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

$s_n(y)=exp{1/2n[-2+ln(1-y)(1-1/y)+ln(1+y)(1+1/y)]},$

(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

$E_n={c_n(f^((n+1))(xi))/((n+1)!) n odd; c_n(f^((n+2))(xi))/((n+2)!) n even,$

(14)

where

$c_n={int_(-1)^1xG_n(x)dx n odd; int_(-1)^1x^2G_n(x)dx n even.$

(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 .