Is Gauss Quadrature Better than Clenshaw–Curtis?

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