Also called Hermite Quadrature. A Gaussian Quadrature over the interval
with
Weighting Function
. The Abscissas for quadrature order
are given by the
roots of the Hermite Polynomials
, 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}](h_1353.gif) |
(1) |
where
is the Coefficient of
in
. For Hermite Polynomials,
![\begin{displaymath}
A_n=2^n,
\end{displaymath}](h_1355.gif) |
(2) |
so
![\begin{displaymath}
{A_{n+1}\over A_n}=2.
\end{displaymath}](h_1356.gif) |
(3) |
Additionally,
![\begin{displaymath}
\gamma_n=\sqrt{\pi}\,2^nn!,
\end{displaymath}](h_1357.gif) |
(4) |
so
Using the Recurrence Relation
![\begin{displaymath}
H_n'(x)=2nH_{n-1}(x)=2xH_n(x)-H_{n+1}(x)
\end{displaymath}](h_1361.gif) |
(6) |
yields
![\begin{displaymath}
H_n'(x_i)=2nH_{n-1}(x_i)=-H_{n+1}(x_i)
\end{displaymath}](h_1362.gif) |
(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}](h_1363.gif) |
(8) |
The error term is
![\begin{displaymath}
E={n!\sqrt{\pi}\over 2^n(2n)!}f^{(2n)}(\xi).
\end{displaymath}](h_1364.gif) |
(9) |
Beyer (1987) gives a table of Abscissas and weights up to
=12.
![$n$](h_63.gif) |
![$x_i$](h_496.gif) |
![$w_i$](h_1365.gif) |
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
.
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.
© 1996-9 Eric W. Weisstein
1999-05-25