From Gaussian Quadrature to Chebyshev Quadrature

An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the points [itex] x_i [/itex] and weights [itex] w_i [/itex] for i = 1,...,n. The domain of integration for such a rule is conventionally taken as [−1, 1], so the rule is stated as

[itex] \int_{-1}^{1} f(x) dx \approx \sum_{i=1}^n w_i f(x_i) [/itex]

...if the integrated function can be written as [itex] f(x) = W(x) g(x) [/itex], where [itex] g(x) [/itex] is approximately polynomial, and [itex] W(x) [/itex] is known, then there are alternative weights [itex] {w'}_i[/itex] such that

[itex] \int_{-1}^1 f(x)\,dx = \int_{-1}^1 W(x) g(x)\,dx \approx \sum_{i=1}^n w_i' g(x_i) [/itex]

Common weighting functions include [itex] W(x)=(1-x^2)^{-1/2} [/itex] (Chebyshev–Gauss)....

A Gaussian quadrature-like formula for numerical estimation of integrals. It uses weighting function W(x)=1 in the interval [-1,1] and forces all the weights to be equal. The general formula is
[itex] \int_{-1}^1 f(x)dx=\frac{2}{n} \sum_{i=1}^n f(x_i)
[/itex]

where the abscissas x_i are found by taking terms up to [itex] y^n [/itex] in the Maclaurin series of
[itex] s_n(y)=exp(1/2n[-2+ln(1-y)(1-\frac{1}{y})+ln(1+y)(1+\frac{1}{y})]) [/itex]

and then defining
[itex] G_n(x)=x^n s_n(\frac{1}{x}) [/itex]

The roots of [itex] G_n(x) [/itex] then give the abscissas.