{"id":"https://openalex.org/W2170078634","doi":"https://doi.org/10.1137/s0895479803436202","title":"Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices","display_name":"Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices","publication_year":2006,"publication_date":"2006-01-01","ids":{"openalex":"https://openalex.org/W2170078634","doi":"https://doi.org/10.1137/s0895479803436202","mag":"2170078634"},"language":"en","primary_location":{"id":"doi:10.1137/s0895479803436202","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479803436202","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/A5075101239","display_name":"Arvind Sankar","orcid":null},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Arvind Sankar","raw_affiliation_strings":["Department of Computer Science Yale University","Department of Mathematics Massachusetts Institute of Technology","Department of Computer Science Boston University and Akamai Technologies Inc"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science Yale University","institution_ids":[]},{"raw_affiliation_string":"Department of Mathematics Massachusetts Institute of Technology","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Department of Computer Science Boston University and Akamai Technologies Inc","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5061644327","display_name":"Daniel A. Spielman","orcid":"https://orcid.org/0000-0002-8958-879X"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Daniel A. Spielman","raw_affiliation_strings":["Department of Computer Science Yale University","Department of Mathematics Massachusetts Institute of Technology","Department of Computer Science Boston University and Akamai Technologies Inc"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science Yale University","institution_ids":[]},{"raw_affiliation_string":"Department of Mathematics Massachusetts Institute of Technology","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Department of Computer Science Boston University and Akamai Technologies Inc","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5102005063","display_name":"Shang\u2010Hua Teng","orcid":"https://orcid.org/0000-0001-5011-4514"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Shang-Hua Teng","raw_affiliation_strings":["Department of Computer Science Boston University and Akamai Technologies Inc","Department of Computer Science Yale University","Department of Mathematics Massachusetts Institute of Technology"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science Boston University and Akamai Technologies Inc","institution_ids":[]},{"raw_affiliation_string":"Department of Computer Science Yale University","institution_ids":[]},{"raw_affiliation_string":"Department of Mathematics Massachusetts Institute of Technology","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5075101239"],"corresponding_institution_ids":["https://openalex.org/I63966007"],"apc_list":null,"apc_paid":null,"fwci":7.0774,"has_fulltext":false,"cited_by_count":213,"citation_normalized_percentile":{"value":0.97598939,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":98,"max":100},"biblio":{"volume":"28","issue":"2","first_page":"446","last_page":"476"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9995999932289124,"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":0.9995999932289124,"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":0.9926000237464905,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9919000267982483,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic 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/gaussian-elimination","display_name":"Gaussian elimination","score":0.8363274335861206},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8344063758850098},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.703084409236908},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.6538568139076233},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5603325366973877},{"id":"https://openalex.org/keywords/condition-number","display_name":"Condition number","score":0.520047664642334},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4975629150867462},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.49262648820877075},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2548069357872009},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.07970866560935974},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.05683031678199768}],"concepts":[{"id":"https://openalex.org/C126312332","wikidata":"https://www.wikidata.org/wiki/Q2658","display_name":"Gaussian elimination","level":3,"score":0.8363274335861206},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8344063758850098},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.703084409236908},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.6538568139076233},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5603325366973877},{"id":"https://openalex.org/C84545080","wikidata":"https://www.wikidata.org/wiki/Q1147936","display_name":"Condition number","level":3,"score":0.520047664642334},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4975629150867462},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.49262648820877075},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2548069357872009},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.07970866560935974},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.05683031678199768},{"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/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1137/s0895479803436202","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479803436202","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:CiteSeerX.psu:10.1.1.165.7729","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.7729","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.yale.edu/homes/spielman/Research/nopivotdas.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.341.5189","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.341.5189","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/cs/0310022v3.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.341.7738","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.341.7738","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/cs/0310022v2.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.342.419","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.342.419","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/cs/0310022v4.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.342.7376","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.342.7376","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/cs/0310022v1.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G101112439","display_name":null,"funder_award_id":"Fellow","funder_id":"https://openalex.org/F4320306151","funder_display_name":"Alfred P. Sloan Foundation"},{"id":"https://openalex.org/G2869385521","display_name":"ITR:  Collaborative Research:  Smoothed Analysis of Algorithms","funder_award_id":"0325630","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4798087266","display_name":"ITR/SY(CISE): Why algorithms work well in practice:  pertubation-based average-case analysis of the simplex algorithm and beyond","funder_award_id":"0112487","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6011228350","display_name":"ITR:  Collaborative Research:  Smoothed Analysis of Algorithms","funder_award_id":"0324914","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6723386178","display_name":null,"funder_award_id":"ITR CCR-0325630","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6894402473","display_name":null,"funder_award_id":"Fellowship","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7234352822","display_name":"The Eigenvalue Problem in Geometry and Combinatorial Optimization","funder_award_id":"9972532","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G848032724","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8835350212","display_name":"Spectral Analysis for Graph Partitioning","funder_award_id":"0311430","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306151","display_name":"Alfred P. Sloan Foundation","ror":"https://ror.org/052csg198"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W63574446","https://openalex.org/W188867022","https://openalex.org/W1523762444","https://openalex.org/W1533410687","https://openalex.org/W1576347883","https://openalex.org/W1577111082","https://openalex.org/W1760551737","https://openalex.org/W1828979180","https://openalex.org/W1974755392","https://openalex.org/W2002728800","https://openalex.org/W2031958534","https://openalex.org/W2034053794","https://openalex.org/W2041364558","https://openalex.org/W2089028133","https://openalex.org/W2100440346","https://openalex.org/W2112016415","https://openalex.org/W2115780241","https://openalex.org/W2118408279","https://openalex.org/W2120316325","https://openalex.org/W2133428703","https://openalex.org/W2140137938","https://openalex.org/W2499636426","https://openalex.org/W2555352757","https://openalex.org/W2916304084","https://openalex.org/W4289920060","https://openalex.org/W4293860921","https://openalex.org/W4300223101","https://openalex.org/W4301491118"],"related_works":["https://openalex.org/W1969557273","https://openalex.org/W2023689870","https://openalex.org/W1731719754","https://openalex.org/W2170271555","https://openalex.org/W2014666842","https://openalex.org/W2952318043","https://openalex.org/W4281713237","https://openalex.org/W4306743306","https://openalex.org/W156253123","https://openalex.org/W4308012852"],"abstract_inverted_index":{"Let":[0],"\u00c5":[1,77],"be":[2,9],"an":[3,79],"arbitrary":[4],"matrix":[5],"and":[6,98],"let":[7],"A":[8,23,39],"a":[10,25],"slight":[11],"random":[12],"perturbation":[13],"of":[14,90],"\u00c5.":[15],"We":[16],"prove":[17,33],"that":[18,22,34,38,55],"it":[19,35],"is":[20,36,73,78],"unlikely":[21,37],"has":[24,40],"large":[26,41],"condition":[27],"number.":[28],"Using":[29],"this":[30],"result,":[31],"we":[32,53],"growth":[42],"factor":[43],"under":[44],"Gaussian":[45,69,91],"elimination":[46,70,92],"without":[47,71,93],"pivoting.":[48],"By":[49],"combining":[50],"these":[51],"results,":[52],"show":[54],"the":[56,87],"smoothed":[57],"precision":[58],"necessary":[59],"to":[60],"solve":[61],"Ax":[62],"=":[63],"b,":[64,67],"for":[65],"any":[66],"using":[68],"pivoting":[72,94],"logarithmic.":[74],"Moreover,":[75],"when":[76],"all-zero":[80],"square":[81],"matrix,":[82],"our":[83],"results":[84],"significantly":[85],"improve":[86],"average-case":[88],"analysis":[89],"performed":[95],"by":[96],"Yeung":[97],"Chan":[99],"(SIAM":[100],"J.":[101],"Matrix":[102],"Anal.":[103],"Appl.,":[104],"18":[105],"(1997),":[106],"pp.":[107],"499-517).":[108]},"counts_by_year":[{"year":2025,"cited_by_count":9},{"year":2024,"cited_by_count":10},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":12},{"year":2021,"cited_by_count":9},{"year":2020,"cited_by_count":12},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":7},{"year":2017,"cited_by_count":9},{"year":2016,"cited_by_count":15},{"year":2015,"cited_by_count":18},{"year":2014,"cited_by_count":12},{"year":2013,"cited_by_count":16},{"year":2012,"cited_by_count":19}],"updated_date":"2026-04-11T08:14:18.477133","created_date":"2025-10-10T00:00:00"}