# divided difference interpolation formula

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

If $f$ is a real function and $x_{0},x_{1},\ldots$ is a sequence of distinct real numbers, then we have, for any integer $n>0$,

 $f(x)=f(x_{0})+(x-x_{0})\Delta f(x_{0},x_{1})+\cdots+(x-x_{0})\cdots(x-x_{n-1})% \Delta^{n}f(x_{0},\ldots x_{n})+R$

where the remainder can be expressed either as

 $R=(x-x_{0})\cdots(x-x_{n})\Delta^{n+1}f(x,x_{1},\ldots,x_{n})$

or as

 $R={1\over(n+1)!}(x-x_{0})\cdots(x-x_{n})f^{(n+1)}(\eta)$

where $\eta$ lies between the smallest and the largest of $x,x_{0},\ldots,x_{n}$.

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

Title divided difference interpolation formula DividedDifferenceInterpolationFormula 2013-03-22 16:19:13 2013-03-22 16:19:13 CWoo (3771) CWoo (3771) 8 CWoo (3771) Theorem msc 39A70