The LR Cholesky algorithm for symmetric hierarchical matrices / Peter Benner, Thomas Mach

Abstract

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 data-sparsity of hierarchical matrices is not well preserved. 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 ℋℓ -matrices is almost preserved and so the LR Cholesky algorithm is of almost quadratic complexity for ℋℓ-matrices.