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：通过求解适当的初值问题，我们确定了泰勒多项式逼近中的拉格朗日函数，进而推导出余项，并利用自然三次样条对其近似。该方法显著提升了泰勒逼近的精度。观测结果表明，逼近精度可提升多个数量级，即便在自变量超出相关收敛半径的情况下亦如此。