## 1 Introduction  
We are interested in computing a real root \(\alpha\) of a function \(\mathbf{F}:\mathrm{D}\subseteq\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}\), that is a vector \(\alpha\) belonging to \(\mathrm{D}\) such that \(\mathbf{F}(\alpha)=0\). The most classical iterative method for solving a system of nonlinear equations, especially in the case of simple zeros, is the Newton method that offers quadratic convergence [1; 2], under certain local regularity conditions. Many researchers have proposed further iterative methods for solving a system of nonlinear equations, that are efficient and with a high order of convergence [3; 4; 5; 6; 7; 8; 9]. However, as far as we are concerned with roots having multiplicity \(\mathrm{m}\) (\(\geq 2\)), the classical Newton method deteriorates and its convergence rate is linear with convergence factor deteriorating when the multiplicity is higher: modified variants of the Newton method can offer a good alternative, by recovering quadratic convergence, under the hypothesis that the multiplicity is known in advance. The most classical modified Newton, for scalar nonlinear equations, can be written as  
\[ \begin{cases}\mathrm{x}_{0}=\mathrm{initial\ guess}\\ \mathrm{x}_{k+1}=\mathrm{x}_{k}-\mathrm{m}\frac{\mathrm{f}(\mathrm{x}_{k})}{ \mathrm{f}(\mathrm{x}_{k})},\ \ \mathrm{k}=0,1,\cdots\end{cases}  \tag{1}\]The straightforward generalization of Equation (1) is presented in [10]  
\[ \begin{cases}\mathbf{x}_{0}=\text{initial guess},\\ \mathbf{x}_{k+1}=\mathbf{x}_{k}-\mathbf{F}^{\prime}(\mathbf{x}_{k})^{-1}\text{ diag}(\mathbf{m})\,\mathbf{F}(\mathbf{x}_{k}),\ \ \mathbf{k}=0,1,\cdots\end{cases}  \tag{2}\]  
where \(\mathbf{m}=[\mathbf{m}_{1},\mathbf{m}_{2},\cdots,\mathbf{m}_{n}]^{\text{T}}\) is a vector of multiplicities for system of nonlinear equations \(\mathbf{F}(\mathbf{x})=\mathbf{0}\) and \(\text{diag}(\cdot)\) represents a diagonal matrix that keeps the input vector at its main diagonal. The proof of quadratic convergence of Equation (2) is established in [10]. [PERSON] [11] proposed a variant of [PERSON] method with the help of an auxiliary function. To complete the reference, we give new details of developments proposed by [PERSON]. Suppose we have a system of nonlinear equations \(\mathbf{F}(\mathbf{x})=\mathbf{0}\) and we define a new system of nonlinear equation that have the same root  
\[\mathbf{U}(\mathbf{x})=\mathrm{e}^{\mathbf{v}\odot\mathbf{x}}\odot\mathbf{F}( \mathbf{x})=\mathbf{0} \tag{3}\]  
where \(\odot\) is the component-wise multiplication of two vectors and \(\mathbf{v}=[\mathbf{v}_{1},\mathbf{v}_{2},\cdots,\mathbf{v}_{n}]^{\text{T}}\). The Frechet derivative of \(\mathbf{U}(\mathbf{x})\) is  
\[ \begin{split}\mathbf{U}^{\prime}(\mathbf{x})&=\text{ diag}\left(\mathrm{e}^{\mathbf{v}\odot\mathbf{x}}\right)\mathbf{F}^{\prime}( \mathbf{x})+\text{diag}\left(\mathbf{F}(\mathbf{x})\right)\mathbf{v}\odot \mathrm{e}^{\mathbf{v}\odot\mathbf{x}}\\ \mathbf{U}^{\prime}(\mathbf{x})&=\text{diag}\left( \mathrm{e}^{\mathbf{v}\odot\mathbf{x}}\right)\left(\mathbf{F}^{\prime}( \mathbf{x})+\text{diag}\left(\mathbf{v}\odot\mathbf{F}(\mathbf{x})\right) \right)\end{split}  \tag{4}\]  
The application of [PERSON] method for Equation (3) is  
\[