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In
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
, Gauss–Hermite quadrature is a form of
Gaussian quadrature In numerical analysis, an -point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree or less by a suitable choice of the nodes and weights for . Th ...
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 In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well a ...
''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 In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is f(x ...
: 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 In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and c ...
, 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

* * * *


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 Free software, libre software, libreware sometimes known as freedom-respecting software is computer software distributed open-source license, under terms that allow users to run the software for any purpose as well as to study, change, distribut ...
in C++, Fortran, and Matlab {{DEFAULTSORT:Gauss-Hermite quadrature Numerical integration Estimation methods