Generalized anti-Gauss quadrature rules

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.