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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) |
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(2) |
where is the coefficient of in . For Hermite polynomials,
(3) |
so
(4) |
Additionally,
(5) |
so
(6) |
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(7) |
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(8) |
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(9) |
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(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.
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