an:05872587
Zbl 1211.65026
Sidje, Roger B.; Saad, Yousef
Rational approximation to the Fermi-Dirac function with applications in density functional theory
EN
Numer. Algorithms 56, No. 3, 455-479 (2011).
00275760
2011
j
65D20 81Q05 65F05 33E20 33F05
diagonal of matrix inverse; electronic structure calculations; density functional theory; continued fraction; numerical examples; Fermi-Dirac matrix function; Hamiltonian matrix; rational Chebyshev approximation; sparse direct methods
Summary: 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.