Decentralized gradient algorithm  for   solution of a linear equation

Brian  D. O.  Anderson; Shaoshuai  Mou; A.  Stephen  Morse; Uwe  Helmke

doi:10.3934/naco.2016014

Article Contents

2016, Volume 6, Issue 3: 319-328. Doi: 10.3934/naco.2016014

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Decentralized gradient algorithm for solution of a linear equation

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

Received: June 30, 2015

Revised: August 31, 2016

Published: August 2016

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Abstract

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.

Keywords:

Autonomous systems; distributed algorithms; linear equations.

Mathematics Subject Classification: Primary: 68Q85 ; Secondary: 11D04.

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