On the equivalence of the Hermitian eigenvalue problem and hypergraph edge elimination Article Swipe
Karsten Kahl
,
Bruno Lang
·
YOU?
·
· 2020
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2003.03145
YOU?
·
· 2020
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2003.03145
It is customary to identify sparse matrices with the corresponding adjacency or incidence graph. For the solution of linear systems of equations using Gaussian elimination, the representation by its adjacency graph allows a symbolic computation that can be used to predict memory footprints and enables the determination of near-optimal elimination orderings based on heuristics. The Hermitian eigenvalue problem on the other hand seems to evade such treatment at first glance due to its inherent iterative nature. In this paper we prove this assertion wrong by showing the equivalence of the Hermitian eigenvalue problem with a symbolic edge elimination procedure. A symbolic calculation based on the incidence graph of the matrix can be used in analogy to the symbolic phase of Gaussian elimination to develop heuristics which reduce memory footprint and computations. Yet, we also show that the question of an optimal elimination strategy remains NP-hard, in analogy to the linear systems case.
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- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2003.03145
- https://arxiv.org/pdf/2003.03145
- OA Status
- green
- References
- 11
- Related Works
- 10
- OpenAlex ID
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https://openalex.org/W3009824939Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2003.03145Digital Object Identifier
- Title
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On the equivalence of the Hermitian eigenvalue problem and hypergraph edge eliminationWork title
- Type
-
preprintOpenAlex work type
- Language
-
enPrimary language
- Publication year
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2020Year of publication
- Publication date
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2020-03-06Full publication date if available
- Authors
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Karsten Kahl, Bruno LangList of authors in order
- Landing page
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https://arxiv.org/abs/2003.03145Publisher landing page
- PDF URL
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https://arxiv.org/pdf/2003.03145Direct link to full text PDF
- Open access
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YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
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https://arxiv.org/pdf/2003.03145Direct OA link when available
- Concepts
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Eigenvalues and eigenvectors, Gaussian elimination, Heuristics, Adjacency matrix, Computation, Mathematics, Hypergraph, Hermitian matrix, Gaussian, Applied mathematics, Graph, Computer science, Mathematical optimization, Algorithm, Discrete mathematics, Pure mathematics, Quantum mechanics, PhysicsTop concepts (fields/topics) attached by OpenAlex
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0Total citation count in OpenAlex
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11Number of works referenced by this work
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10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.identify | 4 |
| abstract_inverted_index.inherent | 73 |
| abstract_inverted_index.matrices | 6 |
| abstract_inverted_index.question | 137 |
| abstract_inverted_index.solution | 16 |
| abstract_inverted_index.strategy | 142 |
| abstract_inverted_index.symbolic | 33, 95, 100, 117 |
| abstract_inverted_index.Hermitian | 55, 90 |
| abstract_inverted_index.adjacency | 10, 29 |
| abstract_inverted_index.assertion | 82 |
| abstract_inverted_index.customary | 2 |
| abstract_inverted_index.equations | 21 |
| abstract_inverted_index.footprint | 128 |
| abstract_inverted_index.incidence | 12, 105 |
| abstract_inverted_index.iterative | 74 |
| abstract_inverted_index.orderings | 50 |
| abstract_inverted_index.treatment | 66 |
| abstract_inverted_index.eigenvalue | 56, 91 |
| abstract_inverted_index.footprints | 42 |
| abstract_inverted_index.heuristics | 124 |
| abstract_inverted_index.procedure. | 98 |
| abstract_inverted_index.calculation | 101 |
| abstract_inverted_index.computation | 34 |
| abstract_inverted_index.elimination | 49, 97, 121, 141 |
| abstract_inverted_index.equivalence | 87 |
| abstract_inverted_index.heuristics. | 53 |
| abstract_inverted_index.elimination, | 24 |
| abstract_inverted_index.near-optimal | 48 |
| abstract_inverted_index.computations. | 130 |
| abstract_inverted_index.corresponding | 9 |
| abstract_inverted_index.determination | 46 |
| abstract_inverted_index.representation | 26 |
| cited_by_percentile_year | |
| countries_distinct_count | 1 |
| institutions_distinct_count | 2 |
| citation_normalized_percentile |