## Communications in Mathematical Sciences

### Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems

#### 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.

#### Article information

Source
Commun. Math. Sci., Volume 7, Number 3 (2009), 755-777.

Dates
First available in Project Euclid: 26 October 2009

https://projecteuclid.org/euclid.cms/1256562822

Mathematical Reviews number (MathSciNet)
MR2569032

Zentralblatt MATH identifier
1182.65072

Subjects
Primary: 65F30: Other matrix algorithms
Secondary: 65Z05: Applications to physics

#### Citation

Lin, Lin; Lu, Jianfeng; Ying, Lexing; Car, Roberto; E, Weinan. Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems. Commun. Math. Sci. 7 (2009), no. 3, 755--777. https://projecteuclid.org/euclid.cms/1256562822