
Mathematics 2012
Quadrature as a leastsquares and minimax problemAbstract: The vector of weights of an interpolatory quadrature rule with $n$ preassigned nodes is shown to be the leastsquares 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 NewtonCotes, Fej\'er, ClenshawCurtis and GaussLegendre rules.
