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we have the following theorem:

Theorem: Let
be points on
the interval . Let be the corresponding values of
at . Let be the polynomial of degree n which
interpolates at the n+1 points
Then if

Ψ(x)=(x-x_{0})(x-x_{1}).....(x-x_{n})

,

for some point in .
Proof: We have from the error formula for the interpolating polynomial

If does not change sign on the interval , then we
can apply the second mean value theorem of integral calculus, we
obtain the desired result. Let us assume that the function
can be computed at a set of n+1 equally spaced points
. Then can be approximated by the Newton
forward difference interpolating polynomial

We integrate over the double interval of width and get

The error of this formula is given by

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