{"id":"https://openalex.org/W2040466831","doi":"https://doi.org/10.1137/0613066","title":"Estimating the Largest Eigenvalue by the Power and Lanczos Algorithms with a Random Start","display_name":"Estimating the Largest Eigenvalue by the Power and Lanczos Algorithms with a Random Start","publication_year":1992,"publication_date":"1992-10-01","ids":{"openalex":"https://openalex.org/W2040466831","doi":"https://doi.org/10.1137/0613066","mag":"2040466831"},"language":"en","primary_location":{"id":"doi:10.1137/0613066","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0613066","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","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.7916/d8b56svv","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5023600398","display_name":"J. Kuczy\u0144ski","orcid":null},"institutions":[{"id":"https://openalex.org/I78577930","display_name":"Columbia University","ror":"https://ror.org/00hj8s172","country_code":"US","type":"education","lineage":["https://openalex.org/I78577930"]},{"id":"https://openalex.org/I4210152232","display_name":"Institute of Informatics of the Slovak Academy of Sciences","ror":"https://ror.org/04jgqpc26","country_code":"SK","type":"facility","lineage":["https://openalex.org/I207624831","https://openalex.org/I4210152232"]},{"id":"https://openalex.org/I4654613","display_name":"University of Warsaw","ror":"https://ror.org/039bjqg32","country_code":"PL","type":"education","lineage":["https://openalex.org/I4654613"]}],"countries":["PL","SK","US"],"is_corresponding":true,"raw_author_name":"J. Kuczy\u0144ski","raw_affiliation_strings":["Department of Computer Science, Columbia University and Institute of Informatics, University of Warsaw"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science, Columbia University and Institute of Informatics, University of Warsaw","institution_ids":["https://openalex.org/I78577930","https://openalex.org/I4654613","https://openalex.org/I4210152232"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5113538000","display_name":"H. Wo\u017aniakowski","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"H. Wo\u017aniakowski","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5023600398"],"corresponding_institution_ids":["https://openalex.org/I4210152232","https://openalex.org/I4654613","https://openalex.org/I78577930"],"apc_list":null,"apc_paid":null,"fwci":4.8056,"has_fulltext":false,"cited_by_count":266,"citation_normalized_percentile":{"value":0.95614245,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":99,"max":100},"biblio":{"volume":"13","issue":"4","first_page":"1094","last_page":"1122"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9984999895095825,"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":0.9984999895095825,"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9962000250816345,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T11716","display_name":"Random Matrices and Applications","score":0.9954000115394592,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/mathematics","display_name":"Mathematics","score":0.7665772438049316},{"id":"https://openalex.org/keywords/unit-vector","display_name":"Unit vector","score":0.583770751953125},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.564077615737915},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5496776700019836},{"id":"https://openalex.org/keywords/lanczos-resampling","display_name":"Lanczos resampling","score":0.5091628432273865},{"id":"https://openalex.org/keywords/randomized-algorithm","display_name":"Randomized algorithm","score":0.49931907653808594},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.49566876888275146},{"id":"https://openalex.org/keywords/lanczos-algorithm","display_name":"Lanczos algorithm","score":0.47166430950164795},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.467478483915329},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.46414437890052795},{"id":"https://openalex.org/keywords/multivariate-random-variable","display_name":"Multivariate random variable","score":0.45497867465019226},{"id":"https://openalex.org/keywords/probabilistic-logic","display_name":"Probabilistic logic","score":0.4246058464050293},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3373553156852722},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.18204990029335022},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1392422318458557},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.10548719763755798},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.07019069790840149}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7665772438049316},{"id":"https://openalex.org/C84316537","wikidata":"https://www.wikidata.org/wiki/Q36255","display_name":"Unit vector","level":2,"score":0.583770751953125},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.564077615737915},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5496776700019836},{"id":"https://openalex.org/C119256216","wikidata":"https://www.wikidata.org/wiki/Q913012","display_name":"Lanczos resampling","level":3,"score":0.5091628432273865},{"id":"https://openalex.org/C128669082","wikidata":"https://www.wikidata.org/wiki/Q583461","display_name":"Randomized algorithm","level":2,"score":0.49931907653808594},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.49566876888275146},{"id":"https://openalex.org/C20501136","wikidata":"https://www.wikidata.org/wiki/Q366640","display_name":"Lanczos algorithm","level":4,"score":0.47166430950164795},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.467478483915329},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.46414437890052795},{"id":"https://openalex.org/C138405894","wikidata":"https://www.wikidata.org/wiki/Q3179949","display_name":"Multivariate random variable","level":3,"score":0.45497867465019226},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.4246058464050293},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3373553156852722},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.18204990029335022},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1392422318458557},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.10548719763755798},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07019069790840149},{"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}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/0613066","is_oa":false,"landing_page_url":"https://doi.org/10.1137/0613066","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:10.1.1.29.4725","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.4725","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.columbia.edu/~library/TR-repository/reports/reports-1992/cucs-026-92.ps.gz","raw_type":"text"},{"id":"pmh:oai:academiccommons.columbia.edu:10.7916/D8B56SVV","is_oa":false,"landing_page_url":"https://doi.org/10.7916/D8B56SVV","pdf_url":null,"source":{"id":"https://openalex.org/S4306402601","display_name":"Columbia Academic Commons (Columbia University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I78577930","host_organization_name":"Columbia University","host_organization_lineage":["https://openalex.org/I78577930"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Reports"},{"id":"doi:10.7916/d8b56svv","is_oa":true,"landing_page_url":"https://doi.org/10.7916/d8b56svv","pdf_url":null,"source":{"id":"https://openalex.org/S4306402601","display_name":"Columbia Academic Commons (Columbia University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I78577930","host_organization_name":"Columbia University","host_organization_lineage":["https://openalex.org/I78577930"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.7916/d8b56svv","is_oa":true,"landing_page_url":"https://doi.org/10.7916/d8b56svv","pdf_url":null,"source":{"id":"https://openalex.org/S4306402601","display_name":"Columbia Academic Commons (Columbia University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I78577930","host_organization_name":"Columbia University","host_organization_lineage":["https://openalex.org/I78577930"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article-journal"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":24,"referenced_works":["https://openalex.org/W141418607","https://openalex.org/W1599373951","https://openalex.org/W1965612527","https://openalex.org/W1976798627","https://openalex.org/W1982573289","https://openalex.org/W2009737642","https://openalex.org/W2012680203","https://openalex.org/W2012728721","https://openalex.org/W2013737143","https://openalex.org/W2022618764","https://openalex.org/W2059586807","https://openalex.org/W2060647548","https://openalex.org/W2065706569","https://openalex.org/W2074320389","https://openalex.org/W2075488055","https://openalex.org/W2569481888","https://openalex.org/W3021898623","https://openalex.org/W3022074680","https://openalex.org/W3022136271","https://openalex.org/W4230941884","https://openalex.org/W4251656402","https://openalex.org/W4255783786","https://openalex.org/W4285719527","https://openalex.org/W4301435544"],"related_works":["https://openalex.org/W1970812947","https://openalex.org/W2156009096","https://openalex.org/W3176451595","https://openalex.org/W2027838537","https://openalex.org/W2058241770","https://openalex.org/W1980579360","https://openalex.org/W4248073715","https://openalex.org/W2095282848","https://openalex.org/W2949972855","https://openalex.org/W3046472007"],"abstract_inverted_index":{"This":[0],"paper":[1],"addresses":[2],"the":[3,10,53,60,80,89,93,97,103,115,123,130,136,143,162,167,185,190,194,197,210,213,220,227,244,262,283,289,301,311,317,320,330,335,338,341,346,349,352],"problem":[4,61,98],"of":[5,13,41,166,189,212,215,273,288,303,319,340,348],"computing":[6],"an":[7,14,181],"approximation":[8],"to":[9,88,232],"largest":[11,322,343,350],"eigenvalue":[12],"$n":[15],"\\times":[16],"n$":[17],"large":[18],"symmetric":[19],"positive":[20],"definite":[21],"matrix":[22,295],"with":[23,86],"relative":[24,132,138,164,187,204,229],"error":[25,133,165,188],"at":[26,248],"most":[27,249],"$\\varepsilon":[28],"$.":[29],"Only":[30],"algorithms":[31,119,291],"that":[32,226],"use":[33],"Krylov":[34],"information":[35],"$[b,Ab,":[36],"\\cdots":[37],",A^k":[38],"b]$":[39],"consisting":[40],"k":[42,159,177,306],"matrix-vector":[43,69,304],"multiplications":[44,70,305],"for":[45,109,218,243,261,268,292,310,329],"some":[46],"unit":[47,94],"vector":[48,54,81,113],"b":[49,55,82,217],"are":[50,71,120,140,149,267],"considered.":[51],"If":[52],"is":[56,76,83,180,199,207,224,297],"chosen":[57,84,112],"deterministically,":[58],"then":[59,96,315],"cannot":[62],"be":[63,100],"solved":[64,101],"no":[65],"matter":[66],"how":[67],"many":[68],"performed":[72],"and":[73,105,117,134,247,285,324,345,351],"what":[74],"algorithm":[75,125,145,198,221,246,313,332],"used.":[77],"If,":[78],"however,":[79],"randomly":[85,111],"respect":[87],"uniform":[90],"distribution":[91,272],"over":[92],"sphere,":[95],"can":[99],"on":[102,129,135,184,278,316,334],"average":[104,131,163,186,284],"probabilistically.":[106],"More":[107],"precisely,":[108],"a":[110,269,293],"b,":[114],"power":[116,124,168,245,312],"Lanczos":[118,144,191,263,331],"studied.":[121],"For":[122,142],"(method),":[126],"sharp":[127],"bounds":[128,148,266,309,328],"probabilistic":[137,195,203,228,286],"failure":[139,230],"proven.":[141],"only":[146],"upper":[147,182],"presented.":[150],"In":[151,193],"particular,":[152],"$\\ln":[153],"(":[154,172,174],"n":[155,157,175,237,252],")/(":[156],")":[158],"$":[160,242,260],"characterizes":[161],"algorithm,":[169],"whereas":[170],"$O(":[171],"\\ln":[173],")/":[176],")^2":[178],")$":[179],"bound":[183],"algorithm.":[192,264],"case,":[196],"characterized":[200],"by":[201],"its":[202],"failure,":[205],"which":[206,219,275],"defined":[208],"as":[209,235,250,300],"measure":[211],"set":[214],"vectors":[216],"fails.":[222],"It":[223],"shown":[225],"goes":[231],"zero":[233],"roughly":[234],"$\\sqrt":[236,251],"(1":[238],"-":[239,254,256],"\\varepsilon":[240,258],")^k":[241],"e^{":[253],"(2k":[255],"1)\\sqrt":[257],"}":[259],"These":[265],"worst":[270],"case":[271],"eigenvalues":[274,323,344],"may":[276],"depend":[277,314,333],"k.":[279],"The":[280,308,327],"behavior":[281],"in":[282],"cases":[287],"two":[290,321,342],"fixed":[294],"A":[296],"also":[298],"studied":[299],"number":[302],"increases.":[307],"ratio":[318,336],"their":[325],"multiplicities.":[326],"between":[337],"difference":[339,347],"smallest":[353],"eigenvalues.":[354]},"counts_by_year":[{"year":2025,"cited_by_count":13},{"year":2024,"cited_by_count":12},{"year":2023,"cited_by_count":11},{"year":2022,"cited_by_count":8},{"year":2021,"cited_by_count":24},{"year":2020,"cited_by_count":25},{"year":2019,"cited_by_count":18},{"year":2018,"cited_by_count":16},{"year":2017,"cited_by_count":13},{"year":2016,"cited_by_count":11},{"year":2015,"cited_by_count":14},{"year":2014,"cited_by_count":11},{"year":2013,"cited_by_count":8},{"year":2012,"cited_by_count":12}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
