{"id":"https://openalex.org/W2134009775","doi":"https://doi.org/10.1137/06067818x","title":"On the Error in QR Integration","display_name":"On the Error in QR Integration","publication_year":2008,"publication_date":"2008-01-01","ids":{"openalex":"https://openalex.org/W2134009775","doi":"https://doi.org/10.1137/06067818x","mag":"2134009775"},"language":"en","primary_location":{"id":"doi:10.1137/06067818x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/06067818x","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"http://hdl.handle.net/1808/17247","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5076407367","display_name":"Luca Dieci","orcid":"https://orcid.org/0000-0002-3618-9293"},"institutions":[{"id":"https://openalex.org/I130701444","display_name":"Georgia Institute of Technology","ror":"https://ror.org/01zkghx44","country_code":"US","type":"education","lineage":["https://openalex.org/I130701444"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Luca Dieci","raw_affiliation_strings":["School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332"],"affiliations":[{"raw_affiliation_string":"School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332","institution_ids":["https://openalex.org/I130701444"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5012033652","display_name":"Erik S. Van Vleck","orcid":"https://orcid.org/0000-0001-9203-8345"},"institutions":[{"id":"https://openalex.org/I146416000","display_name":"University of Kansas","ror":"https://ror.org/001tmjg57","country_code":"US","type":"education","lineage":["https://openalex.org/I146416000"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Erik S. Van Vleck","raw_affiliation_strings":["Department of Mathematics, University of Kansas, Lawrence, KS 66045"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Kansas, Lawrence, KS 66045","institution_ids":["https://openalex.org/I146416000"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5076407367"],"corresponding_institution_ids":["https://openalex.org/I130701444"],"apc_list":null,"apc_paid":null,"fwci":4.115,"has_fulltext":false,"cited_by_count":23,"citation_normalized_percentile":{"value":0.93949192,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"46","issue":"3","first_page":"1166","last_page":"1189"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9994999766349792,"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"}},"topics":[{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9994999766349792,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9994000196456909,"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/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9524999856948853,"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":0.7804378867149353},{"id":"https://openalex.org/keywords/qr-decomposition","display_name":"QR decomposition","score":0.5589997172355652},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4568430185317993},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.4192313551902771},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.180730938911438}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7804378867149353},{"id":"https://openalex.org/C188060507","wikidata":"https://www.wikidata.org/wiki/Q653242","display_name":"QR decomposition","level":3,"score":0.5589997172355652},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4568430185317993},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.4192313551902771},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.180730938911438},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","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},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/06067818x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/06067818x","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:kuscholarworks.ku.edu:1808/17247","is_oa":true,"landing_page_url":"http://hdl.handle.net/1808/17247","pdf_url":null,"source":{"id":"https://openalex.org/S4306400421","display_name":"KU ScholarWorks (The University of Kansas)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I146416000","host_organization_name":"University of Kansas","host_organization_lineage":["https://openalex.org/I146416000"],"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":"","raw_type":"Article"}],"best_oa_location":{"id":"pmh:oai:kuscholarworks.ku.edu:1808/17247","is_oa":true,"landing_page_url":"http://hdl.handle.net/1808/17247","pdf_url":null,"source":{"id":"https://openalex.org/S4306400421","display_name":"KU ScholarWorks (The University of Kansas)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I146416000","host_organization_name":"University of Kansas","host_organization_lineage":["https://openalex.org/I146416000"],"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":"","raw_type":"Article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","score":0.4300000071525574,"display_name":"Reduced inequalities"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":22,"referenced_works":["https://openalex.org/W628074600","https://openalex.org/W1480802543","https://openalex.org/W1976069054","https://openalex.org/W1976858936","https://openalex.org/W1977299049","https://openalex.org/W1988664436","https://openalex.org/W2005528159","https://openalex.org/W2005752655","https://openalex.org/W2014672563","https://openalex.org/W2016221866","https://openalex.org/W2020710505","https://openalex.org/W2023137937","https://openalex.org/W2031202067","https://openalex.org/W2036601166","https://openalex.org/W2055035157","https://openalex.org/W2062908485","https://openalex.org/W2065687090","https://openalex.org/W2069259920","https://openalex.org/W2094879944","https://openalex.org/W2123571060","https://openalex.org/W2162949801","https://openalex.org/W3141151088"],"related_works":["https://openalex.org/W4230623537","https://openalex.org/W2063488590","https://openalex.org/W2351859806","https://openalex.org/W4239376463","https://openalex.org/W1996810073","https://openalex.org/W2089811522","https://openalex.org/W2092244978","https://openalex.org/W2613209927","https://openalex.org/W1983663462","https://openalex.org/W2027224733"],"abstract_inverted_index":{"An":[0],"important":[1],"change":[2,41],"of":[3,21,42,114,123],"variables":[4],"for":[5,53,92,108],"a":[6,48,59],"linear":[7],"time":[8],"varying":[9],"system":[10],"$\\dot":[11],"x=A(t)x,":[12],"t\\ge":[13],"0$,":[14],"is":[15,86,131],"that":[16],"induced":[17],"by":[18,64],"the":[19,22,75,79,93,101,115,121,124,142],"QR-factorization":[20],"underlying":[23],"fundamental":[24],"matrix":[25,50],"solution:":[26],"$X=QR$,":[27],"with":[28],"Q":[29,63],"orthogonal":[30],"and":[31,110],"R":[32],"upper":[33],"triangular":[34],"(with":[35],"positive":[36],"diagonal).":[37],"To":[38],"find":[39],"this":[40,56,84],"variable,":[43],"one":[44,71],"needs":[45],"to":[46,62,73,87,106,138],"solve":[47],"nonlinear":[49],"differential":[51],"equation":[52],"Q.":[54,96],"Practically,":[55],"means":[57],"finding":[58],"numerical":[60],"approximation":[61],"using":[65],"some":[66],"appropriate":[67],"discretization":[68],"scheme,":[69],"whereby":[70],"attempts":[72],"control":[74],"local":[76,102],"error":[77,90,103],"during":[78],"integration.":[80],"Our":[81],"contribution":[82],"in":[83],"work":[85],"obtain":[88],"global":[89],"bounds":[91,98],"numerically":[94],"computed":[95],"These":[97],"depend":[99],"on":[100,111,120,141],"tolerance":[104],"used":[105],"integrate":[107],"Q,":[109],"structural":[112],"properties":[113],"problem":[116],"itself,":[117],"but":[118],"not":[119],"length":[122],"interval":[125],"over":[126],"which":[127],"we":[128],"integrate.":[129],"This":[130],"particularly":[132],"important,":[133],"since\u2014in":[134],"principle\u2014Q":[135],"may":[136],"need":[137],"be":[139],"found":[140],"half-line":[143],"$t\\ge":[144],"0$.":[145]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":6},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}