{"id":"https://openalex.org/W7214926906","doi":"https://doi.org/10.48550/arxiv.2609.38101","title":"Solving Linear Systems in $\\widetilde{O}(mn \\log \\frac\u03ba\u03b5)$ Bit Operations","display_name":"Solving Linear Systems in $\\widetilde{O}(mn \\log \\frac\u03ba\u03b5)$ Bit Operations","publication_year":2026,"publication_date":"2026-09-29","ids":{"openalex":"https://openalex.org/W7214926906","doi":"https://doi.org/10.48550/arxiv.2609.38101"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.38101","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.38101","pdf_url":"https://arxiv.org/pdf/2609.38101","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/2609.38101","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5022068777","display_name":"Jonathan A. Kelner","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Kelner, Jonathan A.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5022068777"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":2,"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":0.4733999967575073,"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":0.4733999967575073,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.22550000250339508,"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/T11435","display_name":"Polynomial and algebraic computation","score":0.22460000216960907,"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/linear-solvers","display_name":"linear system solvers","score":0.6729999780654907},{"id":"https://openalex.org/keywords/sparse-matrices","display_name":"sparse matrices","score":0.7799999713897705},{"id":"https://openalex.org/keywords/bit-complexity","display_name":"bit complexity","score":0.5839999914169312},{"id":"https://openalex.org/keywords/deterministic-algorithms","display_name":"deterministic algorithms","score":0.7319999933242798},{"id":"https://openalex.org/keywords/condition-number","display_name":"condition number","score":0.7770000100135803},{"id":"https://openalex.org/keywords/modular-inverse","display_name":"modular inverse","score":0.671999990940094},{"id":"https://openalex.org/keywords/rational-reconstruction","display_name":"rational reconstruction","score":0.8339999914169312},{"id":"https://openalex.org/keywords/fixed-point-arithmetic","display_name":"fixed-point arithmetic","score":0.7080000042915344},{"id":"https://openalex.org/keywords/floating-point-arithmetic","display_name":"floating-point arithmetic","score":0.9599999785423279},{"id":"https://openalex.org/keywords/conjugate-gradient-method","display_name":"conjugate gradient method","score":0.6299999952316284},{"id":"https://openalex.org/keywords/nonsingular-matrices","display_name":"nonsingular matrices","score":0.42800000309944153}],"concepts":[{"id":"https://openalex.org/C136625980","wikidata":"https://www.wikidata.org/wiki/Q663208","display_name":"Rounding","level":2,"score":0.6550999879837036},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.635699987411499},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5672000050544739},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.5471000075340271},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5446000099182129},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.44110000133514404},{"id":"https://openalex.org/C17349429","wikidata":"https://www.wikidata.org/wiki/Q1049914","display_name":"Matrix multiplication","level":3,"score":0.43779999017715454},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.4348999857902527},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.414000004529953},{"id":"https://openalex.org/C81184566","wikidata":"https://www.wikidata.org/wiki/Q1191895","display_name":"Conjugate gradient method","level":2,"score":0.382099986076355},{"id":"https://openalex.org/C2780595030","wikidata":"https://www.wikidata.org/wiki/Q3860309","display_name":"Multiplication (music)","level":2,"score":0.3619000017642975},{"id":"https://openalex.org/C2078106","wikidata":"https://www.wikidata.org/wiki/Q14906620","display_name":"Accumulator (cryptography)","level":2,"score":0.3499999940395355},{"id":"https://openalex.org/C153269930","wikidata":"https://www.wikidata.org/wiki/Q430001","display_name":"Decidability","level":2,"score":0.3343999981880188},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.33379998803138733},{"id":"https://openalex.org/C2779960059","wikidata":"https://www.wikidata.org/wiki/Q7113681","display_name":"Overhead (engineering)","level":2,"score":0.33309999108314514},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33250001072883606},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.3310999870300293},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.3215000033378601},{"id":"https://openalex.org/C164680029","wikidata":"https://www.wikidata.org/wiki/Q1055293","display_name":"Multiple","level":2,"score":0.319599986076355},{"id":"https://openalex.org/C119677069","wikidata":"https://www.wikidata.org/wiki/Q4846249","display_name":"Integer factorization","level":4,"score":0.314300000667572},{"id":"https://openalex.org/C136886441","wikidata":"https://www.wikidata.org/wiki/Q926129","display_name":"Normalization (sociology)","level":2,"score":0.3140000104904175},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.3125},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.31119999289512634},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.31040000915527344},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.3089999854564667},{"id":"https://openalex.org/C96865113","wikidata":"https://www.wikidata.org/wiki/Q2946816","display_name":"Certificate","level":2,"score":0.29820001125335693},{"id":"https://openalex.org/C137559481","wikidata":"https://www.wikidata.org/wiki/Q1231787","display_name":"Dixon's factorization method","level":5,"score":0.29420000314712524},{"id":"https://openalex.org/C32049820","wikidata":"https://www.wikidata.org/wiki/Q319400","display_name":"Modular arithmetic","level":3,"score":0.2874999940395355},{"id":"https://openalex.org/C195906000","wikidata":"https://www.wikidata.org/wiki/Q1191722","display_name":"Matrix exponential","level":3,"score":0.2874999940395355},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2728999853134155},{"id":"https://openalex.org/C2778775528","wikidata":"https://www.wikidata.org/wiki/Q5135432","display_name":"Closing (real estate)","level":2,"score":0.271699994802475},{"id":"https://openalex.org/C101468663","wikidata":"https://www.wikidata.org/wiki/Q1620158","display_name":"Modular design","level":2,"score":0.2667999863624573},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.2632000148296356},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.26190000772476196},{"id":"https://openalex.org/C520416788","wikidata":"https://www.wikidata.org/wiki/Q5419229","display_name":"Exact solutions in general relativity","level":2,"score":0.2533000111579895},{"id":"https://openalex.org/C87698059","wikidata":"https://www.wikidata.org/wiki/Q1808960","display_name":"LTI system theory","level":3,"score":0.2506999969482422}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.38101","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.38101","pdf_url":"https://arxiv.org/pdf/2609.38101","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.2609.38101","doi":"https://doi.org/10.48550/arxiv.2609.38101","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.38101","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.38101","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.38101","pdf_url":"https://arxiv.org/pdf/2609.38101","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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214926906.pdf","grobid_xml":"https://content.openalex.org/works/W7214926906.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W3116248644","https://openalex.org/W7203702364"],"related_works":["https://openalex.org/W4414179560","https://openalex.org/W2960394841","https://openalex.org/W7119133639","https://openalex.org/W4220829346","https://openalex.org/W7100691371","https://openalex.org/W4292536401","https://openalex.org/W127267591","https://openalex.org/W2286693792","https://openalex.org/W2032080903","https://openalex.org/W2087096265"],"abstract_inverted_index":{"We":[0],"give":[1],"a":[2,7,88,136,163,214],"deterministic":[3],"algorithm":[4,52,60,118,129],"that":[5,110],"solves":[6],"nonsingular":[8],"linear":[9],"system":[10,133,146],"$Ax=b$,":[11],"where":[12],"$A\\in\\mathbb{R}^{n\\times":[13],"n}$":[14],"has":[15,111],"$m$":[16],"nonzero":[17],"entries":[18],"and":[19,63,85,102,157,195,202,213],"condition":[20],"number":[21],"$\u03ba$,":[22],"to":[23,130,200],"any":[24,54],"relative":[25],"residual":[26],"tolerance":[27],"$0<\u03b5\\le1/2$":[28],"using":[29,209],"$\\widetilde{O(}mn\\log(\u03ba/\u03b5))$":[30],"bit":[31],"operations":[32],"for":[33,53,113,135,224],"inputs":[34],"with":[35,46,78,107,167,205],"logarithmically":[36],"many":[37],"bits":[38],"per":[39],"entry.":[40],"For":[41,122],"sparse,":[42],"polynomially":[43],"conditioned":[44],"systems":[45],"$m=\\widetilde{O}(n)$,":[47],"this":[48,73,145],"gives":[49],"an":[50,187],"$\\widetilde{O}(n^2)$":[51],"inverse-polynomial":[55],"accuracy,":[56],"improving":[57],"on":[58,169],"the":[59,79,91,95,103,131,142,178,182,191,198,222],"of":[61,94,144,181,190,227],"Peng":[62],"Vempala,":[64],"as":[65],"sharpened":[66],"by":[67],"Nie,":[68],"whose":[69],"running":[70,104],"time":[71,105],"in":[72,99],"regime":[74],"is":[75,119,147,155,186],"approximately":[76],"$O(n^{2.2707})$":[77],"best":[80],"current":[81],"matrix":[82,143],"multiplication":[83],"exponent,":[84],"largely":[86],"closing":[87],"gap":[89],"between":[90],"idealized":[92],"performance":[93],"conjugate":[96],"gradient":[97],"method":[98],"exact":[100,179],"arithmetic":[101],"achievable":[106],"finite-precision":[108],"computation":[109],"persisted":[112],"over":[114],"70":[115],"years.":[116],"The":[117],"surprisingly":[120],"simple.":[121],"integer":[123,138,211],"inputs,":[124,204],"we":[125],"apply":[126],"Dixon's":[127],"lifting":[128,159],"perturbed":[132,183],"$(A+I/R)x=b$":[134],"suitable":[137],"$R$.":[139],"After":[140],"scaling,":[141],"$RA+I\\equiv":[148],"I\\pmod":[149],"R$,":[150],"so":[151],"its":[152],"modular":[153],"inverse":[154],"trivial,":[156],"each":[158],"step":[160],"needs":[161],"only":[162],"sparse":[164],"matrix-vector":[165],"product":[166],"$A$":[168],"$O(\\log":[170],"R)$-bit":[171],"numbers.":[172],"Fast":[173],"rational":[174],"reconstruction":[175],"then":[176],"recovers":[177],"solution":[180,189],"system,":[184],"which":[185],"$\u03b5$-accurate":[188],"original":[192],"one.":[193],"Normalization":[194],"rounding":[196],"extend":[197],"result":[199],"fixed-point":[201],"floating-point":[203,206],"outputs":[207],"represented":[208],"short":[210],"significands":[212],"common":[215],"encoded":[216],"exponent.":[217],"A":[218],"computable":[219],"certificate":[220],"removes":[221],"need":[223],"prior":[225],"knowledge":[226],"$\u03ba$.":[228]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-01T00:00:00"}
