Localization in Matrix Computations: Theory and Applications

  • Michele BenziEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2173)


Many important problems in mathematics and physics lead to (non-sparse) functions, vectors, or matrices in which the fraction of nonnegligible entries is vanishingly small compared the total number of entries as the size of the system tends to infinity. In other words, the nonnegligible entries tend to be localized, or concentrated, around a small region within the computational domain, with rapid decay away from this region (uniformly as the system size grows). When present, localization opens up the possibility of developing fast approximation algorithms, the complexity of which scales linearly in the size of the problem. While localization already plays an important role in various areas of quantum physics and chemistry, it has received until recently relatively little attention by researchers in numerical linear algebra. In this chapter we survey localization phenomena arising in various fields, and we provide unified theoretical explanations for such phenomena using general results on the decay behavior of matrix functions. We also discuss computational implications for a range of applications.


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I would like to express my sincere gratitude to several friends and collaborators without whose contributions these lecture notes would not have been written, namely, Paola Boito, Matt Challacombe, Nader Razouk, Valeria Simoncini, and the late Gene Golub. I am also grateful for financial support to the Fondazione CIME and to the US National Science Foundation (grants DMS-0810862, DMS-1115692 and DMS-1418889).


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© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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