## 5 Numerical Tests
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(x_{i}\) & \(E_{3}(x_{i},10)\) & \(E_{3}(x_{i},20)\) & \(E_{3}(x_{i},40)\) & \(O_{3}(x_{i},10,20)\) & \(O_{3}(x_{i},10,40)\) \\ \hline
0.0 & \(5.275\times 10^{-4}\) & \(2.780\times 10^{-5}\) & \(1.471\times 10^{-6}\) & 4.2 & 4.2 \\
0.1 & \(7.466\times 10^{-5}\) & \(1.054\times 10^{-6}\) & \(5.170\times 10^{-9}\) & 6.1 & 6.8 \\
0.2 & \(1.955\times 10^{-5}\) & \(2.196\times 10^{-8}\) & \(2.435\times 10^{-9}\) & 9.8 & 6.4 \\
0.3 & \(5.064\times 10^{-6}\) & \(3.244\times 10^{-8}\) & \(2.362\times 10^{-9}\) & 7.3 & 5.5 \\
0.4 & \(4.352\times 10^{-7}\) & \(3.852\times 10^{-8}\) & \(2.592\times 10^{-10}\) & 3.5 & 5.3 \\
0.5 & \(1.209\times 10^{-6}\) & \(4.265\times 10^{-8}\) & \(5.388\times 10^{-9}\) & 4.8 & 3.9 \\
0.6 & \(6.714\times 10^{-7}\) & \(4.702\times 10^{-8}\) & \(1.748\times 10^{-9}\) & 3.8 & 4.2 \\
0.7 & \(7.116\times 10^{-6}\) & \(4.759\times 10^{-8}\) & \(1.095\times 10^{-9}\) & 7.2 & 6.3 \\
0.8 & \(2.705\times 10^{-5}\) & \(4.163\times 10^{-8}\) & \(4.917\times 10^{-9}\) & 9.3 & 6.1 \\
0.9 & \(1.027\times 10^{-4}\) & \(1.950\times 10^{-6}\) & \(9.570\times 10^{-9}\) & 5.7 & 6.6 \\
1.0 & \(8.400\times 10^{-4}\) & \(6.182\times 10^{-5}\) & \(4.229\times 10^{-6}\) & 3.8 & 3.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The absolute errors of the third 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 \(x_{i}\) & \(E_{5}(x_{i},10)\) & \(E_{5}(x_{i},20)\) & \(E_{5}(x_{i},40)\) & \(O_{5}(x_{i},10,20)\) & \(O_{5}(x_{i},10,40)\) \\ \hline
0.1 & \(3.494\times 10^{-2}\) & \(2.340\times 10^{-3}\) & \(3.900\times 10^{-6}\) & 3.9 & 6.5 \\
0.2 & \(1.086\times 10^{-2}\) & \(1.096\times 10^{-4}\) & \(6.875\times 10^{-5}\) & 6.6 & 3.6 \\
0.3 & \(4.136\times 10^{-3}\) & \(2.740\times 10^{-4}\) & \(4.483\times 10^{-5}\) & 3.9 & 3.3 \\
0.4 & \(5.293\times 10^{-4}\) & \(3.101\times 10^{-4}\) & \(6.563\times 10^{-5}\) & 0.8 & 1.5 \\
0.5 & \(1.727\times 10^{-3}\) & \(3.430\times 10^{-4}\) & \(1.049\times 10^{-4}\) & 2.3 & 2.0 \\
0.6 & \(5.522\times 10^{-4}\) & \(3.787\times 10^{-4}\) & \(6.581\times 10^{-5}\) & 0.5 & 1.5 \\
0.7 & \(5.874\times 10^{-3}\) & \(4.064\times 10^{-4}\) & \(8.943\times 10^{-5}\) & 3.9 & 3.0 \\
0.8 & \(1.453\times 10^{-2}\) & \(1.946\times 10^{-4}\) & \(1.241\times 10^{-4}\) & 6.2 & 3.4 \\
0.9 & \(5.871\times 10^{-2}\) & \(4.198\times 10^{-3}\) & \(1.931\times 10^{-5}\) & 3.8 & 5.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The absolute errors of the fifth 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 \(x_{i}\) & \(E_{0}(x_{i},10)\) & \(E_{0}(x_{i},20)\) & \(E_{0}(x_{i},40)\) & \(O_{0}(x_{i},10,20)\) & \(O_{0}(x_{i},10,40)\) \\ \hline \(-0.5\) & \(1.224\times 10^{-8}\) & \(9.184\times 10^{-11}\) & \(7.080\times 10^{-13}\) & 7.1 & 7.0 \\ \(-0.4\) & \(1.748\times 10^{-9}\) & \(4.632\times 10^{-12}\) & \(1.765\times 10^{-14}\) & 8.6 & 8.2 \\ \(-0.3\) & \(5.906\times 10^{-10}\) & \(2.425\times 10^{-12}\) & \(1.460\times 10^{-14}\) & 7.9 & 7.6 \\ \(-0.2\) & \(3.360\times 10^{-10}\) & \(2.273\times 10^{-12}\) & \(5.940\times 10^{-15}\) & 7.2 & 7.8 \\ \(-0.1\) & \(2.934\times 10^{-10}\) & \(2.220\times 10^{-12}\) & \(1.117\times 10^{-14}\) & 7.0 & 7.3 \\
0.0 & \(2.761\times 10^{-10}\) & \(2.167\times 10^{-12}\) & \(6.559\times 10^{-15}\) & 7.0 & 7.6 \\
0.1 & \(2.586\times 10^{-10}\) & \(2.116\times 10^{-12}\) & \(6.231\times 10^{-15}\) & 6.9 & 7.6 \\
0.2 & \(2.150\times 10^{-10}\) & \(2.057\times 10^{-12}\) & \(4.469\times 10^{-15}\) & 6.7 & 7.7 \\
0.3 & \(4.428\times 10^{-11}\) & \(1.895\times 10^{-12}\) & \(4.385\times 10^{-15}\) & 4.5 & 6.6 \\
0.4 & \(1.225\times 10^{-9}\) & \(4.481\times 10^{-13}\) & \(2.498\times 10^{-15}\) & 11.4 & 9.4 \\
0.5 & \(1.194\times 10^{-8}\) & \(9.321\times 10^{-11}\) & \(7.356\times 10^{-13}\) & 7.0 & 6.9 \\ \hline \hline \end{tabular}
\end{table}