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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.

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Metadaten
Author:Kai Diethelm
DOI:https://doi.org/10.1093/imanum/17.3.479
Parent Title (English):IMA Journal of Numerical Analysis
Document Type:Article
Language:English
Year of publication:1997
Release Date:2023/08/09
Volume:17
Issue:3
First Page:479
Last Page:493
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