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Abstract
We discuss an explicit algorithm for solving the WienerHopf factorization problem for matrix polynomials. By an exact solution of the problem, we understand the one constructed by a symbolic computation. Since the problem is, generally speaking, unstable, this requirement is crucial to guarantee that the result following from the explicit algorithm is indeed a solution of the original factorization problem. We prove that a matrix polynomial over the field of Gaussian rational numbers admits the exact WienerHopf factorization if and only if its determinant is exactly factorable. Under such a condition, we adapt the explicit algorithm to the exact calculations and develop the ExactMPF package realized within the Maple Software. The package has been extensively tested. Some examples are presented in the paper, while the listing is provided in the electronic supplementary material. If, however, a matrix polynomial does not admit the exact factorization, we clarify a notion of the numerical (or approximate) factorization that can be constructed by following the explicit factorization algorithm. We highlight possible obstacles on the way and discuss a level of confidence in the final result in the case of an unstable set of partial indices. The full listing of the package ExactMPF is given in the electronic supplementary material.
Original language  English 

Article number  20210941 
Number of pages  22 
Journal  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 
Volume  478 
Issue number  2263 
Early online date  06 Jul 2022 
DOIs  
Publication status  Published  27 Jul 2022 
Keywords
 essential polynomials
 exact computation
 implementation
 matrix polynomials
 Toeplitz matrices
 WienerHopf factorization
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Supplementary material from "An explicit Wiener–Hopf factorization algorithm for matrix polynomials and its exact realizations within ExactMPF package"
Adukov, V. M., Adukova, N. V. & Mishuris, G., Royal Society, 23 Jun 2022
DOI: 10.6084/m9.figshare.c.6060394
Dataset

EffectFact: Effective Factorisation techniques for matrixfunctions: Developing theory, numerical methods and impactful applications
01 Sept 2021 → 31 Aug 2025
Project: Externally funded research

Wolfson Visiting Fellowship  Professor Victor Eremeyev
01 Jul 2021 → 30 Jun 2023
Project: Externally funded research