H 2-matrix arithmetics in linear complexity

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 precondi-tioners based on approximate factorizations or solutions of certain matrix equations. H 2-matrices are a variant of hierarchical matrices which 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 H 2-matrices in a prescribed H 2-matrix format, and we prove that these computations can be accomplished in linear complexity. Numerical experiments demonstrate that the new algorithms are more efficient than the well-known methods for hierarchical matrices.