Fast algorithms for hierarchically semiseparable matrices

Jianlin Xia, Shivkumar Chandrasekaran, Ming Gu, Xiaoye S. Li
2010 Numerical Linear Algebra with Applications  
Semiseparable matrices and many other rank structured matrices have been widely used in developing new fast matrix algorithms. In this paper, we generalize the hierarchically semiseparable (HSS) matrix representations and propose some fast algorithms for HSS matrices. We define HSS matrices in terms of general binary HSS trees and use simplified postordering notation for HSS matrices. Fast HSS algorithms including new HSS structure generation, HSS form Cholesky factorization, and model
more » ... on are developed. Moreover, we provide a new linear complexity explicit U LV factorization algorithm for symmetric positive definite HSS matrices with a low-rank property. The corresponding factors can be used to solve the HSS systems also in linear complexity. Numerical examples demonstrate the efficiency of the solver. All the algorithms also have nice data locality. These algorithms are useful in developing fast structured numerical methods for large discretized PDEs (such as elliptic equations), integral equations, eigenvalue problems, etc.
doi:10.1002/nla.691 fatcat:neof3jzipfdhxevk53ha5jwb2u