## 3 Convergence Analysis
)[\[PERSON]^{\prime}(\alpha)]^{-1}\[PERSON]^{(i)}(\alpha)\), \(i=2,3,\ldots\) and \(E^{(r)}=x^{(r)}-\alpha\). Inverting the above series, we get  
\[[\Phi^{\prime}(x^{(r)})]^{-1}=\left[I+X_{1}E^{(r)}+X_{2}{E^{(r)}}^{2}+X_{3}{E^{ (r)}}^{3}+X_{4}{E^{(r)}}^{4}+X_{5}{E^{(r)}}^{5}\right][\Phi^{\prime}(\alpha)]^ {-1}, \tag{11}\]  
where  
\[X_{1}=-2C_{2},X_{2}=4C_{2}^{2}-3C_{3},\]  
\[X_{3}=-8C_{2}^{3}+6C_{2}C_{3}+6C_{3}C_{2}-4C_{4},\]  
\[X_{4}=-5C_{5}+9C_{3}^{2}+8C_{2}C_{4}+8C_{4}C_{2}+16C_{2}^{4}-12C_{2}^{2}C_{3}- 12C_{3}C_{2}^{2}-12C_{2}C_{3}C_{2}\]  
and  
\[X_{5}=-32C_{2}^{5}+24C_{3}C_{2}^{3}+24C_{2}C_{3}C_{2}^{2}-16C_{4}C_{2}^{2}+24 C_{2}^{2}C_{3}C_{2}-18C_{3}^{2}C_{2}-16C_{2}C_{4}C_{2}+10C_{5}C_{2}+\]  
\[\Phi^{\prime}(\alpha+h)=\Phi^{\prime}(\alpha)\left[I+\sum_{i=2}^{p-1}iC_{i}h^{ i-1}\right]+O(h^{p}), \tag{10}\]0  
Then  
\[\Phi^{\prime}(\alpha+h)=\Phi^{\prime}(\alpha)\left[I+\sum_{i=2}^{p-1}iC_{i}h^{ i-1}\right]+O(h^{p}), \tag{10}\]1 \[\Phi^{\prime}(\alpha+h)=\Phi^{\prime}(\alpha)\left[I+\sum_{i=2}^{p-1}iC_{i}h^{ i-1}\right]+O(h^{p}), \tag{10}\]2 \[\Phi^{\prime}(\alpha+h)=\Phi^{\prime}(\alpha)\left[I+\sum_{i=2}^{p-1}iC_{i}h^{ i-1}\right]+O(h^{p}), \tag{10}\]3 \[\Phi^{\prime}(\alpha+h)=\Phi^{\prime}(\alpha)\left[I+\sum_{i=2}^{p-1}iC_{i}h^{ i-1}\right]+O(h^{p}), \tag{10}\]4Furthermore, we obtain  
\[ \begin{split}& G_{2^{nd}NM}(x^{(r)})=\alpha+C_{2}{E^{(r)}}^{2}+2(-C_ {2}^{2}+C_{3}){E^{(r)}}^{3}+(3C_{4}+4C_{2}^{3}-4C_{2}C_{3}-3C_{3}C_{2}){E^{(r)}} ^{4}+(-6C_{3}^{2}-8C_{2}^{4}\\ &+8C_{2}^{2}C_{3}+6C_{2}C_{3}C_{2}+6C_{3}C_{2}^{2}-6C_{2}C_{4}-4C _{4}C_{2}+4C_{5}){E^{(r)}}^{5}+(5C_{6}+2C_{2}C_{5}+14C_{2}^{2}C_{4}-9C_{3}C_{4}- 16C_{2}^{3}C_{3}\\ &+12C_{3}C_{2}C_{3}+12C_{2}C_{3}^{2}-8C_{4}C_{3}+16C_{2}^{5}-12C_ {3}C_{2}^{3}-12C_{2}C_{3}C_{2}^{2}+8C_{4}C_{2}^{2}-12C_{2}^{3}C_{3}C_{2}+9C_{3}^ {2}C_{2}\\ &+8C_{2}C_{4}C_{2}-5C_{5}C_{2}-10C_{2}C_{4}){E^{(r)}}^{6}+O({E^{( r)}}^{7}).\end{split}  \tag{12}\]  
Equation (12) leads to  
\[ \begin{split}&\Phi(G_{2^{nd}NM}(x^{(r)}))=\Phi^{\prime}(\alpha) \left[{C_{2}}{E^{(r)}}^{2}+2(-C_{2}^{2}+C_{3}){E^{(r)}}^{3}+(3C_{4}+5C_{2}^{3}- 4C_{2}C_{3}-3C_{3}C_{2}){E^{(r)}}^{4}\right.\\ &+(-6C_{3}^{2}-12C_{2}^{4}+12C_{2}^{2}C_{3}+6C_{2}C_{3}C_{2}+6C_ {3}C_{2}^{2}-6C_{2}C_{4}-4C_{4}C_{2}+4C_{5}){E^{(r)}}^{5}\\ &+(5C_{6}+2C_{2}C_{5}+14C_{2}^{2}C_{4}-9C_{3}C_{4}-16C_{2}^{3}C_{ 3}+12C_{3}C_{2}C_{3}+12C_{2}C_{3}^{2}-8C_{4}C_{3}+16C_{2}^{5}\\ &-12C_{3}C_{2}^{3}-12C_{2}C_{3}C_{2}^{2}+8C_{4}C_{2}^{2}-12C_{2}^ {2}C_{3}C_{2}+9C_{3}^{2}C_{2}+8C_{2}C_{4}C_{2}-5C_{5}C_{2}-10C_{2}C_{4}){E^{(r )}}^{6}\Big{]}+O({E^{(r)}}^{7}).\end{split}  \tag{13}\]  
Also,  
\[\Phi^{\prime}(G_{2^{nd}NM}(x^{(r)}))=\left[\Phi^{\prime}(\alpha)\right]\left[ I+{P_{1}}{E^{(r)}}^{2}+{P_{2}}{E^{(r)}}^{3}+{P_{3}}{E^{(r)}}^{4}\right]+O({E^{(r)}} ^{5}), \tag{14}\]  
where \(P_{1}=2C_{2}^{2}\), \(P_{2}=4C_{2}C_{3}-4C_{2}^{3}\) and \(P_{3}=4C_{2}^{3}-4C_{2}C_{3}-3C_{3}C_{2}+3C_{4}+3C_{3}C_{2}^{2}\). Combining (11) and (14), one gets  
\[ \begin{split}\tau(x^{(r)})&=[\Phi^{\prime}(x^{(r)} )]^{-1}\Phi^{\prime}(G_{2^{nd}NM}(x^{(r)})))=I-2C_{2}{E^{(r)}}+(6C_{2}^{2}-3C_ {3}){E^{(r)}}^{2}+(10C_{2}C_{3}\\ &+6C_{3}C_{2}-16C_{2}^{3}-4C_{4}){E^{(r)}}^{3}+(4C_{2}^{3}-4C_{2 }C_{3}-3C_{3}C_{2}+3C_{4}-15C_{3}C_{2}^{2}-20C_{2}^{2}C_{3}+32C_{2}^{4}-5C_{5} \\ &+9C_{3}^{2}+8C_{2}C_{4}+8C_{4}C_{2}-12C_{2}C_{3}C_{2}){E^{(r)}} ^{4}+O({E^{(r)}}^{5}).\end{split}  \tag{15}\]  
Then we get  
\[ \begin{split} H_{1}(x^{(r)})=I+C_{2}^{2}{E^{(r)}}^{2}+\left(3C_{2 }C_{3}-6C_{2}^{3}\right){E^{(r)}}^{3}+\left(25C_{2}^{4}-19C_{2}^{2}C_{3}+\frac {9}{4}C_{3}^{2}-6C_{2}C_{3}C_{2}+4C_{2}C_{4}\right){E^{(r)}}^{4}+O({E^{(r)}}^ {5}).\end{split}  \tag{16}\]  
After calculating inverse of \(\Phi^{\prime}(G_{2^{nd}NM}(x^{(r)}))\) and using (13), we get  
\[