where n is the number of sample points used. The xi are the roots of the physicists' version of the Hermite polynomialHn(x) (i = 1,2,...,n), and the associated weights wi are given by[1]
Example with change of variable
Consider a function h(y), where the variable y is Normally distributed: . The expectation of h corresponds to the following integral:
As this does not exactly correspond to the Hermite polynomial, we need to change variables:
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 ∫ 0 ∞ e − x 2 f ( x ) d x {\displaystyle \textstyle \int _{0}^{\infty }e^{-x^{2}}f(x)dx} and ∫ 0 b e − x 2 f ( x ) d x {\displaystyle \textstyle \int _{0}^{b}e^{-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. Comput. Phys. 41: 309–328. doi:10.1016/0021-9991(81)90099-1.
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