Distributed Algorithm for Solving Linear Algebraic Equations: An Implicit Gradient Neural Network Approach

Abstract

This brief investigates the online solving problem for linear algebraic equation $Ax=b$ by means of the principle of consensus in multi-agent systems, where $Ainmathbb{R}^{mtimes n}$ and $binmathbb{R}^{n}$. To be specific, we choose $m$ autonomous agents and agent $i$ knows only the $i-$th row of $[A~b]$ under a fixed and connected undirected communication topology. Under local interactions, by designing an implicit gradient neural network based algorithm, it is shown that all the agents' states which starting from any different initial conditions can converge exponentially fast to one of the solutions to $Ax=b$, if the matrix $A$ has full row rank. It is worth noting that the proposed algorithm is fully distributed. In addition, it is shown that the proposed algorithm is effective in obtaining least square solutions for no-solution cases. Finally, computer simulations verify and demonstrate the efficiency of the proposed methods for solving linear algebraic equations.