Quadrature as a least-squares and minimax problem
For numerical analysts, this provides a new theoretical perspective on quadrature rules, but the practical impact is incremental as it primarily reframes existing concepts.
The paper reformulates interpolatory quadrature rules as least-squares and minimax problems, showing the relationship between the weight vector and the minimax solution, and introduces parameters like the angle of a rule to assess and compare rules. Tests on Newton-Cotes, Fejér, Clenshaw-Curtis, and Gauss-Legendre rules demonstrate the approach.
The vector of weights of an interpolatory quadrature rule with $n$ preassigned nodes is shown to be the least-squares solution $ω$ of an overdetermined linear system here called {\em the fundamental system} of the rule. It is established the relation between $ω$ 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 $ω$ 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.