牛頓多項式（英語：Newton Polynomial）是數值分析中一種用於插值的多項式，以英格兰數學家暨物理學家牛頓命名。
給定包含 k + 1 {\displaystyle k+1} 個數據點的集合 ( x 0 , y 0 ) , … , ( x k , y k ) {\displaystyle (x_{0},y_{0}),\ldots ,(x_{k},y_{k})} 。
如果對於 ∀ i , j ∈ { 0 , . . . , k } , i ≠ j {\displaystyle \forall i,j\in \left\{0,...,k\right\},i\neq j} ，滿足 x i ≠ x j {\displaystyle x_{i}\neq x_{j}} ，那麼應用牛頓插值公式所得到的牛頓插值多項式為
其中每個 n j ( x ) {\displaystyle n_{j}(x)} 為牛頓基本多項式（或稱插值基函數），其表達式為
其中 j > 0 {\displaystyle j>0} ，並且 n 0 ( x ) ≡ 1 {\displaystyle n_{0}(x)\equiv 1} 。
係數 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，牛頓多項式可以寫作：
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