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

V M Adukov1, N V Adukova1, G Mishuris2

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Proceedings. Mathematical, Physical, and Engineering Sciences
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PubMed
Summary

This study presents an exact algorithm for Wiener-Hopf factorization of matrix polynomials, crucial for unstable problems. A new Maple package, ExactMPF, is developed for exact and approximate solutions.

Keywords:
Toeplitz matricesWiener–Hopf factorizationessential polynomialsexact computationimplementationmatrix polynomials

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Area of Science:

  • Numerical Analysis
  • Symbolic Computation
  • Linear Algebra

Background:

  • The Wiener-Hopf factorization problem for matrix polynomials is essential in various scientific fields.
  • Exact solutions are critical due to the inherent instability of the problem.

Purpose of the Study:

  • To develop an explicit algorithm for the exact Wiener-Hopf factorization of matrix polynomials.
  • To introduce a computational tool for performing these factorizations.
  • To address the challenges of approximate factorization when exact solutions are not feasible.

Main Methods:

  • Development of an explicit algorithm based on symbolic computation.
  • Proof of the condition for exact Wiener-Hopf factorization (factorable determinant).
  • Adaptation of the algorithm for exact calculations using Gaussian rational numbers.

Main Results:

  • An exact condition for the Wiener-Hopf factorization of matrix polynomials is established.
  • The ExactMPF package in Maple software is developed and tested for exact factorization.
  • A framework for numerical factorization is clarified for cases where exact factorization is not possible.

Conclusions:

  • The ExactMPF package provides a reliable tool for exact Wiener-Hopf factorization under specific conditions.
  • The study clarifies the notion of approximate factorization and associated confidence levels.
  • The research contributes to the robust computational solution of matrix polynomial factorization problems.