## 3 Experiments and Validation
### Design of Experiments
\[\alpha=\frac{2l}{n_{\alpha}} n_{\alpha}=\left\lceil\frac{2l}{\alpha}\right\rceil \tag{6}\]  
Taking \(\alpha\) and \(2l\) as known, Equation (6) can be used to compute the required increment number (\(n_{\alpha}\)), which ensures that the experiments are performed under the same conditions with respect to the resolution of the interval. From this, we initially obtain a regular grid on which the functions are to be evaluated. However, it is known from practice that, for instance, when recording measuring points, these values to obtain a regular grid cannot be set exactly. Instead, an error is produced, which leads to a distortion of the regular grid and results in unstructured data points. We additionally consider such an effect and assume a percentage deviation \(\varepsilon\) from the ideal value (\(\varepsilon=0\) just corresponds to the regular grid). This describes a type of tolerance band in which the values can deviate. With the increase in \(\varepsilon\), the grid is distorted, which is qualitatively shown in Figure 9.  
In the course of numerical experiments, we mainly want to focus on the following aspects for both the contour() and RooTri() functions:  
* Quality of the results;
* Number of computed intersection points;
* Time consumption;
* Required memory. The quality \(\Delta f_{i}(\alpha,\varepsilon),i\in\{1,\ldots,4\}\) refers to the function value that is obtained by reinserting the calculated roots into the respective test function. The smaller the deviation from \(d\) is, the more accurately the algorithm solves the problem. Considering a computed intersection point \(\mathbf{s}\), the above is to be calculated as follows:  
\[\Delta f_{i}(\alpha,\varepsilon)=\left|\frac{f_{i}(\mathbf{s})}{d}-1\right| \cdot 100\%. \tag{7}\]  
Furthermore, we want to evaluate the number of intersection points, \(n_{\mathrm{r}}\), computed by the RooTri() function and \(n_{\mathrm{c}}\) of the contour() function. As a rule, the contour lines can be constructed more effectively with a greater number of intersection points. This is especially the situation as soon as the isolines are close together. In case of large data volumes, both required computing time \(t_{\mathrm{r}},t_{\mathrm{c}}\) and required memory \(q_{\mathrm{r}},q_{\mathrm{c}}\) are of significant importance. For this reason, these parameters are also to be evaluated. \begin{table}
\begin{tabular}{l l} \hline \hline
**Name** & **Equation** \\ \hline Parabola function & \(f_{1}(x,y)=x^{2}+y^{2}-10\) \\ Peaks function & \(f_{2}(x,y)=3(1-x)^{2}\mathrm{e}^{-x^{2}-(y+1)^{2}}-10(\frac{x}{5}-x^{3}-y^{5}) \mathrm{e}^{-x^{2}-y^{2}}-\frac{1}{5}\mathrm{e}^{-(x+1)^{2}-y^{2}}\) \\ Rosenbrock function & \(f_{3}(x,y)=(1-x)^{2}+100(y-x^{2})^{2}-50\) \\ Himmelbau’s function & \(f_{4}(x,y)=(x^{2}+y-11)^{2}+(x+y^{2}-7)^{2}-50\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mathematical formulation of the considered test functions. Figure 5: Parabola function. Figure 6: Peaks function. Figure 7: Rosenbrock function. Based on Figure 10, we note that with finer resolution of the grid (i.e., with smaller \(\alpha\)), the arithmetic mean of the percentage deviation from the nominal value decreases when using the RooTri() function. The same applies to the evolution of the standard deviation. For example, taking into account a regular grid (\(\varepsilon=0\)) with \(\alpha=0.025\), we obtain an average percent deviation from the nominal value of \(2.49\%\pm 3.46\%\). It can be seen that the percentage deviation converges to zero with smaller \(\alpha\). This basic tendency remains the same with the increase in \(\varepsilon\). In contrast, we observe an opposite course when applying the contour() function. Paradoxically, the percentage deviation from the nominal value increases with finer resolution of the grid. Investigations show that the function produces outliers that deviate from the nominal value to a high degree. Thus, the standard deviation also increases with the decrease in \(\alpha\). Again, the tendency remains the same with the variation in \(\varepsilon\).