### 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 |
---|---|

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 |

### Fingerprint

### 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

### Cite this

*Linear Algebra and Its Applications*,

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

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

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 439, no. 4, pp. 1150-1166. https://doi.org/10.1016/j.laa.2013.03.001

}

TY - JOUR

T1 - The LR Cholesky algorithm for symmetric hierarchical matrices

AU - Benner, Peter

AU - Mach, Thomas

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Eigenvalues

KW - Hℓ-Matrices

KW - LR Cholesky algorithm

KW - Semiseparable matrices

KW - Symmetric hierarchical matrices

UR - http://www.scopus.com/inward/record.url?scp=84879881157&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84879881157&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2013.03.001

DO - 10.1016/j.laa.2013.03.001

M3 - Article

AN - SCOPUS:84879881157

VL - 439

SP - 1150

EP - 1166

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 4

ER -