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

Original language | English |
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Pages (from-to) | 1150-1166 |

Number of pages | 17 |

Journal | Linear Algebra and Its Applications |

Volume | 439 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2013 |

Externally published | Yes |

### Keywords

- Eigenvalues
- Hℓ-Matrices
- LR Cholesky algorithm
- Semiseparable matrices
- Symmetric hierarchical matrices

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

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## Cite this

*Linear Algebra and Its Applications*,

*439*(4), 1150-1166. https://doi.org/10.1016/j.laa.2013.03.001