{"id":"https://openalex.org/W4225275317","doi":"https://doi.org/10.1137/21m139205x","title":"Deflation for the Symmetric Arrowhead and Diagonal-Plus-Rank-One Eigenvalue Problems","display_name":"Deflation for the Symmetric Arrowhead and Diagonal-Plus-Rank-One Eigenvalue Problems","publication_year":2022,"publication_date":"2022-05-02","ids":{"openalex":"https://openalex.org/W4225275317","doi":"https://doi.org/10.1137/21m139205x"},"language":"en","primary_location":{"id":"doi:10.1137/21m139205x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/21m139205x","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":"https://scholarsphere.psu.edu/resources/c4b2ef87-dfbf-4040-b1bb-04e7f1b9b11b/downloads/26034","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5111535733","display_name":"Jesse L. Barlow","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Jesse L. Barlow","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5110701881","display_name":"Stanley C. Eisenstat","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Stanley C. Eisenstat","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5000811267","display_name":"Nevena Jakov\u010devi\u0107 Stor","orcid":"https://orcid.org/0000-0002-9037-7631"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Nevena Jakov\u010devi\u0107 Stor","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5020011875","display_name":"Ivan Slapni\u010dar","orcid":"https://orcid.org/0000-0002-8741-3988"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ivan Slapnicar","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5111535733"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.3008,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.573852,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"43","issue":"2","first_page":"681","last_page":"709"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9939000010490417,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9833999872207642,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"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/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.7874917984008789},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7483413219451904},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6991970539093018},{"id":"https://openalex.org/keywords/diagonal","display_name":"Diagonal","score":0.5874083638191223},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5409677028656006},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.5193180441856384},{"id":"https://openalex.org/keywords/diagonal-matrix","display_name":"Diagonal matrix","score":0.5059904456138611},{"id":"https://openalex.org/keywords/zero","display_name":"Zero (linguistics)","score":0.48917725682258606},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.45535850524902344},{"id":"https://openalex.org/keywords/symmetric-matrix","display_name":"Symmetric matrix","score":0.4375481903553009},{"id":"https://openalex.org/keywords/heuristics","display_name":"Heuristics","score":0.4270303547382355},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.25302374362945557},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.11319884657859802},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.09832727909088135}],"concepts":[{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.7874917984008789},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7483413219451904},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6991970539093018},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.5874083638191223},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5409677028656006},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5193180441856384},{"id":"https://openalex.org/C113313756","wikidata":"https://www.wikidata.org/wiki/Q332791","display_name":"Diagonal matrix","level":3,"score":0.5059904456138611},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.48917725682258606},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.45535850524902344},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.4375481903553009},{"id":"https://openalex.org/C127705205","wikidata":"https://www.wikidata.org/wiki/Q5748245","display_name":"Heuristics","level":2,"score":0.4270303547382355},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.25302374362945557},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.11319884657859802},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.09832727909088135},{"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/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"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/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/21m139205x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/21m139205x","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:scholarsphere.psu.edu:36ee11c7-ffce-4b75-80a3-dffab410870e","is_oa":true,"landing_page_url":"https://scholarsphere.psu.edu/resources/36ee11c7-ffce-4b75-80a3-dffab410870e","pdf_url":"https://scholarsphere.psu.edu/resources/c4b2ef87-dfbf-4040-b1bb-04e7f1b9b11b/downloads/26034","source":{"id":"https://openalex.org/S4306401507","display_name":"ScholarSphere (Penn State Libraries)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I3130638595","host_organization_name":"Pangasinan State University","host_organization_lineage":["https://openalex.org/I3130638595"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"}],"best_oa_location":{"id":"pmh:oai:scholarsphere.psu.edu:36ee11c7-ffce-4b75-80a3-dffab410870e","is_oa":true,"landing_page_url":"https://scholarsphere.psu.edu/resources/36ee11c7-ffce-4b75-80a3-dffab410870e","pdf_url":"https://scholarsphere.psu.edu/resources/c4b2ef87-dfbf-4040-b1bb-04e7f1b9b11b/downloads/26034","source":{"id":"https://openalex.org/S4306401507","display_name":"ScholarSphere (Penn State Libraries)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I3130638595","host_organization_name":"Pangasinan State University","host_organization_lineage":["https://openalex.org/I3130638595"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3458749958","display_name":null,"funder_award_id":"IP-2020-02-2240","funder_id":"https://openalex.org/F4320322674","funder_display_name":"Hrvatska Zaklada za Znanost"},{"id":"https://openalex.org/G5241798020","display_name":null,"funder_award_id":"IP-2014009-9540","funder_id":"https://openalex.org/F4320322674","funder_display_name":"Hrvatska Zaklada za Znanost"}],"funders":[{"id":"https://openalex.org/F4320322674","display_name":"Hrvatska Zaklada za Znanost","ror":"https://ror.org/03n51vw80"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4225275317.pdf","grobid_xml":"https://content.openalex.org/works/W4225275317.grobid-xml"},"referenced_works_count":29,"referenced_works":["https://openalex.org/W1827065324","https://openalex.org/W1964217357","https://openalex.org/W1976998625","https://openalex.org/W1981152494","https://openalex.org/W1984092937","https://openalex.org/W1996369530","https://openalex.org/W2008652629","https://openalex.org/W2008694421","https://openalex.org/W2014168697","https://openalex.org/W2017641901","https://openalex.org/W2019166112","https://openalex.org/W2023663671","https://openalex.org/W2035881097","https://openalex.org/W2038833207","https://openalex.org/W2038944114","https://openalex.org/W2054188927","https://openalex.org/W2058221770","https://openalex.org/W2072396164","https://openalex.org/W2073357179","https://openalex.org/W2073955633","https://openalex.org/W2079925720","https://openalex.org/W2091859203","https://openalex.org/W2092131542","https://openalex.org/W2105433369","https://openalex.org/W2105505928","https://openalex.org/W2117686912","https://openalex.org/W2141992912","https://openalex.org/W2149709455","https://openalex.org/W2169378749"],"related_works":["https://openalex.org/W1529195008","https://openalex.org/W2070323717","https://openalex.org/W3157489721","https://openalex.org/W38983538","https://openalex.org/W2924468784","https://openalex.org/W2034395027","https://openalex.org/W1525824672","https://openalex.org/W2950949038","https://openalex.org/W2022398590","https://openalex.org/W2083635853"],"abstract_inverted_index":{"We":[0,49,131],"discuss":[1],"the":[2,5,72,119,169,173,187,199,208,232,245],"eigenproblem":[3],"for":[4,39,80,95,106,182,196,244],"symmetric":[6,188,233],"arrowhead":[7,214],"matrix":[8,162,215],"$C":[9],"=":[10],"(\\begin{smallmatrix}":[11],"D":[12],"\\&":[13],"{z}":[14],"{z}^T":[15],"&":[16],"\\alpha":[17],"\\end{smallmatrix})$,":[18],"where":[19],"$D":[20],"\\in":[21,28,32],"\\mathbb{R}^{n":[22],"\\times":[23],"n}$":[24],"is":[25,103,192],"diagonal,":[26],"${z}":[27],"\\mathbb{R}^n$,":[29],"and":[30,158,226,237],"$\\alpha":[31],"\\mathbb{R}$,":[33],"in":[34,167,186],"order":[35],"to":[36,47,67,129,216,231,239],"examine":[37],"criteria":[38],"when":[40,109],"components":[41],"of":[42,55,62,74,91,121,141,149],"${z}$":[43,63],"may":[44,64],"be":[45,65,220],"set":[46],"zero.":[48],"show":[50,132],"that":[51,81,124,133,194,211,218],"whenever":[52],"two":[53,150],"eigenvalues":[54,73,154,164,185],"$C$":[56],"are":[57,125,165,229],"sufficiently":[58],"close,":[59],"some":[60],"component":[61,82],"deflated":[66,221],"zero,":[68],"without":[69],"significantly":[70],"perturbing":[71],"$C$,":[75],"by":[76],"either":[77],"substituting":[78],"zero":[79],"or":[83,143,151],"performing":[84],"a":[85,114,160,178,240],"Givens":[86],"rotation":[87],"on":[88],"each":[89],"side":[90],"$C$.":[92],"The":[93,223],"strategy":[94,243],"this":[96],"deflation":[97,206,224,242],"requires":[98],"${\\mathcal{O}(n^2)}$":[99,209],"comparisons.":[100],"Although":[101],"it":[102,112],"too":[104],"costly":[105],"many":[107],"applications,":[108],"we":[110,116,176],"use":[111],"as":[113,204,207],"benchmark,":[115],"can":[117],"analyze":[118],"effectiveness":[120],"${{O}(n)}$":[122],"heuristics":[123],"more":[126,144,152,179],"practical":[127],"approaches":[128],"deflation.":[130],"one":[134,217],"such":[135],"${\\mathcal{O}(n)}$":[136,174,200],"heuristic":[137,201],"finds":[138,202],"all":[139],"sets":[140,148],"three":[142],"nearby":[145,153],"eigenvalues,":[146],"misses":[147],"under":[155],"limited":[156],"circumstances,":[157],"produces":[159],"reduced":[161],"whose":[163],"distinct":[166],"double":[168],"working":[170],"precision.":[171],"Using":[172],"heuristic,":[175],"develop":[177],"aggressive":[180],"method":[181],"finding":[183],"converged":[184],"Lanczos":[189],"algorithm.":[190],"It":[191],"shown":[193],"except":[195],"pathological":[197],"exceptions,":[198],"nearly":[203],"much":[205],"algorithm":[210],"reduces":[212],"an":[213],"cannot":[219],"further.":[222],"algorithms":[225],"their":[227],"analysis":[228],"extended":[230],"diagonal-plus-rank-one":[234],"eigenvalue":[235],"problem":[236],"lead":[238],"better":[241],"LAPACK":[246],"routine":[247],"dstedc.f.":[248]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1}],"updated_date":"2026-03-10T16:38:18.471706","created_date":"2025-10-10T00:00:00"}
