Gauss–Hermite quadrature
Form of Gaussian quadrature
From Wikipedia, the free encyclopedia
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]
![{\displaystyle w_{i}={\frac {2^{n-1}n!{\sqrt {\pi }}}{n^{2}[H_{n-1}(x_{i})]^{2}}}.}](http://wikimedia.org/api/rest_v1/media/math/render/svg/2e6f152a1e9ecd4ab8ddf912aaa69bb8d0e66a3c)
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:
![{\displaystyle E[h(y)]=\int _{-\infty }^{+\infty }{\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {(y-\mu )^{2}}{2\sigma ^{2}}}\right)h(y)dy}](http://wikimedia.org/api/rest_v1/media/math/render/svg/4aed1ad69a06a54aca7ef1b6cf0d2f1ee8b5462e)
As this does not exactly correspond to the Hermite polynomial, we need to change variables:
Coupled with the integration by substitution, we obtain:
![{\displaystyle E[h(y)]=\int _{-\infty }^{+\infty }{\frac {1}{\sqrt {\pi }}}\exp(-x^{2})h({\sqrt {2}}\sigma x+\mu )dx}](http://wikimedia.org/api/rest_v1/media/math/render/svg/7a9f8ddc4824a2e554c833573d0e02eb907830ef)
leading to:
![{\displaystyle E[h(y)]\approx {\frac {1}{\sqrt {\pi }}}\sum _{i=1}^{n}w_{i}h({\sqrt {2}}\sigma x_{i}+\mu )}](http://wikimedia.org/api/rest_v1/media/math/render/svg/dd381d046b51bf2b8ec4f1368342fc2cec0c13e2)
As an illustration, in the simplest non-trivial case, with , we have and , so the estimate reduces to:
![{\displaystyle E[h(y)]\approx {\frac {1}{2}}(h(\mu -\sigma )+h(\mu +\sigma ))}](http://wikimedia.org/api/rest_v1/media/math/render/svg/f09c7dac955ed2691e0f232d5a26b8f9b9d66190)
– i.e. the average of the function's values one standard deviation below and above the mean.
