近似函数高阶导数的高精度积分逼近方法

High-accuracy Integral Methods to Approximate Higher-order Derivatives of Noise Functions

主要研究稳定计算近似函数的高阶导数的积分逼近方法,方法因由Lanczos提出故也称为Lanczos算法.利用Legendre多项式的正交性,提出了一类逼近近似函数高阶导数的高精度积分方法,即构造出一系列积分算子Dn,h(m)去逼近噪声函数的高阶导数,且这些积分算子具有O(δ(2n+2)/(2n+m+2))的收敛速度,其中δ为近似函数的噪声水平.数值模拟结果表明提出的方法是稳定而有效的.

This paper mainly studies the integral approximation method for stably computing the higher-order derivatives of noise function.This method is also called Lanczos algorithm because it was firstly proposed by Lanczos.By the orthogonality of Legendre polynomials,a class of integral methods with high accuracy are proposed for approximating high-order derivatives of noise function,i.e.,a series of integration operators Dn,h(m) are constructed to approximate higher-order derivatives of noise function with the high convergence rate O(δ(2 n+2)/(2 n+m+2)),where δ is the noise level of the given function.The numerical simulation results show that the method proposed in this paper is stable and effective.