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
|
(1) |
where is the Coefficient of in . For Hermite Polynomials,
|
(2) |
so
|
(3) |
Additionally,
|
(4) |
so
|
(5) |
Since
|
(6) |
the Abscissas are given explicitly by
|
(7) |
Since
where
|
(10) |
all the Weights are
|
(11) |
The explicit Formula is then
|
(12) |
|
|
|
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