A sparse matrix arithmetic based on \({\mathfrak H}\)-matrices. I: Introduction to \({\mathfrak H}\)-matrices. (English) Zbl 0927.65063

Hackbusch, W.

Computing 62, No. 2, 89-108 (1999).

The concept of hierarchical matrices \(\mathfrak H \) is introduced. This paper is the first of a series that will be devoted to the \(\mathfrak H \)-matrices. These matrices are not sparse in the sense that there are only few non-zero entries, but they are data-sparse – the matrices are described by only few data. The other properties of such matrices are that the matrix-vector multiplication is of almost linear complexity, and sums and products of the matrices are not in the same set, but their truncation to the \(\mathfrak H\)-matrix format are of almost linear complexity. The same statement holds for the inverse of an \(\mathfrak H\)-matrix. Two new concepts are introduced. These allow the exact inversion of tridiagonal matrices and the approximation of some discrete integral operators.

Reviewer: R.P.Tewarson (Stony Brook)

MSC:

65F30	Other matrix algorithms (MSC2010)
15B57	Hermitian, skew-Hermitian, and related matrices
65F05	Direct numerical methods for linear systems and matrix inversion
65F50	Computational methods for sparse matrices

Keywords:

hierarchical matrices; hierarchical block partitioning; sparse matrices; matrix inversion; matrix-vector multiplication; tridiagonal matrices; discrete integral operators