Weights versus xi 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 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]

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

## References

1. ^ 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.