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 Abramowitz, M & Stegun, I A, Handbook of Mathematical Functions, 10th printing with corrections (1972), Dover, . Equation 25.4.46.
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:
As an illustration, in the simplest non-trivial case, with , we have and , so the estimate reduces to:
– i.e. the average of the function's values one standard deviation below and above the mean.
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