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Journal of Applied Mathematics
Volume 2014, Article ID 507175, 7 pages
http://dx.doi.org/10.1155/2014/507175
Research Article

Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations

1School of Mathematics, Beijing Institute of Technology, Beijing 100081, China
2School of Science, Dalian Jiaotong University, Dalian 116028, China
3School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China
4School of Materials Science and Engineering, Dalian Jiaotong University, Dalian 116028, China

Received 6 August 2013; Revised 10 December 2013; Accepted 10 December 2013; Published 6 January 2014

Academic Editor: Zhi-Hong Guan

Copyright © 2014 Xiaomin Duan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.