We show how the error term for the Trapezium Rule can be estimated, by solving an initial value problem using a Runge-Kutta method. The error term can then be added to the Trapezium approximation, yielding a much more accurate result. We also show how the risk of singularities in the relevant initial value problem can be mitigated.

翻译：我们展示了如何通过使用龙格-库塔方法求解初值问题来估计梯形法则中的误差项。随后可将该误差项与梯形近似值相加，从而获得精度显著提高的结果。此外，我们还阐述了如何降低相关初值问题中奇异性出现的风险。