Solution of Algebraic Lyapunov Equation on Positive-Definite Hermitian Matrices by Using Extended Hamiltonian Algorithm

This communique is opted to study the approximate solution of the Algebraic Lyapunov equation on the manifold of positive-definite Hermitian matrices. We choose the geodesic distance between -A(H)X - XA and P as the cost function, and put forward the Extended Hamiltonian algorithm (EHA) and Natural gradient algorithm (NGA) for the solution. Finally, several numerical experiments give you an idea about the effectiveness of the proposed algorithms. We also show the comparison between these two algorithms EHA and NGA. Obtained results are provided and analyzed graphically. We also conclude that the extended Hamiltonian algorithm has better convergence speed than the natural gradient algorithm, whereas the trajectory of the solution matrix is optimal in case of Natural gradient algorithm (NGA) as compared to Extended Hamiltonian Algorithm (EHA). The aim of this paper is to show that the Extended Hamiltonian algorithm (EHA) has superior convergence properties as compared to Natural gradient algorithm (NGA). Upto the best of author's knowledge, no approximate solution of the Algebraic Lyapunov equation on the manifold of positive-definite Hermitian matrices is found so far in the literature.