Weighted steepest descent method for solving matrix equations

Abstract

This paper describes iterative methods for solving the general linear matrix equation including the well-known Lyapunov matrix equation, Sylvester matrix equation and some related matrix equations encountered in control system theory, as special cases. We develop the methods from the optimization point of view in the sense that the iterative algorithms are constructed to solve some optimization problems whose solutions are closely related to the unique solution to the linear matrix equation. Actually, two optimization problems are considered and, therefore, two iterative algorithms are proposed to solve the linear matrix equation. To solve the two optimization problems, the steepest descent method is adopted. By means of the so-called weighted inner product that is defined and studied in this paper, the convergence properties of the algorithms are analysed. It is shown that the algorithms converge at least linearly for arbitrary initial conditions. The proposed approaches are expected to be numerically reliable as only matrix manipulation is required. Numerical examples show the effectiveness of the proposed algorithms.

Keywords:

• steepest descent method
• weighted inner product
• general linear matrix equation
• iterative algorithms
• numerical solutions
• linear system theory

2000 AMS Subject Classification::

• 15A06
• 15A24
• 65F10
• 65F30
• 93C05

Acknowledgements

This work was partially supported by the National Natural Science Foundation of China under Grant numbers 60904007 and 61104124 and by the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 2011007).