{"id":"https://openalex.org/W2040721698","doi":"https://doi.org/10.1137/070697926","title":"On the Complexity of Numerical Analysis","display_name":"On the Complexity of Numerical Analysis","publication_year":2009,"publication_date":"2009-01-01","ids":{"openalex":"https://openalex.org/W2040721698","doi":"https://doi.org/10.1137/070697926","mag":"2040721698"},"language":"en","primary_location":{"id":"doi:10.1137/070697926","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070697926","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"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 Computing","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/A5078734824","display_name":"Eric Allender","orcid":"https://orcid.org/0000-0002-0650-028X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Eric Allender","raw_affiliation_strings":["allender@cs.rutgers.edu#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"allender@cs.rutgers.edu#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5029244957","display_name":"Peter B\u00fcrgisser","orcid":"https://orcid.org/0000-0001-8169-0514"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Peter B\u00fcrgisser","raw_affiliation_strings":["[email\u00a0protected]#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"[email\u00a0protected]#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5058444208","display_name":"Johan Kjeldgaard-Pedersen","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Johan Kjeldgaard-Pedersen","raw_affiliation_strings":["johan.kjeldgaard-pedersen@paconsulting.com#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"johan.kjeldgaard-pedersen@paconsulting.com#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5006968747","display_name":"Peter Bro Miltersen","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Peter Bro Miltersen","raw_affiliation_strings":["bromille@cs.au.dk#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"bromille@cs.au.dk#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":13.54,"has_fulltext":false,"cited_by_count":140,"citation_normalized_percentile":{"value":0.99208986,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"38","issue":"5","first_page":"1987","last_page":"2006"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998999834060669,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998999834060669,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9975000023841858,"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"}},{"id":"https://openalex.org/T10237","display_name":"Cryptography and Data Security","score":0.9955000281333923,"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/mathematics","display_name":"Mathematics","score":0.6297078728675842},{"id":"https://openalex.org/keywords/pspace","display_name":"PSPACE","score":0.5653470158576965},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.507250964641571},{"id":"https://openalex.org/keywords/complexity-class","display_name":"Complexity class","score":0.5046857595443726},{"id":"https://openalex.org/keywords/model-of-computation","display_name":"Model of computation","score":0.4996187686920166},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.4960361421108246},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.48334619402885437},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.4485621452331543},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.44761309027671814},{"id":"https://openalex.org/keywords/polynomial-hierarchy","display_name":"Polynomial hierarchy","score":0.4451947808265686},{"id":"https://openalex.org/keywords/computational-complexity-theory","display_name":"Computational complexity theory","score":0.44472256302833557},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4376786947250366},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.41579902172088623},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.40102240443229675},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3738289475440979},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.19693297147750854}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6297078728675842},{"id":"https://openalex.org/C197685441","wikidata":"https://www.wikidata.org/wiki/Q500716","display_name":"PSPACE","level":3,"score":0.5653470158576965},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.507250964641571},{"id":"https://openalex.org/C39637292","wikidata":"https://www.wikidata.org/wiki/Q908207","display_name":"Complexity class","level":3,"score":0.5046857595443726},{"id":"https://openalex.org/C184596265","wikidata":"https://www.wikidata.org/wiki/Q2651576","display_name":"Model of computation","level":3,"score":0.4996187686920166},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.4960361421108246},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.48334619402885437},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.4485621452331543},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.44761309027671814},{"id":"https://openalex.org/C2776013622","wikidata":"https://www.wikidata.org/wiki/Q2103021","display_name":"Polynomial hierarchy","level":3,"score":0.4451947808265686},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.44472256302833557},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4376786947250366},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.41579902172088623},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.40102240443229675},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3738289475440979},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.19693297147750854},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/070697926","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070697926","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"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 Computing","raw_type":"journal-article"},{"id":"pmh:oai:alma.01RUT_INST:11663546570004646","is_oa":false,"landing_page_url":"https://scholarship.libraries.rutgers.edu/esploro/outputs/journalArticle/ON-THE-COMPLEXITY-OF-NUMERICAL-ANALYSIS/991031654558104646","pdf_url":null,"source":{"id":"https://openalex.org/S4210197018","display_name":"View","issn_l":"2688-268X","issn":["2688-268X","2688-3988"],"is_oa":false,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:pure.atira.dk:publications/6916a6a0-fbfe-11dd-8f9a-000ea68e967b","is_oa":false,"landing_page_url":"https://pure.au.dk/portal/en/publications/6916a6a0-fbfe-11dd-8f9a-000ea68e967b","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Miltersen, P B, Allender, E, Burgisser, P & Kjeldgaard-Pedersen, J 2009, 'On the complexity of numerical analysis', S I A M Journal on Computing, vol. 38, no. 5, pp. 1987-2006. https://doi.org/10.1137/070697926","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":36,"referenced_works":["https://openalex.org/W1963893271","https://openalex.org/W1966137992","https://openalex.org/W1968245619","https://openalex.org/W1979572578","https://openalex.org/W1981241857","https://openalex.org/W1984469422","https://openalex.org/W1984534637","https://openalex.org/W1986481727","https://openalex.org/W1989159237","https://openalex.org/W2003699415","https://openalex.org/W2007080245","https://openalex.org/W2007736689","https://openalex.org/W2011758726","https://openalex.org/W2024091095","https://openalex.org/W2026036943","https://openalex.org/W2026143996","https://openalex.org/W2033323875","https://openalex.org/W2043667326","https://openalex.org/W2047206742","https://openalex.org/W2048353152","https://openalex.org/W2053392671","https://openalex.org/W2065034429","https://openalex.org/W2080132708","https://openalex.org/W2082002555","https://openalex.org/W2082560626","https://openalex.org/W2082774629","https://openalex.org/W2107145684","https://openalex.org/W2109873972","https://openalex.org/W2119564192","https://openalex.org/W2123675424","https://openalex.org/W2135282115","https://openalex.org/W2137407595","https://openalex.org/W2159341052","https://openalex.org/W2166717942","https://openalex.org/W2178086598","https://openalex.org/W4246985475"],"related_works":["https://openalex.org/W1504200807","https://openalex.org/W2085484366","https://openalex.org/W2982557863","https://openalex.org/W278676640","https://openalex.org/W4236887310","https://openalex.org/W1520700928","https://openalex.org/W103968499","https://openalex.org/W2970311880","https://openalex.org/W2082636129","https://openalex.org/W2075449032"],"abstract_inverted_index":{"We":[0,63,121,157],"study":[1],"two":[2],"quite":[3],"different":[4],"approaches":[5,69],"to":[6,49,113,134,155],"understanding":[7,75],"the":[8,17,23,31,50,59,72,76,79,99,105,163,175,182,202,206,220,223],"complexity":[9,77,197,221],"of":[10,20,34,41,67,74,78,148,170,195,204,222],"fundamental":[11],"problems":[12],"in":[13,45,58,162,181],"numerical":[14,35,43,60,149],"analysis:":[15],"(a)":[16],"Blum\u2013Shub\u2013Smale":[18,100],"model":[19],"computation":[21,44,103,150],"over":[22,104],"reals;":[24],"and":[25,136],"(b)":[26],"a":[27,87],"problem":[28,179,192],"we":[29,83,137,172,214,232],"call":[30,84],"\u201cgeneric":[32],"task":[33,147],"computation,\u201d":[36],"which":[37,82],"captures":[38],"an":[39,92],"aspect":[40],"doing":[42],"floating":[46],"point,":[47],"similar":[48],"\u201clong":[51],"exponent":[52],"model\u201d":[53],"that":[54,65,123,159,174,234],"has":[55,243],"been":[56],"studied":[57],"computing":[61],"community.":[62],"show":[64,233],"both":[66],"these":[68],"hinge":[70],"on":[71,211,219],"question":[73],"following":[80],"problem,":[81],"PosSLP:":[85],"Given":[86],"division-free":[88],"straight-line":[89],"program":[90],"producing":[91],"integer":[93],"N,":[94],"decide":[95],"whether":[96],"$N>0$.":[97],"In":[98,201,230],"model,":[101],"polynomial-time":[102,111,153],"reals":[106],"(on":[107],"discrete":[108],"inputs)":[109],"is":[110,151,237],"equivalent":[112,154],"PosSLP":[114,160],"when":[115],"there":[116],"are":[117],"only":[118],"algebraic":[119],"constants.":[120],"conjecture":[122],"using":[124],"transcendental":[125],"constants":[126],"provides":[127],"no":[128],"additional":[129],"power,":[130],"beyond":[131],"nonuniform":[132],"reductions":[133],"PosSLP,":[135],"present":[138,215],"some":[139,216],"preliminary":[140],"results":[141,210],"supporting":[142],"this":[143,167,190],"conjecture.":[144],"The":[145],"generic":[146],"also":[152],"PosSLP.":[156],"prove":[158],"lies":[161,180],"counting":[164,183],"hierarchy.":[165],"Combining":[166],"with":[168],"work":[169],"Tiwari,":[171],"obtain":[173],"Euclidean":[176],"traveling":[177],"salesman":[178],"hierarchy\u2014the":[184],"previous":[185],"best":[186],"upper":[187],"bound":[188],"for":[189,208],"important":[191],"(in":[193],"terms":[194],"classical":[196],"classes)":[198],"being":[199],"PSPACE.":[200],"course":[203],"developing":[205],"context":[207],"our":[209],"arithmetic":[212,224],"circuits,":[213],"new":[217],"observations":[218],"circuit":[225],"identity":[226],"testing":[227],"(ACIT)":[228],"problem.":[229],"particular,":[231],"if":[235],"$n!$":[236],"not":[238],"ultimately":[239],"easy,":[240],"then":[241],"ACIT":[242],"subexponential":[244],"complexity.":[245]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":8},{"year":2018,"cited_by_count":6},{"year":2017,"cited_by_count":12},{"year":2016,"cited_by_count":5},{"year":2015,"cited_by_count":10},{"year":2014,"cited_by_count":26},{"year":2013,"cited_by_count":10},{"year":2012,"cited_by_count":13}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}