# Preprint No. MPIMD/12-05

#### Author(s): Peter Benner, Thomas Mach

#### Email: thomas.mach@googlemail.com

#### Date: 2012-02-28

#### 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 ℋ_{ℓ }-matrices is almost preserved and so the LR Cholesky algorithm is of almost quadratic complexity for ℋ_{ℓ}-matrices.

#### BibTeX:

@TECHREPORT{MPIMD12-05,

author = {Peter Benner and Thomas Mach},

title = {The LR Cholesky Algorithm for Symmetric Hierarchical Matrices},

number = {MPIMD/12-05},

month = feb,

year = 2012,

institution = {Max Planck Institute Magdeburg},

type = {Preprint},

note = {Available from \url{http://www.mpi-magdeburg.mpg.de/preprints/}},

}