An explicit Wiener-Hopf factorization algorithm for matrix polynomials and its exact realizations within ExactMPF package

V. M. Adukov*, N. V. Adukova, G. Mishuris

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (SciVal)
37 Downloads (Pure)

Abstract

We discuss an explicit algorithm for solving the Wiener-Hopf 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 Wiener-Hopf 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 languageEnglish
Article number20210941
Number of pages22
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume478
Issue number2263
Early online date06 Jul 2022
DOIs
Publication statusPublished - 27 Jul 2022

Keywords

  • essential polynomials
  • exact computation
  • implementation
  • matrix polynomials
  • Toeplitz matrices
  • Wiener-Hopf factorization

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