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Hermite-Gauss Quadrature

Also called Hermite Quadrature. A Gaussian Quadrature over the interval $(-\infty, \infty)$ with Weighting Function $W(x)=e^{-x^2}$. The Abscissas for quadrature order $n$ are given by the roots of the Hermite Polynomials $H_n(x)$, which occur symmetrically about 0. The Weights are

\begin{displaymath}
w_i=-{A_{n+1}\gamma_n\over A_nH_n'(x_i)H_{n+1}(x_i)}
={A_n\over A_{n-1}}{\gamma_{n-1}\over H_{n-1}(x_i)H_n'(x_i)},
\end{displaymath} (1)

where $A_n$ is the Coefficient of $x^n$ in $H_n(x)$. For Hermite Polynomials,
\begin{displaymath}
A_n=2^n,
\end{displaymath} (2)

so
\begin{displaymath}
{A_{n+1}\over A_n}=2.
\end{displaymath} (3)

Additionally,
\begin{displaymath}
\gamma_n=\sqrt{\pi}\,2^nn!,
\end{displaymath} (4)

so
$\displaystyle w_i$ $\textstyle =$ $\displaystyle -{2^{n+1}n!\sqrt{\pi}\over H_{n+1}(x_i)H_n'(x_i)}$  
  $\textstyle =$ $\displaystyle {2^n(n-1)!\sqrt{\pi}\over H_{n-1}(x_i)H_n'(x_i)}.$ (5)

Using the Recurrence Relation
\begin{displaymath}
H_n'(x)=2nH_{n-1}(x)=2xH_n(x)-H_{n+1}(x)
\end{displaymath} (6)

yields
\begin{displaymath}
H_n'(x_i)=2nH_{n-1}(x_i)=-H_{n+1}(x_i)
\end{displaymath} (7)

and gives
\begin{displaymath}
w_i={2^{n+1}n!\sqrt{\pi}\over[H_n'(x_i)]^2}={2^{n+1}n!\sqrt{\pi}\over[H_{n+1}(x_i)]^2}.
\end{displaymath} (8)

The error term is
\begin{displaymath}
E={n!\sqrt{\pi}\over 2^n(2n)!}f^{(2n)}(\xi).
\end{displaymath} (9)

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

$n$ $x_i$ $w_i$
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 $n$.

$n$ $x_i$ $w_i$
2 $\pm{\textstyle{1\over 2}}\sqrt{2}$ ${\textstyle{1\over 2}}\sqrt{\pi}$
3 0 ${\textstyle{2\over 3}}\sqrt{\pi}$
  $\pm{\textstyle{1\over 2}}\sqrt{6}$ ${\textstyle{1\over 6}}\sqrt{\pi}$
4 $\pm \sqrt{3-\sqrt{6}\over 2}$ ${\sqrt{\pi}\over 4(3-\sqrt{6}\,)}$
  $\pm \sqrt{3+\sqrt{6}\over 2}$ ${\sqrt{\pi}\over 4(3+\sqrt{6}\,)}$


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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© 1996-9 Eric W. Weisstein
1999-05-25