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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. 121-136, 2024/10/18
Many problems in scientific computing require the evaluation of Gauss quadrature rules. It is important to be able to estimate the quadrature error in these rules. Error estimates or error bounds often can be computed by evaluating an additional related Gauss-type formula such as a Gauss-Radau, Gauss-Lobatto, anti-Gauss, averaged Gauss, or optimal averaged Gauss rule. This paper presents software for both the evaluation of a single Gauss quadrature rule and the calculation of a pair of a Gauss rule and a related Gauss-type rule. The software is based on a divide-and-conquer method. This method is compared to both an available and a new implementation of the Golub-Welsch algorithm, which is the classical approach to evaluate a single Gauss quadrature rule. Timings on a laptop computer show the divide-and-conquer method to be competitive except for the computation of a single quadrature rule with very few nodes.
Keywords: quadrature, Gauss rule, Gauss-Radau rule, Gauss-Lobatto rule, averaged Gauss rule, optimal averaged Gauss rule quadrature, divide-and-conquer method, Golub-Welsch algorithm