II型插值条件下三次样条函数误差分析
Error Analysis of Cubic Spline Function Under Type II Interpolation

样条函数是函数逼近理论一个非常活跃的分支，促使了研究人员需要深刻认识样条函数的本质及性质。本文介绍了基于Hermite两点三次公式的三转角插值算法。三转角以插值节点的一阶导数为未知量构建样条函数，在此基础上，研究插值节点均匀分布时，在第二类边界条件下，即II型插值条件下，当边界初值发生扰动时，对应的三次样条函数在插值节点的一阶导数值如何随第二边界初值的扰动而变化，基于Doolittle分解和Crout分解性质，推导出2个定理，即误差估计的表达式，这些定理为三次样条函数在二阶导数边界初值变化时的误差分析提供了可行的方法。
Spline function is a very active branch of function approximation theory, which makes researchers need to deeply understand the essence and properties of spline function. This paper introduces the three-angle interpolation algorithm based on Hermite two-point cubic formula. The three-angle spline function is constructed with the first derivative of the interpolating node as an unknown quantity. On this basis, when interpolating nodes are evenly distributed, under the second type of boundary condition, that is, under the type II interpolation condition, when the initial value of the boundary is disturbed, the corresponding cubic spline function in the interpolating node’s first derivative value changes with the disturbance of the initial value of the second boundary. Based on the properties of Doolittle decomposition and Crout decomposition, two theorems, namely the expression of error estimation, are derived. These theorems provide a feasible method for error analysis of cubic spline function when the initial value of the second derivative boundary changes.