Rates of convergence of Gaussian quadrature for singular integrands

Source 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 ] [ - 1,1] , the error in n -point Gaussian quadrature of f ( x ) = | x − y | − δ f(x) = |x - y{|^{ - \delta }} is O ( n − 2 + 2 δ ) O({n^{ - 2 + 2\delta }}) if y = ± 1 y = \pm 1 and O ( n − 1 + δ ) O({n^{ - 1 + \delta }}) if y ∈ ( − 1 , 1 ) y \in ( - 1,1) , provided we avoid the singularity. If we ignore the singularity y , the error is O ( n − 1 + 2 δ ( log ⁡ n ) δ ( log ⁡ log ⁡ n ) δ ( 1 + ε ) ) 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.