Note on exact factorization algorithm for matrix polynomials

There are two major obstacles for a wide utilisation of the Wiener–Hopf factorization technique for matrix functions used to solve vectorial Riemann boundary problems. The first one reflects the absence of a general explicit factorization method in the matrix case, even though there are some explicit (constructive) factorizations available for specific classes of matrix functions. The second obstacle follows from the fact that the factorization of a matrix function is, generally speaking, not stable operation with respect to a small perturbation of the original function. As a result of the latter, a realisation of any constructive algorithm, even if it exists for the given matrix function, cannot be performed in practice. Moreover, developing explicit methods, authors do not often analyze its numerical implementation, implicitly assuming that all steps of the proposed constructive algorithm can be carried out exactly. In the proposed work, we continue studying a relation between the explicit and exact solutions of the factorization problem in the class of matrix polynomials. The main goal is to obtain an algorithm for the exact evaluation of the so-called indices and essential polynomials of a finite sequence of matrices. This is the cornerstone of the problem of exact factorization of matrix polynomials.