{"id":"https://openalex.org/W2982293719","doi":"https://doi.org/10.1137/18m1197400","title":"Block Modified Gram--Schmidt Algorithms and Their Analysis","display_name":"Block Modified Gram--Schmidt Algorithms and Their Analysis","publication_year":2019,"publication_date":"2019-01-01","ids":{"openalex":"https://openalex.org/W2982293719","doi":"https://doi.org/10.1137/18m1197400","mag":"2982293719"},"language":"en","primary_location":{"id":"doi:10.1137/18m1197400","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1197400","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/A5111535733","display_name":"Jesse L. Barlow","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Jesse L. Barlow","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5111535733"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.8191,"has_fulltext":false,"cited_by_count":20,"citation_normalized_percentile":{"value":0.87140158,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"40","issue":"4","first_page":"1257","last_page":"1290"},"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/T11697","display_name":"Numerical Methods and Algorithms","score":0.9976999759674072,"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/T10054","display_name":"Parallel Computing and Optimization Techniques","score":0.995199978351593,"subfield":{"id":"https://openalex.org/subfields/1708","display_name":"Hardware and Architecture"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7704235315322876},{"id":"https://openalex.org/keywords/qr-decomposition","display_name":"QR decomposition","score":0.5814008116722107},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5660419464111328},{"id":"https://openalex.org/keywords/block","display_name":"Block (permutation group theory)","score":0.5527587532997131},{"id":"https://openalex.org/keywords/factorization","display_name":"Factorization","score":0.5273780822753906},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5019567012786865},{"id":"https://openalex.org/keywords/lanczos-resampling","display_name":"Lanczos resampling","score":0.43725037574768066},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.42672306299209595},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.13843879103660583}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7704235315322876},{"id":"https://openalex.org/C188060507","wikidata":"https://www.wikidata.org/wiki/Q653242","display_name":"QR decomposition","level":3,"score":0.5814008116722107},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5660419464111328},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.5527587532997131},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.5273780822753906},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5019567012786865},{"id":"https://openalex.org/C119256216","wikidata":"https://www.wikidata.org/wiki/Q913012","display_name":"Lanczos resampling","level":3,"score":0.43725037574768066},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.42672306299209595},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.13843879103660583},{"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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","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:10.1137/18m1197400","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1197400","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":36,"referenced_works":["https://openalex.org/W212216823","https://openalex.org/W591632009","https://openalex.org/W1522703524","https://openalex.org/W1558296484","https://openalex.org/W1907157228","https://openalex.org/W1971662204","https://openalex.org/W1982698949","https://openalex.org/W1985389700","https://openalex.org/W1988425770","https://openalex.org/W1999600349","https://openalex.org/W2001120050","https://openalex.org/W2001135345","https://openalex.org/W2002257715","https://openalex.org/W2005498993","https://openalex.org/W2009355066","https://openalex.org/W2014436847","https://openalex.org/W2014724759","https://openalex.org/W2039778940","https://openalex.org/W2040388588","https://openalex.org/W2041519450","https://openalex.org/W2041797552","https://openalex.org/W2042696716","https://openalex.org/W2045635967","https://openalex.org/W2052087587","https://openalex.org/W2056167745","https://openalex.org/W2064122889","https://openalex.org/W2073260424","https://openalex.org/W2085551594","https://openalex.org/W2086351984","https://openalex.org/W2092590148","https://openalex.org/W2112979995","https://openalex.org/W2157237396","https://openalex.org/W2208990223","https://openalex.org/W2240679564","https://openalex.org/W2766487503","https://openalex.org/W3022909171"],"related_works":["https://openalex.org/W2357291060","https://openalex.org/W1484683478","https://openalex.org/W4239321356","https://openalex.org/W2003374977","https://openalex.org/W2743382746","https://openalex.org/W2377578636","https://openalex.org/W1601294834","https://openalex.org/W2031196824","https://openalex.org/W1673649602","https://openalex.org/W2084348954"],"abstract_inverted_index":{"New":[0],"block":[1,102,109,115,128,181],"modified":[2,170,220,252],"Gram--Schmidt":[3,221],"(BMGS)":[4],"methods":[5,27,144,152],"for":[6,141,149,278,315],"the":[7,67,82,90,156,180,217,231,266,289,293,302,308],"Q-R":[8,224,243,269],"factorization":[9],"of":[10,69,103,111,116,118,129,131,169,184,292,327],"a":[11,98,106,114,126,139,238,317],"full":[12],"column":[13,96],"rank":[14],"matrix":[15,91,263],"$X":[16,47],"\\in":[17,32,41],"{{R}}^{m":[18,33,42],"\\times":[19,34,43],"n}$,":[20],"$m":[21],"\\geq":[22],"n$,":[23],"are":[24,145,329],"considered.":[25],"Such":[26],"factor":[28,85],"$X$":[29,119],"into":[30],"$Q":[31],"n}$":[35,44],"and":[36,77,89,125,186,198,222,244,265,281],"an":[37,63,199],"upper":[38],"triangular":[39],"$R":[40],"such":[45,73,174],"that":[46,164,168,235,273,287],"=":[48],"QR,$":[49],"where,":[50],"in":[51,66],"exact":[52],"arithmetic,":[53],"$Q$":[54,86,132],"is":[55,87,92,120,138,165,259,285],"left":[56,83],"orthogonal":[57,84],"(i.e.,":[58],"$Q^T\\;{Q}=I_n$).":[59],"Gram--Schmidt-based":[60],"algorithms":[61,158,229],"play":[62],"important":[64],"role":[65],"implementation":[68],"Krylov":[70,112,151],"space":[71],"methods,":[72,113],"as":[74],"GMRES,":[75],"Arnoldi,":[76],"Lanczos.":[78],"For":[79,108],"these":[80,150],"applications,":[81],"needed":[88],"produced":[93],"either":[94],"one":[95,101],"at":[97,105,122],"time":[99],"or":[100],"columns":[104,117,130],"time.":[107],"implementations":[110],"introduced":[121],"each":[123],"step,":[124],"new":[127,227,313],"must":[133],"then":[134,305],"be":[135],"produced.":[136],"That":[137],"task":[140],"which":[142],"BMGS":[143,157,175,228,256,294,310,318],"ideally":[146],"suited.":[147],"However,":[148],"to":[153,160,167,241,251,301,325],"converge":[154],"properly,":[155],"need":[159],"have":[161,246,274],"numerical":[162],"behavior":[163],"similar":[166,239,250,324],"Gram--Schmidt.":[171,253],"To":[172],"design":[173],"algorithms,":[176],"we":[177],"build":[178],"upon":[179,262],"Householder":[182,223,242],"representation":[183],"Schreiber":[185],"Van":[187],"Loan":[188],"[SIAM":[189,207],"J.":[190,208],"Sci.":[191],"Stat.":[192],"Comput.,":[193],"10":[194],"(1989),":[195],"pp.":[196,214],"53--57]":[197],"observation":[200],"by":[201,205],"Charles":[202],"Sheffield":[203,232,303],"analyzed":[204],"Paige":[206],"Matrix":[209],"Anal.":[210],"Appl.,":[211],"31":[212],"(2009),":[213],"565--583]":[215],"about":[216],"relationship":[218,240],"between":[219],"factorization.":[225],"Our":[226],"exploit":[230],"framework":[233],"so":[234,306],"they":[236],"share":[237],"thus":[245],"error":[247,297,321],"analysis":[248,298,322],"properties":[249,299,323],"The":[254],"last":[255],"algorithm":[257,295,319],"developed":[258],"based":[260],"entirely":[261],"multiplications":[264],"\u201ctall,":[267],"skinny\u201d":[268],"(TSQR)":[270],"factorization---two":[271],"operations":[272],"been":[275],"studied":[276],"extensively":[277],"cache-based":[279],"architectures":[280],"distributed":[282],"architectures.":[283],"It":[284],"shown":[286],"if":[288],"TSQR":[290],"part":[291],"satisfies":[296],"connected":[300],"structure,":[304],"does":[307],"entire":[309],"algorithm.":[311],"Thus":[312],"criteria":[314],"when":[316],"has":[320],"those":[326],"MGS":[328],"proposed.":[330]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":2}],"updated_date":"2026-05-05T08:41:31.759640","created_date":"2025-10-10T00:00:00"}