Legendre quadrature

Legendre-Gauss quadrature is a numerical integration method also called "the" Gaussian quadrature or Legendre quadrature. A Gaussian quadrature over the interval [-1, 1] with weighting function W(x) = 1. The abscissas for quadrature order n are given by the roots of the Legendre polynomials P_n(x), which occur symmetrically about 0. The weights are w_i | = | -(A_(n + 1) γ_n)/(A_n P_n^, (x_i) P_(n + 1)(x_i)) | = | A_n/A_(n - 1) γ_(n - 1)/(P_(n - 1)(x_i) P_n^, (x_i)), where A_n is the coefficient of x^n in P_n(x).

Adrien-Marie Legendre | Carl Friedrich Gauss