{"id":"https://openalex.org/W2075471447","doi":"https://doi.org/10.1002/(sici)1099-1506(199803/04)5:2<79::aid-nla126>3.0.co;2-4","title":"Approximating minimum norm solutions of rank-deficient least squares problems","display_name":"Approximating minimum norm solutions of rank-deficient least squares problems","publication_year":1998,"publication_date":"1998-03-01","ids":{"openalex":"https://openalex.org/W2075471447","doi":"https://doi.org/10.1002/(sici)1099-1506(199803/04)5:2<79::aid-nla126>3.0.co;2-4","mag":"2075471447"},"language":"en","primary_location":{"id":"doi:10.1002/(sici)1099-1506(199803/04)5:2<79::aid-nla126>3.0.co;2-4","is_oa":false,"landing_page_url":"https://doi.org/10.1002/(sici)1099-1506(199803/04)5:2<79::aid-nla126>3.0.co;2-4","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/A5009299181","display_name":"Achiya Dax","orcid":"https://orcid.org/0000-0002-6403-1020"},"institutions":[{"id":"https://openalex.org/I4210100416","display_name":"Israel Water Authority","ror":"https://ror.org/013zszd91","country_code":"IL","type":"government","lineage":["https://openalex.org/I4210100416"]}],"countries":["IL"],"is_corresponding":true,"raw_author_name":"Achiya Dax","raw_affiliation_strings":["Hydrological Service, P.O.B. 6381, Jerusalem 91063, Israel"],"affiliations":[{"raw_affiliation_string":"Hydrological Service, P.O.B. 6381, Jerusalem 91063, Israel","institution_ids":["https://openalex.org/I4210100416"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5076902388","display_name":"Lars Eld\u00e9n","orcid":"https://orcid.org/0000-0003-2281-856X"},"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":"Lars Eld\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"],"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"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5009299181","https://openalex.org/A5076902388"],"corresponding_institution_ids":["https://openalex.org/I102134673","https://openalex.org/I4210100416"],"apc_list":{"value":4430,"currency":"USD","value_usd":4430},"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.20572373,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"5","issue":"2","first_page":"79","last_page":"99"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9994999766349792,"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":0.9994999766349792,"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/T11205","display_name":"Numerical methods in inverse problems","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9973999857902527,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.9132173657417297},{"id":"https://openalex.org/keywords/singular-value","display_name":"Singular value","score":0.660244882106781},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.5785567760467529},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.5583505034446716},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5311274528503418},{"id":"https://openalex.org/keywords/linear-least-squares","display_name":"Linear least squares","score":0.4735504686832428},{"id":"https://openalex.org/keywords/matrix-norm","display_name":"Matrix norm","score":0.46571090817451477},{"id":"https://openalex.org/keywords/low-rank-approximation","display_name":"Low-rank approximation","score":0.45420002937316895},{"id":"https://openalex.org/keywords/least-squares-function-approximation","display_name":"Least-squares function approximation","score":0.42471277713775635},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4155951738357544},{"id":"https://openalex.org/keywords/singular-value-decomposition","display_name":"Singular value decomposition","score":0.4101675748825073},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.25251343846321106},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.20272433757781982},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.1486676037311554},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.09572070837020874},{"id":"https://openalex.org/keywords/hankel-matrix","display_name":"Hankel matrix","score":0.09138056635856628},{"id":"https://openalex.org/keywords/estimator","display_name":"Estimator","score":0.0891319215297699}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9132173657417297},{"id":"https://openalex.org/C109282560","wikidata":"https://www.wikidata.org/wiki/Q4166054","display_name":"Singular value","level":3,"score":0.660244882106781},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5785567760467529},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.5583505034446716},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5311274528503418},{"id":"https://openalex.org/C58663186","wikidata":"https://www.wikidata.org/wiki/Q6553462","display_name":"Linear least squares","level":3,"score":0.4735504686832428},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.46571090817451477},{"id":"https://openalex.org/C90199385","wikidata":"https://www.wikidata.org/wiki/Q6692777","display_name":"Low-rank approximation","level":3,"score":0.45420002937316895},{"id":"https://openalex.org/C9936470","wikidata":"https://www.wikidata.org/wiki/Q6510405","display_name":"Least-squares function approximation","level":3,"score":0.42471277713775635},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4155951738357544},{"id":"https://openalex.org/C22789450","wikidata":"https://www.wikidata.org/wiki/Q420904","display_name":"Singular value decomposition","level":2,"score":0.4101675748825073},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.25251343846321106},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.20272433757781982},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.1486676037311554},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.09572070837020874},{"id":"https://openalex.org/C25023664","wikidata":"https://www.wikidata.org/wiki/Q1575637","display_name":"Hankel matrix","level":2,"score":0.09138056635856628},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.0891319215297699},{"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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"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":1,"locations":[{"id":"doi:10.1002/(sici)1099-1506(199803/04)5:2<79::aid-nla126>3.0.co;2-4","is_oa":false,"landing_page_url":"https://doi.org/10.1002/(sici)1099-1506(199803/04)5:2<79::aid-nla126>3.0.co;2-4","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":33,"referenced_works":["https://openalex.org/W109696764","https://openalex.org/W1564994953","https://openalex.org/W1953936588","https://openalex.org/W1968735445","https://openalex.org/W1973082942","https://openalex.org/W2005140191","https://openalex.org/W2005423095","https://openalex.org/W2006912744","https://openalex.org/W2013737143","https://openalex.org/W2016182223","https://openalex.org/W2017387343","https://openalex.org/W2017439015","https://openalex.org/W2031604650","https://openalex.org/W2032325054","https://openalex.org/W2038497950","https://openalex.org/W2050001231","https://openalex.org/W2066043610","https://openalex.org/W2075533391","https://openalex.org/W2077166865","https://openalex.org/W2078841894","https://openalex.org/W2105111377","https://openalex.org/W2117026915","https://openalex.org/W2135779729","https://openalex.org/W2149846618","https://openalex.org/W2160775101","https://openalex.org/W2333836441","https://openalex.org/W3022136271","https://openalex.org/W4237219696","https://openalex.org/W4244968939","https://openalex.org/W4249851362","https://openalex.org/W6653932534","https://openalex.org/W6669466037","https://openalex.org/W6750968397"],"related_works":["https://openalex.org/W4319586039","https://openalex.org/W4382583540","https://openalex.org/W4094001","https://openalex.org/W4298234822","https://openalex.org/W2782047334","https://openalex.org/W2163255469","https://openalex.org/W1970576574","https://openalex.org/W3017414697","https://openalex.org/W2522507432","https://openalex.org/W605362495"],"abstract_inverted_index":{"In":[0],"this":[1],"paper":[2],"we":[3,167],"consider":[4],"the":[5,16,39,50,58,63,68,80,101,129,143,150,155,169,174,184,210,219,224,232,258,266],"solution":[6,61,104],"of":[7,42,62,82,121,125,141,145,176,205,235,249,260],"linear":[8],"least":[9,64,252],"squares":[10,65,253],"problems":[11],"minx\u2225Ax":[12],"-":[13,93],"b\u222522":[14,94],"where":[15,75],"matrix":[17],"A":[18,138,146],"\u2208":[19],"Rm":[20],"\u00d7":[21],"n":[22],"is":[23,79,107,114,147,208,228],"rank":[24,81,212],"deficient.":[25],"Put":[26],"p":[27,78],"=":[28,34,71,73,90],"min{m,":[29],"n},":[30],"let":[31,45],"\u03c3i,":[32],"i":[33],"1,":[35],"2,\u2026,":[36],"p,":[37],"denote":[38,49],"singular":[40,55,151,220,233],"values":[41,152],"A,":[43],"and":[44,47,53,154,173,241,263],"ui":[46],"vi":[48],"corresponding":[51],"left":[52],"right":[54],"vectors.":[56],"Then":[57],"minimum":[59,102,246],"norm":[60,103,247],"problem":[66],"has":[67],"form":[69],"x*":[70,186,227],"\u222bri":[72],"1(uTib/\u03c3i)vi,":[74],"r":[76],"\u2264":[77],"A.":[83,126,236],"The":[84,112],"Riley\u2013Golub":[85],"iteration,":[86],"xk":[87,188],"+":[88,95],"1":[89],"arg":[91],"minx{\u2225Ax":[92],"\u03bb\u2225x":[96],"\u2212":[97,187],"xk\u222522}":[98],"converges":[99],"to":[100,110,191],"if":[105],"x0":[106],"chosen":[108],"equal":[109],"zero.":[111],"iteration":[113,130],"implemented":[115],"so":[116],"that":[117,148,182,209,265],"it":[118],"takes":[119],"advantage":[120,204],"a":[122,180,192,239,278],"bidiagonal":[123],"decomposition":[124,280],"Thus":[127],"modified,":[128],"requires":[131],"only":[132],"O(p)":[133],"flops":[134],"(floating":[135],"point":[136],"operations).":[137],"further":[139],"gain":[140],"using":[142,218],"bidiagonalization":[144],"both":[149],"\u03c3i":[153],"scalar":[156],"products":[157],"uTib":[158],"can":[159,213],"be":[160,214],"computed":[161],"at":[162],"marginal":[163],"extra":[164],"cost.":[165],"Moreover,":[166],"determine":[168],"regularization":[170],"parameter,":[171],"\u03bb,":[172],"number":[175],"iterations,":[177],"k,":[178],"in":[179],"way":[181],"minimizes":[183],"difference":[185],"with":[189,281],"respect":[190],"certain":[193],"norm.":[194],"Explicit":[195],"rules":[196],"are":[197],"derived":[198],"for":[199,244],"calculating":[200],"these":[201],"parameters.":[202],"One":[203],"our":[206,261],"approach":[207,275],"numerical":[211],"easily":[215],"determined":[216],"by":[217,223],"values.":[221],"Furthermore,":[222],"iterative":[225],"procedure,":[226],"approximated":[229],"without":[230],"computing":[231],"vectors":[234],"This":[237],"gives":[238,269],"fast":[240],"reliable":[242],"method":[243,268],"approximating":[245],"solutions":[248],"well-conditioned":[250],"rank-deficient":[251],"problems.":[254],"Numerical":[255],"experiments":[256],"illustrate":[257],"viability":[259],"ideas,":[262],"demonstrate":[264],"new":[267],"more":[270],"accurate":[271],"approximations":[272],"than":[273],"an":[274],"based":[276],"on":[277],"QR":[279],"column":[282],"pivoting.":[283],"\u00a9":[284],"1998":[285],"John":[286],"Wiley":[287],"&":[288],"Sons,":[289],"Ltd.":[290]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2015,"cited_by_count":1}],"updated_date":"2026-04-09T08:11:56.329763","created_date":"2025-10-10T00:00:00"}