Electronic Transactions on Numerical Analysis (ETNA)

Volume 61, pp. 157-172, 2024.

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, and Stefan M. Spalević

Abstract

Many applications in science and engineering require the approximation of integrals of the form $\int_{- 1}^{1} f (x) d σ (x)$ , where $f$ is an integrand and $d σ$ is a nonnegative measure. Such approximations often are computed by an $ℓ$ -node Gauss quadrature rule $G_{ℓ} (f)$ 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 $m > ℓ$ nodes, and using the difference $Q_{m} (f) - G_{ℓ} (f)$ as an estimate for the error in $G_{ℓ} (f)$ . This paper considers the situation when $d σ$ is a modified Chebyshev measure and shows that two-measure-based quadrature rules ${\hat{Q}}_{2 ℓ + 1}$ 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 $Q_{2 ℓ + 1}$ 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].

Full Text (PDF) [326 KB], BibTeX , DOI: 10.1553/etna_vol61s157

Key words

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

AMS subject classifications

65D30, 65D32, 33C45, 33C47

Links to the cited ETNA articles

[3]	Vol. 61 (2024), pp. 121-136 Carlos F. Borges and Lothar Reichel: Computation of Gauss-type quadrature rules
[9]	Vol. 45 (2016), pp. 405-419 D. Lj. Djukić, L. Reichel, M. M. Spalević, and J. D. Tomanović: Internality of generalized averaged Gauss rules and their truncations for Bernstein-Szegő weights
[15]	Vol. 45 (2016), pp. 371-404 Sotirios E. Notaris: Gauss-Kronrod quadrature formulae - A survey of fifty years of research

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