On the nonlinear matrix equation $ X^{s}+A^{H}F(X)A = Q $
Nonlinear matrix equation often arises in control theory, statistics, dynamic programming, ladder networks, and so on, so it has widely applied background. In this paper, the nonlinear matrix equation $ X^{s}+A^{H}F(X)A = Q $ are discussed, where operator $ F $ are defined in the set of all $ n\time...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-05-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023935?viewType=HTML |
| _version_ | 1827928349060628480 |
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| author | Yajun Xie Changfeng Ma Qingqing Zheng |
| author_facet | Yajun Xie Changfeng Ma Qingqing Zheng |
| author_sort | Yajun Xie |
| collection | DOAJ |
| description | Nonlinear matrix equation often arises in control theory, statistics, dynamic programming, ladder networks, and so on, so it has widely applied background. In this paper, the nonlinear matrix equation $ X^{s}+A^{H}F(X)A = Q $ are discussed, where operator $ F $ are defined in the set of all $ n\times n $ positive semi-definite matrices, and $ Q $ is a positive definite matrix. Sufficient conditions for the existence and uniqueness of a positive semi-definite solution are derived based on some fixed point theorems. It is shown that under suitable conditions an iteration method converges to a positive semi-definite solution. Moreover, we consider the perturbation analysis for the solution of this class of nonlinear matrix equations, and obtain a perturbation bound of the solution. Finally, we give several examples to show how this works in particular cases, and some numerical results to specify the rationality of the results we have obtain. |
| first_indexed | 2024-03-13T06:06:29Z |
| format | Article |
| id | doaj.art-8e1945fc86e34adc810e96edf48c66e7 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-13T06:06:29Z |
| publishDate | 2023-05-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-8e1945fc86e34adc810e96edf48c66e72023-06-12T01:35:05ZengAIMS PressAIMS Mathematics2473-69882023-05-0188183921840710.3934/math.2023935On the nonlinear matrix equation $ X^{s}+A^{H}F(X)A = Q $Yajun Xie0Changfeng Ma1Qingqing Zheng21. School of Big Data & Key Laboratory of Digital Technology and Intelligent Computing, Fuzhou University of International Studies and Trade, Fuzhou 350202, China1. School of Big Data & Key Laboratory of Digital Technology and Intelligent Computing, Fuzhou University of International Studies and Trade, Fuzhou 350202, China2. School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350007, ChinaNonlinear matrix equation often arises in control theory, statistics, dynamic programming, ladder networks, and so on, so it has widely applied background. In this paper, the nonlinear matrix equation $ X^{s}+A^{H}F(X)A = Q $ are discussed, where operator $ F $ are defined in the set of all $ n\times n $ positive semi-definite matrices, and $ Q $ is a positive definite matrix. Sufficient conditions for the existence and uniqueness of a positive semi-definite solution are derived based on some fixed point theorems. It is shown that under suitable conditions an iteration method converges to a positive semi-definite solution. Moreover, we consider the perturbation analysis for the solution of this class of nonlinear matrix equations, and obtain a perturbation bound of the solution. Finally, we give several examples to show how this works in particular cases, and some numerical results to specify the rationality of the results we have obtain.https://www.aimspress.com/article/doi/10.3934/math.2023935?viewType=HTMLnonlinear matrix equationpositive semi-definite solutionfixed point theoremperturbation analysis |
| spellingShingle | Yajun Xie Changfeng Ma Qingqing Zheng On the nonlinear matrix equation $ X^{s}+A^{H}F(X)A = Q $ AIMS Mathematics nonlinear matrix equation positive semi-definite solution fixed point theorem perturbation analysis |
| title | On the nonlinear matrix equation $ X^{s}+A^{H}F(X)A = Q $ |
| title_full | On the nonlinear matrix equation $ X^{s}+A^{H}F(X)A = Q $ |
| title_fullStr | On the nonlinear matrix equation $ X^{s}+A^{H}F(X)A = Q $ |
| title_full_unstemmed | On the nonlinear matrix equation $ X^{s}+A^{H}F(X)A = Q $ |
| title_short | On the nonlinear matrix equation $ X^{s}+A^{H}F(X)A = Q $ |
| title_sort | on the nonlinear matrix equation x s a h f x a q |
| topic | nonlinear matrix equation positive semi-definite solution fixed point theorem perturbation analysis |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023935?viewType=HTML |
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