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Numerical method based on fiber bundle for solving Lyapunov matrix equation

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  • Win, Aung Naing
  • Li, Mingming

Abstract

In this paper, we firstly introduce the origin of Lyapunov matrix equation, and then the geometric methods for solving Lyapunov equation are given by using the Log-Euclidean metric and the fiber bundle-based Riemannian metric based on the manifold of positive definite Hermitian matrices. Then, the solution of Lyapunov matrix equation is presented by providing a natural gradient descent algorithm (NGDA), a Log-Euclidean descent algorithm (LGDA) and a Riemannian gradient algorithm based on fiber bundle (RGA). At last, the convergence speeds of the RGA, the NGDA and the LGDA are compared via two simulation examples. Simulation results show that the convergence speed of the RGA is faster than both of the LGDA and the NGDA.

Suggested Citation

  • Win, Aung Naing & Li, Mingming, 2022. "Numerical method based on fiber bundle for solving Lyapunov matrix equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 556-566.
  • Handle: RePEc:eee:matcom:v:193:y:2022:i:c:p:556-566
    DOI: 10.1016/j.matcom.2021.10.031
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    References listed on IDEAS

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    1. Xiaomin Duan & Huafei Sun & Xinyu Zhao, 2014. "Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-7, January.
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