A Distributed Algorithm for Solving a Linear Algebraic Equation

A distributed algorithm is described for solving a linear algebraic equation of the form Ax = b assuming the equation has at least one solution. The equation is simultaneously solved by m agents assuming each agent knows only a subset of the rows of the partitioned matrix [A b], the current estimates of the equation´s solution generated by its neighbors, and nothing more. Each agent recursively updates its estimate by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-dependent directed graph N(t) 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 sequence of “repeatedly jointly strongly connected graphs” N(t), t = 1, 2, ..., the algorithm causes all agents´ estimates to converge exponentially fast to the same solution to Ax = b. It is also shown that, under mild assumptions, the neighbor graph sequence must actually be repeatedly jointly strongly connected if exponential convergence is to be assured. A worst case convergence rate bound is derived for the case when Ax = b has a unique solution. It is demonstrated that with minor modification, the algorithm can track the solution to Ax = b, even if A and b are changing with time, provided the rates of change of A and bare sufficiently small. It is also shown that in the absence of communication delays, exponential convergence to a solution occurs even if the times at which each agent updates its estimates are not synchronized with the update times of its neighbors. A modification of the algorithm is outlined which enables it to obtain a least squares solution to Ax = b in a distributed manner, even if Ax = b does not have a solution.