| Volume 6, Number 2, 2016, Pages 582-599 DOI：10.11948/2016040 |
| A new version of the Smith method for solving Sylvester equation and discrete-time Sylvester equation |
| Jicheng Li,Ziya Mei,Xu Kong |
| Keywords:Sylvester equation, Discrete-time Sylvester equation, Smith method, Positive definite matrix, Convergence |
| Abstract: |
| Recently, Xue etc. \cite{28}
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 \cite{gao-2010} considered a doubling iteration scheme to
inexactly solve the Sylvester equations. This paper discusses the
iterative error of the standard Smith method used in \cite{gao-2010}
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. % matrices.
We also study the convergence rate of the new Smith method. At last, numerical examples are given to illustrate
the effectiveness of our methods |
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