Numerical Algorithms

, Volume 56, Issue 3, pp 455–479 | Cite as

Rational approximation to the Fermi–Dirac function with applications in density functional theory

Original Paper


We are interested in computing the Fermi–Dirac matrix function in which the matrix argument is the Hamiltonian matrix arising from density functional theory (DFT) applications. More precisely, we are really interested in the diagonal of this matrix function. We discuss rational approximation methods to the problem, specifically the rational Chebyshev approximation and the continued fraction representation. These schemes are further decomposed into their partial fraction expansions, leading ultimately to computing the diagonal of the inverse of a shifted matrix over a series of shifts. We describe Lanczos and sparse direct methods to address these systems. Each approach has advantages and disadvantages that are illustrated with experiments.


Fermi–Dirac Diagonal of matrix inverse Electronic structure calculations Density functional theory Rational Chebyshev Continued fraction 


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© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Department of MathematicsThe University of AlabamaTuscaloosaUSA
  2. 2.Computer Science & EngineeringUniversity of MinnesotaMinneapolisUSA

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