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Is Gauss quadrature better than Clenshaw-Curtis? (English) Zbl 1141.65018

Summary: We compare the convergence behavior of Gauss quadrature with that of its younger brother, the Clenshaw-Curtis quadrature [cf. C. W. Clenshaw and A. R. Curtis, Numer. Math. 2, 197–205 (1960; Zbl 0093.14006)]. 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 H. O’Hara and F. J. Smith [Comput. J. 11, 213–219 (1968; 0165.17901)], 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 $log\left(\left(z+1\right)/\left(z-1\right)\right)$ in the complex plane. Gauss quadrature corresponds to Padé approximation at $z=\infty$. Clenshaw-Curtis quadrature corresponds to an approximation whose order of accuracy at $z=\infty$ is only half as high, but which is nevertheless equally accurate near $\left[-1,1\right]$.

##### MSC:
 65D32 Quadrature and cubature formulas (numerical methods) 41A55 Approximate quadratures 41A20 Approximation by rational functions 41A58 Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
Matlab