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

egin{align}
f[x] &= f(x) otag\
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}
]