A NEW VERSION OF THE SMITH METHOD FOR SOLVING SYLVESTER EQUATION AND DISCRETE-TIME SYLVESTER EQUATION

作者：Li, Jicheng^*; Mei, Ziya; Kong, Xu

来源：Journal of Applied Analysis and Computation, 2016, 6(2): 582-599.

DOI：10.11948/2016040

• Sylvester equation
• Discrete-time Sylvester equation
• Smith method
• Positive definite matrix
• Convergence

Recently, Xue etc. [37] discussed the Smith method for solving Sylvester equation AX + XB = C, where one of the matrices A and B is at least a nonsingular M-matrix and the other is an (singular or nonsingular) M-matrix. Furthermore, in order to find the minimal non-negative solution of a certain class of non-symmetric algebraic Riccati equations, Gao and Bai [ I I] considered a doubling iteration scheme to inexactly solve the Sylvester equations. This paper discusses the iterative error of the standard Smith method used in [ I I] and presents the prior estimations of the accurate solution X for the Sylvester equation. Furthermore, we give a new version of the Smith method for solving discrete-time Sylvester equation or Stein equation AXB + X = C, while the new version of the Smith method can also be used to solve Sylvester equation AX + XB = C, where both A and B are positive definite. We also study the convergence rate of the new Smith method. At last, numerical examples are given to illustrate the effectiveness of our methods.

• 出版日期2016-5
• 单位聊城大学; 西安交通大学