Hermite quadrature

Hermite-Gauss quadrature, also called Hermite quadrature, is a Gaussian quadrature over the interval (-∞, ∞) with weighting function W(x) = e^(-x^2). The abscissas for quadrature order n are given by the roots x_i of the Hermite polynomials H_n(x), which occur symmetrically about 0. The weights are w_i | = | -(A_(n + 1) γ_n)/(A_n H_n^, (x_i) H_(n + 1)(x_i)) | = | A_n/A_(n - 1) γ_(n - 1)/(H_(n - 1)(x_i) H_n^, (x_i)), where A_n is the coefficient of x^n in H_n(x).

Carl Friedrich Gauss | Charles Hermite