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 2n - 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)