Computer methods for calculating tuple solutions of polynomial matrix equations

• Autorzy: Dorożyński J. , Nedashkovskyy M.

Identyfikatory

DOI

10.24425/bpasts.2020.133113

• Języki publikacji: EN

Abstrakty

EN

Schemes are presented for calculating tuples of solutions of matrix polynomial equations using continued fractions. Despite the fact that the simplest matrix equations were solved in the second half of the 19th century, and the problem of multiplier decomposition was then deeply analysed, many tasks in this area have not yet been solved. Therefore, the construction of computer schemes for calculating the sequences of solutions is proposed in this work. The second-order matrix equations can be solved by a matrix chain function or iterative method. The results of the numerical experiment using the MatLab package for a given number of iterations are presented. A similar calculation is done for a symmetric square matrix equation of the 2nd order. Also, for the discrete (time) Riccati equation, as its analytical solution cannot be performed yet, we propose constructing its own special scheme of development of the solution in the matrix continued fraction. Next, matrix equations of the n-th order, matrix polynomial equations of the order of non-canonical form, and finally, the conditions for the termination of the iterative process in solving matrix equations by branched continued fractions and the criteria of convergence of matrix branching chain fractions to solutions are discussed.

Słowa kluczowe

EN

matrix polynomial equations; discrete (time) Riccati equation; tuples of solutions; MATLAB

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2020
• Tom: Vol. 68, nr 2
• Strony: 235--243
• Opis fizyczny: Bibliogr. 10 poz., tab.

Twórcy

autor

Dorożyński J.

• Kazimierz Wielki University, Institute of Computer Science, M. Kopernika 1, 85-074 Bydgoszcz

autor

Nedashkovskyy M.

• Kazimierz Wielki University, Institute of Computer Science, M. Kopernika 1, 85-074 Bydgoszcz

Bibliografia

• [1] N.A. Nedashkovskiy and T.I. Kroshka, “Computational algorithms for linear balance models of intersectoral ecological-economic interaction”, Cybern. Syst. Anal. 46, 14–24 (2010). https://doi.org/10.1007/s10559‒010‒9179‒0
• [2] N.A. Nedashkovskiy and T.I. Kroshka, “Solution of one class of nonlinear balance models of intersectoral ecological-economic interaction”, Cybern. Syst. Anal. 47, 684 (2011). https://doi.org/10.1007/s10559‒011‒9348‒9
• [3] Kh.D. Ikramov, “Numerical solution of matrix equations [Chislennoye resheniye matrichnykh uravneniy]”, M: Nauka, p. 192 (1984). [in Russian]
• [4] P.S. Kazimirskyy, “Schedule of matrix polynomials on multipliers [Rozklad matrychnykh mnohochleniv na mnozhnyky]”, Kyiv: Naukova dumka, p. 247 (1983). [in Ukrainian]
• [5] V.N. Kublanovskaya, “On the spectral problem for polynomial beams of matrices [K spektral’noy zadache dlya polinomial’nykh puchkov matrits]”, Records of scientific seminars of the Leningrad Department of Mathematical Institute of the Academy of Sciences of the USSR, 83–97 (1978). (in Russian)
• [6] I. Matychyn and V. Onyshchenko, “Matrix Mittag‐Leffler function in fractional systems and its computation”, Bull. Pol. Ac.: Tech. 66 (4), 495‒500 (2018), https://doi.org/10.24425/124266.
• [7] W. Jakowluk, “Optimal input signal design for fractional-order system identification”, Bull. Pol. Ac.: Tech. 67 (1), 37‒44 (2019), https://doi.org/10.24425/bpas.2019.127336.
• [8] S. Elloumi, I. Sansa and N. B. Braiek, “On the stability of optimal controlled systems with SDRE approach”, International Multi-Conference on Systems, Signals & Devices, Chemnitz, 2012, pp. 1‐5. https://doi.org/10.1109/SSD.2012.6198040.
• [9] J. Bernat, J. Kołota, S. Stępień, and P. Superczyńska, “Sub-optimal control of nonlinear continuous-time locally positive systems using input-state linearization and SDRE approach”, Bull. Pol. Ac.: Tech. 66(1), 17‒22 (2018), https://doi.org/10.24425/119054.
• [10] M.O. Nedashkovskyy, “Signs of convergence of matrix branching chain fractions [Oznaky zbizhnosti matrychnykh hillyastykh lantsyuhovykh drobiv]”, Mathematical methods and physical-mechanical fields. Lviv, 46 (4), 50‒56 (2003). [in Ukrainian]

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).