## 4 Computational Aspects and Applications
### Applications to Some Real-World Problems
The equations of the motion of the masses are given as follows:_  
\[m_{s}s_{s}+k_{s}(x_{s}-x_{u})+F=0\] \[m_{u}\dot{x}_{u}-k_{s}(x_{s}-x_{u})-k_{t}(x_{r}-x_{u})-F=0, \tag{46}\]  
_where \(m_{s}\) and \(m_{u}\) are over-sprung and under-sprung masses, \(x_{s}\) and \(x_{u}\) are the displacement variables for \(m_{s}\) and \(m_{u}\), respectively, \(x_{r}\) describes disturbances from road bumps and \(k_{s}\) and \(k_{t}\) are coefficients connected to the stiffness of the suspension's system spring and the tire stiffness. The damper force \(F\) in (46) can be fitted by the following polynomial (see Barethive [34]):_  
\[f(z)=-77.14\,z^{4}+23.14\,z^{3}+342.7\,z^{2}+956.7\,z+124.5. \tag{47}\]  
_For the simultaneous approximation of all of the zeros of \(f\), we apply the above mentioned four members of the family (10) with [PERSON]'s initial approximation \(x^{(0)}\in\mathbb{C}^{n}\) defined by ([35])_  
\[x^{(0)}_{j}=-\frac{a_{1}}{n}+r\epsilon^{i\theta_{j}},\qquad\theta_{j}=\frac{ \pi}{n}\left(2j-\frac{3}{2}\right),\qquad j=1,\ldots,n, \tag{48}\]  
_where \(r=14\), \(a_{1}\) is the second coefficient and \(n\) is the degree of the corresponding polynomial._  
One can see from Table 1 that, for \(\alpha=1\) ([PERSON]'s method), the stopping criterion (45) is satisfied at the eighth iteration with an error estimation of less than \(10^{-24}\), and, at the ninth step, the roots are found with an accuracy of \(10^{-74}\). It is also seen that the obtained COC confirms the theoretical order of convergence proven in Section 3. The approximation trajectories for this example calculated for \(\alpha=1\) ([PERSON]'s method) and \(\alpha=0.766+0.484i\) are plotted in Figure 2a,b, where the blue points are the coordinates of the initial approximation and the red ones are the zeros of \(f\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & \(\mathbf{k}\) & \(\mathbf{E_{f}(x^{(k)})}\) & \(\mathbf{\tau}\) & \(\mathbf{\varepsilon_{k}}\) & \(\mathbf{\varepsilon_{k+1}}\) & \(\mathbf{r_{k}}\) \\ \hline  
In this example, we also compare the stability of the considered four methods (see e.g., [22]). For this purpose, the basins of attraction of the roots of (47) are obtained in the rectangle  
\[E_{f}(x^{(k)})<\tau=\frac{1}{(1+\sqrt{n-1})^{2}}\]0  
which contains the root-vector \(\xi=(-0.136743,-1.32692-1.43467i,-1.32692+1.43467i,\)\(3.09056)\) of \(f\). The dynamical planes, using a mesh of \(200\times 200\) points with \(k\) iterations, are obtained and depicted in \[E_{f}(x^{(k)})<\tau=\frac{1}{(1+\sqrt{n-1})^{2}}\]1 The sets of the initial approximations that generate iteration sequences to fulfill the accuracy criterion \(\max_{1\leq i\leq 4}|f(x_{i}^{(k)})|<10^{-10}\) at the preset iterations (\(k=30\) for [PERSON]'s method (\(\alpha=0\)), \(k=10\) for [PERSON]'s method (\(\alpha=1\)) and \(k=18\) for the methods obtained with \(\alpha=1/2\) and \(\alpha=0.7669+0.4847i)\) are green-colored, the initial approximations that generate sequences such that \(\max_{1\leq i\leq 4}|f(x_{i}^{(k)})|\geq 10^{10}\) at the mentioned iterations are red-colored and the rest are in blue. It can be seen from the figure that [PERSON]'s method and the random method behave better than the other two methods. It can also be seen that [PERSON]'s method behaves the worst for this polynomial. \[E_{f}(x^{(k)})<\tau=\frac{1}{(1+\sqrt{n-1})^{2}}\]2 _Cont._  
\[E_{f}(x^{(k)})<\tau=\frac{1}{(1+\sqrt{n-1})^{2}}\]3 Trajectories of approximations for Example 1. (**a**) For \(\alpha=1\) ([PERSON]’s method). (**b**) For \(\alpha=0.7669+0.4847i\). **Example 2** (Thermo-denaturation of milk proteins). : _To study the behavior of the milk polydisperse system and the change in its structure while turning into gel during enzymatic coagulation, [PERSON] [36] (Ch. 4) sequentially measured the optical density and permeability of natural skimmed fresh milk (NSFM).