Gauss rules associated with nearly singular weights

Abstract

We consider the problem of evaluating View the MathML source, when f is smooth and G is nearly singular and non-negative. For this we construct a Gauss quadrature formula w.r.t. the weight G(x)(1−x2)−1/2. Once the factor G has been chosen, the procedure is relatively simple and mainly involves the application of FFT to compute a finite number of coefficients of the Chebyshev series expansion of G which in turn are used to calculate modified moments. It is shown that this approach is very effective when the complexity of f is high, or when f is parametric and the integral must be calculated for many values of the parameters. For this, there is presented a selection of numerical examples which allows comparison with other methods. In particular, there is considered the evaluation of Hadamard finite part integrals when the regular part of the integrand is nearly singular.