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 ∫10x−pf(x) dx where p∈ 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∉ and n uniformly distributed nodes, the error behaves as O(np–s–1 for f∈Cs[0,1], p–1 <s ≤d+1. In a previous paper we have shown that this is not true for p∈
As an improvement, we consider the case of non-uniformly distributed nodes. Here, we show that for all p ≥ I and f∈Cs[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 staled explicitly. In practice, this graded mesh causes stability problems which are computationally expensive to overcome.