Gauss–Kronrod quadrature formula: Difference between revisions

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Article said the resulting rule is of order 2n+1, but it is of order 3n+1 (see first reference).
 
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[[File:Gauss-Hermite quadrature weights.svg|thumb||Weights versus ''x<sub>i</sub>'' for four choices of ''n'']]
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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>
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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 ''x''<sub>''i''</sub> are the roots of the (physicists' version of the) [[Hermite polynomial]] ''H''<sub>''n''</sub>(''x'') (''i'' = 1,2,...,''n''), and the associated weights ''w''<sub>''i''</sub> are given by
<ref name="AS">[[Abramowitz and Stegun|Abramowitz, M & Stegun, I A, ''Handbook of Mathematical Functions'']], 10th printing with corrections (1972), Dover, ISBN 978-0-486-61272-0. Equation 25.4.46.</ref>
 
:<math>w_i = \frac {2^{n-1} n! \sqrt{\pi}} {n^2[H_{n-1}(x_i)]^2}.</math>
 
 
==Example with change of variable==
Let's take a function ''h'' which variable ''y'' is Normally distributed <math>\mathcal{N}(\mu,\sigma^2)</math>. The [[Expected value|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 doesn't exactly correspond to the Hermite polynomial, we need a change of variable:
 
<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==
<references/>
* {{dlmf| id = 3.5#P6 § 3.5 | title=Quadrature: Gauss–Hermite Formula}}
* {{cite journal|first1=T. S. | last1=Shao | first2=T. C. | last2=Chen | first3= R. M. | last3=Frank
|title=Tables of zeros and Gaussian weights of certain associated Laguerre polynomials and the related generalized Hermite polynomials
|year=1964
|journal=Math. Comp. | mr=0166397 | volume=18 | number=88 | pages=598–616 |doi=10.1090/S0025-5718-1964-0166397-1}}
* {{cite journal|first1=N. M. | last1=Steen | first2=G. D. | last2=Byrne
|first3=E. M. | last3=Gelbard | title = Gaussian quadratures for the integrals int_0^infty exp(-x^2) f(x) dx
|journal = Math. Comp. | year=1969
|volume=23 | number=107 | pages=661–671 | mr=0247744 | doi=10.1090/S0025-5718-1969-0247744-3 }}
* {{cite journal|first1= B. | last1=Shizgal | title = A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems
|journal=J. Comp. Phys. | volume=41 | pages=309–328 | year=1981
|doi=10.1016/0021-9991(81)90099-1 }}
* {{cite web
|url=http://vixra.org/abs/1303.0013 | first1= Richard J. | last1=Mathar
|title = Gauss-Laguerre and Gauss-Hermite Quadrature on 64, 96 and 128 Nodes
|year=2013 }}
 
==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.
* [http://people.sc.fsu.edu/~jburkardt/cpp_src/gen_hermite_rule/gen_hermite_rule.html Generalized Gauss–Hermite quadrature], [[free software]] in C++, Fortran, and Matlab
 
{{DEFAULTSORT:Gauss-Hermite quadrature}}
[[Category:Numerical integration (quadrature)]]

Latest revision as of 16:33, 24 July 2014

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