## 4 Convergence
\begin{split}&\mathbf{F}^{\prime}(x)\mathbf{F}(x)= \operatorname{diag}\Bigl{(}m\odot(x-\omega)^{2m-1}\odot\mathbf{H}(x)\Bigr{)} \,\Bigl{(}\mathbf{I}+\operatorname{O}(x-\omega)\Bigr{)}\\ &\mathbf{F}^{\prime}(x)^{2}-\mathbf{F}^{\prime\prime}(x)\mathbf{F} (x)=\operatorname{diag}\Bigl{(}m\odot(x-\omega)^{2m-2}\odot\mathbf{H}(x) \Bigr{)}\,\Bigl{(}\mathbf{I}+\operatorname{O}(x-\omega)\Bigr{)}\\ &\Bigl{(}\mathbf{F}^{\prime}(x)^{2}-\mathbf{F}^{\prime\prime}(x) \mathbf{F}(x)\Bigr{)}^{-1}\,\mathbf{F}^{\prime}(x)\mathbf{F}(x)=(x-\omega)+ \operatorname{O}\Bigl{(}(x-\omega)^{2}\Bigr{)}\end{split}  \tag{24}\]  
**Theorem 1**. : _Let \(\mathbf{F}:\,\mathbf{D}\subseteq\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}\) and \(\boldsymbol{\alpha}=[\alpha_{1},\alpha_{2},\alpha_{3},\cdots,\alpha_{n}]^{T}\in \mathbf{D}\) is a root of \(\mathbf{F}(x)=(x-\omega)^{m}\odot\mathbf{H}(x)=0\) with corresponding multiplicities vector \(\mathbf{m}=[m_{1},m_{2},\cdots,m_{n}]^{T}\) and \(\mathbf{H}(\omega)=[h_{1}(\omega),h_{2}(\omega),\cdots,h_{n}(\omega)]^{T}\
eq 0\) with \(h_{i}(x)\in C^{2}\,(\mathbf{D})\). Then there exists a subset \(S\subseteq\mathbf{D}\) such that, if we choose \(\mathbf{x}_{0}\in S\), the iterative method in Equation (20) has quadratic convergence in \(S\)._  
Proof. : We can write Equation (20) as  
\[\mathbf{x}_{k+1}=\mathbf{R}(\mathbf{x}_{k})=\mathbf{x}_{k}-\Bigl{(}\mathbf{F}^{ \prime}(\mathbf{x}_{k})^{2}-\mathbf{F}^{\prime\prime}(\mathbf{x}_{k})\,\mathbf{ F}(\mathbf{x}_{k})\Bigr{)}^{-1}\,\mathbf{F}^{\prime}(\mathbf{x}_{k})\,\mathbf{F}( \mathbf{x}_{k}) \tag{25}\]  
By dropping the index \(k\), we obtain  
\[\mathbf{R}(\mathbf{x})=\mathbf{x}-\Bigl{(}\mathbf{F}^{\prime}(\mathbf{x})^{2}- \mathbf{F}^{\prime\prime}(\mathbf{x})\,\mathbf{F}(\mathbf{x})\Bigr{)}^{-1}\, \mathbf{F}^{\prime}(\mathbf{x})\,\mathbf{F}(\mathbf{x}) \tag{26}\]  
By using Equations (24) and (26), the simplified expression for \(\mathbf{R}(\mathbf{x})\) is  
\[ \begin{split}&\mathbf{R}(\mathbf{x})=\mathbf{x}-(\mathbf{x}- \omega)+\operatorname{O}\Bigl{(}(\mathbf{x}-\omega)^{2}\Bigr{)}\\ &\mathbf{R}^{\prime}(\mathbf{x})=\mathbf{O}+\operatorname{O} \Bigl{(}\operatorname{diag}(\mathbf{x}-\omega)\Bigr{)}\end{split}  \tag{27}\]By substituting \(\mathbf{x}=\mathbf{\alpha}\) in Equation (27), we deduce the crucial relationships  
\[ \begin{split}\mathbf{R}(\mathbf{\alpha})=\mathbf{\alpha}\\ \mathbf{R}^{\prime}(\mathbf{\alpha})=\mathbf{O}\end{split}  \tag{28}\]  
and from Equation (28) we conclude that the iterative method in Equation (20) has at least quadratic convergence. Finally, the quadratic convergence of the method reported in Equation (19) can be proven as follows. Let \(\mathbf{e}=\mathbf{x}-\mathbf{\alpha}\) and after applying few simplifications, we can write Equation (19) in the form  
\[ \begin{split}\mathbf{M}(\mathbf{x})=\mathrm{diag}(\mathbf{G}( \mathbf{x}))^{-1}\mathbf{F}^{\prime}(\mathbf{x})^{-1}\mathbf{F}^{\prime\prime }(\mathbf{x})\left(\mathbf{F}(\mathbf{x})\odot\mathbf{G}(\mathbf{x})\right)\\ \mathbf{L}(\mathbf{x})=\mathbf{F}^{\prime}(\mathbf{x})+ \mathrm{diag}\big{(}\mathbf{F}^{\prime}(\mathbf{x})\big{)}\mathrm{diag}( \mathbf{G}(\mathbf{x}))^{-1}\mathbf{G}^{\prime}(\mathbf{x})\\ \mathbf{e}_{k+1}=\mathbf{e}_{k}-\Big{(}\mathbf{L}(\mathbf{x})- \mathbf{M}(\mathbf{x})\Big{)}^{-1}\mathbf{F}(\mathbf{x})\\ \mathbf{e}_{k+1}=\mathbf{e}_{k}-\Big{(}\mathbf{I}-\mathbf{L}( \mathbf{x})^{-1}\mathbf{M}(\mathbf{x})\Big{)}^{-1}\mathbf{L}(\mathbf{x})^{-1} \mathbf{F}(\mathbf{x})\\ \mathbf{e}_{k+1}\approx\mathbf{e}_{k}-\Big{(}\mathbf{I}+\mathbf{ L}(\mathbf{x})^{-1}\mathbf{M}(\mathbf{x})\Big{)}\mathbf{L}(\mathbf{x})^{-1} \mathbf{F}(\mathbf{x})\\ \mathbf{e}_{k+1}\approx\Big{(}\mathbf{e}_{k}-\mathbf{L}(\mathbf{x })^{-1}\mathbf{F}(\mathbf{x})\Big{)}-\mathbf{L}(\mathbf{x})^{-1}\mathbf{M}( \mathbf{x})\mathbf{L}(\mathbf{x})^{-1}\mathbf{F}(\mathbf{x})\end{split}