Gaussian quadrature

Summary

Gaussian quadrature is a quadrature rule with $n$ nodes in which both the nodes and the weights are free. They are chosen to make the rule exact for all polynomials of degree up to $2n - 1$. This is the highest degree of exactness that an $n$-point rule can reach. The nodes are the zeros of the degree $n$ orthogonal polynomial for the weight of the integral, and the weights are positive. For smooth integrands it converges very fast, and it is one of the standard tools of numerical integration.