{"id":"https://openalex.org/W2738805548","doi":"https://doi.org/10.1109/med.2017.7984145","title":"Tridiagonal companion matrices and their use for computing orthogonal and nonorthogonal polynomial zeros","display_name":"Tridiagonal companion matrices and their use for computing orthogonal and nonorthogonal polynomial zeros","publication_year":2017,"publication_date":"2017-07-01","ids":{"openalex":"https://openalex.org/W2738805548","doi":"https://doi.org/10.1109/med.2017.7984145","mag":"2738805548"},"language":"en","primary_location":{"id":"doi:10.1109/med.2017.7984145","is_oa":false,"landing_page_url":"https://doi.org/10.1109/med.2017.7984145","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2017 25th Mediterranean Conference on Control and Automation (MED)","raw_type":"proceedings-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/A5002959264","display_name":"Matthew S. H\u00f6lzel","orcid":"https://orcid.org/0000-0002-6200-4600"},"institutions":[{"id":"https://openalex.org/I180437899","display_name":"University of Bremen","ror":"https://ror.org/04ers2y35","country_code":"DE","type":"education","lineage":["https://openalex.org/I180437899"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Matthew Holzel","raw_affiliation_strings":["University of Bremen, Bremen, Germany"],"affiliations":[{"raw_affiliation_string":"University of Bremen, Bremen, Germany","institution_ids":["https://openalex.org/I180437899"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5002959264"],"corresponding_institution_ids":["https://openalex.org/I180437899"],"apc_list":null,"apc_paid":null,"fwci":0.2343,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.54677462,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":null,"issue":null,"first_page":"364","last_page":"369"},"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/T11416","display_name":"Numerical methods for differential equations","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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9979000091552734,"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":0.947891354560852},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.7749453783035278},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5858640074729919},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.5617163777351379},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5488791465759277},{"id":"https://openalex.org/keywords/polynomial-matrix","display_name":"Polynomial matrix","score":0.48016777634620667},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4527483582496643},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.4334315061569214},{"id":"https://openalex.org/keywords/tridiagonal-matrix-algorithm","display_name":"Tridiagonal matrix algorithm","score":0.4130088984966278},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.38247251510620117},{"id":"https://openalex.org/keywords/matrix-polynomial","display_name":"Matrix polynomial","score":0.36904990673065186},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3422425389289856},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.33145904541015625},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.272365003824234},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12369954586029053}],"concepts":[{"id":"https://openalex.org/C51647924","wikidata":"https://www.wikidata.org/wiki/Q1755277","display_name":"Tridiagonal matrix","level":3,"score":0.947891354560852},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.7749453783035278},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5858640074729919},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.5617163777351379},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5488791465759277},{"id":"https://openalex.org/C126352355","wikidata":"https://www.wikidata.org/wiki/Q2562273","display_name":"Polynomial matrix","level":4,"score":0.48016777634620667},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4527483582496643},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.4334315061569214},{"id":"https://openalex.org/C176603272","wikidata":"https://www.wikidata.org/wiki/Q1819156","display_name":"Tridiagonal matrix algorithm","level":4,"score":0.4130088984966278},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.38247251510620117},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.36904990673065186},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3422425389289856},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33145904541015625},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.272365003824234},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12369954586029053},{"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},{"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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/med.2017.7984145","is_oa":false,"landing_page_url":"https://doi.org/10.1109/med.2017.7984145","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2017 25th Mediterranean Conference on Control and Automation (MED)","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/1","display_name":"No poverty","score":0.6800000071525574}],"awards":[],"funders":[{"id":"https://openalex.org/F4320321602","display_name":"Deutsches Zentrum f\u00fcr Luft- und Raumfahrt","ror":"https://ror.org/04bwf3e34"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":14,"referenced_works":["https://openalex.org/W83533993","https://openalex.org/W400320946","https://openalex.org/W413121783","https://openalex.org/W1987694238","https://openalex.org/W2020328455","https://openalex.org/W2078776235","https://openalex.org/W2095817904","https://openalex.org/W2102884619","https://openalex.org/W2133438931","https://openalex.org/W2148685136","https://openalex.org/W2221321873","https://openalex.org/W2320322137","https://openalex.org/W4240790602","https://openalex.org/W4242722506"],"related_works":["https://openalex.org/W2157863322","https://openalex.org/W2360550119","https://openalex.org/W2057923237","https://openalex.org/W2001381587","https://openalex.org/W1980786482","https://openalex.org/W1606194289","https://openalex.org/W4289860553","https://openalex.org/W3477132","https://openalex.org/W4212973497","https://openalex.org/W2074560815"],"abstract_inverted_index":{"In":[0],"this":[1,90],"paper,":[2],"we":[3,28,49],"show":[4,29],"that":[5,30],"poor":[6],"conditioning":[7,33],"can":[8,63],"arise":[9],"in":[10,85,92],"trivial":[11],"polynomials,":[12],"making":[13],"it":[14],"difficult":[15],"to":[16,88],"compute":[17],"zeros":[18,44],"with":[19],"standard":[20,70],"companion":[21,56,80],"matrix":[22],"approaches.":[23],"Furthermore,":[24],"using":[25],"these":[26],"methods,":[27],"although":[31],"the":[32,55,69,93],"of":[34,58,77],"an":[35,86],"orthogonal":[36,59],"polynomial":[37],"sequence":[38],"may":[39],"degrade":[40],"gracefully,":[41],"its":[42],"numerically-computed":[43],"generally":[45],"do":[46],"not.":[47],"Hence":[48],"introduce":[50,74],"a":[51,75],"tridiagonal":[52,79],"form":[53],"for":[54,82],"matrices":[57,81],"polynomials":[60,84],"whose":[61],"eigenvalues":[62],"be":[64],"computed":[65],"more":[66],"accurately":[67],"than":[68],"form.":[71],"We":[72],"then":[73],"method":[76],"obtaining":[78],"arbitrary":[83],"attempt":[87],"exploit":[89],"advantage":[91],"general":[94],"case.":[95]},"counts_by_year":[{"year":2019,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}