#### Volume 61, pp. 121-136, 2024.

## Computation of Gauss-type quadrature rules

Carlos F. Borges and Lothar Reichel

### Abstract

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.

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### Key words

quadrature, Gauss rule, Gauss-Radau rule, Gauss-Lobatto rule, averaged Gauss rule, optimal averaged Gauss rule quadrature, divide-and-conquer method, Golub-Welsch algorithm

### AMS subject classifications

65D30, 65D32

### Links to the cited ETNA articles

[1] | Vol. 9 (1999), pp. 26-38 G. S. Ammar, D. Calvetti, and L. Reichel: Computation of Gauss-Kronrod quadrature rules with non-positive weights |

[14] | Vol. 53 (2020), pp. 352-382 D. Lj. Djukić, R. M. Mutavdžić Djukić, A. V. Pejčev, and M. M. Spalević: Error estimates of Gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results |

[18] | 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 |

[32] | Vol. 55 (2022), pp. 424-437 D. R. Jandrlić, D. M. Krtinić, Lj. V. Mihić, A. V. Pejčev, and M. M. Spalević: Error bounds for Gaussian quadrature formulae with Legendre weight function for analytic integrands |

[36] | Vol. 45 (2016), pp. 371-404 Sotirios E. Notaris: Gauss-Kronrod quadrature formulae - A survey of fifty years of research |

[37] | Vol. 59 (2023), pp. 89-98 Aleksandar V. Pejčev: A note on “Error bounds of Gaussian quadrature formulae with Legendre weight function for analytic integrands” by M. M. Spalević et al. |

### Additional resources for this document

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