Decentralized gradient algorithm for solution of a linear equation

The paper develops a technique for solving a linear equation Ax = b with a square and nonsingular matrix A, using a decentralized gradient algorithm. In the language of control theory, there are n agents, each storing at time t an n-vector, call it x_i(t), and a graphical structure associating with each agent a vertex of a fixed, undirected and connected but otherwise arbitrary graph G with vertex set and edge set V and E respectively. We provide differential equation update laws for the x_i with the property that each x_i converges to the solution of the linear equation exponentially fast. The equation for x_i includes additive terms weighting those x_j for which vertices in G corresponding to the i-th and j-th agents are adjacent. The results are extended to the case where A is not square but has full row rank, and bounds are given on the convergence rate.