{"id":"https://openalex.org/W7213915019","doi":"https://doi.org/10.48550/arxiv.2609.21540","title":"Element-wise Convergence Behavior of Subspace Iteration","display_name":"Element-wise Convergence Behavior of Subspace Iteration","publication_year":2026,"publication_date":"2026-09-18","ids":{"openalex":"https://openalex.org/W7213915019","doi":"https://doi.org/10.48550/arxiv.2609.21540"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.21540","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.21540","pdf_url":"https://arxiv.org/pdf/2609.21540","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.21540","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5152379839","display_name":"Mingyang Zhao","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Zhao, Mingyang","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5152379839"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9990000128746033,"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":0.9990000128746033,"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11697","display_name":"Numerical Methods and Algorithms","score":9.999999747378752e-05,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/subspace-iteration","display_name":"subspace iteration","score":0.9990000128746033},{"id":"https://openalex.org/keywords/asymptotic-analysis","display_name":"asymptotic analysis","score":0.5519999861717224},{"id":"https://openalex.org/keywords/ritz-vectors","display_name":"Ritz vectors","score":0.871999979019165},{"id":"https://openalex.org/keywords/diagonal-matrices","display_name":"diagonal matrices","score":0.5889999866485596},{"id":"https://openalex.org/keywords/upper-triangular-matrices","display_name":"upper triangular matrices","score":0.7789999842643738},{"id":"https://openalex.org/keywords/lu-decomposition","display_name":"LU decomposition","score":0.7789999842643738},{"id":"https://openalex.org/keywords/eigenvalue-problem","display_name":"eigenvalue problem","score":0.4650000035762787},{"id":"https://openalex.org/keywords/krylov-subspace-methods","display_name":"Krylov subspace methods","score":0.7419999837875366}],"concepts":[{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.839900016784668},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7645999789237976},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6075999736785889},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5831000208854675},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.5670999884605408},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5453000068664551},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.5396999716758728},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.513700008392334},{"id":"https://openalex.org/C147060835","wikidata":"https://www.wikidata.org/wiki/Q1757151","display_name":"Krylov subspace","level":3,"score":0.4966000020503998},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.43630000948905945},{"id":"https://openalex.org/C96214315","wikidata":"https://www.wikidata.org/wiki/Q727103","display_name":"Unitary matrix","level":3,"score":0.424699991941452},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.36959999799728394},{"id":"https://openalex.org/C44292817","wikidata":"https://www.wikidata.org/wiki/Q333871","display_name":"Orthogonal matrix","level":3,"score":0.31630000472068787},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3003000020980835},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.29739999771118164},{"id":"https://openalex.org/C106135958","wikidata":"https://www.wikidata.org/wiki/Q7291993","display_name":"Random subspace method","level":3,"score":0.2849999964237213},{"id":"https://openalex.org/C85817219","wikidata":"https://www.wikidata.org/wiki/Q884772","display_name":"Block matrix","level":3,"score":0.28439998626708984},{"id":"https://openalex.org/C67820243","wikidata":"https://www.wikidata.org/wiki/Q179164","display_name":"Unitary state","level":2,"score":0.2827000021934509},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.2822999954223633},{"id":"https://openalex.org/C162443888","wikidata":"https://www.wikidata.org/wiki/Q1426504","display_name":"Power iteration","level":3,"score":0.28130000829696655},{"id":"https://openalex.org/C113313756","wikidata":"https://www.wikidata.org/wiki/Q332791","display_name":"Diagonal matrix","level":3,"score":0.27799999713897705},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.2734000086784363},{"id":"https://openalex.org/C48216909","wikidata":"https://www.wikidata.org/wiki/Q6889156","display_name":"Modes of convergence (annotated index)","level":5,"score":0.2599000036716461},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.25850000977516174},{"id":"https://openalex.org/C132954091","wikidata":"https://www.wikidata.org/wiki/Q194397","display_name":"Banach space","level":2,"score":0.257999986410141},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.25679999589920044}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.21540","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.21540","pdf_url":"https://arxiv.org/pdf/2609.21540","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.21540","doi":"https://doi.org/10.48550/arxiv.2609.21540","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.21540","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.21540","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.21540","pdf_url":"https://arxiv.org/pdf/2609.21540","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213915019.pdf","grobid_xml":"https://content.openalex.org/works/W7213915019.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2141193768","https://openalex.org/W2522981378","https://openalex.org/W2156553331","https://openalex.org/W7095335384","https://openalex.org/W7095147800","https://openalex.org/W2276291465","https://openalex.org/W7010109566","https://openalex.org/W3098728793","https://openalex.org/W7097046016","https://openalex.org/W2113068912"],"abstract_inverted_index":{"This":[0],"paper":[1],"studies":[2],"the":[3,28,34,46,53,67,71,82,96,112,121,146,162,176],"element-wise":[4,117,159],"convergence":[5,163],"behavior":[6,164],"of":[7,30,33,84,99,148,161,165,174],"subspace":[8,40,68,166],"iteration.":[9,167],"All":[10],"results":[11,156,177],"are":[12],"established":[13,106],"for":[14,64,107,120,127],"both":[15,42],"\\(\\mathbb{F}=\\mathbb{R}\\)":[16],"and":[17,44,81,145],"\\(\\mathbb{F}=\\mathbb{C}\\).":[18],"For":[19,111],"a":[20],"diagonal":[21],"matrix":[22,36,55],"\\(\u039b=\\mathrm{diag}(\u03bb_1,\\ldots,\u03bb_n)\\)":[23],"with":[24,45,75,91,131,139],"\\(|\u03bb_1|>\\cdots>|\u03bb_n|>0\\),":[25],"we":[26,58,115],"analyze":[27],"modulus":[29],"each":[31],"entry":[32],"iterative":[35],"sequence":[37],"generated":[38],"by":[39,95],"iteration,":[41],"without":[43,70],"Rayleigh--Ritz":[47,72,113],"procedure.":[48],"Under":[49],"mild":[50],"assumptions":[51],"on":[52],"initial":[54],"\\(X\\in\\mathbb{F}^{n\\times":[56],"m}\\),":[57],"first":[59],"derive":[60],"exact":[61],"asymptotic":[62,118],"expressions":[63],"\\(|Q_k(i,j)|\\)":[65],"in":[66],"iteration":[69],"procedure:":[73],"entries":[74,130,138],"\\(i\\neq":[76],"j\\)":[77],"decay":[78,134,142],"as":[79,89,135,143,153],"\\(|\u03bb_{\\max\\{i,j\\}}/\u03bb_{\\min\\{i,j\\}}|^k\\),":[80],"deviation":[83,147],"\\(|Q_k(j,j)|\\)":[85],"from":[86,150],"\\(1\\)":[87,151],"decays":[88,152],"\\(\\max\\{|\u03bb_j/\u03bb_{j-1}|,|\u03bb_{j+1}/\u03bb_j|\\}^{2k}\\),":[90],"explicit":[92],"coefficients":[93],"determined":[94],"LU":[97],"factorization":[98],"\\(X\\).":[100],"Similar":[101],"conclusions":[102],"could":[103,178],"also":[104],"be":[105,179],"upper":[108],"triangular":[109],"matrices.":[110],"variant,":[114],"obtain":[116],"formulas":[119],"normalized":[122],"Ritz":[123],"vectors":[124],"\\(Z_k\\).":[125],"Specifically,":[126],"\\(|Z_k(i,j)|\\),":[128],"off-diagonal":[129,137],"\\(i\\geq":[132],"m+1\\)":[133],"\\(|\u03bb_i/\u03bb_j|^k\\),":[136],"\\(i\\leq":[140],"m\\)":[141],"\\(|\u03bb_{m+1}^2/(\u03bb_i\u03bb_j)|^k\\),":[144],"\\(|Z_k(j,j)|\\)":[149],"\\(|\u03bb_{m+1}/\u03bb_j|^{2k}\\).":[154],"These":[155],"give":[157],"an":[158,169],"description":[160],"After":[168],"orthogonal":[170],"or":[171],"unitary":[172],"change":[173],"basis,":[175],"further":[180],"generalized.":[181]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-09-22T00:00:00"}