# Two Efficient Derivative-Free Iterative Methods for Solving Nonlinear Systems
## 4 Numerical Examples
The results shown in Table 4 confirm the first assertion of Theorem 2 and assertion 1 of Theorem 3 for \(m=3\). Namely, \(CEI_{4}>CEI_{2}\) and \(CEI_{5}>CEI_{4}\) for \(\mu>4.7859\). Table 4 shows that sixth-order method \(\psi_{5}\) is the most efficient iterative method in both computational time and \(CEI\). **Example 3** Now, considering the following large scale nonlinear systems [17]:  
\[\left\{ \begin{array}{ll}x_{i}x_{i+1}-1=0,\;1\leq i\leq m-1\\ x_{m}x_{1}-1=0\end{array} \right.\]  
The initial vector is \(x^{(0)}=\{1.5,1.5,\cdots,1.5\}^{t}\) for the solution \(\alpha=\{1,1,\cdots,1\}^{t}\). The stopping criterion is \(||x^{(k)}-x^{(k-1)}||<10^{-100}\). _Application in Integral Equations_  
The Chandrasekhar integral [18] equation comes from radiative transfer theory, which is given by  
\[F(P,c)=0,\;P:[0,1]\to R\]  
with the operator \(F\) and parameter \(c\) as  
\[F(P,c)(u)=P(u)-\left(1-\frac{c}{2}\int_{0}^{1}\frac{uP(v)}{u+v}dv\right)^{-1}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & \(\mathbf{k}\) & \(||\mathbf{x^{(k)}-x^{(k-1)}}||\) & \(||\mathbf{F(x^{(k)})}||\) & \(\mathbf{\rho}\) & \(\mathbf{C}\) & \(\mathbf{CEI}\) & **Time(s)** \\ \hline \(\psi_{1}\) & 10 & 4.993e\(-\)150 & 7.480e\(-\)299 & 2.00000 & 2,745,802 & 1.000000252 & 95.940 \\ \(\psi_{2}\) & 5 & 3.013e\(-\)212 & 1.210e\(-\)423 & 4.00000 & 5,649,610 & 1.000000245 & 126.438 \\ \(\psi_{3}\) & 4 & 3.922e\(-\)556 & 2.197e\(-\)833 & 5.99998 & 5,571,005 & 1.000000322 & 77.111 \\ \(\psi_{4}\) & 5 & 1.404e\(-\)269 & 9.850e\(-\)538 & 4.00000 & 2,944,404 & 1.000000471 & 81.042 \\ \(\psi_{5}\) & 4 & 5.298e\(-\)208 & 2.231e\(-\)621 & 5.99976 & 3,063,804 & 1.000000585 & **64.818** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The computational time (in second) for Example 3 by the methods. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Method** & \(\mathbf{k}\) & \(||\mathbf{x^{(k)}-x^{(k-1)}}||\) & \(||\mathbf{F(x^{(k)})}||\) & \(\mathbf{\rho}\) & \(\mathbf{C}\) & \(\mathbf{CEI}\) & **Time(s)** \\ \hline \(\psi_{1}\) & 9 & 2.136e\(-\)302 & 5.945e\(-\)604 & 2 & 449.6 & 1.00154289 & 0.514 \\ \(\psi_{2}\) & 5 & 2.439e\(-\)675 & 2.703e\(-\)1350 & 4 & 1032.1 & 1.00134408 & 0.592 \\ \(\psi_{3}\) & 4 & 1.414e\(-\)1080 & 7.020e\(-\)1620 & 6 & 1023.1 & 1.00175284 & 0.561 \\ \(\psi_{4}\) & 5 & 4.123e\(-\)699 & 9.73957e\(-\)1397 & 4 & 915.2 & 1.00151589 & 0.561 \\ \(\psi_{5}\) & 4 & 9.097e\(-\)550 & 8.57708e\(-\)1647 & 6 & 1054.1 & 1.00170125 & **0.483** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance of methods for Example 2. We approximate the integrals by the composite midpoint rule:  
\[\int_{0}^{1}f(t)dt=\frac{1}{m}\sum_{j=1}^{m}f(t_{j})\]  
where \(t_{j}=\left(j-1/2\right)/m\) for \(1\leq j\leq m\). We obtain the resulting discrete problem is  
\[F_{i}(u,c)=u_{i}-\left(1-\frac{c}{2m}\sum_{j=1}^{m}\frac{t_{i}u_{j}}{t_{i}+t_{j }}\right)^{-1},1\leq i\leq m\]  
The initial vector is \(x^{(0)}=\left\{1.5,1.5,\cdots,1.5\right\}^{t}\), \(c=0.9\). Tables 7 and 8 show the numerical results of this problem. \(||F(x^{(k)})||<10^{-200}\) is the stopping criterion of this problem. The results shown in Table 5 confirm the assertion of Theorem 2 and Theorem 3 for \(m=199\). Namely, \(CEI_{4}>CEI_{2},CEI_{5}>CEI_{3}\), and \(CEI_{4}>CEI_{1}\). From the Table 6, we remark that the computational time of our fourth-order method \(\psi_{4}\) is less than that of the sixth order method \(\psi_{3}\) for \(m=299\). Tables 5-7 show that, as the nonlinear system is big-sized, our new methods \(\psi_{i}\) (\(i=4,5\)) remarkably reduce the computational time. The numerical results shown in Tables 3-8 are in concordance with the theory developed in this paper. The new methods require less number of iterations to obtain higher accuracy in the contrast to the other methods. The most important is that our methods have higher \(CEI\) and lower computational time than other methods in this paper. The sixth-order method \(\psi_{5}\) is the most efficient iterative methods in both \(CEI\) and computational time.