Matrix iterative methods for solving the Sylvester-transpose and periodic Sylvester matrix equations

The problem of solving matrix equations has many applications in control and system theory. This paper is concerned with the iterative solutions of the Sylvester-transpose matrix equation ∑ i = 1 k ( A i XB i + C i X T D i ) = E , and the periodic Sylvester matrix equation A ^ j X ^ j B ^ j + C ^ j X ^ j + 1 D ^ j = E ^ j for j = 1 , 2 , … , λ . The basic idea is to develop the conjugate gradients squared (CGS) and bi-conjugate gradient stabilized (Bi-CGSTAB) methods for obtaining matrix iterative methods for solving the Sylvester-transpose and periodic Sylvester matrix equations. Numerical test results are given to compare matrix iterative methods with other well-known methods.