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

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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.

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Acknowledgments

The authors are indebted to the referees for the careful reading of the manuscript and their suggestions which have improved the paper. This work was supported in part by the Serbian Ministry of Education, Science and Technological Development [Research Projects: “Methods of numerical and nonlinear analysis with applications” (#174002) and “Approximation of integral and differential operators and applications” (#174015)].

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  1. Department of Mathematics, Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120, Belgrade 35, Serbia

    M. M. Spalević & A. S. Cvetković

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  1. M. M. Spalević
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  2. A. S. Cvetković
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Correspondence to M. M. Spalević.

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Communicated by Lothar Reichel.

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Spalević, M.M., Cvetković, A.S. Estimating the error of Gaussian quadratures with simple and multiple nodes by using their extensions with multiple nodes. Bit Numer Math 56, 357–374 (2016). https://doi.org/10.1007/s10543-015-0551-3

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