A NEW VERSION OF THE SMITH METHOD FOR SOLVING SYLVESTER EQUATION AND DISCRETE-TIME SYLVESTER EQUATION
Jicheng Li, Ziya Mei, Xu Kong · Journal of Applied Analysis & Computation · 2016
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[17] considered a doubling iteration scheme to inexactly solve the Sylvester equations. This paper discusses the iterative error of the standard Smith method used in[17] 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.