## 5 Numerical Tests
\begin{table}
\begin{tabular}{c 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 \(-0.5\) & \(3.940\times 10^{-4}\) & \(2.394\times 10^{-5}\) & \(1.498\times 10^{-6}\) & 4.0 & 4.0 \\ \(-0.4\) & \(5.180\times 10^{-5}\) & \(8.104\times 10^{-7}\) & \(5.789\times 10^{-10}\) & 5.9 & 8.2 \\ \(-0.3\) & \(1.362\times 10^{-5}\) & \(1.640\times 10^{-8}\) & \(4.348\times 10^{-9}\) & 9.6 & 5.8 \\ \(-0.2\) & \(3.555\times 10^{-6}\) & \(2.343\times 10^{-8}\) & \(2.673\times 10^{-9}\) & 7.2 & 5.1 \\ \(-0.1\) & \(3.114\times 10^{-7}\) & \(2.567\times 10^{-8}\) & \(8.109\times 10^{-11}\) & 3.6 & 5.9 \\ \(0.0\) & \(7.246\times 10^{-7}\) & \(2.607\times 10^{-8}\) & \(2.454\times 10^{-9}\) & 4.7 & 4.1 \\ \(0.1\) & \(3.225\times 10^{-7}\) & \(2.594\times 10^{-8}\) & \(1.845\times 10^{-9}\) & 3.6 & 3.7 \\ \(0.2\) & \(3.624\times 10^{-6}\) & \(2.396\times 10^{-8}\) & \(1.008\times 10^{-9}\) & 7.2 & 5.9 \\ \(0.3\) & \(1.387\times 10^{-5}\) & \(1.768\times 10^{-8}\) & \(2.106\times 10^{-9}\) & 9.6 & 6.3 \\ \(0.4\) & \(5.279\times 10^{-5}\) & \(8.602\times 10^{-7}\) & \(6.609\times 10^{-10}\) & 5.9 & 8.1 \\ \(0.5\) & \(4.043\times 10^{-4}\) & \(2.555\times 10^{-5}\) & \(1.637\times 10^{-6}\) & 3.9 & 3.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: The absolute errors of the third order derivatives of \(y_{2}\) at the knots and the numerical convergence orders. \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 \(-0.5\) & \(5.824\times 10^{-3}\) & \(7.682\times 10^{-4}\) & \(1.098\times 10^{-4}\) & 2.9 & 2.8 \\ \(-0.4\) & \(1.743\times 10^{-3}\) & \(7.014\times 10^{-5}\) & \(2.500\times 10^{-5}\) & 4.6 & 3.0 \\ \(-0.3\) & \(1.364\times 10^{-4}\) & \(1.006\times 10^{-4}\) & \(1.931\times 10^{-5}\) & 0.4 & 1.4 \\ \(-0.2\) & \(6.769\times 10^{-5}\) & \(8.285\times 10^{-5}\) & \(1.238\times 10^{-5}\) & \(-0.3\) & 1.2 \\ \(-0.1\) & \(3.940\times 10^{-4}\) & \(6.240\times 10^{-5}\) & \(7.654\times 10^{-6}\) & 2.6 & 2.8 \\ \(0.0\) & \(3.315\times 10^{-4}\) & \(4.161\times 10^{-5}\) & \(2.369\times 10^{-6}\) & 2.9 & 3.5 \\ \(0.1\) & \(2.691\times 10^{-4}\) & \(2.078\times 10^{-5}\) & \(3.074\times 10^{-6}\) & 3.6 & 3.2 \\ \(0.2\) & \(5.980\times 10^{-4}\) & \(7.186\times 10^{-8}\) & \(8.315\times 10^{-6}\) & 13.0 & 3.0 \\ \(0.3\) & \(5.313\times 10^{-4}\) & \(1.816\times 10^{-5}\) & \(1.326\times 10^{-5}\) & 4.8 & 2.6 \\ \(0.4\) & \(2.448\times 10^{-3}\) & \(1.408\times 10^{-5}\) & \(1.853\times 10^{-5}\) & 7.4 & 3.5 \\ \(0.5\) & \(5.332\times 10^{-3}\) & \(7.329\times 10^{-4}\) & \(1.119\times 10^{-4}\) & 2.8 & 2.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: The absolute errors of the fourth order derivatives of \(y_{2}\) at the knots and the numerical convergence orders. has satisfactory approximation abilities in approximating a function and its first order to fifth order derivatives. Hence, the new method is very effective for integro interpolation. This work was supported by the National Natural Science Foundation of China (Grant No. 11501533). The author appreciates the reviewers and the editors very much for their careful reading, professional review and valuable suggestions. The author declares no conflict of interest.