{"id":"https://openalex.org/W1987449469","doi":"https://doi.org/10.1137/140995702","title":"Relative Perturbation Analysis for Eigenvalues and Singular Values of Totally Nonpositive Matrices","display_name":"Relative Perturbation Analysis for Eigenvalues and Singular Values of Totally Nonpositive Matrices","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W1987449469","doi":"https://doi.org/10.1137/140995702","mag":"1987449469"},"language":"en","primary_location":{"id":"doi:10.1137/140995702","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140995702","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/A5103115805","display_name":"Rong Huang","orcid":"https://orcid.org/0000-0002-4756-0182"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Rong Huang","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5102966756","display_name":"Delin Chu","orcid":"https://orcid.org/0000-0001-7736-7490"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Delin Chu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.3376,"has_fulltext":false,"cited_by_count":22,"citation_normalized_percentile":{"value":0.81852763,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"36","issue":"2","first_page":"476","last_page":"495"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":1.0,"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":1.0,"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/T10046","display_name":"Stability and Control of Uncertain Systems","score":0.9794999957084656,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9786999821662903,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8871910572052002},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.8004881143569946},{"id":"https://openalex.org/keywords/invertible-matrix","display_name":"Invertible matrix","score":0.7660846710205078},{"id":"https://openalex.org/keywords/singular-value","display_name":"Singular value","score":0.7392823696136475},{"id":"https://openalex.org/keywords/perturbation","display_name":"Perturbation (astronomy)","score":0.5834947228431702},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5398740768432617},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5152507424354553},{"id":"https://openalex.org/keywords/singular-perturbation","display_name":"Singular perturbation","score":0.506020724773407},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.45692941546440125},{"id":"https://openalex.org/keywords/normal-matrix","display_name":"Normal matrix","score":0.45677465200424194},{"id":"https://openalex.org/keywords/condition-number","display_name":"Condition number","score":0.4457821249961853},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.43691107630729675},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.42619192600250244},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.33088111877441406},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.05587896704673767}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8871910572052002},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.8004881143569946},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.7660846710205078},{"id":"https://openalex.org/C109282560","wikidata":"https://www.wikidata.org/wiki/Q4166054","display_name":"Singular value","level":3,"score":0.7392823696136475},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.5834947228431702},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5398740768432617},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5152507424354553},{"id":"https://openalex.org/C165160513","wikidata":"https://www.wikidata.org/wiki/Q7524249","display_name":"Singular perturbation","level":2,"score":0.506020724773407},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.45692941546440125},{"id":"https://openalex.org/C175380299","wikidata":"https://www.wikidata.org/wiki/Q6852339","display_name":"Normal matrix","level":3,"score":0.45677465200424194},{"id":"https://openalex.org/C84545080","wikidata":"https://www.wikidata.org/wiki/Q1147936","display_name":"Condition number","level":3,"score":0.4457821249961853},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.43691107630729675},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.42619192600250244},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.33088111877441406},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.05587896704673767},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","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},{"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":1,"locations":[{"id":"doi:10.1137/140995702","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140995702","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W1966156785","https://openalex.org/W1975899256","https://openalex.org/W1981463369","https://openalex.org/W1984921424","https://openalex.org/W1987744915","https://openalex.org/W1989660476","https://openalex.org/W1990886712","https://openalex.org/W2004973911","https://openalex.org/W2005928200","https://openalex.org/W2020905775","https://openalex.org/W2024379026","https://openalex.org/W2035693766","https://openalex.org/W2053270331","https://openalex.org/W2054260755","https://openalex.org/W2061521346","https://openalex.org/W2062035226","https://openalex.org/W2062949261","https://openalex.org/W2067475861","https://openalex.org/W2071653647","https://openalex.org/W2094302535","https://openalex.org/W2099781564","https://openalex.org/W2100510181","https://openalex.org/W2127877445","https://openalex.org/W2138193981","https://openalex.org/W2142907115","https://openalex.org/W2158200065"],"related_works":["https://openalex.org/W2409943374","https://openalex.org/W2038812594","https://openalex.org/W98194072","https://openalex.org/W2081802535","https://openalex.org/W2068844053","https://openalex.org/W1979976672","https://openalex.org/W2161439117","https://openalex.org/W1987449469","https://openalex.org/W2017577338","https://openalex.org/W4240972162"],"abstract_inverted_index":{"A":[0],"real":[1],"square":[2],"matrix":[3,65],"is":[4,66,85],"said":[5],"to":[6],"be":[7],"totally":[8,32,51],"nonpositive":[9,33,52],"if":[10,62],"all":[11,89],"of":[12,31,49,54,74,99,110,118,129],"its":[13],"minors":[14],"are":[15,24,116],"nonpositive.":[16],"In":[17],"this":[18],"paper,":[19],"strong":[20],"relative":[21,79,97],"perturbation":[22],"bounds":[23],"developed":[25],"for":[26],"eigenvalues":[27,91],"and":[28,92,112,123],"singular":[29,93],"values":[30,94],"matrices.":[34],"We":[35,58],"first":[36],"show":[37,60],"that":[38,45,61,72],"there":[39],"exist":[40],"exactly":[41],"$n^{2}$":[42],"independent":[43,109,117],"variables":[44],"parameterize":[46],"the":[47,90,103,107,127,130],"set":[48],"nonsingular":[50],"matrices":[53],"size":[55,128],"$n\\times":[56],"n$.":[57],"then":[59,88],"such":[63,69],"a":[64,70,78],"perturbed":[67],"in":[68,106],"way":[71],"each":[73],"these":[75,114],"parameters":[76],"has":[77],"error":[80],"bounded":[81],"by":[82],"$\\epsilon$":[83],"which":[84],"small":[86,96],"enough,":[87],"have":[95],"errors":[98,115],"order":[100],"$O(n^{3}\\epsilon)$,":[101],"with":[102,126],"constant":[104],"involved":[105],"big-$O$":[108],"$n$,":[111],"therefore,":[113],"any":[119],"conventional":[120],"condition":[121],"number":[122],"increase":[124],"polynomially":[125],"problem.":[131]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":3},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":2},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":2}],"updated_date":"2026-07-18T07:39:51.176621","created_date":"2025-10-10T00:00:00"}