## 1 Introduction
They defined a preconditioned system of nonlinear equations as follows:  
\[\mathbf{\Psi}(\mathbf{z})=\mathbf{\Lambda}(\mathbf{z})\odot\mathbf{\Phi}( \mathbf{z})=\mathbf{0}, \tag{7}\]  
where \(\mathbf{\Lambda}(\mathbf{z})\
eq\mathbf{0}\). Notice that the roots of \(\mathbf{\Psi}(\mathbf{z})=\mathbf{0}\) and \(\mathbf{\Phi}(\mathbf{z})=\mathbf{0}\) are the same because \(\mathbf{\Lambda}(\mathbf{z})\
eq\mathbf{0}\) for all \(\mathbf{z}\). The first order Frechet derivative of (7) can be computed as  
\[ \begin{split}&\Psi_{i}(\mathbf{z})=\Phi_{i}(\mathbf{z})\;\Lambda_{i}( \mathbf{z})\\ &\
abla\Psi_{i}(\mathbf{z})^{T}=\Phi_{i}(\mathbf{z})\;\
abla \Lambda_{i}(\mathbf{z})^{T}+\Lambda_{i}(\mathbf{z})\;\
abla\Phi_{i}(\mathbf{z })^{T},\;\;i=1,2,\cdots,n\\ &\begin{bmatrix}\
abla\Psi_{1}(\mathbf{z})^{T}\\ \
abla\Psi_{2}(\mathbf{z})^{T}\\ \
abla\Psi_{3}(\mathbf{z})^{T}\\ \vdots\\ \
abla\Psi_{n}(\mathbf{z})^{T}\end{bmatrix}=\begin{bmatrix}\Phi_{1}(\mathbf{z })&0&\cdots&0\\ 0&\Phi_{2}(\mathbf{z})&\cdots&0\\ 0&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&\Phi_{n}(\mathbf{z})\end{bmatrix}\begin{bmatrix}\
abla\Lambda_{1} (\mathbf{z})^{T}\\ \
abla\Lambda_{2}(\mathbf{z})^{T}\\ \
abla\Lambda_{3}(\mathbf{z})^{T}\\ \vdots\\ 0&\cdots&\Lambda_{n}(\mathbf{z})\end{bmatrix}+\begin{bmatrix}\Lambda_{1}( \mathbf{z})&0&\cdots&0\\ 0&\Lambda_{2}(\mathbf{z})&\cdots&0\\ 0&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&\Lambda_{n}(\mathbf{z})\end{bmatrix}\begin{bmatrix}\
abla\Phi_{1 }(\mathbf{z})^{T}\\ \
abla\Phi_{2}(\mathbf{z})^{T}\\ \
abla\Phi_{3}(\mathbf{z})^{T}\\ \vdots\\ \
abla\Phi_{n}(\mathbf{z})^{T}\end{bmatrix}\\ \end{split}  \tag{8}\]  
From (8), the Frechet derivative of \(\mathbf{\Phi}(\mathbf{z})\odot\mathbf{\Lambda}(\mathbf{z})\) is  
\[ \begin{split}&(\mathbf{\Phi}(\mathbf{z})\odot\mathbf{\Lambda}( \mathbf{z}))^{\prime}=\operatorname{diag}(\mathbf{\Phi}(\mathbf{z}))\mathbf{ \Lambda}^{\prime}(\mathbf{z})+\operatorname{diag}(\mathbf{\Lambda}(\mathbf{z }))\mathbf{\Phi}^{\prime}(\mathbf{z})\\ &\mathbf{\Psi}^{\prime}(\mathbf{z})=\operatorname{diag}(\mathbf{ \Lambda}(\mathbf{z}))\mathbf{\Phi}^{\prime}(\mathbf{z})+\operatorname{diag}( \mathbf{\Phi}(\mathbf{z}))\mathbf{\Lambda}^{\prime}(\mathbf{z})\\ &\mathbf{\Psi}^{\prime}(\mathbf{z})=\operatorname{diag}(\mathbf{ \Lambda}(\mathbf{z}))\left(\mathbf{\Phi}^{\prime}(\mathbf{z})+\operatorname{ diag}(\mathbf{\Phi}(\mathbf{z}))\operatorname{diag}(\mathbf{\Lambda}(\mathbf{z }))^{-1}\mathbf{\Lambda}^{\prime}(\mathbf{z})\right).\end{split}  \tag{9}\]  
If we apply the Newton method to (7), we obtain  
\[ \begin{split}&\mathbf{z}_{k+1}=\mathbf{z}_{k}-\left(\mathbf{\Phi }^{\prime}(\mathbf{z})+\operatorname{diag}(\mathbf{\Phi}(\mathbf{z})) \operatorname{diag}(\mathbf{\Lambda}(\mathbf{z}))^{-1}\mathbf{\Lambda}^{ \prime}(\mathbf{z})\right)^{-1}\;\operatorname{diag}(\mathbf{\Lambda}( \mathbf{z}))^{-1}\mathbf{\Lambda}(\mathbf{z})\odot\mathbf{\Phi}(\mathbf{z}) \\ &\mathbf{z}_{k+1}=\mathbf{z}_{k}-\left(\mathbf{\Phi}^{\prime}( \mathbf{z})+\operatorname{diag}(\mathbf{\Phi}(\mathbf{z}))\operatorname{ diag}(\mathbf{\Lambda}(\mathbf{z}))^{-1}\mathbf{\Lambda}^{\prime}(\mathbf{z}) \right)^{-1}\mathbf{\Phi}(\mathbf{z}).\end{split}  \tag{10}\]  
The convergence order of (10) is two. The iterative method (6) with a general preconditioner can be written as  
\[\mathbf{z}_{k+1}=\mathbf{z}_{k}-\left(\mathbf{\Phi}^{\prime}(\mathbf{z})+ \operatorname{diag}(\mathbf{\Phi}(\mathbf{z}))\operatorname{diag}(\mathbf{ \Lambda}(\mathbf{z}))^{-1}\mathbf{\Lambda}^{\prime}(\mathbf{z})\right)^{-1} \operatorname{diag}(\mathbf{m})\mathbf{\Phi}(\mathbf{z}). \tag{11}\]  
The convergence order of (11) is also two. The modified Newton method [17, 21, 22] for solving nonlinear equations with unknown multiplicity can be developed in this way. We define a new function  
\[s(z)=\frac{\phi(z)}{\phi^{\prime}(z)}. \tag{12}\]  
The application of the Newton method to (12) gives  
\[