(An) automatic quadrature rule for Hadamard finite part integral

Abstract

An automatic quadrature rules are developed for computing Hadamard finite part integrals I(f;x)=∫_{-1}^1f(t)/$(t-x)^2dt$, -1 ＜ x ＜ 1, for smooth function f(t). After transforming the fp integral, using a change variable and substracting out the singularity, we approximate the function f(t) and its derivative by a sum of Chebyshev polynomials whose coefficients are computed using the FFT. The evaluation of I(f;x) for a set of values of x in (-1,1) are efficiently accomplished with the same number of function evaluation. Numerical examples are also given.