{"id":"https://openalex.org/W1993483669","doi":"https://doi.org/10.1137/s0036144503424439","title":"On the Occurrence of Superlinear Convergence of Exact and Inexact Krylov Subspace Methods","display_name":"On the Occurrence of Superlinear Convergence of Exact and Inexact Krylov Subspace Methods","publication_year":2005,"publication_date":"2005-01-01","ids":{"openalex":"https://openalex.org/W1993483669","doi":"https://doi.org/10.1137/s0036144503424439","mag":"1993483669"},"language":"en","primary_location":{"id":"doi:10.1137/s0036144503424439","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036144503424439","pdf_url":null,"source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"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 Review","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/A5081841768","display_name":"Valeria Simoncini","orcid":"https://orcid.org/0000-0003-0795-5865"},"institutions":[{"id":"https://openalex.org/I9360294","display_name":"University of Bologna","ror":"https://ror.org/01111rn36","country_code":"IT","type":"education","lineage":["https://openalex.org/I9360294"]},{"id":"https://openalex.org/I2802536511","display_name":"Italian Aerospace Research Centre","ror":"https://ror.org/01cqx0m58","country_code":"IT","type":"other","lineage":["https://openalex.org/I2802536511"]},{"id":"https://openalex.org/I4210137033","display_name":"Istituto di Matematica Applicata e Tecnologie Informatiche","ror":"https://ror.org/03m0n3c07","country_code":"IT","type":"education","lineage":["https://openalex.org/I4210137033","https://openalex.org/I4210155236"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Valeria Simoncini","raw_affiliation_strings":["CIRSA, Ravenna, Italy","IMATI-CNR, Pavia, Italy;","Dipartimento di Matematica, Universit\u00e0 di Bologna, Piazza di Porta S. Donato, 5, I-40127 Bologna, Italy;"],"affiliations":[{"raw_affiliation_string":"CIRSA, Ravenna, Italy","institution_ids":["https://openalex.org/I2802536511"]},{"raw_affiliation_string":"IMATI-CNR, Pavia, Italy;","institution_ids":["https://openalex.org/I4210137033"]},{"raw_affiliation_string":"Dipartimento di Matematica, Universit\u00e0 di Bologna, Piazza di Porta S. Donato, 5, I-40127 Bologna, Italy;","institution_ids":["https://openalex.org/I9360294"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5047875137","display_name":"Daniel B. Szyld","orcid":"https://orcid.org/0000-0001-8010-0391"},"institutions":[{"id":"https://openalex.org/I84392919","display_name":"Temple University","ror":"https://ror.org/00kx1jb78","country_code":"US","type":"education","lineage":["https://openalex.org/I84392919"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Daniel B. Szyld","raw_affiliation_strings":["Department of Mathematics, Temple University (038-16), 1805 N. Broad Street, Philadelphia, PA 19122-6094"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Temple University (038-16), 1805 N. Broad Street, Philadelphia, PA 19122-6094","institution_ids":["https://openalex.org/I84392919"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5081841768"],"corresponding_institution_ids":["https://openalex.org/I2802536511","https://openalex.org/I4210137033","https://openalex.org/I9360294"],"apc_list":null,"apc_paid":null,"fwci":6.7335,"has_fulltext":false,"cited_by_count":72,"citation_normalized_percentile":{"value":0.9752796,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"47","issue":"2","first_page":"247","last_page":"272"},"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/T11233","display_name":"Advanced Adaptive Filtering Techniques","score":0.991599977016449,"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"}},{"id":"https://openalex.org/T11263","display_name":"Electromagnetic Simulation and Numerical Methods","score":0.9915000200271606,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"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/krylov-subspace","display_name":"Krylov subspace","score":0.9511784315109253},{"id":"https://openalex.org/keywords/generalized-minimal-residual-method","display_name":"Generalized minimal residual method","score":0.9441919922828674},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7235023975372314},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.7049000263214111},{"id":"https://openalex.org/keywords/invariant-subspace","display_name":"Invariant subspace","score":0.6846718192100525},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.6835250854492188},{"id":"https://openalex.org/keywords/subspace-topology","display_name":"Subspace topology","score":0.6514016389846802},{"id":"https://openalex.org/keywords/conjugate-residual-method","display_name":"Conjugate residual method","score":0.6451222896575928},{"id":"https://openalex.org/keywords/conjugate-gradient-method","display_name":"Conjugate gradient method","score":0.6226784586906433},{"id":"https://openalex.org/keywords/block","display_name":"Block (permutation group theory)","score":0.42985352873802185},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.2868109941482544},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.27700233459472656},{"id":"https://openalex.org/keywords/linear-subspace","display_name":"Linear subspace","score":0.23724287748336792},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.22865182161331177},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.18147066235542297},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.13464903831481934},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.08814841508865356},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.03936833143234253}],"concepts":[{"id":"https://openalex.org/C147060835","wikidata":"https://www.wikidata.org/wiki/Q1757151","display_name":"Krylov subspace","level":3,"score":0.9511784315109253},{"id":"https://openalex.org/C155332342","wikidata":"https://www.wikidata.org/wiki/Q1432976","display_name":"Generalized minimal residual method","level":3,"score":0.9441919922828674},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7235023975372314},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.7049000263214111},{"id":"https://openalex.org/C2777059694","wikidata":"https://www.wikidata.org/wiki/Q2706744","display_name":"Invariant subspace","level":3,"score":0.6846718192100525},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.6835250854492188},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.6514016389846802},{"id":"https://openalex.org/C91394653","wikidata":"https://www.wikidata.org/wiki/Q5161159","display_name":"Conjugate residual method","level":4,"score":0.6451222896575928},{"id":"https://openalex.org/C81184566","wikidata":"https://www.wikidata.org/wiki/Q1191895","display_name":"Conjugate gradient method","level":2,"score":0.6226784586906433},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.42985352873802185},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2868109941482544},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.27700233459472656},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.23724287748336792},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.22865182161331177},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.18147066235542297},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.13464903831481934},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.08814841508865356},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.03936833143234253},{"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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"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":4,"locations":[{"id":"doi:10.1137/s0036144503424439","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036144503424439","pdf_url":null,"source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"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 Review","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.129.7291","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.129.7291","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.temple.edu/~szyld/papers/sirev42443.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.8.6860","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.8.6860","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.temple.edu/~szyld/reports/sup_rep.ps","raw_type":"text"},{"id":"pmh:oai:cris.unibo.it:11585/10665","is_oa":false,"landing_page_url":"http://hdl.handle.net/11585/10665","pdf_url":null,"source":{"id":"https://openalex.org/S4306402579","display_name":"Archivio istituzionale della ricerca (Alma Mater Studiorum Universit\u00e0 di Bologna)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210117483","host_organization_name":"Istituto di Ematologia di Bologna","host_organization_lineage":["https://openalex.org/I4210117483"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities","score":0.5099999904632568}],"awards":[{"id":"https://openalex.org/G3115803529","display_name":"Flexible Krylov Methods and Schwarz Preconditioners","funder_award_id":"0207525","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6507447573","display_name":null,"funder_award_id":"several","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G848032724","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320323553","display_name":"Universit\u00e0 di Bologna","ror":"https://ror.org/01111rn36"},{"id":"https://openalex.org/F4320334079","display_name":"Gruppo Nazionale per il Calcolo Scientifico","ror":null}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":46,"referenced_works":["https://openalex.org/W83404773","https://openalex.org/W263597469","https://openalex.org/W658559791","https://openalex.org/W1506342804","https://openalex.org/W1522512418","https://openalex.org/W1539593706","https://openalex.org/W1551860266","https://openalex.org/W1560263223","https://openalex.org/W1563653928","https://openalex.org/W1569661565","https://openalex.org/W1575147392","https://openalex.org/W1970585283","https://openalex.org/W1977061454","https://openalex.org/W1981745143","https://openalex.org/W1984263725","https://openalex.org/W1984840642","https://openalex.org/W1990351032","https://openalex.org/W2006753818","https://openalex.org/W2010315317","https://openalex.org/W2012547784","https://openalex.org/W2013737143","https://openalex.org/W2015953024","https://openalex.org/W2032576830","https://openalex.org/W2040425799","https://openalex.org/W2048752392","https://openalex.org/W2055116131","https://openalex.org/W2061813658","https://openalex.org/W2063683773","https://openalex.org/W2064616889","https://openalex.org/W2067028470","https://openalex.org/W2068062961","https://openalex.org/W2068543774","https://openalex.org/W2083232784","https://openalex.org/W2083715026","https://openalex.org/W2084363278","https://openalex.org/W2085551594","https://openalex.org/W2094553393","https://openalex.org/W2108291106","https://openalex.org/W2116284739","https://openalex.org/W2140153041","https://openalex.org/W2167262293","https://openalex.org/W2798909945","https://openalex.org/W3022136271","https://openalex.org/W4205624883","https://openalex.org/W4230094279","https://openalex.org/W4302564868"],"related_works":["https://openalex.org/W2129625121","https://openalex.org/W4205386174","https://openalex.org/W1848849174","https://openalex.org/W2035892225","https://openalex.org/W4283656204","https://openalex.org/W1976157534","https://openalex.org/W4378945386","https://openalex.org/W2798286561","https://openalex.org/W2896548519","https://openalex.org/W1993483669"],"abstract_inverted_index":{"Krylov":[0],"subspace":[1,25,60],"methods":[2],"often":[3],"exhibit":[4],"superlinear":[5,16,48],"convergence.":[6],"We":[7,21],"present":[8],"a":[9,43],"general":[10],"analytic":[11],"model":[12],"which":[13],"describes":[14],"this":[15],"convergence,":[17],"when":[18],"it":[19],"occurs.":[20],"take":[22],"an":[23],"invariant":[24],"approach,":[26],"so":[27],"that":[28],"our":[29],"results":[30],"apply":[31],"also":[32],"to":[33,37],"inexact":[34,59],"methods,":[35],"and":[36,58],"nondiagonalizable":[38],"matrices.":[39],"Thus,":[40],"we":[41],"provide":[42],"unified":[44],"treatment":[45],"of":[46,50,56],"the":[47,65],"convergence":[49],"GMRES,":[51],"conjugate":[52],"gradients,":[53],"block":[54],"versions":[55],"these,":[57],"methods.":[61],"Numerical":[62],"experiments":[63],"illustrate":[64],"bounds":[66],"obtained.":[67]},"counts_by_year":[{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":4}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}
