{"id":"https://openalex.org/W4414179029","doi":"https://doi.org/10.3390/a18090582","title":"A Hybrid Steffensen\u2013Genetic Algorithm for Finding Multi-Roots of Nonlinear Equations and Applications to Biomedical Engineering","display_name":"A Hybrid Steffensen\u2013Genetic Algorithm for Finding Multi-Roots of Nonlinear Equations and Applications to Biomedical Engineering","publication_year":2025,"publication_date":"2025-09-13","ids":{"openalex":"https://openalex.org/W4414179029","doi":"https://doi.org/10.3390/a18090582"},"language":"en","primary_location":{"id":"doi:10.3390/a18090582","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a18090582","pdf_url":"https://www.mdpi.com/1999-4893/18/9/582/pdf?version=1757908534","source":{"id":"https://openalex.org/S190629608","display_name":"Algorithms","issn_l":"1999-4893","issn":["1999-4893"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1999-4893/18/9/582/pdf?version=1757908534","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5033766089","display_name":"Fiza Zafar","orcid":"https://orcid.org/0000-0003-0552-2783"},"institutions":[{"id":"https://openalex.org/I127670440","display_name":"Bahauddin Zakariya University","ror":"https://ror.org/05x817c41","country_code":"PK","type":"education","lineage":["https://openalex.org/I127670440"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Fiza Zafar","raw_affiliation_strings":["Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan"],"raw_orcid":"https://orcid.org/0000-0003-0552-2783","affiliations":[{"raw_affiliation_string":"Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan","institution_ids":["https://openalex.org/I127670440"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5085887256","display_name":"Alicia Cordero","orcid":"https://orcid.org/0000-0002-7462-9173"},"institutions":[{"id":"https://openalex.org/I60053951","display_name":"Universitat Polit\u00e8cnica de Val\u00e8ncia","ror":"https://ror.org/01460j859","country_code":"ES","type":"education","lineage":["https://openalex.org/I60053951"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"Alicia Cordero","raw_affiliation_strings":["I.U. de Matematica Multidisciplinar, Universitat Polit\u00e8cnica de Val\u00e8ncia, Cami de Vera s/n, 46022 Valencia, Spain"],"raw_orcid":"https://orcid.org/0000-0002-7462-9173","affiliations":[{"raw_affiliation_string":"I.U. de Matematica Multidisciplinar, Universitat Polit\u00e8cnica de Val\u00e8ncia, Cami de Vera s/n, 46022 Valencia, Spain","institution_ids":["https://openalex.org/I60053951"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Sadia Mujtaba","orcid":null},"institutions":[{"id":"https://openalex.org/I127670440","display_name":"Bahauddin Zakariya University","ror":"https://ror.org/05x817c41","country_code":"PK","type":"education","lineage":["https://openalex.org/I127670440"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Sadia Mujtaba","raw_affiliation_strings":["Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan","institution_ids":["https://openalex.org/I127670440"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5049721976","display_name":"Juan R. Torregrosa","orcid":"https://orcid.org/0000-0002-9893-0761"},"institutions":[{"id":"https://openalex.org/I60053951","display_name":"Universitat Polit\u00e8cnica de Val\u00e8ncia","ror":"https://ror.org/01460j859","country_code":"ES","type":"education","lineage":["https://openalex.org/I60053951"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Juan R. Torregrosa","raw_affiliation_strings":["I.U. de Matematica Multidisciplinar, Universitat Polit\u00e8cnica de Val\u00e8ncia, Cami de Vera s/n, 46022 Valencia, Spain"],"raw_orcid":"https://orcid.org/0000-0002-9893-0761","affiliations":[{"raw_affiliation_string":"I.U. de Matematica Multidisciplinar, Universitat Polit\u00e8cnica de Val\u00e8ncia, Cami de Vera s/n, 46022 Valencia, Spain","institution_ids":["https://openalex.org/I60053951"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5085887256"],"corresponding_institution_ids":["https://openalex.org/I60053951"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.33555317,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"18","issue":"9","first_page":"582","last_page":"582"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12661","display_name":"Iterative Methods for Nonlinear Equations","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/T12661","display_name":"Iterative Methods for Nonlinear Equations","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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9807999730110168,"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/T10998","display_name":"Heat Transfer and Optimization","score":0.9376999735832214,"subfield":{"id":"https://openalex.org/subfields/2210","display_name":"Mechanical 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/convergence","display_name":"Convergence (economics)","score":0.6474999785423279},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6212000250816345},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.5788000226020813},{"id":"https://openalex.org/keywords/genetic-algorithm","display_name":"Genetic algorithm","score":0.49939998984336853},{"id":"https://openalex.org/keywords/hybrid-algorithm","display_name":"Hybrid algorithm (constraint satisfaction)","score":0.4564000070095062},{"id":"https://openalex.org/keywords/local-search","display_name":"Local search (optimization)","score":0.41659998893737793},{"id":"https://openalex.org/keywords/local-convergence","display_name":"Local convergence","score":0.37700000405311584},{"id":"https://openalex.org/keywords/global-optimization","display_name":"Global optimization","score":0.3407999873161316}],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6725999712944031},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6474999785423279},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6212000250816345},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.6085000038146973},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.5788000226020813},{"id":"https://openalex.org/C8880873","wikidata":"https://www.wikidata.org/wiki/Q187787","display_name":"Genetic algorithm","level":2,"score":0.49939998984336853},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4932999908924103},{"id":"https://openalex.org/C62469222","wikidata":"https://www.wikidata.org/wiki/Q17092103","display_name":"Hybrid algorithm (constraint satisfaction)","level":5,"score":0.4564000070095062},{"id":"https://openalex.org/C135320971","wikidata":"https://www.wikidata.org/wiki/Q1868524","display_name":"Local search (optimization)","level":2,"score":0.41659998893737793},{"id":"https://openalex.org/C117898588","wikidata":"https://www.wikidata.org/wiki/Q6664310","display_name":"Local convergence","level":3,"score":0.37700000405311584},{"id":"https://openalex.org/C164752517","wikidata":"https://www.wikidata.org/wiki/Q5570875","display_name":"Global optimization","level":2,"score":0.3407999873161316},{"id":"https://openalex.org/C50897621","wikidata":"https://www.wikidata.org/wiki/Q2665508","display_name":"Hybrid system","level":2,"score":0.32109999656677246},{"id":"https://openalex.org/C2993955422","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Science and engineering","level":2,"score":0.3151000142097473},{"id":"https://openalex.org/C89109886","wikidata":"https://www.wikidata.org/wiki/Q1535924","display_name":"Trust region","level":3,"score":0.3012999892234802},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.29840001463890076},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.2881999909877777},{"id":"https://openalex.org/C105902424","wikidata":"https://www.wikidata.org/wiki/Q1197129","display_name":"Evolutionary computation","level":2,"score":0.2842000126838684},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.2759000062942505},{"id":"https://openalex.org/C4935549","wikidata":"https://www.wikidata.org/wiki/Q6822261","display_name":"Meta-optimization","level":3,"score":0.26899999380111694},{"id":"https://openalex.org/C2987595161","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Optimization algorithm","level":2,"score":0.2669999897480011},{"id":"https://openalex.org/C159149176","wikidata":"https://www.wikidata.org/wiki/Q14489129","display_name":"Evolutionary algorithm","level":2,"score":0.2574000060558319},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.2529999911785126},{"id":"https://openalex.org/C106516650","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm design","level":2,"score":0.25130000710487366}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/a18090582","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a18090582","pdf_url":"https://www.mdpi.com/1999-4893/18/9/582/pdf?version=1757908534","source":{"id":"https://openalex.org/S190629608","display_name":"Algorithms","issn_l":"1999-4893","issn":["1999-4893"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithms","raw_type":"journal-article"},{"id":"pmh:oai:riunet.upv.es:10251/229913","is_oa":true,"landing_page_url":"https://riunet.upv.es/handle/10251/229913","pdf_url":null,"source":{"id":"https://openalex.org/S4306400639","display_name":"RiuNet (Universitat Polit\u00e8cnica de Val\u00e8ncia)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I60053951","host_organization_name":"Universitat Polit\u00e8cnica de Val\u00e8ncia","host_organization_lineage":["https://openalex.org/I60053951"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:doaj.org/article:ecfd3ba96064430fbc70a11e6294f584","is_oa":true,"landing_page_url":"https://doaj.org/article/ecfd3ba96064430fbc70a11e6294f584","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":"Algorithms, Vol 18, Iss 9, p 582 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/a18090582","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a18090582","pdf_url":"https://www.mdpi.com/1999-4893/18/9/582/pdf?version=1757908534","source":{"id":"https://openalex.org/S190629608","display_name":"Algorithms","issn_l":"1999-4893","issn":["1999-4893"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Algorithms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4414179029.pdf","grobid_xml":"https://content.openalex.org/works/W4414179029.grobid-xml"},"referenced_works_count":29,"referenced_works":["https://openalex.org/W570534109","https://openalex.org/W1544329015","https://openalex.org/W1966536904","https://openalex.org/W1979056867","https://openalex.org/W2025026078","https://openalex.org/W2031629670","https://openalex.org/W2046133214","https://openalex.org/W2050221432","https://openalex.org/W2062786256","https://openalex.org/W2075341408","https://openalex.org/W2075659701","https://openalex.org/W2076634712","https://openalex.org/W2121230124","https://openalex.org/W3021906145","https://openalex.org/W3033262208","https://openalex.org/W3036761082","https://openalex.org/W3113176096","https://openalex.org/W3159382260","https://openalex.org/W3162412663","https://openalex.org/W4220887481","https://openalex.org/W4224231919","https://openalex.org/W4285606813","https://openalex.org/W4285653871","https://openalex.org/W4312389694","https://openalex.org/W4327968230","https://openalex.org/W4388127691","https://openalex.org/W4389200778","https://openalex.org/W4401167180","https://openalex.org/W4410904706"],"related_works":[],"abstract_inverted_index":{"A":[0],"new":[1],"hybrid":[2,122],"of":[3,17,19,58,68,100,110,120],"a":[4,46,69,106],"Steffensen-type":[5],"method":[6],"and":[7,29,80,87],"genetic":[8,60],"algorithm":[9,52,61,104],"is":[10],"developed":[11],"for":[12,45],"the":[13,43,54,59,64,92,97,103,118],"efficient":[14],"simultaneous":[15],"computation":[16],"roots":[18,113],"nonlinear":[20],"equations,":[21],"particularly":[22],"in":[23,127],"all":[24,112],"cases":[25],"involving":[26],"non-differentiable":[27],"functions":[28],"multiple":[30],"roots.":[31],"Traditional":[32],"numerical":[33],"methods":[34],"often":[35],"fail":[36],"to":[37,84],"handle":[38],"these":[39],"complexities":[40],"effectively,":[41],"highlighting":[42],"need":[44],"more":[47],"robust":[48],"solution.":[49],"The":[50],"proposed":[51],"combines":[53],"global":[55,94],"search":[56],"strength":[57],"(GA)":[62],"with":[63,96],"local":[65,98],"refinement":[66,99],"capabilities":[67],"derivative-free":[70],"optimal":[71],"fourth-order":[72],"Steffensen":[73],"method.":[74],"This":[75,115],"integration":[76],"enhances":[77],"both":[78],"exploration":[79],"exploitation":[81],"capabilities,":[82],"leading":[83],"improved":[85],"convergence":[86],"computational":[88],"accuracy.":[89],"By":[90],"uniting":[91],"GA\u2019s":[93],"optimization":[95],"iterative":[101],"solvers,":[102],"forms":[105],"higher-order":[107],"framework":[108],"capable":[109],"locating":[111],"concurrently.":[114],"study":[116],"validates":[117],"performance":[119],"this":[121],"strategy":[123],"through":[124],"diverse":[125],"applications":[126],"biomedical":[128],"engineering":[129],"problems.":[130]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-09-16T00:00:00"}
