Trefethen, Lloyd N. (2006) *Is Gauss quadrature better than Clenshaw-Curtis?* Technical Report. Unspecified. (Submitted)

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## 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 in the complex plane. Gauss quadrature corresponds to Pade approximation at . Clenshaw-Curtis quadrature corresponds to an approximation whose order of accuracy at is only half as high, but which is nevertheless equally accurate near .

Item Type: | Technical Report (Technical Report) |
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Subjects: | A - C > Approximations and expansions H - N > Numerical analysis |

Research Groups: | Numerical Analysis Group |

ID Code: | 1116 |

Deposited By: | Lotti Ekert |

Deposited On: | 11 May 2011 10:56 |

Last Modified: | 11 May 2011 10:56 |

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