# 牛顿多项式

## 定義

${\displaystyle N(x):=\sum _{j=0}^{k}a_{j}n_{j}(x)}$

${\displaystyle n_{j}(x):=\prod _{i=0}^{j-1}(x-x_{i})}$

${\displaystyle 0}$階差商 ${\displaystyle 1}$階差商 ${\displaystyle 2}$階差商 ${\displaystyle 3}$階差商 ${\displaystyle \ldots }$ ${\displaystyle k-1}$階差商
${\displaystyle x_{0}}$ ${\displaystyle f[x_{0}]}$
${\displaystyle x_{1}}$ ${\displaystyle f[x_{1}]}$ ${\displaystyle f[x_{0},x_{1}]}$
${\displaystyle x_{2}}$ ${\displaystyle f[x_{2}]}$ ${\displaystyle f[x_{1},x_{2}]}$ ${\displaystyle f[x_{0},x_{1},x_{2}]}$
${\displaystyle x_{3}}$ ${\displaystyle f[x_{3}]}$ ${\displaystyle f[x_{2},x_{3}]}$ ${\displaystyle f[x_{1},x_{2},x_{3}]}$ ${\displaystyle f[x_{0},x_{1},x_{2},x_{3}]}$
${\displaystyle \ldots }$ ${\displaystyle \ldots }$ ${\displaystyle \ldots }$ ${\displaystyle \ldots }$ ${\displaystyle \ldots }$ ${\displaystyle \ldots }$
${\displaystyle x_{k}}$ ${\displaystyle f[x_{k}]}$ ${\displaystyle f[x_{k-1},x_{k}]}$ ${\displaystyle f[x_{k-2},x_{k-1},x_{k}]}$ ${\displaystyle f[x_{k-3},x_{k-2},x_{k-1},x_{k}]}$ ${\displaystyle \ldots }$ ${\displaystyle f[x_{0},\ldots ,x_{k}]}$

${\displaystyle N(x)=[y_{0}]+[y_{0},y_{1}](x-x_{0})+\cdots +[y_{0},\ldots ,y_{k}](x-x_{0})(x-x_{1})\cdots (x-x_{k-1})}$