## 4 Comparison with Previous Results  
As it was mentioned, in [9] [PERSON] proved the following local and semilocal convergence theorems about the modified Weierstrass method (3):  
**Theorem 5** ([9], Theorem 3). : _Let \(f\in\mathbb{C}[z]\) be a monic polynomial of degree \(n\geq 2\) with only simple roots and such that \(f(0)\
eq 0\). Let also \(\xi\in\mathbb{C}^{n}\) be a root vector of \(f\) and \(x^{(0)}\in\mathbb{C}^{n}\) be an initial guess with distinct nonzero components satisfying_  
\[\widetilde{E}(x^{(0)})=\frac{\|x^{(0)}-\xi\|_{\infty}}{\widetilde{\Delta}(x^{ (0)})}\leq\mu=\frac{\sqrt[n+1]{2}-1}{4\sqrt[n+1]{2}-3}\,, \tag{38}\]  
_where the function \(\widetilde{\Delta}\) is defined by (9). Then, the modified Weierstrass iteration (3) is well defined and converges quadratically to the root vector \(\xi\) of \(f\) with error estimates_  
\[\|x^{(k+1)}-\xi\|_{\infty}\leq\widetilde{\lambda}^{2k}\|x^{(k)}-\xi\|_{\infty }\quad\text{and}\quad\|x^{(k)}-\xi\|_{\infty}\leq\widetilde{\lambda}^{2k-1}\| x^{(0)}-\xi\|_{\infty}, \tag{39}\]  
_where \(\widetilde{\lambda}=\widetilde{E}(x^{(0)})\,/\,\widetilde{\mu}\) with \(\widetilde{\mu}=\left(\sqrt[n+1]{2}-1\right)/\left(2\sqrt[n+1]{2}-1\right)\)._  
**Theorem 6** ([9], Theorem 4). : _Let \(f\in\mathbb{C}[z]\) be a monic polynomial of degree \(n\geq 2\) and let there exist a vector \(x^{(0)}\in\mathbb{C}^{n}\) with distinct nonzero components such that_\[\frac{\|W_{f}(x^{(0)})\|_{\infty}}{\widetilde{\Delta}(x^{(0)})}\leq\frac{\mu(1- \mu)}{1+(n-2)\mu}\,, \tag{40}\]  
_where \(\widetilde{\Delta}\) is defined by (9) and \(\mu\) is defined in (38). Then \(f\) has only simple zeros and the iteration (3) is well defined and converges quadratically to the root vector \(\xi\) of \(f\) with error estimates (39)._  
**Remark 1**. : _Theorem 1 generalizes Theorem 5 since the initial condition (38) implies (12). Indeed, taking \(p=\infty\), we have \(a=n-1\) and \(b=2\), and so_  
\[\Phi(\mu)=(1+4\mu)\,(1+\mu)^{n-1}=\left(1+4\,\frac{\sqrt[n+1]{2}-1}{4\,\sqrt[n +1]{2}-3}\right)\left(1+\frac{\sqrt[n+1]{2}-1}{4\,\sqrt[n+1]{2}-3}\right)^{n-1} \leq 2\]  
_which follows from the fact that the left hand side of the inequality increases on \(n\geq 2\) and tends to \(2\) as \(n\to\infty\)._  
_Theorem 1 improves and complements Theorem 5 with the guaranteed \(Q\)-quadratic convergence of the method (3) and the improved error estimates._  
_Theorem 1 generalizes Theorem 6 because it is a direct consequence of Theorem 5._  
**Remark 2**. : _Theorem 2 improves and complements Theorem 6 with the guaranteed \(Q\)-quadratic convergence of the method (3)._  
Very recently, the following semilocal convergence theorem about the modified Weierstrass method (3) has been proven in [11]:  
**Theorem 7** ([11], Theorem 9). : _Let \(f\in\mathbb{C}[z]\) be a polynomial of degree \(n\geq 2\) and \(x^{(0)}\in\mathbb{C}^{n}\) be an initial guess with pairwise distinct nonzero components satisfying_  
\[E_{f}(x^{(0)})=\left\|\frac{W_{f}(x^{(0)})}{\Delta(x^{(0)})}\right\|_{p}<R= \frac{\
u(1+(b-1)\,\
u)}{(1+b\
u)(1+(a+b-1)\,\
u)}\,, \tag{41}\]  
_for some \(1\leq p\leq\infty\), where \(W_{f}\) is Weierstrass' correction defined by (2) and \(\
u\) is defined by_  
\[\
u=\frac{\sqrt[n-1]{l}-1}{b\,\left(\sqrt[n-1]{l}-1\right)+a/(n-1)}\,, \tag{42}\]  
_where \(a,b\) are defined by (6) and \(l\) is defined by_  
\[l=\frac{3b-a-1+\sqrt{(3b-a-1)^{2}+8(b+1)(a+1-b)}}{2(b+1)}\,. \tag{43}\]  
_Then \(f\) has only simple roots and the iteration (3) is well defined and converges quadratically to a root vector \(\xi\) of \(f\)_  
**Remark 3**. : _Theorem 2 enlarges the convergence domain of Theorem 7 since every vector \(x^{(0)}\) that satisfies the initial condition (41) satisfies (17) but not vice versa.