Abstract
Accurate numerical integration of singular functions usually requires either adaptivity or product integration. Both interfere with fast summation techniques and thus hamper large-scale computations.This paper presents a method for computing highly accurate quadrature formulas for singular functions which combine well with fast summation methods. Given the singularity and the N nodes, we first construct weights which integrate smooth functions with order-k accuracy. Then we locally correct a small number of weights near the singularity, to achieve order-k accuracy on singular functions as well. The method is highly efficient and runs in $O(Nk^{2d} + N\log ^2 N)$ time and $O(k^{2d} + N)$ space. We derive precise error bounds and time estimates and confirm them with numerical results which demonstrate the accuracy and efficiency of the method in large-scale computations. As part of our implementation, we also construct a new adaptive multidimensional product Gauss quadrature routine with an effective error ...
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