Hermite-Gauss Quadrature

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)
			(8)
			(9)
			(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

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 890, 1972.

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.