# On the Kung-Traub Conjecture for Iterative Methods for Solving Quadratic Equations
## 5 Numerical Experiments
### Systems of Quadratic Equations  
For our numerical experiments in this section, the approximate solutions are calculated correct to 1000 digits by using variable precision arithmetic in MATLAB. We use the following stopping criterion for the numerical scheme:  
\[\|\mathbf{x}^{(k+1)}-\mathbf{x}^{(k)}\|_{2}<1e-50 \tag{23}\]  
For a system of equations, we used the approximated computational order of convergence \(p_{c}\) given by (see [13])  
\[p_{c}\approx\frac{\log\left(\|\mathbf{x}^{(k+1)}-\mathbf{x}^{(k)}\|_{2}/\| \mathbf{x}^{(k)}-\mathbf{x}^{(k-1)}\|_{2}\right)}{\log\left(\|\mathbf{x}^{(k)} -\mathbf{x}^{(k-1)}\|_{2}/\|\mathbf{x}^{(k-1)}-\mathbf{x}^{(k-2)}\|_{2}\right)} \tag{24}\]  
We consider the **Test Problem 3** (TP3) which is a system of 2 equations:  
\[ \begin{array}{l}x_{1}^{2}+x_{2}^{2}-7=0\\ x_{1}-x_{2}+1=0\end{array}  \tag{25}\]  
Using the substitution method, Equation (25) reduces to the quadratic equation  
\(x_{2}^{2}-x_{2}-3=0\) whose positive root is given by \(x_{2}^{*}=\dfrac{1+\sqrt{13}}{2}=2.302775638\).. Therefore \(x_{1}^{*}=x_{2}^{*}-1=\dfrac{\sqrt{13}}{2}=1.302775638\)..  
We use \(\mathbf{x}^{(0)}=(1,2)^{T}\) as starting vector and apply our Equation (16), \(r=1,2,...,6\) to calculate the approximate solutions of Equation (25). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Error** & **3 rdBQIM** & **4 thBQIM** & **5 thBQIM** & **6 thBQIM** & **7 thBQIM** & **8 thBQIM** \\ \hline \(\|\mathbf{x}^{(1)}-\mathbf{x}^{(0)}\|_{2}\) & 4.2e\(-\)1 & 4.3e\(-\)1 & 4.3e\(-\)1 & 4.3e\(-\)1 & 4.3e\(-\)1 & 4.3e\(-\)1 \\ \(\|\mathbf{x}^{(2)}-\mathbf{x}^{(1)}\|_{2}\) & 9.2e\(-\)3 & 2.5e\(-\)3 & 7.6e\(-\)4 & 2.5e\(-\)4 & 8.6e\(-\)5 & 3.1e\(-\)5 \\ \(\|\mathbf{x}^{(3)}-\mathbf{x}^{(2)}\|_{2}\) & 6.0e\(-\)8 & 1.4e\(-\)12 & 5.1e\(-\)18 & 2.9e\(-\)24 & 2.7e\(-\)31 & 4.0e\(-\)39 \\ \(\|\mathbf{x}^{(4)}-\mathbf{x}^{(3)}\|_{2}\) & 1.6e\(-\)23 & 1.5e\(-\)49 & 7.5e\(-\)89 & 7.4e\(-\)144 & 7.0e\(-\)217 & 3.0e\(-\)310 \\ \(\|\mathbf{x}^{(5)}-\mathbf{x}^{(4)}\|_{2}\) & 3.4e\(-\)70 & 1.9e\(-\)197 & - & - & - & - \\ \(p_{c}\) & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of the TP3 for the 3 rdBQIM, 4 thBQIM, 5 thBQIM, 6 thBQIM, 7 thBQIM and 8 thBQIM I.F.s. Figure 2: Polynomiographs of 6 thBQIM, 7 thBQIM and 8 thBQIM I.F.s. for \(f(z)=z^{2}-1\). (**a**) 6 thBQIM; (**b**) 7 thBQIM; (**c**) 8 thBQIM. Table 3 shows that as the order of the methods increase the methods converge in less iterations (4 iterations) and with a smaller error. Similarly, as in the case for scalar equations, the computational order of convergence for this system of 2 equations agree with the theoretical one. We next consider the **Test Problem 4** (TP4) [14]  
\[ \begin{array}{l}x_{1}^{2}+x_{2}^{2}-1=0\\ x_{1}^{2}-x_{2}^{2}-0.5=0\end{array}  \tag{26}\]  
Using the elimination method, Equation (26) reduces to the simple quadratic equation  
\(2x_{2}^{2}-1.5=0\) whose positive root is given by \(x_{2}^{*}=\dfrac{\sqrt{3}}{2}=0.866025403..\) and therefore \(x_{1}^{*}=\dfrac{1}{2}\). Using \(\mathbf{x}^{(0)}=(2,3)^{T}\) as starting vector far from the root, we apply our methods (16), \(r=1,2,...,6\) to find the numerical solutions of Equation (26). In Table 4, with the starting vector distant from the root, we observe that the methods take more iterations to converge. As from the third iteration, the iterate of the methods are close to the root and they converge to the root at their respective rate of convergence. We next consider the **Test Problem 5** (TP5) which is a system of 4 equations [15]. \[ \begin{array}{l}x_{2}x_{3}+x_{4}(x_{2}+x_{3})=0\\ x_{1}x_{3}+x_{4}(x_{1}+x_{3})=0\\ x_{1}x_{2}+x_{4}(x_{1}+x_{2})=0\\ x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}=1\end{array}