Hierarchical Matrices and H-Matrix software

Hierarchical Matrices (or short H-Matrices) are matrices in some special, very efficient data-sparse, block structured format. The term "hierarchical" indicates that the block structure of the matrices has been build up hierarchically. Many types of sparse and dense matrices arising in practical applications (such as discretized integral equations or elliptic boundary value problems) can be approximated very well by H-Matrices.

The basic idea of approximating a given matrix by an H-Matrix is to split the given matrix into a hierarchy of rectangular blocks and approximate each of these blocks by a low-rank matrix. Based on this structure, efficient algorithms with matrix arithmetics, inversion and preconditioning for solving the linear systems and even matrix equations can be applied that work in almost linear complexity.

Summarizing, the techniques of H-matrices can be applied to those linear systems of equations whose matrices can be efficiently approximated by H-matrices. In particular, efficient solvers are available for corresponding applications. The software library HLIBpro provides all necessary routines for the construction of hierarchical matrix structures, arithmetic algorithms that perform certain approximative matrix operations (such as addition, multiplication, factorization and inversion), as well as conversion routines that turn sparse and dense matrices into H-format.