{"id":"https://openalex.org/W2118863844","doi":"https://doi.org/10.1137/040607009","title":"Computationally Efficient Approximations of the Joint Spectral Radius","display_name":"Computationally Efficient Approximations of the Joint Spectral Radius","publication_year":2005,"publication_date":"2005-01-01","ids":{"openalex":"https://openalex.org/W2118863844","doi":"https://doi.org/10.1137/040607009","mag":"2118863844"},"language":"en","primary_location":{"id":"doi:10.1137/040607009","is_oa":false,"landing_page_url":"https://doi.org/10.1137/040607009","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5073875709","display_name":"Vincent D. Blondel","orcid":null},"institutions":[{"id":"https://openalex.org/I95674353","display_name":"UCLouvain","ror":"https://ror.org/02495e989","country_code":"BE","type":"education","lineage":["https://openalex.org/I95674353"]}],"countries":["BE"],"is_corresponding":false,"raw_author_name":"Vincent D. Blondel","raw_affiliation_strings":["Center for Operations Research and Econometrics (CORE), Catholic University of Louvain (UCL), 34 voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium"],"affiliations":[{"raw_affiliation_string":"Center for Operations Research and Econometrics (CORE), Catholic University of Louvain (UCL), 34 voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium","institution_ids":["https://openalex.org/I95674353"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5067792459","display_name":"Yurii Nesterov","orcid":"https://orcid.org/0000-0002-0542-8757"},"institutions":[{"id":"https://openalex.org/I95674353","display_name":"UCLouvain","ror":"https://ror.org/02495e989","country_code":"BE","type":"education","lineage":["https://openalex.org/I95674353"]}],"countries":["BE"],"is_corresponding":true,"raw_author_name":"Yurii Nesterov","raw_affiliation_strings":["Center for Operations Research and Econometrics (CORE), Catholic University of Louvain (UCL), 34 voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium"],"affiliations":[{"raw_affiliation_string":"Center for Operations Research and Econometrics (CORE), Catholic University of Louvain (UCL), 34 voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium","institution_ids":["https://openalex.org/I95674353"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5067792459"],"corresponding_institution_ids":["https://openalex.org/I95674353"],"apc_list":null,"apc_paid":null,"fwci":10.0989,"has_fulltext":false,"cited_by_count":143,"citation_normalized_percentile":{"value":0.98814552,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":91,"max":100},"biblio":{"volume":"27","issue":"1","first_page":"256","last_page":"272"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9991999864578247,"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":0.9991999864578247,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9962999820709229,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11236","display_name":"Control Systems and Identification","score":0.9821000099182129,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8480224609375},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6308903694152832},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.5542035102844238},{"id":"https://openalex.org/keywords/spectral-radius","display_name":"Spectral radius","score":0.5359267592430115},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.46135473251342773},{"id":"https://openalex.org/keywords/kronecker-product","display_name":"Kronecker product","score":0.432686448097229},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.42094987630844116},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3790847361087799},{"id":"https://openalex.org/keywords/kronecker-delta","display_name":"Kronecker delta","score":0.37031567096710205},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.23210784792900085},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.17019397020339966},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10515439510345459}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8480224609375},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6308903694152832},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.5542035102844238},{"id":"https://openalex.org/C140532419","wikidata":"https://www.wikidata.org/wiki/Q249748","display_name":"Spectral radius","level":3,"score":0.5359267592430115},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.46135473251342773},{"id":"https://openalex.org/C46030957","wikidata":"https://www.wikidata.org/wiki/Q1238125","display_name":"Kronecker product","level":3,"score":0.432686448097229},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.42094987630844116},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3790847361087799},{"id":"https://openalex.org/C39482219","wikidata":"https://www.wikidata.org/wiki/Q192826","display_name":"Kronecker delta","level":2,"score":0.37031567096710205},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.23210784792900085},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.17019397020339966},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10515439510345459},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","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},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/040607009","is_oa":false,"landing_page_url":"https://doi.org/10.1137/040607009","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:cor:louvrp:-1800","is_oa":false,"landing_page_url":"http://doi.org/10.1137/040607009","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"preprint"},{"id":"pmh:oai:dial.uclouvain.be:boreal:23381","is_oa":false,"landing_page_url":"http://hdl.handle.net/2078.1/23381","pdf_url":null,"source":{"id":"https://openalex.org/S4306401902","display_name":"Digital Access to Libraries (Universit\u00e9 catholique de Louvain (UCL), l'Universit\u00e9 de Namur (UNamur) and the Universit\u00e9 Saint-Louis (USL-B))","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I95674353","host_organization_name":"UCLouvain","host_organization_lineage":["https://openalex.org/I95674353"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"S I A M Journal on Matrix Analysis and Applications, Vol. 27, no. 1, p. 256-272 (2005)","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":34,"referenced_works":["https://openalex.org/W102983431","https://openalex.org/W1488435683","https://openalex.org/W1597116596","https://openalex.org/W1705381972","https://openalex.org/W1967400268","https://openalex.org/W1978242229","https://openalex.org/W1980368307","https://openalex.org/W1991656001","https://openalex.org/W2001393739","https://openalex.org/W2008442517","https://openalex.org/W2023729441","https://openalex.org/W2032326995","https://openalex.org/W2042664819","https://openalex.org/W2048103084","https://openalex.org/W2055472587","https://openalex.org/W2070972632","https://openalex.org/W2076566814","https://openalex.org/W2079227289","https://openalex.org/W2093818546","https://openalex.org/W2097086116","https://openalex.org/W2098678476","https://openalex.org/W2103966928","https://openalex.org/W2113789941","https://openalex.org/W2116656166","https://openalex.org/W2133763708","https://openalex.org/W2143181390","https://openalex.org/W2163553871","https://openalex.org/W2319661631","https://openalex.org/W2911681356","https://openalex.org/W3210839039","https://openalex.org/W4206042551","https://openalex.org/W4233004553","https://openalex.org/W4238907387","https://openalex.org/W4249667877"],"related_works":["https://openalex.org/W123417539","https://openalex.org/W2079407403","https://openalex.org/W2060299328","https://openalex.org/W4375957393","https://openalex.org/W4233239985","https://openalex.org/W2117336295","https://openalex.org/W2073634779","https://openalex.org/W3102722572","https://openalex.org/W2090103374","https://openalex.org/W577870507"],"abstract_inverted_index":{"The":[0,241],"joint":[1,60,339],"spectral":[2,61,129,340],"radius":[3,62,130,341],"of":[4,7,12,26,38,63,67,81,87,99,108,118,131,187,221,252,268,284,313,327,337,362],"a":[5,10,36,55,64,211,310,325,333],"set":[6,66],"matrices":[8,27,68,83,88,100,132,190,285],"is":[9,42,76,136,161,230,281,288,306,359],"measure":[11],"the":[13,30,59,79,82,85,90,96,128,139,167,178,183,218,222,245,250,253,258,282,338,360],"maximal":[14],"asymptotic":[15],"growth":[16],"rate":[17],"that":[18,111,127,166,206,236,308,332],"can":[19,207,260],"be":[20,208,261],"obtained":[21],"by":[22,138],"forming":[23],"long":[24],"products":[25],"taken":[28],"from":[29,256,295],"set.":[31],"This":[32],"quantity":[33],"appears":[34],"in":[35,52,78,210,272,275,317,320],"number":[37,86,283,361],"application":[39],"contexts":[40],"but":[41,247],"notoriously":[43],"difficult":[44],"to":[45,48,115,164],"compute":[46],"and":[47,89,286,315,322],"approximate.":[49],"We":[50],"introduce":[51,198],"this":[53],"paper":[54],"procedure":[56,75,201,223],"for":[57],"approximating":[58],"finite":[65],"with":[69,101,133,191],"arbitrary":[70,119,192],"high":[71,120],"accuracy.":[72,121],"Our":[73,263],"approximation":[74,200,226,259,264,304,336],"polynomial":[77,274,319],"size":[80],"once":[84],"desired":[91],"accuracy":[92,229,246,270,312,354],"are":[93,293],"fixed.":[94],"For":[95,189,214],"special":[97],"case":[98],"nonnegative":[102,134],"entries":[103,135,193],"we":[104,112,197,298,330],"give":[105],"elementary":[106],"proofs":[107],"simple":[109,140],"inequalities":[110,124],"then":[113],"use":[114],"obtain":[116],"approximations":[117,267],"From":[122],"these":[123],"it":[125,160],"follows":[126],"given":[137],"expression":[141],"$$":[142,158],"\\rho(A_1,":[143],"\\ldots,":[144],"A_m)=":[145],"\\lim_{k":[146],"\\rightarrow":[147],"\\infty}":[148],"\\rho^{1/k}":[149],"(":[150],"A_1^{\\otimes":[151],"k}":[152],"+":[153,155],"\\cdots":[154],"A_m^{\\otimes":[156],"k}),":[157],"where":[159,279,357],"somewhat":[162],"surprising":[163],"notice":[165],"right-hand":[168],"side":[169],"does":[170],"not":[171],"directly":[172],"involve":[173],"any":[174],"mixed":[175],"product":[176],"between":[177],"matrices.":[179,363],"($A^{\\otimes":[180],"k}$":[181],"denotes":[182],"kth":[184],"Kronecker":[185],"power":[186],"A.)":[188],"(not":[194],"necessarily":[195],"nonnegative),":[196],"an":[199,225],"based":[202,342],"on":[203,343,349],"semidefinite":[204],"liftings":[205],"implemented":[209],"recursive":[212],"way.":[213],"two":[215],"matrices,":[216],"even":[217],"first":[219],"step":[220],"gives":[224],"whose":[227],"relative":[228,269,311,353],"at":[231],"least":[232],"${1":[233],"/":[234],"\\sqrt{2}}$,":[235],"is,":[237],"more":[238],"than":[239],"$70\\%$.":[240],"subsequent":[242],"steps":[243],"improve":[244],"also":[248],"increase":[249],"dimension":[251],"auxiliary":[254],"problems":[255],"which":[257],"found.":[262],"procedures":[265],"provide":[266],"$1-\\epsilon$":[271,314],"time":[273,318],"$n^{(\\ln":[276],"m)":[277],"/\\epsilon}$,":[278],"m":[280,358],"n":[287,321],"their":[289],"size.":[290],"These":[291],"bounds":[292],"close":[294],"optimality":[296],"since":[297],"show":[299],"that,":[300],"unless":[301],"P=NP,":[302],"no":[303],"algorithm":[305],"possible":[307],"provides":[309],"runs":[316],"$1/\\epsilon$.":[323],"As":[324],"by-product":[326],"our":[328],"results":[329],"prove":[331],"widely":[334],"used":[335],"common":[344],"quadratic":[345],"Lyapunov":[346],"functions":[347],"(or":[348],"ellipsoid":[350],"norms)":[351],"has":[352],"$1/\\sqrt":[355],"m$,":[356]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":7},{"year":2020,"cited_by_count":9},{"year":2019,"cited_by_count":9},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":7},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":9},{"year":2013,"cited_by_count":9},{"year":2012,"cited_by_count":13}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
