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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: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis, pp. 157-172, 2024/12/10
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