## 3 Proposed Method  
For the purpose of motivation, we present some developments for single nonlinear equations and subsequently we establish results for the multi-dimensional case. Recently [PERSON] _et al._[PERSON] _et al._ (2015) constructed a family of methods for solving nonlinear equations with unknown multiplicities and this represents a crucial improvement wit respect to procedures requiring this information. What they have established is the following. Let g(x) be a non-zero function and let us define a new function  
\[\mathrm{q}(\mathrm{x})=\frac{\mathrm{f}(\mathrm{x})\,\mathrm{g}(\mathrm{x})}{ \mathrm{f}^{\prime}(\mathrm{x})} \tag{14}\]The application of a classical Newton method to the equation \(q(x)=0\) leads to the iteration  
\[x_{k+1} =x_{k}-\frac{q(x_{k})}{q^{\prime}(x_{k})} \tag{15}\] \[x_{k+1} =x_{k}-\frac{f^{\prime}(x_{k})\,f(x_{k})\,g(x_{k})}{f^{\prime}(x_{ k})(f(x_{k})\,g(x_{k}))^{\prime}-\frac{f^{\prime\prime}(x_{k})\,f(x_{k})\,g(x_{k})}{ }}\]  
The order of convergence of Equation (15) is two, under suitable regularity assumptions both on \(f(\cdot)\) and \(g(\cdot)\). We are interested in developing a possible multidimensional version of Equation (15). Let \(\mathbf{F}(x)=0\) be a system of nonlinear equations and having a root of unknown multiplicities. With the help of a non-zero auxiliary function \(\mathbf{G}(x)\), we define a new function \(\mathbf{Q}(x)\).  
\[\mathbf{Q}(x)=\mathbf{F}^{\prime}(x)^{-1}\,\left(\mathbf{G}(x)\odot\mathbf{F} (x)\right)=0 \tag{16}\]  
The first order Frechet derivative of Equation (16) can be written as  
\[\mathbf{Q}^{\prime}(x)=\left(\mathbf{F}^{\prime}(x)^{-1}\right)^{2}\left( \mathbf{F}^{\prime}(x)\,\left(\mathbf{F}(x)\odot\mathbf{G}(x)\right)^{\prime }-\mathbf{F}^{\prime}(x)\mathbf{F}^{\prime\prime}(x)\mathbf{F}^{\prime}(x)^{ -1}\,\left(\mathbf{F}(x)\odot\mathbf{G}(x)\right)\right) \tag{17}\]  
Further simplification of \(\mathbf{Q}^{\prime}(x)^{-1}\,\mathbf{Q}(x)\) gives  
\[\mathbf{Q}^{\prime}(x)^{-1}\,\mathbf{Q}(x)=\left(\mathbf{F}^{\prime}(x)\, \left(\mathbf{F}(x)\odot\mathbf{G}(x)\right)^{\prime}-\,\frac{\mathbf{F}^{ \prime}(x)\mathbf{F}^{\prime\prime}(x)\mathbf{F}^{\prime}(x)^{-1}\,\left( \mathbf{F}(x)\odot\mathbf{G}(x)\right)}{}\,\right)^{-1}\,\mathbf{F}^{\prime}( x)\,\left(\mathbf{F}(x)\odot\mathbf{G}(x)\right) \tag{18}\]  
When comparing the underlined expressions in Equations (15) and (18), we clearly see that it is not possible to simplify the expression \(\mathbf{F}^{\prime}(x)\mathbf{F}^{\prime\prime}(x)\mathbf{F}^{\prime}(x)^{-1}\, \left(\mathbf{F}(x)\odot\mathbf{G}(x)\right)\), simply because \(\mathbf{F}^{\prime}(x)\) and \(\mathbf{F}^{\prime\prime}(x)\) do not commute in general. Clearly in the scalar case the elimination of \(f^{\prime}(x)\) is possible due to commutativity.  
Our idea amounts in artificially eliminating \(\mathbf{F}^{\prime}(x)\) and \(\mathbf{F}^{\prime}(x)^{-1}\) from the expression \(\mathbf{F}(x)^{\prime}\mathbf{F}^{\prime\prime}(x)\mathbf{F}^{\prime}(x)^{-1}\, \left(\mathbf{F}(x)\odot\mathbf{G}(x)\right)\): in this way we obtain a new iterative method for solving system of nonlinear equations with unknown multiplicities  
\[\mathbf{x}_{k+1}=\mathbf{x}_{k}-\left(\mathbf{F}^{\prime}(x_{k})\,\left( \mathbf{F}(x_{k})\odot\mathbf{G}(x_{k})\right)^{\prime}-\mathbf{F}^{\prime \prime}(x_{k})\,\left(\mathbf{F}(x_{k})\odot\mathbf{G}(x_{k})\right)\right)^ {-1}\,\mathbf{F}^{\prime}(x_{k})\,\left(\mathbf{F}(x_{k})\odot\mathbf{G}(x_{ k})\right) \tag{19}\]  
We clearly state that our proposed scheme _i.e._, the iterative method in Equation (19) is not the application of [PERSON] method to Equation (16). However, our procedure is simpler and preserves the same quadratic convergence as the [PERSON] method, under the very same regularity assumptions. In the next section, we will establish the proof of quadratic convergence for Equation (19).