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Abstract and Applied Analysis
Volume 2014, Article ID 436164, 10 pages
http://dx.doi.org/10.1155/2014/436164
Research Article

On Fast and Stable Implementation of Clenshaw-Curtis and Fejér-Type Quadrature Rules

1School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China
2School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

Received 27 June 2014; Accepted 5 August 2014; Published 27 August 2014

Academic Editor: Robert A. Van Gorder

Copyright © 2014 Shuhuang Xiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Based upon the fast computation of the coefficients of the interpolation polynomials at Chebyshev-type points by FFT, together with the efficient evaluation of the modified moments by forward recursions or by Oliver’s algorithms, this paper presents fast and stable interpolating integration algorithms, by using the coefficients and modified moments, for Clenshaw-Curtis, Fejér’s first- and second-type rules for Jacobi weights or Jacobi weights multiplied by a logarithmic function. Numerical examples illustrate the stability, efficiency, and accuracy of these quadratures.