Gauss–Hermite Quadrature

In numerical analysis, Gauss–Hermite quadrature is a form of Gaussian quadrature for approximating the value of integrals of the following kind:

:$\int_^ e^ f(x)\,dx.$

In this case

:$\int_^ e^ f(x)\,dx \approx \sum_^n w_i f(x_i)$

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
Abramowitz, M & Stegun, I A, ''Handbook of Mathematical Functions'', 10th printing with corrections (1972), Dover, . Equation 25.4.46.

:$w_i = \frac .$

Example with change of variable

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

E (y)= \int_^ \frac \exp \left( -\frac \right) h(y) dy

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

$x = \frac \Leftrightarrow y = \sqrt \sigma x + \mu$

Coupled with the integration by substitution, we obtain:

E (y)= \int_^ \frac \exp(-x^2) h(\sqrt \sigma x + \mu) dx

leading to:

E (y)\approx \frac \sum_^n w_i h(\sqrt \sigma x_i + \mu)

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

E (y)\approx \frac(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

{{DEFAULTSORT:Gauss-Hermite quadrature
Numerical integration
Estimation methods