## 2 Iterative Methods for Solving Matrix-Vector Quadratic Equations Presented in [3]
\] \[\boldsymbol{\phi}_{6}=&\,\mathbf{e}_{0}-23/3\,\mathbf{C}_{2} \,\mathbf{e}_{0}^{2}+418/9\,\mathbf{C}_{2}^{2}\,\mathbf{e}_{0}^{3}-6448/27\, \mathbf{C}_{2}^{3}\,\mathbf{e}_{0}^{4}+89168/81\,\mathbf{C}_{2}^{4}\,\mathbf{e}_ {0}^{5}\\ &-1137712/243\,\mathbf{C}_{2}^{5}\,\mathbf{e}_{0}^{6}+4543840/243 \,\mathbf{C}_{2}^{6}\,\mathbf{e}_{0}^{7}-5747840/81\,\mathbf{C}_{2}^{7}\, \mathbf{e}_{0}^{8}+O\left(\mathbf{e}_{0}^{9}\right)\end{split}\] \[\boldsymbol{\phi}_{7}= \,\mathbf{e}_{0}-9\,\mathbf{C}_{2}\,\mathbf{e}_{0}^{2}+182/3\, \mathbf{C}_{2}^{2}\,\mathbf{e}_{0}^{3}-9236/27\We obtain expression for \(\mathbf{e}_{2}\) by using Equation (14)  
\[ \begin{split}\mathbf{e}_{2}=&\Big{(}1+\alpha_{2,1}+ \alpha_{2,2}+\alpha_{2,3}+\alpha_{2,4}+\alpha_{2,5}+\alpha_{2,6}+\alpha_{2,7} \Big{)}\mathbf{e}_{0}\\ &+\Big{(}-\alpha_{2,1}-7/3\,\alpha_{2,2}-11/3\,\alpha_{2,3}-5\, \alpha_{2,4}-19/3\,\alpha_{2,5}-23/3\,\alpha_{2,6}-9\,\alpha_{2,7}\Big{)}\, \mathbf{C}_{2}\,\mathbf{e}_{0}^{2}\\ &+\Big{(}2\,\alpha_{2,1}+22/3\,\alpha_{2,2}+130/9\,\alpha_{2,3}+7 0/3\,\alpha_{2,4}+34\,\alpha_{2,5}\,\mathbf{C}_{2}^{2}+418/9\,\alpha_{2,6}\\ &+182/3\,\alpha_{2,7}\Big{)}\,\mathbf{C}_{2}^{2}\,\mathbf{e}_{0}^ {3}+\Big{(}-4\,\alpha_{2,1}-64/3\,\alpha_{2,2}-460/9\,\alpha_{2,3}-2584/27\, \alpha_{2,4}-4252/27\,\alpha_{2,5}\\ &-6448/27\,\alpha_{2,6}-9236/27\,\alpha_{2,7}\Big{)}\,\mathbf{C}_ {2}^{3}\,\mathbf{e}_{0}^{4}+\Big{(}8\,\alpha_{2,1}+176/3\,\alpha_{2,2}+168\, \alpha_{2,3}+9712/27\,\alpha_{2,4}\\ &+53512/81\,\alpha_{2,5}+89168/81\,\alpha_{2,6}+46264/27\,\alpha _{2,7}\Big{)}\,\mathbf{C}_{2}^{4}\,\mathbf{e}_{0}^{5}+\Big{(}-16\,\alpha_{2,1} -464/3\,\alpha_{2,2}\\ &-1568/3\,\alpha_{2,3}-34208/27\,\alpha_{2,4}-208624/81\,\alpha_ {2,5}-1137712/243\,\alpha_{2,6}\\ &-637376/81\,\alpha_{2,7}\Big{)}\,\mathbf{C}_{2}^{5}\,\mathbf{e}_ {0}^{6}+\Big{(}32\,\alpha_{2,1}+1184/3\,\alpha_{2,2}+4672/3\,\alpha_{2,3}+11449 6/27\,\alpha_{2,4}\\ &+767968/81\,\alpha_{2,5}+4543840/243\,\alpha_{2,6}+24618880/729 \,\alpha_{2,7}\Big{)}\,\mathbf{C}_{2}^{6}\,\mathbf{e}_{0}^{7}+\Big{(}-64\, \alpha_{2,1}\\ &-2944/3\,\alpha_{2,2}-4480\,\alpha_{2,3}-367616/27\,\alpha_{2,4} -2697920/81\,\alpha_{2,5}-5747840/81\,\alpha_{2,6}\\ &-100011520/729\,\alpha_{2,7}\Big{)}\,\mathbf{C}_{2}^{7}\, \mathbf{e}_{0}^{8}+O\Big{(}\mathbf{e}_{0}^{9}\Big{)}\,.\end{split}  \tag{15}\]  
By equating the coefficients of powers of \(\mathbf{e}_{0}\) in Equation (15), we get system of seven equations  
\[ \begin{split} eq_{1}:=& 1+\alpha_{2,1}+\alpha_{2,2}+ \alpha_{2,3}+\alpha_{2,4}+\alpha_{2,5}+\alpha_{2,6}+\alpha_{2,7}=0\\ eq_{2}:=&-\alpha_{2,1}-7/3\,\alpha_{2,2}-11/3\, \alpha_{2,3}-5\,\alpha_{2,4}-19/3\,\alpha_{2,5}-23/3\,\alpha_{2,6}-9\,\alpha_{ 2,7}=0\\ eq_{3}:=& 2\,\alpha_{2,1}+22/3\,\alpha_{2,2}+130/9\, \alpha_{2,3}+70/3\,\alpha_{2,4}+34\,\alpha_{2,5}+418/9\,\alpha_{2,6}+182/3\, \alpha_{2,7}=0\\ eq_{4}:=&-4\,\alpha_{2,1}-64/3\,\alpha_{2,2}-460/9\, \alpha_{2,3}-2584/27\,\alpha_{2,4}-4252/27\,\alpha_{2,5}-6448/27\,\alpha_{2,6} \\ &-9236/27\,\alpha_{2,7}=0\\ eq_{5}:=& 8\,\alpha_{2,1}+176/3\,\alpha_{2,2}+168\, \alpha_{2,3}+9712/27\,\alpha_{2,4}+53512/81\,\alpha_{2,5}+89168/81\,\alpha_{2,6} \\ &+46264/27\,\alpha_{2,7}=0\\ eq_{6}:=&-16\,\alpha_{2,1}-464/3\,\alpha_{2,2}-1568/3\, \alpha_{2,3}-34208/27\,\alpha_{2,4}-208624/81\,\alpha_{2,5}\\ &-1137712/243\,\alpha_{2,6}-637376/81\,\alpha_{2,7}=0\\ eq_{7}:=& 32\,\alpha_{2,1}+1184/3\,\alpha_{2,2}+4672/3\, \alpha_{2,3}+114496/27\,\alpha_{2,4}+767968/81\,\alpha_{2,5}\\ &+4543840/243\,\alpha_{2,6}+24618880/729\,\alpha_{2,7}=0\,. \end{split}  \tag{16}\]  
The solution set of seven equations is  
\[ \begin{split} sol=\left\{\alpha_{2,1}&=-\frac{43903} {1024},\alpha_{2,2}=\frac{55767}{256},\alpha_{2,3}=-\frac{497763}{1024},\alpha_ {2,4}=\frac{37719}{64},\alpha_{2,5}=-\frac{416421}{1024},\\ &\alpha_{2,6}=\frac{38637}{256},\alpha_{2,7}=-\frac{24057}{1024} \right\}.\end{split}  \tag{17}\]  
After simplification the error equation we get  
\[\mathbf{e}_{2}=429\,\mathbf{C}_{2}^{7}\,\mathbf{e}_{0}^{8}+O\Big{(}\mathbf{e}_ {0}^{9}\Big{)}\,. \tag{18}\]  
\(\square\)\[