## 5 Numerical Examples
Furthermore, we propose the following algorithm for the experimental study of the dynamics of the simultaneous methods:  
1. Generate [PERSON]'s initial approximation \(x^{(0)}\in\mathbb{C}^{n}\) defined by [27] \[x_{j}^{(0)}=-\frac{a_{1}}{n}+re^{i\theta_{j}},\qquad\theta_{j}=\frac{\pi}{n} \left(2j-\frac{3}{2}\right),\qquad j\in I_{n},\] (46) where \(a_{1}\) and \(n\) are the second coefficient and the degree of \(f\) while \(r\) is chosen in accordance with a well-known Cauchy bound theorem (see, e.g., ([12], Theorem 11.4)). Namely, it is defined by \(r=1+\max_{i\in I_{n}}|a_{i}/a_{0}|\). 2. Take a square containing all \(n\) zeros of \(f\) and, using a mesh of \(m\times m\) points and up to \(K\) iterations, 'cover' the square by picking one by one random number from any cell and using it as a coordinate of the initial vector \(x^{(0)}\) instead of one of the coordinates of Aberth's initial vector. Thus, the basins of attraction and the _convergence areas_, i.e., for preset \(\epsilon\), the areas with number of iterations \(k\leq K\) needed to meet the criterion \[\max_{i\in I_{n}}|f(x_{i}^{(k)})|<\epsilon,\] (47) are obtained. In the following examples, we use a mesh of \(400\times 400\) points, \(K=80\) and \(\epsilon=10^{-6}\) in squares selected to contain all zeros of \(f\) and to ensure a good view on them and their basins. The areas with initial points that generate iterative sequences such that \(\max_{i\in I_{n}}|f(x_{i}^{(k)})|\in[10^{-6},10^{12}]\) are white colored and shell be called _had areas_. In other words, if the 'crawling' coordinate is in a _bad area_; then, the method behaves very badly or even diverges. The red stars depict the exact zeros of the corresponding polynomial. In the following example, we consider a polynomial with very close complex zeros. **Example 1** (Van der Waals Equation (1873)). : _A well-known equation that describes the behavior of real gases is the so-called Van der Waals equation. In the case of Oxygen, it reads:_  
\[100000x^{3}-5622.002x^{2}+1.382x-0.04403052=0. \tag{48}\]  
_This equation has the following simple zeros \(\xi_{1,2}=0.000053\pm 0.002800i\) and \(\xi_{3}=0.056113\). [PERSON]'s initial guess, with \(r=1\), is_  
\[x^{(0)}=(1874.91\ldots+0.52\ldots i,1873.08\ldots+0.52\ldots i,1874.00\ldots-1.05\ldots i).\]  
_The basins of attraction and the convergence areas are obtained by \(MWM\) and \(WM\) in the square_  
\[\{x\in\mathbb{C}\colon|Re(x)|\leq 0.15\text{ and }|Im(x)|\leq 0.15\}\]and plotted in Figures 1 and 2. Together with \(x^{(0)}\), in this example, we also use the following initial vectors:_  
\[y^{(0)} = (-0.06\ldots-0.13\ldots i,1873.08\ldots+0.52\ldots i,1874.00\ldots -1.05\ldots i);\] \[z^{(0)} = (1874.91\ldots+0.52\ldots i,0.12\ldots-0.08\ldots i,1874.00\ldots -1.05\ldots i);\] \[t^{(0)} = (1874.91\ldots+0.52\ldots i,1873.08\ldots+0.52\ldots i,-0.03\ldots +0.03\ldots i)\]  
_one coordinate of any of which is randomly taken from the'red', 'green' and 'blue' areas in Figure 1 instead of the first, second and third coordinates of \(x^{(0)}\), respectively._  
\[\xi_{f}(x^{(k)})=\left\|\frac{W_{f}(x^{(k)})}{d(x^{(k)})}\right\|_{\infty}<\tau_{ n}\quad\text{and}\quad\epsilon_{k}<\epsilon, \tag{45}\]0 Dynamical planes for MWM of Example 1. Table 2 shows that starting from \(x^{(0)}\)\(MWM\) diverges but in all other cases it converges no matter the rough choice of the initial guess. Namely, starting from \(y^{(0)}\)\(MWM\) meets the convergence criterion (44) at the eighteenth iteration and the accuracy criterion (45) at the twenty-first one, where the zeros are found with accuracy at least \(10^{-6}\). On the other hand, \(WM\) converges in any case but behaves quite better when starts from \(y^{(0)}\), \(z^{(0)}\) and \(t^{(0)}\). So, the view on the basins of attraction gives us an opportunity to correct or even to restore the convergence of any simultaneous method. The basins of attraction and the convergence areas are plotted in Figures 1 and 2, where the color scales show the number of iterations that are needed for any method to converge.