### Abstract

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 language | English |
---|---|

Pages (from-to) | 9899-9905 |

Number of pages | 7 |

Journal | Applied Mathematics and Computation |

Volume | 219 |

Issue number | 19 |

DOIs | |

Publication status | Published - 2013 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics

### Cite this

*Applied Mathematics and Computation*,

*219*(19), 9899-9905. https://doi.org/10.1016/j.amc.2013.03.119

**A natural gradient descent algorithm for the solution of discrete algebraic Lyapunov equations based on the geodesic distance.** / Duan, Xiaomin; Sun, Huafei; Peng, Linyu; Zhao, Xinyu.

Research output: Contribution to journal › Article

*Applied Mathematics and Computation*, vol. 219, no. 19, pp. 9899-9905. https://doi.org/10.1016/j.amc.2013.03.119

}

TY - JOUR

T1 - A natural gradient descent algorithm for the solution of discrete algebraic Lyapunov equations based on the geodesic distance

AU - Duan, Xiaomin

AU - Sun, Huafei

AU - Peng, Linyu

AU - Zhao, Xinyu

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Discrete Lyapunov equation

KW - Geodesic distance

KW - Natural gradient

KW - Riemannian manifold

KW - Riemannian metric

UR - http://www.scopus.com/inward/record.url?scp=84877345914&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84877345914&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2013.03.119

DO - 10.1016/j.amc.2013.03.119

M3 - Article

AN - SCOPUS:84877345914

VL - 219

SP - 9899

EP - 9905

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

IS - 19

ER -