“Hermite-Gauss Quadrature”的概念、定义、习题解题方法及翻译-数学辞典

单词

Hermite-Gauss Quadrature

释义

Hermite-Gauss Quadrature

Also called Hermite Quadrature. A Gaussian Quadrature over the interval withWeighting Function . The Abscissas for quadrature order are given by theroots of the Hermite Polynomials , which occur symmetrically about 0. TheWeights are

(1)

where

is the Coefficient of

. For Hermite Polynomials,

(2)

(3)

Additionally,

(4)


			(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.

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