Gauss–Hermite quadrature

Weights versus x_i 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:

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

In this case

${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{e}^{-x^2}f(x)dx\approx \sum _{i=1}^n{w}_{i}f(x_i)$

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

$w_i=\frac{2^{n-1}n!\sqrt{\pi }}{n^2[H_{n-1}(x_i){]}^2}.$

Example with change of variable

Consider a function h(y), where the variable y is normally distributed: $y\sim 𝒩(\mu ,{\sigma }^2)$. The expectation of h corresponds to the following integral:

$E[h(y)]={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{1}{\sigma \sqrt{2\pi }}\exp (-\frac{(y-\mu {)}^2}{2{\sigma }^2})h(y)dy$

As this does not exactly correspond to the Hermite polynomial, we need to change variables:

$x=\frac{y-\mu }{\sqrt{2}\sigma }\iff y=\sqrt{2}\sigma x+\mu$

Coupled with the integration by substitution, we obtain:

$E[h(y)]={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{1}{\sqrt{\pi }}\exp (-x^2)h(\sqrt{2}\sigma x+\mu )dx$

leading to:

$E[h(y)]\approx \frac{1}{\sqrt{\pi }}\sum _{i=1}^n{w}_{i}h(\sqrt{2}\sigma x_i+\mu )$

As an illustration, in the simplest non-trivial case, with $n=2$, we have $x_1=-\frac{1}{\sqrt{2}},x_2=\frac{1}{\sqrt{2}}$ and $w_1=w_2=\frac{\sqrt{\pi }}{2}$, so the estimate reduces to:

$E[h(y)]\approx \frac{1}{2}(h(\mu -\sigma )+h(\mu +\sigma ))$

– i.e. the average of the function's values one standard deviation below and above the mean.

References

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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