Numerical algorithms for solving discrete Lyapunov tensor equation

Abstract

In this paper, we present several numerical algorithms to solve the discrete Lyapunov tensor equation, which is the generalized form of discrete Lyapunov matrix equation. Based on the structure of the tensor equation, we firstly propose a simple iterative (SI) method to consider the numerical solution of the discrete Lyapunov tensor equation. Then we extend the gradient based iterative (GBI) method and the residual norm steepest descent (RNSD) iterative method to study the tensor equation, respectively. Furthermore, we develop a residual norm conjugate gradient (RNCG) iterative method to accelerate the convergence speed of the RNSD method. Convergence analysis shows that the proposed methods converge to an exact solution for any initial value. Finally, some numerical examples are provided to illustrate the efficiency and validity of these methods proposed.