A symmetric preserving iterative method for generalized Sylvester equation

Abstract

The generalized Sylvester matrix equation AX + YB = C is encountered in many systems and control applications, and also has several applications relating to the problem of image restoration, and the numerical solution of implicit ordinary differential equations. In this paper, we construct a symmetric preserving iterative method, basing on the classic Conjugate Gradient Least Squares (CGLS) method, for AX + YB = C with the unknown matrices X, Y having symmetric structures. With this method, for any arbitrary initial symmetric matrix pair, a desired solution can be obtained within finitely iterate steps. The unique optimal (least norm) solution can also be obtained by choosing a special kind of initial matrix. We also consider the matrix nearness problem. Some numerical results confirm the efficiency of these algorithms. It is more important that some numerical stability analysis on the matrix nearness problem is given combined with numerical examples, which is not given in the earlier papers. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society