{"id":"https://openalex.org/W2053529276","doi":"https://doi.org/10.1137/050628283","title":"The Conditioning of Linearizations of Matrix Polynomials","display_name":"The Conditioning of Linearizations of Matrix Polynomials","publication_year":2006,"publication_date":"2006-01-01","ids":{"openalex":"https://openalex.org/W2053529276","doi":"https://doi.org/10.1137/050628283","mag":"2053529276"},"language":"en","primary_location":{"id":"doi:10.1137/050628283","is_oa":false,"landing_page_url":"https://doi.org/10.1137/050628283","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/A5069339254","display_name":"Nicholas J. Higham","orcid":"https://orcid.org/0000-0001-5956-4976"},"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":"Nicholas J. Higham","raw_affiliation_strings":["Applied Mathematics"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Applied Mathematics","institution_ids":["https://openalex.org/I4210131439"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014969883","display_name":"D. Steven Mackey","orcid":"https://orcid.org/0000-0001-7906-6317"},"institutions":[{"id":"https://openalex.org/I28407311","display_name":"University of Manchester","ror":"https://ror.org/027m9bs27","country_code":"GB","type":"education","lineage":["https://openalex.org/I28407311"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"D. Steven Mackey","raw_affiliation_strings":["University of Manchester\u00a0"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Manchester\u00a0","institution_ids":["https://openalex.org/I28407311"]}]},{"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"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Applied Mathematics","institution_ids":["https://openalex.org/I4210131439"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":10.4559,"has_fulltext":false,"cited_by_count":135,"citation_normalized_percentile":{"value":0.98727634,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"28","issue":"4","first_page":"1005","last_page":"1028"},"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":0.9969000220298767,"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/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9968000054359436,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"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":0.8558544516563416},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.804556667804718},{"id":"https://openalex.org/keywords/matrix-polynomial","display_name":"Matrix polynomial","score":0.6701745986938477},{"id":"https://openalex.org/keywords/linearization","display_name":"Linearization","score":0.631062924861908},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.6172993183135986},{"id":"https://openalex.org/keywords/scaling","display_name":"Scaling","score":0.5507885813713074},{"id":"https://openalex.org/keywords/companion-matrix","display_name":"Companion matrix","score":0.5092297792434692},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4952467978000641},{"id":"https://openalex.org/keywords/polynomial-matrix","display_name":"Polynomial matrix","score":0.47218266129493713},{"id":"https://openalex.org/keywords/matrix-pencil","display_name":"Matrix pencil","score":0.4399604797363281},{"id":"https://openalex.org/keywords/pencil","display_name":"Pencil (optics)","score":0.41824066638946533},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.41236215829849243},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.39457249641418457},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.31362998485565186},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.10689345002174377},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.05861780047416687}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8558544516563416},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.804556667804718},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.6701745986938477},{"id":"https://openalex.org/C11210021","wikidata":"https://www.wikidata.org/wiki/Q1520713","display_name":"Linearization","level":3,"score":0.631062924861908},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.6172993183135986},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.5507885813713074},{"id":"https://openalex.org/C141495983","wikidata":"https://www.wikidata.org/wiki/Q814361","display_name":"Companion matrix","level":5,"score":0.5092297792434692},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4952467978000641},{"id":"https://openalex.org/C126352355","wikidata":"https://www.wikidata.org/wiki/Q2562273","display_name":"Polynomial matrix","level":4,"score":0.47218266129493713},{"id":"https://openalex.org/C1576492","wikidata":"https://www.wikidata.org/wiki/Q3739929","display_name":"Matrix pencil","level":3,"score":0.4399604797363281},{"id":"https://openalex.org/C134949993","wikidata":"https://www.wikidata.org/wiki/Q2068617","display_name":"Pencil (optics)","level":2,"score":0.41824066638946533},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.41236215829849243},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.39457249641418457},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.31362998485565186},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.10689345002174377},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.05861780047416687},{"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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.1137/050628283","is_oa":false,"landing_page_url":"https://doi.org/10.1137/050628283","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/e30a8b54-7c0c-4e87-ab80-6ba247cdb149","is_oa":false,"landing_page_url":"https://research.manchester.ac.uk/en/publications/e30a8b54-7c0c-4e87-ab80-6ba247cdb149","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":"Higham, N J, Mackey, D S & Tisseur, F 2006, 'The conditioning of linearizations of matrix polynomials', SIAM Journal on Matrix Analysis and Applications, vol. 28, no. 4, pp. 1005-1028. https://doi.org/10.1137/050628283","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:669","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:CiteSeerX.psu:10.1.1.589.1888","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.589.1888","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://eprints.ma.man.ac.uk/71/01/covered/MIMS_ep2005_23.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.658.7394","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.658.7394","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://eprints.ma.man.ac.uk/669/01/hmt06c.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.66.6284","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.66.6284","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://eprints.ma.man.ac.uk/669/01/covered/MIMS_ep2005_23.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.66.7246","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.66.7246","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.maths.man.ac.uk/~nareports/narep465.pdf","raw_type":"text"},{"id":"pmh:oai:pure.atira.dk:publications/e30a8b54-7c0c-4e87-ab80-6ba247cdb149","is_oa":false,"landing_page_url":"https://www.research.manchester.ac.uk/portal/en/publications/the-conditioning-of-linearizations-of-matrix-polynomials(e30a8b54-7c0c-4e87-ab80-6ba247cdb149).html","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Higham, N J, Mackey, D S & Tisseur, F 2006, 'The conditioning of linearizations of matrix polynomials', SIAM Journal on Matrix Analysis and Applications, vol. 28, no. 4, pp. 1005-1028. https://doi.org/10.1137/050628283","raw_type":"info:eu-repo/semantics/publishedVersion"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W137390859","https://openalex.org/W658559791","https://openalex.org/W1965573117","https://openalex.org/W1970606629","https://openalex.org/W1981159057","https://openalex.org/W1999834714","https://openalex.org/W2003693652","https://openalex.org/W2013737143","https://openalex.org/W2035300576","https://openalex.org/W2037002891","https://openalex.org/W2082684197","https://openalex.org/W2110442969","https://openalex.org/W2128978199","https://openalex.org/W2140715711","https://openalex.org/W2163224973","https://openalex.org/W2163336887","https://openalex.org/W3022136271","https://openalex.org/W3110374254","https://openalex.org/W3150353689"],"related_works":["https://openalex.org/W2905919007","https://openalex.org/W2575230999","https://openalex.org/W1489986405","https://openalex.org/W2335071938","https://openalex.org/W2099412787","https://openalex.org/W2016290153","https://openalex.org/W2143986475","https://openalex.org/W4244361680","https://openalex.org/W2100313195","https://openalex.org/W4206014518"],"abstract_inverted_index":{"The":[0,189,222],"standard":[1,278],"way":[2],"of":[3,9,54,61,86,106,180,191,199,216,220,240,275,286,310],"solving":[4],"the":[5,28,52,76,82,87,117,126,146,166,177,181,192,206,210,213,217,238,241,249,265,277,283,298,308],"polynomial":[6],"eigenvalue":[7,30,47,151,178],"problem":[8,118,167,299],"degree":[10],"m":[11],"in":[12,22,102,273,300],"$n\\times":[13],"n$":[14],"matrices":[15,25,243],"is":[16,119,162,171,201,225],"to":[17,19,96,139,164,168,197,203,227,256,312],"\u201clinearize\u201d":[18],"a":[20,33,57,159],"pencil":[21],"$mn\\times":[23],"mn$":[24],"and":[26,41,65,71,80,124,185,212,219,252,282,288,314],"solve":[27],"generalized":[29],"problem.":[31],"For":[32,149],"given":[34],"polynomial,":[35],"P,":[36,235],"infinitely":[37],"many":[38],"linearizations":[39,55,101],"exist":[40],"they":[42],"can":[43],"have":[44,257],"widely":[45,182],"varying":[46],"condition":[48,259,280,284,305],"numbers.":[49,260],"We":[50,73,174],"investigate":[51],"conditioning":[53,83,179,190],"from":[56],"vector":[58],"space":[59],"$\\mathbb{DL}(P)$":[60,103],"pencils":[62,92,127,136],"recently":[63],"identified":[64],"studied":[66],"by":[67],"Mackey,":[68,69],"Mehl,":[70],"Mehrmann.":[72],"look":[74],"for":[75,104,264,297],"best":[77],"conditioned":[78,144,233],"linearization":[79,195,218],"compare":[81],"with":[84],"that":[85,116,125,153,170,198],"original":[88,147],"polynomial.":[89,148],"Two":[90],"particular":[91],"are":[93,128,137,154,244,254,271],"shown":[94,138,163,202],"always":[95],"be":[97,140,228],"almost":[98],"optimal":[99],"over":[100],"eigenvalues":[105],"modulus":[107],"greater":[108],"than":[109,112,234],"or":[110],"less":[111],"1,":[113],"respectively,":[114],"provided":[115],"not":[120,155],"too":[121,156],"badly":[122],"scaled":[123],"linearizations.":[129,188],"Moreover,":[130],"under":[131],"this":[132,303],"scaling":[133,161],"assumption,":[134],"these":[135],"about":[141],"as":[142,145],"well":[143,172],"quadratic":[150],"problems":[152],"heavily":[157],"damped,":[158],"simple":[160],"convert":[165],"one":[169],"scaled.":[173],"also":[175],"analyze":[176],"used":[183],"first":[184,193],"second":[186,266],"companion":[187,194,223,250,267],"relative":[196,279],"P":[200,253],"depend":[204],"on":[205],"coefficient":[207,242],"matrix":[208],"norms,":[209],"eigenvalue,":[211],"left":[214],"s":[215],"P.":[221],"form":[224,251],"found":[226],"potentially":[229],"much":[230],"more":[231],"ill":[232],"but":[236],"if":[237],"2\u2010norms":[239],"all":[245],"approximately":[246],"1":[247],"then":[248],"guaranteed":[255],"similar":[258],"Analogous":[261],"results":[262,270],"hold":[263],"form.":[268],"Our":[269],"phrased":[272],"terms":[274],"both":[276],"number":[281,285,306],"Dedieu":[287],"Tisseur":[289],"[Linear":[290],"Algebra":[291],"Appl.,":[292],"358":[293],"(2003),":[294],"pp.":[295],"71\u201394]":[296],"homogeneous":[301],"form,":[302],"latter":[304],"having":[307],"advantage":[309],"applying":[311],"zero":[313],"infinite":[315],"eigenvalues.":[316]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":7},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":9},{"year":2018,"cited_by_count":9},{"year":2017,"cited_by_count":10},{"year":2016,"cited_by_count":7},{"year":2015,"cited_by_count":5},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":9},{"year":2012,"cited_by_count":3}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
