Distributed Algorithms for Linear Equations Over General Directed Networks

This article deals with linear equations of the form $Ax = b$ . By reformulating the original problem as an unconstrained optimization problem, we first provide a gradient-based distributed continuous-time algorithm over weight-balanced directed graphs, in which each agent only knows partial rows of the augmented matrix $(A\; b)$ . The algorithm is also applicable to time-varying networks. By estimating a right-eigenvector corresponding to 0 eigenvalue of the out-Laplacian matrix in finite time, we further propose a distributed algorithm over weight-unbalanced communication networks. It is proved that each solution of the designed algorithms converges exponentially to an equilibrium point. Moreover, the convergence rate is given out clearly. For linear equations without solution, these algorithms are used to obtain a least-squares solution in approximate sense. These theoretical results are illustrated by four numerical examples.