## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 455-479 |

Number of pages | 25 |

Journal | Numerical Algorithms |

Volume | 56 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2011 |

### Bibliographical note

Funding Information:Work supported by NSF under grant 0325218, by DOE under grant DE-FG02-03ER25585, and by the Minnesota Supercomputing Institute.

## Keywords

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