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
NewtonCotes 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
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 P
_{y}(x) by:
we have
where

is the error value.