• Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

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Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.

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ETNA - Electronic Transactions on Numerical Analysis



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Verlag der Österreichischen Akademie der Wissenschaften
Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2,
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400
https://verlag.oeaw.ac.at, e-mail: bestellung.verlag@oeaw.ac.at
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Internality of two-measure-based generalized Gauss quadrature rules for modified Chebyshev measures

    Dušan Lj. Djukić, Rada M. Mutavdžić Djukić, Aleksandar V. Pejčev, Lothar Reichel, Miodrag M. Spalević, Stefan M. Spalević

ETNA - Electronic Transactions on Numerical Analysis, pp. 157-172, 2024/12/10

doi: 10.1553/etna_vol61s157

doi: 10.1553/etna_vol61s157


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doi:10.1553/etna_vol61s157



doi:10.1553/etna_vol61s157

Abstract

Many applications in science and engineering require the approximation of integrals of the form $\int_{-1}^1 f(x)d\sigma(x)$, where $f$ is an integrand and $d\sigma$ is a nonnegative measure. Such approximations often are computed by an $\ell$-node Gauss quadrature rule $G_\ell(f)$ that is determined by the measure. It is important to be able to estimate the quadrature error in these approximations. Error estimates can be computed by applying another quadrature rule, $Q_m(f)$, with $m>\ell$ nodes, and using the difference $Q_m(f)-G_\ell(f)$ as an estimate for the error in $G_\ell(f)$. This paper considers the situation when $d\sigma$ is a modified Chebyshev measure and shows that two-measure-based quadrature rules $\widehat{Q}_{2\ell+1}$ exist, have positive weights, and have distinct nodes in the interval $[-1,1]$. The last property makes them applicable also when the integrand $f$ only is defined in $[-1,1]$. Comparisons with other choices of quadrature formulas $Q_{2\ell+1}$ are presented. This paper extends the investigation of two-measure-based quadrature rules for Jacobi and generalized Laguerre measures initiated in A. V. Pejčev et. al [Appl. Numer. Math., 204 (2024), pp. 206–221].

Keywords: Gauss quadrature rule, averaged Gauss rule, generalized averaged Gauss rule, modified Chebyshev measure