Английская Википедия: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:
- <math>\int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx.</math>
In this case
- <math>\int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)</math>
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]
- <math>w_i = \frac {2^{n-1} n! \sqrt{\pi}} {n^2[H_{n-1}(x_i)]^2}.</math>
Example with change of variable
Consider a function h(y), where the variable y is Normally distributed: <math> y \sim \mathcal{N}(\mu,\sigma^2)</math>. The expectation of h corresponds to the following integral:
<math>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</math>
As this does not exactly correspond to the Hermite polynomial, we need to change variables:
<math>x = \frac{y-\mu}{\sqrt{2} \sigma} \Leftrightarrow y = \sqrt{2} \sigma x + \mu</math>
Coupled with the integration by substitution, we obtain:
<math>E[h(y)] = \int_{-\infty}^{+\infty} \frac{1}{\sqrt{\pi}} \exp(-x^2) h(\sqrt{2} \sigma x + \mu) dx</math>
leading to:
<math>E[h(y)] \approx \frac{1}{\sqrt{\pi}} \sum_{i=1}^n w_i h(\sqrt{2} \sigma x_i + \mu)</math>
References
- ↑ Abramowitz, M & Stegun, I A, Handbook of Mathematical Functions, 10th printing with corrections (1972), Dover, Шаблон:ISBN. Equation 25.4.46.
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
