  Hierarchical matrices (usually abbreviated ${\mathcal H}$-matrices) are
frequently used to construct preconditioners for systems of linear equations.
Since it is possible to compute approximate inverses or $LU$ factorizations in
${\mathcal H}$-matrix representation using only ${\mathcal O}(n \log^2 n)$
operations, these preconditioners can be very efficient.
  Here we consider an algorithm that allows us to solve a linear system of
equations given in a simple ${\mathcal H}$-matrix format \emph{exactly} using
${\mathcal O}(n \log^2 n)$ operations. The central idea of our approach is to
avoid computing the inverse and instead use an efficient representation of the
$LU$ factorization based on low-rank updates performed with the well-known
Sherman-Morrison-Woodbury equation.
