We determine the Lagrange function in Taylor polynomial approximation by solving an appropriate initial-value problem. Hence, we determine the remainder term which we then approximate by means of a natural cubic spline. This results in a significant improvement in the quality of the Taylor approximation. We observe improvements in the accuracy of the approximation of many orders of magnitude, including a case when the independent variable x lies beyond the relevant radius of convergence.

翻译：我们通过求解恰当的初值问题来确定泰勒多项式近似中的拉格朗日函数，进而确定余项，并利用自然三次样条对该余项进行近似。这一方法显著提升了泰勒近似的质量。我们观察到近似精度提升了多个数量级，包括自变量x超出相关收敛半径的情形。