Talk:Clenshaw–Curtis quadrature

From Wikipedia, the free encyclopedia
Jump to: navigation, search

It would be useful to have a comparison with Gaussian quadrature. This paper makes some interesting observations. Fredrik Johansson 21:08, 16 September 2006 (UTC)

The first paragraph says that the accuracy is comparable to Gaussian quadrature, which is basically what Trefethen's conclusion is. As Trefethen explains in his paper, he's by no means the first person to observe this fact. I'm reluctant to cite Trefethen's paper until it is published, however, since Wikipedia's citation rules strongly discourage the use of self-published preprints. —Steven G. Johnson 21:10, 16 September 2006 (UTC)
I think there is room for some elaboration. If I understand it correctly, Clenshaw-Curtis is theoretically supposed to be only half as accurate at the same number of evaluation points, but performs better than this for certain types of functions, and benefits from being faster than Gauss. (If I didn't understand that correctly, it's a sign that the article should be expanded :-) Trefethen mentions the paper "Error estimation in the Clenshaw-Curtis quadrature formula" by H. O'Hara and F. J. Smith — perhaps there is more information to be found in there; however, I don't have access to it. Fredrik Johansson 08:31, 17 September 2006 (UTC)
You didn't quite understand it correctly.  :-) C-C exactly integrates polynomials with half the degree of Gaussian quadrature with the same number of points. Most functions aren't polynomials, however, so this is not as much of a practical advantage as some people think. Okay, I'll add some elaboration when I get a chance. —Steven G. Johnson 14:44, 17 September 2006 (UTC)

Thanks for the addition! This is a very nice article. Fredrik Johansson 05:41, 24 September 2006 (UTC)

common weight functions w(x)[edit]

if I understand this great article, then the common weight functions w(x) could be density functions of bounded distributions, such as beta. then integration of w(cos(theta))cos(k*theta)*sin(theta) could be done beforehand with some specialized methods. Is there a reference for this too? Patiobarbecue 04:31, 6 February 2007 (UTC)

This could certainly be done, although I haven't seen any specific references on this. You might want to check the Evans and Webster paper to see if they have more references on other weight functions. —Steven G. Johnson 18:42, 6 February 2007 (UTC)

Number of points[edit]

There seems to be some small confusion about the number of points in the derivations: ".. are accurately computed by the N equally-spaced and equally-weighted points θn = nπ / N for n = 0,\ldots,N". If n goes from 0 to N we have N+1 points. It's harder than it should be to implement Clenshaw-Curtis from this article. 130.37.28.128 (talk) 09:14, 16 March 2010 (UTC)

Fixed. — Steven G. Johnson (talk) 15:32, 16 March 2010 (UTC)