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Hermite-Gauss Quadrature


Hermite-Gauss quadrature, also called Hermite quadrature, is a Gaussian quadrature over the interval (-infty,infty) with weighting function W(x)=e^(-x^2) (Abramowitz and Stegun 1972, p. 890). The abscissas for quadrature order n are given by the roots x_i of the Hermite polynomials H_n(x), which occur symmetrically about 0. The weights are

w_i=-(A_(n+1)gamma_n)/(A_nH_n^'(x_i)H_(n+1)(x_i))
(1)
=(A_n)/(A_(n-1))(gamma_(n-1))/(H_(n-1)(x_i)H_n^'(x_i)),
(2)

where A_n is the coefficient of x^n in H_n(x). For Hermite polynomials,

 A_n=2^n,
(3)

so

 (A_(n+1))/(A_n)=2.
(4)

Additionally,

 gamma_n=sqrt(pi)2^nn!,
(5)

so

w_i=-(2^(n+1)n!sqrt(pi))/(H_(n+1)(x_i)H_n^'(x_i))
(6)
=(2^n(n-1)!sqrt(pi))/(H_(n-1)(x_i)H_n^'(x_i))
(7)
=(2^(n+1)n!sqrt(pi))/([H_n^'(x_i)]^2)
(8)
=(2^(n+1)n!sqrt(pi))/([H_(n+1)(x_i)]^2)
(9)
=(2^(n-1)n!sqrt(pi))/(n^2[H_(n-1)(x_i)]^2),
(10)

where (8) and (9) follow using the recurrence relation

 H_n^'(x)=2nH_(n-1)(x)=2xH_n(x)-H_(n+1)(x)
(11)

to obtain

 H_n^'(x_i)=2nH_(n-1)(x_i)=-H_(n+1)(x_i),
(12)

and (10) is from Abramowitz and Stegun (1972 p. 890).

The error term is

 E=(n!sqrt(pi))/(2^n(2n)!)f^((2n))(xi).
(13)

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

nx_iw_i
2+/-0.7071070.886227
301.18164
+/-1.224740.295409
4+/-0.5246480.804914
+/-1.650680.0813128
500.945309
+/-0.9585720.393619
+/-2.020180.0199532

The abscissas and weights can be computed analytically for small n.

nx_iw_i
2+/-1/2sqrt(2)1/2sqrt(pi)
302/3sqrt(pi)
+/-1/2sqrt(6)1/6sqrt(pi)
4+/-sqrt((3-sqrt(6))/2)(sqrt(pi))/(4(3-sqrt(6)))
+/-sqrt((3+sqrt(6))/2)(sqrt(pi))/(4(3+sqrt(6)))

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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.

Referenced on Wolfram|Alpha

Hermite-Gauss Quadrature

Cite this as:

Weisstein, Eric W. "Hermite-Gauss Quadrature." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Hermite-GaussQuadrature.html

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