## 2 Some Generalizations  
The original idea of using a auxiliary function was proposed in [11]. The auxiliary function employed in Equation (3) is the exponential function. The question is, why we choose the exponential function? The answer is: it is a non-zeros function with non-zero derivative. So generalization is straightforward, we can choose any function that is non-zero everywhere in the vicinity of the root and in this way we will ensure the roots of \(\mathbf{F}(\mathbf{x})=\mathbf{0}\) are not affected by the auxiliary function multiplicity. Let \(\mathbf{G}(\mathbf{x})\) be a non-zero auxiliary function in the neighborhood of the root of the system of nonlinear equations with unknown multiplicity and define a new system of nonlinear equations associated with \(\mathbf{F}(\mathbf{x})\) as below  
\[\mathbf{U}(\mathbf{x})=\mathbf{G}(\mathbf{x})\odot\mathbf{F}(\mathbf{x})= \mathbf{0} \tag{8}\]Notice that the roots of \(\mathbf{U}(\mathbf{x})=\mathbf{0}\) and \(\mathbf{F}(\mathbf{x})=\mathbf{0}\) are the same because \(\mathbf{G}(\mathbf{x})\
eq\mathbf{0}\) for all \(\mathbf{x}\) in the neighborhood of root. The first order Frechet derivative of Equation (8) can be computed as  
\[ \begin{split}&\mathrm{U}_{\mathrm{i}}(\mathbf{x})=\mathrm{F}_{ \mathrm{i}}(\mathbf{x})\;\mathrm{G}_{\mathrm{i}}(\mathbf{x})\\ &\
abla\mathrm{U}_{\mathrm{i}}(\mathbf{x})^{\mathrm{T}}=\mathrm{F }_{\mathrm{i}}(\mathbf{x})\;\
abla\mathrm{G}_{\mathrm{i}}(\mathbf{x})^{ \mathrm{T}}+\mathrm{G}_{\mathrm{i}}(\mathbf{x})\;\
abla\mathrm{F}_{\mathrm{i}}( \mathbf{x})^{\mathrm{T}},\;\;\mathrm{i}=1,2,\cdots,\mathrm{n}\\ &\begin{bmatrix}\
abla\mathrm{U}_{1}(\mathbf{x})^{\mathrm{T}}\\ \
abla\mathrm{U}_{2}(\mathbf{x})^{\mathrm{T}}\\ \
abla\mathrm{U}_{3}(\mathbf{x})^{\mathrm{T}}\\ \vdots\\ \
abla\mathrm{U}_{\mathrm{n}}(\mathbf{x})^{\mathrm{T}}\end{bmatrix}=\begin{bmatrix} \mathrm{F}_{1}(\mathbf{x})&0&\cdots&0\\ 0&\mathrm{F}_{2}(\mathbf{x})&\cdots&0\\ 0&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&\mathrm{G}_{\mathrm{n}}(\mathbf{x})\end{bmatrix}+\begin{bmatrix} \mathrm{G}_{1}(\mathbf{x})&0&\cdots&0\\ 0&\mathrm{G}_{2}(\mathbf{x})&\cdots&0\\ 0&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&\mathrm{G}_{\mathrm{n}}(\mathbf{x})\end{bmatrix}\begin{bmatrix} \
abla\mathrm{F}_{1}(\mathbf{x})^{\mathrm{T}}\\ \
abla\mathrm{F}_{2}(\mathbf{x})^{\mathrm{T}}\\ \
abla\mathrm{F}_{3}(\mathbf{x})^{\mathrm{T}}\\ \vdots\\ \
abla\mathrm{F}_{n}(\mathbf{x})^{\mathrm{T}}\end{bmatrix}\end{split}  \tag{9}\]  
From Equation (9), the Frechet derivative of \(\mathbf{F}(\mathbf{x})\odot\mathbf{G}(\mathbf{x})\) is  
\[(\mathbf{F}(\mathbf{x})\odot\mathbf{G}(\mathbf{x}))^{\prime}=\mathrm{diag}\, (\mathbf{F}(\mathbf{x}))\,\mathbf{G}^{\prime}(\mathbf{x})+\mathrm{diag}\,( \mathbf{G}(\mathbf{x}))\,\mathbf{F}^{\prime}(\mathbf{x}) \tag{10}\]  
\[ \begin{split}\mathbf{U}^{\prime}(\mathbf{x})&=\mathrm{diag}\,( \mathbf{G}(\mathbf{x}))\;\mathbf{F}^{\prime}(\mathbf{x})+\mathrm{diag}\,( \mathbf{F}(\mathbf{x}))\;\mathbf{G}^{\prime}(\mathbf{x})\\ \mathbf{U}^{\prime}(\mathbf{x})&=\mathrm{diag}\,(\mathbf{G}( \mathbf{x}))\,\left(\mathbf{F}^{\prime}(\mathbf{x})+\mathrm{diag}\,(\mathbf{F }(\mathbf{x}))\;\mathrm{diag}\,(\mathbf{G}(\mathbf{x}))^{-1}\;\mathbf{G}^{ \prime}(\mathbf{x})\right)\end{split}  \tag{11}\]  
If we apply the Newton method to the system in Equation (8), then we obtain  
\[ \begin{split}&\mathbf{x}_{\mathrm{k+1}}=\mathbf{x}_{\mathrm{k}}- \left(\mathbf{F}^{\prime}(\mathbf{x})+\mathrm{diag}\,(\mathbf{F}(\mathbf{x}) )\;\mathrm{diag}\,(\mathbf{G}(\mathbf{x}))^{-1}\;\mathbf{G}^{\prime}(\mathbf{x })\right)^{-1}\;\mathrm{diag}\,(\mathbf{G}(\mathbf{x}))^{-1}\;\mathbf{G}( \mathbf{x})\odot\mathbf{F}(\mathbf{x})\\ &\mathbf{x}_{\mathrm{k+1}}=\mathbf{x}_{\mathrm{k}}-\left(\mathbf{F }^{\prime}(\mathbf{x})+\mathrm{diag}\,(\mathbf{F}(\mathbf{x}))\;\mathrm{diag}\,( \mathbf{G}(\mathbf{x}))^{-1}\;\mathbf{G}^{\prime}(\mathbf{x})\right)^{-1}\; \mathbf{F}(\mathbf{x})\end{split}  \tag{12}\]  
The convergence order of Equation (12) is two, under the usual regularity assumptions.