Rates of convergence of Gauss, Lobatto, and Radau integration rules for singular integrands

Abstract:

Rates of convergence (or divergence) are obtained in the application of Gauss, Lobatto, and Radau integration rules to 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 generalized Jacobi weight function on $[ - 1,1]$, the error in applying an n-point rule to $f(x) = |x - y{|^{ - \delta }}$ 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.