{"id":"https://openalex.org/W2015320521","doi":"https://doi.org/10.1137/s0895479803428795","title":"The Generalized Eigenvalue Problem for Nonsquare Pencils Using a Minimal Perturbation Approach","display_name":"The Generalized Eigenvalue Problem for Nonsquare Pencils Using a Minimal Perturbation Approach","publication_year":2005,"publication_date":"2005-01-01","ids":{"openalex":"https://openalex.org/W2015320521","doi":"https://doi.org/10.1137/s0895479803428795","mag":"2015320521"},"language":"en","primary_location":{"id":"doi:10.1137/s0895479803428795","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479803428795","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/A5070671573","display_name":"Gregory Boutry","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Gregory Boutry","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5020279598","display_name":"Michael Elad","orcid":"https://orcid.org/0000-0001-8131-6928"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Michael Elad","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5110003399","display_name":"Gene H. Golub","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Gene H. Golub","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5002085979","display_name":"Peyman Milanfar","orcid":"https://orcid.org/0000-0003-1455-7662"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Peyman Milanfar","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5070671573"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.3762,"has_fulltext":false,"cited_by_count":89,"citation_normalized_percentile":{"value":0.94754271,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"27","issue":"2","first_page":"582","last_page":"601"},"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9900000095367432,"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"}},{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.9771000146865845,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.8774445652961731},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8155291080474854},{"id":"https://openalex.org/keywords/lambda","display_name":"Lambda","score":0.6701092720031738},{"id":"https://openalex.org/keywords/factorization","display_name":"Factorization","score":0.6504724621772766},{"id":"https://openalex.org/keywords/perturbation","display_name":"Perturbation (astronomy)","score":0.584839940071106},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.533224880695343},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4361236095428467},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.34154003858566284},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3252184987068176},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.32204940915107727},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.17441466450691223},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06098741292953491}],"concepts":[{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.8774445652961731},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8155291080474854},{"id":"https://openalex.org/C2778113609","wikidata":"https://www.wikidata.org/wiki/Q10897","display_name":"Lambda","level":2,"score":0.6701092720031738},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.6504724621772766},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.584839940071106},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.533224880695343},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4361236095428467},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.34154003858566284},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3252184987068176},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.32204940915107727},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.17441466450691223},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06098741292953491},{"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/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1137/s0895479803428795","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479803428795","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:CiteSeerX.psu:10.1.1.10.9068","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.10.9068","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.technion.ac.il/~elad/Journals/17_GEP_SIMAX.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.14.2535","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.14.2535","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.soe.ucsc.edu/~milanfar/GEP.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.4.9172","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.4.9172","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www-sccm.stanford.edu/pub/sccm/sccm03-14.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.58.4660","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.58.4660","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cse.ucsc.edu/~milanfar/GEP_Updated.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.79.4323","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.79.4323","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cse.ucsc.edu/~milanfar/GEP_Final.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W1522512418","https://openalex.org/W1524677621","https://openalex.org/W1573181415","https://openalex.org/W1965683017","https://openalex.org/W1981745143","https://openalex.org/W1985852691","https://openalex.org/W1993068751","https://openalex.org/W2008256634","https://openalex.org/W2015224767","https://openalex.org/W2019609727","https://openalex.org/W2032604276","https://openalex.org/W2039333245","https://openalex.org/W2045896870","https://openalex.org/W2053980552","https://openalex.org/W2056509440","https://openalex.org/W2069400331","https://openalex.org/W2075690086","https://openalex.org/W2079854271","https://openalex.org/W2106533107","https://openalex.org/W2122860281","https://openalex.org/W2128217535","https://openalex.org/W2135631911","https://openalex.org/W2142518232","https://openalex.org/W2164349943","https://openalex.org/W2167193748","https://openalex.org/W4205696294","https://openalex.org/W4230094279","https://openalex.org/W4242975128"],"related_works":["https://openalex.org/W2966956638","https://openalex.org/W2951695430","https://openalex.org/W2318687609","https://openalex.org/W91590591","https://openalex.org/W2806574890","https://openalex.org/W2092711613","https://openalex.org/W3136711762","https://openalex.org/W2009034807","https://openalex.org/W2062690431","https://openalex.org/W2070382194"],"abstract_inverted_index":{"This":[0],"work":[1],"focuses":[2],"on":[3],"nonsquare":[4,26,126],"matrix":[5],"pencils":[6],"$A":[7],"-":[8,31],"\\lambda":[9,32],"B$,":[10],"where":[11],"$A,B":[12],"\\in":[13],"{\\cal":[14],"M}^{m":[15],"\\times":[16],"n}$":[17],"and":[18,75,84,128],"$m>":[19],"n$.":[20],"Traditional":[21],"methods":[22],"for":[23,56,124],"solving":[24],"such":[25,64,125],"generalized":[27],"eigenvalue":[28],"problems":[29,109],"$(A":[30],"B)\\underline{v}":[33],"=":[34],"\\underline{0}$":[35],"are":[36,68,73],"expected":[37],"to":[38,40,60,131],"lead":[39],"no":[41],"solutions":[42,67],"in":[43],"most":[44],"cases.":[45],"In":[46],"this":[47,99],"paper":[48,100],"we":[49],"propose":[50],"a":[51,121],"different":[52],"treatment:":[53],"We":[54,118],"search":[55],"the":[57,61,78,88,92,104,107,132],"minimal":[58],"perturbation":[59],"pair":[62],"$(A,B)$":[63],"that":[65],"these":[66],"indeed":[69],"possible.":[70],"Two":[71],"cases":[72],"considered":[74],"analyzed:":[76],"(i)":[77],"case":[79,90],"when":[80],"$n=1$":[81],"(vector":[82],"pencils);":[83],"(ii)":[85],"more":[86],"generally,":[87],"$n>1$":[89],"with":[91,111],"existence":[93],"of":[94,106,134],"one":[95],"eigenpair.":[96],"For":[97],"both,":[98],"proposes":[101],"insight":[102],"into":[103],"characteristics":[105],"described":[108],"along":[110],"practical":[112],"numerical":[113],"algorithms":[114],"toward":[115],"their":[116],"solution.":[117],"also":[119],"present":[120],"simplifying":[122],"factorization":[123],"pencils,":[127],"some":[129],"relations":[130],"notion":[133],"pseudospectra.":[135]},"counts_by_year":[{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":7},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":6},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":8},{"year":2012,"cited_by_count":5}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}