Rational symbolic cubature rules over the first quadrant in a Cartesian plane

Abstract

. In this paper we introduce a new symbolic Gaussian formula for the evaluation of an integral over the first quadrant in a Cartesian plane, 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 ∈ R 2+ . It integrates exactly a class of homogeneous Laurent polynomials with coefficients in the commutative field of rational functions in two variables. It is derived using the connection between orthogonal polynomials, two-point Padé approximants, and Gaussian cubatures. We also discuss the connection to two-point Padé-type approximants in order to establish symbolic cubature formulas of interpolatory type. Numerical examples are presented to illustrate the different formulas developed in the paper.