Factorisation of the matrix polynomial: what does exact factorisation mean?

Let A(t) be a matrix function from the matrix Wiener algebra Wp×p(T) that is invertible on the unit circle T. The representation A(t) = A+(t)D(t)A−(t), t ∈ T is called a left Wiener–Hopf factorisation of A(t). Here A±(t) ∈ GWp×p ± (T), where GWp×p ± (T) are the groups of invertible elements of the subalgebras W p×p ± (T), D(t) is the diagonal matrix D(t) = diag tλ1, . . . , tλp, where integers λ1 ≥ . . . ≥ λp are the left partial indices of A(t). This problem has extensive applications in different areas of mathematics, mechanics and physics. Unfortunately, there are two main obstacles for its wider applications: there does not exist a general method for explicit factorisation of matrix functions, while possible factorisation is not always stable in other words the problem is ill-posed. As a result, even if an explicit algorithm for factorisation of a class of matrix functions has been developed, its implementation into a software requires approximate calculations, which may not be properly realised due to the aforementioned instability. To overcome this issue, obe can utilise exact calculations, that is, calculations in rational arithmetic. Unfortunately, this is not always possible. In the talk, for the class of Laurent matrix polynomials, a criterion for the existence of exact solution of the factorisation problem is found [1]. Based on the well-known explicit algorithm [2], an exact solution is constructed and implemented as a package ExactMPF in Maple [1, 3, 4]. Numerical experiments are presented.