{"id":"https://openalex.org/W2082559624","doi":"https://doi.org/10.1137/070681478","title":"A Fast Newton's Method for a Nonsymmetric Algebraic Riccati Equation","display_name":"A Fast Newton's Method for a Nonsymmetric Algebraic Riccati Equation","publication_year":2008,"publication_date":"2008-01-01","ids":{"openalex":"https://openalex.org/W2082559624","doi":"https://doi.org/10.1137/070681478","mag":"2082559624"},"language":"en","primary_location":{"id":"doi:10.1137/070681478","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070681478","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/A5023356208","display_name":"Dario A. Bini","orcid":"https://orcid.org/0000-0002-5885-9411"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Dario A. Bini","raw_affiliation_strings":["bini@mail.dm.unipi.it#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"bini@mail.dm.unipi.it#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5036085823","display_name":"Bruno Iannazzo","orcid":"https://orcid.org/0000-0002-3098-3824"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bruno Iannazzo","raw_affiliation_strings":["bruno.iannazzo@uninsubria.it#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"bruno.iannazzo@uninsubria.it#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5031634329","display_name":"Federico Poloni","orcid":"https://orcid.org/0000-0003-2291-990X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Federico Poloni","raw_affiliation_strings":["[email\u00a0protected]#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"[email\u00a0protected]#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":7.0731,"has_fulltext":false,"cited_by_count":64,"citation_normalized_percentile":{"value":0.97050369,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"30","issue":"1","first_page":"276","last_page":"290"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9995999932289124,"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":0.9995999932289124,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9951000213623047,"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/T11206","display_name":"Model Reduction and Neural Networks","score":0.9921000003814697,"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":0.8433263301849365},{"id":"https://openalex.org/keywords/riccati-equation","display_name":"Riccati equation","score":0.6762128472328186},{"id":"https://openalex.org/keywords/algebraic-riccati-equation","display_name":"Algebraic Riccati equation","score":0.6176475286483765},{"id":"https://openalex.org/keywords/jacobian-matrix-and-determinant","display_name":"Jacobian matrix and determinant","score":0.5385656356811523},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.5316525101661682},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4604073464870453},{"id":"https://openalex.org/keywords/newtons-method","display_name":"Newton's method","score":0.45651766657829285},{"id":"https://openalex.org/keywords/singularity","display_name":"Singularity","score":0.425246000289917},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.4247053265571594},{"id":"https://openalex.org/keywords/linear-algebra","display_name":"Linear algebra","score":0.4197940528392792},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.40449365973472595},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.34413859248161316},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.28937309980392456},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2696019411087036},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.20166224241256714},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.1929481327533722},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10270822048187256},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.09483152627944946}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8433263301849365},{"id":"https://openalex.org/C45473103","wikidata":"https://www.wikidata.org/wiki/Q851503","display_name":"Riccati equation","level":3,"score":0.6762128472328186},{"id":"https://openalex.org/C13847129","wikidata":"https://www.wikidata.org/wiki/Q4723989","display_name":"Algebraic Riccati equation","level":4,"score":0.6176475286483765},{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.5385656356811523},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5316525101661682},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4604073464870453},{"id":"https://openalex.org/C85189116","wikidata":"https://www.wikidata.org/wiki/Q374195","display_name":"Newton's method","level":3,"score":0.45651766657829285},{"id":"https://openalex.org/C16171025","wikidata":"https://www.wikidata.org/wiki/Q863349","display_name":"Singularity","level":2,"score":0.425246000289917},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4247053265571594},{"id":"https://openalex.org/C139352143","wikidata":"https://www.wikidata.org/wiki/Q82571","display_name":"Linear algebra","level":2,"score":0.4197940528392792},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.40449365973472595},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.34413859248161316},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.28937309980392456},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2696019411087036},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.20166224241256714},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.1929481327533722},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10270822048187256},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.09483152627944946},{"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","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},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/070681478","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070681478","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"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.511.5470","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.511.5470","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://fph.altervista.org/acad/files/bip_preprint.pdf","raw_type":"text"},{"id":"pmh:oai:arpi.unipi.it:11568/180598","is_oa":false,"landing_page_url":"http://hdl.handle.net/11568/180598","pdf_url":null,"source":{"id":"https://openalex.org/S4377196265","display_name":"CINECA IRIS Institutial research information system (University of Pisa)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I108290504","host_organization_name":"University of Pisa","host_organization_lineage":["https://openalex.org/I108290504"],"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":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W1500921805","https://openalex.org/W1981576702","https://openalex.org/W1983184346","https://openalex.org/W1984265227","https://openalex.org/W1997228246","https://openalex.org/W2013108889","https://openalex.org/W2017636104","https://openalex.org/W2022067462","https://openalex.org/W2023471726","https://openalex.org/W2036932727","https://openalex.org/W2056589646","https://openalex.org/W2102352719","https://openalex.org/W2114349947","https://openalex.org/W2127238600","https://openalex.org/W2166298394","https://openalex.org/W2167493384","https://openalex.org/W2170005034","https://openalex.org/W2798909945","https://openalex.org/W4243081751"],"related_works":["https://openalex.org/W2073802676","https://openalex.org/W2949824413","https://openalex.org/W2114783212","https://openalex.org/W1975873608","https://openalex.org/W2599212506","https://openalex.org/W582220655","https://openalex.org/W2893597281","https://openalex.org/W1494550340","https://openalex.org/W2678466775","https://openalex.org/W2059893998"],"abstract_inverted_index":{"A":[0],"special":[1],"instance":[2],"of":[3,27,54,127,136,191],"the":[4,10,24,36,52,55,81,90,97,109,112,128,137,150,155,159,164,189,192],"algebraic":[5],"Riccati":[6,147],"equation":[7,141,148],"$XCX-XE-AX+B=0$":[8],"where":[9,108,149],"$n\\times":[11],"n$":[12],"matrix":[13,156],"coefficients":[14,157],"$A,B,C,E$":[15],"are":[16,195],"rank":[17],"structured":[18],"matrices":[19],"is":[20,32,48,93,115,142,152],"considered.":[21],"Relying":[22],"on":[23],"structural":[25],"properties":[26],"Cauchy-like":[28],"matrices,":[29],"an":[30,125,145],"algorithm":[31,56,92,173],"designed":[33],"for":[34],"performing":[35],"customary":[37,98],"Newton":[38,99],"iteration":[39],"in":[40,61,80,131,163],"$O(n^2)$":[41,73,176],"arithmetic":[42],"operations":[43],"(ops).":[44],"The":[45,139],"same":[46,160],"technique":[47,130],"used":[49],"to":[50,72,96,121,133,169],"reduce":[51],"cost":[53],"proposed":[57],"by":[58,101],"L.-Z.":[59],"Lu":[60],"[Numer.":[62],"Linear":[63],"Algebra":[64],"Appl.,":[65],"12":[66],"(2005),":[67],"pp.":[68],"191\u2013200]":[69],"from":[70],"$O(n^3)$":[71],"ops":[74],"while":[75,154],"still":[76],"preserving":[77],"quadratic":[78,118],"convergence":[79,119],"generic":[82],"case.":[83],"As":[84],"a":[85,170],"byproduct":[86],"we":[87,123],"show":[88],"that":[89],"latter":[91],"closely":[94],"related":[95],"method":[100],"simple":[102],"formal":[103],"relations.":[104],"In":[105],"critical":[106],"cases":[107],"Jacobian":[110],"at":[111],"required":[113],"solution":[114],"singular":[116],"and":[117,185],"turns":[120],"linear,":[122],"provide":[124],"adaptation":[126],"shift":[129],"order":[132],"get":[134],"rid":[135],"singularity.":[138],"original":[140,165],"transformed":[143],"into":[144],"equivalent":[146],"singularity":[151],"removed":[153],"maintain":[158],"structure":[161],"as":[162],"equation.":[166],"This":[167],"leads":[168],"quadratically":[171],"convergent":[172],"with":[174,180],"complexity":[175],"which":[177,187],"provides":[178],"approximations":[179],"full":[181],"precision.":[182],"Numerical":[183],"experiments":[184],"comparisons":[186],"confirm":[188],"effectiveness":[190],"new":[193],"approach":[194],"reported.":[196]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":10}],"updated_date":"2026-07-15T18:14:33.161393","created_date":"2025-10-10T00:00:00"}
