An Algebraic Approach for H-matrix Preconditioners∗

Hierarchical matrices (H-matrices) approximate matrices in a data-sparse way, and the approximate arithmetic for H-matrices is almost optimal. In this paper we present an algebraic approach to constructing H-matrices which combines multilevel clustering methods with the H-matrix arithmetic to compute the H-inverse, H-LU, and the H-Cholesky factors of a matrix. Then the H-inverse, H-LU or H-Cholesky factors can be used as preconditioners in iterative methods to solve systems of linear equations. The numerical results show that this method is efficient and speeds up convergence greatly against other approaches such as JOR or AMG to solving some large, sparse linear systems, and it is comparable to other H-matrix constructions based on Nested Dissection.