# Approximating Integrals with the Gaussian Quadrature (w/ example)

This is more like a memory dump so I will have a backup in case I’ll ever need it again. And if someone else finds this information useful, the better it is.

## Intro

The **Gaussian Quadrature** is a method used to approximate the value of a given integral by choosing a set of points (**x _{1}, x_{2}, x_{3}, … x_{n}**) that will maximize the accuracy. Basically the integral can be approximated using coefficients and known values of our function. It’s a pretty neat method since it doesn’t require many points and it works for a set of integrals - not only for one.

On the example above, we have an integral that we want to approximate and on the right side of the equal sign is the **Gaussian Quadrature**.

**f(x)** is our function (it’s not required to know how it looks like), **w(x)** is a **weight function**.

As you noticed we’ll be working within the interval **[-1;1]** - and we’ll consider **w(x) = 1** (Legendre).

*- But…what if the integral has a different interval?*; use this little trick:

Our purpose is to find a set of coefficients (**A _{1}, A_{2}, … A_{n}**) and a set of points (

**x**) that will make this quadrature as exact as possible, for a set of integrals.

_{1}, x_{2}, … x_{n}To keep everything simple we’ll take **n=2** so our equation will look like this:

From now on I’ll be working with this example, however it’s the same method when **n** has a different value.

## How to find the ‘x’ values

In order to calculate the coefficients (**A _{1}, A_{2}**) we need to know the values of

**x**and

_{1}**x**. If these are not given, we will have to calculate them using some theory implying

_{2}**orthogonal polynomials**.

__Now here's how you do this:__

Create a polynomial (usually noted **π**), that has **n=2** roots: **x _{1}** and

**x**.

_{2}It will look like this:

Here you need to know this little part of theory related to **orthogonal polynomials**:

- polynomials
**P**and**Q**are orthogonal if their**inner product**is**0**(**<p, q>= 0**) - a polynomial of degree
**n**is orthogonal to any other polynomial of degree lower than**n**

These properties are required in order to produce a system of equations which will provide us the values for **x _{1}** and

**x**.

_{2}We define that **inner product** that I was talking about earlier:

where **P** and **Q** are 2 polynomials and **w(x)** is the **weight function**.

Ok enough with the theory, moving back to our example: we had that **π** polynomial.

Now we pick the first **n=2** terms from the **monomial basis** (**1**, **x**, x^{2}, x^{3}…).

Notice that those 2 “polynomials” are of degree **0** and **1** - both lower than **n=2**(the degree of our **π** polynomial) => these are **orthogonal** to **π**. Knowing this we can create the following system of equations:

Note that in there I already did the substitution **w(x)=1**.

By solving the integrals…

So we get these values:

\[x_1 = -\frac{\sqrt{3}}{3}\] \[x_2 = \frac{\sqrt{3}}{3}\]## Finding the coefficients

This is the easy part - if you managed to get here then you’re almost done.

To find the values of the coefficients, we’ll use the same **Gauss Quadrature**, but this time on these functions:

**f _{1}(x) = 1
f_{2}(x) = x**

where **1** and **x** are the monomials we chose before.

If we rewrite the **Gaussian Quadrature** for these 2 functions we get:

(substituted **w(x) = 1**)

So **A _{1} = A_{2} = 1**, hence the

__final equation__:

## Precision

The degree of precision for this example is **2*n-1 = 3**. This means if the integrand is a polynomial of degree **3** (or lower) the result will be exact. If not, the result will be an approximation.

Note that this also works for non-polynomial functions; if we integrate **cos(x)** from **-1** to **1**

**original**result:**2*sin(1) ~ 1.6829**- result approximated with the
**Gaussian Quadrature**:**cos(-sqrt(3)/3) + cos(sqrt(3)/3) = 1.6758**

Not the best result, but will do - usually adding more points helps gaining precision.