{"id":"https://openalex.org/W2198179779","doi":"https://doi.org/10.3390/a9010005","title":"A Family of Iterative Methods for Solving Systems of Nonlinear Equations Having Unknown Multiplicity","display_name":"A Family of Iterative Methods for Solving Systems of Nonlinear Equations Having Unknown Multiplicity","publication_year":2015,"publication_date":"2015-12-31","ids":{"openalex":"https://openalex.org/W2198179779","doi":"https://doi.org/10.3390/a9010005","mag":"2198179779"},"language":"en","primary_location":{"id":"doi:10.3390/a9010005","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a9010005","pdf_url":"https://www.mdpi.com/1999-4893/9/1/5/pdf?version=1451566178","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/1999-4893/9/1/5/pdf?version=1451566178","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5090575478","display_name":"Fayyaz Ahmad","orcid":"https://orcid.org/0000-0003-2634-5130"},"institutions":[{"id":"https://openalex.org/I115752224","display_name":"University of Insubria","ror":"https://ror.org/00s409261","country_code":"IT","type":"education","lineage":["https://openalex.org/I115752224"]},{"id":"https://openalex.org/I9617848","display_name":"Universitat Polit\u00e8cnica de Catalunya","ror":"https://ror.org/03mb6wj31","country_code":"ES","type":"education","lineage":["https://openalex.org/I9617848"]}],"countries":["ES","IT"],"is_corresponding":true,"raw_author_name":"Fayyaz Ahmad","raw_affiliation_strings":["Departament de F\u00edsica i Enginyeria Nuclear, Universitat Polit\u00e8cnica de Catalunya, Comte d\u2019Urgell 187, 08036 Barcelona, Spain","Dipartimento di Scienza e Alta Tecnologia, Universita dell\u2019Insubria, Via Valleggio 11, Como 22100, Italy","Dipartimento di Scienza e Alta Tecnologia, Universita dell'Insubria, Via Valleggio 11, Como 22100, Italy"],"affiliations":[{"raw_affiliation_string":"Departament de F\u00edsica i Enginyeria Nuclear, Universitat Polit\u00e8cnica de Catalunya, Comte d\u2019Urgell 187, 08036 Barcelona, Spain","institution_ids":["https://openalex.org/I9617848"]},{"raw_affiliation_string":"Dipartimento di Scienza e Alta Tecnologia, Universita dell\u2019Insubria, Via Valleggio 11, Como 22100, Italy","institution_ids":["https://openalex.org/I115752224"]},{"raw_affiliation_string":"Dipartimento di Scienza e Alta Tecnologia, Universita dell'Insubria, Via Valleggio 11, Como 22100, Italy","institution_ids":["https://openalex.org/I115752224"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5053268193","display_name":"Stefano Serra\u2010Capizzano","orcid":"https://orcid.org/0000-0001-9477-109X"},"institutions":[{"id":"https://openalex.org/I123387679","display_name":"Uppsala University","ror":"https://ror.org/048a87296","country_code":"SE","type":"education","lineage":["https://openalex.org/I123387679"]},{"id":"https://openalex.org/I115752224","display_name":"University of Insubria","ror":"https://ror.org/00s409261","country_code":"IT","type":"education","lineage":["https://openalex.org/I115752224"]}],"countries":["IT","SE"],"is_corresponding":false,"raw_author_name":"S. Serra-Capizzano","raw_affiliation_strings":["Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden","Dipartimento di Scienza e Alta Tecnologia, Universita dell\u2019Insubria, Via Valleggio 11, Como 22100, Italy","Dipartimento di Scienza e Alta Tecnologia, Universita dell'Insubria, Via Valleggio 11, Como 22100, Italy"],"affiliations":[{"raw_affiliation_string":"Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden","institution_ids":["https://openalex.org/I123387679"]},{"raw_affiliation_string":"Dipartimento di Scienza e Alta Tecnologia, Universita dell\u2019Insubria, Via Valleggio 11, Como 22100, Italy","institution_ids":["https://openalex.org/I115752224"]},{"raw_affiliation_string":"Dipartimento di Scienza e Alta Tecnologia, Universita dell'Insubria, Via Valleggio 11, Como 22100, Italy","institution_ids":["https://openalex.org/I115752224"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5006450000","display_name":"Malik Zaka Ullah","orcid":"https://orcid.org/0000-0003-2944-0352"},"institutions":[{"id":"https://openalex.org/I185163786","display_name":"King Abdulaziz University","ror":"https://ror.org/02ma4wv74","country_code":"SA","type":"education","lineage":["https://openalex.org/I185163786"]},{"id":"https://openalex.org/I115752224","display_name":"University of Insubria","ror":"https://ror.org/00s409261","country_code":"IT","type":"education","lineage":["https://openalex.org/I115752224"]}],"countries":["IT","SA"],"is_corresponding":false,"raw_author_name":"Malik Ullah","raw_affiliation_strings":["Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia","Dipartimento di Scienza e Alta Tecnologia, Universita dell\u2019Insubria, Via Valleggio 11, Como 22100, Italy","Dipartimento di Scienza e Alta Tecnologia, Universita dell'Insubria, Via Valleggio 11, Como 22100, Italy"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia","institution_ids":["https://openalex.org/I185163786"]},{"raw_affiliation_string":"Dipartimento di Scienza e Alta Tecnologia, Universita dell\u2019Insubria, Via Valleggio 11, Como 22100, Italy","institution_ids":["https://openalex.org/I115752224"]},{"raw_affiliation_string":"Dipartimento di Scienza e Alta Tecnologia, Universita dell'Insubria, Via Valleggio 11, Como 22100, Italy","institution_ids":["https://openalex.org/I115752224"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5032529951","display_name":"A. S. Al-Fhaid","orcid":null},"institutions":[{"id":"https://openalex.org/I185163786","display_name":"King Abdulaziz University","ror":"https://ror.org/02ma4wv74","country_code":"SA","type":"education","lineage":["https://openalex.org/I185163786"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"A. Al-Fhaid","raw_affiliation_strings":["Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia","institution_ids":["https://openalex.org/I185163786"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5090575478"],"corresponding_institution_ids":["https://openalex.org/I115752224","https://openalex.org/I9617848"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.5039,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.66614359,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"9","issue":"1","first_page":"5","last_page":"5"},"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":0.9955999851226807,"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/T11435","display_name":"Polynomial and algebraic computation","score":0.9894999861717224,"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/nonlinear-system","display_name":"Nonlinear system","score":0.7520231008529663},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7218833565711975},{"id":"https://openalex.org/keywords/multiplicity","display_name":"Multiplicity (mathematics)","score":0.7086238861083984},{"id":"https://openalex.org/keywords/jacobian-matrix-and-determinant","display_name":"Jacobian matrix and determinant","score":0.6650069952011108},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.549383282661438},{"id":"https://openalex.org/keywords/singularity","display_name":"Singularity","score":0.5484663844108582},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.46945351362228394},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.4531919062137604},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.36544740200042725},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.2041052281856537},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06301924586296082}],"concepts":[{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.7520231008529663},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7218833565711975},{"id":"https://openalex.org/C156004811","wikidata":"https://www.wikidata.org/wiki/Q2228257","display_name":"Multiplicity (mathematics)","level":2,"score":0.7086238861083984},{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.6650069952011108},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.549383282661438},{"id":"https://openalex.org/C16171025","wikidata":"https://www.wikidata.org/wiki/Q863349","display_name":"Singularity","level":2,"score":0.5484663844108582},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.46945351362228394},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.4531919062137604},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.36544740200042725},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2041052281856537},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06301924586296082},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/a9010005","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a9010005","pdf_url":"https://www.mdpi.com/1999-4893/9/1/5/pdf?version=1451566178","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:doaj.org/article:bb5cb1d4d6aa42a0ae294808cb08a00d","is_oa":true,"landing_page_url":"https://doaj.org/article/bb5cb1d4d6aa42a0ae294808cb08a00d","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"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 9, Iss 1, p 5 (2015)","raw_type":"article"},{"id":"pmh:oai:irinsubria.uninsubria.it:11383/2047967","is_oa":false,"landing_page_url":"http://hdl.handle.net/11383/2047967","pdf_url":null,"source":{"id":"https://openalex.org/S4377196351","display_name":"IrInSubria (University of Insubria)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I115752224","host_organization_name":"University of Insubria","host_organization_lineage":["https://openalex.org/I115752224"],"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"},{"id":"pmh:oai:mdpi.com:/1999-4893/9/1/5/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/a9010005","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":"Algorithms; Volume 9; Issue 1; Pages: 5","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/a9010005","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a9010005","pdf_url":"https://www.mdpi.com/1999-4893/9/1/5/pdf?version=1451566178","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":[{"display_name":"Reduced inequalities","score":0.7400000095367432,"id":"https://metadata.un.org/sdg/10"}],"awards":[{"id":"https://openalex.org/G2831977395","display_name":null,"funder_award_id":"KAW 2013.0341","funder_id":"https://openalex.org/F4320320936","funder_display_name":"Royal Swedish Academy of Sciences"}],"funders":[{"id":"https://openalex.org/F4320311030","display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","ror":"https://ror.org/01vx64p53"},{"id":"https://openalex.org/F4320320936","display_name":"Royal Swedish Academy of Sciences","ror":"https://ror.org/00j62qv07"},{"id":"https://openalex.org/F4320334079","display_name":"Gruppo Nazionale per il Calcolo Scientifico","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2198179779.pdf","grobid_xml":"https://content.openalex.org/works/W2198179779.grobid-xml"},"referenced_works_count":13,"referenced_works":["https://openalex.org/W562544820","https://openalex.org/W1983230696","https://openalex.org/W2005999511","https://openalex.org/W2011023522","https://openalex.org/W2032825696","https://openalex.org/W2045871123","https://openalex.org/W2048632019","https://openalex.org/W2067572998","https://openalex.org/W2090963365","https://openalex.org/W2156565866","https://openalex.org/W2158858295","https://openalex.org/W2235325310","https://openalex.org/W2321924776"],"related_works":["https://openalex.org/W2007405763","https://openalex.org/W3136087161","https://openalex.org/W2379794025","https://openalex.org/W2166615337","https://openalex.org/W2358210647","https://openalex.org/W2144801384","https://openalex.org/W2158743457","https://openalex.org/W1985623390","https://openalex.org/W2355238070","https://openalex.org/W2375378023"],"abstract_inverted_index":{"The":[0,23,69],"singularity":[1],"of":[2,17,20,25,36,51,54,66,71,83,90],"Jacobian":[3],"happens":[4],"when":[5],"we":[6,31],"are":[7],"looking":[8],"for":[9,56],"a":[10,18,34,52,72,105],"root,":[11],"with":[12,43,60,97],"multiplicity":[13],"greater":[14],"than":[15],"one,":[16],"system":[19,65],"nonlinear":[21,58,67],"equations.":[22,68],"purpose":[24],"this":[26],"article":[27],"is":[28,77],"two-fold.":[29],"Firstly,":[30],"will":[32,47],"present":[33],"modification":[35],"an":[37],"existing":[38],"method":[39,93],"that":[40],"computes":[41],"roots":[42,96],"known":[44],"multiplicities.":[45,99],"Secondly,":[46],"propose":[48],"the":[49,64,78,84,91,109],"generalization":[50],"family":[53],"methods":[55],"solving":[57],"equations":[59],"unknown":[61,98],"multiplicities,":[62],"to":[63,94],"inclusion":[70],"nonzero":[73],"multi-variable":[74],"auxiliary":[75,85],"function":[76,86],"key":[79],"idea.":[80],"Different":[81],"choices":[82],"give":[87],"different":[88],"families":[89],"iterative":[92],"find":[95],"Few":[100],"illustrative":[101],"numerical":[102],"experiments":[103],"and":[104],"critical":[106],"discussion":[107],"end":[108],"paper.":[110]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2017,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2016-06-24T00:00:00"}
