Gauss-Legendre quadrature

Gaussian quadrature is a general recipe; Gauss-Legendre is its most common concrete realization — the version for an ordinary, unweighted integral over a finite interval. It is the rule most people mean when they say 'Gauss quadrature', and it is the integration engine buried inside finite-element solvers and scientific libraries.

It targets the plain integral of f over [-1, 1] (any finite [a, b] is mapped onto [-1, 1] by a simple linear change of variable). The n nodes are the n roots of the Legendre polynomial P_n, the orthogonal polynomials for constant weight on [-1, 1]; the matching weights come from a closed formula involving P_n'. The result inherits the Gaussian guarantee: an n-point Gauss-Legendre rule integrates every polynomial up to degree 2n - 1 exactly. The nodes are symmetric about 0 and cluster gently toward the endpoints. In practice the nodes and weights are precomputed once (classically via the Golub-Welsch eigenvalue algorithm, which reads them off a small symmetric tridiagonal matrix) and reused for every integral of that order.

Gauss-Legendre is the default high-accuracy quadrature for smooth functions on a bounded interval, converging exponentially fast when f is analytic. The same honest limits as general Gaussian quadrature apply: the spectacular accuracy assumes smoothness, so a singularity or kink demands a different tool; the classical nodes are not nested, so you cannot cheaply reuse work when increasing n (this motivates Gauss-Kronrod extensions and the nested Clenshaw-Curtis rule); and remember the affine remapping introduces a Jacobian factor (b - a)/2 that you must not forget when integrating over a general [a, b].