Distributed solution to linear equations from arbitrary initializations
Peng Wang, Wei Ren, Zhisheng Duan · 2017
A discrete-time distributed algorithm to solve a system of linear equations Ax = b is proposed with the M-Fejer mappings in this paper. The algorithm can find a solution of Ax = b from arbitrary initializations at a geometric rate when Ax = b has either unique or multiple solutions. The geometric convergence rate of the algorithm is first proved by analyzing the mixed norm of homogeneous M-Fejer mappings when Ax = b has a unique solution. Then when Ax = b has multiple solutions, the algorithm is proved to converge at a geometric rate through orthogonal decompositions of the agents' estimates onto the row space and null space of A, and the relationship between the initializations and the final convergence point is also specified.