## 5 Numerical Tests  
In this section, we test the approximation properties of the new integro quintic spline. Our tests are performed by MATLAB. We take:  
\[y_{1}=e^{x},\quad x\in[0,1],\]  
and:  
\[y_{2}=\left\{ \begin{array}{ll}\sin x,&\mbox{if }x\in[-0.5,0)\,,\\ x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!},&\mbox{if }x\in[0,0.5]\,, \end{array} \right.\]  
as two illustrative examples. Furthermore, \(y_{1}\) will be used in the comparison of our method with some other methods. The absolute errors at the knots are defined as follows:  
\[E_{k}(x_{i},n):=|y^{(k)}(x_{i})-s^{(k)}(x_{i})|,\quad\ k=0,1,2,3,4,\quad i=0, 1,\ldots,n,\]  
and:  
\[E_{5}(x_{i},n):=|y^{(5)}(x_{i})-\frac{s^{(4)}(x_{i+1})-s^{(4)}(x_{i-1})}{2h}|,\quad i=1,2,\ldots,n-1.\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(x_{i}\) & \(E_{4}(x_{i},10)\) & \(E_{4}(x_{i},20)\) & \(E_{4}(x_{i},40)\) & \(O_{4}(x_{i},10,20)\) & \(O_{4}(x_{i},10,40)\) \\ \hline
0.0 & \(6.139\times 10^{-3}\) & \(5.105\times 10^{-4}\) & \(2.006\times 10^{-5}\) & 3.6 & 4.0 \\
0.1 & \(1.166\times 10^{-3}\) & \(2.577\times 10^{-4}\) & \(5.397\times 10^{-5}\) & 2.2 & 2.2 \\
0.2 & \(1.219\times 10^{-3}\) & \(2.171\times 10^{-4}\) & \(5.919\times 10^{-5}\) & 2.5 & 2.0 \\
0.3 & \(1.414\times 10^{-3}\) & \(2.406\times 10^{-4}\) & \(6.602\times 10^{-5}\) & 2.6 & 2.1 \\
0.4 & \(8.417\times 10^{-4}\) & \(2.700\times 10^{-4}\) & \(7.006\times 10^{-5}\) & 1.6 & 1.8 \\
0.5 & \(1.806\times 10^{-3}\) & \(3.026\times 10^{-4}\) & \(8.012\times 10^{-5}\) & 2.6 & 2.2 \\
0.6 & \(1.046\times 10^{-3}\) & \(3.387\times 10^{-4}\) & \(8.963\times 10^{-5}\) & 1.6 & 1.7 \\
0.7 & \(2.303\times 10^{-3}\) & \(3.783\times 10^{-4}\) & \(9.521\times 10^{-5}\) & 2.6 & 2.2 \\
0.8 & \(5.430\times 10^{-4}\) & \(4.165\times 10^{-4}\) & \(1.067\times 10^{-4}\) & 0.4 & 1.1 \\
0.9 & \(5.952\times 10^{-3}\) & \(3.446\times 10^{-4}\) & \(1.174\times 10^{-4}\) & 4.1 & 2.8 \\
1.0 & \(1.311\times 10^{-2}\) & \(2.204\times 10^{-3}\) & \(3.719\times 10^{-4}\) & 2.6 & 2.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The absolute errors of the fourth order derivatives of \(y_{1}\) at the knots and the numerical convergence orders. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(x_{i}\) & \(E_{2}(x_{i},10)\) & \(E_{2}(x_{i},20)\) & \(E_{2}(x_{i},40)\) & \(O_{2}(x_{i},10,20)\) & \(O_{2}(x_{i},10,40)\) \\ \hline
0.0 & \(2.647\times 10^{-5}\) & \(7.099\times 10^{-7}\) & \(1.949\times 10^{-8}\) & 5.2 & 5.2 \\
0.1 & \(4.869\times 10^{-7}\) & \(1.820\times 10^{-9}\) & \(3.831\times 10^{-10}\) & 8.1 & 5.1 \\
0.2 & \(2.978\times 10^{-7}\) & \(1.957\times 10^{-9}\) & \(4.236\times 10^{-10}\) & 7.2 & 4.7 \\
0.3 & \(5.713\times 10^{-7}\) & \(3.211\times 10^{-9}\) & \(5.213\times 10^{-10}\) & 7.5 & 5.0 \\
0.4 & \(1.569\times 10^{-7}\) & \(4.450\times 10^{-9}\) & \(3.785\times 10^{-10}\) & 5.1 & 4.3 \\
0.5 & \(5.861\times 10^{-7}\) & \(5.800\times 10^{-9}\) & \(6.002\times 10^{-10}\) & 6.7 & 4.9 \\
0.6 & \(9.784\times 10^{-8}\) & \(7.297\times 10^{-9}\) & \(7.447\times 10^{-10}\) & 3.7 & 3.5 \\
0.7 & \(6.007\times 10^{-7}\) & \(8.972\times 10^{-9}\) & \(5.578\times 10^{-10}\) & 6.1 & 5.0 \\
0.8 & \(1.011\times 10^{-7}\) & \(1.108\times 10^{-8}\) & \(7.138\times 10^{-10}\) & 3.2 & 3.5 \\
0.9 & \(7.946\times 10^{-7}\) & \(1.825\times 10^{-8}\) & \(8.472\times 10^{-10}\) & 5.4 & 4.9 \\
1.0 & \(4.041\times 10^{-5}\) & \(1.462\times 10^{-6}\) & \(4.772\times 10^{-8}\) & 4.8 & 4.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The absolute errors of the second order derivatives of \(y_{1}\) at the knots and the numerical convergence orders.