2008/01/01 by Lloyd N. Trefethen · 42 citations
Mathematics · #Mathematical functions and polynomials #Mathematical Approximation and Integration #Iterative Methods for Nonlinear Equations
paper · doi:10.1137/060659831
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].