## 5 Numerical Testing  
It is important to test the computational convergence order (COC) of the iterative methods discussed so far. In all our simulations, we adopt the following definition of COC  
\[ \begin{split}\mathrm{COC}=\frac{\log\big{(}\mid\mathbf{F}( \mathbf{x}_{k+1})\mid\mid_{\infty}/\mid\mathbf{F}(\mathbf{x}_{k})\mid\mid_{ \infty}\big{)}}{\log\big{(}\mid\mathbf{F}(\mathbf{x}_{k})\mid\mid_{\infty}/ \mid\mathbf{F}(\mathbf{x}_{k-1})\mid\mid_{\infty}\big{)}}\,\,\,\,\mathrm{or}\, \,\,\frac{\log\big{(}\mid\mathbf{x}_{k+1}-\mathbf{\alpha}\mid\mid_{\infty}/\mid \mathbf{x}_{k-1}-\mathbf{\alpha}\mid\mid_{\infty}\big{)}}{\log\big{(}\mid\mathbf{ x}_{k}-\mathbf{\alpha}\mid\mid_{\infty}/\mid\mathbf{x}_{k-1}-\mathbf{\alpha}\mid\mid_{\infty} \big{)}}\end{split}  \tag{31}\]  
Next, we explain how to compute the term \(\mathbf{F}^{\prime\prime}(\mathbf{x})(\mathbf{G}(\mathbf{x})\odot\mathbf{F}( \mathbf{x}))\). Suppose, we have a system of three nonlinear equations  
\[ \begin{split}\mathrm{Problem}\,1=\begin{cases}&\mathrm{F}_{1}( \mathbf{x})=(\mathrm{x}_{1}-1)^{4}\exp(\mathrm{x}_{2})=0\\ &\mathrm{F}_{2}(\mathbf{x})=(\mathrm{x}_{2}-2)^{5}\,(\mathrm{x}_{1} \,\mathrm{x}_{2}-1)=0\\ &\mathrm{F}_{3}(\mathbf{x})=(\mathrm{x}_{3}+4)^{6}=0\end{cases}\end{split}  \tag{32}\]The Jacobian \(\mathbf{F}^{\prime}(\mathbf{x})\) of Equation (32) is  
\[\mathbf{F}^{\prime}(\mathbf{x})= \begin{bmatrix}4\exp(x_{2})\,(x_{1}-1)^{3}&\exp(x _{2})\,(x_{1}-1)^{4}&0\\ &x_{2}\,(x_{2}-2)^{5}&x_{1}\,(x_{2}-2)^{5}+5\,(x_{1}\,x_{2}-1)\,(x_{2}-2)^{4}&0 \\ &0&0&6\,(x_{3}+4)^{5}\end{bmatrix}  \tag{33}\]  
Now we take a constant vector \(\mathbf{w}=[w_{1},w_{2},w_{3}]^{\mathrm{T}}\). By multiplying \(\mathbf{F}^{\prime}(\mathbf{x})\) and \(\mathbf{w}\), we get  
\[\mathbf{F}^{\prime}(\mathbf{x})\,\mathbf{w}= \begin{bmatrix}4\,w_{1}\exp(x_{2} )\,(x_{1}-1)^{3}+w_{2}\exp(x_{2})\,(x_{1}-1)^{4}\\ w_{2}\,(x_{1}\,(x_{2}-2)^{5}+5\,(x_{1}\,x_{2}-1)\,(x_{2}-2)^{4})+w_{1}\,x_{2}\,( x_{2}-2)^{5}\\ &6\,w_{3}\,(x_{3}+4)^{5}\end{bmatrix}  \tag{34}\]  
Now again, we take the Jacobian of \(\mathbf{F}^{\prime}(\mathbf{x})\,\mathbf{w}\)  
\[\mathbf{F}^{\prime\prime}(\mathbf{x})\,\mathbf{w}=\left(\mathbf{F}^{\prime}( \mathbf{x})\,\mathbf{w}\right)^{\prime}= \tag{35}\]  
By replacing \(\mathbf{w}\) with \(\mathbf{F}(\mathbf{x})\odot\mathbf{G}(\mathbf{x})=[f_{1},f_{2},f_{3}]^{\mathrm{T}}\) in Equation (35), we obtain  
\[\mathbf{F}^{\prime\prime}(\mathbf{x})\left(\mathbf{F}(\mathbf{x})\odot \mathbf{G}(\mathbf{x})\right)= \tag{36}\] \[ \begin{bmatrix}12\,f_{i}\,\exp(x_{3})\,(x_{1}-1)^{2}+4\,f_{2}\, \exp(x_{3})\,(x_{1}-1)^{2}&4\,f_{1}\,\exp(x_{3})\,(x_{1}-1)^{4}&0\\ &f_{2}\,(5\,x_{2}\,(x_{2}-2)^{4}+(x_{2}-2)^{5})&f_{1}\,(x_{2}-2)^{5}+f_{2}\,(1 0\,x_{1}\,(x_{2}-2)^{4}+20\,(x_{1}\,x_{2}-1)\,(x_{2}-2)^{5})+5\,f_{1}\,x_{2}\, (x_{2}-2)^{4}&0\\ &0&0&30\,f_{5}\,(x_{3}+4)^{4}\end{bmatrix} \]  
For a large system of nonlinear equations, it is not practical that we compute symbolically second order Frechet derivatives. There is a way to approximate the second order Frechet derivative numerically by using the history of iterations [13]. However, then, it is hard to keep the quadratic convergence of the iterative method. On the other hand, the methods that do not use the second order Frechet derivative, require the knowledge of multiplicities of roots. Again practically, it is hard to have the knowledge of root multiplicities for a general system of nonlinear equations. For single nonlinear equations, many authors propose some recipes to approximate the multiplicity of roots iteratively. Finally, we have two kinds of iterative method with quadratic convergence and with the inclusion of auxiliary function \(\mathbf{G}(\mathbf{x})\).