Gauss–Hermite quadrature

File:Gauss-Hermite quadrature weights.svg

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:

$\int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx.$

In this case

$\int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)$

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

Let's take a function h which variable y is Normally distributed $\mathcal{N}(\mu,\sigma^2)$. The expectation of h corresponds to the following integral:

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

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

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

Coupled with the integration by substitution, we obtain:

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

leading to:

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

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