Journal of Applied Mathematics

Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations

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 $Q=X+{\sum }_{i=1}^{m}{A}_{i}^{T}X{A}_{i}$. 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.

Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 507175, 7 pages.

Dates
First available in Project Euclid: 26 March 2014

https://projecteuclid.org/euclid.jam/1395855275

Digital Object Identifier
doi:10.1155/2014/507175

Mathematical Reviews number (MathSciNet)
MR3166768

Zentralblatt MATH identifier
07010658

Citation

Duan, Xiaomin; Sun, Huafei; Zhao, Xinyu. Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations. J. Appl. Math. 2014 (2014), Article ID 507175, 7 pages. doi:10.1155/2014/507175. https://projecteuclid.org/euclid.jam/1395855275

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