An imbedded family of cubature formulae for \(n\)-dimensional product regions
An imbedded family of cubature formulae for \(n\)-dimensional fully symmetric product regions is developed in the manner of \textit{A. C. Genz} and \textit{A. A. Malik} [SIAM J. Numer. Anal. 20, 580-588 (1983; Zbl 0541.65012)]. A formula of degree \(2m + 1\) is based on \(m + 1\) generators obtained recursively. The \(n\)-dimensional cube and the whole of \(n\)- dimensional space with a Gaussian weight function are studied explicitly. Symbolic computation is used to obtain closed formulae for the (parameter-dependent) weights and points, enabling an investigation of the existence of formulae having positive weights and all points inside the region. Weights for the \(n\)-cube with degree \(\leq 7\), and for \(n\)- space with degree \(\leq 9\), are listed in tabular form.
scientific article; zbMATH DE number 644311