GaussHermite

Gauss-Hermite refers to a family of orthogonal polynomials with respect to the weight function e^{-x^2} on the real line and to the corresponding Gauss-Hermite quadrature rule. The polynomials are proportional to the classical Hermite polynomials H_n(x) and satisfy orthogonality ∫_{-∞}^{∞} e^{-x^2} H_m(x) H_n(x) dx = √π 2^n n! δ_mn. The Gauss-Hermite polynomials provide an efficient basis for approximating integrals weighted by a Gaussian factor.

Gauss-Hermite quadrature is a numerical integration method for integrals of the form ∫_{-∞}^{∞} e^{-x^2} f(x) dx. The integral is approximated by a weighted sum ∑_{i=1}^n w_i f(x_i), where the nodes x_i are the real zeros of H_n(x) and the weights w_i are given by w_i = √π 2^{n-1} / (n! [H_{n-1}(x_i)]^2). This construction yields exact results for all polynomials f of degree up to 2n−1 when multiplied by e^{-x^2}.

Practical use of Gauss-Hermite quadrature includes evaluating expectations under Gaussian distributions, quantum mechanics integrals, and other problems involving Gaussian weight factors. The nodes and weights can be precomputed and stored, or generated via the three-term recurrence relation for Hermite polynomials and their derivatives. The approach is particularly effective when the integrand contains a Gaussian factor or decays rapidly at infinity.

Gauss-Hermite is related to, but distinct from, other Gaussian quadrature schemes and is a standard tool in numerical analysis, physics, and statistics for high-accuracy integration of functions with Gaussian weighting.