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 $x_i(t)$, and a graphical structure associating with each agent a vertex of a fixed, undirected and connected but otherwise arbitrary graph $\mathcal G$ with vertex set and edge set $\mathcal V$ and $\mathcal E$ 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 $\mathcal G$ 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.