{"id":"https://openalex.org/W1971440257","doi":"https://doi.org/10.1002/(sici)1099-1506(199811/12)5:6<483::aid-nla156>3.0.co;2-7","title":"High quality preconditioning of a general symmetric positive definite matrix based on its UTU\u2005+\u2005UTR\u2005+\u2005RTU-decomposition","display_name":"High quality preconditioning of a general symmetric positive definite matrix based on its UTU\u2005+\u2005UTR\u2005+\u2005RTU-decomposition","publication_year":1998,"publication_date":"1998-11-01","ids":{"openalex":"https://openalex.org/W1971440257","doi":"https://doi.org/10.1002/(sici)1099-1506(199811/12)5:6<483::aid-nla156>3.0.co;2-7","mag":"1971440257"},"language":"en","primary_location":{"id":"doi:10.1002/(sici)1099-1506(199811/12)5:6<483::aid-nla156>3.0.co;2-7","is_oa":false,"landing_page_url":"https://doi.org/10.1002/(sici)1099-1506(199811/12)5:6<483::aid-nla156>3.0.co;2-7","pdf_url":null,"source":{"id":"https://openalex.org/S60324941","display_name":"Numerical Linear Algebra with Applications","issn_l":"1070-5325","issn":["1070-5325","1099-1506"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Linear Algebra with 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/A5075272796","display_name":"Igor Kaporin","orcid":"https://orcid.org/0000-0002-1451-1632"},"institutions":[{"id":"https://openalex.org/I1313323035","display_name":"Russian Academy of Sciences","ror":"https://ror.org/05qrfxd25","country_code":"RU","type":"funder","lineage":["https://openalex.org/I1313323035"]}],"countries":["RU"],"is_corresponding":true,"raw_author_name":"Igor E. Kaporin","raw_affiliation_strings":["Centre for Supercomputer and Massively Parallel Applications, Computing Centre of Russian Academy of Sciences, Vavilova 40, Moscow 117967, Russia"],"affiliations":[{"raw_affiliation_string":"Centre for Supercomputer and Massively Parallel Applications, Computing Centre of Russian Academy of Sciences, Vavilova 40, Moscow 117967, Russia","institution_ids":["https://openalex.org/I1313323035"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5075272796"],"corresponding_institution_ids":["https://openalex.org/I1313323035"],"apc_list":{"value":4430,"currency":"USD","value_usd":4430},"apc_paid":null,"fwci":4.0695,"has_fulltext":false,"cited_by_count":115,"citation_normalized_percentile":{"value":0.93703253,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":91,"max":99},"biblio":{"volume":"5","issue":"6","first_page":"483","last_page":"509"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":1.0,"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":1.0,"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/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9976000189781189,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"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/cholesky-decomposition","display_name":"Cholesky decomposition","score":0.8282123804092407},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7220292687416077},{"id":"https://openalex.org/keywords/positive-definite-matrix","display_name":"Positive-definite matrix","score":0.6583507657051086},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.591161847114563},{"id":"https://openalex.org/keywords/matrix-decomposition","display_name":"Matrix decomposition","score":0.5459836721420288},{"id":"https://openalex.org/keywords/triangular-matrix","display_name":"Triangular matrix","score":0.5320948958396912},{"id":"https://openalex.org/keywords/incomplete-cholesky-factorization","display_name":"Incomplete Cholesky factorization","score":0.520315408706665},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.48638418316841125},{"id":"https://openalex.org/keywords/singular-value-decomposition","display_name":"Singular value decomposition","score":0.47229912877082825},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4625987112522125},{"id":"https://openalex.org/keywords/symmetric-matrix","display_name":"Symmetric matrix","score":0.44535544514656067},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.42140650749206543},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4094894528388977},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3383399248123169},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2885967195034027},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.25357967615127563},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.23069527745246887},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.147409588098526},{"id":"https://openalex.org/keywords/invertible-matrix","display_name":"Invertible matrix","score":0.12068480253219604}],"concepts":[{"id":"https://openalex.org/C34727166","wikidata":"https://www.wikidata.org/wiki/Q515375","display_name":"Cholesky decomposition","level":3,"score":0.8282123804092407},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7220292687416077},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.6583507657051086},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.591161847114563},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.5459836721420288},{"id":"https://openalex.org/C104528550","wikidata":"https://www.wikidata.org/wiki/Q506265","display_name":"Triangular matrix","level":3,"score":0.5320948958396912},{"id":"https://openalex.org/C44363057","wikidata":"https://www.wikidata.org/wiki/Q6015160","display_name":"Incomplete Cholesky factorization","level":4,"score":0.520315408706665},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.48638418316841125},{"id":"https://openalex.org/C22789450","wikidata":"https://www.wikidata.org/wiki/Q420904","display_name":"Singular value decomposition","level":2,"score":0.47229912877082825},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4625987112522125},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.44535544514656067},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.42140650749206543},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4094894528388977},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3383399248123169},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2885967195034027},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.25357967615127563},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.23069527745246887},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.147409588098526},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.12068480253219604},{"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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","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/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1002/(sici)1099-1506(199811/12)5:6<483::aid-nla156>3.0.co;2-7","is_oa":false,"landing_page_url":"https://doi.org/10.1002/(sici)1099-1506(199811/12)5:6<483::aid-nla156>3.0.co;2-7","pdf_url":null,"source":{"id":"https://openalex.org/S60324941","display_name":"Numerical Linear Algebra with Applications","issn_l":"1070-5325","issn":["1070-5325","1099-1506"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Linear Algebra with Applications","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321079","display_name":"Russian Foundation for Basic Research","ror":"https://ror.org/02mh1ke95"},{"id":"https://openalex.org/F4320321800","display_name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek","ror":"https://ror.org/04jsz6e67"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W1546004968","https://openalex.org/W1969468690","https://openalex.org/W1974421439","https://openalex.org/W1985968347","https://openalex.org/W1986310461","https://openalex.org/W2006309149","https://openalex.org/W2013336356","https://openalex.org/W2024001269","https://openalex.org/W2029023151","https://openalex.org/W2052527407","https://openalex.org/W2057194215","https://openalex.org/W2071371660","https://openalex.org/W2074190947","https://openalex.org/W2087666987","https://openalex.org/W2110290705","https://openalex.org/W2134482604","https://openalex.org/W2164374667","https://openalex.org/W4300694379","https://openalex.org/W6646867389"],"related_works":["https://openalex.org/W2394244817","https://openalex.org/W2075453197","https://openalex.org/W2040413248","https://openalex.org/W2093365458","https://openalex.org/W2057318281","https://openalex.org/W2924783966","https://openalex.org/W1980238345","https://openalex.org/W2813261456","https://openalex.org/W2087974793","https://openalex.org/W3006906848"],"abstract_inverted_index":{"A":[0,7,59,118],"new":[1],"matrix":[2,24,41,58,99],"decomposition":[3,62],"of":[4,51,81,88,120,123,145,160,171,178,191],"the":[5,28,46,79,97,110,136,141,151,158,179],"form":[6],"=":[8,101],"UTU":[9],"+":[10,12],"UTR":[11],"RTU":[13],"is":[14,20,35,72,76,94,107,127,131,135,144,154,183],"proposed":[15,113,142,152],"and":[16,33,45,65,75,91,104,116,129,167,185],"investigated,":[17],"where":[18],"U":[19],"an":[21,54,82,124,188],"upper":[22,38],"triangular":[23,39],"(an":[25],"approximation":[26,121],"to":[27,157],"exact":[29],"Cholesky":[30],"factor":[31],"U0),":[32],"R":[34],"a":[36,61,105],"strictly":[37],"error":[40],"(with":[42],"small":[43],"elements":[44],"fill-in":[47],"limited":[48],"by":[49,78,114,187],"that":[50,133],"U0).":[52],"For":[53],"arbitrary":[55],"symmetric":[56],"positive":[57,173],"such":[60],"always":[63],"exists":[64],"can":[66],"be":[67],"efficiently":[68],"constructed;":[69],"however":[70],"it":[71,130],"not":[73],"unique,":[74],"determined":[77],"choice":[80],"involved":[83],"truncation":[84],"rule.":[85],"An":[86],"analysis":[87,159],"both":[89],"spectral":[90],"K-condition":[92],"numbers":[93],"given":[95],"for":[96,163],"preconditioned":[98],"M":[100],"U\u2212T":[102],"AU\u22121":[103],"comparison":[106],"made":[108],"with":[109],"RIC":[111,134],"preconditioning":[112,180],"Ajiz":[115],"Jennings.":[117],"concept":[119],"order":[122,138],"incomplete":[125],"factorization":[126],"introduced":[128],"shown":[132],"first":[137],"method,":[139],"whereas":[140],"method":[143,153],"second":[146],"order.":[147],"The":[148],"idea":[149],"underlying":[150],"also":[155],"applicable":[156],"CGNE-type":[161],"methods":[162],"general":[164],"non-singular":[165],"matrices":[166],"approximate":[168],"LU":[169],"factorizations":[170],"non-symmetric":[172],"definite":[174],"matrices.":[175],"Practical":[176],"use":[177],"techniques":[181],"developed":[182],"discussed":[184],"illustrated":[186],"extensive":[189],"set":[190],"numerical":[192],"examples.":[193],"Copyright":[194],"\u00a9":[195],"1999":[196],"John":[197],"Wiley":[198],"&":[199],"Sons,":[200],"Ltd.":[201]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":9},{"year":2020,"cited_by_count":7},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":9},{"year":2016,"cited_by_count":5},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":7},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":8}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}
