{"id":"https://openalex.org/W4391767365","doi":"https://doi.org/10.1145/3666000.3669715","title":"Computing Krylov iterates in the time of matrix multiplication","display_name":"Computing Krylov iterates in the time of matrix multiplication","publication_year":2024,"publication_date":"2024-02-12","ids":{"openalex":"https://openalex.org/W4391767365","doi":"https://doi.org/10.1145/3666000.3669715"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2402.07345","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2402.07345","pdf_url":"https://arxiv.org/pdf/2402.07345","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"type":"preprint","indexed_in":["arxiv"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2402.07345","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5016685971","display_name":"Vincent Neiger","orcid":"https://orcid.org/0000-0002-8311-9490"},"institutions":[{"id":"https://openalex.org/I4210159265","display_name":"CS Communication & Syst\u00e8mes (France)","ror":"https://ror.org/053hexc75","country_code":"FR","type":"company","lineage":["https://openalex.org/I4210159265"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Neiger, Vincent","raw_affiliation_strings":["PolSys - Polynomial Systems (France)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"PolSys - Polynomial Systems (France)","institution_ids":["https://openalex.org/I4210159265"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Pernet, Cl\\'ement","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Pernet, Cl\\'ement","raw_affiliation_strings":["CASC - Calcul Alg\u00e9brique et Symbolique, S\u00e9curit\u00e9, Syst\u00e8mes Complexes, Codes et Cryptologie (France)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"CASC - Calcul Alg\u00e9brique et Symbolique, S\u00e9curit\u00e9, Syst\u00e8mes Complexes, Codes et Cryptologie (France)","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5021399584","display_name":"Gilles Villard","orcid":"https://orcid.org/0000-0003-1936-7155"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Villard, Gilles","raw_affiliation_strings":["ARIC - Arithm\u00e9tiques des ordinateurs, m\u00e9thodes formelles, g\u00e9n\u00e9ration de code (46 All\u00e9e d'Italie 69364 Lyon France - France)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"ARIC - Arithm\u00e9tiques des ordinateurs, m\u00e9thodes formelles, g\u00e9n\u00e9ration de code (46 All\u00e9e d'Italie 69364 Lyon France - France)","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5016685971"],"corresponding_institution_ids":["https://openalex.org/I4210159265"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.02445466,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9886999726295471,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9886999726295471,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9470999836921692,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.9365000128746033,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/iterated-function","display_name":"Iterated function","score":0.8301048278808594},{"id":"https://openalex.org/keywords/multiplication","display_name":"Multiplication (music)","score":0.6926551461219788},{"id":"https://openalex.org/keywords/matrix-multiplication","display_name":"Matrix multiplication","score":0.6469478607177734},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5623802542686462},{"id":"https://openalex.org/keywords/parallel-computing","display_name":"Parallel computing","score":0.45839935541152954},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4453681707382202},{"id":"https://openalex.org/keywords/computational-science","display_name":"Computational science","score":0.403396338224411},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.38322287797927856},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.38023921847343445},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3326086401939392},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.214150071144104},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1308860182762146},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.09578055143356323},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.08704209327697754},{"id":"https://openalex.org/keywords/materials-science","display_name":"Materials science","score":0.06225535273551941}],"concepts":[{"id":"https://openalex.org/C140479938","wikidata":"https://www.wikidata.org/wiki/Q5254619","display_name":"Iterated function","level":2,"score":0.8301048278808594},{"id":"https://openalex.org/C2780595030","wikidata":"https://www.wikidata.org/wiki/Q3860309","display_name":"Multiplication (music)","level":2,"score":0.6926551461219788},{"id":"https://openalex.org/C17349429","wikidata":"https://www.wikidata.org/wiki/Q1049914","display_name":"Matrix multiplication","level":3,"score":0.6469478607177734},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5623802542686462},{"id":"https://openalex.org/C173608175","wikidata":"https://www.wikidata.org/wiki/Q232661","display_name":"Parallel computing","level":1,"score":0.45839935541152954},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4453681707382202},{"id":"https://openalex.org/C459310","wikidata":"https://www.wikidata.org/wiki/Q117801","display_name":"Computational science","level":1,"score":0.403396338224411},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.38322287797927856},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.38023921847343445},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3326086401939392},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.214150071144104},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1308860182762146},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.09578055143356323},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.08704209327697754},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.06225535273551941},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"pmh:oai:arXiv.org:2402.07345","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2402.07345","pdf_url":"https://arxiv.org/pdf/2402.07345","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2402.07345","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2402.07345","pdf_url":"https://arxiv.org/pdf/2402.07345","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1484044075","display_name":null,"funder_award_id":"22-PECY-0010","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"},{"id":"https://openalex.org/G3726857880","display_name":null,"funder_award_id":"ANR-22-CE91-0007","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"},{"id":"https://openalex.org/G6791052079","display_name":"Fast reconstruction of multivariate algebraic relations","funder_award_id":"ANR-23-CE48-0003","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"},{"id":"https://openalex.org/G7022847571","display_name":null,"funder_award_id":"ANR-22-PECY-0010","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"},{"id":"https://openalex.org/G8136746263","display_name":null,"funder_award_id":"FA8665-20-1-7029","funder_id":"https://openalex.org/F4320338503","funder_display_name":"European Office of Aerospace Research and Development"},{"id":"https://openalex.org/G8781429262","display_name":null,"funder_award_id":"ANR-21-CE39-0006","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"}],"funders":[{"id":"https://openalex.org/F4320320883","display_name":"Agence Nationale de la Recherche","ror":"https://ror.org/00rbzpz17"},{"id":"https://openalex.org/F4320321181","display_name":"Austrian Science Fund","ror":"https://ror.org/013tf3c58"},{"id":"https://openalex.org/F4320338279","display_name":"Air Force Office of Scientific Research","ror":"https://ror.org/011e9bt93"},{"id":"https://openalex.org/F4320338503","display_name":"European Office of Aerospace Research and Development","ror":null}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4391767365.pdf"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W3099313426","https://openalex.org/W4287593139","https://openalex.org/W2036954759","https://openalex.org/W2057797376","https://openalex.org/W2090319426","https://openalex.org/W752783541","https://openalex.org/W2506252583","https://openalex.org/W1506547947","https://openalex.org/W2048249848","https://openalex.org/W1987422982"],"abstract_inverted_index":{"Krylov":[0,33,75,127],"methods":[1],"rely":[2,199],"on":[3,200],"iterated":[4],"matrix-vector":[5],"products":[6],"$A^k":[7,24],"u_j$":[8,25],"for":[9,113,145,187,194,203],"an":[10,122],"$n\\times":[11],"n$":[12],"matrix":[13,114,164,174],"$A$":[14,101],"and":[15,65,102,137,172,215],"vectors":[16,56],"$u_1,\\ldots,u_m$.":[17],"The":[18,77],"space":[19],"spanned":[20],"by":[21],"all":[22],"iterates":[23,39],"admits":[26],"a":[27,58,110,151,162],"particular":[28],"basis":[29,59,128],"--":[30,35],"the":[31,41,54,117,125,156,180],"\\emph{maximal":[32],"basis}":[34],"which":[36],"consists":[37],"of":[38,40,100,208],"first":[42],"vector":[43],"$u_1,":[44],"Au_1,":[45],"A^2u_1,\\ldots$,":[46],"until":[47,57,85],"reaching":[48],"linear":[49],"dependency,":[50],"then":[51],"iterating":[52],"similarly":[53],"subsequent":[55],"is":[60,69,109,197],"obtained.":[61],"Finding":[62],"minimal":[63,216],"polynomials":[64],"Frobenius":[66,157],"normal":[67,158],"forms":[68],"closely":[70],"related":[71],"to":[72,80,198],"computing":[73,124],"maximal":[74,126],"bases.":[76,218],"fastest":[78],"way":[79],"produce":[81],"these":[82,195],"bases":[83],"was,":[84],"this":[86],"paper,":[87],"Keller-Gehrig's":[88],"1985":[89],"algorithm":[90,123],"whose":[91],"complexity":[92,182],"bound":[93],"$O(n^\\omega":[94,170],"\\log(n))$":[95],"comes":[96],"from":[97],"repeated":[98],"squarings":[99],"logarithmically":[103],"many":[104],"Gaussian":[105],"eliminations.":[106],"Here":[107],"$\\omega>2$":[108],"feasible":[111],"exponent":[112],"multiplication":[115],"over":[116],"base":[118],"field.":[119],"We":[120],"present":[121],"in":[129,169,179],"$O(n^\\omega\\log\\log(n))$":[130],"field":[131],"operations":[132],"when":[133],"$m":[134],"\\in":[135,185],"O(n)$,":[136],"even":[138],"$O(n^\\omega)$":[139],"as":[140,142],"soon":[141],"$m\\in":[143],"O(n/\\log(n)^c)$":[144],"some":[146,188],"fixed":[147,189],"real":[148],"$c>0$.":[149],"As":[150],"consequence,":[152],"we":[153],"show":[154],"that":[155],"form":[159],"together":[160],"with":[161],"transformation":[163],"can":[165,176],"be":[166,177],"computed":[167],"deterministically":[168],"(\\log\\log(n))^2)$,":[171],"therefore":[173],"exponentiation~$A^k$":[175],"performed":[178],"latter":[181],"if":[183],"$\\log(k)":[184],"O(n^{\\omega-1-\\varepsilon})$":[186],"$\\varepsilon>0$.":[190],"A":[191],"key":[192],"idea":[193],"improvements":[196],"fast":[201],"algorithms":[202],"$m\\times":[204],"m$":[205],"polynomial":[206],"matrices":[207],"average":[209],"degree":[210],"$n/m$,":[211],"involving":[212],"high-order":[213],"lifting":[214],"kernel":[217]},"counts_by_year":[],"updated_date":"2026-04-28T14:05:53.105641","created_date":"2024-02-13T00:00:00"}