Gauss–Legendre quadrature

Summary

Gauss–Legendre quadrature is Gaussian quadrature for the constant weight $1$ on the interval $[-1, 1]$. Its $n$ nodes are the zeros of the Legendre polynomial $P_n$, and it integrates every polynomial of degree at most $2n - 1$ exactly. An integral over any bounded interval $[a, b]$ is handled by a linear change of variable. It is the default Gaussian rule for smooth functions on a bounded interval.