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 |
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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
The LR Cholesky algorithm for symmetric hierarchical matrices. / Benner, Peter; Mach, Thomas.
In: Linear Algebra and Its Applications, Vol. 439, No. 4, 2013, p. 1150-1166.Research output: Contribution to journal › Article
}
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 -