Relaxed gradient-based iterative solutions to coupled Sylvester-conjugate transpose matrix equations of two unknowns

PurposeThis article proposes a relaxed gradient iterative (RGI) algorithm to solve coupled Sylvester-conjugate transpose matrix equations (CSCTME) with two unknowns.Design/methodology/approachThis article proposes a RGI algorithm to solve CSCTME with two unknowns.FindingsThe introduced (RGI) algorithm is more efficient than the gradient iterative (GI) algorithm presented in Bayoumi (2014), where the author's method exhibits quick convergence behavior.Research limitations/implicationsThe introduced (RGI) algorithm is more efficient than the GI algorithm presented in Bayoumi (2014), where the author's method exhibits quick convergence behavior.Practical implicationsIn systems and control, Lyapunov matrix equations, Sylvester matrix equations and other matrix equations are commonly encountered.Social implicationsIn systems and control, Lyapunov matrix equations, Sylvester matrix equations and other matrix equations are commonly encountered.Originality/valueThis article proposes a relaxed gradient iterative (RGI) algorithm to solve coupled Sylvester conjugate transpose matrix equations (CSCTME) with two unknowns. For any initial matrices, a sufficient condition is derived to determine whether the proposed algorithm converges to the exact solution. To demonstrate the effectiveness of the suggested method and to compare it with the gradient-based iterative algorithm proposed in [6] numerical examples are provided.

Automated summary

This article proposes a relaxed gradient iterative (RGI) algorithm to solve coupled Sylvester-conjugate transpose matrix equations (CSCTME) with two unknowns. The introduced algorithm is more efficient than the gradient iterative (GI) algorithm presented in Bayoumi (2014), where the author's method exhibits quick convergence behavior.