-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 preconditioners based on approximate factorizations or solutions of certain matrix equations.-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 -matrices in a prescribed -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.

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