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

(1)

where the abscissas  are found by taking terms up to  in the Maclaurin series of

(2)

and then defining

(3)

The roots of  then give the abscissas. The first few values are

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)

(OEIS A002680 and A101270).

Because the roots are all real for  and  only (Hildebrand 1956), these are the only permissible orders for Chebyshev quadrature. The error term is

(14)

where

(15)

The first few values of  are 2/3, 8/45, 1/15, 32/945, 13/756, and 16/1575 (Hildebrand 1956). Beyer (1987) gives abscissas up to  and Hildebrand (1956) up to .

2
3	0
4
5	0
6
7	0
9	0

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.

Salzer, H. E. "Tables for Facilitating the Use of Chebyshev's Quadrature Formula." J. Math. Phys. 26, 191-194, 1947.

Sloane, N. J. A. Sequences A002680/M2261 and A101270 in "The On-Line Encyclopedia of Integer Sequences."