Quadrature as a least-squares and minimax problem

The vector of weights of an interpolatory quadrature rule with \(n\) preassigned nodes is shown to be the least-squares solution \(\omega\) of an overdetermined linear system here called {\em the fundamental system} of the rule. It is established the relation between \(\omega\) and the minimax solut...

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Bibliographic Details
Published in:arXiv.org 2012-06
Main Author: Graça, Mário M
Format: Article
Language:English
Subjects:
Least squares
Minimax technique
Norms
Parameters
Online Access:Get full text
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Summary:The vector of weights of an interpolatory quadrature rule with \(n\) preassigned nodes is shown to be the least-squares solution \(\omega\) of an overdetermined linear system here called {\em the fundamental system} of the rule. It is established the relation between \(\omega\) and the minimax solution \(\stackrel{\ast}{z}\) of the fundamental system, and shown the constancy of the \(\infty\)-norms of the respective residual vectors which are equal to the {\em principal moment} of the rule. Associated to \(\omega\) and \(\stackrel{\ast}{z}\) we define several parameters, such as the angle of a rule, in order to assess the main properties of a rule or to compare distinct rules. These parameters are tested for some Newton-Cotes, Fejér, Clenshaw-Curtis and Gauss-Legendre rules.
ISSN:2331-8422