Gradient-based maximal convergence rate iterative method for solving linear matrix equations

Abstract

This paper is concerned with numerical solutions to general linear matrix equations including the well-known Lyapunov matrix equation and Sylvester matrix equation as special cases. Gradient based iterative algorithm is proposed to approximate the exact solution. A necessary and sufficient condition guaranteeing the convergence of the algorithm is presented. A sufficient condition that is easy to compute is also given. The optimal convergence factor such that the convergence rate of the algorithm is maximized is established. The proposed approach not only gives a complete understanding on gradient based iterative algorithm for solving linear matrix equations, but can also be served as a bridge between linear system theory and numerical computing. Numerical example shows the effectiveness of the proposed approach.

Keywords:

• discrete-time linear system theory
• gradient-based iterative algorithm
• lyapunov matrix equations
• numerical computing
• Sylvester matrix equations

2000 AMS Subject Classification :

• 65F10
• 65N22
• 15A09
• 15A12
• 15A24

Acknowledgements

The work of Bin Zhou and Guang-Ren Duan was partially supported by the Major Program of National Natural Science Foundation of China under grant No. 60710002 and Program for Changjiang Scholars and Innovative Research Team in University. The work of James Lam was partially supported by RGC HKU 7029/05P.