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Coordination of Groups of Mobile Autonomous Agents Using Nearest Neighbor Rules
, 2002
"... In a recent Physical Review Letters paper, Vicsek et. al. propose a simple but compelling discretetime model of n autonomous agents fi.e., points or particlesg all moving in the plane with the same speed but with dierent headings. Each agent's heading is updated using a local rule based on ..."
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In a recent Physical Review Letters paper, Vicsek et. al. propose a simple but compelling discretetime model of n autonomous agents fi.e., points or particlesg all moving in the plane with the same speed but with dierent headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its \neighbors." In their paper, Vicsek et. al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models.
Inexact Inverse Iteration for Symmetric Matrices
"... In this paper we analyse inexact inverse iteration for the real symmetric eigenvalue problem Av = v. Our analysis is designed to apply to the case when A is large and sparse and where iterative methods are used to solve the shifted linear systems (A I)y = x which arise. We present a convergence th ..."
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In this paper we analyse inexact inverse iteration for the real symmetric eigenvalue problem Av = v. Our analysis is designed to apply to the case when A is large and sparse and where iterative methods are used to solve the shifted linear systems (A I)y = x which arise. We present a convergence theory that is independent of the nature of the inexact solver used, and, though the use of the Rayleigh quotient is emphasised, our analysis also extends to quite general choices for shift and inexact solver strategies. Additionally, the convergence framework allows us to treat both standard preconditioning and to present a new analysis of the variation introduced by Simoncini and Elden (BIT, Vol 42, pp.159182, 2002). Also, we provide an analysis of the performance of inner iteration solves when preconditioned MINRES is used as the inexact solver. This analysis provides descriptive bounds which are shown to predict well the actual behaviour observed in practice. Also, it explains the improvement in performance of the modication introduced by Simoncini and Elden over the standard preconditioned form. Importantly, our analysis shows that letting the shift tend to the eigenvalue, as is the case if the Rayleigh quotient is used, does not harm signicantly the performance of the iterative method for the shifted systems. Throughout the paper numerical results are given to illustrate the theory.