[Paper Review] Convergence rate and acceleration of Clenshaw-Curtis quadrature for functions with endpoint singularities

Haiyong Wang|arXiv (Cornell University)|Jan 3, 2014

Mathematical functions and polynomials18 references3 citations

TL;DR

This paper establishes that Clenshaw-Curtis quadrature achieves a convergence rate of $\mathcal{O}(n^{-s-2})$ for functions with endpoint singularities in $X^s$, outperforming prior estimates by one power of $n$. It derives an asymptotic error expansion enabling Richardson extrapolation to further accelerate convergence, particularly advantageous due to the nested node structure and FFT-efficient weights.

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.

Motivation & Objective

• Investigate the convergence rate of Clenshaw-Curtis quadrature for integrands with algebraic or algebraic-logarithmic endpoint singularities.
• Address the gap in convergence rate estimates for functions in $X^s$ with such singularities, particularly when standard bounds fall short.
• Develop an asymptotic error expansion to enable convergence acceleration using extrapolation techniques.
• Demonstrate that Clenshaw-Curtis quadrature, due to its nested nodes and FFT-computable weights, offers advantages over Gauss-Legendre quadrature for singular integrands.

Proposed method

• Define $X^s$ as the space of functions whose Chebyshev coefficients decay as $a_k = \mathcal{O}(k^{-s-1})$, linking smoothness to decay rate.
• Analyze the convergence rate of $(n+1)$-point Clenshaw-Curtis quadrature for $f(x) = (1-x)^\alpha(1+x)^\beta g(x)$ with $\alpha, \beta \geq 0$ not both integers and $g \in C^\infty[-1,1]$.
• Derive an asymptotic error expansion in inverse powers of $n$ for even $n$, valid for both algebraic and algebraic-logarithmic singularities.
• Apply Richardson extrapolation to the quadrature results using the derived asymptotic expansion, yielding accelerated convergence rates of $\mathcal{O}(n^{-d_q - 1})$.
• Use the inverse FFT to compute Clenshaw-Curtis weights efficiently in $\mathcal{O}(n\log n)$ operations.
• Validate theoretical results numerically using test functions with known singularities and compare with Gauss-Legendre quadrature.

Experimental results

Research questions

• RQ1 What is the optimal convergence rate of Clenshaw-Curtis quadrature for functions in $X^s$ with endpoint singularities?
• RQ2 How does the presence of algebraic or algebraic-logarithmic singularities affect the convergence rate compared to smooth functions?
• RQ3 Can an asymptotic error expansion be derived for Clenshaw-Curtis quadrature that enables effective convergence acceleration?
• RQ4 To what extent does Richardson extrapolation improve the convergence rate when applied to Clenshaw-Curtis quadrature with singular integrands?
• RQ5 How does the convergence behavior of Clenshaw-Curtis compare to Gauss-Legendre quadrature for singular integrands, especially considering node nesting and computational efficiency?

Key findings

• The convergence rate of $(n+1)$-point Clenshaw-Curtis quadrature for functions with endpoint singularities in $X^s$ is $\mathcal{O}(n^{-s-2})$, improving upon the prior bound of $\mathcal{O}(n^{-s-1})$.
• For functions with algebraic-logarithmic singularities $f(x) = (1-x)^\alpha(1+x)^\beta \log(1-x) g(x)$, where $\alpha$ is a positive integer, the convergence rate remains $\mathcal{O}(n^{-s-2})$ if $f \in X^s$.
• An asymptotic error expansion in negative powers of $n$ is derived for even $n$, enabling the use of extrapolation techniques such as Richardson extrapolation.
• The Richardson extrapolation scheme $R(q,n)$ achieves a convergence rate of $\mathcal{O}(n^{-d_q - 1})$, where $d_q$ depends on the singularity strength and is explicitly determined for different cases.
• Numerical examples confirm that the convergence rate of Clenshaw-Curtis quadrature matches the theoretical $\mathcal{O}(n^{-s-2})$ estimate for various $\alpha$, $\beta$, and $s$ values.
• Despite similar asymptotic convergence rates, Clenshaw-Curtis quadrature is more advantageous than Gauss-Legendre for singular integrands due to its nested nodes and efficient FFT-based weight computation.