of higher degree. Simpson's rule belong to the family of rules derived from
. The most common is a quadratic polynomial interpolating at
Proof
We want to have our polynomial on the form:
Assume we have the function values , and . The situation will look like this, with our sampled function values at , and :
As this Simpson's rule apply to equidistant points, we know that and that . This means we may transport our solution to the intervals formed by such that
We need to interpolate these values and function values with a polynomial and form our equations:
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Which yields:
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We then integrate our polynomial:
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Substitute back our original values:
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Q.E.D
Error of Simpson's Rule
To examine the accuracy of the rule, take , so
Using integration by parts we get:
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and
where α and β are constants that we can choose. Adding these expressions, we get:
Let's take α and β, so as to get Simpson's Rule:
and defining the function Py(x) by:
we have
where
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is the error value.