{"id":"https://openalex.org/W1977911260","doi":"https://doi.org/10.1002/(sici)1099-1506(199703/04)4:2<69::aid-nla98>3.0.co;2-f","title":"Asymptotic Convergence of Conjugate Gradient Methods for the Partial Symmetric Eigenproblem","display_name":"Asymptotic Convergence of Conjugate Gradient Methods for the Partial Symmetric Eigenproblem","publication_year":1997,"publication_date":"1997-03-01","ids":{"openalex":"https://openalex.org/W1977911260","doi":"https://doi.org/10.1002/(sici)1099-1506(199703/04)4:2<69::aid-nla98>3.0.co;2-f","mag":"1977911260"},"language":"en","primary_location":{"id":"doi:10.1002/(sici)1099-1506(199703/04)4:2<69::aid-nla98>3.0.co;2-f","is_oa":false,"landing_page_url":"https://doi.org/10.1002/(sici)1099-1506(199703/04)4:2<69::aid-nla98>3.0.co;2-f","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/A5089013612","display_name":"Luca Bergamaschi","orcid":"https://orcid.org/0000-0001-8273-9674"},"institutions":[{"id":"https://openalex.org/I138689650","display_name":"University of Padua","ror":"https://ror.org/00240q980","country_code":"IT","type":"education","lineage":["https://openalex.org/I138689650"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Luca Bergamaschi","raw_affiliation_strings":["Dip. di Metodi e Modelli Matematici per le Scienze Applicate, University of Padua, Via Belzoni 7, 35131 Padua, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dip. di Metodi e Modelli Matematici per le Scienze Applicate, University of Padua, Via Belzoni 7, 35131 Padua, Italy","institution_ids":["https://openalex.org/I138689650"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5084886129","display_name":"Giuseppe Gambolati","orcid":null},"institutions":[{"id":"https://openalex.org/I138689650","display_name":"University of Padua","ror":"https://ror.org/00240q980","country_code":"IT","type":"education","lineage":["https://openalex.org/I138689650"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Giuseppe Gambolati","raw_affiliation_strings":["Dip. di Metodi e Modelli Matematici per le Scienze Applicate, University of Padua, Via Belzoni 7, 35131 Padua, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dip. di Metodi e Modelli Matematici per le Scienze Applicate, University of Padua, Via Belzoni 7, 35131 Padua, Italy","institution_ids":["https://openalex.org/I138689650"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5017227319","display_name":"Giorgio Pini","orcid":null},"institutions":[{"id":"https://openalex.org/I138689650","display_name":"University of Padua","ror":"https://ror.org/00240q980","country_code":"IT","type":"education","lineage":["https://openalex.org/I138689650"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Giorgio Pini","raw_affiliation_strings":["Dip. di Metodi e Modelli Matematici per le Scienze Applicate, University of Padua, Via Belzoni 7, 35131 Padua, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dip. di Metodi e Modelli Matematici per le Scienze Applicate, University of Padua, Via Belzoni 7, 35131 Padua, Italy","institution_ids":["https://openalex.org/I138689650"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5089013612"],"corresponding_institution_ids":["https://openalex.org/I138689650"],"apc_list":{"value":4430,"currency":"USD","value_usd":4430},"apc_paid":null,"fwci":1.5542,"has_fulltext":false,"cited_by_count":46,"citation_normalized_percentile":{"value":0.81170599,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"4","issue":"2","first_page":"69","last_page":"84"},"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/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9937000274658203,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9894999861717224,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8692548274993896},{"id":"https://openalex.org/keywords/conjugate-gradient-method","display_name":"Conjugate gradient method","score":0.801399827003479},{"id":"https://openalex.org/keywords/hessian-matrix","display_name":"Hessian matrix","score":0.7973202466964722},{"id":"https://openalex.org/keywords/rayleigh-quotient","display_name":"Rayleigh quotient","score":0.738418459892273},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.6382911801338196},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.60845547914505},{"id":"https://openalex.org/keywords/conjugate","display_name":"Conjugate","score":0.4653162956237793},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.46492695808410645},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4594138562679291},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4556177258491516},{"id":"https://openalex.org/keywords/linear-subspace","display_name":"Linear subspace","score":0.4508703052997589},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4375406801700592},{"id":"https://openalex.org/keywords/derivation-of-the-conjugate-gradient-method","display_name":"Derivation of the conjugate gradient method","score":0.4254136085510254},{"id":"https://openalex.org/keywords/conjugate-residual-method","display_name":"Conjugate residual method","score":0.3542138338088989},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.21777567267417908},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.19542041420936584},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.12095966935157776},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.07600858807563782}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8692548274993896},{"id":"https://openalex.org/C81184566","wikidata":"https://www.wikidata.org/wiki/Q1191895","display_name":"Conjugate gradient method","level":2,"score":0.801399827003479},{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.7973202466964722},{"id":"https://openalex.org/C2778158742","wikidata":"https://www.wikidata.org/wiki/Q1665115","display_name":"Rayleigh quotient","level":3,"score":0.738418459892273},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.6382911801338196},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.60845547914505},{"id":"https://openalex.org/C197336794","wikidata":"https://www.wikidata.org/wiki/Q5161150","display_name":"Conjugate","level":2,"score":0.4653162956237793},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.46492695808410645},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4594138562679291},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4556177258491516},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.4508703052997589},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4375406801700592},{"id":"https://openalex.org/C198562538","wikidata":"https://www.wikidata.org/wiki/Q5262612","display_name":"Derivation of the conjugate gradient method","level":5,"score":0.4254136085510254},{"id":"https://openalex.org/C91394653","wikidata":"https://www.wikidata.org/wiki/Q5161159","display_name":"Conjugate residual method","level":4,"score":0.3542138338088989},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.21777567267417908},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.19542041420936584},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.12095966935157776},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07600858807563782},{"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/C119599485","wikidata":"https://www.wikidata.org/wiki/Q43035","display_name":"Electrical engineering","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/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"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/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1002/(sici)1099-1506(199703/04)4:2<69::aid-nla98>3.0.co;2-f","is_oa":false,"landing_page_url":"https://doi.org/10.1002/(sici)1099-1506(199703/04)4:2<69::aid-nla98>3.0.co;2-f","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"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.56.5074","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.5074","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.dmsa.unipd.it/~berga/Research/Eigen/nlaa.ps.gz","raw_type":"text"},{"id":"pmh:oai:www.research.unipd.it:11577/106847","is_oa":false,"landing_page_url":"http://hdl.handle.net/11577/106847","pdf_url":null,"source":{"id":"https://openalex.org/S4377196283","display_name":"Research Padua  Archive (University of Padua)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I138689650","host_organization_name":"University of Padua","host_organization_lineage":["https://openalex.org/I138689650"],"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":[{"id":"https://openalex.org/F4320322892","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":33,"referenced_works":["https://openalex.org/W119872124","https://openalex.org/W141418607","https://openalex.org/W1969468690","https://openalex.org/W1973391382","https://openalex.org/W1979974817","https://openalex.org/W1980357388","https://openalex.org/W1980396943","https://openalex.org/W1988662661","https://openalex.org/W1989926740","https://openalex.org/W1992208469","https://openalex.org/W2009737642","https://openalex.org/W2011000353","https://openalex.org/W2013737143","https://openalex.org/W2031605731","https://openalex.org/W2032371871","https://openalex.org/W2035864055","https://openalex.org/W2059400898","https://openalex.org/W2059586807","https://openalex.org/W2071128523","https://openalex.org/W2071371660","https://openalex.org/W2103270577","https://openalex.org/W2117686912","https://openalex.org/W2138543857","https://openalex.org/W2290679954","https://openalex.org/W2312462199","https://openalex.org/W2328789213","https://openalex.org/W2330913512","https://openalex.org/W3022136271","https://openalex.org/W4240648535","https://openalex.org/W4251656402","https://openalex.org/W6653932534","https://openalex.org/W6658458732","https://openalex.org/W6665700644"],"related_works":["https://openalex.org/W2353110519","https://openalex.org/W2356365850","https://openalex.org/W2098528027","https://openalex.org/W2375597358","https://openalex.org/W3159557833","https://openalex.org/W3159907817","https://openalex.org/W2383960336","https://openalex.org/W2129625121","https://openalex.org/W3123621400","https://openalex.org/W2410048189"],"abstract_inverted_index":{"Recently":[0],"an":[1,180],"efficient":[2],"method":[3],"(DACG)":[4],"for":[5,205],"the":[6,10,22,28,46,54,57,62,73,87,101,105,122,126,132,138,174,192,201,206],"partial":[7],"solution":[8,208],"of":[9,27,35,45,51,56,91,100,147,157,183,195],"symmetric":[11],"generalized":[12],"eigenproblem":[13],"Ax":[14],"=":[15],"\u03b4Bx":[16],"has":[17,179],"been":[18],"developed,":[19],"based":[20],"on":[21],"conjugate":[23],"gradient":[24],"(CG)":[25],"minimization":[26],"Rayleigh":[29,93],"quotient":[30,94],"over":[31],"successive":[32],"deflated":[33],"subspaces":[34],"decreasing":[36],"size.":[37],"The":[38],"present":[39],"paper":[40],"provides":[41],"a":[42,92,154,171],"numerical":[43],"analysis":[44],"asymptotic":[47,181],"convergence":[48,184],"rate":[49,182],"\u03c1j":[50,78],"DACG":[52,148],"in":[53,96,197],"calculation":[55],"eigenpair":[58],"\u03bbj,":[59],"uj,":[60],"when":[61,72],"scheme":[63],"is":[64,69,79,86,114,119,150],"preconditioned":[65],"with":[66,165,167,169,200],"A\u22121.":[67],"It":[68,113],"shown":[70,116],"that,":[71],"search":[74,159],"direction":[75],"are":[76,162],"A-conjugate,":[77],"well":[80],"approximated":[81],"by":[82,104,214],"4/\u03bej,":[83],"where":[84],"\u03bej":[85],"Hessian":[88],"condition":[89],"number":[90],"defined":[95],"appropriate":[97],"oblique":[98],"complements":[99],"space":[102],"spanned":[103],"leftmost":[106],"eigenvectors":[107],"u1,":[108],"u2,\u2026,":[109],"uj\u22121":[110],"already":[111],"calculated.":[112],"also":[115],"that":[117],"1/\u03bej":[118],"equal":[120],"to":[121,187,191,209],"relative":[123],"separation":[124],"between":[125],"eigenvalue":[127],"\u03bbj":[128,142],"currently":[129],"sought":[130],"and":[131,137],"next":[133,139],"higher":[134,140],"one":[135,141],"\u03bbj+1":[136],"+":[143],"1.":[144],"A":[145],"modification":[146],"(MDACG)":[149],"studied,":[151],"which":[152,161,185],"involves":[153],"new":[155],"set":[156],"CG":[158,207],"directions":[160],"made":[163],"M-conjugate,":[164,166,168],"M":[170],"matrix":[172],"approximating":[173],"Hessian.":[175],"By":[176],"distinction,":[177],"MDACG":[178],"appears":[186],"be":[188],"inversely":[189],"proportional":[190],"square":[193],"root":[194],"\u03bej,":[196],"complete":[198],"agreement":[199],"theoretical":[202],"results":[203],"known":[204],"linear":[210],"systems.":[211],"\u00a9":[212],"1997":[213],"John":[215],"Wiley":[216],"&":[217],"Sons,":[218],"Ltd.":[219]},"counts_by_year":[{"year":2025,"cited_by_count":4},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":3}],"updated_date":"2026-05-01T08:36:08.643496","created_date":"2025-10-10T00:00:00"}
