# Some Inexact Version of the Iterative Exponential Method for Solving Nonsmooth Equations
## 4 Numerical Experiments  
Now, we present some numerical results which help us to confirm the theoretical properties of the algorithm. We solved a few different problems. Algorithm 1 was implementedin C++ with double precision arithmetic and performed using Apache NetBeans. In all examples, the crucial parameters were set to be alike as follows: \(\theta=0.999\), \(\eta_{k}=0.5\) for all \(k\) and \(\tau_{1}=\tau_{2}=0.5\). The stopping criterion was the condition \(\|F(x^{(k)})\|_{2}\leq 1.0\times 10^{-7}\). A flowchart of Algorithm 1 is presented in Figure 1. In our opinion, the results of the numerical computations were satisfactory, as we will now demonstrate. In all examples, \(x^{(0)}\) is the starting approximation, and \(N_{IEM}\) is the number of iterations performed by Algorithm 1. For comparison of the efficiency of the new method, we also show the results obtained in tests by the inexact generalized Newton (IGM) method presented in [20] (\(N_{IGM}\)), which also has superlinear convergence. **Example 1**. : _As the first (double) test problem, we chose the usual Equation (1) with two different nonsmooth functions:_  
\[F_{1}(x) = 0.2x|x-1|+e^{x-0.5}-1.05,\] \[F_{2}(x) = \left[ \begin{array}{l}|x_{1}|+(x_{2}-1)^{2}-1\\ (x_{1}-1)^{2}+|x_{2}|-1\end{array} \right]\]  
Figure 1: Flowchart for Algorithm 1. The equation with function \(F_{1}\) has one solution \(x^{*}=0.5\), while the equation with function \(F_{2}\) has two solutions: \((0,0)\) and \((1,1)\). Both equations have one nondifferentiability point each: \(x=1\) for \(F_{1}\) and \(x=(0,0)\) for \(F_{2}\). Both functions \(F_{1}\) and \(F_{2}\) are locally Lipschitz at their solution points. \(F_{1}\) is \(C^{1}\) at \(x^{*}\) because  
\[F_{1}^{\prime}(x)=\left\{ \begin{array}{l}-0.2(x-2)+e^{x-0.5}\text{ for }x<1,\\ 0.2(x-2)+e^{x-0.5}\text{ for }x>1,\end{array} \right.\]  
Therefore, \(F_{1}\) is also semismooth at \(x^{*}\). In turn, \(Qi\) and [PERSON] proved in [14] that the function is semismooth at some point if and only if all of its component functions are semismooth at this point. And the component functions of \(F_{2}\) are semismooth as the sum of the semismooth functions (the local Lipschitz continuities of the absolute value function and quadratic function are easy to check). Moreover, the Bouligand subdifferential of \(F_{1}\) at \((0,0)\) has the form  
\[\partial_{B}F((0,0)=\left\{\left[ \begin{array}{cc}-1&-2\\ -2&-1\end{array} \right]\right. }^{*}\left[ \begin{array}{cc}-1&-2\\ -2&1\end{array} \right]\right.^{*}\left[ \begin{array}{cc}1&-2\\ -2&-1\end{array} \right]\right\}.\]  
The equations can be solved by Algorithm 1. The only point of nondifferentiability of the function \(F_{1}\) is not a solution, and function \(F_{2}\) satisfies Assumption 1 (it follows from Remark 3(i)) and is BD-regular at \(x^{*}=(0,0)\). The exemplary results are shown in Tables 1 and 2. _with_  
\[g_{i}(x)=i-\sum_{j=1}^{i}\left\{\cos(x_{j}-1)+j\big{[}1-\cos(x_{j}-1)\big{]}-\sin( x_{j}-1)\right\}\,.\]  
_Obviously, if \(c_{1}=c_{2}\), then function \(F\) is differentiable. Therefore, the value \(|c_{1}-c_{2}|\) may be treated as the degree of nondifferentiability of function \(F\)[22]. Equation (1) has an infinite number of solutions \((1+2k_{1}\pi,\ldots,1+2k_{n}\pi)^{T}\), where \(k_{1},\ldots,k_{n}\in Z\) are arbitrary. First, the starting approximation was \((0,\ldots,0)\). We used Algorithm 1 to solve one series of smooth problems (with \(c_{1}=c_{2}=1\)) and three series of nonsmooth problems with various parameters \(c_{1}=-c_{2}=1,10,100\)._  
_Function \(F\) is semismooth as a composite of semismooth functions (as mentioned in Example 1), and all functions \(f_{i}\) are semismooth as piecewise smooth functions (see [13])._  
_In the smooth case, Algorithm 1 (IEM) stopped after six iterations, while Algorithm IGM generated seven successive approximations.