# Two Efficient Derivative-Free Iterative Methods for Solving Nonlinear Systems
## 4 Numerical Examples  
In this section, we compare the performance of related methods by mathematical experiments. The numerical experiments have been carried out using Maple 14 computer algebra system with 2048 digits. The computer specifications are Microsoft Windows 7 Intel(R), Core(TM) i3-2350M CPU, 1.79 GHz with 2 GB of RAM. Figure 1: The boundary function \(\mathrm{H}_{5,4}\) in\((m,\mu)-\)plain. According to the Equation (29), the factor \(\mu\) is claimed by expressing the cost of the evaluation of elementary functions in terms of products [15]. Table 2 gives an estimation of the cost of the elementary functions in amount of equivalent products, where the running time of one product is measured in milliseconds. Tables 3-8 show the following information of the methods \(\psi_{i}\)\((i=1,2,\cdots,5)\): the number of iterations \(k\) needed to converge to the solution, the norm of function \(F(x^{(k)})\)at the last step, the value of the stopping factors at the last step,the computational cost \(C\), the computational time \(Time(s)\), the computational efficiency indices \(CEI\) and the computational order of convergence \(\rho\). Using the commond time( ) in Maple 14, we can obtain the computational time of different methods. The computational order of convergence \(\rho\) is defined by [16]:  
\[\rho\approx\frac{\ln(||x^{(k+1)}-x^{(k)}||/||x^{(k)}-x^{(k-1)}||)}{\ln(||x^{(k )}-x^{(k-1)}||/||x^{(k-1)}-x^{(k-2)}||)} \tag{38}\]  
The following problems are chosen for numerical tests:  
**Example 1** Considering the following system  
\[\left\{ \begin{array}{l}x_{1}+e^{x_{1}}-\cos(x_{2})=0\\ 3x_{1}-x_{2}-\sin(x_{2})=0\end{array} \right.\]  
where \((m,\;\mu)=(2,\frac{53+110+112+1}{2})=(2,\;138)\) are the values used in Equation (29). \(x^{(0)}=(0.5,\;0.5)^{T}\) is the initial point and \(\alpha\approx(0,\;0)^{T}\) is the solution of the Example 1. \(||x^{(k)}-x^{(k-1)}||<10^{-100}\) is the stopping criterion. The results shown in Table 3 confirm the first assertion of Theorem 2 and the first assertion of Theorem 3 for \(m\doteq 2\). Namely, \(CEI_{5}>CEI_{4}\) for \(\mu>4.7859\). The new sixth-order method \(\psi_{5}\) spends minimum time for finding the numerical solution. The 'nc' denotes that the method does not converge in the Table 3.  
**Example 2** The second system is defined by [11]  
\[\left\{ \begin{array}{l}x_{2}+x_{3}-e^{-x_{1}}=0\\ x_{1}+x_{3}-e^{-x_{3}}=0\\ x_{1}+x_{2}-e^{-x_{3}}=0\end{array} \right.\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Digits** & \(\mathbf{x}\cdot\mathbf{y}\) & \(\mathbf{x}/\mathbf{y}\) & \(\sqrt{x}\) & \(\mathbf{\exp(x)}\) & \(\mathbf{\ln(x)}\) & \(\mathbf{\sin(x)}\) & \(\mathbf{\cos(x)}\) & \(\mathbf{\arctan(x)}\) \\ \hline
2048 & 0.109 ms & 1 & 5 & 53 & 12 & 112 & 110 & 95 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Estimation of computational cost of elementary functions computed with Maple 14 and using a processor Intel® Core(TM) i3-2350M CPU, 1.79 GHz (32-bit Machine) Microsoft Windows 7 Professional, where \(x=\sqrt{3}-1\) and \(y=\sqrt{5}\). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Method** & \(\mathbf{k}\) & \(||\mathbf{x^{(k)}-x^{(k-1)}}||\) & \(||\mathbf{F(x^{(k)})}||\) & \(\rho\) & \(C\) & \(CEI\) & **Time(s)** \\ \hline \(\psi_{1}\) & 13 & 1.792e\(-\)161 & 3.748e\(-\)322 & 2.00000 & 838 & 1.0008275 & 1.127 \\ \(\psi_{2}\) & nc & & & & & & & \\ \(\psi_{3}\) & 4 & 3.558e\(-\)743 & 8.245e\(-\)496 & 6.00314 & 1960 & 1.0009148 & 0.780 \\ \(\psi_{4}\) & 5 & 4.086e\(-\)211 & 5.330e\(-\)421 & 4.00015 & 1686 & 1.0008226 & 0.836 \\ \(\psi_{5}\) & 4 & 6.240e\(-\)164 & 2.389e\(-\)489 & 6.00420 & 1978 & 1.0009063 & **0.546** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance of methods for Example 1.  
where \((m,\;\mu)=(3,\frac{53+53}{3})=(3,\;35.3)\). The initial point is \(x^{(0)}=(0.5,\;0.5,\;0.5)\). \(||x^{(k)}-x^{(k-1)}||<10^{-200}\) is the stopping criterion.The solution is \(\alpha\approx(0.3517337,0.3517337,0.3517337)\).