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 P_y(x) by:

we have

where

is the error value.