Newton 插值法

定义

\(f(x)\) 关于 \(x_0, x_1, \dots, x_k\) 的 \(k\) 阶均差（差商）记做 $ f [x_0, x_1, \dots, x_k] $，均差是递归定义的，有两种等价定义

\begin{align}
f[x] &= f(x)\notag\\
f[x_0,x_1,\dots,x_k] &=\frac{f[x_0, x_1, \dots, x_{k-2}, x_{k-1}] - f[x_1, x_2, \dots, x_{k-1}, x_{k}]}{x_0 - x_k}\label{E:1}\\
&= \frac{ f[x_0, x_1, \dots, x_{k-2}, x_{k-1}] - f [x_0, x_1, \dots, x_{k-2}, x_{k}] } { x_{k-1} - x_{k} }
\end{align}

编程实现时，\eqref{E:1} 式更为方便。令 $d_{i,j} = f [x_i, x_{i+1}, \dots, x_j] $，则有

\[
d_{i,j} = \frac{d_{i,j-1} - d_{i+1, j} } {x_i - x_j}
\]