On fast multiplication of a matrix by its transpose
We present a non-commutative algorithm for the multiplication of a 2x2-block-matrix by its transpose using 5 block products (3 recursive calls and 2 general products) over C or any finite field.We use geometric considerations on the space of bilinear forms describing 2x2 matrix products to obtain th...
Gespeichert in:
| Veröffentlicht in: | arXiv.org 2020-06 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | |
|---|---|
| container_issue | |
| container_start_page | |
| container_title | arXiv.org |
| container_volume | |
| creator | Dumas, Jean-Guillaume Pernet, Clement Sedoglavic, Alexandre |
| description | We present a non-commutative algorithm for the multiplication of a 2x2-block-matrix by its transpose using 5 block products (3 recursive calls and 2 general products) over C or any finite field.We use geometric considerations on the space of bilinear forms describing 2x2 matrix products to obtain this algorithm and we show how to reduce the number of involved additions.The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its transpose to general matrix product, improving by a constant factor previously known reductions.Finally we propose schedules with low memory footprint that support a fast and memory efficient practical implementation over a finite field.To conclude, we show how to use our result in LDLT factorization. |
| doi_str_mv | 10.48550/arxiv.2001.04109 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_arxiv</sourceid><recordid>TN_cdi_arxiv_primary_2001_04109</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2337684681</sourcerecordid><originalsourceid>FETCH-LOGICAL-a521-8842db06d327a28a087f88593e3bb49bbd349d062901af09d6885c78bcafb3383</originalsourceid><addsrcrecordid>eNotj8tqwzAUREWh0JDmA7qqoGu7V7qWLS9L6CMQyCZ7c-UHKPhVSS7J31dNuprFDDNzGHsSkGZaKXgld7Y_qQQQKWQCyju2kogi0ZmUD2zj_QkAZF5IpXDF8DDyjnzgw9IHO_e2pmCnkU8dJz5QcPbMzYXb4HlwNPp58u0ju--o9-3mX9fs-PF-3H4l-8Pnbvu2T0jJuBcHGwN5g7IgqQl00WmtSmzRmKw0psGsbCCXJQjqoGzy6NaFNjV1BlHjmj3faq9E1ezsQO5S_ZFVV7KYeLklZjd9L60P1Wla3Bg_VRG5yHWWa4G_nTxPYg</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2337684681</pqid></control><display><type>article</type><title>On fast multiplication of a matrix by its transpose</title><source>arXiv.org</source><source>Free E- Journals</source><creator>Dumas, Jean-Guillaume ; Pernet, Clement ; Sedoglavic, Alexandre</creator><creatorcontrib>Dumas, Jean-Guillaume ; Pernet, Clement ; Sedoglavic, Alexandre</creatorcontrib><description>We present a non-commutative algorithm for the multiplication of a 2x2-block-matrix by its transpose using 5 block products (3 recursive calls and 2 general products) over C or any finite field.We use geometric considerations on the space of bilinear forms describing 2x2 matrix products to obtain this algorithm and we show how to reduce the number of involved additions.The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its transpose to general matrix product, improving by a constant factor previously known reductions.Finally we propose schedules with low memory footprint that support a fast and memory efficient practical implementation over a finite field.To conclude, we show how to use our result in LDLT factorization.</description><identifier>EISSN: 2331-8422</identifier><identifier>DOI: 10.48550/arxiv.2001.04109</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Algorithms ; Computer Science - Symbolic Computation ; Fields (mathematics) ; Multiplication ; Schedules</subject><ispartof>arXiv.org, 2020-06</ispartof><rights>2020. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,777,781,882,27906</link.rule.ids><backlink>$$Uhttps://doi.org/10.48550/arXiv.2001.04109$$DView paper in arXiv$$Hfree_for_read</backlink><backlink>$$Uhttps://doi.org/10.1145/3373207.3404021$$DView published paper (Access to full text may be restricted)$$Hfree_for_read</backlink></links><search><creatorcontrib>Dumas, Jean-Guillaume</creatorcontrib><creatorcontrib>Pernet, Clement</creatorcontrib><creatorcontrib>Sedoglavic, Alexandre</creatorcontrib><title>On fast multiplication of a matrix by its transpose</title><title>arXiv.org</title><description>We present a non-commutative algorithm for the multiplication of a 2x2-block-matrix by its transpose using 5 block products (3 recursive calls and 2 general products) over C or any finite field.We use geometric considerations on the space of bilinear forms describing 2x2 matrix products to obtain this algorithm and we show how to reduce the number of involved additions.The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its transpose to general matrix product, improving by a constant factor previously known reductions.Finally we propose schedules with low memory footprint that support a fast and memory efficient practical implementation over a finite field.To conclude, we show how to use our result in LDLT factorization.</description><subject>Algorithms</subject><subject>Computer Science - Symbolic Computation</subject><subject>Fields (mathematics)</subject><subject>Multiplication</subject><subject>Schedules</subject><issn>2331-8422</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GOX</sourceid><recordid>eNotj8tqwzAUREWh0JDmA7qqoGu7V7qWLS9L6CMQyCZ7c-UHKPhVSS7J31dNuprFDDNzGHsSkGZaKXgld7Y_qQQQKWQCyju2kogi0ZmUD2zj_QkAZF5IpXDF8DDyjnzgw9IHO_e2pmCnkU8dJz5QcPbMzYXb4HlwNPp58u0ju--o9-3mX9fs-PF-3H4l-8Pnbvu2T0jJuBcHGwN5g7IgqQl00WmtSmzRmKw0psGsbCCXJQjqoGzy6NaFNjV1BlHjmj3faq9E1ezsQO5S_ZFVV7KYeLklZjd9L60P1Wla3Bg_VRG5yHWWa4G_nTxPYg</recordid><startdate>20200619</startdate><enddate>20200619</enddate><creator>Dumas, Jean-Guillaume</creator><creator>Pernet, Clement</creator><creator>Sedoglavic, Alexandre</creator><general>Cornell University Library, arXiv.org</general><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>HCIFZ</scope><scope>L6V</scope><scope>M7S</scope><scope>PIMPY</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>AKY</scope><scope>GOX</scope></search><sort><creationdate>20200619</creationdate><title>On fast multiplication of a matrix by its transpose</title><author>Dumas, Jean-Guillaume ; Pernet, Clement ; Sedoglavic, Alexandre</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a521-8842db06d327a28a087f88593e3bb49bbd349d062901af09d6885c78bcafb3383</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Algorithms</topic><topic>Computer Science - Symbolic Computation</topic><topic>Fields (mathematics)</topic><topic>Multiplication</topic><topic>Schedules</topic><toplevel>online_resources</toplevel><creatorcontrib>Dumas, Jean-Guillaume</creatorcontrib><creatorcontrib>Pernet, Clement</creatorcontrib><creatorcontrib>Sedoglavic, Alexandre</creatorcontrib><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Engineering Collection</collection><collection>Engineering Database</collection><collection>Publicly Available Content Database</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection><collection>arXiv Computer Science</collection><collection>arXiv.org</collection><jtitle>arXiv.org</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Dumas, Jean-Guillaume</au><au>Pernet, Clement</au><au>Sedoglavic, Alexandre</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On fast multiplication of a matrix by its transpose</atitle><jtitle>arXiv.org</jtitle><date>2020-06-19</date><risdate>2020</risdate><eissn>2331-8422</eissn><abstract>We present a non-commutative algorithm for the multiplication of a 2x2-block-matrix by its transpose using 5 block products (3 recursive calls and 2 general products) over C or any finite field.We use geometric considerations on the space of bilinear forms describing 2x2 matrix products to obtain this algorithm and we show how to reduce the number of involved additions.The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its transpose to general matrix product, improving by a constant factor previously known reductions.Finally we propose schedules with low memory footprint that support a fast and memory efficient practical implementation over a finite field.To conclude, we show how to use our result in LDLT factorization.</abstract><cop>Ithaca</cop><pub>Cornell University Library, arXiv.org</pub><doi>10.48550/arxiv.2001.04109</doi><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | EISSN: 2331-8422 |
| ispartof | arXiv.org, 2020-06 |
| issn | 2331-8422 |
| language | eng |
| recordid | cdi_arxiv_primary_2001_04109 |
| source | arXiv.org; Free E- Journals |
| subjects | Algorithms Computer Science - Symbolic Computation Fields (mathematics) Multiplication Schedules |
| title | On fast multiplication of a matrix by its transpose |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-20T18%3A49%3A35IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_arxiv&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=On%20fast%20multiplication%20of%20a%20matrix%20by%20its%20transpose&rft.jtitle=arXiv.org&rft.au=Dumas,%20Jean-Guillaume&rft.date=2020-06-19&rft.eissn=2331-8422&rft_id=info:doi/10.48550/arxiv.2001.04109&rft_dat=%3Cproquest_arxiv%3E2337684681%3C/proquest_arxiv%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=2337684681&rft_id=info:pmid/&rfr_iscdi=true |