The modified conjugate gradient methods for solving a class of generalized coupled Sylvester-transpose matrix equations

Abstract

In this paper, we consider the iteration solutions of generalized coupled Sylvester-transpose matrix equations: A1XB1+C1YTD1=F1, A2YB2+C2XTD2=F2. When the coupled matrix equations are consistent, we propose a modified conjugate gradient method to solve the equations and prove that a solution (X∗,Y∗) can be obtained within finite iterative steps in the absence of roundoff-error for any initial value. Furthermore, we show that the minimum-norm solution can be got by choosing a special kind of initial matrices. When the coupled matrix equations are inconsistent, we present another modified conjugate gradient method to find the least-squares solution with the minimum-norm. Finally, some numerical examples are given to show the behavior of the considered algorithms.