# Note on Exact Factorization Algorithm for Matrix Polynomials

V.M. Adukov, N.V. Adukova, G. MishurisThere 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.Full text

- Keywords
- Wiener-Hopf factorization; toeplitz matrices; essential polynomials of sequence.
- References
- 1. Lawrie J.B., Abrahams I.D. A Brief Historical Perspective of the Wiener-Hopf Technique. Journal of Engineering Mathematics, 2007, vol. 59, pp. 351-358. DOI: 10.1007/s10665-007-9195-x

2. Daniele V.G., Zich R.S. The Wiener-Hopf Method in Electromagnetics. ISMB Series. New York, SciTech Publishing, Edison, 2014.

3. Abrahams I.D. On the Application of the Wiener-Hopf Technique to Problems in Dynamic Elasticity. Wave Motion, 2002, vol. 36, pp. 311-333.

4. Kisil A.V., Abrahams I.D., Mishuris G. et al. The Wiener-Hopf Technique, its Generalizations and Applications: Constructive and Approximate Methods. Proceedings of the Royal Society A, 2021, vol. 477, no. 2254, article ID: 20210533, 32 p. DOI: 10.1098/rspa.2021.0533

5. Gohberg I.C., Feldman I.A. Convolution Equations and Projection Methods for their Solution. Providence, American Mathematical Society, 1974.

6. Clancey K., Gohberg I. Factorization of Matrix Functions and Singular Integral Operators. Basel, Boston, Birk"auser, 1987.

7. Zakharov V.E., Manakov S.V., Novikov S.P. et.al. Soliton Theory: Inverse Scattering Method, 1980, Nauka, Moscow.

8. Gohberg I.C., M.A. Kaashoek M.A., Spitkovsky I.M. An Overview of Matrix Factorization Theory and Operator Applications, Operator Theory: Advances and Applications, 2003, vol. 141, pp. 1-102.

9. Ephremidze L., Janashia G., Lagvilava E., A New Method of Matrix Spectral Factorization. IEEE Transactions on Information Theory, 2011, vol. 57, no. 4, pp. 2318 - 2326.

10. Gohberg I.C., Lerer L., Rodman L. Factorization Indices for Matrix Polynomials. Bulletin of the American Mathematical Society, 1978, vol. 84, no. 2, pp. 275-277.

11. Adukov V.M. Factorization of Analytic Matrix-Valued Functions. Theoretical and Mathematical Physics, 1999, vol. 118, no. 3, pp. 255-263. DOI: 10.4213/tmf704

12. Adukov V.M. Wiener-Hopf Factorization of Meromorphic Matrix-Valued Functions. St. Petersburg Mathematical Journal, 1993, vol. 4, no. 1, pp. 51-69.

13. Rogosin S.V., Mishuris G. Constructive Methods for Factorization of Matrix Functions. IMA Journal of Applied Mathematics, 2016, vol. 81, no. 2, pp. 365-391. DOI: 10.1093/imamat/hxv038

14. Giorgadze G, Manjavidze N. On Some Constructive Methods for the Matrix Riemann-Hilbert Boundary Value Problem. Journal of Mathematical Sciences, 2013, vol. 195, no.2, pp. 146-174. DOI: 10.1007/s10958-013-1571-7

15. Kisil A.V. Stability Analysis of Matrix Wiener-Hopf Factorization of Daniele-Khrapkov Class and Reliable Approximate Factorization. Proceedings of the Royal Society A, 2015, vol. 471, article ID: 20150146, 15 p. DOI: 10.1098/rspa.2015.0146

16. Adukov V.M., Adukova N.V., Mishuris G. An Explicit Wiener-Hopf Factorization Algorithm for Matrix Polynomials and Its Exact Realizations within ExactMPF Package. Proceedings of the Royal Society A, 2022, vol. 478, no. 2263, article ID: 20210941, 22 p. DOI: 10.1098/rspa.2021.0941

17. Adukov V.M. Generalized Inversion of Block Toeplitz Matrices. Linear Algebra and Its Applications, 1998, vol. 274, pp. 85-124.

18. Adukova N.V. ExactMPF Package for Constructing the Exact Wiener-Hopf Factorization of Matrix Polynomials in SCM Maple. Proceedings of the XXII International Scientific Conference ''Computer Mathematics Systems and their Applications'', Smolensk, 2021, vol. 22, pp. 20-27. (in Russian)