## 2 Iterative Methods for Solving Matrix-Vector Quadratic Equations Presented in [3]
 \tag{9}\]  
We compute \(\boldsymbol{\phi}_{1}\) using Equations (9) and (7)  
\[ \begin{split}\boldsymbol{\phi}_{1}=\mathbf{e}_{0}-\mathbf{C}_{2} \,\mathbf{e}_{0}^{2}+2\,\mathbf{C}_{2}^{2}\,\mathbf{e}_{0}^{3}-4\,\mathbf{C}_ {2}^{3}\,\mathbf{e}_{0}^{4}+8\,\mathbf{C}_{2}^{4}\,\mathbf{e}_{0}^{5}-16\, \mathbf{C}_{2}^{5}\,\mathbf{e}_{0}^{6}+32\,\mathbf{C}_{2}^{6}\,\mathbf{e}_{0 }^{7}-64\,\mathbf{C}_{2}^{7}\,\mathbf{e}_{0}^{8}+O\left(\mathbf{e}_{0}^{9} \right).\end{split}  \tag{10}\]  
The expression for \(\mathbf{e}_{1}\) is  
\[ \begin{split}\mathbf{e}_{1}=& 1/3\left(\mathbf{e}_{0}+2\, \mathbf{C}_{2}\,\mathbf{e}_{0}^{2}-4\,\mathbf{C}_{2}^{2}\,\mathbf{e}_{0}^{3}+ 8\,\mathbf{C}_{2}^{3}\,\mathbf{e}_{0}^{4}-16\,\mathbf{C}_{2}^{4}\,\mathbf{e}_ {0}^{5}+32\,\mathbf{C}_{2}^{5}\,\mathbf{e}_{0}^{6}+64\,\mathbf{C}_{2}^{6}\, \mathbf{e}_{0}^{7}+128\,\mathbf{C}_{2}^{7}\,\mathbf{e}_{0}^{8}\right)\\ &+O\left(\mathbf{e}_{0}^{9}\right)\end{split}  \tag{11}\]  
First order Frechet derivative of \(\mathbf{F}(\mathbf{x})\) at \(\mathbf{x}_{1}\) is  
\[ \begin{split}\mathbf{F}^{\prime}(\mathbf{x}_{1})=& \,\mathbf{C}_{1}\left(\mathbf{I}+1/3\left(2\,\mathbf{C}_{2}\, \mathbf{e}_{0}+4\,\mathbf{C}_{2}^{2}\,\mathbf{e}_{0}^{2}-8\,\mathbf{C}_{2}^{3 }\,\mathbf{e}_{0}^{3}+16\,\mathbf{C}_{2}^{4}\,\mathbf{e}_{0}^{4}-32\,\mathbf{ C}_{2}^{5}\,\mathbf{e}_{0}^{5}+64\,\mathbf{C}_{2}^{6}\,\mathbf{e}_{0}^{6}\right. \\ &\left.-128\,\mathbf{C}_{2}^{7}\,\mathbf{e}_{0}^{7}+256\,\mathbf{ C}_{2}^{8}\,\mathbf{e}_{0}^{8}\right)\right)+O\left(\mathbf{e}_{0}^{9}\right) \end{split}  \tag{12}\]  
Next we compute \(\mathbf{M}=\mathbf{F}^{\prime}(\mathbf{x}_{0})^{-1}\mathbf{F}^{\prime}( \mathbf{x}_{1})\)  
\[ \begin{split}\mathbf{M}=&\,\mathbf{I}-4/3\,\mathbf{C} _{2}\,\mathbf{e}_{0}+4\,\mathbf{C}_{2}^{2}\,\mathbf{e}_{0}^{2}-32/3\,\mathbf{ C}_{2}^{3}\,\mathbf{e}_{0}^{3}+80/3\,\mathbf{C}_{2}^{4}\,\mathbf{e}_{0}^{4}-64\, \mathbf{C}_{2}^{5}\,\mathbf{e}_{0}^{5}+448/3\,\mathbf{C}_{2}^{6}\,\mathbf{e}_ {0}^{6}\\ &-1024/3\,\mathbf{C}_{2}^{7}\,\mathbf{e}_{0}^{7}+768\,\mathbf{C }_{2}^{8}\,\mathbf{e}_{0}^{8}+O\left(\mathbf{e}_{0}^{9}\right)\end{split}  \tag{13}\]  
with help of Equation (13), we obtain the expressions for \(\boldsymbol{\phi}_{i}\) for \(i\in\{2,3,4,5,6,7\}\)  
\[ \begin{split}\boldsymbol{\phi}_{2}=&\,\mathbf{e}_{0} -7/3\,\mathbf{C}_{2}\,\mathbf{e}_{0}^{2}+22/3\,\mathbf{C}_{2}^{2}\,\mathbf{e}_{0 }^{3}-64/3\,\mathbf{C}_{2}^{3}\,\mathbf{e}_{0}^{4}+176/3\,\mathbf{C}_{2}^{4} \,\mathbf{e}_{0}^{5}-464/3\,\mathbf{C}_{2}^{5}\,\mathbf{e}_{0}^{6}+1184/3\, \mathbf{C}_{2}^{6}\,\mathbf{e}_{0}^{7}\\ &-2944/3\,\mathbf{C}_{2}^{7}\,\mathbf{e}_{0}^{8}+O\left(\mathbf{e }_{0}^{9}\right)\end{split}  \tag{14}\]  
\[ \begin{split}\boldsymbol{\phi}_{3}=&\,\mathbf{e}_{0} -11/3\,\mathbf{C}_{2}\,\mathbf{e}_{0}^{2}+130/9\,\mathbf{C}_{2}^{2}\,\mathbf{e}_{0 }^{3}-460/9\,\mathbf{C}_{2}^{3}\,\mathbf{e}_{0}^{4}+168\,\mathbf{C}_{2}^{4} \,\mathbf{e}_{0}^{5}-1568/3\,\mathbf{C}_{2}^{5}\,\mathbf{e}_{0}^{6}\\ &+4672/3\,\mathbf{C}_{2}^{6}\,\mathbf{e}_{0}^{7}-4480\,\mathbf{C }_{2}^{7}\,\mathbf{e}_{0}^{8}+O\left(\mathbf{e}_{0}^{9}\right)\end{split} \] \[ \begin{split}\boldsymbol{\phi}_{4}=&\,\mathbf{e}_{0} -5\,\mathbf{C}_{2}\,\mathbf{e}_{0}^{2}+70/3\,\mathbf{C}_{2}^{2}\,\mathbf{e}_{0 }^{3}-2584/27\,\mathbf{C}_{2}^{3}\,\mathbf{e}_{0}^{4}+9712/27\,\mathbf{C}_{2}^{4} \,\mathbf{e}_{0}^{5}-34208/27\,\mathbf{C}_{2}^{5}\,\mathbf{e}_{0}^{6}\\ &+114496/27\,\mathbf{C}_{2}^{6}\,\mathbf{e}_{0}^{7}-367616/27\, \mathbf{C}_{2}^{7}\,\mathbf{e}_{0}^{8}+O\left(\mathbf{e}_{0}^{9}\right) \end{split} \] \[ \begin{split}\boldsymbol{\phi}_{5}=&\,\mathbf{e}_{0} -19/3\,\mathbf{C}_{2}\,\mathbf{e}_{0}^{2}+34\,\mathbf{C}_{2}^{2}\,\mathbf{e}_{0 }^{3}-4252/27\,\mathbf{C}_{2}^{3}\,\mathbf{e}_{0}^{4}+53512/81\,\mathbf{C}_{2}^{4} \,\mathbf{e}_{0}^{5}-208624/81\,\mathbf{C}_{2}^{5}\,\mathbf{e}_{0}^{6}\\ &+767968/81\,\mathbf{C}_{2}^{6}\,\mathbf{e}_{0}^{7}-2697920/81\, \mathbf{C}_{2}^{7}\,\mathbf{e}_{0}^{8}+O\left(\mathbf{e}_{0}^{9}\right) \end{split}