Also called Chebyshev Quadrature. A Gaussian Quadrature over the interval
with Weighting Function
. The Abscissas for quadrature order
are given by the roots of the
Chebyshev Polynomial of the First Kind
, which occur
symmetrically about 0. The Weights are
![\begin{displaymath}
w_i=-{A_{n+1}\gamma_n\over A_nT_n'(x_i)T_{n+1}(x_i)}={A_n\over A_{n-1}}{\gamma_{n-1}\over T_{n-1}(x_i)T_n'(x_i)},
\end{displaymath}](c1_1318.gif) |
(1) |
where
is the Coefficient of
in
. For Hermite Polynomials,
![\begin{displaymath}
A_n=2^{n-1},
\end{displaymath}](c1_1319.gif) |
(2) |
so
![\begin{displaymath}
{A_{n+1}\over A_n}=2.
\end{displaymath}](c1_1320.gif) |
(3) |
Additionally,
![\begin{displaymath}
\gamma_n={\textstyle{1\over 2}}\pi,
\end{displaymath}](c1_1321.gif) |
(4) |
so
![\begin{displaymath}
w_i=-{\pi\over T_{n+1}(x_i)T_n'(x_i)}.
\end{displaymath}](c1_1322.gif) |
(5) |
Since
![\begin{displaymath}
T_n(x)=\cos(n\cos^{-1}x),
\end{displaymath}](c1_1323.gif) |
(6) |
the Abscissas are given explicitly by
![\begin{displaymath}
x_i=\cos\left[{(2i-1)\pi\over 2n}\right].
\end{displaymath}](c1_1324.gif) |
(7) |
Since
where
![\begin{displaymath}
\alpha_i={(2i-1)\pi\over 2n},
\end{displaymath}](c1_1329.gif) |
(10) |
all the Weights are
![\begin{displaymath}
w_i={\pi\over n}.
\end{displaymath}](c1_1330.gif) |
(11) |
The explicit Formula is then
![\begin{displaymath}
\int_{-1}^1 {f(x)\,dx\over\sqrt{1-x^2}}={\pi\over n}\sum_{k=...
...2n}\pi}\right)}\right]+ {2\pi\over 2^{2n}(2n)!} f^{(2n)}(\xi).
\end{displaymath}](c1_1331.gif) |
(12) |
![$n$](c1_20.gif) |
![$x_i$](c1_908.gif) |
![$w_i$](c1_1332.gif) |
2 |
± 0.707107 |
1.5708 |
3 |
0 |
1.0472 |
|
± 0.866025 |
1.0472 |
4 |
± 0.382683 |
0.785398 |
|
± 0.92388 |
0.785398 |
5 |
0 |
0.628319 |
|
± 0.587785 |
0.628319 |
|
± 0.951057 |
0.628319 |
References
Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 330-331, 1956.
© 1996-9 Eric W. Weisstein
1999-05-26