We determine the pointwise error in Hermite interpolation by numerically solving an appropriate differential equation, derived from the error term itself. We use this knowledge to approximate the error term by means of a polynomial, which is then added to the original Hermite polynomial to form a more accurate approximation. An example demonstrates that improvements in accuracy are significant.

翻译：我们通过数值求解由误差项本身导出的微分方程，确定了埃尔米特插值中的逐点误差。利用这一结果，我们以多项式形式近似误差项，并将其添加到原埃尔米特多项式中，从而形成更高精度的逼近。算例表明，精度提升效果显著。