Hermite-Gauss Quadrature

Also called Hermite Quadrature. A Gaussian Quadrature over the interval $(-\infty, \infty)$ with Weighting Function $W(x)=e^{-x^2}$. The Abscissas for quadrature order $n$ are given by the roots of the Hermite Polynomials $H_n(x)$, which occur symmetrically about 0. The Weights are

$$w_i=-{A_{n+1}\gamma_n\over A_nH_n'(x_i)H_{n+1}(x_i)} ={A_n\over A_{n-1}}{\gamma_{n-1}\over H_{n-1}(x_i)H_n'(x_i)},$$ (1)

where $A_n$ is the Coefficient of $x^n$ in $H_n(x)$. For Hermite Polynomials,

$$A_n=2^n,$$ (2)

so

$${A_{n+1}\over A_n}=2.$$ (3)

Additionally,

$$\gamma_n=\sqrt{\pi}\,2^nn!,$$ (4)

so

$$w_i = -{2^{n+1}n!\sqrt{\pi}\over H_{n+1}(x_i)H_n'(x_i)}$$

$$= {2^n(n-1)!\sqrt{\pi}\over H_{n-1}(x_i)H_n'(x_i)}.$$ (5)

Using the Recurrence Relation

$$H_n'(x)=2nH_{n-1}(x)=2xH_n(x)-H_{n+1}(x)$$ (6)

yields

$$H_n'(x_i)=2nH_{n-1}(x_i)=-H_{n+1}(x_i)$$ (7)

and gives

$$w_i={2^{n+1}n!\sqrt{\pi}\over[H_n'(x_i)]^2}={2^{n+1}n!\sqrt{\pi}\over[H_{n+1}(x_i)]^2}.$$ (8)

The error term is

$$E={n!\sqrt{\pi}\over 2^n(2n)!}f^{(2n)}(\xi).$$ (9)

Beyer (1987) gives a table of Abscissas and weights up to $n$=12.

$n$	$x_i$	$w_i$
2	± 0.707107	0.886227
3	0	1.18164
	± 1.22474	0.295409
4	± 0.524648	0.804914
	± 1.65068	0.0813128
5	0	0.945309
	± 0.958572	0.393619
	± 2.02018	0.0199532

The Abscissas and weights can be computed analytically for small $n$.

$n$	$x_i$	$w_i$
2	$\pm{\textstyle{1\over 2}}\sqrt{2}$	${\textstyle{1\over 2}}\sqrt{\pi}$
3	0	${\textstyle{2\over 3}}\sqrt{\pi}$
	$\pm{\textstyle{1\over 2}}\sqrt{6}$	${\textstyle{1\over 6}}\sqrt{\pi}$
4	$\pm \sqrt{3-\sqrt{6}\over 2}$	${\sqrt{\pi}\over 4(3-\sqrt{6}\,)}$
	$\pm \sqrt{3+\sqrt{6}\over 2}$	${\sqrt{\pi}\over 4(3+\sqrt{6}\,)}$

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 464, 1987.

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 327-330, 1956.