Computer methods for calculating tuple solutions of polynomial matrix equations
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...
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| Format: | Article |
| Language: | English |
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Polish Academy of Sciences
2020-04-01
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| Series: | Bulletin of the Polish Academy of Sciences: Technical Sciences |
| Subjects: | |
| Online Access: | https://journals.pan.pl/Content/116291/PDF/08D_235-243_01318_Bpast.No.68-2_17.04.20_K2A_SS.pdf |
| _version_ | 1811343552183009280 |
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| author | J. Dorożyński M. Nedashkovskyy |
| author_facet | J. Dorożyński M. Nedashkovskyy |
| author_sort | J. Dorożyński |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-13T19:31:00Z |
| format | Article |
| id | doaj.art-566cd680ca984897a6768bb3b900a3a4 |
| institution | Directory Open Access Journal |
| issn | 2300-1917 |
| language | English |
| last_indexed | 2024-04-13T19:31:00Z |
| publishDate | 2020-04-01 |
| publisher | Polish Academy of Sciences |
| record_format | Article |
| series | Bulletin of the Polish Academy of Sciences: Technical Sciences |
| spelling | doaj.art-566cd680ca984897a6768bb3b900a3a42022-12-22T02:33:10ZengPolish Academy of SciencesBulletin of the Polish Academy of Sciences: Technical Sciences2300-19172020-04-0168No. 2 (i.a. Special Section on Computational Intelligence in Communications)235243https://doi.org/10.24425/bpasts.2020.133113Computer methods for calculating tuple solutions of polynomial matrix equationsJ. DorożyńskiM. NedashkovskyySchemes 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.https://journals.pan.pl/Content/116291/PDF/08D_235-243_01318_Bpast.No.68-2_17.04.20_K2A_SS.pdfmatrix polynomial equationsdiscrete (time) riccati equationtuples of solutionsmatlab |
| spellingShingle | J. Dorożyński M. Nedashkovskyy Computer methods for calculating tuple solutions of polynomial matrix equations Bulletin of the Polish Academy of Sciences: Technical Sciences matrix polynomial equations discrete (time) riccati equation tuples of solutions matlab |
| title | Computer methods for calculating tuple solutions of polynomial matrix equations |
| title_full | Computer methods for calculating tuple solutions of polynomial matrix equations |
| title_fullStr | Computer methods for calculating tuple solutions of polynomial matrix equations |
| title_full_unstemmed | Computer methods for calculating tuple solutions of polynomial matrix equations |
| title_short | Computer methods for calculating tuple solutions of polynomial matrix equations |
| title_sort | computer methods for calculating tuple solutions of polynomial matrix equations |
| topic | matrix polynomial equations discrete (time) riccati equation tuples of solutions matlab |
| url | https://journals.pan.pl/Content/116291/PDF/08D_235-243_01318_Bpast.No.68-2_17.04.20_K2A_SS.pdf |
| work_keys_str_mv | AT jdorozynski computermethodsforcalculatingtuplesolutionsofpolynomialmatrixequations AT mnedashkovskyy computermethodsforcalculatingtuplesolutionsofpolynomialmatrixequations |