# 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 \(r_{R}\) & \(r_{1}\) & \(r_{2}\) & \(r_{3}\) & \(r_{q}\) & \(r_{p_{0}}\) & \(r\) \\ \hline
0.22856 & 0.22856 & 0.089459 & 0.086166 & 0.11255 & 0.26191 & 0.086166 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Different values of parameters which satisfy Theorem 1. We calculate the computational order of convergence, asymptotic errors constant and other constants up to several numbers of significant digits (minimum 1000 significant digits) to minimise the round off errors. In the context of systems of nonlinear equations, we also consider a nonlinear systems in Example 3 to check the proposed theoretical results for nonlinear systems. In this regard, we displayed the number of iteration indexes \((n)\), residual errors of the corresponding function \((\|F(x_{n})\|)\), errors \(\|e_{n}\|\) (where \(e_{n}=x_{n}-x^{*}\)), \(\left\|\frac{e_{n}}{e_{n-1}}\right\|\) and the asymptotic errors constant \(\eta=\lim\limits_{n\to\infty}\left\|\frac{e_{n}}{e_{n-1}^{p}}\right\|\) in the Tables 7 and 8. As we mentioned in the earlier paragraph, we calculate the values of all the constants and functional residuals up to several numbers of significant digits but, due to the limited paper space, we display the values of \(x_{n}\) up to 15 significant digits. Further, the values of other constants, namely, \(\xi(COC)\) up to 5 significant digits and the values \(\left|\frac{e_{n}}{e_{n-1}}\right|\), \(\eta\) and \(\left\|\frac{e_{n}}{e_{n-1}^{p}}\right\|\) are up to 10 significant digits. Furthermore, the residual errors in the function/systems of nonlinear functions (\(|F(x_{n})|\) or \(\|F(x_{n})\|\)) and the errors \(|e_{n}|\) or \(\|e_{n}\|\) are displayed up to 2 significant digits with exponent power which are mentioned in the following tables corresponding to the test function. However, a minimum of 1000 significant digits are available with us for every value. Furthermore, we consider the approximated zero of test functions when the exact zero is not available, which is corrected up to 1000 significant digits to calculate \(\|x_{n}-x^{*}\|\). During the current numerical experiments with programming language Mathematica (Version 9), all computations have been done with multiple precision arithmetic, which minimises round-off errors. \begin{table}
\begin{tabular}{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 & 1.7e\(-\)1 & 1.6e\(-\)1 & & \\  & 1 & 7.2e\(-\)12 & 7.2 e\(-\)12 & & 0.00001412834175 & 0.001157325063 \\  & 2 & 3.0e\(-\)93 & 3.0e\(-\)93 & 7.8569 & 0.0004283379739 & \\  & 3 & _7.9e\(-\)744_ & 7.9e\(-\)744 & 7.9947 & 0.001157325063 & \\ \hline \multirow{4}{*}{\(M_{2}\)} & 0 & 1.7e\(-\)1 & 1.6e\(-\)1 & & \\  & 1 & 1.9e\(-\)5 & 1.9e\(-\)5 & 0.02642235158 & 0.2812313492 \\  & 2 & 2.8e\(-\)20 & 2.8e\(-\)20 & 3.7651 & 0.2223712525 & \\  & 3 & 1.7e\(-\)79 & 1.7e\(-\)79 & 3.9931 & 0.2812313492 & \\ \hline \multirow{4}{*}{\(M_{3}\)} & 0 & 1.7e\(-\)1 & 1.6e\(-\)1 & & \\  & 1 & 2.8e\(-\)5 & 2.5e\(-\)5 & 0.03967215686 & 0.4999499551 \\ \cline{1-1}  & 2 & 2.5e\(-\)19 & 2.5e\(-\)19 & 3.7373 & 0.3860243040 & \\ \cline{1-1}  & 3 & 1.9e\(-\)75 & 1.9e\(-\)75 & 3.9920 & 0.4999499551 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Convergence behavior of different cases on Example 3.