From Gaussian Quadrature to Chebyshev Quadrature

[CFXMSC]

Hi,

I'm studying about Chebyshev Quadrature and i found so little and confused information about this.
I don't know if Gauss-Chebyshev Quadrature is the same of Chebyshev Quadrature.
The only good information that i found was from Wolfram:

http://mathworld.wolfram.com/ChebyshevQuadrature.html

And there is write Chebyshev Quadrature is a simplification of Gaussian quadrature. So here is my question: How can i simplify from Gaussian Quadrature to Chebyshev Quadrature?

 

[Stephen Tashi]

You ask an interesting question about terminology. I don't know the answer, but I think it would help to state the question explicitly rather than expecting readers to follow links.

The Wikipedia article on Gaussian Quadrature states:

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)...

The question is whether that definition is equivalent to the one on the Wolfram site which defines Chebyshev Quadrature as:

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.

I had to do the LaTex manually instead of a straight cut-and-past. I hope I haven't introduced any typos.

 

[Stephen Tashi]

The most plausible definition I found for Chebyshev quadrature is on page 2 of this PDF of lecture notes: http://www.google.com/url?sa=t&rct=...sg=AFQjCNG4ANcllrpzdgevxooA35CzZdaFdA&cad=rja

It says Chebyshev quadrature is based on using Chebyshev polynomials.

 

[CFXMSC]

Thanks Stephen Tashi.

Finally i found the proof. Chebyshev quadrature is really hard to find because always when you google it other similar topics appears. So the book i found this information is: Introduction to Numerical Analysis - F. B. Hildebrand