A distributed algorithm for solving general linear equations over networks

In this paper, we develop a continuous-time distributed algorithm that allows nodes in an undirected, connected network to cooperatively solve a general system of linear equations, where the only assumption is that each equation is known to at least one node. We show that the algorithm enables the nodes to asymptotically agree on a solution when there are infinitely many solutions, determine the solution when there is exactly one, and discover that no solution exists when there are none. In addition, we prove that the algorithm is globally exponentially convergent, derive an explicit lower bound on its convergence rate, and show that under certain conditions, the larger the network's algebraic connectivity, or the further away from being singular the system of equations, the larger this lower bound.