# Gaussian quadrature

A

**quadrature integration rule** is a method of numerical approximation of the definite

integral of a

function, particularly as a weighted sum of function values at quadrature points within the domain of integration:

**Gaussian quadrature** rules attempt to give the most accurate possible formulae by choosing the quadrature points

*x*_{i} and weights

*w*_{i} to give exact results for polynomials of the highest degree possible. For quadrature of a function of one variable,

*n* Gaussian quadrature points will give accurate integrals for all polynomials of degree up to 2

*n* - 1.

In one dimension, on the domain (-1, 1), some low order polynomials can be integrated as follows:

1-D Gaussian Quadrature Rules
Number of points |
Quadrature weights |
Quadrature points |

1 |
2 |
0 |

2 |
1, 1 |
-1/√3, 1/√3 |

3 |
5/9, 8/9, 5/9 |
-√(3/5), 0, √(3/5) |