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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 xi and weights wi 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)