![]() |
![]() |
ETNA - Electronic Transactions on Numerical Analysis
|
![]() |
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 |
![]() |
|
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)
|
ETNA - Electronic Transactions on Numerical Analysis, pp. 432-449, 2023/05/02
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.
Keywords: homogeneous orthogonal polynomials, homogeneous two-point Padé, symbolic Gaussian cubature