Vincent E. Sacksteder
Published 2003-12-01, updated 2004-04-30Version 2
The past thirteen years have seen the development of many algorithms for approximating matrix functions in O(N) time, where N is the basis size. These O(N) algorithms rely on assumptions about the spatial locality of the matrix function; therefore their validity depends very much on the argument of the matrix function. In this article I carefully examine the validity of certain O(N) algorithms when applied to hamiltonians of disordered systems. I focus on the prototypical disordered system, the Anderson model. I find that O(N) algorithms for the density matrix function can be used well below the Anderson transition (i.e. in the metallic phase;) they fail only when the coherence length becomes large. This paper also includes some experimental results about the Anderson model's behavior across a range of disorders.
Comments: 12 pages, 5 figures. All code and configuration files necessary to reproduce the results will be made available at http://www.sacksteder.com . Accepted by Numerical Linear Algebra with Applications. Changes in version two include correction of a substantial error in the error estimates section, changes of both the error estimates section and the numerical results section to give for the first time a good way of estimating the errors incurred by an O(N) algorithm, and other less important refinements
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