Legendre-Gauss-Lobatto节点的一个注记

A Note to Legendre-Gauss-Lobatto Nodes

以Legendre-Gauss-Lobatto点为节点的Lagrange插值基函数,构造N阶插值多项式P_N(x)。对P_N(x)分别求一阶和二阶导数,得到一阶和二阶微分矩阵。利用Legendre-Gauss-Lobatto点的性质导出一阶和二阶微分矩阵的关系,由此可利用Lagrange插值多项式数值求解微分方程。

In this paper,Legendre-Gauss-Lobatto nodes were used to construct the N degree Lagrange interpolation polynomial P_N(x).First-order differentiation matrices and second order differentiation matrices were given by taking first-order derivative and second-order derivative of the P_N(x),respectively.The relations of the first-order differentiation matrices and second order differentiation matrices were derived by the properties of Legendre-Gauss-Lobatto nodes.It is much easier to approximate the solution of differential equations by Lagrange interpolation polynomial.