{"id":"https://openalex.org/W2156010245","doi":"https://doi.org/10.1137/s0895479803429788","title":"The Ehrlich--Aberth Method for the Nonsymmetric Tridiagonal Eigenvalue Problem","display_name":"The Ehrlich--Aberth Method for the Nonsymmetric Tridiagonal Eigenvalue Problem","publication_year":2005,"publication_date":"2005-01-01","ids":{"openalex":"https://openalex.org/W2156010245","doi":"https://doi.org/10.1137/s0895479803429788","mag":"2156010245"},"language":"en","primary_location":{"id":"doi:10.1137/s0895479803429788","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479803429788","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/A5023356208","display_name":"Dario A. Bini","orcid":"https://orcid.org/0000-0002-5885-9411"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Dario A. Bini","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5085341608","display_name":"Luca Gemignani","orcid":"https://orcid.org/0000-0001-8000-4906"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Luca Gemignani","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5059791596","display_name":"Fran\u00e7oise Tisseur","orcid":"https://orcid.org/0000-0002-1011-2570"},"institutions":[{"id":"https://openalex.org/I4210131439","display_name":"Applied Mathematics (United States)","ror":"https://ror.org/03seew607","country_code":"US","type":"company","lineage":["https://openalex.org/I4210131439"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Fran\u00e7oise Tisseur","raw_affiliation_strings":["Applied Mathematics"],"affiliations":[{"raw_affiliation_string":"Applied Mathematics","institution_ids":["https://openalex.org/I4210131439"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5023356208"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.0305,"has_fulltext":false,"cited_by_count":41,"citation_normalized_percentile":{"value":0.91964286,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"27","issue":"1","first_page":"153","last_page":"175"},"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":0.9983000159263611,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9976999759674072,"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/tridiagonal-matrix","display_name":"Tridiagonal matrix","score":0.8493366241455078},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.747810423374176},{"id":"https://openalex.org/keywords/lambda","display_name":"Lambda","score":0.7251005172729492},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.6372532844543457},{"id":"https://openalex.org/keywords/subroutine","display_name":"Subroutine","score":0.5639805197715759},{"id":"https://openalex.org/keywords/factorization","display_name":"Factorization","score":0.5577730536460876},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.5434146523475647},{"id":"https://openalex.org/keywords/divide-and-conquer-algorithms","display_name":"Divide and conquer algorithms","score":0.5132424831390381},{"id":"https://openalex.org/keywords/tridiagonal-matrix-algorithm","display_name":"Tridiagonal matrix algorithm","score":0.4419832229614258},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.440204381942749},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4187619090080261},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4084793031215668},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3917256295681},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.36125120520591736},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2327737808227539},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.17394879460334778}],"concepts":[{"id":"https://openalex.org/C51647924","wikidata":"https://www.wikidata.org/wiki/Q1755277","display_name":"Tridiagonal matrix","level":3,"score":0.8493366241455078},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.747810423374176},{"id":"https://openalex.org/C2778113609","wikidata":"https://www.wikidata.org/wiki/Q10897","display_name":"Lambda","level":2,"score":0.7251005172729492},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6372532844543457},{"id":"https://openalex.org/C96147967","wikidata":"https://www.wikidata.org/wiki/Q190686","display_name":"Subroutine","level":2,"score":0.5639805197715759},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.5577730536460876},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.5434146523475647},{"id":"https://openalex.org/C71559656","wikidata":"https://www.wikidata.org/wiki/Q671298","display_name":"Divide and conquer algorithms","level":2,"score":0.5132424831390381},{"id":"https://openalex.org/C176603272","wikidata":"https://www.wikidata.org/wiki/Q1819156","display_name":"Tridiagonal matrix algorithm","level":4,"score":0.4419832229614258},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.440204381942749},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4187619090080261},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4084793031215668},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3917256295681},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.36125120520591736},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2327737808227539},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.17394879460334778},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","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":4,"locations":[{"id":"doi:10.1137/s0895479803429788","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479803429788","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:pure.atira.dk:openaire_cris_publications/4d67a668-7b32-4cba-9d19-873eb9e1d19f","is_oa":false,"landing_page_url":"https://research.manchester.ac.uk/en/publications/4d67a668-7b32-4cba-9d19-873eb9e1d19f","pdf_url":null,"source":{"id":"https://openalex.org/S4306400662","display_name":"Research Explorer (The University of Manchester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28407311","host_organization_name":"University of Manchester","host_organization_lineage":["https://openalex.org/I28407311"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Bini, D A, Gemignani, L & Tisseur, F 2006, 'The Ehrlich-aberth method for the nonsymmetric tridiagonal eigenvalue problem', SIAM Journal on Matrix Analysis and Applications, vol. 27, no. 1, pp. 153-175. https://doi.org/10.1137/S0895479803429788","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:977","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306400452","display_name":"MIMS EPrints (University of Southampton)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I43439940","host_organization_name":"University of Southampton","host_organization_lineage":["https://openalex.org/I43439940"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:arpi.unipi.it:11568/180612","is_oa":false,"landing_page_url":"http://hdl.handle.net/11568/180612","pdf_url":null,"source":{"id":"https://openalex.org/S4377196265","display_name":"CINECA IRIS Institutial research information system (University of Pisa)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I108290504","host_organization_name":"University of Pisa","host_organization_lineage":["https://openalex.org/I108290504"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"1.1 Articolo in rivista"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":42,"referenced_works":["https://openalex.org/W633035909","https://openalex.org/W1542574673","https://openalex.org/W1832583097","https://openalex.org/W1963616334","https://openalex.org/W1964799071","https://openalex.org/W1970606629","https://openalex.org/W1972133117","https://openalex.org/W1979738559","https://openalex.org/W1980445106","https://openalex.org/W1981745143","https://openalex.org/W1997396389","https://openalex.org/W2001038069","https://openalex.org/W2019589422","https://openalex.org/W2020804487","https://openalex.org/W2023413246","https://openalex.org/W2025026078","https://openalex.org/W2033359201","https://openalex.org/W2035814233","https://openalex.org/W2036063980","https://openalex.org/W2038471095","https://openalex.org/W2042818381","https://openalex.org/W2043237762","https://openalex.org/W2044008067","https://openalex.org/W2046133214","https://openalex.org/W2057400951","https://openalex.org/W2059586807","https://openalex.org/W2061771449","https://openalex.org/W2064468023","https://openalex.org/W2065205300","https://openalex.org/W2069403372","https://openalex.org/W2070596272","https://openalex.org/W2073084447","https://openalex.org/W2073943554","https://openalex.org/W2098739181","https://openalex.org/W2103943585","https://openalex.org/W2108735528","https://openalex.org/W2112932899","https://openalex.org/W2133893622","https://openalex.org/W2798909945","https://openalex.org/W3022909171","https://openalex.org/W3161820760","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2162598472","https://openalex.org/W2071826812","https://openalex.org/W2088506018","https://openalex.org/W2361045733","https://openalex.org/W1971890403","https://openalex.org/W2214925497","https://openalex.org/W3046472166","https://openalex.org/W2530756931","https://openalex.org/W2001038069","https://openalex.org/W2156010245"],"abstract_inverted_index":{"An":[0],"algorithm":[1,28,90,119],"based":[2,76],"on":[3,59,77],"the":[4,10,13,30,40,47,89,98,101,104,112],"Ehrlich--Aberth":[5],"iteration":[6],"is":[7,20,36,91,120],"presented":[8],"for":[9,122],"computation":[11],"of":[12,15,32,43,50,55,103],"zeros":[14],"$p(\\lambda)=\\det(T-\\lambda":[16],"I)$,":[17],"where":[18],"T":[19],"a":[21,60,72,78],"real":[22],"irreducible":[23],"nonsymmetric":[24],"tridiagonal":[25],"matrix.":[26],"The":[27,53],"requires":[29],"evaluation":[31],"$p(\\lambda)/p'(\\lambda)=-1/\\mathrm{trace}(T-\\lambda":[33],"I)^{-1}$,":[34],"which":[35],"done":[37],"by":[38],"exploiting":[39],"QR":[41],"factorization":[42],"$T-\\lambda":[44],"I$":[45],"and":[46,63,93,100],"semiseparable":[48],"structure":[49],"$(T-\\lambda":[51],"I)^{-1}$.":[52],"choice":[54],"initial":[56],"approximations":[57],"relies":[58],"divide-and-conquer":[61],"strategy,":[62],"some":[64],"results":[65],"motivating":[66],"this":[67],"strategy":[68],"are":[69,82,106],"given.":[70],"Guaranteed":[71],"posteriori":[73],"error":[74,80],"bounds":[75],"running":[79],"analysis":[81],"proved.":[83],"A":[84],"Fortran":[85],"95":[86],"module":[87],"implementing":[88],"provided":[92],"numerical":[94],"experiments":[95],"that":[96,117],"confirm":[97],"effectiveness":[99],"robustness":[102],"approach":[105],"presented.":[107],"In":[108],"particular,":[109],"comparisons":[110],"with":[111],"LAPACK":[113],"subroutine":[114],"\\texttt{dhseqr}":[115],"show":[116],"our":[118],"faster":[121],"large":[123],"dimensions.":[124]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":5},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":4}],"updated_date":"2026-04-19T08:26:33.389920","created_date":"2025-10-10T00:00:00"}