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: In this case where n is the number of sample points used. The xi are the roots of the physicists' version of the Hermite polynomial Hn(x) (i = 1,2,...,n), and the associated weights wi are given by Example with change of variable Consider a function h(y), where the variable y is Normally distributed: . The expectation of h corresponds to the following integral: As this does not exactly correspond to the Hermite polynomial, we need to change variables: Coupled with the integration by substitution, we obtain: leading to: References External links For tables of Gauss-Hermite abscissae and weights up to order n = 32 see http://www.efunda.com/math/num_integration/findgausshermite.cfm. Generalized Gauss–Hermite quadrature, free software in C++, Fortran, and Matlab Numerical integration (quadrature) Estimation methods