Simpson's rule
In computer science, in the field of numerical analysis, Simpson's Rule is a way to get an approximation of an integral:
using an interpolating polynomial of higher degree. Simpson's rule belong to the family of rules derived from Newton-Cotes formulas. The most common is a quadratic polynomial interpolating at a, (a+b)/2, and b which gives us the polynomial:
From this Simpson's Rule is:
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:
Which yields:
We then integrate our polynomial:
Substitute back our original values:
Q.E.D.
Error of Simpson's Rule
To examine the accuracy of the rule, take  , so
Using integration by parts we get:
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
is the error value.
Referenced By
Linear interpolation | List of calculus topics | List of mathematical proofs | List of mathematical topics (S-U) | List of numerical analysis topics | List of proofs | Newton-Cotes formula | Newton-Cotes formulas | Numerical integration | Simpson
|
