## 2 Developments of the Methods
using weight function to obtain a three-point fifth order I.F. (\(5^{th}3 pVS\)):  
\[e_{n+1}=C\;e_{n}^{p}+O\Big{(}e_{n}^{p+1}\Big{)}=O\Big{(}e_{n}^{p}\Big{)}. \tag{1}\]1  
It is remarkable that with the same number of functional evaluations we have improved the efficiency index from \(IE_{3^{rd}3 pVS}=1.316\) to \(IE_{5^{th}3 pVS}=5^{1/4}=1.495\). However, the maximum order we could achieve I.F.s with \(1\) function and \(3\) first derivative evaluations is five. [PERSON] [2] developed a technique to improve the order of old methods. **Theorem 3** ([PERSON]'s theorem for improving the order of old methods). : _[_2_]_ _Let a sufficiently smooth function \(f:D\subset\mathbb{R}\rightarrow\mathbb{R}\) has a simple root \(x^{*}\) in the open interval \(D\). Let \(\psi_{old}(x)\) be an Iteration Function (I.F.) of order \(p\). Then the I.F. defined as \(\psi_{new}(x)=\psi_{old}(x)-G\times f(\psi_{old}(x))\) is of local convergence of order \(p+q\) if \(G\) is a function satisfying the error equation_  
\[e_{n+1}=C\;e_{n}^{p}+O\Big{(}e_{n}^{p+1}\Big{)}=O\Big{(}e_{n}^{p}\Big{)}. \tag{1}\]2  
_where \(C_{G}\) is a constant._  
_Suppose that the error equation of the old I.F. is given by_  
\[e_{n+1}=C\;e_{n}^{p}+O\Big{(}e_{n}^{p+1}\Big{)}=O\Big{(}e_{n}^{p}\Big{)}. \tag{1}\]3  
_Then, the error equation of the new I.F. is given by_  
\[e_{n+1}=C\;e_{n}^{p}+O\Big{(}e_{n}^{p+1}\Big{)}=O\Big{(}e_{n}^{p}\Big{)}. \tag{1}\]4  
_where \(c_{j}=\frac{f^{(j)}(x^{*})}{j!f^{\prime}(x^{*})},\;j=1,2,3...\)_  
Usually, \(G\) is a weight function or an approximation to \(\frac{1}{f^{\prime}(\psi_{old}(x))}\) obtained from polynomial interpolation. Using [PERSON]'s theorem and applying Newton I.F., we obtain a 10 th order I.F. (\(10^{th}\)_VS_):  
\[e_{n+1}=C\;e_{n}^{p}+O\Big{(}e_{n}^{p+1}\Big{)}=O\Big{(}e_{n}^{p}\Big{)}. \tag{1}\]5  
However, we need to compute two more function evaluations. So we estimate \(f^{\prime}(\psi_{5^{th}3 pVS}(x))\) by the following polynomial:  
\[e_{n+1}=C\;e_{n}^{p}+O\Big{(}e_{n}^{p+1}\Big{)}=O\Big{(}e_{n}^{p}\Big{)}. \tag{1}\]6  
which satisfies the following conditions  
\[ \begin{split}& q_{1}(\psi_{5^{th}3 pVS}(x))=f(\psi_{5^{th}3 pVS}(x)),\,q^{\prime}_{1}(\psi_{2^{nd}NR}(x))=f^{\prime}(\psi_{2^{nd}NR}(x)),\\ & q^{\prime}_{1}\left(\frac{x+\psi_{2^{nd}NR}(x)}{2}\right)=f^{ \prime}\left(\frac{x+\psi_{2^{nd}NR}(x)}{2}\right).\end{split}  \tag{14}\]  
Let  
\[ \begin{split}& A_{1}=\psi_{5^{th}3 pVS}(x)-x,\,A_{2}=\psi_{2^{nd}NR} (x)-x,\,A_{3}=\frac{x+\psi_{2^{nd}NR}(x)}{2}-x,\\ & B_{1}=f[\psi_{5^{th}3 pVS}(x),x,x],\,B_{2}=f^{\prime}[\psi_{2^{ nd}NR}(x),x],\,B_{3}=f^{\prime}\left[\frac{x+\psi_{2^{nd}NR}(x)}{2},x\right], \end{split} \]  
where we define the divided differences:  
\[f[y,x]=\frac{f(y)-f(x)}{y-x},\,f^{\prime}[y,x]=\frac{f^{\prime}(y)-f^{\prime} (x)}{y-x},\,f[y,x,x]=\frac{f[y,x]-f^{\prime}(x)}{y-x}\]Equation (14) reduces, after using the divided differences, to a system of 3 linear equations in matrix form:  
\[\left( \begin{array}{cccc}1&A_{1}&{A_{1}}^{2}\\ 2&3A_{2}&4{A_{2}}^{2}\\ 2&3A_{3}&4{A_{3}}^{2}\end{array} \right)\,\left( \begin{array}{c}a_{1}\\ a_{2}\\ a_{3}\end{array} \right)=\left( \begin{array}{c}B_{1}\\ B_{2}\\ B_{3}\end{array}