Gauss–Hermite quadrature

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Weights versus xi for four choices of n

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 [1]

Example with change of variable[edit]

Consider a function h(y), where the variable y is Normally distributed: . The expectation of h corresponds to the following integral:

As this doesn't exactly correspond to the Hermite polynomial, we need to change variables:

Coupled with the integration by substitution, we obtain:

leading to:


  1. ^ Abramowitz, M & Stegun, I A, Handbook of Mathematical Functions, 10th printing with corrections (1972), Dover, ISBN 978-0-486-61272-0. Equation 25.4.46.

External links[edit]