On the properties of Sard kernels and multiple error estimates for bounded linear functionals of bivariate functions with application to non-product cubature
type: Item
instance of: scholarly article
title: On the properties of Sard kernels and multiple error estimates for bounded linear functionals of bivariate functions with application to non-product cubature
zbMATH Open document ID: 1259.65052
author: Chin-Yun Chen
published in: Numerische Mathematik
publication date: 2012-12-20T00:00:00Z
review text: Sard kernel theorems concern the result of applying a bounded linear functional to a Taylor expansion of a bivariate, sufficiently smooth function defined on a rectangle. \textit{A. H. Stroud} [Approximate calculation of multiple integrals. Englewood Cliffs, New Jersey: Prentice-Hall, Inc. (1971; Zbl 0379.65013)] has used Sard kernels to estimate the errors of certain tensor product and non-product cubature rules. In this paper, the author extends these results and presents a verified method for computing Sard error constants. The flexibility and superiority of the Sard kernel method for certain tensor product and non-product cubature rules are shown. First the author discusses some basic properties of Sard kernels that are useful for the verified computation. Then relevant properties of Sard kernels for cubature rules are examined. The ideas for computing Sard error constants by interval arithmetic are described and corresponding algorithms are presented. The error constants for a non-product cubature rule are calculated by the proposed algorithms. Numerical examples are given.
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 65G30
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 6117691
zbMATH Keywords: cubature rules
zbMATH Keywords: numerical integration
zbMATH Keywords: bivariate function
zbMATH Keywords: Sard kernels
zbMATH Keywords: cubature errors
zbMATH Keywords: Sard error constants
zbMATH Keywords: verified computation
zbMATH Keywords: interval arithmetic
zbMATH Keywords: tensor product cubature rule
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
full work available at URL: s00211-012-0471-y
OpenAlex ID: W1997261822
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DOI: 10.1007/S00211-012-0471-Y
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description: scientific article; zbMATH DE number 6117691