几类经典插值函数截断误差的简单证明

Simple Proof of Truncation Error of Classical Interpolation Functions

截断误差估计关系到逼近方法的收敛性及精度的刻画,是数值逼近中十分重要的研究课题。在前人研究工作的基础上,应用积分中值定理、Rolle定理和构造一些函数特殊不等式,对插值余项定理、Lagrange插值、Newton插值、Hermite插值、分段线性插值和三次样条插值的误差计算公式给予简单证明,结果对数值逼近理论与实践均有一定的价值和意义。

The estimates of truncatederrors is significant for approximation methods,which conc-erns their convergence and precision.Based on previous research,the interpolation remainder theo-rem and the truncated error formulas of Lagrange interpolation,Newton interpolation,Hermiteint-erpolation,Piecewise linear interpolation,and cubic spline interpolation were concisely proved b-y using mean value theorem for integrals,Rolle theorem and some specific inequalities.The result-s are valuable and meaningful in both theory and practice for studying numerical approximation.