{"id":"https://openalex.org/W2052832983","doi":"https://doi.org/10.1137/040606636","title":"Computing the Condition Number of Tridiagonal and Diagonal-Plus-Semiseparable Matrices in Linear Time","display_name":"Computing the Condition Number of Tridiagonal and Diagonal-Plus-Semiseparable Matrices in Linear Time","publication_year":2005,"publication_date":"2005-01-01","ids":{"openalex":"https://openalex.org/W2052832983","doi":"https://doi.org/10.1137/040606636","mag":"2052832983"},"language":"en","primary_location":{"id":"doi:10.1137/040606636","is_oa":false,"landing_page_url":"https://doi.org/10.1137/040606636","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/A5041794956","display_name":"Gareth I. Hargreaves","orcid":null},"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":true,"raw_author_name":"Gareth I. Hargreaves","raw_affiliation_strings":["School of Mathematics, University of Manchester, Manchester, M13 9PL, England"],"affiliations":[{"raw_affiliation_string":"School of Mathematics, University of Manchester, Manchester, M13 9PL, England","institution_ids":["https://openalex.org/I28407311"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5041794956"],"corresponding_institution_ids":["https://openalex.org/I28407311"],"apc_list":null,"apc_paid":null,"fwci":0.6733,"has_fulltext":false,"cited_by_count":12,"citation_normalized_percentile":{"value":0.71514977,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"27","issue":"3","first_page":"801","last_page":"820"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9995999932289124,"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":0.9995999932289124,"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":0.9983999729156494,"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":0.998199999332428,"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/tridiagonal-matrix","display_name":"Tridiagonal matrix","score":0.7271853685379028},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.725814700126648},{"id":"https://openalex.org/keywords/factorization","display_name":"Factorization","score":0.5923953652381897},{"id":"https://openalex.org/keywords/qr-decomposition","display_name":"QR decomposition","score":0.5566005706787109},{"id":"https://openalex.org/keywords/arithmetic-underflow","display_name":"Arithmetic underflow","score":0.5526448488235474},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5308241248130798},{"id":"https://openalex.org/keywords/condition-number","display_name":"Condition number","score":0.5195779800415039},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.48236769437789917},{"id":"https://openalex.org/keywords/diagonal","display_name":"Diagonal","score":0.4759630560874939},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.4405818581581116},{"id":"https://openalex.org/keywords/matrix-norm","display_name":"Matrix norm","score":0.42476123571395874},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.3521149754524231},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.19381454586982727}],"concepts":[{"id":"https://openalex.org/C51647924","wikidata":"https://www.wikidata.org/wiki/Q1755277","display_name":"Tridiagonal matrix","level":3,"score":0.7271853685379028},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.725814700126648},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.5923953652381897},{"id":"https://openalex.org/C188060507","wikidata":"https://www.wikidata.org/wiki/Q653242","display_name":"QR decomposition","level":3,"score":0.5566005706787109},{"id":"https://openalex.org/C3265923","wikidata":"https://www.wikidata.org/wiki/Q669129","display_name":"Arithmetic underflow","level":2,"score":0.5526448488235474},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5308241248130798},{"id":"https://openalex.org/C84545080","wikidata":"https://www.wikidata.org/wiki/Q1147936","display_name":"Condition number","level":3,"score":0.5195779800415039},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.48236769437789917},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.4759630560874939},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.4405818581581116},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.42476123571395874},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.3521149754524231},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.19381454586982727},{"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},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1137/040606636","is_oa":false,"landing_page_url":"https://doi.org/10.1137/040606636","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:eprints.maths.manchester.ac.uk.MIMS.EPrints:157","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:10.1.1.495.1935","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.495.1935","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/157/02/covered/MIMS_ep2006_15.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.66.8922","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.66.8922","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/narep447.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.660.2318","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.660.2318","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/157/02/narep447%5B1%5D.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":20,"referenced_works":["https://openalex.org/W1964080902","https://openalex.org/W1964799071","https://openalex.org/W1972133117","https://openalex.org/W1983483574","https://openalex.org/W1999194916","https://openalex.org/W2002864633","https://openalex.org/W2020804487","https://openalex.org/W2030992481","https://openalex.org/W2032604276","https://openalex.org/W2035814233","https://openalex.org/W2037200674","https://openalex.org/W2067878034","https://openalex.org/W2070201622","https://openalex.org/W2073943554","https://openalex.org/W2088694512","https://openalex.org/W2135622428","https://openalex.org/W2136434495","https://openalex.org/W2156010245","https://openalex.org/W2170013390","https://openalex.org/W3022909171"],"related_works":["https://openalex.org/W3096021259","https://openalex.org/W2978021564","https://openalex.org/W2888367694","https://openalex.org/W2057555084","https://openalex.org/W2018239771","https://openalex.org/W1762420609","https://openalex.org/W3007862902","https://openalex.org/W2134887131","https://openalex.org/W2592387312","https://openalex.org/W2294724139"],"abstract_inverted_index":{"For":[0],"an":[1,78,110],"$n":[2,79],"\\times":[3,80],"n$":[4,81],"tridiagonal":[5],"matrix":[6,107],"we":[7],"exploit":[8,101],"the":[9,22,53,59,102,106,114],"structure":[10],"of":[11,52,105],"its":[12],"QR":[13,103],"factorization":[14,104],"to":[15,64,77,93,108],"devise":[16],"two":[17],"new":[18],"algorithms":[19,30,40,88],"for":[20,46,85],"computing":[21],"1-norm":[23,115],"condition":[24,116],"number":[25,117],"in":[26,67,96,118],"$O(n)$":[27,97,119],"operations.":[28,98,120],"The":[29],"avoid":[31],"underflow":[32],"and":[33,35],"overflow,":[34],"are":[36,43],"simpler":[37],"than":[38],"existing":[39,70],"since":[41],"tests":[42],"not":[44],"required":[45],"degenerate":[47],"cases.":[48],"An":[49],"error":[50],"analysis":[51],"first":[54],"algorithm":[55,61,111],"is":[56,62],"given,":[57],"while":[58],"second":[60],"shown":[63],"be":[65],"competitive":[66],"speed":[68],"with":[69],"algorithms.":[71],"We":[72,99],"then":[73],"turn":[74],"our":[75],"attention":[76],"diagonal-plus-semiseparable":[82],"matrix,":[83],"A,":[84],"which":[86],"several":[87],"have":[89],"recently":[90],"been":[91],"developed":[92],"solve":[94],"$Ax=b$":[95],"again":[100],"present":[109],"that":[112],"computes":[113]},"counts_by_year":[{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}
