An Asynchronous Distributed Algorithm for Solving a Linear Algebraic Equation

A distributed algorithm is described for solving a linear algebraic equation of the form Ax = b where A is a matrix for which the equation has at least one solution. The equation is simultaneously and asynchronously solved by m agents assuming each agent knows only a subset of the rows of the partitioned matrix [A b], the estimates of the equation's solution generated by its neighbors, and nothing more. Each agent recursively updates its estimate of a solution at its own event times by utilizing estimates generated by each of its neighbors which are transmitted with delays. Each agent has its own event time sequence and the event time sequences of different agents are not assumed to be synchronized. Neighbor relations are characterized by a time-dependent directed graph whose vertices correspond to agents and whose arcs depict neighbor relations. It is shown that for any matrix A for which the equation has a solution and any repeatedly jointly strongly connected sequence of neighbor graphs defined on the merged sequence of all agents' event times, the algorithm causes all agents' estimates to converge exponentially fast to the same solution to Ax = b