Generalized compound quadrature formulae for finite-part integrals

We investigate the error term of the dth degree compound quadrature formulae for finite-part integrals of the form (sic)(0)(1)x(-p)f(x)dr where p is an element of R and p greater than or equal to 1. We are mainly interested in error bounds of the form \R[f]\ less than or equal to c\\f((s))\\(infinity) with best possible constants c. It is shown that, for p is not an element of N and n uniformly distributed nodes, the error behaves as O(n(p-s-1)) for f is an element of C-s[0, 1], p-1 < s less than or equal to d + 1. In a previous paper we have shown that this is not true for p is an element of N. As an improvement, we consider the case of non-uniformly distributed nodes. Here, we show that for all p greater than or equal to 1 and f is an element of C-s[0, 1], an O(n(-s)) error estimate can be obtained in theory by a suitable choice of the nodes. A set of nodes with this property is stated explicitly. In practice, this graded mesh causes stability problems which are computationally expensive to overcome.