Is Gauss quadrature better than Clenshaw-Curtis?

Author: 

Trefethen, LN

Journal: 

SIAM Review

Publication Date: 

1 March 2008

Last Updated: 

2018-12-08T23:14:49.613+00:00

Issue: 

1

DOI: 

10.1137/060659831

Volume: 

50

page: 

67-87

abstract: 

We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw-Curtis. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following O'Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log((z + 1)/(Z - 1)) in the complex plane. Gauss quadrature corresponds to Padé approximation at z = ∞. Clenshaw-Curtis quadrature corresponds to an approximation whose order of accuracy at z = ∞ is only half as high, but which is nevertheless equally accurate near [-1, 1]. © 2008 Society for Industrial and Applied Mathematics.

Symplectic id: 

188096

Submitted to ORA: 

Not Submitted

Publication Type: 

Journal Article