*(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 $log\left(\right(z+1)/(z-1\left)\right)$ 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 $[-1,1]$.

##### 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) |