Is Gauss Quadrature Better than Clenshaw–Curtis?

Abstract. We consider the question of whether Gauss quadrature, which is very famous, is more powerful than the much simpler Clenshaw–Curtis quadrature, which is less well-known. 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 Elliott and 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 Pade ́ 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.