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Homedivided difference interpolation formula

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# 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.

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## Mathematics Subject Classification

39A70*no label found*

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