Optimal Averaged Gauss–Radau Quadrature for Functions with Endpoint Singularities

The averaged Gauss–Radau quadrature rule assigns equal weights to the left and right Gauss–Radau formulas regardless of the behavior of the integrand. This choice is generally not optimal for integrands possessing algebraic endpoint singularities. In this paper, we introduce an oracle-weighted averaging strategy in which the averaging parameter is chosen to minimize the quadrature error. For a class of integrands consisting of an algebraic endpoint singularity multiplied by a smooth regular factor, we establish an asymptotic invariance property of the oracle averaging parameter. Its leading asymptotic behavior is determined by the endpoint singularity exponent, while the regular factor contributes only to higher-order terms. Using the established Gauss–Radau remainder estimates, together with the boundedness of the oracle parameter associated with the principal singular component, we show that the difference between the oracle parameters associated with the general integrand and its principal singular component is O(n−2) as the quadrature order increases. Motivated by these results, we construct a simple rational approximation of the oracle averaging parameter depending solely on the endpoint exponent. Numerical experiments support the theoretical analysis and show that the resulting model-weighted averaged Gauss–Radau rule improves upon equal weighting for the test problems considered, without requiring knowledge of the exact integral.