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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 $\int^1_0 x^{-p} f(x)dx$ where $p\in \bbfR$ and $p\ge 1$. We are mainly interested in error bounds of the form $|R[f] |\le c|f^{(s)} |_\infty$ with best possible constant $c$. It is shown that, for $p\not \in\bbfN$ and $n$ uniformly distributed nodes, the error behaves as $O(n^{p-s-1})$ for $f\in C^s[0,1]$, $p-1< s\le d +1$. In a previous paper we have shown that this is not true for $p\in\bbfN$. As an improvement, we consider the case of non-uniformly distributed nodes. Here, we show that for all $p\ge 1$ and $f\in C^s [0,1]$, an $O(n^{-s})$ 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.

41A55Approximate quadratures
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