Product integration of logarithmic singular integrands based on cubic splines

摘要

This paper is concerned with the practical evaluation of the product integral ∫1− 1f(x)k(x)dx for the case when k(x) = In|x - λ|, λϵ (−1, +1) and f is bounded in [−1, +1]. The approximation is a quadrature rule where the weights {wn,n,i} are chosen to be exact when f is given by a linear combination of a chosen set of functions {φn,j}. In this paper the functions {φn,j} are chosen to be cubic B-splines. An error bound for product quadrature rules based on cubic splines is provided. Examples that test the performance of the product quadrature rules for different choices of the function are given. A comparison is made with product quadrature rules based on first kind Chebyshev polynomials.