Gradient descent-based parameter-free methods for solving coupled matrix equations and studying an application in dynamical systems

Abstract

This paper explores advanced gradient descent-based parameter-free methods for solving coupled matrix equations and examines their applications in dynamical systems. We focus on the coupled matrix equations$\{\begin{array}{c}AX+YB=C,\\ DX+YE=F,\end{array}$ where $A,D\in R^{m\times m},B,E\in R^{n\times n},C,F\in R^{m\times n}$ are given matrices, and $X,Y\in R^{m\times n}$ are the unknown matrices to be determined. We propose a novel gradient descent-based approach with parameter-free, enhancing convergence through an accelerated technique related to momentum methods. A comprehensive analysis of the convergence and characteristics of these methods is provided. Our convergence analysis demonstrates that if the spectrum of a block matrix constructed from the matrices A, B, D, and E is confined within a horizontal ellipse in the complex plane, centered at $(0,0)$ with a major axis length of 3 and a minor axis length of 1, then the accelerated momentum method will converge to the exact solution of the discussed model. The numerical results indicate that proposed methods significantly improve efficiency, showing faster convergence and reduced computational time compared to traditional approaches. Additionally, we apply these methods to linear dynamic systems, demonstrating their effectiveness in real-world scenarios.