Generalized compound quadrature formulae for finite-part integrals

Abstract

We investigate the error term of the dth degree compound quadrature formulae for finite-part integrals of the form f(x -p f(x)dx where p ∈ R and p ≥ 1. We are mainly interested in error bounds of the form (R[f]( ≤ c∥f (s) ∥∞ with best possible constants c. It is shown that, for p ∈ N and n uniformly distributed nodes, the error behaves as O(n p-s-1 ) for f ∈ C 5 [0,1], p - 1 < s ≤ d + 1. In a previous paper we have shown that this is not true for p ∈ N. As an improvement, we consider the case of non-uniformly distributed nodes. Here, we show that for all p ≥ 1 and f ∈ C 5 [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.