The LR Cholesky algorithm for symmetric hierarchical matrices

Abstract

We investigate the application of the LR Cholesky algorithm to symmetric hierarchical matrices, symmetric simple structured hierarchical matrices and symmetric hierarchically semiseparable (HSS) matrices. The data-sparsity of these matrices make the otherwise expensive LR Cholesky algorithm applicable, as long as the data-sparsity is preserved. We will see in an example that the ranks of the low rank blocks grow and the data-sparsity gets lost.

We will explain this behavior by applying a theorem on the structure preservation of diagonal plus semiseparable matrices under LR Cholesky transformations. Therefore we have to give a new more constructive proof for the theorem. We will show that the structure of ${\mathcal{H}}_{\ell }$-matrices is almost preserved and so the LR Cholesky algorithm is of almost quadratic complexity for ${\mathcal{H}}_{\ell }$-matrices.