Gaussian quadrature

A quadrature integration rule ismethodnumerical approximation ofdefinite integral offunction, particularly asweighted sumfunction values at quadrature points withindomainintegration:

Gaussian quadrature rules attemptgivemost accurate possible formulae by choosingquadrature points x_iweights w_igive exact resultspolynomials ofhighest degree possible. For quadrature offunctionone variable, n Gaussian quadrature points will give accurate integralsall polynomialsdegree up2n - 1.

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

1-D Gaussian Quadrature Rules

Numberpoints	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)