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arXiv · 1206.0281

Quadrature as a least-squares and minimax problem

Mário M. Graça
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Abstract

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, Fej\'er, Clenshaw-Curtis and Gauss-Legendre rules.

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Mário M. Graça. 2012-06-01. Quadrature as a least-squares and minimax problem. https://arxiv.org/abs/1206.0281

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