{"id":"https://openalex.org/W2490132387","doi":null,"title":"Perturbation analysis of some matrix factorizations","display_name":"Perturbation analysis of some matrix factorizations","publication_year":1997,"publication_date":"1997-01-01","ids":{"openalex":"https://openalex.org/W2490132387","mag":"2490132387"},"language":"en","primary_location":{"id":"mag:2490132387","doi":null,"is_oa":false,"landing_page_url":"http://dl.acm.org/citation.cfm?id=927553","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null},"volume":null,"type":"article","is_traditional":true,"indexed_in":[],"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/A5113669453","display_name":"Chris C. Paige","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Chris Paige","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5062087251","display_name":"Xiao-Wen Chang","orcid":"https://orcid.org/0000-0001-9251-2959","observed_orcids":["https://orcid.org/0000-0001-9251-2959"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xiao-Wen Chang","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0001-9251-2959","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.1374,"has_fulltext":false,"cited_by_count":21,"citation_normalized_percentile":{"value":0.8494311,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":87,"max":97},"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":0.9990000128746033,"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":0.9990000128746033,"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/T11697","display_name":"Numerical Methods and Algorithms","score":0.00039999998989515007,"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/T13487","display_name":"Statistical and numerical algorithms","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/perturbation-analysis","display_name":"perturbation analysis","score":0.9359999895095825},{"id":"https://openalex.org/keywords/matrix-factorization","display_name":"matrix factorization","score":0.9240000247955322},{"id":"https://openalex.org/keywords/cholesky-decomposition","display_name":"Cholesky decomposition","score":0.9129999876022339},{"id":"https://openalex.org/keywords/qr-decomposition","display_name":"QR decomposition","score":0.9990000128746033},{"id":"https://openalex.org/keywords/lu-decomposition","display_name":"LU decomposition","score":1.0},{"id":"https://openalex.org/keywords/condition-number","display_name":"condition number","score":0.8939999938011169},{"id":"https://openalex.org/keywords/pivoting-strategies","display_name":"pivoting strategies","score":0.7760000228881836},{"id":"https://openalex.org/keywords/perturbation-bounds","display_name":"perturbation bounds","score":0.6890000104904175}],"concepts":[{"id":"https://openalex.org/C34727166","wikidata":"https://www.wikidata.org/wiki/Q515375","display_name":"Cholesky decomposition","level":3,"score":0.8035156726837158},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5738754868507385},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.5593342185020447},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.5406444072723389},{"id":"https://openalex.org/C139352143","wikidata":"https://www.wikidata.org/wiki/Q82571","display_name":"Linear algebra","level":2,"score":0.5319098830223083},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5206329226493835},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.5036405920982361},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.5011587142944336},{"id":"https://openalex.org/C84545080","wikidata":"https://www.wikidata.org/wiki/Q1147936","display_name":"Condition number","level":3,"score":0.48437127470970154},{"id":"https://openalex.org/C44363057","wikidata":"https://www.wikidata.org/wiki/Q6015160","display_name":"Incomplete Cholesky factorization","level":4,"score":0.46956300735473633},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4382247030735016},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.4166214168071747},{"id":"https://openalex.org/C163834973","wikidata":"https://www.wikidata.org/wiki/Q2004891","display_name":"Numerical linear algebra","level":3,"score":0.41078051924705505},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.35058531165122986},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2845228314399719},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.2759469747543335},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.21404528617858887},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.12724557518959045},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.09825459122657776},{"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},{"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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"mag:2490132387","doi":null,"is_oa":false,"landing_page_url":"http://dl.acm.org/citation.cfm?id=927553","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":6,"referenced_works":["https://openalex.org/W1983686222","https://openalex.org/W1989314419","https://openalex.org/W2014042806","https://openalex.org/W2059084675","https://openalex.org/W2065687090","https://openalex.org/W2087399108"],"related_works":["https://openalex.org/W7070782502","https://openalex.org/W1982894432","https://openalex.org/W2047418515","https://openalex.org/W2056568837","https://openalex.org/W2065687090","https://openalex.org/W2949211572","https://openalex.org/W2113465019","https://openalex.org/W2056728100","https://openalex.org/W2059084675","https://openalex.org/W4404224119"],"abstract_inverted_index":{"Matrix":[0],"factorizations":[1,18,206],"are":[2,19,139,180],"among":[3],"the":[4,51,56,73,80,88,103,123,155,185,201,211,216,220,260],"most":[5,186],"important":[6,22,237],"and":[7,37,69,76,79,98,115,223,254],"basic":[8],"tools":[9],"in":[10,23,30,34,50,219],"numerical":[11],"linear":[12],"algebra.":[13],"Perturbation":[14],"analyses":[15,43],"of":[16,41,151,165,173,203,263],"matrix":[17,246,265],"not":[20,117],"only":[21],"their":[24],"own":[25],"right,":[26],"but":[27,142,184],"also":[28,190],"useful":[29,241],"many":[31,204],"applications,":[32],"e.g.":[33],"estimation,":[35],"control":[36],"statistics.":[38],"The":[39,163],"aim":[40],"such":[42,205],"is":[44,60,207,233,239,249],"to":[45,72,94,134,144,230,256,259],"show":[46,199],"what":[47],"effects":[48],"changes":[49],"data":[52],"will":[53],"have":[54],"on":[55,112],"factors.":[57],"This":[58,146,235],"thesis":[59],"concerned":[61],"with":[62],"developing":[63],"new":[64,86],"general":[65,252],"purpose":[66],"perturbation":[67,107,136,178,194,261],"analyses,":[68],"applying":[70],"them":[71],"Cholesky,":[74],"QR":[75],"LU":[77,221],"factorizations,":[78],"Cholesky":[81],"downdating":[82],"problem.\r\nWe":[83],"develop":[84],"a":[85,170,250],"approach,":[87,93],"so":[89,124],"called":[90,125],"'matrix-vector":[91],"equation'":[92,127],"obtain":[95],"sharp":[96],"results":[97],"true":[99,156],"condition":[100,157,202],"numbers":[101,158],"for":[102,154,182,196,242],"above":[104],"problems.":[105,175],"Our":[106],"bounds":[108,137,179,195],"give":[109,191],"significant":[110],"improvements":[111],"previous":[113],"results,":[114],"could":[116],"be":[118,257],"sharper.":[119],"Also":[120],"we":[121,189],"use":[122],"'matrix":[126],"approach":[128,147,248],"originated":[129],"by":[130,160,210],"G.":[131],"W.":[132],"Stewart":[133],"derive":[135],"that":[138,200],"usually":[140],"weaker":[141],"easier":[143],"interpret.":[145],"allows":[148],"efficient":[149],"computation":[150],"satisfactory":[152,181],"estimates":[153],"derived":[159],"our":[161],"approach.":[162],"combination":[164],"these":[166,174],"two":[167],"approaches":[168],"gives":[169],"powerful":[171,251],"understanding":[172],"Although":[176],"first-order":[177],"all":[183],"delicate":[187],"work,":[188],"some":[192,197],"rigorous":[193],"factorizations.\r\nWe":[198],"significantly":[208],"improved":[209],"standard":[212],"pivoting":[213],"strategies":[214],"(except":[215],"L":[217],"factor":[218],"factorization),":[222],"provide":[224],"firmly":[225],"based":[226],"theoretical":[227],"explanations":[228],"as":[229],"why":[231],"this":[232],"so.":[234],"extremely":[236],"information":[238],"very":[240],"designing":[243],"more":[244],"reliable":[245],"algorithms.\r\nOur":[247],"tool,":[253],"appears":[255],"applicable":[258],"analysis":[262],"any":[264],"factorization.":[266]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":3},{"year":2017,"cited_by_count":2},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":1},{"year":2012,"cited_by_count":3}],"updated_date":"2026-09-22T06:50:58.402377","created_date":"2016-08-23T00:00:00"}
