Conjugate gradient least squares algorithm for solving the generalized coupled Sylvester matrix equations

This paper discusses the conjugate gradient least squares algorithm for solving the generalized coupled Sylvester matrix equations ∑j=1qAijXjBij=Fi, i=1,2,…,p. We prove that if this system is consistent then the iterative solution converges to the exact solution and if this system is inconsistent then the iterative solution converges to the least squares solution within the finite iteration steps in the absence of the roundoff errors. Also by setting the initial iterative value properly we prove that the iterative solution converges to the least squares and minimum-norm solution.