Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations
A Riemannian gradient algorithm based on geometric structures of a manifold consisting of all positive definite matrices is proposed to calculate the numerical solution of the linear matrix equation Q=X+∑i=1mAiTXAi. In this algorithm, the geodesic distance on the curved Riemannian manifold is taken...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2014/507175 |
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| _version_ | 1832554083855106048 |
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| author | Xiaomin Duan Huafei Sun Xinyu Zhao |
| author_facet | Xiaomin Duan Huafei Sun Xinyu Zhao |
| author_sort | Xiaomin Duan |
| collection | DOAJ |
| description | A Riemannian gradient algorithm based on geometric structures of a manifold consisting of all positive definite matrices is proposed to calculate the numerical solution of the linear matrix equation Q=X+∑i=1mAiTXAi. In this algorithm, the geodesic distance on the curved Riemannian manifold is taken as an objective function and the geodesic curve is treated as the convergence path. Also the optimal variable step sizes corresponding to the minimum value of the objective function are provided in order to improve the convergence speed. Furthermore, the convergence speed of the Riemannian gradient algorithm is compared with that of the traditional conjugate gradient method in two simulation examples. It is found that the convergence speed of the provided algorithm is faster than that of the conjugate gradient method. |
| format | Article |
| id | doaj-art-5df40aa6cee34494ae49b42ccbfbecaa |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-5df40aa6cee34494ae49b42ccbfbecaa2025-02-03T05:52:19ZengWileyJournal of Applied Mathematics1110-757X1687-00422014-01-01201410.1155/2014/507175507175Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix EquationsXiaomin Duan0Huafei Sun1Xinyu Zhao2School of Mathematics, Beijing Institute of Technology, Beijing 100081, ChinaSchool of Mathematics, Beijing Institute of Technology, Beijing 100081, ChinaSchool of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, ChinaA Riemannian gradient algorithm based on geometric structures of a manifold consisting of all positive definite matrices is proposed to calculate the numerical solution of the linear matrix equation Q=X+∑i=1mAiTXAi. In this algorithm, the geodesic distance on the curved Riemannian manifold is taken as an objective function and the geodesic curve is treated as the convergence path. Also the optimal variable step sizes corresponding to the minimum value of the objective function are provided in order to improve the convergence speed. Furthermore, the convergence speed of the Riemannian gradient algorithm is compared with that of the traditional conjugate gradient method in two simulation examples. It is found that the convergence speed of the provided algorithm is faster than that of the conjugate gradient method.http://dx.doi.org/10.1155/2014/507175 |
| spellingShingle | Xiaomin Duan Huafei Sun Xinyu Zhao Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations Journal of Applied Mathematics |
| title | Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations |
| title_full | Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations |
| title_fullStr | Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations |
| title_full_unstemmed | Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations |
| title_short | Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations |
| title_sort | riemannian gradient algorithm for the numerical solution of linear matrix equations |
| url | http://dx.doi.org/10.1155/2014/507175 |
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