插值节点具有高阶导数信息的多项式插值算法

Polynomial Interpolation Algorithm with Higher-Order Derivative Information for Interpolation Nodes

摘要: 目前大部分对插值的研究都只研究函数的多项式插值，少数文章给出了一阶或二阶导数的多项式插值，本文给出了对函数及其高阶导数的多项式插值的理论算法，构造高阶导数的多项式插值听起来是复杂的，一般的算法也是庞大的，本文采取与经典研究方法不同的思路，运用行列式函数求导法则直接待定插值多项式，并给出证明，最后通过K值法和罗尔定理计算了Lagrange型插值余项，便于误差的分析，至此，要求在各插值节点上插值多项式函数与被插函数的函数值相等，且任意阶导数值也相等的问题在理论上解决了。

Abstract: At present, most of the researches on interpolation only study the polynomial interpolation of func-tion. A few papers give the polynomial interpolation of the first or second derivative. This paper gives the theoretical algorithm of polynomial interpolation of function and its higher derivative, the construction of polynomial interpolation of high order derivative sounds complicated, and the gen-eral algorithm is huge. This paper takes a different idea from the classical research methods. Finally, the residual terms of Lagrange-type interpolation are calculated by K-value method and Rohr’s theorem, which is convenient for error analysis. So far, the problems that the interpolation polyno-mial function and the interpolated function are equal on each interpolation node, and the derivative values of any order are equal, have been solved theoretically.