TY - JOUR T1 - A Natural Gradient Descent Algorithm for the Solution of Lyapunov Equations Based on the Geodesic Distance JO - Journal of Computational Mathematics VL - 1 SP - 93 EP - 106 PY - 2014 DA - 2014/02 SN - 32 DO - http://doi.org/10.4208/jcm.1310-m4225 UR - https://global-sci.org/intro/article_detail/jcm/9871.html KW - Lyapunov equation, Geodesic distance, Natural gradient descent algorithm. AB -

A new framework based on the curved Riemannian manifold is proposed to calculate the numerical solution of the Lyapunov matrix equation by using a natural gradient descent algorithm and taking the geodesic distance as the objective function. Moreover, a gradient descent algorithm based on the classical Euclidean distance is provided to compare with this natural gradient descent algorithm. Furthermore, the behaviors of two proposed algorithms and the conventional modified conjugate gradient algorithm are compared and demonstrated by two simulation examples. By comparison, it is shown that the convergence speed of the natural gradient descent algorithm is faster than both of the gradient descent algorithm and the conventional modified conjugate gradient algorithm in solving the Lyapunov equation.