{"id":"https://openalex.org/W4309622546","doi":"https://doi.org/10.1137/21m1451440","title":"Verified Error Bounds for All Eigenvalues and Eigenvectors of a Matrix","display_name":"Verified Error Bounds for All Eigenvalues and Eigenvectors of a Matrix","publication_year":2022,"publication_date":"2022-11-22","ids":{"openalex":"https://openalex.org/W4309622546","doi":"https://doi.org/10.1137/21m1451440"},"language":"en","primary_location":{"id":"doi:10.1137/21m1451440","is_oa":false,"landing_page_url":"https://doi.org/10.1137/21m1451440","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/A5081963598","display_name":"Siegfried M. Rump","orcid":"https://orcid.org/0000-0002-4779-4800"},"institutions":[{"id":"https://openalex.org/I150744194","display_name":"Waseda University","ror":"https://ror.org/00ntfnx83","country_code":"JP","type":"education","lineage":["https://openalex.org/I150744194"]},{"id":"https://openalex.org/I159176309","display_name":"Universit\u00e4t Hamburg","ror":"https://ror.org/00g30e956","country_code":"DE","type":"education","lineage":["https://openalex.org/I159176309"]},{"id":"https://openalex.org/I884043246","display_name":"Hamburg University of Technology","ror":"https://ror.org/04bs1pb34","country_code":"DE","type":"education","lineage":["https://openalex.org/I884043246"]}],"countries":["DE","JP"],"is_corresponding":true,"raw_author_name":"Siegfried M. Rump","raw_affiliation_strings":["Institute for Reliable Computing, Hamburg University of Technology, 21073 Hamburg, Germany, and Visiting Professor at Waseda University, Faculty of Science and Engineering, Shinjuku-ku, Tokyo 169\u20138555, Japan ()"],"raw_orcid":"https://orcid.org/0000-0002-4779-4800","affiliations":[{"raw_affiliation_string":"Institute for Reliable Computing, Hamburg University of Technology, 21073 Hamburg, Germany, and Visiting Professor at Waseda University, Faculty of Science and Engineering, Shinjuku-ku, Tokyo 169\u20138555, Japan ()","institution_ids":["https://openalex.org/I150744194","https://openalex.org/I159176309","https://openalex.org/I884043246"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5081963598"],"corresponding_institution_ids":["https://openalex.org/I150744194","https://openalex.org/I159176309","https://openalex.org/I884043246"],"apc_list":null,"apc_paid":null,"fwci":0.3017,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.60986074,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"43","issue":"4","first_page":"1736","last_page":"1754"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","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/T11697","display_name":"Numerical Methods and Algorithms","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/T11435","display_name":"Polynomial and algebraic computation","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/T10792","display_name":"Matrix Theory and Algorithms","score":0.998199999332428,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.8195928931236267},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6566505432128906},{"id":"https://openalex.org/keywords/numerical-linear-algebra","display_name":"Numerical linear algebra","score":0.5910325646400452},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.5639040470123291},{"id":"https://openalex.org/keywords/linear-algebra","display_name":"Linear algebra","score":0.5560083389282227},{"id":"https://openalex.org/keywords/invariant-subspace","display_name":"Invariant subspace","score":0.5182766318321228},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5008702278137207},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.47445425391197205},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.46738162636756897},{"id":"https://openalex.org/keywords/linear-subspace","display_name":"Linear subspace","score":0.4613495171070099},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4140508472919464},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.37842297554016113},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.300975501537323},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.1849420964717865},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12136432528495789}],"concepts":[{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.8195928931236267},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6566505432128906},{"id":"https://openalex.org/C163834973","wikidata":"https://www.wikidata.org/wiki/Q2004891","display_name":"Numerical linear algebra","level":3,"score":0.5910325646400452},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.5639040470123291},{"id":"https://openalex.org/C139352143","wikidata":"https://www.wikidata.org/wiki/Q82571","display_name":"Linear algebra","level":2,"score":0.5560083389282227},{"id":"https://openalex.org/C2777059694","wikidata":"https://www.wikidata.org/wiki/Q2706744","display_name":"Invariant subspace","level":3,"score":0.5182766318321228},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5008702278137207},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.47445425391197205},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.46738162636756897},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.4613495171070099},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4140508472919464},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.37842297554016113},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.300975501537323},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.1849420964717865},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12136432528495789},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"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/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"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":2,"locations":[{"id":"doi:10.1137/21m1451440","is_oa":false,"landing_page_url":"https://doi.org/10.1137/21m1451440","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:tore.tuhh.de:11420/14721","is_oa":false,"landing_page_url":"http://hdl.handle.net/11420/14721","pdf_url":null,"source":{"id":"https://openalex.org/S4306401751","display_name":"tub.dok (Hamburg University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I884043246","host_organization_name":"Hamburg University of Technology","host_organization_lineage":["https://openalex.org/I884043246"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Journal Article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":21,"referenced_works":["https://openalex.org/W139350644","https://openalex.org/W178772886","https://openalex.org/W364859224","https://openalex.org/W412163067","https://openalex.org/W1497910784","https://openalex.org/W1975956496","https://openalex.org/W1982605007","https://openalex.org/W2000736912","https://openalex.org/W2005734078","https://openalex.org/W2045738033","https://openalex.org/W2069092055","https://openalex.org/W2070179402","https://openalex.org/W2070381457","https://openalex.org/W2072500636","https://openalex.org/W2077030276","https://openalex.org/W2090092878","https://openalex.org/W2123022662","https://openalex.org/W2139330196","https://openalex.org/W2170345902","https://openalex.org/W2978215563","https://openalex.org/W3022136271"],"related_works":["https://openalex.org/W4235906205","https://openalex.org/W145755097","https://openalex.org/W2522245970","https://openalex.org/W3163773192","https://openalex.org/W4301478900","https://openalex.org/W4205760564","https://openalex.org/W102084567","https://openalex.org/W2963446266","https://openalex.org/W2118409924","https://openalex.org/W2948021207"],"abstract_inverted_index":{"A":[0],"verification":[1,52,72,143],"method":[2,73],"is":[3,120,158,172,210],"presented":[4],"to":[5,199],"compute":[6,200],"error":[7,32],"bounds":[8,33,180,189],"for":[9,34,92,100,109,190],"all":[10,86,203],"eigenvectors":[11],"and":[12,51,133,154,188],"eigenvalues,":[13],"including":[14],"clustered":[15],"and/or":[16],"multiple":[17],"ones":[18],"of":[19,28,42,159,162,164,177,185,202,205],"a":[20,29,43,67,71],"general,":[21],"real,":[22],"or":[23,82],"complex":[24],"matrix.":[25],"In":[26,122,141],"case":[27],"narrow":[30],"cluster,":[31],"an":[35,206],"invariant":[36,192],"subspace":[37],"are":[38,61,87,114,194],"computed":[39,59],"because":[40,116],"computation":[41],"single":[44],"eigenvector":[45],"may":[46,74,90,126,134],"be":[47],"ill-posed.":[48],"Computer":[49],"algebra":[50,69,111,130],"methods":[53,112,131,144],"have":[54],"in":[55,76,104],"common":[56],"that":[57,79,125,150,163,174],"the":[58,77,101,138,160,175,178,182,186],"results":[60],"correct":[62],"with":[63],"mathematical":[64],"certainty.":[65],"Unlike":[66],"computer":[68,110,129],"method,":[70],"fail":[75],"sense":[78],"only":[80],"partial":[81],"no":[83],"inclusions":[84,201],"at":[85],"computed.":[88,195],"That":[89,106],"happen":[91,108],"very":[93],"ill":[94],"conditioned":[95],"problems":[96],"being":[97],"too":[98],"sensitive":[99],"arithmetical":[102],"precision":[103,119],"use.":[105],"cannot":[107],"which":[113],"\u201cnever-failing\u201d":[115],"potentially":[117],"infinite":[118],"used.":[121],"turn,":[123],"however,":[124],"slow":[127],"down":[128],"significantly":[132],"impose":[135],"limitations":[136],"on":[137],"problem":[139,156,170],"size.":[140],"contrast,":[142],"solely":[145],"use":[146],"floating-point":[147],"operations":[148],"so":[149],"their":[151],"computing":[152],"time":[153],"treatable":[155],"size":[157],"order":[161],"purely":[165],"numerical":[166],"algorithms.":[167],"For":[168],"our":[169],"it":[171],"proved":[173],"union":[176],"eigenvalue":[179],"contains":[181],"whole":[183],"spectrum":[184],"matrix,":[187],"corresponding":[191],"subspaces":[193],"The":[196],"computational":[197],"complexity":[198],"eigenpairs":[204],"n":[207],"\u00d7":[208],"nmatrix":[209],"O(n3).":[211]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}