龙格现象难题破解之系数与阶次双确定方法

龙格现象指出，使用基于等距节点的高阶插值多项式逼近龙格函数时，插值多项式在逼近区间两端会产生明显的振荡现象。因此，传统认为，不适宜用基于等距节点的高阶多项式逼近龙格函数。针对龙格现象，展示一种新型的多项式系数与阶次双确定方法。该方法可快速构造出基于等距节点的不会振荡且有较高逼近精度的高阶多项式，良好地逼近龙格函数。计算机数值实验表明该方法是有效的，即运用基于等距节点的高阶多项式可以很好地消解龙格现象。

Solving the problem of Runge phenomenon by coefficients-and-orderdetermination method

The Runge phenomenon demonstrates that it is not suitable to use high-order interpolation polynomials with equidistant nodes to approximate the Runge function, as oscillation occurs near the ends of the interpolation interval. Nevertheless, this paper presents an innovative method called Coefficients-And-Order-Determination(CAOD)method to solve the problem of the Runge phenomenon. This method can efficiently determine the coefficients and the order of the optimal polynomial that approximates the target function. By such a CAOD method, high-order optimal polynomials are constructed for different numbers of equidistant nodes, which all approximate the Runge function without causing oscillation. Thus, such constructed optimal polynomials can achieve high approximation accuracy(i.e., eliminate the Runge phenomenon). Numerical experiment results further substantiate the efficacy and accuracy of the CAOD method.