# Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative
## 3 Numerical Example and Applications
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Methods** & \(n\) & \(\|F(x_{n})\|\) & \(\|e_{n}\|\) & \(\xi\) & \(\left|\frac{e_{n}}{e_{n-1}}\right|\) & \(\eta\) \\ \hline \multirow{4}{*}{\(M_{1}\)} & 0 & 0.317 & 3.0e\(-\)4 & 1.3e\(-\)3 & & \\  & 1 & 0.318309886183791 & 3.8e\(-\)18 & 1.6e\(-\)17 & & 1.881774410e+6 & 1.746669349e+6 \\  & 2 & 0.318309886183791 & 2.1e\(-\)129 & 8.7e\(-\)129 & 8.0023 & 1.746669349e+6 & \\  & 3 & 0.318309886183791 & 1.4e\(-\)1019 & 6.0e\(-\)1019 & 8.0000 & 1.746669349e+6 & \\ \hline \multirow{4}{*}{\(M_{2}\)} & 0 & 0.317 & 3.0e\(-\)4 & 1.3e\(-\)3 & & \\  & 1 & 0.318309886183791 & 5.5e\(-\)17 & 2.3e\(-\)16 & & 2.711175817e+7 & 2.406103772e+7 \\  & 2 & 0.318309886183791 & 5.3e\(-\)119 & 2.2e\(-\)118 & 8.0041 & 2.406103772e+7 & \\ \cline{1-1}  & 3 & 0.318309886183791 & 3.5e\(-\)935 & 1.5e\(-\)934 & 8.0000 & 2.406103772e+7 & \\ \hline \multirow{4}{*}{\(M_{3}\)} & 0 & 0.317 & 3.0e\(-\)4 & 1.3e\(-\)3 & & \\  & 1 & 0.318309886183790 & 1.7e\(-\)16 & 7.2 e\(-\)16 & & 8.333410395e+7 & 7.204280059e+7 \\ \cline{1-1}  & 2 & 0.318309886183791 & 1.3e\(-\)114 & 5.3e\(-\)114 & 8.0052 & 7.204280059e+7 & \\ \cline{1-1}  & 3 & 0.318309886183791 & 1.1e\(-\)899 & 4.7e\(-\)899 & 8.0000 & 7.204280059e+7 & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Convergence behavior of different cases on Example 4. Further, we use \(\lambda=2\) and function \(g_{2}\) defined by Equation (35) in all the examples. **Example 1**. : _Let \(\mathbb{X}=\mathbb{Y}=\mathbb{R}^{3},\ \mathbb{D}=\Omega(0,\ 1).\) Define \(F\) on \(\mathbb{D}\) for \(v=(x,\ y,\ z)^{T}\) by_  
\[F(v)=(10x+\sin(x+y)-1,8y-\cos^{2}(z-y)-1,\ 12z+\sin z-1)^{T}. \tag{42}\]  
_Then the Frechet-derivative is given by_  
\[F^{\prime}(v)= \begin{bmatrix}10+\cos(x+y)&\cos(x+y)&0\\ 0&8+\sin 2(z-y)&\sin 2(z-y)\\ 0&0&12+\cos z\end{bmatrix} .\]  
_Then, for \(x^{*}=(0.068978\dots,\ 0.246442\dots,\ 0.076929\dots)\), we get \(L=L_{0}=38.9911\), \(K=M=168.962\) and \(K_{0}=K_{1}=\frac{L_{0}}{2}\). We calculate the radius of convergence, the residual errors of the corresponding function \((\|F(x_{n})\|)\) (by considering initial approximation \((0.072,\ 0.25,\ 0.080)\) in the methods \(M_{1},\ M_{2}\) and \(M_{3}\)), errors \(\|e_{n}\|\) (where \(e_{n}=x_{n}-x^{*}\)), \(\left\|\frac{e_{n}}{e_{n-1}^{p}}\right\|\) and the asymptotic errors constant \(\eta=\lim\limits_{n\to\infty}\left\|\frac{e_{n}}{e_{n-1}^{p}}\right\|\), which are displayed in the Tables 1 and 5._  
**Example 2**. : _Let \(\mathbb{X}=\mathbb{Y}=\mathbb{R},\ \mathbb{D}=[-1,1]\) and define function \(F\) on \(\mathbb{D}\) by_  
\[F(x)=e^{x}-4x^{2}. \tag{43}\]  
_Then, for \(x^{*}=0.714806\dots\), we obtain that \(L_{0}=L=2.91681,\ M=K=1.82827\) and \(K_{0}=K_{1}=\frac{L_{0}}{2}\). Hence, we obtain all the values of constants and residual errors in the functions, etc., which we have described earlier in the Tables 2 and 6._  
**Example 3**. : _Let \(\mathbb{X}=\mathbb{Y}=\mathbb{R}^{3},\ \mathbb{D}=\tilde{U}(0,\ 1).\) Define \(F\) on \(\mathbb{D}\) for \(v=(x,\ y,\ z)^{T}\) by_  
\[F(v)=\left(e^{x}-1,\ \frac{e-1}{2}y^{2}+y,\ z\right)^{T}. \tag{44}\]  
_Then the Frechet-derivative is given by_  
\[F^{\prime}(v)= \begin{bmatrix}e^{x}&0&0\\ 0&(e-1)y+1&0\\ 0&0&1\end{bmatrix} .\]  
_Notice that \(x^{*}=(0,\ 0,\ 0),\ F^{\prime}(x^{*})=F^{\prime}(x^{*})^{-1}=diag\{1,\ 1,\ 1\}\), \(L_{0}=e-1<L=1.78957239,\ K_{0}=K_{1}=\frac{e-1}{2}\) and \(K=M=1.7896\). Hence, we calculate the radius of convergence, the residual errors of the corresponding function \((\|F(x_{n})\|)\) (by considering initial approximation \((0.094,\ 0.094,\ 0.095)\) in the methods \(M_{1},\ M_{2}\) and \(M_{3}\)), errors \(\|e_{n}\|\) (where \(e_{n}=x_{n}-x^{*}\)) and other constants, which are given in the Tables 3 and 7._  
**Example 4**. : _Returning back to the motivational example in the introduction of this paper, we have \(L=L_{0}=\frac{2}{2\pi+1}(80+16\pi+(11+12\log 2)\pi^{2})\), \(M=K=2\) an \(K_{0}=K_{1}=\frac{L_{0}}{2}\).