{"id":"https://openalex.org/W1487828487","doi":"https://doi.org/10.1155/2015/961487","title":"A Derivative-Free Conjugate Gradient Method and Its Global Convergence for Solving Symmetric Nonlinear Equations","display_name":"A Derivative-Free Conjugate Gradient Method and Its Global Convergence for Solving Symmetric Nonlinear Equations","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W1487828487","doi":"https://doi.org/10.1155/2015/961487","mag":"1487828487"},"language":"en","primary_location":{"id":"doi:10.1155/2015/961487","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2015/961487","pdf_url":"http://downloads.hindawi.com/journals/ijmms/2015/961487.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"http://downloads.hindawi.com/journals/ijmms/2015/961487.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5059042253","display_name":"Mohammed Yusuf Waziri","orcid":"https://orcid.org/0000-0001-7112-659X"},"institutions":[{"id":"https://openalex.org/I919958821","display_name":"Bayero University Kano","ror":"https://ror.org/049pzty39","country_code":"NG","type":"education","lineage":["https://openalex.org/I919958821"]}],"countries":["NG"],"is_corresponding":true,"raw_author_name":"Mohammed Yusuf Waziri","raw_affiliation_strings":["Department of Mathematical Sciences, Faculty of Science, Bayero University Kano, Kano, Nigeria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Faculty of Science, Bayero University Kano, Kano, Nigeria","institution_ids":["https://openalex.org/I919958821"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5073495551","display_name":"Jamilu Sabi\u2019u","orcid":"https://orcid.org/0000-0002-7262-490X"},"institutions":[{"id":"https://openalex.org/I4210099431","display_name":"Maitama Sule University Kano","ror":"https://ror.org/011wymc20","country_code":"NG","type":"education","lineage":["https://openalex.org/I4210099431"]}],"countries":["NG"],"is_corresponding":false,"raw_author_name":"Jamilu Sabi\u2019u","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Northwest University Kano, Kano, Nigeria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Northwest University Kano, Kano, Nigeria","institution_ids":["https://openalex.org/I4210099431"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5059042253"],"corresponding_institution_ids":["https://openalex.org/I919958821"],"apc_list":{"value":1050,"currency":"USD","value_usd":1050},"apc_paid":{"value":1050,"currency":"USD","value_usd":1050},"fwci":2.1934,"has_fulltext":true,"cited_by_count":37,"citation_normalized_percentile":{"value":0.86210788,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"2015","issue":null,"first_page":"1","last_page":"8"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9994000196456909,"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"}},"topics":[{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9994000196456909,"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/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9973999857902527,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9805999994277954,"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/mathematics","display_name":"Mathematics","score":0.7997479438781738},{"id":"https://openalex.org/keywords/conjugate-gradient-method","display_name":"Conjugate gradient method","score":0.7903503775596619},{"id":"https://openalex.org/keywords/derivation-of-the-conjugate-gradient-method","display_name":"Derivation of the conjugate gradient method","score":0.7290768027305603},{"id":"https://openalex.org/keywords/jacobian-matrix-and-determinant","display_name":"Jacobian matrix and determinant","score":0.705452024936676},{"id":"https://openalex.org/keywords/nonlinear-conjugate-gradient-method","display_name":"Nonlinear conjugate gradient method","score":0.6942046880722046},{"id":"https://openalex.org/keywords/conjugate-residual-method","display_name":"Conjugate residual method","score":0.666034460067749},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6582415699958801},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5936274528503418},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5809467434883118},{"id":"https://openalex.org/keywords/benchmark","display_name":"Benchmark (surveying)","score":0.5277534127235413},{"id":"https://openalex.org/keywords/derivative","display_name":"Derivative (finance)","score":0.48970678448677063},{"id":"https://openalex.org/keywords/biconjugate-gradient-method","display_name":"Biconjugate gradient method","score":0.4511985182762146},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3954462707042694},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.2657036781311035},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.14466175436973572},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.11843594908714294},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.060896068811416626}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7997479438781738},{"id":"https://openalex.org/C81184566","wikidata":"https://www.wikidata.org/wiki/Q1191895","display_name":"Conjugate gradient method","level":2,"score":0.7903503775596619},{"id":"https://openalex.org/C198562538","wikidata":"https://www.wikidata.org/wiki/Q5262612","display_name":"Derivation of the conjugate gradient method","level":5,"score":0.7290768027305603},{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.705452024936676},{"id":"https://openalex.org/C26362088","wikidata":"https://www.wikidata.org/wiki/Q17086453","display_name":"Nonlinear conjugate gradient method","level":4,"score":0.6942046880722046},{"id":"https://openalex.org/C91394653","wikidata":"https://www.wikidata.org/wiki/Q5161159","display_name":"Conjugate residual method","level":4,"score":0.666034460067749},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6582415699958801},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5936274528503418},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5809467434883118},{"id":"https://openalex.org/C185798385","wikidata":"https://www.wikidata.org/wiki/Q1161707","display_name":"Benchmark (surveying)","level":2,"score":0.5277534127235413},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.48970678448677063},{"id":"https://openalex.org/C89408827","wikidata":"https://www.wikidata.org/wiki/Q855169","display_name":"Biconjugate gradient method","level":5,"score":0.4511985182762146},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3954462707042694},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2657036781311035},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.14466175436973572},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.11843594908714294},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.060896068811416626},{"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},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"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/C205649164","wikidata":"https://www.wikidata.org/wiki/Q1071","display_name":"Geography","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/C13280743","wikidata":"https://www.wikidata.org/wiki/Q131089","display_name":"Geodesy","level":1,"score":0.0},{"id":"https://openalex.org/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1155/2015/961487","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2015/961487","pdf_url":"http://downloads.hindawi.com/journals/ijmms/2015/961487.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:hin:jijmms:961487","is_oa":false,"landing_page_url":"http://downloads.hindawi.com/journals/IJMMS/2015/961487.xml","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"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":"article"},{"id":"pmh:oai:doaj.org/article:c00a24c92d1c4e3eac32289053e12161","is_oa":true,"landing_page_url":"https://doaj.org/article/c00a24c92d1c4e3eac32289053e12161","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences, Vol 2015 (2015)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2015/961487","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2015/961487","pdf_url":"http://downloads.hindawi.com/journals/ijmms/2015/961487.pdf","source":{"id":"https://openalex.org/S151851439","display_name":"International Journal of Mathematics and Mathematical Sciences","issn_l":"0161-1712","issn":["0161-1712","1687-0425"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Mathematics and Mathematical Sciences","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1487828487.pdf","grobid_xml":"https://content.openalex.org/works/W1487828487.grobid-xml"},"referenced_works_count":17,"referenced_works":["https://openalex.org/W1977459204","https://openalex.org/W1978621794","https://openalex.org/W1987583991","https://openalex.org/W1991271057","https://openalex.org/W1999855318","https://openalex.org/W2012341322","https://openalex.org/W2018215034","https://openalex.org/W2024674718","https://openalex.org/W2032316963","https://openalex.org/W2059012394","https://openalex.org/W2065471299","https://openalex.org/W2085694061","https://openalex.org/W2102183118","https://openalex.org/W2136780394","https://openalex.org/W2149454052","https://openalex.org/W2333529718","https://openalex.org/W2494779131"],"related_works":["https://openalex.org/W2798336637","https://openalex.org/W2895534382","https://openalex.org/W3010456047","https://openalex.org/W2368719500","https://openalex.org/W2060532089","https://openalex.org/W2146877103","https://openalex.org/W2380780519","https://openalex.org/W2096999732","https://openalex.org/W1487828487","https://openalex.org/W2342236909"],"abstract_inverted_index":{"We":[0],"suggest":[1],"a":[2],"conjugate":[3],"gradient":[4,18],"(CG)":[5],"method":[6,33,61,75],"for":[7],"solving":[8],"symmetric":[9],"systems":[10],"of":[11,23,30,59],"nonlinear":[12],"equations":[13],"without":[14],"computing":[15],"Jacobian":[16],"and":[17],"via":[19],"the":[20,24,31,56,73],"special":[21],"structure":[22],"underlying":[25],"function.":[26],"This":[27],"derivative-free":[28],"feature":[29],"proposed":[32,74],"gives":[34],"it":[35],"advantage":[36],"to":[37,49],"solve":[38],"relatively":[39],"large-scale":[40],"problems":[41,70],"(500,000":[42],"variables)":[43],"with":[44],"lower":[45],"storage":[46],"requirement":[47],"compared":[48],"some":[50,67],"existing":[51],"methods.":[52],"Under":[53],"appropriate":[54],"conditions,":[55],"global":[57],"convergence":[58],"our":[60],"is":[62,76],"reported.":[63],"Numerical":[64],"results":[65],"on":[66],"benchmark":[68],"test":[69],"show":[71],"that":[72],"practically":[77],"effective.":[78]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":7},{"year":2020,"cited_by_count":9},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":3}],"updated_date":"2026-07-31T08:31:51.225901","created_date":"2025-10-10T00:00:00"}