Hypergraph edge elimination - A symbolic phase for Hermitian eigensolvers based on rank-1 modifications Article Swipe
Karsten Kahl
,
Bruno Lang
·
YOU?
·
· 2020
· Open Access
·
· DOI: https://doi.org/10.1553/etna_vol54s51
YOU?
·
· 2020
· Open Access
·
· DOI: https://doi.org/10.1553/etna_vol54s51
It is customary to identify sparse matrices with the corresponding adjacency or incidence graphs. For the solution of a linear system of equations using Gaussian elimination, the representation by its adjacency graph allows a symbolic factorization 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 revealing a tight connection of Hermitian eigensolvers based on rank-1 modifications 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-complete, in analogy to the linear systems case.
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- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1553/etna_vol54s51
- https://epub.oeaw.ac.at/0xc1aa5576_0x003bfef6.pdf
- OA Status
- bronze
- References
- 18
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W3099194693
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- OpenAlex ID
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https://openalex.org/W3099194693Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.1553/etna_vol54s51Digital Object Identifier
- Title
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Hypergraph edge elimination - A symbolic phase for Hermitian eigensolvers based on rank-1 modificationsWork title
- Type
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articleOpenAlex work type
- Language
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enPrimary language
- Publication year
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2020Year of publication
- Publication date
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2020-01-01Full publication date if available
- Authors
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Karsten Kahl, Bruno LangList of authors in order
- Landing page
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https://doi.org/10.1553/etna_vol54s51Publisher landing page
- PDF URL
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https://epub.oeaw.ac.at/0xc1aa5576_0x003bfef6.pdfDirect link to full text PDF
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YesWhether a free full text is available
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bronzeOpen access status per OpenAlex
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https://epub.oeaw.ac.at/0xc1aa5576_0x003bfef6.pdfDirect OA link when available
- Concepts
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Gaussian elimination, Heuristics, Hypergraph, Eigenvalues and eigenvectors, Mathematics, Adjacency matrix, Computer science, Gaussian, Algorithm, Graph, Mathematical optimization, Discrete mathematics, Computational chemistry, Chemistry, Physics, Quantum mechanicsTop concepts (fields/topics) attached by OpenAlex
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0Total citation count in OpenAlex
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10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.used | 39, 116 |
| abstract_inverted_index.with | 7, 97 |
| abstract_inverted_index.based | 52, 93, 106 |
| abstract_inverted_index.case. | 155 |
| abstract_inverted_index.evade | 65 |
| abstract_inverted_index.first | 69 |
| abstract_inverted_index.graph | 31, 110 |
| abstract_inverted_index.other | 61 |
| abstract_inverted_index.paper | 79 |
| abstract_inverted_index.phase | 122 |
| abstract_inverted_index.prove | 81 |
| abstract_inverted_index.seems | 63 |
| abstract_inverted_index.tight | 88 |
| abstract_inverted_index.using | 23 |
| abstract_inverted_index.which | 129 |
| abstract_inverted_index.wrong | 84 |
| abstract_inverted_index.allows | 32 |
| abstract_inverted_index.glance | 70 |
| abstract_inverted_index.linear | 19, 153 |
| abstract_inverted_index.matrix | 113 |
| abstract_inverted_index.memory | 42, 131 |
| abstract_inverted_index.rank-1 | 95 |
| abstract_inverted_index.reduce | 130 |
| abstract_inverted_index.sparse | 5 |
| abstract_inverted_index.system | 20 |
| abstract_inverted_index.analogy | 118, 150 |
| abstract_inverted_index.develop | 127 |
| abstract_inverted_index.enables | 45 |
| abstract_inverted_index.graphs. | 13 |
| abstract_inverted_index.nature. | 76 |
| abstract_inverted_index.optimal | 144 |
| abstract_inverted_index.predict | 41 |
| abstract_inverted_index.problem | 58 |
| abstract_inverted_index.remains | 147 |
| abstract_inverted_index.systems | 154 |
| abstract_inverted_index.Gaussian | 24, 124 |
| abstract_inverted_index.identify | 4 |
| abstract_inverted_index.inherent | 74 |
| abstract_inverted_index.matrices | 6 |
| abstract_inverted_index.question | 141 |
| abstract_inverted_index.solution | 16 |
| abstract_inverted_index.strategy | 146 |
| abstract_inverted_index.symbolic | 34, 99, 104, 121 |
| abstract_inverted_index.Hermitian | 56, 91 |
| abstract_inverted_index.adjacency | 10, 30 |
| abstract_inverted_index.assertion | 83 |
| abstract_inverted_index.customary | 2 |
| abstract_inverted_index.equations | 22 |
| abstract_inverted_index.footprint | 132 |
| abstract_inverted_index.incidence | 12, 109 |
| abstract_inverted_index.iterative | 75 |
| abstract_inverted_index.orderings | 51 |
| abstract_inverted_index.revealing | 86 |
| abstract_inverted_index.treatment | 67 |
| abstract_inverted_index.connection | 89 |
| abstract_inverted_index.eigenvalue | 57 |
| abstract_inverted_index.footprints | 43 |
| abstract_inverted_index.heuristics | 128 |
| abstract_inverted_index.procedure. | 102 |
| abstract_inverted_index.calculation | 105 |
| abstract_inverted_index.elimination | 50, 101, 125, 145 |
| abstract_inverted_index.heuristics. | 54 |
| abstract_inverted_index.NP-complete, | 148 |
| abstract_inverted_index.eigensolvers | 92 |
| abstract_inverted_index.elimination, | 25 |
| abstract_inverted_index.near-optimal | 49 |
| abstract_inverted_index.computations. | 134 |
| abstract_inverted_index.corresponding | 9 |
| abstract_inverted_index.determination | 47 |
| abstract_inverted_index.factorization | 35 |
| abstract_inverted_index.modifications | 96 |
| abstract_inverted_index.representation | 27 |
| cited_by_percentile_year | |
| countries_distinct_count | 1 |
| institutions_distinct_count | 2 |
| citation_normalized_percentile.value | 0.17094237 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | False |