等距节点插值公式
将 Newton 差商插值多项式中各阶差商用相应差分代替，就可得到各种形式的等距节点插值公式，例如常用的前插公式与后插公式。
Newton 前插公式
如果节点 x k = x 0 + k h ( k = 0 , 1 , ⋅ ⋅ ⋅ , n ) x_k=x_0+kh (k=0,1,\cdotp\cdotp\cdotp,n) xk=x0+kh(k=0,1,⋅⋅⋅,n),要计算 x 0 x_0 x0附近点 x x x的函数 f ( x ) f(x) f(x)的值，可令 x = x 0 + t h x=x_0+th x=x0+th ， 0 ⩽ t ⩽ 1 0\leqslant t\leqslant1 0⩽t⩽1，于是
ω k + 1 ( x ) = ∏ j = 0 k ( x − x j ) = t ( t − 1 ) ⋅ ⋅ ⋅ ( t − k ) h k + 1 . \omega_{k+1}(x)=\prod_{j=0}^k(x-x_j)=t(t-1)\cdotp\cdotp\cdotp(t-k)h^{k+1}. ωk+1(x)=j=0∏k(x−xj)=t(t−1)⋅⋅⋅(t−k)hk+1.
代入Newton插值公式，则有
N n ( x 0 + t h ) = f 0 + t Δ f 0 + t ( t − 1 ) 2 ! Δ 2 f 0 + ⋯ + t ( t − 1 ) ⋯ ( t − n + 1 ) n ! Δ n f 0 . N_n(x_0+th)=f_0+t\Delta f_0+\frac{t(t-1)}{2!}\Delta^2f_0+\cdots+\frac{t(t-1)\cdots(t-n+1)}{n!}\Delta^nf_0. Nn(x0+th)=f0+tΔf0+2!t(t−1)Δ2f0+⋯+n!t(t−1)⋯(t−n+1)Δnf0.
上式称为 Newton 前插公式，其余项为
R n ( x ) = t ( t − 1 ) ⋯ ( t − n ) ( n + 1 ) ! h n + 1 f ( n + 1 ) ( ξ ) , ξ ∈ ( x 0 , x n ) . R_n(x)=\frac{t(t-1)\cdots(t-n)}{(n+1)!}h^{n+1}f^{(n+1)}(\xi),\quad\xi\in(x_0,x_n). Rn(x)=(n+1)!t(t−1)⋯(t−n)hn+1f(n+1)(ξ),ξ∈(x0,xn).
Newton 后插公式
如果要求表示函数在 x n x_n xn附近的值 f ( x ) f(x) f(x),应用 Newton 插值公式,插值点应按 x n , x n − 1 , ⋅ ⋅ ⋅ , x 0 x_n,x_{n-1},\cdotp\cdotp\cdotp,x_0 xn,xn−1,⋅⋅⋅,x0的次序排列，有
N n ( x ) = f ( x n ) + f [ x n , x n − 1 ] ( x − x n ) + f [ x n , x n − 1 , x n − 2 ] ( x − x n ) ( x − x n − 1 ) + ⋯ + f [ x n , x n − 1 , ⋅ ⋅ ⋅ , x 0 ] ( x − x n ) • ⋅ ⋅ ( x − x 1 ) . N_n(x)=f(x_n)+f[x_n,x_{n-1}](x-x_n)+f[x_n,x_{n-1},x_{n-2}](x-x_n)(x-x_{n-1})+\cdots+f[x_n,x_{n-1},\cdotp\cdotp\cdotp,x_0](x-x_n)•\cdotp\cdotp(x-x_1). Nn(x)=f(xn)+f[xn,xn−1](x−xn)+f[xn,xn−1,xn−2](x−xn)(x−xn−1)+⋯+f[xn,xn−1,⋅⋅⋅,x0](x−xn)•⋅⋅(x−x1).
作变换 x = x n + t h x= x_n+ th x=xn+th ( − 1 ⩽ t ⩽ 0 ) ( - 1\leqslant t\leqslant 0) (−1⩽t⩽0),代入上式得
N n ( x n + t h ) = f n + t ∇ f n + t ( t + 1 ) 2 ! ∇ 2 f n + ⋯ + t ( t + 1 ) ⋯ ( t + n − 1 ) n ! ∇ n f n . \begin{aligned}N_n(x_n+th)&=f_n+t\:\nabla\:f_n+\frac{t(t+1)}{2!}\:\nabla^2\:f_n+\cdots+\frac{t(t+1)\cdots(t+n-1)}{n!}\:\nabla^nf_n.\end{aligned} Nn(xn+th)=fn+t∇fn+2!t(t+1)∇2fn+⋯+n!t(t+1)⋯(t+n−1)∇nfn.
上式称为 Newton 后插公式，其余项为
R n ( x ) = f ( x ) − N n ( x n + t h ) = t ( t + 1 ) ⋯ ( t + n ) h n + 1 f ( n + 1 ) ( ξ ) ( n + 1 ) ! , ξ ∈ ( x 0 , x n ) . R_n(x)=f(x)-N_n(x_n+th)=\frac{t(t+1)\cdots(t+n)h^{n+1}f^{(n+1)}(\xi)}{(n+1)!},\quad\xi\in(x_0,x_n). Rn(x)=f(x)−Nn(xn+th)=(n+1)!t(t+1)⋯(t+n)hn+1f(n+1)(ξ),ξ∈(x0,xn).