{"id":"https://openalex.org/W2072004616","doi":"https://doi.org/10.1080/00207160412331289065","title":"Multiple search direction conjugate gradient method II: theory and numerical experiments","display_name":"Multiple search direction conjugate gradient method II: theory and numerical experiments","publication_year":2004,"publication_date":"2004-10-01","ids":{"openalex":"https://openalex.org/W2072004616","doi":"https://doi.org/10.1080/00207160412331289065","mag":"2072004616"},"language":"en","primary_location":{"id":"doi:10.1080/00207160412331289065","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160412331289065","pdf_url":null,"source":{"id":"https://openalex.org/S124867444","display_name":"International Journal of Computer Mathematics","issn_l":"0020-7160","issn":["0020-7160","1026-7425","1029-0265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computer Mathematics","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/A5100858883","display_name":"Tong-Xiang Gu","orcid":"https://orcid.org/0000-0002-6175-7061"},"institutions":[{"id":"https://openalex.org/I126792128","display_name":"Institute of Applied Physics","ror":"https://ror.org/01km14z09","country_code":"UA","type":"facility","lineage":["https://openalex.org/I126792128","https://openalex.org/I149851306"]},{"id":"https://openalex.org/I4210108629","display_name":"Computer Network Information Center","ror":"https://ror.org/01s0wyf50","country_code":"CN","type":"facility","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210108629"]},{"id":"https://openalex.org/I4210145278","display_name":"Institute of Applied Physics and Computational Mathematics","ror":"https://ror.org/03sxpbt26","country_code":"CN","type":"facility","lineage":["https://openalex.org/I4210145278"]}],"countries":["CN","UA"],"is_corresponding":true,"raw_author_name":"Tongxiang Gu","raw_affiliation_strings":["Laboratory of Computational Physics , Institute of Applied Physics and Computational Mathematics , P.O. Box 8009, Beijing, 100088, P.R. China ; Supercomputing Center of Computer Network Information Center , Chinese Academy of Science , P.O. Box 349, Beijing, 100080, P.R. China","Supercomputing Center of Computer Network Information Center , Chinese Academy of Science , P.O. Box 349, Beijing, 100080, P.R. China","Laboratory of Computational Physics , Institute of Applied Physics and Computational Mathematics , P.O. Box 8009, Beijing, 100088, P.R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratory of Computational Physics , Institute of Applied Physics and Computational Mathematics , P.O. Box 8009, Beijing, 100088, P.R. China ; Supercomputing Center of Computer Network Information Center , Chinese Academy of Science , P.O. Box 349, Beijing, 100080, P.R. China","institution_ids":["https://openalex.org/I126792128","https://openalex.org/I4210108629"]},{"raw_affiliation_string":"Supercomputing Center of Computer Network Information Center , Chinese Academy of Science , P.O. Box 349, Beijing, 100080, P.R. China","institution_ids":["https://openalex.org/I4210108629"]},{"raw_affiliation_string":"Laboratory of Computational Physics , Institute of Applied Physics and Computational Mathematics , P.O. Box 8009, Beijing, 100088, P.R. China","institution_ids":["https://openalex.org/I4210145278"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5059821211","display_name":"Xingping Liu","orcid":"https://orcid.org/0000-0003-1831-9290"},"institutions":[{"id":"https://openalex.org/I4210145278","display_name":"Institute of Applied Physics and Computational Mathematics","ror":"https://ror.org/03sxpbt26","country_code":"CN","type":"facility","lineage":["https://openalex.org/I4210145278"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xingping Liu","raw_affiliation_strings":["Laboratory of Computational Physics , Institute of Applied Physics and Computational Mathematics , P.O. Box 8009, Beijing, 100088, P.R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratory of Computational Physics , Institute of Applied Physics and Computational Mathematics , P.O. Box 8009, Beijing, 100088, P.R. China","institution_ids":["https://openalex.org/I4210145278"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5054583215","display_name":"Zeyao Mo","orcid":"https://orcid.org/0000-0003-3280-5682"},"institutions":[{"id":"https://openalex.org/I4210145278","display_name":"Institute of Applied Physics and Computational Mathematics","ror":"https://ror.org/03sxpbt26","country_code":"CN","type":"facility","lineage":["https://openalex.org/I4210145278"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zeyao Mo","raw_affiliation_strings":["Laboratory of Computational Physics , Institute of Applied Physics and Computational Mathematics , P.O. Box 8009, Beijing, 100088, P.R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratory of Computational Physics , Institute of Applied Physics and Computational Mathematics , P.O. Box 8009, Beijing, 100088, P.R. China","institution_ids":["https://openalex.org/I4210145278"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5073200513","display_name":"Xuebin Chi","orcid":"https://orcid.org/0000-0002-0150-5815"},"institutions":[{"id":"https://openalex.org/I4210108629","display_name":"Computer Network Information Center","ror":"https://ror.org/01s0wyf50","country_code":"CN","type":"facility","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210108629"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xuebin Chi","raw_affiliation_strings":["Supercomputing Center of Computer Network Information Center , Chinese Academy of Science , P.O. Box 349, Beijing, 100080, P.R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Supercomputing Center of Computer Network Information Center , Chinese Academy of Science , P.O. Box 349, Beijing, 100080, P.R. China","institution_ids":["https://openalex.org/I4210108629"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5100858883"],"corresponding_institution_ids":["https://openalex.org/I126792128","https://openalex.org/I4210108629","https://openalex.org/I4210145278"],"apc_list":null,"apc_paid":null,"fwci":0.4454,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.66242578,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"81","issue":"10","first_page":"1289","last_page":"1307"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9994000196456909,"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"}},"topics":[{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9994000196456909,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9976999759674072,"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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9882000088691711,"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/conjugate-gradient-method","display_name":"Conjugate gradient method","score":0.9059916734695435},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7102415561676025},{"id":"https://openalex.org/keywords/consistency","display_name":"Consistency (knowledge bases)","score":0.6730852723121643},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6444917321205139},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.6242479085922241},{"id":"https://openalex.org/keywords/nonlinear-conjugate-gradient-method","display_name":"Nonlinear conjugate gradient method","score":0.6192311644554138},{"id":"https://openalex.org/keywords/derivation-of-the-conjugate-gradient-method","display_name":"Derivation of the conjugate gradient method","score":0.5627048015594482},{"id":"https://openalex.org/keywords/conjugate-residual-method","display_name":"Conjugate residual method","score":0.5608554482460022},{"id":"https://openalex.org/keywords/conjugate","display_name":"Conjugate","score":0.560761034488678},{"id":"https://openalex.org/keywords/domain-decomposition-methods","display_name":"Domain decomposition methods","score":0.5339049100875854},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.475965678691864},{"id":"https://openalex.org/keywords/domain","display_name":"Domain (mathematical analysis)","score":0.4728921949863434},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4665006995201111},{"id":"https://openalex.org/keywords/gradient-method","display_name":"Gradient method","score":0.45288893580436707},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3790454566478729},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.25843894481658936},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.22330212593078613},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.18750813603401184},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.14911282062530518},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.11556726694107056},{"id":"https://openalex.org/keywords/key","display_name":"Key (lock)","score":0.06009310483932495},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.05701911449432373}],"concepts":[{"id":"https://openalex.org/C81184566","wikidata":"https://www.wikidata.org/wiki/Q1191895","display_name":"Conjugate gradient method","level":2,"score":0.9059916734695435},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7102415561676025},{"id":"https://openalex.org/C2776436953","wikidata":"https://www.wikidata.org/wiki/Q5163215","display_name":"Consistency (knowledge bases)","level":2,"score":0.6730852723121643},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6444917321205139},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.6242479085922241},{"id":"https://openalex.org/C26362088","wikidata":"https://www.wikidata.org/wiki/Q17086453","display_name":"Nonlinear conjugate gradient method","level":4,"score":0.6192311644554138},{"id":"https://openalex.org/C198562538","wikidata":"https://www.wikidata.org/wiki/Q5262612","display_name":"Derivation of the conjugate gradient method","level":5,"score":0.5627048015594482},{"id":"https://openalex.org/C91394653","wikidata":"https://www.wikidata.org/wiki/Q5161159","display_name":"Conjugate residual method","level":4,"score":0.5608554482460022},{"id":"https://openalex.org/C197336794","wikidata":"https://www.wikidata.org/wiki/Q5161150","display_name":"Conjugate","level":2,"score":0.560761034488678},{"id":"https://openalex.org/C198880260","wikidata":"https://www.wikidata.org/wiki/Q5289813","display_name":"Domain decomposition methods","level":3,"score":0.5339049100875854},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.475965678691864},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.4728921949863434},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4665006995201111},{"id":"https://openalex.org/C115680565","wikidata":"https://www.wikidata.org/wiki/Q5977448","display_name":"Gradient method","level":2,"score":0.45288893580436707},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3790454566478729},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.25843894481658936},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.22330212593078613},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.18750813603401184},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.14911282062530518},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.11556726694107056},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.06009310483932495},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.05701911449432373},{"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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"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/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"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/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00207160412331289065","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160412331289065","pdf_url":null,"source":{"id":"https://openalex.org/S124867444","display_name":"International Journal of Computer Mathematics","issn_l":"0020-7160","issn":["0020-7160","1026-7425","1029-0265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computer Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":20,"referenced_works":["https://openalex.org/W138506437","https://openalex.org/W1506342804","https://openalex.org/W1968131776","https://openalex.org/W1980677368","https://openalex.org/W1981452262","https://openalex.org/W1987551307","https://openalex.org/W1998366068","https://openalex.org/W2009215380","https://openalex.org/W2014344607","https://openalex.org/W2017085690","https://openalex.org/W2037390210","https://openalex.org/W2040425799","https://openalex.org/W2045809113","https://openalex.org/W2053157162","https://openalex.org/W2066692739","https://openalex.org/W2081465995","https://openalex.org/W2090314242","https://openalex.org/W2167698346","https://openalex.org/W2316564661","https://openalex.org/W2588081979"],"related_works":["https://openalex.org/W2005717169","https://openalex.org/W2098528027","https://openalex.org/W2183734858","https://openalex.org/W3159557833","https://openalex.org/W4319027779","https://openalex.org/W4291138102","https://openalex.org/W2357791884","https://openalex.org/W3159907817","https://openalex.org/W2388344948","https://openalex.org/W3082608044"],"abstract_inverted_index":{"In":[0],"this":[1,45],"article,":[2],"we":[3],"give":[4],"the":[5,10,41,60],"convergence":[6,42],"and":[7,19,32,39,56,65],"consistency":[8],"of":[9,44,62],"MSD-CG":[11],"method":[12,29,64],"[Gu,":[13],"T.-X.,":[14],"Liu,":[15],"X.-P.,":[16],"Mo,":[17],"Z.-Y.":[18],"Chi,":[20],"X.-B.":[21],"(in":[22],"press).":[23],"Multiple":[24],"search":[25],"direction":[26],"conjugate":[27],"gradient":[28],"I:":[30],"Methods":[31],"their":[33],"propositions.":[34],"Int.":[35],"J.":[36],"Comput.":[37],"Math.]":[38],"estimate":[40],"rate":[43],"method.":[46],"Numerical":[47],"experiments":[48],"on":[49],"two":[50],"distributed":[51],"parallel":[52],"computers,":[53],"Dawning":[54],"3000":[55],"P-II":[57],"Cluster,":[58],"show":[59,66],"efficiency":[61],"our":[63],"that":[67],"it":[68],"compares":[69],"favorably":[70],"with":[71],"some":[72],"domain":[73],"decomposition":[74],"methods.":[75]},"counts_by_year":[{"year":2017,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}