A natural gradient descent algorithm for the solution of discrete algebraic Lyapunov equations based on the geodesic distance

A new framework to calculate the numerical solution of the discrete algebraic Lyapunov equation is proposed by using the geometric structures on the Riemannian manifold. Specifically, two algorithms based on the manifold of positive definite symmetric matrices are provided. One is a gradient descent algorithm with an objective function of the classical Euclidean distance. The other is a natural gradient descent algorithm with an objective function of the geodesic distance on the curved Riemannian manifold. Furthermore, these two algorithms are compared with a traditional iteration method. Simulation examples show that the convergence speed of the natural gradient descent algorithm is the fastest one among three algorithms.