Distributed Algorithms for Solving a Least-Squares Solution of Linear Algebraic Equations

This article proposes three distributed algorithms for solving linear algebraic equations to seek a least-squares (LS) solution via multiagent networks. We consider that each agent has only access to a small and incomplete block of linear equations rather than the complete row or column in the existing results. First, we focus on the case of a homogeneous partition of linear equations. A distributed algorithm is proposed via a single-layered grid network, in which each agent only needs to control three scalar states. Second, we consider the case of heterogeneous partitions of linear equations. Two distributed algorithms with a doubled-layered network are developed, which allow each agent's states to have different dimensions and can be applied to heterogeneous agents with different storage and computation capabilities. Rigorous proofs show that the proposed distributed algorithms collaboratively obtain an LS solution with exponential convergence and also own a solvability verification property, i.e., a criterion to verify whether the obtained solution is an exact solution. Finally, some simulation examples are provided to demonstrate the effectiveness of the proposed algorithms.