Numerical Algorithms

, Volume 56, Issue 3, pp 455–479

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

Original Paper

DOI: 10.1007/s11075-010-9397-6

Cite this article as:
Sidje, R.B. & Saad, Y. Numer Algor (2011) 56: 455. doi:10.1007/s11075-010-9397-6


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–DiracDiagonal of matrix inverseElectronic structure calculationsDensity functional theoryRational ChebyshevContinued fraction

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

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