Gradient-based iterative algorithm for the extended coupled Sylvester matrix equations

A gradient-based iterative algorithm is established for solving the extended coupled Sylvester matrix equations A
₁
XB
₁
+ A
₂
Y B
₂
= F
₁
and C
₁
XD
₁
+ C
₂
Y D
₂
= F
₂
by using the hierarchical identification principle. We prove that the iterative solution converges fast to the exact one for any initial iterative value if the coupled matrix equations has a unique solution. A numerical example is offered to illustrate the efficiency of the proposed algorithm.