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On large‐scale diagonalization techniques for the Anderson model of localization
Author(s) -
Schenk Olaf,
Bollhöfer Matthias,
Römer Rudolf A.
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700768
Subject(s) - eigenvalues and eigenvectors , computation , complement (music) , algebraic number , mathematics , scale (ratio) , anderson impurity model , algebra over a field , pure mathematics , algorithm , mathematical analysis , physics , quantum mechanics , biochemistry , chemistry , complementation , gene , phenotype , electron
We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large‐scale sparse real and symmetric indefinite matrices of the Anderson model of localization. Our preconditioning approaches for the shift‐and‐invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multi‐level incomplete LDL T factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill‐conditioned symmetric indefinite Anderson matrices. Our numerical examples reveal that recent algebraic multi‐level preconditioning solvers can accelerate the computation of a large‐scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)