Symbolic–numeric Gaussian cubature rules

Near-minimal cubature formulae on the disk

“…Note that the vector (cos θ , sin θ) spans the one-dimensional subspace {(z cos θ , z sin θ) : z ∈ R} of R 2 . Then under some simple and obvious conditions (Benouahmane & Cuyt, 2000;Cuyt et al, 2004Cuyt et al, , 2011, B 2 π m (x cos θ + y sin θ)w ( (x, y) 2 ) dx dy = where π m ∈ P m,θ and the ξ i,n (θ ) are the zero curves of the polynomial P n ∈ P m,θ satisfying the orthogonality conditions…”

“…So a θ -parameterized multivariate polynomial (the result also exists in higher dimensions) of degree m = 2n − 1 is integrated exactly over the unit ball (the result holds in other norms) by a linear combination of n θ -parameterized weights and evaluations of the polynomial integrand along zero curves of the spherical orthogonal polynomial of degree n (Cuyt et al, 2004(Cuyt et al, , 2011. In the next section, we convert this symbolic integration result into a numeric rule of the classical form (1.1), and this for general n. In this way, we will rediscover all the minimal and near-minimal rules listed in Table 1.…”