Gauss–Hermite quadrature

In numerical analysis, Gauss–Hermite quadrature is a form of Gaussian quadrature for approximating the value of integrals of the following kind:

${\displaystyle \int _{-\infty }^{+\infty }e^{-x^{2}}f(x)\,dx.}$

In this case

${\displaystyle \int _{-\infty }^{+\infty }e^{-x^{2}}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i})}$

where n is the number of sample points used. The x_i are the roots of the physicists' version of the Hermite polynomial H_n(x) (i = 1,2,...,n), and the associated weights w_i are given by

${\displaystyle w_{i}={\frac {2^{n-1}n!{\sqrt {\pi }}}{n^{2}[H_{n-1}(x_{i})]^{2}}}.}$