A distributed algorithm for efficiently solving linear equations and its applications (Special Issue JCW)

Abstract A distributed algorithm is proposed for solving a linear algebraic equation A x = b over a multi-agent network, where A ∈ R n × n and the equation has a unique solution x ∗ ∈ R n . Each agent knows only a subset of the rows of [ A b ] , controls a state vector x i ( t ) of size smaller than n and is able to receive information from its nearby neighbors. Neighbor relations are characterized by time-dependent directed graphs. It is shown that for a large class of time-varying networks, the proposed algorithm enables each agent to recursively update its own state by only using its neighbors’ states such that all x i ( t ) converge exponentially fast to a specific part of x ∗ of interest to agent i . Applications of the proposed algorithm include solving the least square solution problem and the network localization problem.