三次样条插值

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局部插值方法

　　指插值函数满足$s\kh{x}=\sum_{i=1}^nl_i\kh{x}f\kh{x_i}$，其中$l_i\kh{x_j}=\delta_{ij}$，且$l_i$仅在$x_i$某邻域内非零。例如分片线性、局部分片三次多项式插值。

三次样条插值

　　给定插值点均匀分布，间隔为$h$。要求$s$为有界分片三次函数，且插值函数二次导数连续。则有$s$每个分片$s_i$

$$\left\{ \begin{align*} s_j\kh{x}={}&f\kh{x_j}+s'\kh{x_j}\kh{x-x_j}+c^2_j\kh{x-x_j}^2+c^3_j\kh{x-x_j}^3\\ s_j\kh{x_{j+1}}={}&f\kh{x_{j+1}}\\ s''_j\kh{x_{j+1}}={}&s''_{j+1}\kh{x_{j+1}} \end{align*} \right.$$

得

$$\left\{ \begin{align*} c^2_j={}&3\frac{f\kh{x_{j+1}}-f\kh{x_j}}{h^2}-\frac{2s'\kh{x_{j}}-s'\kh{x_{j+1}}}{h}\\ c^3_j={}&2\frac{f\kh{x_{j}}-f\kh{x_{j+1}}}{h^3}+\frac{s'\kh{x_{j}}+s'\kh{x_{j+1}}}{h^2} \end{align*} \right.$$

即

$$c_{j+1}^2=c_j^2+3c_j^3h\Longrightarrow s'\kh{x_{j-1}}+4s'\kh{x_{j}}+s'\kh{x_{j+1}}=\frac{3}{h}\zkh{f\kh{x_{j+1}}-f\kh{x_{j-1}}}.$$

　　代入$s\kh{x}=\sum_{i=1}^nl_i\kh{x}f\kh{x_i}$，其中$l_i\kh{x_j}=\delta_{ij}$。则有

$$l'_i\kh{x_{j-1}}+4l'_i\kh{x_{j}}+l'_i\kh{x_{j+1}}=\frac{3}{h}\zkh{\delta_{i,j+1}-\delta_{i,j-1}}.$$

　　由上式右端当$j>i+1$时为0，易得对$j>i$，$l'_i\kh{x_j}=\alpha\kh{-2+\sqrt{3}}^{j-i}+\beta\kh{-2-\sqrt{3}}^{j-i}$。$l_i$有界，故$\beta=0$。$j<i$同理。从而代入得

$$l'_i\kh{x_j}=\begin{cases} -\frac{3}{h}\kh{-2+\sqrt{3}}^{j-i}&j<i\\ 0&j=i\\ \frac{3}{h}\kh{-2+\sqrt{3}}^{j-i}&j>i \end{cases}$$

结合$l_i\kh{x_j}=\delta_{ij}$可解出$l_i$方程。

$$\begin{align*} &\max_{\substack{x\in\zkh{0,h}\\\normm{f}_\infty\leq1}}\norm{\sum_{j\in\bbN}l_j\kh{x}f\kh{x_j}}\\ ={}&\max_{x\in\zkh{0,h}}\sum_{j\in\bbN}\norm{l_j\kh{x}}\\ ={}&\max_{x\in\zkh{0,h}}\sum_{j\in\bbZ}\kh{-1}^j\zkh{l_{-j}\kh{x}+l_{j+1}\kh{x}}\\ :={}&\max_{x\in\zkh{0,h}}p\kh{x}\\ ={}&\frac{1+3\sqrt{3}}{4}. \end{align*}$$