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divided difference interpolation formula

(Theorem)

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

If is a real function and $x_0, x_1, \ldots$ is a sequence of distinct real numbers, then we have, for any integer ,

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

where the remainder can be expressed either as

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

or as

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

where $\eta$ lies between the smallest and the largest of $x, x_0, \ldots, x_n$ .

Remark. If is a polynomial of degree , then vanishes.

"divided difference interpolation formula" is owned by CWoo. [ full author list (2) | owner history (1) ]

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Attachments:: example of divided difference interpolaton (Example) by rspuzio
solution of equations by divided difference interpolaton (Application) by rspuzio

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39A70 (Difference and functional equations :: Difference equations :: Difference operators)

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