Decentralized gradient algorithm for solution of a linear equation

Abstract

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 xi(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 xi with the property that each xi converges to the solution of the linear equation exponentially fast. The equation for xi includes additive terms weighting those xj 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.