Also called Hermite Quadrature. A Gaussian Quadrature over the interval with Weighting Function . The Abscissas for quadrature order are given by the roots of the Hermite Polynomials , which occur symmetrically about 0. The Weights are

 (1)

where is the Coefficient of in . For Hermite Polynomials,
 (2)

so
 (3)

 (4)

so
 (5)

Using the Recurrence Relation
 (6)

yields
 (7)

and gives
 (8)

The error term is
 (9)

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

 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 .

 2 3 0 4

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.