Rational symbolic cubature rules over the first quadrant in a Cartesian plane. ETNA - Electronic Transactions on Numerical Analysis
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2 mai 2023
- handle: 10670/1.ispjcu
- issn: 1068-9613
- doi: 10.1553/etna_vol58s432
info:eu-repo/semantics/openAccess
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Brahim Benouahmane et al., « Rational symbolic cubature rules over the first quadrant in a Cartesian plane. ETNA - Electronic Transactions on Numerical Analysis », Elektronisches Publikationsportal der Österreichischen Akademie der Wissenschafte, ID : 10.1553/etna_vol58s432
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Résumé
In this paper we introduce a new symbolic Gaussian formula for theevaluation of an integral over the first quadrant in a Cartesianplane, in particular with respect to the weight function$w(x)=\exp(-x^T x-1/x^T x)$, where $x=(x_1,x_2)^T\in \mathbb{R}^2_+$. Itintegrates exactly a class of homogeneous Laurent polynomials withcoefficients in the commutative field of rational functions in twovariables. It is derived using the connection between orthogonalpolynomials, two-point Padé approximants, and Gaussian cubatures.We also discuss the connection to two-point Padé-typeapproximants in order to establish symbolic cubature formulas ofinterpolatory type. Numerical examples are presented toillustrate the different formulas developed in the paper.