Generalized anti-Gauss quadrature rules

Abstract

Gauss quadrature is a popular approach to approximate the value of a desired integral determined by a measure with support on the real axis. Laurie proposed an $(n+1)$-point quadrature rule that gives an error of the same magnitude and of opposite sign as the associated $n$-point Gauss quadrature rule for all polynomials of degree up to $2n+1$. This rule is referred to as an anti-Gauss rule. It is useful for the estimation of the error in the approximation of the desired integral furnished by the $n$-point Gauss rule. This paper describes a modification of the $(n+1)$-point anti-Gauss rule, that has $n+k$ nodes and gives an error of the same magnitude and of opposite sign as the associated $n$-point Gauss quadrature rule for all polynomials of degree up to $2n+2k-1$ for some $k>1$. We refer to this rule as a generalized anti-Gauss rule. An application to error estimation of matrix functionals is presented.