Linearisations of vector nonlinearities

Univariate matrix polynomials are intimately related to matrices, called linearizations. The best known example is the Campanion matrix for determining the roots of a polynomial. The concept of linearization of a matrix polynomial is well understood and has shown its use for solving applications such as the nonlinear eigenvalue problem and model order reduction of nonlinear models in the Laplace domain. The characterizing property is that a matrix polynomial is mapped to a linearization by a transformation that preserves the determinant. In this project, we consider linearizations for multivariate matrix polynomials. In particular, we consider systems whose matrix does not only depend on the Laplace variable but also on the state vector. In general, there is no map to a linearization as for matrix polynomials. The Carleman linearisation, however, provides an approximation to such nonlinearities. This project analyses various choices of linearization and polynomial bases in order to improve the approximation properties.