# Some Inexact Version of the Iterative Exponential Method for Solving Nonsmooth Equations
## 4 Numerical Experiments
The most interesting results we selected are presented in Table 3._  
_In these tests, we noticed that the choice of starting point did not affect the number of iterations very much when Algorithm 1 generated a convergent sequence. Therefore, we omitted the results for other starting points. The number of iterations was actually very similar for a small number of variables. However, Algorithm 1 showed its advantage for larger problems (i.e., when a system of nonlinear equations has at least a dozen unknowns)._  
**Example 3**. : _The last test problem was the Walrasian production-price problem [23] with the demand function \(\xi\) defined in the following form:_  
\[\xi_{j}(\pi)=\sum_{i=1}^{l}\frac{a_{ij}\phi_{i}(\pi)}{\pi_{j}^{b_{i}}},\,\,j=1,\ldots,m.\]  
_Here, \(\pi\in R^{m}\) is the price vector, \(a_{ij}\) measures the \(i\)-th consumer's intensity of demand for commodity \(j\), and \(b_{i}\) is the \(i\)-th consumer's elasticity of substitution for \(i=1,\ldots,l\). Furthermore, the following_  
\[\phi_{i}(\pi)=\frac{\sum_{k=1}^{m}\pi_{k}\omega_{ik}}{\sum_{k=1}^{m}a_{ik}\pi _{k}^{1-b_{i}}}\]  
_is a function of the price chosen in such way to satisfy the budget constraint for each unit. Let \(B\in R^{m\times n}\) denote the technology, \(w\in R^{m}\) denote the total endowment of the economy prior to production and \(y\in R^{n}\) denote the production plan._  
_Then, the Walrasian model can be written as the nonlinear complementarity problem (NCP) in the form_  
\[\min(g(x),f(x))=0\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{\(c_{1}=-c_{2}=1\)} & \multicolumn{2}{c}{\(c_{1}=-c_{2}=10\)} & \multicolumn{2}{c}{\(c_{1}=-c_{2}=100\)} \\ \(n\) & \(N_{IEM}\) & \(N_{IGM}\) & \(N_{IEM}\) & \(N_{IGM}\) & \(N_{IEM}\) & \(N_{IGM}\) \\ \hline
2 & 5 & 6 & 5 & 6 & 5 & 6 \\
3 & 5 & 6 & 5 & 7 & 6 & 7 \\
4 & 6 & 7 & 7 & 8 & 8 & 8 \\
5 & 6 & 8 & 7 & 8 & 8 & 9 \\
8 & 8 & 10 & 9 & 9 & 9 & 10 \\
10 & 9 & 10 & 9 & 10 & 10 & 11 \\
12 & 10 & 11 & 10 & 11 & 11 & 11 \\
15 & 19 & 21 & 23 & 27 & 25 & 30 \\
20 & 17 & 19 & 19 & 23 & 18 & 21 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The numerical results obtained for the equation from Example 2 (\(n\) is the number of component functions.). with the following functions:_  
\[g(\pi,y)=\left[ \begin{array}{c}\pi\\ y\end{array} \right]\text{ and }f(\pi,y)=\left[ \begin{array}{c}\omega-\xi(\pi)+By\\ -B^{T}\pi\end{array} \right],\]  
_where the operator \"min\" denotes the component-wise minimum of the vectors._  
_Function \(F(x)=\min(g(x),f(x))\) is locally Lipschitzian, semismooth and BD-regular, so we can use Algorithm 1 to solve nonsmooth Equation (1)._  
_An approximate solution of such a problem (NCP) with \(m=6\), \(n=8\) and \(l=5\) is_  
\[\tilde{x}=(0.220,0.251,0.161,0.055,0.106,0.207,\ 0.463,0,3.940,0.006,0,0,0.438,0).\]  
_We solved this problem using the inexact exponential method with various starting approximations from the neighborhood of \(\tilde{x}\) in the form_  
\[x^{(0)}=\tilde{x}+a_{1}\dot{e}+a_{2}\tilde{x},\]  
_where \(\dot{e}=(1,1,1,1,1,1,0,\ldots,0)\in R^{14}\) and \(\ddot{e}=(0,0,0,0,0,1,\ldots,1)\in R^{14}\). The exemplary results are shown in Table 4._  
_The Broyden-like method (from [24]) generated somewhat comparable results, which was not surprising since all these methods have the same order of convergence. The number of iterations was equal to 27, 9, \(-\), 66 and 263 for the same initial points as in Table 4. In this test problem, the convergence significantly depended on the location of the starting point (parameters \(a_{1}\) and \(a_{2}\)). Here, Algorithm 1 showed its advantage for initial points further from the solution._