## 4 Computational Aspects and Applications
### Applications to Some Real-World Problems  
In this part, some computer implementations are performed to show the applicability of four particular methods of the family (10) to some important real-world problems, as well as to emphasize the convergence behavior of the chosen methods. Suppose that \(f\in\mathbb{C}[z]\) is a polynomial of degree \(n\geq 2\) and \(x^{(0)}\in\mathbb{C}^{n}\) is an initial approximation with pairwise distinct coordinates. In order to compute all of the zeros of \(f\) simultaneously, in the examples below, we apply four members of the family (10), namely the ones obtained for \(\alpha=0\) ([PERSON]'s method), \(\alpha=1\) ([PERSON]'s method), \(\alpha=1/2\) and \(\alpha=0.766+0.484i\). The penultimate value of \(\alpha\) was chosen because of its 'border' role in the proof of the theoretical results, whereas the last one was randomly chosen from the square  
\[\{z\in\mathbb{C}\colon\left|Re(z)\right|\leq 1\quad\text{and}\quad\left|Im(z) \right|\leq 1\}.\]  
In the performed examples, we used the following theorem of [27] as a stopping criterion. **Theorem 5** ([27] (Theorem 5.1)). : _Let \(f\in\mathbb{C}[z]\) be a polynomial of degree \(n\geq 2\) and \((x^{(k)})_{k=0}^{\infty}\) be an iterative sequence in \(\mathbb{C}^{n}\) consisting of vectors with pairwise distinct components. Then, for every \(k\geq 0\), there exists a root-vector \(\xi\in\mathbb{C}^{n}\) of \(f\) such that_  
\[E_{f}(x^{(k)})<\tau=\frac{1}{(1+\sqrt{n-1})^{2}}\]  
_implies_  
\[\|\,x^{(k)}-\xi\,\|_{\infty}\leq\varepsilon_{k}=\alpha(E_{f}(x^{(k)}))\,\|W_{ f}(x^{(k)})\|_{\infty}\]  
_where the function \(E_{f}\) is defined by \(E_{f}(x)=\left\|W(x)/d(x)\right\|_{\infty}\) and \(\alpha\colon[0,\tau]\to\mathbb{R}_{+}\) is defined by_  
\[\alpha(t)=\frac{2}{1-(n-2)t+\sqrt{(1-(n-2)t)^{2}-4t}}\,. \tag{43}\]  
In 2000, [PERSON] and [PERSON] [28] (Definition 2.2) gave a definition for the _computational order of convergence (COC)_ of an iterative sequence. A main drawback of this definition is the involvement of the exact roots of \(f\). In order to avoid this problem, [PERSON] and [PERSON] [29], and later on [PERSON] et al. [30] (Definition 2), used some _approximations_ of COC that do not involve the roots of \(f\). Following these ideas, together with Theorem 5, we made use of the following practically applicable COC (see also [25]):  
\[r_{k}=\frac{\ln(\varepsilon_{k+1}/\varepsilon_{k})}{\ln(\varepsilon_{k}/ \varepsilon_{k-1})}. \tag{44}\]In the considered examples, we calculated the smallest \(k\) that satisfies the following stopping criterion:  
\[E_{f}(x^{(k)})<\tau\quad\text{and}\quad\varepsilon_{k}<10^{-10} \tag{45}\]  
In the tables below, for any of the tested polynomials, we exhibit the values of \(k\), \(E_{f}(x^{(k)})\), \(\tau\), \(\varepsilon_{k}\), \(\varepsilon_{k+1}\) and \(r_{k}\) with at least six decimal digits. **Example 1** (Quarter car suspension model). : _The shock absorber is an element of the suspension system that is also used to control the transient behavior of the vehicle mass and the suspension mass (see [PERSON] [31], [PERSON] [32]). It is one of the most complex parts of the suspension system due to its nonlinear behavior. In fact, the damping force of the dampers is described by an asymmetric nonlinear hysteresis loop ([PERSON] [33]). For the purposes of the present example, the vehicle characteristics are represented by a two-degrees-of-freedom quarter-car model to study the damper effect based on linear and nonlinear damping characteristics. Since simpler models such as linear and piecewise linear models are ineffective in describing the behavior of the damper, it is essential to develop a damper model that describes the nonlinear hysteresis characteristics of the damper such as a polynomial model.