Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals
- We show that, iff∈Ck[−1,1] (k≥2), the error term of every modified positive interpolatory quadrature rule for Cauchy principal value integrals of the type∫−1−1w(x)f(x)x−λdx ,λ∈(−1,1), fulfills Rn[f;λ]=O(n−klnn) uniformly for allλ∈(−1,1), and hence it is of optimal order of magnitude in the classesCk[−1,1] (k=2,3,4,…). Here, w is a weight function with the property0≤w(x)1−x2−−−−−√≤C . We give explicit upper bounds for the Peano-type error constants of such rules. This improves and completes earlier results by Criscuolo and Mastroianni (Calcolo 22 (1985), 391–441 and Numer. Math. 54 (1989), 445–461) and Ioakimidis (Math. Comp. 44 (1985), 191–198). For the special case of the Gaussian rule, we show that the restrictionk≥2 can be dropped. The results are based on a new representation of the Peano kernels of these formulae via the Peano kernels of the underlying classical quadrature formulae. This representation may also be useful in connection with some different problems.
Author: | Kai Diethelm |
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URN: | urn:nbn:de:bvb:863-opus-20560 |
Persistent identifier: | https://doi.org/10.1007/s002110050183 |
Parent Title (English): | Numerische Mathematik |
Document Type: | Article |
Language: | English |
Year of publication: | 1996 |
Publishing Institution: | Hochschule für Angewandte Wissenschaften Würzburg-Schweinfurt |
Release Date: | 2022/07/20 |
Issue: | 73 |
First Page: | 53 |
Last Page: | 63 |
Comments: | Accepted version des Artikels. Published source: Diethelm, K. Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals . Numer. Math. 73, 53–63 (1996). https://doi.org/10.1007/s002110050183 |
Open access colour: | Grün (Zweitveröffentlichung) |