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Zbl 1141.65018
Trefethen, Lloyd N.
Is Gauss quadrature better than Clenshaw-Curtis? (English)
[J] SIAM Rev. 50, No. 1, 67-87 (2008). ISSN 0036-1445; ISSN 1095-7200/e

Summary: We compare the convergence behavior of Gauss quadrature with that of its younger brother, the Clenshaw-Curtis quadrature [cf. {\it C. W. Clenshaw} and {\it 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 {\it H. O'Hara} and {\it 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 $\log((z+1)/(z-1))$ 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 2000:: *65D32 Quadrature formulas (numerical methods)
41A55 Approximate quadratures
41A20 Approximation by rational functions
41A58 Series expansions

Keywords: Gauss quadrature; Chebyshev expansion; rational approximation; spectral methods; Newton-Cotes quadrature; numerical examples; convergence; MATLAB codes; Padé approximation; Clenshaw-Curtis quadrature

Citations: Zbl 0093.14006

Cited in: Zbl 1201.65040 Zbl 1186.65030

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