{"id":"https://openalex.org/W4392667163","doi":"https://doi.org/10.1109/tsp.2024.3376088","title":"Chaotic Convergence of Newton's Method","display_name":"Chaotic Convergence of Newton's Method","publication_year":2024,"publication_date":"2024-01-01","ids":{"openalex":"https://openalex.org/W4392667163","doi":"https://doi.org/10.1109/tsp.2024.3376088"},"language":"en","primary_location":{"id":"doi:10.1109/tsp.2024.3376088","is_oa":true,"landing_page_url":"https://doi.org/10.1109/tsp.2024.3376088","pdf_url":"https://ieeexplore.ieee.org/ielx7/78/4359509/10466642.pdf","source":{"id":"https://openalex.org/S168680287","display_name":"IEEE Transactions on Signal Processing","issn_l":"1053-587X","issn":["1053-587X","1941-0476"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Signal Processing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://ieeexplore.ieee.org/ielx7/78/4359509/10466642.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5086635324","display_name":"Jont B. Allen","orcid":"https://orcid.org/0000-0003-3106-1191"},"institutions":[{"id":"https://openalex.org/I157725225","display_name":"University of Illinois Urbana-Champaign","ror":"https://ror.org/047426m28","country_code":"US","type":"education","lineage":["https://openalex.org/I157725225"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Jont B. Allen","raw_affiliation_strings":["Electrical and Computer Engineering Department, University of Illinois, Urbana, IL, USA"],"raw_orcid":"https://orcid.org/0000-0003-3106-1191","affiliations":[{"raw_affiliation_string":"Electrical and Computer Engineering Department, University of Illinois, Urbana, IL, USA","institution_ids":["https://openalex.org/I157725225"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5086635324"],"corresponding_institution_ids":["https://openalex.org/I157725225"],"apc_list":null,"apc_paid":null,"fwci":2.8229,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.89164217,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":96,"max":98},"biblio":{"volume":"72","issue":null,"first_page":"1683","last_page":"1690"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9988999962806702,"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":0.9988999962806702,"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":0.9890999794006348,"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/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9817000031471252,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/mathematics","display_name":"Mathematics","score":0.7140173316001892},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5888503193855286},{"id":"https://openalex.org/keywords/local-convergence","display_name":"Local convergence","score":0.5790361762046814},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5638430714607239},{"id":"https://openalex.org/keywords/convergence-tests","display_name":"Convergence tests","score":0.5590126514434814},{"id":"https://openalex.org/keywords/normal-convergence","display_name":"Normal convergence","score":0.5561769008636475},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.5467988848686218},{"id":"https://openalex.org/keywords/newtons-method","display_name":"Newton's method","score":0.4983065128326416},{"id":"https://openalex.org/keywords/uniform-convergence","display_name":"Uniform convergence","score":0.48935431241989136},{"id":"https://openalex.org/keywords/newtons-method-in-optimization","display_name":"Newton's method in optimization","score":0.48144644498825073},{"id":"https://openalex.org/keywords/chaotic","display_name":"Chaotic","score":0.46759718656539917},{"id":"https://openalex.org/keywords/aliasing","display_name":"Aliasing","score":0.4529385566711426},{"id":"https://openalex.org/keywords/root-finding-algorithm","display_name":"Root-finding algorithm","score":0.4520260989665985},{"id":"https://openalex.org/keywords/compact-convergence","display_name":"Compact convergence","score":0.4491356611251831},{"id":"https://openalex.org/keywords/divergence","display_name":"Divergence (linguistics)","score":0.4451538026332855},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.43651413917541504},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.43163660168647766},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3323444724082947},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.2534782588481903},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.2464028298854828},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.19023597240447998},{"id":"https://openalex.org/keywords/undersampling","display_name":"Undersampling","score":0.09861385822296143}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7140173316001892},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5888503193855286},{"id":"https://openalex.org/C117898588","wikidata":"https://www.wikidata.org/wiki/Q6664310","display_name":"Local convergence","level":3,"score":0.5790361762046814},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5638430714607239},{"id":"https://openalex.org/C538505008","wikidata":"https://www.wikidata.org/wiki/Q3979448","display_name":"Convergence tests","level":4,"score":0.5590126514434814},{"id":"https://openalex.org/C14928880","wikidata":"https://www.wikidata.org/wiki/Q1512559","display_name":"Normal convergence","level":4,"score":0.5561769008636475},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.5467988848686218},{"id":"https://openalex.org/C85189116","wikidata":"https://www.wikidata.org/wiki/Q374195","display_name":"Newton's method","level":3,"score":0.4983065128326416},{"id":"https://openalex.org/C157709441","wikidata":"https://www.wikidata.org/wiki/Q1411887","display_name":"Uniform convergence","level":3,"score":0.48935431241989136},{"id":"https://openalex.org/C179127668","wikidata":"https://www.wikidata.org/wiki/Q17086396","display_name":"Newton's method in optimization","level":4,"score":0.48144644498825073},{"id":"https://openalex.org/C2777052490","wikidata":"https://www.wikidata.org/wiki/Q5072826","display_name":"Chaotic","level":2,"score":0.46759718656539917},{"id":"https://openalex.org/C4069607","wikidata":"https://www.wikidata.org/wiki/Q868732","display_name":"Aliasing","level":3,"score":0.4529385566711426},{"id":"https://openalex.org/C194777461","wikidata":"https://www.wikidata.org/wiki/Q1085860","display_name":"Root-finding algorithm","level":3,"score":0.4520260989665985},{"id":"https://openalex.org/C5871375","wikidata":"https://www.wikidata.org/wiki/Q1780715","display_name":"Compact convergence","level":4,"score":0.4491356611251831},{"id":"https://openalex.org/C207390915","wikidata":"https://www.wikidata.org/wiki/Q1230525","display_name":"Divergence (linguistics)","level":2,"score":0.4451538026332855},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.43651413917541504},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.43163660168647766},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3323444724082947},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2534782588481903},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.2464028298854828},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.19023597240447998},{"id":"https://openalex.org/C136536468","wikidata":"https://www.wikidata.org/wiki/Q1225894","display_name":"Undersampling","level":2,"score":0.09861385822296143},{"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/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.0},{"id":"https://openalex.org/C2776257435","wikidata":"https://www.wikidata.org/wiki/Q1576430","display_name":"Bandwidth (computing)","level":2,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"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/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tsp.2024.3376088","is_oa":true,"landing_page_url":"https://doi.org/10.1109/tsp.2024.3376088","pdf_url":"https://ieeexplore.ieee.org/ielx7/78/4359509/10466642.pdf","source":{"id":"https://openalex.org/S168680287","display_name":"IEEE Transactions on Signal Processing","issn_l":"1053-587X","issn":["1053-587X","1941-0476"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Signal Processing","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1109/tsp.2024.3376088","is_oa":true,"landing_page_url":"https://doi.org/10.1109/tsp.2024.3376088","pdf_url":"https://ieeexplore.ieee.org/ielx7/78/4359509/10466642.pdf","source":{"id":"https://openalex.org/S168680287","display_name":"IEEE Transactions on Signal Processing","issn_l":"1053-587X","issn":["1053-587X","1941-0476"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Signal Processing","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4392667163.pdf","grobid_xml":"https://content.openalex.org/works/W4392667163.grobid-xml"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W1486073615","https://openalex.org/W1985726041","https://openalex.org/W1990096926","https://openalex.org/W2017736962","https://openalex.org/W2068641562","https://openalex.org/W2084038240","https://openalex.org/W2094086293","https://openalex.org/W3088733666","https://openalex.org/W4241820619"],"related_works":["https://openalex.org/W2034145661","https://openalex.org/W2028595206","https://openalex.org/W2099365622","https://openalex.org/W2482464444","https://openalex.org/W2060809327","https://openalex.org/W2042111348","https://openalex.org/W2102183453","https://openalex.org/W2180233726","https://openalex.org/W2389658966","https://openalex.org/W2088484500"],"abstract_inverted_index":{"Problem":[0],"statement:":[1],"In":[2],"1680":[3],"Newton":[4],"proposed":[5],"an":[6],"algorithm":[7],"for":[8,159],"finding":[9],"roots":[10],"of":[11,26,49,85,92,110,134,145,169,177],"polynomials.":[12,67],"His":[13],"method":[14,56,87,117,158],"has":[15,29],"since":[16],"evolved,":[17],"but":[18],"the":[19,47,78,90,132,146,155,170,184,217,233],"core":[20],"concept":[21],"remains":[22],"intact.":[23],"The":[24,83,167,193],"convergence":[25,84,111,161],"Newton\u2019s":[27,55,86,116],"Method":[28],"been":[30],"widely":[31],"challenged":[32],"to":[33,60,112,130,143],"be":[34,58,212],"unstable":[35],"or":[36],"even":[37],"chaotic.":[38],"Here":[39,154],"we":[40,139],"briefly":[41],"review":[42],"this":[43,135],"evolution,":[44],"and":[45,125,157,222],"consider":[46],"question":[48],"stable":[50,191],"convergence.":[51,192,204],"<p":[52,95,164],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[53,96,165],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">Methods:":[54],"may":[57],"applied":[59],"any":[61],"complex":[62,120],"analytic":[63,121],"function,":[64],"such":[65],"as":[66,152],"Its":[68],"derivation":[69],"is":[70,118,128,141,173,196,236],"based":[71],"on":[72,89],"a":[73,107,113,189,202,208,237],"Taylor":[74],"series":[75],"expansion":[76],"in":[77,106,175,188,201,226],"Laplace":[79],"frequency":[80],"<inline-formula><tex-math":[81],"notation=\"LaTeX\">$s=\\sigma+j\\omega$</tex-math></inline-formula>.":[82],"depends":[88],"<i>R</i>egion":[91],"Convergence":[93],"(RoC).":[94],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">Findings:":[97],"Under":[98],"certain":[99],"conditions,":[100],"non-linear":[101],"(NL)":[102],"limit-cycles":[103],"appear,":[104],"resulting":[105],"reduced":[108],"rate":[109],"root.":[114],"Since":[115],"inherently":[119],"(that":[122],"is,":[123],"linear":[124],"convergent),":[126],"it":[127,198,220,225,230],"important":[129],"establish":[131],"source":[133,168],"NL":[136],"divergence,":[137],"which":[138],"show":[140],"due":[142],"violations":[144],"Nyquist":[147],"Sampling":[148],"theorem,":[149],"also":[150],"known":[151],"aliasing.":[153,178],"conditions":[156],"uniform":[160],"are":[162],"explored.":[163],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">Conclusions:":[166],"nonlinear":[171],"limit-cycle":[172],"explained":[174],"terms":[176],"We":[179],"numerically":[180],"demonstrate":[181],"that":[182,197,207],"reducing":[183],"step-size":[185,210,218,235],"always":[186,199],"results":[187,200],"more":[190],"down":[194],"side":[195],"sub-optimal":[203],"It":[205],"follows":[206],"dynamic":[209],"would":[211],"ideal,":[213],"by":[214],"slowly":[215],"increasing":[216],"until":[219,229],"fails,":[221],"then":[223],"decreasing":[224],"small":[227],"steps":[228],"converges.":[231],"Finding":[232],"optimal":[234],"reasonable":[238],"solution.":[239]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}