text{fzxx}=f[y_{k,2},y_{k,0},y_{k,0}],\;\text{fxy}=f[y_{k,0},y_{k,1}],\text{fzxxy}=f[y_{k,2},y_{k,0},y _{k,0},y_{k,1}].\]  
**Program** (written in Mathematica)  
fx=fla*(e+c2*e^2+c3*e^3+c4*e^4+c5*e^5+c6*e^6+c7*e^7+c8*e^8);  
dfx=D[fx,e];t=Series[fx/(L*fx+dfx),{e,0,8}]; d=Series[e-t,{e,0,8}]//Simplify fy=Series[fla*(d+c2*d^2+c3*d^3+c4*d^4),{e,0,8}]; fxy=Series[(fy-fx)/(d-e),{e,0,8}]; z=Series[d-fy/(2*fxy-dfx),{e,0,8}]//Simplify fz=Series[fla*(z+c2*z^2),{e,0,8}]; fyz=Series[(fy-fx)/(d-z),{e,0,8}]; fxz=Series[(fx-fz)/(e-z),{e,0,8}]; fzxy=Series[(dfx-fxzy)/(e-d),{e,0,8}]; fzxx=Series[(dfx-fxz)/(e-z),{e,0,8}]; fzxxy=Series[(fzxx-fzy)/(z-d),{e,0,8}]; fzxy=Series[(fxz-fyz)/(e-d),{e,0,8}]; e1=Series[z-fz/(fyz+fzxy*(z-d)+fzxxy*(z-e)*(z-d)),{e,0,8}]//Simplify  
\[\text{Out[d]}=(c_{2}+L)e^{2}+O[e]^{3} \tag{5}\]  
\[\text{Out[z]}=(c_{2}+L)(c_{2}^{2}-c_{3}+c_{2}L)e^{4}+O[e]^{5} \tag{6}\]  
\[\text{Out[e1]}=(c_{2}+L)^{2}(c_{2}^{2}-c_{3}+c_{2}L)(c_{2}^{3}-c_{2}c_{3}+c_{4 }+c_{2}^{2}L)e^{8}+O[e]^{9} \tag{7}\]  
We obtain the asymptotic error constants of \(n\)-point methods Equation (4) with \(n=1,2,3\). Altogether, we can state the following theorem. **Theorem 1. ** Let \(I\) be an open interval and \(a\in I\) a simple zero point of a sufficiently differentiable function \(f:\ I\to R\). Then the new method defined by Equation (4) \((n=2)\)is fourth order, and satisfies the error equation  
\[e_{k+1}=(c_{2}+\lambda)d_{2}e_{k}^{4}+O(e_{k}^{5}) \tag{8}\]  
the Equation (4) (\(n=3\)) is eighth-order and satisfies the error equation  
\[e_{k+1}=(c_{2}+\lambda)^{2}d_{3}e_{k}^{8}+O(e_{k}^{9}) \tag{9}\]  
where \(e_{k}=x_{k}-a,\ d_{0}=1,\ d_{1}=1,\ d_{2}=c_{2}^{2}+c_{2}\lambda-c_{3}\) and \(d_{3}=d_{2}(c_{2}d_{2}+c_{4}d_{1}d_{0})\). The order of the convergence of the Equation (4) is analyzed in the following theorem. **Theorem 2. ** Let \(I\) be an open interval and \(a\in I\) a simple zero point of a sufficiently differentiable function \(f:\ I\to R\).