{"id":"https://openalex.org/W1983997964","doi":"https://doi.org/10.1137/12088642x","title":"Accurate Solution of Structured Least Squares Problems via Rank-Revealing Decompositions","display_name":"Accurate Solution of Structured Least Squares Problems via Rank-Revealing Decompositions","publication_year":2013,"publication_date":"2013-01-01","ids":{"openalex":"https://openalex.org/W1983997964","doi":"https://doi.org/10.1137/12088642x","mag":"1983997964"},"language":"en","primary_location":{"id":"doi:10.1137/12088642x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/12088642x","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":true,"oa_status":"green","oa_url":"http://hdl.handle.net/10016/22066","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5058307001","display_name":"N. Castro-Gonz\u00e1lez","orcid":null},"institutions":[{"id":"https://openalex.org/I88060688","display_name":"Universidad Polit\u00e9cnica de Madrid","ror":"https://ror.org/03n6nwv02","country_code":"ES","type":"education","lineage":["https://openalex.org/I88060688"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Nieves Castro-Gonz\u00e1lez","raw_affiliation_strings":["Facultad de Inform\u00e1tica, Universidad Polit\u00e9cnica de Madrid, 28660 Boadilla del Monte, Madrid, Spain"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Facultad de Inform\u00e1tica, Universidad Polit\u00e9cnica de Madrid, 28660 Boadilla del Monte, Madrid, Spain","institution_ids":["https://openalex.org/I88060688"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5004322016","display_name":"Johan Ceballos","orcid":"https://orcid.org/0000-0001-8039-8060"},"institutions":[{"id":"https://openalex.org/I50357001","display_name":"Universidad Carlos III de Madrid","ror":"https://ror.org/03ths8210","country_code":"ES","type":"education","lineage":["https://openalex.org/I50357001"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Johan Ceballos","raw_affiliation_strings":["Departamento de Matem\u00e1ticas, Universidad Carlos III de Madrid, 28911 Legan\u00e9s, Spain"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departamento de Matem\u00e1ticas, Universidad Carlos III de Madrid, 28911 Legan\u00e9s, Spain","institution_ids":["https://openalex.org/I50357001"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5076369577","display_name":"Froil\u00e1n M. Dopico","orcid":"https://orcid.org/0000-0001-7779-8960"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Froil\u00e1n M. Dopico","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5082168257","display_name":"Juan M. Molera","orcid":"https://orcid.org/0000-0002-2690-2470"},"institutions":[{"id":"https://openalex.org/I4210157032","display_name":"Institute of Mathematical Sciences","ror":"https://ror.org/05e9bn444","country_code":"ES","type":"facility","lineage":["https://openalex.org/I4210157032"]},{"id":"https://openalex.org/I50357001","display_name":"Universidad Carlos III de Madrid","ror":"https://ror.org/03ths8210","country_code":"ES","type":"education","lineage":["https://openalex.org/I50357001"]},{"id":"https://openalex.org/I63634437","display_name":"Universidad Aut\u00f3noma de Madrid","ror":"https://ror.org/01cby8j38","country_code":"ES","type":"education","lineage":["https://openalex.org/I63634437"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Juan M. Molera","raw_affiliation_strings":["Instituto de Ciencias Matem\u00e1ticas CSIC-UAM-UC3M-UCM and Departamento de Matem\u00e1ticas, Universidad Carlos III de Madrid, 28911 Legan\u00e9s, Spain","Departamento de Matem\u00e1ticas, Universidad Carlos III de Madrid, 28911 Legan\u00e9s, Spain"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Instituto de Ciencias Matem\u00e1ticas CSIC-UAM-UC3M-UCM and Departamento de Matem\u00e1ticas, Universidad Carlos III de Madrid, 28911 Legan\u00e9s, Spain","institution_ids":["https://openalex.org/I4210157032","https://openalex.org/I50357001","https://openalex.org/I63634437"]},{"raw_affiliation_string":"Departamento de Matem\u00e1ticas, Universidad Carlos III de Madrid, 28911 Legan\u00e9s, Spain","institution_ids":["https://openalex.org/I50357001"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.0971,"has_fulltext":false,"cited_by_count":21,"citation_normalized_percentile":{"value":0.93773949,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"34","issue":"3","first_page":"1112","last_page":"1128"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9997000098228455,"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":0.9997000098228455,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"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":0.9993000030517578,"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":0.787532389163971},{"id":"https://openalex.org/keywords/moore\u2013penrose-pseudoinverse","display_name":"Moore\u2013Penrose pseudoinverse","score":0.6281639337539673},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6220861077308655},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.5402623414993286},{"id":"https://openalex.org/keywords/least-squares-function-approximation","display_name":"Least-squares function approximation","score":0.5284014940261841},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4655486047267914},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.46439462900161743},{"id":"https://openalex.org/keywords/flops","display_name":"FLOPS","score":0.4632291793823242},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.4625588655471802},{"id":"https://openalex.org/keywords/low-rank-approximation","display_name":"Low-rank approximation","score":0.45683544874191284},{"id":"https://openalex.org/keywords/matrix-norm","display_name":"Matrix norm","score":0.412566214799881},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3741154074668884},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.35291406512260437},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.13383793830871582},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.1306101381778717},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12640735507011414},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10848736763000488},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.10023468732833862}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.787532389163971},{"id":"https://openalex.org/C21556879","wikidata":"https://www.wikidata.org/wiki/Q43219517","display_name":"Moore\u2013Penrose pseudoinverse","level":3,"score":0.6281639337539673},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6220861077308655},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5402623414993286},{"id":"https://openalex.org/C9936470","wikidata":"https://www.wikidata.org/wiki/Q6510405","display_name":"Least-squares function approximation","level":3,"score":0.5284014940261841},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4655486047267914},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.46439462900161743},{"id":"https://openalex.org/C3826847","wikidata":"https://www.wikidata.org/wiki/Q188768","display_name":"FLOPS","level":2,"score":0.4632291793823242},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.4625588655471802},{"id":"https://openalex.org/C90199385","wikidata":"https://www.wikidata.org/wiki/Q6692777","display_name":"Low-rank approximation","level":3,"score":0.45683544874191284},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.412566214799881},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3741154074668884},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.35291406512260437},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.13383793830871582},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.1306101381778717},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12640735507011414},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10848736763000488},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.10023468732833862},{"id":"https://openalex.org/C25023664","wikidata":"https://www.wikidata.org/wiki/Q1575637","display_name":"Hankel matrix","level":2,"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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C173608175","wikidata":"https://www.wikidata.org/wiki/Q232661","display_name":"Parallel computing","level":1,"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/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/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political 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/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/12088642x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/12088642x","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:dnet:earchivo____::0ca9919e24cd03d845429fc273765928","is_oa":true,"landing_page_url":"http://hdl.handle.net/10016/22066","pdf_url":null,"source":{"id":"https://openalex.org/S4306402641","display_name":"LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Cient\u00edficas)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4383465926","host_organization_name":"LA Referencia","host_organization_lineage":["https://openalex.org/I4383465926"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:dnet:earchivo____::0ca9919e24cd03d845429fc273765928","is_oa":true,"landing_page_url":"http://hdl.handle.net/10016/22066","pdf_url":null,"source":{"id":"https://openalex.org/S4306402641","display_name":"LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Cient\u00edficas)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4383465926","host_organization_name":"LA Referencia","host_organization_lineage":["https://openalex.org/I4383465926"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3677564837","display_name":null,"funder_award_id":"MTM-2012-32542","funder_id":"https://openalex.org/F4320321837","funder_display_name":"Ministerio de Econom\u00eda y Competitividad"}],"funders":[{"id":"https://openalex.org/F4320321837","display_name":"Ministerio de Econom\u00eda y Competitividad","ror":"https://ror.org/034900433"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":51,"referenced_works":["https://openalex.org/W5857843","https://openalex.org/W180879127","https://openalex.org/W658559791","https://openalex.org/W1558617950","https://openalex.org/W1568240140","https://openalex.org/W1594234351","https://openalex.org/W1760551737","https://openalex.org/W1965192858","https://openalex.org/W1966156785","https://openalex.org/W1984302541","https://openalex.org/W1987753018","https://openalex.org/W1989242344","https://openalex.org/W1996488362","https://openalex.org/W1996522765","https://openalex.org/W2003137566","https://openalex.org/W2003670573","https://openalex.org/W2009871339","https://openalex.org/W2010315317","https://openalex.org/W2013442644","https://openalex.org/W2020298878","https://openalex.org/W2020828929","https://openalex.org/W2021604182","https://openalex.org/W2024558657","https://openalex.org/W2039937805","https://openalex.org/W2054260755","https://openalex.org/W2054643050","https://openalex.org/W2061479838","https://openalex.org/W2068844053","https://openalex.org/W2075665712","https://openalex.org/W2077166865","https://openalex.org/W2093149052","https://openalex.org/W2099781564","https://openalex.org/W2100510181","https://openalex.org/W2109624367","https://openalex.org/W2114458102","https://openalex.org/W2117419843","https://openalex.org/W2119233169","https://openalex.org/W2130485496","https://openalex.org/W2134099700","https://openalex.org/W2142907115","https://openalex.org/W2143866207","https://openalex.org/W2150200785","https://openalex.org/W2166488918","https://openalex.org/W2329633813","https://openalex.org/W2342316458","https://openalex.org/W2798909945","https://openalex.org/W3022909171","https://openalex.org/W4233692749","https://openalex.org/W4237384699","https://openalex.org/W4244681853","https://openalex.org/W4302564868"],"related_works":["https://openalex.org/W2952655975","https://openalex.org/W2141025287","https://openalex.org/W2963310150","https://openalex.org/W1495927536","https://openalex.org/W3097005701","https://openalex.org/W4287606344","https://openalex.org/W2951291947","https://openalex.org/W2098538831","https://openalex.org/W1966777587","https://openalex.org/W1967690903"],"abstract_inverted_index":{"Least":[0],"squares":[1,76,99,154,223],"problems":[2,31,100,242],"$\\min_x":[3],"\\|b":[4],"-":[5,107],"Ax\\|_2$":[6],"where":[7,116],"the":[8,58,120,125,146,194,221,246,258],"matrix":[9],"$A\\in":[10],"\\mathbb{C}^{m\\times":[11],"n}$":[12],"($m\\geq":[13],"n$)":[14],"has":[15,174],"some":[16],"particular":[17],"structure":[18],"arise":[19],"frequently":[20],"in":[21,158,162,178,193,230,245],"applications.":[22],"Polynomial":[23],"data":[24],"fitting":[25],"is":[26,57,119,140,225],"a":[27,83,215],"well-known":[28],"instance":[29],"of":[30,61,74,93,96,124,127,136,169,199,220,248,256],"that":[32,85,173,210],"yield":[33],"highly":[34],"structured":[35,44,97,184],"matrices,":[36,186,257,268],"but":[37],"many":[38,94],"other":[39,254],"examples":[40],"exist.":[41],"Very":[42],"often,":[43],"matrices":[45,275],"have":[46],"huge":[47],"condition":[48],"numbers":[49],"$\\kappa_2":[50,128],"(A)":[51],"=":[52,113],"\\|A\\|_2":[53],"\\,":[54],"\\|A^\\dagger\\|_2$":[55],"($A^\\dagger$":[56],"Moore--Penrose":[59],"pseudoinverse":[60],"$A$)":[62],"and":[63,191,197,239,243,266,269],"therefore":[64],"standard":[65,150],"algorithms":[66,151],"fail":[67],"to":[68,88,181,208,262],"compute":[69,89],"accurate":[70,138,166,211],"minimum":[71,90],"2-norm":[72,91],"solutions":[73,92,198,212],"least":[75,98,153,222],"problems.":[77,155],"In":[78,206],"this":[79,159,231],"work,":[80],"we":[81],"introduce":[82],"framework":[84],"allows":[86],"us":[87],"classes":[95],"accurately,":[101],"i.e.,":[102,144],"with":[103,202],"errors":[104],"$\\|":[105],"\\widehat{x}_0":[106],"x_0":[108],"\\|_2":[109,112],"/":[110],"\\|x_0":[111],"O({\\tt":[114],"u})$,":[115],"${\\tt":[117],"u}$":[118],"unit":[121],"roundoff,":[122],"independently":[123],"magnitude":[126],"(A)$":[129],"for":[130,152,183,235,273],"most":[131],"vectors":[132],"$b$.":[133],"The":[134,156,227],"cost":[135,148],"these":[137],"computations":[139],"$O(n^2":[141],"m)$":[142],"flops,":[143],"roughly":[145],"same":[147],"as":[149],"approach":[157],"work":[160],"relies":[161],"computing":[163],"first":[164],"an":[165,171],"rank-revealing":[167],"decomposition":[168],"$A$,":[170],"idea":[172],"been":[175],"widely":[176],"used":[177],"recent":[179],"decades":[180],"compute,":[182],"ill-conditioned":[185],"singular":[187],"value":[188],"decompositions,":[189],"eigenvalues,":[190],"eigenvectors":[192],"Hermitian":[195],"case":[196,247],"linear":[200,250],"systems":[201,251],"high":[203],"relative":[204],"accuracy.":[205],"order":[207],"prove":[209],"are":[213,233,276],"computed,":[214],"new":[216,259],"multiplicative":[217],"perturbation":[218],"theory":[219],"problem":[224],"needed.":[226],"results":[228],"presented":[229],"paper":[232],"valid":[234],"both":[236],"full":[237],"rank":[238,240],"deficient":[241],"also":[244],"underdetermined":[249],"($m<n$).":[252],"Among":[253],"types":[255],"method":[260],"applies":[261],"rectangular":[263],"Cauchy,":[264],"Vandermonde,":[265],"graded":[267],"detailed":[270],"numerical":[271],"tests":[272],"Cauchy":[274],"presented.":[277]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
