Is Gauss quadrature better than Clenshaw-Curtis?
Dear NA Digest friends, I have been surprised to discover that Clenshaw-Curtis quadrature (based on Chebyshev points) seems to be about as accurate as Gauss quadrature (based on optimal points) rather than half as accurate as many of us are trained to expect. Since the N-point C-C rule can be implemented by the FFT in just O(NlogN) operations, this makes it a powerful formula indeed. Zachary Battles and I use C-C quadrature with thousands or even millions of points very happily in the “chebfun” system. The notion that C-C is more or less as good as Gauss gets some mention in papers in the 1960s by O’Hara & Smith and Elliott, but doesn’t seem generally known now. Anyone knowledgeable on this subject — I’d be most grateful for advice you can give me in response to the paper I’ve written with the above title: see my home page or http://www.comlab.ox.ac.uk/nick.trefethen/CC_trefethen_submitted.pdf. (Don’t miss Figures 6.1 and 6.2!) Many thanks – Nick Trefethen, Oxford University ———
