Rates of convergence of Gaussian quadrature for singular integrands

Abstract:

The authors obtain the rates of convergence (or divergence) of Gaussian quadrature on functions with an algebraic or logarithmic singularity inside, or at an endpoint of, the interval of integration. A typical result is the following: For a bounded smooth weight function on $[ - 1,1]$, the error in n-point Gaussian quadrature of $f(x) = |x - y{|^{ - \delta }}$ is $O({n^{ - 2 + 2\delta }})$ if $y = \pm 1$ and $O({n^{ - 1 + \delta }})$ if $y \in ( - 1,1)$, provided we avoid the singularity. If we ignore the singularity y, the error is $O({n^{ - 1 + 2\delta }}{(\log n)^\delta }{(\log \log n)^{\delta (1 + \varepsilon )}})$ for almost all choices of y. These assertions are sharp with respect to order.