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:

P(x) = αx² + βx + γ

Assume we have the function values a = x₀,   and b = x₂. The situation will look like this, with our sampled function values at f(a),   and f(b):

As this Simpson's rule apply to equidistant points, we know that x₀ < x₁ < x₂ and that x₁ - x₀ = x₂ - x₁. This means we may transport our solution to the intervals formed by - h,0,h such that

We need to interpolate these values and function values with a polynomial and form our equations:

f( - h) = αh² - βh + γ

f(0) = γ

f(h) = αh² + βh + γ

Which yields:

 

 

γ = f(0)

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.