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

Xiaomin Duan, Huafei Sun, Linyu Peng, Xinyu Zhao

Research output: Contribution to journalArticle

12 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)9899-9905
Number of pages7
JournalApplied Mathematics and Computation
Issue number19
Publication statusPublished - 2013
Externally publishedYes



  • Discrete Lyapunov equation
  • Geodesic distance
  • Natural gradient
  • Riemannian manifold
  • Riemannian metric

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics

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