The Mathematical Institute, University of Oxford, Eprints Archive

The Kink Phenomenon in Fejér and Clenshaw-Curtis Quadrature

Weideman, J. A. C. and Trefethen, Lloyd N. (2006) The Kink Phenomenon in Fejér and Clenshaw-Curtis Quadrature. Technical Report. Unspecified. (Submitted)



The Fejér and Clenshaw-Curtis rules for numerical integration exhibit a curious phenomenon when applied to certain analytic functions. When N, (the number of points in the integration rule) increases, the error does not decay to zero evenly but does so in two distinct stages. For N less than a critical value, the error behaves like $O(\varrho^{-2N})$, where $\varrho$ is a constant greater than 1. For these values of N the accuracy of both the Fejér and Clenshaw-Curtis rules is almost indistinguishable from that of the more celebrated Gauss-Legendre quadrature rule. For larger N, however, the error decreases at the rate $O(\varrho^{-N})$, i.e., only half as fast as before. Convergence curves typically display a kink where the convergence rate cuts in half. In this paper we derive explicit as well as asymptotic error formulas that provide a complete description of this phenomenon.

This work was supported by the Royal Society of the UK and the National Research Foundation of South Africa under the South Africa-UK Science Network Scheme. The first author also acknowledges grant FA2005032300018 of the NRF.

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1107
Deposited By:Lotti Ekert
Deposited On:11 May 2011 10:58
Last Modified:11 May 2011 10:58

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