Distributed Algorithms for Solving a Linear Equation Under a Directed Graph

In this paper, we explore to solve the linear equation problem $A\theta= b$ under a directed graph. A “consen-sus+innovation” type distributed algorithm is proposed. In order to promote the convergence speed, we make the weight of consensus and innovation be different. By using the proposed algorithm, we can solve any linear equation with at least one solution at a geometric rate from arbitrary initializations. We also propose a new “consensus+residual” algorithm which replace the innovation by the residual and can speed up the iterations.