Gauss–Hermite quadrature

Weights versus x_i for four choices of n

In numerical analysis, Gauss–Hermite quadrature is 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

$w_i = \frac {2^{n-1} n! \sqrt{\pi}} {n^2[H_{n-1}(x_i)]^2}.$

References

• Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., eds. (2010), § 3.5 "Quadrature: Gauss–Hermite Formula", NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248 
• Shao, T. S.; Chen, T. C.; Frank, R. M. (1964). "Tables of zeros and Gaussian weights of certain associated Laguerre polynomials and the related generalized Hermite polynomials". Math. Comp. 18 (88): 598–616. doi:10.1090/S0025-5718-1964-0166397-1. MR 0166397. 
• Steen, N. M.; Byrne, G. D.; Gelbard, E. M. (1969). "Gaussian quadratures for the integrals int_0^infty exp(-x^2) f(x) dx". Math. Comp. 23 (107): 661–671. doi:10.1090/S0025-5718-1969-0247744-3. MR 0247744. 
• Shizgal, B. (1981). "A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems". J. Comp. Phys. 41: 309–328. doi:10.1016/0021-9991(81)90099-1. 
• Mathar, Richard J. (2013). "Gauss-Laguerre and Gauss-Hermite Quadrature on 64, 96 and 128 Nodes". 

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