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Hierarchical matrices (usually abbreviated ${\mathcal H}$-matrices) arefrequently 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 ofequations 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 toavoid computing the inverse and instead use an efficient representation of the$LU$ factorization based on low-rank updates performed with the well-knownSherman-Morrison-Woodbury equation.

An exact solver for simple ${\mathcal H}$-matrix systems