{"id":"https://openalex.org/W1524943583","doi":"https://doi.org/10.1137/140983434","title":"Fast and Backward Stable Computation of Roots of Polynomials","display_name":"Fast and Backward Stable Computation of Roots of Polynomials","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W1524943583","doi":"https://doi.org/10.1137/140983434","mag":"1524943583"},"language":"en","primary_location":{"id":"doi:10.1137/140983434","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140983434","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://nur.nu.edu.kz/handle/123456789/2225","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5091548806","display_name":"Jared L. Aurentz","orcid":"https://orcid.org/0000-0001-7822-6374"},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Jared L. Aurentz","raw_affiliation_strings":["University of Oxford"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Oxford","institution_ids":["https://openalex.org/I40120149"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5074334527","display_name":"Thomas Mach","orcid":"https://orcid.org/0000-0001-7155-2556"},"institutions":[{"id":"https://openalex.org/I99464096","display_name":"KU Leuven","ror":"https://ror.org/05f950310","country_code":"BE","type":"education","lineage":["https://openalex.org/I99464096"]}],"countries":["BE"],"is_corresponding":false,"raw_author_name":"Thomas Mach","raw_affiliation_strings":[", KU Leuven"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":", KU Leuven","institution_ids":["https://openalex.org/I99464096"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5039999347","display_name":"Raf Vandebril","orcid":"https://orcid.org/0000-0003-2119-8696"},"institutions":[{"id":"https://openalex.org/I99464096","display_name":"KU Leuven","ror":"https://ror.org/05f950310","country_code":"BE","type":"education","lineage":["https://openalex.org/I99464096"]}],"countries":["BE"],"is_corresponding":false,"raw_author_name":"Raf Vandebril","raw_affiliation_strings":[", KU Leuven"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":", KU Leuven","institution_ids":["https://openalex.org/I99464096"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5006021673","display_name":"David S. Watkins","orcid":null},"institutions":[{"id":"https://openalex.org/I72951846","display_name":"Washington State University","ror":"https://ror.org/05dk0ce17","country_code":"US","type":"education","lineage":["https://openalex.org/I72951846"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"David S. Watkins","raw_affiliation_strings":["Washington State University\u2014Pullman"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Washington State University\u2014Pullman","institution_ids":["https://openalex.org/I72951846"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":5.6814,"has_fulltext":false,"cited_by_count":60,"citation_normalized_percentile":{"value":0.96363567,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"36","issue":"3","first_page":"942","last_page":"973"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9993000030517578,"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":0.9993000030517578,"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/T11697","display_name":"Numerical Methods and Algorithms","score":0.9987999796867371,"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/T11206","display_name":"Model Reduction and Neural Networks","score":0.9969000220298767,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7817426919937134},{"id":"https://openalex.org/keywords/flops","display_name":"FLOPS","score":0.7653855085372925},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.6614382266998291},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.620834231376648},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.6036286354064941},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4533419609069824},{"id":"https://openalex.org/keywords/representation","display_name":"Representation (politics)","score":0.4465197026729584},{"id":"https://openalex.org/keywords/qr-decomposition","display_name":"QR decomposition","score":0.4315153956413269},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4089079797267914},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3253432512283325},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.21786877512931824},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.11512693762779236}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7817426919937134},{"id":"https://openalex.org/C3826847","wikidata":"https://www.wikidata.org/wiki/Q188768","display_name":"FLOPS","level":2,"score":0.7653855085372925},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.6614382266998291},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.620834231376648},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6036286354064941},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4533419609069824},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.4465197026729584},{"id":"https://openalex.org/C188060507","wikidata":"https://www.wikidata.org/wiki/Q653242","display_name":"QR decomposition","level":3,"score":0.4315153956413269},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4089079797267914},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3253432512283325},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.21786877512931824},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.11512693762779236},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","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},{"id":"https://openalex.org/C94625758","wikidata":"https://www.wikidata.org/wiki/Q7163","display_name":"Politics","level":2,"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/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1137/140983434","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140983434","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:10.1.1.564.4577","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.564.4577","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.wsu.edu/math/faculty/watkins/pdfiles/TW654.pdf","raw_type":"text"},{"id":"pmh:oai:testnur.nu.edu.kz:123456789/2225","is_oa":true,"landing_page_url":"http://nur.nu.edu.kz/handle/123456789/2225","pdf_url":null,"source":{"id":"https://openalex.org/S4377196417","display_name":"Nazarbayev University Repository (Nazarbayev University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I60559429","host_organization_name":"Nazarbayev University","host_organization_lineage":["https://openalex.org/I60559429"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-sa","license_id":"https://openalex.org/licenses/cc-by-nc-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:lirias2repo.kuleuven.be:123456789/461961","is_oa":true,"landing_page_url":"http://www.cs.kuleuven.be/publicaties/rapporten/tw/TW654.abs.html","pdf_url":null,"source":{"id":"https://openalex.org/S7407055369","display_name":"Lirias","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"TW Reports, Report No. TW654","raw_type":"info:eu-repo/semantics/report"},{"id":"pmh:oai:lirias2repo.kuleuven.be:123456789/494876","is_oa":true,"landing_page_url":"http://epubs.siam.org/doi/abs/10.1137/140983434","pdf_url":null,"source":{"id":"https://openalex.org/S7407055369","display_name":"Lirias","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications, vol. 36 (3), Art.No. 3, (942-973)","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:lirias2repo.kuleuven.be:123456789/516890","is_oa":false,"landing_page_url":"http://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=21361","pdf_url":null,"source":{"id":"https://openalex.org/S7407055369","display_name":"Lirias","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Conference on Applied Linear Algebra, Atlanta, Georgia, USA, 26-30 October 2015","raw_type":"info:eu-repo/semantics/publishedVersion"}],"best_oa_location":{"id":"pmh:oai:testnur.nu.edu.kz:123456789/2225","is_oa":true,"landing_page_url":"http://nur.nu.edu.kz/handle/123456789/2225","pdf_url":null,"source":{"id":"https://openalex.org/S4377196417","display_name":"Nazarbayev University Repository (Nazarbayev University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I60559429","host_organization_name":"Nazarbayev University","host_organization_lineage":["https://openalex.org/I60559429"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-sa","license_id":"https://openalex.org/licenses/cc-by-nc-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":34,"referenced_works":["https://openalex.org/W296618920","https://openalex.org/W1568240140","https://openalex.org/W1569795718","https://openalex.org/W1832583097","https://openalex.org/W1967884132","https://openalex.org/W1977765413","https://openalex.org/W1985540142","https://openalex.org/W1992856421","https://openalex.org/W1995854415","https://openalex.org/W2003412314","https://openalex.org/W2003844998","https://openalex.org/W2008652629","https://openalex.org/W2013737143","https://openalex.org/W2020804487","https://openalex.org/W2024786759","https://openalex.org/W2025783125","https://openalex.org/W2035180740","https://openalex.org/W2035754171","https://openalex.org/W2042893763","https://openalex.org/W2043237762","https://openalex.org/W2067225260","https://openalex.org/W2069415575","https://openalex.org/W2069524360","https://openalex.org/W2087263172","https://openalex.org/W2093911470","https://openalex.org/W2101150179","https://openalex.org/W2112277662","https://openalex.org/W2124986435","https://openalex.org/W2133549820","https://openalex.org/W2165797890","https://openalex.org/W2171583203","https://openalex.org/W2183160780","https://openalex.org/W3100648733","https://openalex.org/W4231407710"],"related_works":["https://openalex.org/W2009193081","https://openalex.org/W2391023104","https://openalex.org/W1993821330","https://openalex.org/W2023965391","https://openalex.org/W2361403716","https://openalex.org/W2904061867","https://openalex.org/W4289115725","https://openalex.org/W2070382194","https://openalex.org/W2967798957","https://openalex.org/W3161672423"],"abstract_inverted_index":{"A":[0,30],"stable":[1],"algorithm":[2,121,138,156,169],"to":[3,114,159,171],"compute":[4],"the":[5,17,20,45,85,97,100,107,116,165,172,177],"roots":[6,12],"of":[7,19,47,61,80,150],"polynomials":[8],"is":[9,33,39,44,66,75,93,103,111,139,144,153,157,180,193],"presented.":[10,154],"The":[11,73,137,155],"are":[13,57,70],"found":[14],"by":[15,24,59],"computing":[16],"eigenvalues":[18],"associated":[21],"companion":[22,31,173],"matrix":[23,32,37,50,74,174],"Francis's":[25,62,120],"implicitly":[26],"shifted":[27],"QR":[28,168],"algorithm.":[29],"an":[34,147],"upper":[35],"Hessenberg":[36],"that":[38,41,69,186],"unitary-plus-rank-one,":[40],"is,":[42],"it":[43,65,110],"sum":[46],"a":[48,52,78],"unitary":[49],"and":[51,64,132,146,190],"rank-one":[53,86,101,117],"matrix.":[54],"These":[55],"properties":[56,68],"preserved":[58],"iterations":[60],"algorithm,":[63],"these":[67],"exploited":[71],"here.":[72],"represented":[76],"as":[77,162,164],"product":[79],"3n-1":[81],"Givens":[82],"rotators":[83],"plus":[84],"part,":[87],"so":[88,109],"only":[89,127],"$O(n)$":[90],"storage":[91],"space":[92],"required.":[94],"In":[95],"fact,":[96],"information":[98],"about":[99,161],"part":[102,118],"also":[104],"encoded":[105],"in":[106],"rotators,":[108],"not":[112],"necessary":[113],"store":[115],"explicitly.":[119],"implemented":[122],"on":[123],"this":[124],"representation":[125],"requires":[126],"O(n)":[128],"flops":[129,135],"per":[130],"iteration":[131],"thus":[133],"$O(n^{2})$":[134],"overall.":[136],"described,":[140],"normwise":[141],"backward":[142],"stability":[143],"proved,":[145],"extensive":[148],"set":[149],"numerical":[151],"experiments":[152],"shown":[158],"be":[160],"accurate":[163],"(slow)":[166],"Francis":[167],"applied":[170],"without":[175],"exploiting":[176],"structure.":[178],"It":[179],"faster":[181],"than":[182],"other":[183],"fast":[184],"methods":[185],"have":[187],"been":[188],"proposed,":[189],"its":[191],"accuracy":[192],"comparable":[194],"or":[195],"better.":[196]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":9},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":9},{"year":2020,"cited_by_count":7},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":6},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":1}],"updated_date":"2026-08-11T07:18:39.950985","created_date":"2025-10-10T00:00:00"}
