Abstract

In this talk, we discuss a notion of exact solution of the Wiener–Hopf factorisation problem for matrix polynomials. By the exact solution, we understand the fulfilment of the following two conditions: 1) the input data belongs to the Gaussian field Q(i) of complex rational numbers and 2) all (finite) steps of the explicit algorithm can be performed in the rational arithmetic. Since the factorisation is generally speaking unstable with respect to small perturbation, those requirements are crucial to guarantee that the instability issue does not arise. Unfortunately, even the conditions 1) – 2) are not sufficient for the exact solution to exist. We have proven the following necessary and sufficient condition: a matrix polynomial over the field of Gaussian rational numbers admits the exact Wiener–Hopf factorisation if and only if its determinant is exactly factorable. For the factorisation, we use the explicit algorithm based on the method of essential polynomials. It has been proven already its efficiency (it provides both left and right factorisation simultaneously) but is rather technical. To help possible users, we develop its realisation within an ExactMPF package in Maple Software. We illustrate its performance presenting several examples.