## 5 Numerical Examples  
Here, we show that Theorem 2 can be applied to some very important practical problems. Namely, we use it for a numerical proof of the \(Q\)-quadratic convergence of the modified Weierstrass method (MWM) (3) and for a numerical guarantee of the accuracy of approximation at each iteration. Also, we suggest a method for the experimental study of the dynamics of simultaneous methods and apply it to the modified Weierstrass method and the famous Weierstrass' method (WM) (1). For our proposes, we apply the above-mentioned iteration methods to two very known physics problems and to a polynomial for which [PERSON] and [PERSON] [26] have failed to find a suitable initial vector to use WM. For the sake of brevity, we shall consider only the case \(p=\infty\). Let \(f(z)=a_{0}\,z^{n}+a_{1}\,z^{n-1}+\cdots+a_{n}\) be a polynomial of degree \(n\geq 2\) with coefficients in \(\mathbb{C}\) and \(x^{(0)}\in\mathbb{C}^{n}\) be an initial guess. We utilize \(MWM\) and \(WM\) for computing all the zeros of \(f\) simultaneously. Furthermore, from Theorems 2 and 4, we obtain the following two criteria:  
1. Convergence criterion. If there exists an integer \(m\geq 0\) such that \[E_{f}(x^{(m)})<\tau_{n}=\frac{1}{(1+\sqrt{n-1})^{2}}\quad\text{ and }\quad\quad\Omega(E_{f}(x^{(m)}))\leq 2,\] (44) where \(E_{f}\) is defined by (10) and the function \(\Omega\) is defined by (16), then \(f\) has only simple zeros and the iteration (3) is well defined and \(Q\)-quadratically convergent to a root vector \(\xi\) of \(f\). 2. Accuracy criterion. If, for a preset accuracy, \(\varepsilon>0\), there exists an integer \(k\geq 0\) for which \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{\(\boldsymbol{p=1}\)} & \multicolumn{2}{c}{\(\boldsymbol{p=2}\)} & \multicolumn{2}{c}{\(\boldsymbol{p=\infty}\)} \\ \hline \(n\) & \(R\) & \(\Omega(R)\) & \(R\) & \(\Omega(R)\) & \(R\) & \(\Omega(R)\) \\ \hline
2 & 0.147476 & 1.816214 & 0.136294 & 1.809102 & 0.122449 & 1.795918 \\ \hline
3 & 0.142763 & 1.790711 & 0.115166 & 1.798972 & 0.090245 & 1.801232 \\ \hline
4 & 0.141210 & 1.782394 & 0.103661 & 1.798553 & 0.072327 & 1.813296 \\ \hline
5 & 0.140437 & 1.778270 & 0.095868 & 1.800342 & 0.060653 & 1.825738 \\ \hline
10 & 0.139155 & 1.771449 & 0.075611 & 1.813043 & 0.034149 & 1.872578 \\ \hline
15 & 0.138790 & 1.769513 & 0.065704 & 1.823715 & 0.023943 & 1.900192 \\ \hline
20 & 0.138617 & 1.768597 & 0.059349 & 1.832223 & 0.018471 & 1.918019 \\ \hline
25 & 0.138517 & 1.768063 & 0.054767 & 1.839215 & 0.015047 & 1.930447 \\ \hline
30 & 0.138451 & 1.767714 & 0.051233 & 1.845121 & 0.012699 & 1.939602 \\ \hline
50 & 0.138322 & 1.767030 & 0.042279 & 1.862230 & 0.007827 & 1.960427 \\ \hline
100 & 0.138227 & 1.766530 & 0.032167 & 1.885650 & 0.003999 & 1.978739 \\ \hline
1000 & 0.138144 & 1.766088 & 0.011831 & 1.949246 & 0.000408 & 1.997720 \\ \hline
10,000 & 0.138135 & 1.766044 & 0.003971 & 1.981548 & 0.000040 & 1.999770 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical comparison between Theorem 2 and Theorem 7. \[\xi_{f}(x^{(k)})=\left\|\frac{W_{f}(x^{(k)})}{d(x^{(k)})}\right\|_{\infty}<\tau_{ n}\quad\text{and}\quad\epsilon_{k}<\epsilon, \tag{45}\]  
where \(\epsilon_{k}=\alpha(\xi_{f}(x^{(k)}))\left\|W_{f}(x^{(k)})\right\|_{\infty}\) with \(\alpha\) defined by (16), then the root vector \(\xi\) of \(f\) is calculated with accuracy \(\varepsilon\). At the \(k\)th iteration, the guaranteed accuracy is \(\epsilon_{k}\). In any example, we calculate the smallest integer \(m\geq 0\) that satisfies the convergence criterion (44) and the smallest integer \(k\geq 0\) that satisfies the accuracy criterion (45) with accuracy \(\varepsilon=10^{-6}\). In the tables below, we give the values of \(m\), \(\Omega(E_{f}(x^{(m)}))\), \(k\), \(\xi_{f}(x^{(k)})\), \(\tau_{n}\) and \(\epsilon_{k}\) with at least six decimal places. Since the convergence criterion concerns only MWM, we use the abbreviation \(N/A\) for WM.