牛顿多项式（英语：Newton Polynomial）是数值分析中一种用于插值的多项式，以英格兰数学家暨物理学家牛顿命名。
给定包含 k + 1 {\displaystyle k+1} 个数据点的集合 ( x 0 , y 0 ) , … , ( x k , y k ) {\displaystyle (x_{0},y_{0}),\ldots ,(x_{k},y_{k})} 。
如果对于 ∀ i , j ∈ { 0 , . . . , k } , i ≠ j {\displaystyle \forall i,j\in \left\{0,...,k\right\},i\neq j} ，满足 x i ≠ x j {\displaystyle x_{i}\neq x_{j}} ，那么应用牛顿插值公式所得到的牛顿插值多项式为
其中每个 n j ( x ) {\displaystyle n_{j}(x)} 为牛顿基本多项式（或称插值基函数），其表达式为
其中 j > 0 {\displaystyle j>0} ，并且 n 0 ( x ) ≡ 1 {\displaystyle n_{0}(x)\equiv 1} 。
系数 a j := [ y 0 , … , y j ] {\displaystyle a_{j}:=[y_{0},\ldots ,y_{j}]} ，而 [ y 0 , … , y j ] {\displaystyle [y_{0},\ldots ,y_{j}]} 表示差商。
因此，牛顿多项式可以写作：
阶差商
阶差商
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阶差商
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