Survey on the Technique of Hierarchical Matrices

Usually, one avoids numerical algorithms involving operations with large, fully populated matrices. Instead, one tries to reduce all algorithms to matrix-vector multiplications involving only sparse matrices. The reason is the large number of floating point operations; e.g., $\mathcal {O}(n^{3})$ for the multiplication of two general n × n matrices. The hierarchical matrix ( $\mathcal {H}$ -matrix) technique provides tools to perform the matrix operations approximately in almost linear work $\mathcal {O}(n\log ^{\ast }n)$ . The approximation errors are nevertheless acceptable, since large-scale matrices are usually obtained from discretisations which anyway contain a discretisation error. Adjusting the approximation error to the discretisation error yields the factor $\mathcal {O}(\log ^{\ast }n).$ The operations enabled by the $\mathcal {H}$ -matrix technique are not only the matrix addition and multiplication but also the matrix inversion and the LU or Cholesky decomposition. The positive statements from above do not hold for all matrices, but they are valid for the important class of matrices originating from standard discretisations of elliptic partial differential equations or the related integral equations. An important aspect is the fact that the algorithms can be applied in a black-box fashion. Having all matrix operations available, a much larger class of problems can be treated than by the restriction to matrix-vector multiplications. The LU decomposition can be used to construct fast iterations for solving linear systems. Also eigenvalue problems can be treated. The computation of matrix-valued functions is possible (e.g., the matrix exponential function) as well as the solution of matrix equations (e.g., of the Riccati equation).