A Distributed Algorithm for Solving Time-Varying Linear Equations based on the Newton Methods

This study introduces a distributed algorithm based on the Newton methods, which is designed to collaboratively solve time-varying linear equations with a unique solution through the cooperation of multiple agents. Each agent possesses only a portion of the global equation's information, specifically, a subset of the rows of the augmented matrix. Through communication among agents, the algorithm converges towards the global solution. When the derivative of the global solution of the time-varying equations is bounded, the algorithm ensures that the local solutions converge within a certain range of the global solution, thereby achieving tracking. The algorithm is also capable of rapidly solving time-invariant linear equations, quickly reaching consensus and determining the global solution of the equations. Efficacy and efficiency of the algorithm have been substantiated through a simulation experiment.