Gradient based iterative algorithms for solving a class of matrix equations

TLDR

The paper applies a hierarchical identification principle to solve Sylvester and Lyapunov matrix equations. It introduces a gradient iterative algorithm that treats the unknown matrix as system parameters, minimizes a criterion function, extends to general linear equations, and is implemented on a computer. The algorithm converges to the true solution from any starting point, its rate improves with an appropriate step‑size, requires less storage than existing methods, and numerical tests confirm the theory.

Abstract

In this note, we apply a hierarchical identification principle to study solving the Sylvester and Lyapunov matrix equations. In our approach, we regard the unknown matrix to be solved as system parameters to be identified, and present a gradient iterative algorithm for solving the equations by minimizing certain criterion functions. We prove that the iterative solution consistently converges to the true solution for any initial value, and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factor (or step-size) appropriately. Furthermore, the iterative method is extended to solve general linear matrix equations. The algorithms proposed require less storage capacity than the existing numerical ones. Finally, the algorithms are tested on computer and the results verify the theoretical findings.