Chebyshev Quadrature
A Gaussian Quadrature-like Formula for numerical estimation of integrals. It uses Weighting Function
in the interval 
and forces all the weights to be equal. The general Formula is
The Abscissas are found by taking terms up to
in the Maclaurin Series of
and then defining
The Roots of
then give the Abscissas. The first few values are
Because the Roots are all Real for
and 
only (Hildebrand 1956), these are the only permissible orders forChebyshev quadrature. The error term is
where
The first few values of
are 2/3, 8/45, 1/15, 32/945, 13/756, and 16/1575 (Hildebrand 1956). Beyer (1987) givesabscissas up to 
and Hildebrand (1956) up to . |  |
| 2 | ± 0.57735 |
| 3 | 0 |
| ± 0.707107 | |
| 4 | ± 0.187592 |
| ± 0.794654 | |
| 5 | 0 |
| ± 0.374541 | |
| ± 0.832497 | |
| 6 | ± 0.266635 |
| ± 0.422519 | |
| ± 0.866247 | |
| 7 | 0 |
| ± 0.323912 | |
| ± 0.529657 | |
| ± 0.883862 | |
| 9 | 0 |
| ± 0.167906 | |
| ± 0.528762 | |
| ± 0.601019 | |
| ± 0.911589 |
The Abscissas and weights can be computed analytically for small .
| |
| 2 |  |
| 3 | 0 |
 | |
| 4 |  |
 | |
| 5 | 0 |
 | |
 |
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 466, 1987. Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 345-351, 1956.
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