MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

${\mathcal H}^2$-matrix arithmetics in linear complexity

Steffen Börm


For hierarchical matrices, approximations of the matrix-matrix sum and product can be computed in almost linear complexity, and using these matrix operations it is possible to construct the matrix inverse, efficient preconditioners or solutions of certain matrix equations.

${\mathcal H}^2$-matrices are a variant of hierarchical matrices that allow us to perform certain operations, like the matrix-vector product, in "true" linear complexity, but until now it was not clear whether matrix arithmetic operations could also reach this, in some sense optimal, complexity.

We present algorithms that compute the best-approximation of the sum and product of two ${\mathcal H}^2$-matrices in a prescribed ${\mathcal H}^2$-matrix format, and we prove that this computation can be accomplished in linear complexity.

MSC Codes:
hierarchical matrices, formatted matrix operations

Related publications

2006 Repository Open Access
Steffen Börm

\(\mathscr {H}^2\)-matrix arithmetics in linear complexity

In: Computing, 77 (2006) 1, pp. 1-28