On the numerical evaluation of singular integrals

Product-integration rules of the form _–1 ¹ k(x)f(x)dx _i ⁼¹ⁿ w _ni f(x _ni ) are studied, with the points {w _ni } chosen to be the zeros of certain orthogonal polynomials, and the weights {w _ni } chosen to make the rule exact iff is any polynomial of degree less thann. If, in particular, the points are the Chebyshev points, and ifk L _p –1, 1] for somep>1, then it is shown that the rule converges to the exact result for all continuous functionsf. With this choice of points, the practical application of the rule is shown to be straightforward in many cases, and to yield satisfactory rates of convergence. The casek(x)=|–x|, >–1, is studied in detail. Results of a similar, but weaker, kind are also obtained for other choices of the points {x _ni }.