### Abstract

We propose an algorithm for extracting the diagonal of the inverse matrices arising from electronic structure calculation. The proposed algorithm uses a hierarchical decomposition of the computational domain. It first constructs hierarchical Schur complements of the interior points for the blocks of the domain in a bottom-up pass and then extracts the diagonal entries efficiently in a top-down pass by exploiting the hierarchical local dependence of the inverse matrices. The overall cost of our algorithm is O(N^{3/2}) for a two dimensional problem with N degrees of freedom. Numerical results in electronic structure calculation illustrate the efficiency and accuracy of the proposed algorithm.

Original language | English (US) |
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Pages (from-to) | 755-777 |

Number of pages | 23 |

Journal | Communications in Mathematical Sciences |

Volume | 7 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2009 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Keywords

- Diagonal extraction
- Electronic structure calculation
- Hierarchical Schur complement