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Generalized compound quadrature formulae for finite-part integrals. (English) Zbl 0871.41021
Summary: We investigate the error term of the $d$th degree compound quadrature formulae for finite part integrals of the form ${\int }_{0}^{1}{x}^{-p}f\left(x\right)dx$ where $p\in ℝ$ and $p\ge 1$. We are mainly interested in error bounds of the form $|R\left[f\right]|\le c|{f}^{\left(s\right)}{|}_{\infty }$ with best possible constant $c$. It is shown that, for $p\notin ℕ$ and $n$ uniformly distributed nodes, the error behaves as $O\left({n}^{p-s-1}\right)$ for $f\in {C}^{s}\left[0,1\right]$, $p-1. In a previous paper we have shown that this is not true for $p\in ℕ$. As an improvement, we consider the case of non-uniformly distributed nodes. Here, we show that for all $p\ge 1$ and $f\in {C}^{s}\left[0,1\right]$, an $O\left({n}^{-s}\right)$ error estimates 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.