Gauss–Hermite quadrature

Remarks

This is the Gaussian quadrature rule for the weight $e^{-x^2}$ on $\R$, whose orthogonal polynomials are the $H_n$. For $p$ of degree at most $2n - 1$, write $p = q H_n + r$ with $\deg q, \deg r \le n - 1$. Orthogonality kills $\int e^{-x^2} q H_n$, and $r$ is integrated exactly by the interpolatory weights. The closed form of $w_i$ comes from the Christoffel–Darboux formula. Example: for $n = 2$, $H_2 = 4x^2 - 2$ has zeros $\pm 1/\sqrt{2}$, and $H_1(x) = 2x$ gives $w_1 = w_2 = \frac{4\sqrt{\pi}}{4 \cdot 2} = \frac{\sqrt{\pi}}{2}$. Check with $p(x) = x^2$: the rule gives $2 \cdot \frac{\sqrt{\pi}}{2} \cdot \frac12 = \frac{\sqrt{\pi}}{2} = \int_{\R} x^2 e^{-x^2} dx$. For $f \in C^{2n}(\R)$ with suitable growth, the error is $\frac{n! \sqrt{\pi}}{2^n} \frac{f^{(2n)}(\xi)}{(2n)!}$ for some real $\xi$. The degree $2n - 1$ is the best possible: $p = H_n^2$ has degree $2n$, the rule gives $0$, but the integral is positive.

Going deeper

For a student

The rule computes expectations under a normal distribution: if $Z$ is standard normal, then $\mathbb{E}[g(Z)] = \frac{1}{\sqrt{\pi}} \int_{\R} e^{-x^2} g(\sqrt{2} x) \, dx$. It is used in statistics, in quantum mechanics with harmonic oscillator states, and in uncertainty quantification. For functions that are not smooth or that grow fast, the nodes spread out and accuracy can be poor, and a change of variable may help.