Section: New Results

A moment matrix approach to computing symmetric cubatures

Participants : Mathieu Collowald, Evelyne Hubert.

A quadrature is an approximation of the definite integral of a function by a weighted sum of function values at specified points, or nodes, within the domain of integration. Gaussian quadratures are constructed to yield exact results for any polynomial of degree $2r-1$ or less by a suitable choice of $r$ nodes and weights. Cubature is a generalization of quadrature in higher dimension. Constructing a cubature amounts to find a linear form

                      ${\displaystyle \Lambda :ℝ\left[x\right]\to ℝ,p\mapsto \sum _{j=1}^r{a}_{j}p\left({\xi }_j\right)}$

from the knowledge of its restriction to $ℝ{\left[x\right]}_{\le d}$. The unknowns are the number of nodes $r$, the weights $a_j$ and the nodes ${\xi }_j$.

In [7] we use a basis-free version of an approach to cubatures based on moment matrices in terms of the Hankel operator $\mathscr{H}$ associated to $\Lambda$. The existence of a cubature of degree $d$ with $r$ nodes boils down to conditions of ranks and positive semidefiniteness on $\mathscr{H}$. We then recognize the nodes as the solutions of a generalized eigenvalue problem.

Standard domains of integration are symmetric under the action of a finite group. It is natural to look for cubatures that respect this symmetry. Introducing adapted bases obtained from representation theory, the symmetry constraint allows to block diagonalize the Hankel operator $\mathscr{H}$. We then deal with smaller-sized matrices both for securing the existence of the cubature and computing the nodes. The sizes of the blocks are furthermore explicitly related to the orbit types of the nodes with the new concept of the matrix of multiplicities of a finite group. It provides preliminary criteria of existence of a cubature with a given organisation of the nodes in orbit types.

The Maple implementation of the presented algorithms allows to determine, with moderate computational efforts, all the symmetric cubatures of a given degree. We present new relevant cubatures.