Chebyshev-Gauss Quadrature

Also called Chebyshev Quadrature. A Gaussian Quadrature over the interval $[-1,1]$ with Weighting Function $W(x)=1/\sqrt{1-x^2}$. The Abscissas for quadrature order $n$ are given by the roots of the Chebyshev Polynomial of the First Kind $T_n(x)$, which occur symmetrically about 0. The Weights are

$$w_i=-{A_{n+1}\gamma_n\over A_nT_n'(x_i)T_{n+1}(x_i)}={A_n\over A_{n-1}}{\gamma_{n-1}\over T_{n-1}(x_i)T_n'(x_i)},$$ (1)

where $A_n$ is the Coefficient of $x^n$ in $T_n(x)$. For Hermite Polynomials,

$$A_n=2^{n-1},$$ (2)

so

$${A_{n+1}\over A_n}=2.$$ (3)

Additionally,

$$\gamma_n={\textstyle{1\over 2}}\pi,$$ (4)

so

$$w_i=-{\pi\over T_{n+1}(x_i)T_n'(x_i)}.$$ (5)

Since

$$T_n(x)=\cos(n\cos^{-1}x),$$ (6)

the Abscissas are given explicitly by

$$x_i=\cos\left[{(2i-1)\pi\over 2n}\right].$$ (7)

Since

$$T_n'(x_i) = {(-1)^{i+1}n\over\alpha_i}$$ (8)

$$T_{n+1}(x_i) = (-1)^i\sin\alpha_i,$$ (9)

where

$$\alpha_i={(2i-1)\pi\over 2n},$$ (10)

all the Weights are

$$w_i={\pi\over n}.$$ (11)

The explicit Formula is then

$$\int_{-1}^1 {f(x)\,dx\over\sqrt{1-x^2}}={\pi\over n}\sum_{k=... ...2n}\pi}\right)}\right]+ {2\pi\over 2^{2n}(2n)!} f^{(2n)}(\xi).$$ (12)

$n$	$x_i$	$w_i$
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.