## 3 Convergence Analysis
\begin{split}&[\Phi^{\prime}(G_{2^{nd}NM}(x^{(r)}))]^{-1}\Phi(G_{ 2^{nd}NM}(x^{(r)}))=\left({C_{2}}{E^{(r)}}^{2}+2\left(C_{3}-C_{2}^{2}\right){E^{ (r)}}^{3}+\left(3C_{4}+3C_{2}^{3}-3C_{3}C_{2}-4C_{2}C_{3}\right){E^{(r)}}^{4} \\ &+\left(4C_{5}-6C_{2}C_{4}-4C_{4}C_{2}+6C_{2}^{2}C_{3}+4C_{2}C_{3 }C_{2}+6C_{3}C_{2}^{2}-6C_{3}^{2}-4C_{2}^{4}\right)\right)+O({E^{(r)}}^{5}). \end{split}  \tag{17}\]  
We get the required error estimate by using (12), (16) and (17) in (7)  
\[ \begin{split} E^{(r+1)}&=G_{5^{th}Pj}(x^{(r)})-\alpha =\left(4C_{2}^{4}-C_{2}C_{3}C_{2}\right){E^{(r)}}^{5}+O({E^{(r)}}^{6}).\end{split} \]  
The above result agrees with fifth-order convergence. Theorem 4.2: _Let \(\Phi:D\subseteq\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}\) be sufficiently Frechet differentiable at each point of an open convex neighborhood \(D\) of \(\alpha\in\mathbb{R}^{n}\). Choose \(x^{(0)}\) close enough to \(\alpha\), where \(\alpha\) is a solution of the system \(\Phi(x)=0\) and \(\Phi^{\prime}(x)\) is continuous and nonsingular at \(\alpha\). The iterative formula (8) producing the sequence \(\{x^{(r)}\}\) converges to a locally with order \(3k+5\), where \(k\) is a positive integer and \(k\geq 1\)._  
Proof: [PERSON]'s expansion of \(\Phi(\mu_{j-1}(x^{(r)}))\) around \(\alpha\) gives  
\[\Phi(\mu_{j-1}(x^{(r)}))=\Phi^{\prime}(\alpha)\left[(\mu_{j-1}(x^{(r)})-\alpha)+C_ {2}(\mu_{j-1}(x^{(r)})-\alpha)^{2}+...\right] \tag{18}\]By expanding  
\[H_{2}(x^{(r)})=I+\frac{1}{2}(\eta(x^{(r)})-I)^{2}=I+2C_{2}^{2}{E^{(r)}}^{2}+(6C_{ 2}C_{3}-12C_{2}^{3}){E^{(r)}}^{3}+.... \tag{19}\]  
After evaluating \([\[PERSON]^{\prime}(G_{2^{nd}NM}(x^{(r)})]^{-1}\) and combining with (19) yields  
\[H_{2}(x^{(r)})[\[PERSON]^{\prime}(G_{2^{nd}NM}(x^{(r)})]^{-1}=\Big{[}I+L_{2}\;{E^{( r)}}^{3}+...\Big{]}\;[\[PERSON]^{\prime}(\alpha)]^{-1},\quad L_{2}=2C_{2}C_{3}-8C_{2}^{3}. \tag{20}\]  
Equations (18) and (20) leads to  
\[H_{2}(x^{(r)})[\[PERSON]^{\prime}(G_{2^{nd}NM}(x^{(r)})]^{-1}\Phi(\mu _{j-1}(x^{(r)}))=\Big{(}I+L_{2}\;{E^{(r)}}^{3}+...\Big{)}\Big{(}(\mu_{j-1}(x^{ (r)})-\alpha)+C_{2}(\mu_{j-1}(x^{(r)})-\alpha)^{2}+...\Big{)}\] \[=\Big{[}\mu_{j-1}(x^{(r)})-\alpha+L_{2}\;{E^{(r)}}^{3}(\mu_{j-1}(x ^{(r)})-\alpha)+C_{2}(\mu_{j-1}(x^{(r)})-\alpha)^{2}+...\Big{]} \tag{21}\]  
Substituting (21) in (8) and calculating the error term, we get  
\[\mu_{j}(x^{(r)})-\alpha = \big{(}\mu_{j-1}(x^{(r)})-\alpha\big{)}-\Big{(}(\mu_{j-1}(x^{(r)} )-\alpha)+L_{2}\;{E^{(r)}}^{3}(\mu_{j-1}(x^{(r)})-\alpha) \tag{22}\] \[+C_{2}(\mu_{j-1}(x^{(r)})-\alpha)^{2}+...\Big{)}\] \[= -L_{2}\;{E^{(r)}}^{3}(\mu_{j-1}(x^{(r)})-\alpha)+...\]  
It is proved that \(\mu_{0}(x^{(r)})-\alpha=L_{1}{E^{(r)}}^{5}+O({E^{(r)}}^{6})\). Therefore, for j = 1, 2,... in (22), we get  
\[\mu_{1}(x^{(r)})-\alpha = -L_{2}({E^{(r)}}^{(3)})\Big{(}\mu_{0}(x^{(r)})-\alpha\Big{)}+...\] \[= -L_{2}L_{1}{E^{(r)}}^{8}+...\] \[H_{2}(x^{(r)})=I+\frac{1}{2}(\eta(x^{(r)})-I)^{2}=I+2C_{2}^{2}{E^{(r)}}^{2}+(6C_{ 2}C_{3}-12C_{2}^{3}){E^{(r)}}^{3}+.... \tag{19}\]0 \[H_{2}(x^{(r)})=I+\frac{1}{2}(\eta(x^{(r)})-I)^{2}=I+2C_{2}^{2}{E^{(r)}}^{2}+(6C_{ 2}C_{3}-12C_{2}^{3}){E^{(r)}}^{3}+.... \tag{19}\]1 \[H_{2}(x^{(r)})=I+\frac{1}{2}(\eta(x^{(r)})-I)^{2}=I+2C_{2}^{2}{E^{(r)}}^{2}+(6C_{ 2}C_{3}-12C_{2}^{3}){E^{(r)}}^{3}+.... \tag{19}\]2  
Proceeding by induction, we get  
\[H_{2}(x^{(r)})=I+\frac{1}{2}(\eta(x^{(r)})-I)^{2}=I+2C_{2}^{2}{E^{(r)}}^{2}+(6C_{ 2}C_{3}-12C_{2}^{3}){E^{(r)}}^{3}+.... \tag{19}\]3  
The above result shows that the method has \((3k+5)\) order convergence.