divided difference interpolation formula

Newton’s divided difference interpolation formula is the analogueof the Gregory-Newton and Taylor series for divided differences.

If $f$ is a real function and $x_0,x_1,\mathrm{\dots }$ is a sequence of distinctreal numbers, then we have, for any integer $n>0$,

$$f(x)=f(x_0)+(x-x_0)\mathrm{\Delta }f(x_0,x_1)+\mathrm{\cdots }+(x-x_0)\mathrm{\cdots }(x-x_{n-1}){\mathrm{\Delta }}^{n}f(x_0,\mathrm{\dots }{x}_n)+R$$

where the remainder can be expressed either as

$$R=(x-x_0)\mathrm{\cdots }(x-x_n){\mathrm{\Delta }}^{n+1}f(x,x_1,\mathrm{\dots },x_n)$$

or as

$$R=\frac{1}{(n+1)!}(x-x_0)\mathrm{\cdots }(x-x_n)f^{(n+1)}(\eta )$$

where $\eta$ lies between the smallest and the largest of $x,x_0,\mathrm{\dots },x_n$.

Remark. If $f$ is a polynomial of degree $n$, then $R$ vanishes.