# On the Kung-Traub Conjecture for Iterative Methods for Solving Quadratic Equations
## 5 Numerical Experiments
### Application  
As an application, we consider the quadratic integral equation of the type:  
\[x(s)=g(s)+\lambda x(s)\int_{0}^{1}\ K(s,t)\ x(t)\ dt \tag{28}\]  
Equation (28) appears in [16] and is known as [PERSON]'s integral equation. It arises from the study of the radiative transfer theory, the transport of neutrons and the kinetic theory of the gases. It is studied in [17] and, under certain conditions for the kernel, in [18; 19].  
We define the kernel \(K(s,t)\) as a continuous function in \(s,t\in[0,1]\) such that \(0<K(s,t)<1\) and \(K(s,t)+K(t,s)=1\). Moreover, we assume that \(g(s)\in C[0,1]\) is a given function and \(\lambda\) is a real constant. The solution of Equation (28) is equivalent to solving the equation \(F(x)=0\), where \(F:C[0,1]\to C[0,1]\) and  
\[F(x)(s)=x(s)-g(s)-\lambda\ x(s)\int_{0}^{1}\ K(s,t)\ x(t)\ dt,\ x\in C[0,1],\ s\in[0,1]\]  
We choose \(g(s)=1\) and \(K(s,t)=\dfrac{s}{s+t}\) so that we are required to solve the following equation:  
\[F(x)(s)=x(s)-1-\lambda\ x(s)\int_{0}^{1}\dfrac{s}{s+t}\ x(t)\ dt,\ x\in C[0,1],\ s\in[0,1] \tag{29}\]  
If we discretize the integral given in Equation (29) using the Mid-point Integration Rule with \(n\) grid points  
\[\int_{0}^{1}\dfrac{s}{s+t}\ x(t)\ dt=\dfrac{1}{n}\sum_{j=1}^{n}\ \dfrac{t_{j}}{t_{i}+t_{j}}\ x_{j},\quad x_{j}=x(t_{j}),\quad t_{j}=(j-0.5)h,\quad h=\dfrac{1}{n},\quad 1\leq j\leq n\]  
we obtain the resulting system of non-linear equations:  
\[f_{i}(\mathbf{x})=x_{i}-\lambda\ \dfrac{x_{i}}{n}\sum_{j=1}^{n}\ \dfrac{t_{j}}{t_{i}+t_{j}}\ x_{j},\quad 1 \leq i\leq n \tag{30}\]  
The \(\lambda\) are equally spaced with \(\Delta\lambda=0.01\) in the interval \(\lambda\in(0,0.5)\). We choose \(n=100\) and \((1,1,....,1)^{T}\) as the starting vector. In this case, for each \(\lambda\), we let \(M_{\lambda}\) be the minimum number of iterations for which the infinity norm between the successive approximations \(\|\mathbf{x}^{(k+1)}-\mathbf{x}^{(k)}\|_{\infty}<1e-13\), where the approximation \(\mathbf{x}^{(k)}\) is calculated correct to 16 digits (double precision in MATLAB). Let \(\overline{M_{\lambda}}\) be the mean of iteration number for the 49 \(\lambda\)'s.  
All methods converge for all 49 values of \(\lambda\). The results are given in Table 6 which shows that all methods converge in less than five iterations. It is the 8_thBQIM_ I.F. which has the greatest number of \(\lambda\) converging in two or three iterations and the smallest mean iteration number. We also observe  
\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 \(\|\mathbf{x}^{(1)}-\mathbf{x}^{(0)}\|_{2}\) & 1.4e\(-1\) & 1.4e\(-1\) & 1.4e\(-1\) & 1.4e\(-1\) & 1.4e\(-1\) & 1.4e\(-1\) \\ \(\|\mathbf{x}^{(2)}-\mathbf{x}^{(1)}\|_{2}\) & 1.7e\(-3\) & 3.5e\(-4\) & 8.1e\(-5\) & 2.0e\(-5\) & 5.2e\(-6\) & 1.4e\(-6\) \\ \(\|\mathbf{x}^{(3)}-\mathbf{x}^{(2)}\|_{2}\) & 2.4e\(-9\) & 8.6e\(-15\) & 2.6e\(-21\) & 7.1e\(-29\) & 1.7e\(-37\) & 3.9e\(-47\) \\ \(\|\mathbf{x}^{(4)}-\mathbf{x}^{(3)}\|_{2}\) & 6.5e\(-27\) & 3.1e\(-57\) & 9.7e\(-104\) & 1.4e\(-169\) & 7.9e\(-258\) & 0 \\ \(\|\mathbf{x}^{(5)}-\mathbf{x}^{(4)}\|_{2}\) & 1.3e\(-79\) & - & - & - & - & - \\ \(p_{c}\) & 3 & 4 & 5 & 6 & 7 & 8.1 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Results of TP5 for the 3_rdBQIM_, 4_thBQIM_, 5_thBQIM_, 6_thBQIM_, 7_thBQIM_ and 8_thBQIM_ I.F.s.  
that there is a small difference in the mean iteration number between the 7_thBQIM_ and 8_thBQIM_ I.F.s. Developing 9 th or higher order I.F.s would not be necessary for this application.