Three types of biconjugate residual method for general periodic matrix equations over generalized bisymmetric periodic matrices

Abstract

As is well known, periodic matrix equations have wide applications in many areas of control and system theory. This paper is devoted to a study of the numerical solutions of a general type of periodic matrix equations. We present three types of biconjugate residual (BCR) method to find the generalized bisymmetric periodic solutions $(X_1,Y_1,X_2,Y_2,\dots ,X_{\eta },Y_{\eta })$ of general periodic matrix equations

$$\sum _{s=0}^{\eta -1}(A_{i,s}{X}_{i+s}{B}_{i,s}+C_{i,s}{Y}_{i+s}{D}_{i,s})=E_i,i=1,2,\dots ,\eta$$

The main theorems of this paper show that the presented methods can compute the generalized bisymmetric periodic solutions in a finite number of steps in the absence of round-off errors. We give two numerical examples to illustrate and interpret the theoretical results.