Hierarchical matrices (H-matrices) form a subset of the set of all matrices. They have been introduced by Hackbusch to enable matrix operations of almost linear complexity. “The underlying idea is to decompose these matrices hierarchically into subblocks and to approximate most of the off-diagonal blocks by matrices of low rank.” The concept is especially suitable for the discretization of integral operators or the inverse of elliptic partial differential equations. Efficient implementations of matrix-vector multiplication and matrix addition, multiplication, and inversion have been presented previously [1,2].

This paper studies “the solution of block triangular linear equations in H-arithmetic, the Cholesky factorization of symmetric positive definite H-matrices, and the LDL^T factorization” in H-arithmetic. Especially, the LDL^T factorization of H-matrices is used for computing eigenvalues and eigenvectors for the H-representation of the discrete two-dimensional (2D) Laplacian. The extended H-arithmetic is applied to a 2D wave equation and a 2D heat equation.

The paper is especially interesting for numerical analysts in the area of numerical linear algebra. Although the paper is self-contained, it is suitable to know about H-arithmetic as presented by Hackbusch [1] and Hackbusch and Khoromskij [2].