## 3 Convergence Analysis
### Local Convergence of the Second Kind  
In the present section, we study the convergence of the iteration (10) under the function of initial conditions \(E\colon\mathcal{D}\subset\mathbb{C}^{n}\to\mathbb{R}_{+}\) defined by (18).  
Let \(a\) and \(b\) be the numbers (14). For the needs of the main result, we define the real functions \(\gamma,\eta,\
u,\mu\colon[0,1)\to[1,\infty)\) by  
\[\gamma(t)=\left(1+\frac{a\,t}{n-1}\right)^{n-1},\quad\eta(t)=\frac{a\,t^{2}}{ 1-t}\,,\quad\
u(t)=a\,t\,\gamma(t)\quad\mbox{and}\quad\mu(t)=(\gamma(t)-1)^{2 }+\gamma(t)^{2}\,\eta(t). \tag{38}\]  
Using these functions, we define \(\beta_{\alpha}\) by  
\[\beta_{\alpha}(t)=\left\{\begin{array}{ll}\frac{|1-\alpha|\,\mu(t)+|\alpha| \,\gamma(t)\,\eta(t)}{|1-\alpha|-|\alpha|\,\gamma(t)\,(1+\eta(t))}&\mbox{for }\ Re(\alpha)<1/2,\\ \frac{|\alpha|\,\
u(t)^{2}+\eta(t)\,(1+|\alpha|\,\
u(t))}{(1- \eta(t))(1-|\alpha|\,\
u(t))}&\mbox{for }\ Re(\alpha)=1/2,\\ \frac{|1-\alpha|\,\mu(t)+|\alpha|\,\eta(t)\,(1-t)^{n-1}}{|\alpha|\,(1- \eta(t))(1-t)^{n-1}-|1-\alpha|}&\mbox{for }\ Re(\alpha)>1/2.\end{array}\right. \tag{39}\]  
The following lemma shows that \(T_{\alpha}\) is an iteration function of second kind at a point \(\xi\).  
**Lemma 6**.: _Let \(f\in\mathbb{C}[z]\) be a polynomial of degree \(n\geq 2\) that has only simple zeros in \(\mathbb{C}\), and \(\xi\in\mathbb{C}^{n}\) be a root-vector of \(f\). Then, \(T_{\alpha}\colon D\subset\mathbb{C}^{n}\to\mathbb{C}^{n}\) defined by (11) is an iteration function of second kind at \(\xi\) with control function \(\beta_{\alpha}\) defined by (39)._  
Proof.: The proof is performed the same way as the one of Lemma 5. One just has to use the inequalities  
\[|B_{i}|\leq\frac{a\,E(x)^{2}}{1-E(x)}=\eta(E(x))\quad\mbox{and}\quad|C_{i}| \leq a\gamma(E(x))E(x)=\
u(E(x))\]  
instead of (32). The first of these inequalities can be found in [24] (Lemma 6), whereas the second one follows from Lemma 1 and the known inequalities  
\[|W_{j}|\leq\gamma(E(x))|x_{j}-\xi_{j}|\quad\mbox{and}\quad\sum_{j\
eq i}\frac {x_{j}-\xi_{j}}{x_{i}-x_{j}}\leq aE(x).\]  
Now, we are ready to introduce our second main result.  
**Theorem 4**.: _Let \(f\in\mathbb{C}[z]\) be a polynomial of degree \(n\geq 2\) with only simple zeros and \(\xi\in\mathbb{C}^{n}\) be a root-vector of \(f\). Suppose that \(x^{(0)}\in\mathbb{C}^{n}\) is an initial approximation satisfying_  
\[E(x^{(0)})=\left\|\frac{x^{(0)}-\xi}{d(x^{(0)})}\right\|_{p}<1\quad\text{and} \quad\Psi_{\alpha}(E(x^{(0)}))\geq 0, \tag{40}\]  
_where the function \(\Psi_{\alpha}\colon J\to\mathbb{R}\) is defined by_  
\[\Psi_{\alpha}(t)=1-bt-\beta_{\alpha}(t)(1+bt) \tag{41}\]  
_with \(\beta_{\alpha}\) defined by (39). Then, the iteration (10) is well defined and \(Q\)-cubically convergent to \(\xi\) with the following error estimates for all \(k\geq 0\):_  
\[\|x^{(k)}-\xi\,\|\preceq\theta^{k}\lambda^{(3^{k}-1)/2}\,\|x^{(0)}-\xi\,\|\quad \text{and}\quad\|x^{(k+1)}-\xi\,\|\preceq\theta\lambda^{3^{k}}\,\|x^{(k)}-\xi\,\|\]  
_where \(\lambda=\phi(E(x^{(0)}))\), \(\theta=\psi(E(x^{(0)}))\) and \(\psi\) and \(\phi\) are defined by_  
\[\psi(t)=1-bt(1+\beta_{\alpha}(t))\quad\text{and}\quad\phi(t)=\beta_{\alpha}(t )/\psi(t). \tag{42}\]  
Proof.: The conclusions of the theorem follow directly from Lemma 6 and Theorem 2.