Estimating the error of Gaussian quadratures with simple and multiple nodes by using their extensions with multiple nodes

Abstract

The estimation of the error in a quadrature formula is an important problem. A simple and effective procedure for estimating the error of Gaussian quadrature formulas using their extensions with multiple nodes will be presented. Our method works for estimating the error of any interpolatory quadrature formula with simple or multiple nodes. We concentrate the most of our attention to the estimation of the error of standard Gauss quadratures, as the most known and popular ones. In that sense we offer an adequate alternative to Gauss–Kronrod quadratures, which have been intensively investigated in the last five decades, both from the theoretical and computational point of view. We found and a few cases of optimal, i.e., Kronrod extensions with multiple nodes for some Gauss quadrature formulas; we are not aware of any results of this kind in the mathematical literature. Numerical results are presented, in order to demonstrate the efficiency of the proposed method.