Volume 45, pp. 371-404, 2016.

Gauss-Kronrod quadrature formulae - A survey of fifty years of research

Sotirios E. Notaris

Abstract

Kronrod in 1964, trying to estimate economically the error of the $n$-point Gauss quadrature formula for the Legendre weight function, developed a new formula by adding to the $n$ Gauss nodes $n+1$ new ones, which are determined, together with all weights, such that the new formula has maximum degree of exactness. It turns out that the new nodes are zeros of a polynomial orthogonal with respect to a variable-sign weight function, considered by Stieltjes in 1894, without though making any reference to quadrature. We survey the considerable research work that has been emerged on this subject, during the past fifty years, after Kronrod's original idea.

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Key words

Gauss quadrature formula, Gauss-Kronrod quadrature formula, Stieltjes polynomials

AMS subject classifications

65D32, 33C45

Links to the cited ETNA articles

[2]	Vol. 9 (1999), pp. 26-38 G. S. Ammar, D. Calvetti, and L. Reichel: Computation of Gauss-Kronrod quadrature rules with non-positive weights
[35]	Vol. 41 (2014), pp. 1-12 Aleksandar S. Cvetković and Miodrag M. Spalević: Estimating the error of Gauss-Turán quadrature formulas using their extensions
[83]	Vol. 9 (1999), pp. 65-76 Walter Gautschi: Orthogonal polynomials and quadrature
[87]	Vol. 25 (2006), pp. 129-137 Walter Gautschi: The circle theorem and related theorems for Gauss-type quadrature rules
[129]	Vol. 28 (2007-2008), pp. 168-173 Dirk Laurie: Variable-precision arithmetic considered perilous - a detective story

ETNA articles which cite this article

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

Vol. 47 (2017), pp. 197-205 Nasim Eshghi, Lothar Reichel, and Miodrag M. Spalević: Enhanced matrix function approximation

Vol. 50 (2018), pp. 20-35 Rada M. Mutavdžić, Aleksandar V. Pejčev, and Miodrag M. Spalević: Error bounds for Kronrod extension of generalizations of Micchelli-Rivlin quadrature formula for analytic functions

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

Vol. 59 (2023), pp. 145-156 Dusan Lj. Djukić, Rada M. Mutavdžić Djukić, Lothar Reichel, and Miodrag M. Spalević: Optimal averaged Padé-type approximants

Vol. 59 (2023), pp. 230-249 Jelena Tomanović: Gauss-type quadrature rules with respect to external zeros of the integrand

Vol. 61 (2024), pp. 92-104 Jelena Tomanović: Error estimates for Gauss-type quadrature rules for variable-sign weight functions

Vol. 61 (2024), pp. 121-136 Carlos F. Borges and Lothar Reichel: Computation of Gauss-type quadrature rules

Vol. 61 (2024), pp. 157-172 Dušan Lj. Djukić, Rada M. Mutavdžić Djukić, Aleksandar V. Pejčev, Lothar Reichel, Miodrag M. Spalević, and Stefan M. Spalević: Internality of two-measure-based generalized Gauss quadrature rules for modified Chebyshev measures