样条插值

三次样条有另外一个非常重要的解释，实际上它是在索伯列夫空间${\displaystyle H^{2}([a;b])}$最小化泛函

${\displaystyle J(f)=\int _{a}^{b}|f''(x)|^{2}dx}$

的函数。

泛函${\displaystyle J}$包含对于函数${\displaystyle f(x)}$全部曲率${\displaystyle \left|{\frac {f''(x)}{(1+f'(x)^{2})^{\frac {3}{2}}}}\right|}$的近似，样条是${\displaystyle f(x)}$最小曲率的近似。

由于弹性条的总体能量与曲率成比例，所以样条是受到${\displaystyle n}$个点约束的弹性条的最小能量形状。样条也是基于弹性条设计的工具。

它可以定义为

${\displaystyle S_{i}(x)={\frac {z_{i+1}(x-x_{i})^{3}+z_{i}(x_{i+1}-x)^{3}}{6h_{i}}}+\left({\frac {y_{i+1}}{h_{i}}}-{\frac {h_{i}}{6}}z_{i+1}\right)(x-x_{i})+\left({\frac {y_{i}}{h_{i}}}-{\frac {h_{i}}{6}}z_{i}\right)(x_{i+1}-x)}$

以及

${\displaystyle h_{i}=x_{i+1}-x_{i}\,\!}$.

通过解下面的方程可以得到它的系数。

${\displaystyle \left\{{\begin{matrix}z_{0}=0\\h_{i-1}z_{i-1}+2(h_{i-1}+h_{i})z_{i}+h_{i}z_{i+1}=6\left({\frac {y_{i+1}-y_{i}}{h_{i}}}-{\frac {y_{i}-y_{i-1}}{h_{i-1}}}\right)\\z_{n}=0\end{matrix}}\right.}$

假设要为带有节点

${\displaystyle (x_{0},f(x_{0}))=(x_{0},y_{0})=\left(-1,\ e^{-1}\right)\,\!}$

${\displaystyle (x_{1},f(x_{1}))=(x_{1},y_{1})=\left(-{\frac {1}{2}},\ e^{-{\frac {1}{4}}}\right)\,\!}$

${\displaystyle (x_{2},f(x_{2}))=(x_{2},y_{2})=\left(0,\ 1\right)\,\!}$

${\displaystyle (x_{3},f(x_{3}))=(x_{3},y_{3})=\left({\frac {1}{2}},\ e^{-{\frac {1}{4}}}\right)\,\!}$

${\displaystyle (x_{4},f(x_{4}))=(x_{4},y_{4})=\left(1,\ e^{-1}\right)\,\!}$

的函数

${\displaystyle f(x)=e^{-x^{2}}}$

找一个线性样条。直接代入样条公式，我们得到如下样条：

${\displaystyle S(x)=\left\{{\begin{matrix}e^{-1}+2(e^{-{\frac {1}{4}}}-e^{-1})(x+1)&x\in [-1,-{\frac {1}{2}}]\\e^{-{\frac {1}{4}}}+2(1-e^{-{\frac {1}{4}}})(x+{\frac {1}{2}})&x\in [-{\frac {1}{2}},0]\\1+2(e^{-{\frac {1}{4}}}-1)x&x\in [0,{\frac {1}{2}}]\\e^{-{\frac {1}{4}}}+2(e^{-1}-e^{-{\frac {1}{4}}})(x-{\frac {1}{2}})&x\in [{\frac {1}{2}},1]\\\end{matrix}}\right.}$

样条函数（蓝线）以及所近似的函数（红点）如下图所示：

下图是一个k=4的样条函数（蓝线）与所近似的函数（红线）的例子：