Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations

A Riemannian gradient algorithm based on geometric structures of a manifold consisting of all positive definite matrices is proposed to calculate the numerical solution of the linear matrix equation . In this algorithm, the geodesic distance on the curved Riemannian manifold is taken as an objective function and the geodesic curve is treated as the convergence path. Also the optimal variable step sizes corresponding to the minimum value of the objective function are provided in order to improve the convergence speed. Furthermore, the convergence speed of the Riemannian gradient algorithm is compared with that of the traditional conjugate gradient method in two simulation examples. It is found that the convergence speed of the provided algorithm is faster than that of the conjugate gradient method. 1. Introduction The linear matrix equation where are arbitrary real matrices, is a nonnegative integer, and denotes the transpose of the matrix , arises from many fields, such as the control theory, the dynamic programming, and the stochastic filtering [1–4]. In the past decades, there has been increasing interest in the solution problems of this equation. In the case of , some numerical methods, such as Bartels-Stewart method, Hessenberg-Schur method, and Schur and QR decompositions method, were proposed in [5, 6]. Based on the Kronecker product and the fixed point theorem in partially ordered sets, some sufficient conditions for the existence of a unique symmetric positive definite solution are given in [7, 8]. Ran and Reurings ([7, Theorem 3.3] and [9, Theorem 3.1]) pointed out that if is a positive definite matrix, then there exists a unique solution and it is symmetric positive definite. Recently, under the condition that (1) is consistent, Su and Chen presented an efficient numerical iterative method based on the conjugate gradient method (CGM) [10]. In addition, based on geometric structures on a Riemannian manifold, Duan et al. proposed a natural gradient descent algorithm to solve algebraic Lyapunov equations [11, 12]. Following them, we investigate the solution problem of (1) in the view of Riemannian manifolds. Note that this solution of (1) is a symmetric positive definite matrix and the set of all the symmetric positive definite matrices can be considered as a manifold. Thus, it is more convenient to investigate the solution problem with the help of these geometric structures on this manifold. To address such a need, a new framework is presented in this paper to calculate the numerical solution, which is based on the geometric structures on the