## 3 Numerical Example and Applications  
In this section, we shall check the effectiveness and validity of our theoretical results which we have proposed in Section 2 on the scheme proposed by [PERSON] and [PERSON] [PERSON] and [PERSON] (2015). For this purpose, we shall choose a variety of nonlinear equations which are mentioned in the following examples including motivational example. At this point, we chose the following eighth order methods proposed by [PERSON] and [PERSON] [PERSON] and [PERSON] (2015)  
\[\left\{ \begin{aligned} & y_{n}=x_{n}-F^{\prime}(x_{n})^{-1}F(x_{n}),\\ & z_{n}=y_{n}-\left(2[y_{n},\ x_{n};\ F]-F^{\prime}(x_{n})\right) ^{-1}F(y_{n}),\\ & x_{n+1}=\phi_{8}(x_{n},y_{n},z_{n}),\end{aligned} \right. \tag{38}\]  
\[\left\{ \begin{aligned} & y_{n}=x_{n}-F^{\prime}(x_{n})^{-1}F(x_{n}),\\ & z_{n}=y_{n}-\left([y_{n},\ x_{n};\ F]^{2}\right)^{-1}F^{\prime}( x_{n})F(y_{n}),\\ & x_{n+1}=\phi_{8}(x_{n},y_{n},z_{n})\end{aligned} \right. \tag{39}\]  
and  
\[\left\{ \begin{aligned} & y_{n}=x_{n}-F^{\prime}(x_{n})^{-1}F(x_{n}),\\ & z_{n}=y_{n}-\left(2[y_{n},\ x_{n};\ F]^{-1}-F^{\prime}(x_{n})^{- 1}\right)F(y_{n}),\\ & x_{n+1}=\phi_{8}(x_{n},y_{n},z_{n}),\end{aligned} \right. \tag{40}\]  
denoted by \(M_{1}\), \(M_{2}\) and \(M_{3}\), respectively. The initial guesses \(x_{0}\) are selected with in the range of convergence domain which gives guarantee for convergence of the iterative methods. Due to the pages limit, all the values of parameters are done for only 5 significant digits and displayed in the Tables 1-3 and examples Equations (1)-(3), although \(100\) significant digits are available. The considered test examples with corresponding initial guess, radius of convergence and necessary number of iterations (\(n\)) for getting the desired accuracy are displayed in Tables 1-3. In addition, we also want to verify the theoretical order of convergence of Methods (38)-(40). Therefore, we calculate the computational order of convergence (COC) Haire and Arora (2015) approximated by using the following formulas  
\[\rho=\frac{ln\frac{\left\|x_{n+2}-x^{*}\right\|}{\left\|x_{n+1}-x^{*}\right\|} }{ln\frac{\left\|x_{n+1}-x^{*}\right\|}{\left\|x_{n}-x^{*}\right\|}},\quad\text { for each }n=0,1,2,\ldots \tag{41}\]  
or the approximate computational order of convergence (ACOC) [PERSON] and [PERSON] (2015)  
\[\rho^{*}=\frac{ln\frac{\left\|x_{n+2}-x_{n+1}\right\|}{\left\|x_{n+1}-x_{n} \right\|}}{ln\frac{\left\|x_{n+1}-x_{n}\right\|}{\left\|x_{n}-x_{n-1}\right\|} },\quad\text{ for each }n=1,2,\ldots \tag{42}\]  
During the current numerical experiments with programming language Mathematica (Version 9), all computations have been done with multiple precision arithmetic, which minimize round-off errors. We use \(\epsilon=10^{-200}\) as a tolerance error. The following stopping criteria are used for computer programs:  
\((i)|x_{n+1}-x_{n}|<\epsilon\) and \((ii)|f(x_{n+1})|<\epsilon\). Further, we use \(\lambda=2\) and function \(g_{2}\) as defined above Equation (37) in all the examples. **Example 1**. : _Let \(S=\mathbb{R},D=[-1,1],x^{*}=0\) and define function \(F\) on \(D\) by_  
\[F(x)=\sin x. \tag{43}\]  
_Then, we get \(L_{0}=L=M=K=1\) and \(K_{0}=K_{1}=\frac{L_{0}}{2}\). We obtain different radius of convergence, COC (\(\rho\)) and \(n\) in the following Table 1._  
**Example 2**. : _Let \(\mathbb{X}=\mathbb{Y}=C[0,\ 1]\), the space of continuous functions defined on \([0,\ 1]\) be and equipped with the max norm. Let \(\mathbb{D}=\bar{U}(0,\ 1)\). Define function \(F\) on \(\mathbb{D}\) by_  
\[F(\varphi)(x)=\varphi(x)-5\int_{0}^{1}x\tau\varphi(\tau)^{3}d\tau, \tag{44}\]  
_we have that_  
\[F^{\prime}\big{(}\varphi(\xi)\big{)}(x)=\xi(x)-15\int_{0}^{1}x\tau\varphi(\tau )^{2}\xi(\tau)d\tau,\text{ for each }\xi\in\mathbb{D}. \tag{45}\]  
_Then, for \(x^{*}=0,\) we obtain that \(L_{0}=7.5,\ L=15,\ M=K=2\) and \(K_{0}=K_{1}=\frac{L_{0}}{2}\). We obtain different radius of convergence in the following Table 2._  
**Example 3**.