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2016, 6(3): 319-328. doi: 10.3934/naco.2016014

## Decentralized gradient algorithm for solution of a linear equation

 1 College of Engineering and Computer Science, The Australian National University, Canberra, Australia 2 School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, United States 3 Department of Electrical Engineering, Yale University, New Haven, CT, United States 4 Institute for Mathematics, University of Würzburg, Emil-Fischer Straße 40, 97074 Würzburg

Received  July 2015 Revised  September 2016 Published  September 2016

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.
Citation: Brian D. O. Anderson, Shaoshuai Mou, A. Stephen Morse, Uwe Helmke. Decentralized gradient algorithm for solution of a linear equation. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 319-328. doi: 10.3934/naco.2016014
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