Convergence rate and acceleration of Clenshaw-Curtis quadrature for functions with endpoint singularities

2014/01/03 by Haiyong Wang, Wang, Haiyong
Mathematics · #Mathematical functions and polynomials #Mathematical Approximation and Integration #Iterative Methods for Nonlinear Equations

paper · pdf · doi:10.48550/arxiv.1401.0638

Abstract

In this paper, we investigate the rate of convergence of Clenshaw-Curtis quadrature and its acceleration for functions with endpoint singularities in X^s, where X^s denotes the space of functions whose Chebyshev coefficients decay asymptotically as a_k = O(k^-s-1) for some positive s. For such unctions, we show that the convergence rate of (n + 1)-point Clenshaw-Curtis quadrature is O(n^-s-2). Furthermore, an asymptotic error expansion for Clenshaw-Curtis quadrature is presented which enables us to employ some extrapolation techniques to accelerate its convergence. Numerical examples are provided to confirm our analysis.

Citations

• Is Gauss Quadrature Better than Clenshaw–Curtis?

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