## 2 Iterative Methods for Solving Matrix-Vector Quadratic Equations Presented in [3]  
In this direction, a manuscript [3] was posted on 4 May 2015 on Researchgate in which the author provided models of three iterative methods, with their respective convergence orders, for computing the solution of matrix-vector quadratic equations. As the proposed iterative methods are valid for solving systems of nonlinear equations with quadratic nonlinearity, they are also valid for scalar quadratic equations. In the article [3] the model of iterative Method II can be written as:  
\[ \begin{cases}\mathbf{x}_{0}=\text{initial guess}\\ \mathbf{F}^{\prime}(\mathbf{x}_{0})\,\boldsymbol{\phi}_{1}=\mathbf{F}(\mathbf{x }_{0})\\ \mathbf{x}_{1}=\mathbf{x}_{0}+\alpha_{1,1}\,\boldsymbol{\phi}_{1}\\ \text{for}\,\,\,i=1,m\\ \quad\quad\mathbf{F}^{\prime}(\mathbf{x}_{0})\,\boldsymbol{\phi}_{i+1}= \mathbf{F}^{\prime}(\mathbf{x}_{1})\,\boldsymbol{\phi}_{i}\\ \text{end}\\ \mathbf{x}_{2}=\mathbf{x}_{0}+\sum\limits_{j=1}^{m}\alpha_{2,j}\,\boldsymbol{ \phi}_{j}\\ \mathbf{x}_{2}=\mathbf{x}_{0}\end{cases}  \tag{5}\]  
where \(\mathbf{F}(\mathbf{x})=\mathbf{0}\) is a quadratic equation _i.e._, \(\mathbf{F}^{\prime\prime}(\mathbf{x})=\mathbf{B}\) is a bilinear form and \(\mathbf{F}^{(s)}(\mathbf{x})=\mathbf{O}\) (zero tensor) for \(s\geq 3\) and the convergence order is \(m+2\). The scalar version of iterative method Equation (5) can be written as  
\[ \begin{cases}x_{0}=\text{initial guess}\\ f^{\prime}(x_{0})\,\phi_{1}=f(x_{0})\\ x_{1}=x_{0}+\alpha_{1,1}\,\phi_{1}\\ \text{for}\,\,\,i=1,m\\ \quad\quad f^{\prime}(x_{0})\,\phi_{i+1}=f^{\prime}(x_{1})\,\phi_{i}\\ \text{end}\\ \begin{split} x_{2}=x_{0}+\sum\limits_{j=1}^{m}\alpha_{2,j}\,\phi_{ j}\\ x_{0}=x_{2}\end{cases}  \tag{6}\]  
where \(f(x)=a\,x^{2}+b\,x+c\) with \(a\
eq 0\) and convergence order is \(m+2\). The convergence proofs of different iterative methods are established in Figures 1-4. We can see that the error equations in all cases are the same. The Figure 4 shows that the iterative method (1) is a particular case of iterative method Equation (6) for \(\alpha_{1,1}=-2/3\). Finally we provide the convergence proof of the iterative method Equation (5). **Theorem 2.1**. : _Let \(\mathbf{F}:\mathbf{D}\subseteq\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}\) be a function with all continuous Frechet derivatives, \(\mathbf{F}^{\prime\prime}(\mathbf{x})=\mathbf{B}\) and \(\mathbf{F}^{(j)}(\mathbf{x})=\mathbf{O}\) for \(j\geq 3\), where \(\mathbf{B}\) is a bilinear form and \(\mathbf{D}\) is convex open subset of \(\mathbb{R}^{n}\). If we take \(\mathbf{x}_{0}\) in the vicinity of a simple root \(\mathbf{x}^{*}\) of \(\mathbf{F}(\mathbf{x})=\mathbf{0}\) then the sequence of successive approximation generated by iterative method Equation (5) for \(m=6\) and \(\alpha_{1,1}=-2/3\) converges to \(\mathbf{x}^{*}\) with convergence order eight._  
Proof. : We denote \(\mathbf{e}_{k}=\mathbf{x}_{k}-\mathbf{x}^{*}\), \(\mathbf{C}_{1}=\mathbf{F}^{\prime}(\mathbf{x}^{*})\) and \(\mathbf{C}_{2}=\mathbf{F}^{\prime}(\mathbf{x}^{*})^{-1}\mathbf{F}^{\prime\prime }(\mathbf{x}^{*})/2\). By expanding \(\mathbf{F}^{\prime}(\mathbf{x}_{0})\) around \(\mathbf{x}^{*}\) we get  
\[\mathbf{F}(\mathbf{x}_{0})=\mathbf{C}_{1}\left(\mathbf{e}_{0}+\mathbf{C}_{2} \,\mathbf{e}_{0}^{2}\right) \tag{7}\]  
The Frechet derivative of Equation (7) with respective to \(\mathbf{e}_{0}\) is  
\[\mathbf{F}^{\prime}(\mathbf{x}_{0})=\mathbf{C}_{1}\left(\mathbf{I}+2\,\mathbf{ C}_{2}\,\mathbf{e}_{0}\right) \tag{8}\]The inverse of Equation (8) is  
\[ \begin{split}\mathbf{F}^{\prime}(\mathbf{x}_{0})^{-1}=& \left(I-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}-128\,\mathbf{C}_{2}^{7}\,\mathbf{e}_{0} ^{7}\\ &+256\,\mathbf{C}_{2}^{8}\,\mathbf{e}_{0}^{8}\Big{)}\mathbf{C}_{1 }^{-1}+O\left(\mathbf{e}_{0}^{9}\right)\end{split}