{"id":"https://openalex.org/W4377137297","doi":"https://doi.org/10.3390/sym15051116","title":"Extension of King\u2019s Iterative Scheme by Means of Memory for Nonlinear Equations","display_name":"Extension of King\u2019s Iterative Scheme by Means of Memory for Nonlinear Equations","publication_year":2023,"publication_date":"2023-05-19","ids":{"openalex":"https://openalex.org/W4377137297","doi":"https://doi.org/10.3390/sym15051116"},"language":"en","primary_location":{"id":"doi:10.3390/sym15051116","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15051116","pdf_url":"https://www.mdpi.com/2073-8994/15/5/1116/pdf?version=1684491055","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/15/5/1116/pdf?version=1684491055","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5062977264","display_name":"Saima Akram","orcid":"https://orcid.org/0000-0001-6434-7650"},"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"]},{"id":"https://openalex.org/I4210158239","display_name":"Government College Women University Faisalabad","ror":"https://ror.org/05rq0zy06","country_code":"PK","type":"education","lineage":["https://openalex.org/I4210158239"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Saima Akram","raw_affiliation_strings":["Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya, Multan 60000, Pakistan","Department of Mathematics, Government College Women University Faisalabad, Faisalabad 38000, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya, Multan 60000, Pakistan","institution_ids":["https://openalex.org/I127670440"]},{"raw_affiliation_string":"Department of Mathematics, Government College Women University Faisalabad, Faisalabad 38000, Pakistan","institution_ids":["https://openalex.org/I4210158239"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5023959233","display_name":"Maira Khalid","orcid":"https://orcid.org/0000-0002-6135-2734"},"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":"Maira Khalid","raw_affiliation_strings":["Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya, Multan 60000, Pakistan"],"raw_orcid":"https://orcid.org/0000-0002-6135-2734","affiliations":[{"raw_affiliation_string":"Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya, Multan 60000, Pakistan","institution_ids":["https://openalex.org/I127670440"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5079820359","display_name":"Moin\u2010ud\u2010Din Junjua","orcid":"https://orcid.org/0000-0002-1251-1532"},"institutions":[{"id":"https://openalex.org/I4210116806","display_name":"Ghazi University","ror":"https://ror.org/023a7t361","country_code":"PK","type":"education","lineage":["https://openalex.org/I4210116806"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Moin-ud-Din Junjua","raw_affiliation_strings":["Department of Mathematics, Ghazi University, Dera Ghazi Khan 32200, Pakistan"],"raw_orcid":"https://orcid.org/0000-0002-1251-1532","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Ghazi University, Dera Ghazi Khan 32200, Pakistan","institution_ids":["https://openalex.org/I4210116806"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5111288616","display_name":"Shazia Altaf","orcid":null},"institutions":[{"id":"https://openalex.org/I4210096473","display_name":"Muhammad Nawaz Shareef University of Agriculture","ror":"https://ror.org/00vmr6593","country_code":"PK","type":"education","lineage":["https://openalex.org/I4210096473"]},{"id":"https://openalex.org/I4391012582","display_name":"Institute of Southern Punjab","ror":"https://ror.org/05fvxrd95","country_code":"PK","type":"education","lineage":["https://openalex.org/I4391012582"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Shazia Altaf","raw_affiliation_strings":["Department of Mathematics and Statistics, Institute of Southern Punjab, Multan 60800, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, Institute of Southern Punjab, Multan 60800, Pakistan","institution_ids":["https://openalex.org/I4210096473","https://openalex.org/I4391012582"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5004397459","display_name":"Sunil Kumar","orcid":"https://orcid.org/0000-0002-8471-5139"},"institutions":[{"id":"https://openalex.org/I101407740","display_name":"Chandigarh University","ror":"https://ror.org/05t4pvx35","country_code":"IN","type":"education","lineage":["https://openalex.org/I101407740"]}],"countries":["IN"],"is_corresponding":true,"raw_author_name":"Sunil Kumar","raw_affiliation_strings":["Department of Mathematics, University Centre for Research and Development, Chandigarh University, Mohali 140413, India"],"raw_orcid":"https://orcid.org/0000-0002-8471-5139","affiliations":[{"raw_affiliation_string":"Department of Mathematics, University Centre for Research and Development, Chandigarh University, Mohali 140413, India","institution_ids":["https://openalex.org/I101407740"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5004397459"],"corresponding_institution_ids":["https://openalex.org/I101407740"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":3.3569,"has_fulltext":true,"cited_by_count":10,"citation_normalized_percentile":{"value":0.93064241,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"15","issue":"5","first_page":"1116","last_page":"1116"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":1.0,"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":1.0,"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":0.972000002861023,"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":0.9395999908447266,"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/convergence","display_name":"Convergence (economics)","score":0.628343939781189},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5784311890602112},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5632953643798828},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.5613580346107483},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.5437040328979492},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.5323607921600342},{"id":"https://openalex.org/keywords/interpolation","display_name":"Interpolation (computer graphics)","score":0.5242262482643127},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.512344241142273},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5107588768005371},{"id":"https://openalex.org/keywords/variable","display_name":"Variable (mathematics)","score":0.48467448353767395},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4695972204208374},{"id":"https://openalex.org/keywords/extension","display_name":"Extension (predicate logic)","score":0.42062562704086304},{"id":"https://openalex.org/keywords/scheme","display_name":"Scheme (mathematics)","score":0.4130995273590088},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.18079930543899536},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.07448700070381165}],"concepts":[{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.628343939781189},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5784311890602112},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5632953643798828},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.5613580346107483},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.5437040328979492},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.5323607921600342},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.5242262482643127},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.512344241142273},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5107588768005371},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.48467448353767395},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4695972204208374},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.42062562704086304},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.4130995273590088},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.18079930543899536},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.07448700070381165},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","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/C104114177","wikidata":"https://www.wikidata.org/wiki/Q79782","display_name":"Motion (physics)","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},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"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/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym15051116","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15051116","pdf_url":"https://www.mdpi.com/2073-8994/15/5/1116/pdf?version=1684491055","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:eab616f97b1a443d9e78120abaab4845","is_oa":false,"landing_page_url":"https://doaj.org/article/eab616f97b1a443d9e78120abaab4845","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 15, Iss 5, p 1116 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/15/5/1116/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym15051116","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym15051116","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15051116","pdf_url":"https://www.mdpi.com/2073-8994/15/5/1116/pdf?version=1684491055","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Affordable and clean energy","id":"https://metadata.un.org/sdg/7","score":0.8700000047683716}],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4377137297.pdf","grobid_xml":"https://content.openalex.org/works/W4377137297.grobid-xml"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W598649262","https://openalex.org/W658872159","https://openalex.org/W1972417972","https://openalex.org/W1999734576","https://openalex.org/W2008211319","https://openalex.org/W2026376036","https://openalex.org/W2043570811","https://openalex.org/W2055308255","https://openalex.org/W2134748397","https://openalex.org/W2172712181","https://openalex.org/W2483406971","https://openalex.org/W2741763219","https://openalex.org/W2782600048","https://openalex.org/W2784616802","https://openalex.org/W2792138164","https://openalex.org/W2796930124","https://openalex.org/W2902485869","https://openalex.org/W2922292190","https://openalex.org/W6618132482","https://openalex.org/W6661177502","https://openalex.org/W6682429911","https://openalex.org/W6685412182"],"related_works":["https://openalex.org/W2075768550","https://openalex.org/W3022218857","https://openalex.org/W2369178846","https://openalex.org/W2370289839","https://openalex.org/W2082716031","https://openalex.org/W2933494595","https://openalex.org/W4313347705","https://openalex.org/W1568049691","https://openalex.org/W1590096425","https://openalex.org/W2161448262"],"abstract_inverted_index":{"We":[0],"developed":[1,152],"a":[2,31,35],"new":[3],"family":[4,42],"of":[5,15,57,79,109,111,126,131,141,158],"optimal":[6],"eighth-order":[7],"derivative-free":[8],"iterative":[9,46,114],"methods":[10],"for":[11,102,128],"finding":[12],"simple":[13],"roots":[14],"nonlinear":[16],"equations":[17],"based":[18],"on":[19],"King\u2019s":[20],"scheme":[21,47],"and":[22,30,65,82,99,107,134],"Lagrange":[23],"interpolation.":[24],"By":[25],"incorporating":[26],"four":[27],"self-accelerating":[28],"parameters":[29],"weight":[32],"function":[33],"in":[34,138,164],"single":[36],"variable,":[37],"we":[38],"extend":[39],"the":[40,55,66,77,80,92,112,129,139,161],"proposed":[41,81,113],"to":[43,63,73],"an":[44],"efficient":[45],"with":[48],"memory.":[49,144],"Without":[50],"performing":[51],"additional":[52],"functional":[53],"evaluations,":[54],"order":[56],"convergence":[58,110,159],"is":[59,69],"boosted":[60],"from":[61,71],"8":[62],"15.51560,":[64],"efficiency":[67],"index":[68],"raised":[70],"1.6817":[72],"1.9847.":[74],"To":[75],"compare":[76],"performance":[78],"existing":[83,162],"schemes,":[84],"some":[85],"real-world":[86],"problems":[87],"are":[88,116],"selected,":[89],"such":[90,121],"as":[91,122],"eigenvalue":[93],"problem,":[94,98],"continuous":[95],"stirred-tank":[96],"reactor":[97],"energy":[100],"distribution":[101],"Planck\u2019s":[103],"radiation.":[104],"The":[105,145],"stability":[106,146],"regions":[108,157],"schemes":[115,133,142,153,163],"investigated":[117],"through":[118],"graphical":[119],"tools,":[120],"2D":[123],"symmetric":[124,156],"basins":[125],"attractions":[127],"case":[130,140],"memory-based":[132],"3D":[135],"stereographic":[136],"projections":[137],"without":[143],"analysis":[147],"demonstrates":[148],"that":[149],"our":[150],"newly":[151],"have":[154],"wider":[155],"than":[160],"their":[165],"respective":[166],"domains.":[167]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":2}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}