Generalized Averaged Gauss Quadrature Rules: A Survey

Dušan L. Djukić, Rada M. Mutavdžić Djukić, Lothar Reichel and Miodrag M. Spalević ()
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Dušan L. Djukić: Department of Mathematics, Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 35, 11120 Belgrade, Serbia
Rada M. Mutavdžić Djukić: Department of Mathematics, Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 35, 11120 Belgrade, Serbia
Lothar Reichel: Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA
Miodrag M. Spalević: Department of Mathematics, Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 35, 11120 Belgrade, Serbia

Mathematics, 2025, vol. 13, issue 19, 1-17

Abstract: Consider the problem of approximating an integral of a real-valued integrand on a real interval by a Gauss quadrature rule. The classical approach to estimate the quadrature error of a Gauss rule is to evaluate an associated Gauss–Kronrod rule and compute the difference between the value of the Gauss–Kronrod rule and that of the Gauss rule. However, for a variety of measures and a number of nodes of interest, Gauss–Kronrod rules do not have real nodes or positive weights. This makes these rules impossible to apply when the integrand is defined on a real interval only. This has spurred the development of several averaged Gauss quadrature rules for estimating the quadrature error of Gauss rules. A significant advantage of the averaged Gauss rules is that they have real nodes and positive weights also in situations when Gauss–Kronrod rules do not. The most popular averaged rules include Laurie’s averaged Gauss quadrature rules, optimal averaged Gauss quadrature rules, weighted averaged Gauss quadrature rules, and two-measure-based generalized Gauss quadrature rules. This paper reviews the accuracy, numerical construction, and internality of averaged Gauss rules.

Keywords: Gauss quadrature rule; averaged Gauss quadrature rule; precision; convergence; numerical construction; error estimate; matrix function; matrix functional (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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