Abstract
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 nd 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 denite. We also study the convergence rate of the new Smith method. At last, numerical examples are given to illustrate the eectiveness of our methods.
Highlights
The Sylvester equation and the discrete-time Sylvester equation or Stein equation take the forms AX + XB = C, (1.1) and AXB + X = C, (1.2)respectively, where A, B, and C are known complex matrices of size m × m, n × n, and m × n, respectively, and X is the unknown matrix of size m × n
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 M -matrix
We will study the convergence error of the Smith method for solving the Sylvester equation, while the convergence error of the Smith method for solving the discrete-time Sylvester equation can be discussed in a similar way
Summary
The Sylvester equation and the discrete-time Sylvester equation or Stein equation take the forms. The Krylov-subspace Galerkin and minimal residual algorithms [22] for solving the Sylvester equation (1.1) were presented by Hu & Reichel. Direct methods for solving the matrix equation (1.2) such as those proposed in [18, 19] are attractive if both matrices A and B are of small size When both A and B are large and sparse, the iterative solution of (1.2) by the alternating-direction-implicit (ADI) method might be more attractive, see [8, 33]. The Krylov-subspace Galerkin and minimal residual algorithms for solving the discrete-time Sylvester equation (1.2) were presented in [2].
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