# Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative
## 3 Numerical Example and Applications  
In this section, we will verify the validity and effectiveness of our theoretical results which we have proposed in Section 2 based on the scheme proposed by [PERSON] et al. [PERSON] et al. (2013). For this purpose, we shall choose a variety of nonlinear equations and systems of nonlinear equations which are mentioned in the following examples including motivational example. At this point, we will chose the following methods  
\[\left\{ \begin{array}{rl}&y_{n}=x_{n}-F^{\prime}(x_{n})^{-1}F(x_{n}),\\ &z_{n}=y_{n}-\left(2[y_{n},\ x_{n};\ F]-F^{\prime}(x_{n})\right)^{-1}F(y_{n}),\\ &x_{n+1}=\phi_{8}(x_{n},y_{n},z_{n}),\end{array} \right. \tag{37}\]  
\[\left\{ \begin{array}{rl}&y_{n}=x_{n}-F^{\prime}(x_{n})^{-1}F(x_{n}),\\ &z_{n}=y_{n}-\left(2[y_{n},\ x_{n};\ F]^{-1}-F^{\prime}(x_{n})^{-1}\right)F(y_ {n}),\\ &x_{n+1}=\phi_{8}(x_{n},y_{n},z_{n})\end{array} \right. \tag{38}\]  
and  
\[\left\{ \begin{array}{rl}&y_{n}=x_{n}-F^{\prime}(x_{n})^{-1}F(x_{n}),\\ &z_{n}=y_{n}-\left(3I-2F^{\prime}(x_{n})^{-1}[y_{n},\ x_{n},\ F]\right)F^{ \prime}(x_{n})^{-1}F(y_{n}),\\ &x_{n+1}=\phi_{8}(x_{n},y_{n},z_{n})\end{array} \right. \tag{39}\]  
denoted by \(M_{1}\), \(M_{2}\) and \(M_{3}\), respectively. First of all, we require the initial guesses \(x_{0}\) which gives the guarantee for convergence of the iterative methods. Therefore, we shall calculate the values of \(r_{R}\), \(r_{1}\), \(r_{2}\), \(r_{q}\), \(r_{3}\), \(r_{p_{0}}\) and \(r\) to find the range of convergence domain, which are displayed in the Tables 1-4 up to 5 significant digits. However, we have the values of these constants up to several number of significant digits. Then, we will also verify the theoretical order of convergence of these methods for scalar equations on the basis of the results obtained from computational order of convergence and \(\left|\frac{e_{n+1}}{e_{n}^{p}}\right|\). In the Tables 5 and 6, we display the number of iteration indexes \((n)\), approximated zeros \((x_{n})\), residual errors of the corresponding function \((|F(x_{n})|)\), errors \(|e_{n}|\) (where \(e_{n}=x_{n}-x^{*}\)), \(\left|\frac{e_{n+1}}{e_{n}^{p}}\right|\) and the asymptotic errors constant \(\eta=\lim\limits_{n\to\infty}\left|\frac{e_{n+1}}{e_{n}^{p}}\right|\). In addition, We will use the formulas proposed by [PERSON] et al. in [PERSON] et al. (2013) to calculate the computational order of convergence (COC), which is given by  
\[\rho=\frac{In\frac{\|x_{n+2}-x^{*}\|}{\|x_{n+1}-x^{*}\|}}{In\frac{\|x_{n+1}-x^ {*}\|}{\|x_{n}-x^{*}\|}},\quad\text{ for each }n=0,1,2,\ldots \tag{40}\]  
or the approximate computational order of convergence (ACOC) [PERSON] et al. (2013)  
\[\rho^{*}=\frac{In\frac{\|x_{n+2}-x_{n+1}\|}{\|x_{n+1}-x_{n}\|}}{In\frac{\|x_{n +2}-x_{n-1}\|}{\|x_{n}-x_{n-1}\|}},\quad\text{ for each }n=1,2,\ldots \tag{41}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Methods** & \(n\) & \(|F(x_{n})|\) & \(|e_{n}|\) & \(\xi\) & \(\frac{e_{n}}{e_{n-1}}\) & \(\eta\) \\ \hline \multirow{4}{*}{\(M_{1}\)} & 0 & 6.1e\(-\)2 & 5.6e\(-\)3 & & & \\  & 1 & 1.6e\(-\)17 & 1.8e\(-\)18 & & 0.0000605014094 & 2.153729894e\(+\)49 & \\  & 2 & 1.0e\(-\)93 & 9.1e\(-\)95 & 4.9289 & 2.314984786e\(+\)12 & \\  & 3 & 1.4e\(-\)514 & 1.3e\(-\)515 & 5.5155 & 2.153729894e\(+\)49 & \\ \hline \multirow{4}{*}{\(M_{2}\)} & 0 & 6.1e\(-\)2 & 5.6e\(-\)3 & & & \\  & 1 & 2.3e\(-\)13 & 2.1e\(-\)14 & & 0.00002153200773 & 4.146304252e\(-\)6 & \\  & 2 & 2.3e\(-\)59 & 2.1e\(-\)60 & 4.0264 & 0.00001076045277 & \\  & 3 & 8.6e\(-\)244 & 7.9e\(-\)245 & 4.0090 & 4.146304252e\(-\)6 & \\ \hline \multirow{4}{*}{\(M_{3}\)} & 0 & 6.1e\(-\)2 & 5.6e\(-\)3 & & & \\  & 1 & 3.6e\(-\)13 & 3.3e\(-\)14 & & 0.00003395540062 & 9.787194795e\(-\)6 & \\ \cline{1-1}  & 2 & 2.2e\(-\)58 & 2.0e\(-\)59 & 4.0278 & 0.00001654426326 & \\ \cline{1-1}  & 3 & 1.7e\(-\)239 & 1.5e\(-\)240 & 4.0050 & 9.787194795e\(-\)6 & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Convergence behavior of different cases on Example 1.