{"id":"https://openalex.org/W2078643711","doi":"https://doi.org/10.1002/nla.1680020308","title":"Optimal backward perturbation bounds for the linear least squares problem","display_name":"Optimal backward perturbation bounds for the linear least squares problem","publication_year":1995,"publication_date":"1995-05-01","ids":{"openalex":"https://openalex.org/W2078643711","doi":"https://doi.org/10.1002/nla.1680020308","mag":"2078643711"},"language":"en","primary_location":{"id":"doi:10.1002/nla.1680020308","is_oa":false,"landing_page_url":"https://doi.org/10.1002/nla.1680020308","pdf_url":null,"source":{"id":"https://openalex.org/S60324941","display_name":"Numerical Linear Algebra with Applications","issn_l":"1070-5325","issn":["1070-5325","1099-1506"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Linear Algebra with 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/A5066032886","display_name":"Bertil Wald\u00e9n","orcid":null},"institutions":[{"id":"https://openalex.org/I102134673","display_name":"Link\u00f6ping University","ror":"https://ror.org/05ynxx418","country_code":"SE","type":"education","lineage":["https://openalex.org/I102134673"]}],"countries":["SE"],"is_corresponding":true,"raw_author_name":"Bertil Wald\u00e9n","raw_affiliation_strings":["Department of Mathematics, Linko\u0308ping University, S-581 83 Linko\u0308ping, Sweden","Department of Mathematics Link\u00f6ping University S\u2010581 83 Link\u00f6ping Sweden"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Linko\u0308ping University, S-581 83 Linko\u0308ping, Sweden","institution_ids":["https://openalex.org/I102134673"]},{"raw_affiliation_string":"Department of Mathematics Link\u00f6ping University S\u2010581 83 Link\u00f6ping Sweden","institution_ids":["https://openalex.org/I102134673"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5000488953","display_name":"Rune Karlson","orcid":null},"institutions":[{"id":"https://openalex.org/I102134673","display_name":"Link\u00f6ping University","ror":"https://ror.org/05ynxx418","country_code":"SE","type":"education","lineage":["https://openalex.org/I102134673"]}],"countries":["SE"],"is_corresponding":false,"raw_author_name":"Rune Karlson","raw_affiliation_strings":["Department of Mathematics, Linko\u0308ping University, S-581 83 Linko\u0308ping, Sweden","Department of Mathematics Link\u00f6ping University S\u2010581 83 Link\u00f6ping Sweden"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Linko\u0308ping University, S-581 83 Linko\u0308ping, Sweden","institution_ids":["https://openalex.org/I102134673"]},{"raw_affiliation_string":"Department of Mathematics Link\u00f6ping University S\u2010581 83 Link\u00f6ping Sweden","institution_ids":["https://openalex.org/I102134673"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5054679759","display_name":"Jiguang Sun","orcid":"https://orcid.org/0000-0003-4105-8463"},"institutions":[{"id":"https://openalex.org/I90267481","display_name":"Ume\u00e5 University","ror":"https://ror.org/05kb8h459","country_code":"SE","type":"education","lineage":["https://openalex.org/I90267481"]}],"countries":["SE"],"is_corresponding":false,"raw_author_name":"Ji\u2010Guang Sun","raw_affiliation_strings":["Institute of Information Processing, University of Umea\u030a, S-901 87 Umea\u030a, Sweden","Institute of Information Processing, University of Ume\u00e5, S-901 87 Ume\u00e5, Sweden#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Information Processing, University of Umea\u030a, S-901 87 Umea\u030a, Sweden","institution_ids":["https://openalex.org/I90267481"]},{"raw_affiliation_string":"Institute of Information Processing, University of Ume\u00e5, S-901 87 Ume\u00e5, Sweden#TAB#","institution_ids":["https://openalex.org/I90267481"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5066032886"],"corresponding_institution_ids":["https://openalex.org/I102134673"],"apc_list":{"value":4430,"currency":"USD","value_usd":4430},"apc_paid":null,"fwci":5.5676,"has_fulltext":false,"cited_by_count":63,"citation_normalized_percentile":{"value":0.96637324,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"2","issue":"3","first_page":"271","last_page":"286"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13487","display_name":"Statistical and numerical algorithms","score":0.9998999834060669,"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"}},"topics":[{"id":"https://openalex.org/T13487","display_name":"Statistical and numerical algorithms","score":0.9998999834060669,"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"}},{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9983000159263611,"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":0.9860000014305115,"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":0.8599060773849487},{"id":"https://openalex.org/keywords/matrix-norm","display_name":"Matrix norm","score":0.6925619840621948},{"id":"https://openalex.org/keywords/perturbation","display_name":"Perturbation (astronomy)","score":0.6334893107414246},{"id":"https://openalex.org/keywords/condition-number","display_name":"Condition number","score":0.6163864135742188},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.6087010502815247},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.517707347869873},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4884721338748932},{"id":"https://openalex.org/keywords/linear-least-squares","display_name":"Linear least squares","score":0.44768619537353516},{"id":"https://openalex.org/keywords/least-squares-function-approximation","display_name":"Least-squares function approximation","score":0.4246724843978882},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.33952388167381287},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.303409218788147},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.23528903722763062},{"id":"https://openalex.org/keywords/linear-model","display_name":"Linear model","score":0.10747423768043518},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.05657115578651428},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.05237627029418945}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8599060773849487},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.6925619840621948},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.6334893107414246},{"id":"https://openalex.org/C84545080","wikidata":"https://www.wikidata.org/wiki/Q1147936","display_name":"Condition number","level":3,"score":0.6163864135742188},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.6087010502815247},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.517707347869873},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4884721338748932},{"id":"https://openalex.org/C58663186","wikidata":"https://www.wikidata.org/wiki/Q6553462","display_name":"Linear least squares","level":3,"score":0.44768619537353516},{"id":"https://openalex.org/C9936470","wikidata":"https://www.wikidata.org/wiki/Q6510405","display_name":"Least-squares function approximation","level":3,"score":0.4246724843978882},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.33952388167381287},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.303409218788147},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.23528903722763062},{"id":"https://openalex.org/C163175372","wikidata":"https://www.wikidata.org/wiki/Q3339222","display_name":"Linear model","level":2,"score":0.10747423768043518},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.05657115578651428},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.05237627029418945},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"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/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1002/nla.1680020308","is_oa":false,"landing_page_url":"https://doi.org/10.1002/nla.1680020308","pdf_url":null,"source":{"id":"https://openalex.org/S60324941","display_name":"Numerical Linear Algebra with Applications","issn_l":"1070-5325","issn":["1070-5325","1099-1506"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Linear Algebra with Applications","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.57.1526","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.57.1526","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://math.liu.se/~bewal/pertnew8.ps","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320322581","display_name":"Vetenskapsr\u00e5det","ror":"https://ror.org/03zttf063"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":5,"referenced_works":["https://openalex.org/W54390761","https://openalex.org/W83404773","https://openalex.org/W204634507","https://openalex.org/W658559791","https://openalex.org/W2026927759"],"related_works":["https://openalex.org/W2978021564","https://openalex.org/W2057555084","https://openalex.org/W2888367694","https://openalex.org/W1762420609","https://openalex.org/W1634450618","https://openalex.org/W2493516920","https://openalex.org/W2077855410","https://openalex.org/W2090329912","https://openalex.org/W3022039156","https://openalex.org/W2078643711"],"abstract_inverted_index":{"Abstract":[0],"Let":[1],"A":[2,44,56],"be":[3,10,16],"an":[4,11],"m":[5,12],"\u00d7":[6],"n":[7],"matrix,":[8],"b":[9,26,53],"\u2010vector,":[13],"and":[14,86,91],"x\u0303":[15,49],"a":[17],"purported":[18],"solution":[19],"to":[20],"the":[21,34,39,47,74,79,92],"problem":[22,65],"of":[23,43,82],"minimizing":[24],"\u2016":[25,29,52,61],"\u2014":[27,54],"Ax":[28],"2":[30,62],".":[31,63],"We":[32],"consider":[33],"following":[35],"open":[36],"problem:":[37],"find":[38],"smallest":[40],"perturbation":[41],"E":[42,58,70,83],"such":[45],"that":[46],"vector":[48],"exactly":[50],"minimizes":[51],"(":[55],"+":[57],")":[59],"x":[60],"This":[64],"is":[66,71,94],"completely":[67],"solved":[68],"when":[69],"measured":[72],"in":[73],"Frobenius":[75],"norm.":[76],"When":[77],"using":[78],"spectral":[80],"norm":[81],",":[84],"upper":[85],"lower":[87],"bounds":[88],"are":[89],"given,":[90],"optimum":[93],"found":[95],"under":[96],"certain":[97],"conditions.":[98]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":2}],"updated_date":"2026-05-28T09:10:13.091523","created_date":"2025-10-10T00:00:00"}