## 1 Introduction  
The design of an iterative method for solving nonlinear equations and systems of nonlinear equations is an active area of research. Many researchers have proposed iterative methods for solving nonlinear and systems of nonlinear equations for finding simple zeros or zeros with multiplicity greater than one [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]. The classical iterative method for solving nonlinear and systems of nonlinear equations to find simple zeros is the Newton method, which offers quadratic convergence [16; 17] under certain conditions. When we are talking about the iterative method for solving nonlinear equations or systems of nonlinear equations to find zeros with multiplicities greater than one, the classical Newtonmethod requires a modification. The modified Newton method for finding zeros with multiplicity greater than one for nonlinear equations can be written as  
\[ \begin{cases}z_{0}=\text{initial guess}\\ z_{k+1}=z_{k}-m\,\frac{\phi(z_{k})}{\phi^{\prime}(z_{k})},\,\,\,k=0,1,\cdots, \end{cases}  \tag{1}\]  
where \(\phi(z_{k})=0\) is the nonlinear equation. [PERSON] et al. (2018) proposed the multidimensional version of of (1) as  
\[ \begin{cases}\mathbf{z}_{0}=\text{initial guess}\\ \mathbf{z}_{k+1}=\mathbf{z}_{k}-\mathbf{\Phi}^{\prime}(\mathbf{z}_{k})^{-1} \,\text{diag}(\mathbf{m})\,\mathbf{\Phi}(\mathbf{z}_{k}),\,\,\,k=0,1,\cdots, \end{cases}  \tag{2}\]  
where \(\mathbf{m}=[m_{1},m_{2},\cdots,m_{n}]^{T}\) is a vector of multiplicities for a system of nonlinear equations \(\mathbf{\Phi}(\mathbf{z})=\mathbf{0}\), and \(\text{diag}(\cdot)\) represents a diagonal matrix that keeps the vector at its main diagonal. The proof of the quadratic convergence of (2) is provided in [PERSON] et al. (2018). [PERSON] (2019) proposed a variant of the [PERSON] method with the help of an auxiliary or a preconditioner exponential function. Suppose we have a system of nonlinear equations \(\mathbf{\Phi}(\mathbf{z})=\mathbf{0}\), and we define a new system of nonlinear equations with a nonlinear preconditioner function that has the same root  
\[\mathbf{\Psi}(\mathbf{z})=e^{\mathbf{v}\odot\mathbf{z}}\odot\mathbf{\Phi}( \mathbf{z})=\mathbf{0}, \tag{3}\]  
where \(\odot\) is the element-wise multiplication of two vectors. The application of the [PERSON] method for (3) is  
\[ \begin{split}\mathbf{z}_{k+1}&=\mathbf{z}_{k}- \mathbf{\Psi}^{\prime}(\mathbf{z}_{k})^{-1}\mathbf{\Psi}(\mathbf{z}_{k})\\ \mathbf{z}_{k+1}&=\mathbf{z}_{k}-\left(\text{diag} \big{(}e^{\mathbf{v}\odot\mathbf{z}}\big{)}\left(\mathbf{\Phi}^{\prime}( \mathbf{z})+\text{diag}(\mathbf{v}\odot\mathbf{\Phi}(\mathbf{z}))\right) \right)^{-1}\,e^{\mathbf{v}\odot\mathbf{z}}\odot\mathbf{\Phi}(\mathbf{z})\\ \mathbf{z}_{k+1}&=\mathbf{z}_{k}-\left(\mathbf{\Phi}^{ \prime}(\mathbf{z})+\text{diag}(\mathbf{v}\odot\mathbf{\Phi}(\mathbf{z})) \right)^{-1}\,\text{diag}\big{(}e^{\mathbf{v}\odot\mathbf{z}}\big{)}^{-1}e^{ \mathbf{v}\odot\mathbf{z}}\odot\mathbf{\Phi}(\mathbf{z})\\ \mathbf{z}_{k+1}&=\mathbf{z}_{k}-\left(\mathbf{\Phi}^{ \prime}(\mathbf{z})+\text{diag}(\mathbf{v}\odot\mathbf{\Phi}(\mathbf{z})) \right)^{-1}\,\mathbf{\Phi}(\mathbf{z}).\end{split}  \tag{4}\]  
The rate of convergence of (4) is quadratic. A modification [PERSON] et al. (2018) in (1) is proposed by using a exponential preconditioner  
\[\mathbf{\Psi}(\mathbf{z})=e^{\mathbf{v}\odot\mathbf{z}}\odot\mathbf{\Phi}( \mathbf{z})^{1/\mathbf{m}}=\mathbf{0}, \tag{5}\]  
where \(1/\mathbf{m}=[1/m_{1},1/m_{2},\cdots,1/m_{n}]^{T}\) and power of \(\mathbf{\Phi}(\mathbf{z})\) is component-wise. The application of the Newton method to (5) gives  
\[\mathbf{z}_{k+1}=\mathbf{z}_{k}-\left(\mathbf{\Phi}^{\prime}(\mathbf{z})+ \text{diag}(\mathbf{v}\odot\mathbf{\Phi}(\mathbf{z}))\right)^{-1}\,\text{diag }(\mathbf{m})\,\mathbf{\Phi}(\mathbf{z}). \tag{6}\]  
The original idea of a nonlinear preconditioner function was proposed in [PERSON] et al. (2019). [PERSON] et al. (2019) proposed a [PERSON] method with a general preconditioner.