Hermite-Gauss quadrature, also called Hermite quadrature, is a Gaussian quadrature over the interval 
with weighting
function 
(Abramowitz and Stegun 1972, p. 890). The abscissas
for quadrature order 
are given by the roots 
of the Hermite polynomials 
,
which occur symmetrically about 0. The weights are
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is the coefficient of 
in 
. For Hermite polynomials,
 |
(3)
|
so
 |
(4)
|
Additionally,
 |
(5)
|
so
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  | ![(2^(n+1)n!sqrt(pi))/([H_n^'(x_i)]^2)](/images/equations/Hermite-GaussQuadrature/Inline23.svg) |
(8)
|
 |  | ![(2^(n+1)n!sqrt(pi))/([H_(n+1)(x_i)]^2)](/images/equations/Hermite-GaussQuadrature/Inline26.svg) |
(9)
|
 |  | ![(2^(n-1)n!sqrt(pi))/(n^2[H_(n-1)(x_i)]^2),](/images/equations/Hermite-GaussQuadrature/Inline29.svg) |
(10)
|
where (8) and (9) follow using the recurrence relation
 |
(11)
|
to obtain
 |
(12)
|
and (10) is from Abramowitz and Stegun (1972 p. 890).
The error term is
 |
(13)
|
Beyer (1987) gives a table of abscissas and weights up to 
.
 |  |  |
| 2 |  | 0.886227 |
| 3 | 0 | 1.18164 |
 | 0.295409 | |
| 4 |  | 0.804914 |
 | 0.0813128 | |
| 5 | 0 | 0.945309 |
 | 0.393619 | |
 | 0.0199532 |
The abscissas and weights can be computed analytically for small 
.
 |  |  |
| 2 |  |  |
| 3 | 0 |  |
 |  | |
| 4 |  |  |
 |  |
