## 3 Convergence  
In the following theorem, we established the proof of quadratic convergence of (21). **Theorem 1**. : _Let \(\mathbf{\Phi}:D\subseteq\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}\) and \(\mathbf{\kappa}=[\kappa_{1},\kappa_{2},\kappa_{3},\cdots,\kappa_{n}]^{T}\in D\) is a root of \(\mathbf{\Phi}(\mathbf{z})=(\mathbf{z}-\mathbf{\kappa})^{\mathbf{m}}\odot\mathbf{P}( \mathbf{z})=\mathbf{0}\) with corresponding multiplicities vector \(\mathbf{m}=[m_{1},m_{2},\cdots,m_{n}]^{T}\) and non-zero function \(\mathbf{P}=[p_{1}(\mathbf{z}),p_{2}(\mathbf{z}),\cdots,p_{n}(\mathbf{z})]^{T}\) with \(p_{i}(\mathbf{z})(\
eq\mathbf{0})\in C^{2}\left(D\right)\). Then, there exists a subset \(E\subseteq D\) such that--if we choose \(\mathbf{z}_{0}\in E\)--the iterative method (21) has quadratic convergence in \(E\)._  
Proof. : Let \(\mathbf{\epsilon}=\mathbf{z}-\mathbf{\kappa}\) then \(\mathbf{\Phi}(\mathbf{z})=\mathbf{\epsilon}^{\mathbf{m}}\odot\mathbf{P}(\mathbf{z})\). Whenever we take vector power of a vector, it is always component-wise. So, \(\mathbf{\epsilon}^{\mathbf{m}}=\left[e_{1}^{m_{1}},e_{2}^{m_{2}},\cdots,e_{n}^{m_ {n}}\right]^{T}\). The first-order Frechet derivative of \(\mathbf{\Phi}(\mathbf{z})\) is  
\[\mathbf{\Phi}^{\prime}(\mathbf{z})=\text{diag}\Big{(}\mathbf{m}\odot\mathbf{ \epsilon}^{\mathbf{m}-\mathbf{1}}\odot\mathbf{P}(\mathbf{z})\Big{)}+\text{ diag}(\mathbf{\epsilon}^{\mathbf{m}})\mathbf{P}^{\prime}(\mathbf{z}). \tag{22}\]  
The expressions for terms in (21) are computed as follows. \[\left(\mathbf{\Omega}(\mathbf{z})\odot\mathbf{\Phi}(\mathbf{z})\right)^{\prime} =\text{diag}\Big{(}\mathbf{m}\odot\mathbf{\epsilon}^{\mathbf{m}- \mathbf{1}}\odot\mathbf{P}(\mathbf{z})\odot\mathbf{\Omega}(\mathbf{z})\Big{)}+ \text{diag}(\mathbf{\epsilon}^{\mathbf{m}}\odot\mathbf{\Omega}(\mathbf{z})) \mathbf{P}^{\prime}(\mathbf{z})+\text{diag}(\mathbf{\epsilon}^{\mathbf{m}}\odot \mathbf{P}(\mathbf{z}))\mathbf{\Omega}^{\prime}(\mathbf{z}) \tag{23}\] \[\left(\mathbf{\Lambda}(\mathbf{z})\odot\mathbf{\Phi}(\mathbf{z})\right)^{ \prime} =\text{diag}\Big{(}\mathbf{m}\odot\mathbf{\epsilon}^{\mathbf{m}- \mathbf{1}}\odot\mathbf{P}(\mathbf{z})\odot\mathbf{\Lambda}(\mathbf{z})\Big{)}+ \text{diag}(\mathbf{\epsilon}^{\mathbf{m}}\odot\mathbf{\Lambda}(\mathbf{z}))\mathbf{P} ^{\prime}(\mathbf{z})+\text{diag}(\mathbf{\epsilon}^{\mathbf{m}}\odot\mathbf{P}( \mathbf{z}))\mathbf{\Lambda}^{\prime}(\mathbf{z}) \tag{24}\]  
By using (23) and (24), we can write the product of \(\left(\mathbf{\Omega}(\mathbf{z})\odot\mathbf{\Phi}(\mathbf{z})\right)^{\prime}\) and \(\left(\mathbf{\Lambda}(\mathbf{z})\odot\mathbf{\Phi}(\mathbf{z})\right)^{\prime}\) as  
\[\left(\mathbf{\Omega}(\mathbf{z})\odot\mathbf{\Phi}(\mathbf{z})\right)^{\prime}\left( \mathbf{\Lambda}(\mathbf{z})\odot\mathbf{\Phi}(\mathbf{z})\right)^{\prime}=\Big{(} \mathbf{m}^{2}\odot\mathbf{\epsilon}^{\mathbf{2m}-\mathbf{2}}\odot\mathbf{P}( \mathbf{z})^{2}\odot\mathbf{\Omega}(\mathbf{z})\odot\mathbf{\Lambda}(\mathbf{z}) \Big{)}+O\Big{(}\text{diag}\Big{(}\mathbf{\epsilon}^{\mathbf{2m}-\mathbf{1}} \Big{)}\Big{)}. \tag{25}\]  
Next, we compute the second order Frechet derivative of \(\left(\mathbf{\Omega}(\mathbf{z})\odot\mathbf{\Phi}(\mathbf{z})\right)^{\prime}\). Let \(\mathbf{\phi}\) be a scalar vector of length \(n\)  
\[\left(\mathbf{\Omega}(\mathbf{z})\odot\mathbf{\Phi}(\mathbf{z})\right)^{\prime}\mathbf{ \phi}=\mathbf{m}\odot\mathbf{\epsilon}^{\mathbf{m}-\mathbf{1}}\odot\mathbf{P}( \mathbf{z})\odot\mathbf{\Omega}(\mathbf{z})\odot\mathbf{\phi}+\mathbf{A}(\mathbf{\phi}), \tag{26}\]  
where \(\mathbf{A}(\mathbf{\phi})=\text{diag}(\mathbf{\epsilon}^{\mathbf{m}}\odot\mathbf{\Omega }(\mathbf{z}))\mathbf{P}^{\prime}(\mathbf{z})\mathbf{\phi}+\text{diag}(\mathbf{ \epsilon}^{\mathbf{m}}\odot\mathbf{P}(\mathbf{z}))\mathbf{\Omega}^{\prime}( \mathbf{z})\mathbf{\phi}\).