{"id":"https://openalex.org/W3207570649","doi":"https://doi.org/10.3390/sym13101971","title":"Improving the Convergence of Interval Single-Step Method for Simultaneous Approximation of Polynomial Zeros","display_name":"Improving the Convergence of Interval Single-Step Method for Simultaneous Approximation of Polynomial Zeros","publication_year":2021,"publication_date":"2021-10-19","ids":{"openalex":"https://openalex.org/W3207570649","doi":"https://doi.org/10.3390/sym13101971","mag":"3207570649"},"language":"en","primary_location":{"id":"doi:10.3390/sym13101971","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13101971","pdf_url":"https://www.mdpi.com/2073-8994/13/10/1971/pdf?version=1634819498","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/2073-8994/13/10/1971/pdf?version=1634819498","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5004960425","display_name":"Nur Raidah Salim","orcid":"https://orcid.org/0000-0002-7941-1878"},"institutions":[{"id":"https://openalex.org/I130343225","display_name":"Universiti Putra Malaysia","ror":"https://ror.org/02e91jd64","country_code":"MY","type":"education","lineage":["https://openalex.org/I130343225"]}],"countries":["MY"],"is_corresponding":true,"raw_author_name":"Nur Raidah Salim","raw_affiliation_strings":["Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"],"raw_orcid":"https://orcid.org/0000-0002-7941-1878","affiliations":[{"raw_affiliation_string":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","institution_ids":["https://openalex.org/I130343225"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5046716047","display_name":"Chuei Yee Chen","orcid":"https://orcid.org/0000-0003-1768-8668"},"institutions":[{"id":"https://openalex.org/I130343225","display_name":"Universiti Putra Malaysia","ror":"https://ror.org/02e91jd64","country_code":"MY","type":"education","lineage":["https://openalex.org/I130343225"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Chuei Yee Chen","raw_affiliation_strings":["Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"],"raw_orcid":"https://orcid.org/0000-0003-1768-8668","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","institution_ids":["https://openalex.org/I130343225"]},{"raw_affiliation_string":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","institution_ids":["https://openalex.org/I130343225"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5065211899","display_name":"Zahari Mahad","orcid":"https://orcid.org/0000-0001-6954-8494"},"institutions":[{"id":"https://openalex.org/I130343225","display_name":"Universiti Putra Malaysia","ror":"https://ror.org/02e91jd64","country_code":"MY","type":"education","lineage":["https://openalex.org/I130343225"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Zahari Mahad","raw_affiliation_strings":["Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"],"raw_orcid":"https://orcid.org/0000-0001-6954-8494","affiliations":[{"raw_affiliation_string":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","institution_ids":["https://openalex.org/I130343225"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5057400728","display_name":"Siti Hasana Sapar","orcid":"https://orcid.org/0000-0002-0060-482X"},"institutions":[{"id":"https://openalex.org/I130343225","display_name":"Universiti Putra Malaysia","ror":"https://ror.org/02e91jd64","country_code":"MY","type":"education","lineage":["https://openalex.org/I130343225"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Siti Hasana Sapar","raw_affiliation_strings":["Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","institution_ids":["https://openalex.org/I130343225"]},{"raw_affiliation_string":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia","institution_ids":["https://openalex.org/I130343225"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5004960425"],"corresponding_institution_ids":["https://openalex.org/I130343225"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.21206704,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"13","issue":"10","first_page":"1971","last_page":"1971"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9997000098228455,"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":0.9997000098228455,"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/T11697","display_name":"Numerical Methods and Algorithms","score":0.9988999962806702,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9969000220298767,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/midpoint","display_name":"Midpoint","score":0.9247552156448364},{"id":"https://openalex.org/keywords/interval","display_name":"Interval (graph theory)","score":0.7850853204727173},{"id":"https://openalex.org/keywords/midpoint-method","display_name":"Midpoint method","score":0.784663736820221},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.7242779731750488},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.6276189684867859},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6252928376197815},{"id":"https://openalex.org/keywords/interval-arithmetic","display_name":"Interval arithmetic","score":0.5457824468612671},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.515056848526001},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.49181726574897766},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3672299385070801},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3439618945121765},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.27059072256088257},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2632611393928528},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.18034633994102478},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.11191561818122864}],"concepts":[{"id":"https://openalex.org/C34130140","wikidata":"https://www.wikidata.org/wiki/Q1645406","display_name":"Midpoint","level":2,"score":0.9247552156448364},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.7850853204727173},{"id":"https://openalex.org/C85807060","wikidata":"https://www.wikidata.org/wiki/Q17103851","display_name":"Midpoint method","level":2,"score":0.784663736820221},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.7242779731750488},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.6276189684867859},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6252928376197815},{"id":"https://openalex.org/C191252586","wikidata":"https://www.wikidata.org/wiki/Q1671453","display_name":"Interval arithmetic","level":3,"score":0.5457824468612671},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.515056848526001},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.49181726574897766},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3672299385070801},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3439618945121765},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.27059072256088257},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2632611393928528},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.18034633994102478},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.11191561818122864},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.0},{"id":"https://openalex.org/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym13101971","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13101971","pdf_url":"https://www.mdpi.com/2073-8994/13/10/1971/pdf?version=1634819498","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:13b19371efae41169cc3fdeb7ba47a49","is_oa":true,"landing_page_url":"https://doaj.org/article/13b19371efae41169cc3fdeb7ba47a49","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":"Symmetry, Vol 13, Iss 10, p 1971 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/13/10/1971/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym13101971","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/sym13101971","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13101971","pdf_url":"https://www.mdpi.com/2073-8994/13/10/1971/pdf?version=1634819498","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":[],"awards":[{"id":"https://openalex.org/G4301499819","display_name":null,"funder_award_id":"GP-IPM/2021/9699800","funder_id":"https://openalex.org/F4320322714","funder_display_name":"Universiti Putra Malaysia"}],"funders":[{"id":"https://openalex.org/F4320322714","display_name":"Universiti Putra Malaysia","ror":"https://ror.org/02e91jd64"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3207570649.pdf","grobid_xml":"https://content.openalex.org/works/W3207570649.grobid-xml"},"referenced_works_count":31,"referenced_works":["https://openalex.org/W626618571","https://openalex.org/W1518139620","https://openalex.org/W1580809199","https://openalex.org/W1755235811","https://openalex.org/W1875302918","https://openalex.org/W1980848524","https://openalex.org/W2019656110","https://openalex.org/W2031629670","https://openalex.org/W2040610859","https://openalex.org/W2062927510","https://openalex.org/W2076169133","https://openalex.org/W2080960729","https://openalex.org/W2111247991","https://openalex.org/W2149454052","https://openalex.org/W2163015871","https://openalex.org/W2205265587","https://openalex.org/W2392188227","https://openalex.org/W2407441603","https://openalex.org/W2542167254","https://openalex.org/W2594817031","https://openalex.org/W2781810605","https://openalex.org/W2895116250","https://openalex.org/W2916725184","https://openalex.org/W2963212354","https://openalex.org/W3008764586","https://openalex.org/W3126862472","https://openalex.org/W3175012356","https://openalex.org/W6660526799","https://openalex.org/W6665988834","https://openalex.org/W6687905279","https://openalex.org/W6711414207"],"related_works":["https://openalex.org/W2391598905","https://openalex.org/W2330791370","https://openalex.org/W2072226187","https://openalex.org/W37565017","https://openalex.org/W2357395917","https://openalex.org/W3091779247","https://openalex.org/W2391440809","https://openalex.org/W3018722875","https://openalex.org/W1988919041","https://openalex.org/W2054870455"],"abstract_inverted_index":{"This":[0],"paper":[1],"describes":[2],"the":[3,13,17,27,39,42,48,53,64,68,80,102,114,118,124,128],"extended":[4],"method":[5,70,116],"of":[6,41,44,47,75],"solving":[7],"real":[8],"polynomial":[9,65],"zeros":[10],"problems":[11],"using":[12,106],"single-step":[14,22],"method,":[15,24],"namely,":[16],"interval":[18,46,130],"trio":[19],"midpoint":[20,28,43],"symmetric":[21],"(ITMSS)":[23],"which":[25,126],"updates":[26],"at":[29,77,87],"each":[30,45],"forward-backward-forward":[31],"step.":[32],"The":[33,109],"proposed":[34,69,115],"algorithm":[35],"will":[36],"constantly":[37],"update":[38],"value":[40],"previous":[49],"roots":[50],"before":[51],"entering":[52],"preceding":[54],"steps;":[55],"hence,":[56],"it":[57],"always":[58],"generate":[59],"intervals":[60],"that":[61,113],"decrease":[62],"toward":[63],"zeros.":[66],"Theoretically,":[67],"possesses":[71],"a":[72],"superior":[73],"rate":[74],"convergence":[76,133],"16,":[78],"while":[79],"existing":[81],"methods":[82,121],"are":[83,104],"known":[84],"to":[85],"have,":[86],"most,":[88],"9.":[89],"To":[90],"validate":[91],"its":[92],"efficiency,":[93],"we":[94],"perform":[95],"numerical":[96,110],"experiments":[97],"on":[98],"52":[99],"polynomials,":[100],"and":[101],"results":[103,111],"presented,":[105],"performance":[107],"profiles.":[108],"indicate":[112],"surpasses":[117],"other":[119],"three":[120],"by":[122],"fine-tuning":[123],"midpoint,":[125],"reduces":[127],"final":[129],"width":[131],"upon":[132],"with":[134],"fewer":[135],"iterations.":[136]},"counts_by_year":[],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}
