{"id":"https://openalex.org/W2033539266","doi":"https://doi.org/10.1145/1576702.1576750","title":"An iterative method for calculating approximate GCD of univariate polynomials","display_name":"An iterative method for calculating approximate GCD of univariate polynomials","publication_year":2009,"publication_date":"2009-07-28","ids":{"openalex":"https://openalex.org/W2033539266","doi":"https://doi.org/10.1145/1576702.1576750","mag":"2033539266"},"language":"en","primary_location":{"id":"doi:10.1145/1576702.1576750","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1576702.1576750","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 2009 international symposium on Symbolic and algebraic computation","raw_type":"proceedings-article"},"type":"conference-paper","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"http://hdl.handle.net/2241/103423","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5023800217","display_name":"Akira Terui","orcid":"https://orcid.org/0000-0003-0846-3643"},"institutions":[{"id":"https://openalex.org/I146399215","display_name":"University of Tsukuba","ror":"https://ror.org/02956yf07","country_code":"JP","type":"education","lineage":["https://openalex.org/I146399215"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Akira Terui","raw_affiliation_strings":["University of Tsukuba, Tsukuba, Japan",", University of Tsukuba, Tsukuba, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Tsukuba, Tsukuba, Japan","institution_ids":["https://openalex.org/I146399215"]},{"raw_affiliation_string":", University of Tsukuba, Tsukuba, Japan","institution_ids":["https://openalex.org/I146399215"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5023800217"],"corresponding_institution_ids":["https://openalex.org/I146399215"],"apc_list":null,"apc_paid":null,"fwci":2.1113,"has_fulltext":false,"cited_by_count":16,"citation_normalized_percentile":{"value":0.8666986,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":null,"issue":null,"first_page":"351","last_page":"358"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9994999766349792,"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"}},"topics":[{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9994999766349792,"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/T11697","display_name":"Numerical Methods and Algorithms","score":0.9993000030517578,"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/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9962000250816345,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"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/greatest-common-divisor","display_name":"Greatest common divisor","score":0.8604528903961182},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7752575874328613},{"id":"https://openalex.org/keywords/univariate","display_name":"Univariate","score":0.5983731746673584},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.5903950333595276},{"id":"https://openalex.org/keywords/sylvester-matrix","display_name":"Sylvester matrix","score":0.574516236782074},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.5685325264930725},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.5012593269348145},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4412071704864502},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.43364182114601135},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.37278881669044495},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3562858998775482},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.23090216517448425},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.13091346621513367},{"id":"https://openalex.org/keywords/polynomial-matrix","display_name":"Polynomial matrix","score":0.10380399227142334},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.08660715818405151}],"concepts":[{"id":"https://openalex.org/C125257309","wikidata":"https://www.wikidata.org/wiki/Q131752","display_name":"Greatest common divisor","level":2,"score":0.8604528903961182},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7752575874328613},{"id":"https://openalex.org/C199163554","wikidata":"https://www.wikidata.org/wiki/Q1681619","display_name":"Univariate","level":3,"score":0.5983731746673584},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.5903950333595276},{"id":"https://openalex.org/C20160815","wikidata":"https://www.wikidata.org/wiki/Q1482452","display_name":"Sylvester matrix","level":5,"score":0.574516236782074},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.5685325264930725},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.5012593269348145},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4412071704864502},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.43364182114601135},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.37278881669044495},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3562858998775482},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.23090216517448425},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.13091346621513367},{"id":"https://openalex.org/C126352355","wikidata":"https://www.wikidata.org/wiki/Q2562273","display_name":"Polynomial matrix","level":4,"score":0.10380399227142334},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.08660715818405151},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C161584116","wikidata":"https://www.wikidata.org/wiki/Q1952580","display_name":"Multivariate statistics","level":2,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","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}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1145/1576702.1576750","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1576702.1576750","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 2009 international symposium on Symbolic and algebraic computation","raw_type":"proceedings-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.507.5417","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.507.5417","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"https://www.tulips.tsukuba.ac.jp/dspace/bitstream/2241/103423/1/issac2009.pdf","raw_type":"text"},{"id":"pmh:oai:tsukuba.repo.nii.ac.jp:00018169","is_oa":true,"landing_page_url":"http://hdl.handle.net/2241/103423","pdf_url":null,"source":{"id":"https://openalex.org/S4306402076","display_name":"Terrestrial Environment Research Center (University of Tsukuba)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I146399215","host_organization_name":"University of Tsukuba","host_organization_lineage":["https://openalex.org/I146399215"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"pmh:oai:tsukuba.repo.nii.ac.jp:00018169","is_oa":true,"landing_page_url":"http://hdl.handle.net/2241/103423","pdf_url":null,"source":{"id":"https://openalex.org/S4306402076","display_name":"Terrestrial Environment Research Center (University of Tsukuba)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I146399215","host_organization_name":"University of Tsukuba","host_organization_lineage":["https://openalex.org/I146399215"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":24,"referenced_works":["https://openalex.org/W1480547232","https://openalex.org/W1602290632","https://openalex.org/W1760551737","https://openalex.org/W1965692635","https://openalex.org/W1990436029","https://openalex.org/W2011703168","https://openalex.org/W2017333591","https://openalex.org/W2020706127","https://openalex.org/W2026909646","https://openalex.org/W2036474546","https://openalex.org/W2060143865","https://openalex.org/W2061311874","https://openalex.org/W2068937275","https://openalex.org/W2069065624","https://openalex.org/W2077315987","https://openalex.org/W2130331374","https://openalex.org/W2137868850","https://openalex.org/W2143101939","https://openalex.org/W2264885058","https://openalex.org/W3029645440","https://openalex.org/W3151570429","https://openalex.org/W4301014524","https://openalex.org/W4301491118","https://openalex.org/W6842097421"],"related_works":["https://openalex.org/W2187918584","https://openalex.org/W2055039533","https://openalex.org/W2142335183","https://openalex.org/W1480547232","https://openalex.org/W2464284689","https://openalex.org/W1993227164","https://openalex.org/W2968119093","https://openalex.org/W2290956096","https://openalex.org/W2317736117","https://openalex.org/W4237949527"],"abstract_inverted_index":{"We":[0,98,143],"present":[1],"an":[2],"iterative":[3],"algorithm":[4,29,102,148],"for":[5],"calculating":[6],"approximate":[7,68,104],"greatest":[8],"common":[9],"divisor":[10],"(GCD)":[11],"of":[12,23,33,39,67,89],"univariate":[13],"polynomials":[14,24,34],"with":[15,62,79,106,140,154],"the":[16,40,51,55,63,90,95,118],"real":[17],"coefficients.":[18],"For":[19],"a":[20,26,31,37,73,80,87,114],"given":[21,41],"pair":[22,32],"and":[25,43],"degree,":[27],"our":[28,101,126,147],"finds":[30],"which":[35,85],"has":[36],"GCD":[38,69,105,155],"degree":[42],"whose":[44],"coefficients":[45],"are":[46],"perturbed":[47],"from":[48],"those":[49,111],"in":[50],"original":[52],"inputs,":[53],"making":[54],"perturbations":[56,107],"as":[57,59,108,110],"small":[58,109,157],"possible,":[60],"along":[61],"GCD.":[64],"The":[65],"problem":[66],"is":[70,86],"transfered":[71],"to":[72,136],"constrained":[74],"minimization":[75],"problem,":[76],"then":[77],"solved":[78],"so-called":[81],"modified":[82],"Newton":[83],"method,":[84,92,124],"generalization":[88],"gradient-projection":[91],"by":[93,113,133],"searching":[94],"solution":[96],"iteratively.":[97],"demonstrate":[99],"that":[100,146],"calculates":[103],"calculated":[112],"method":[115,127],"based":[116],"on":[117],"structured":[119],"total":[120],"least":[121],"norm":[122],"(STLN)":[123],"while":[125],"runs":[128],"significantly":[129],"faster":[130],"than":[131],"theirs":[132],"approximately":[134],"up":[135],"30":[137],"times,":[138],"compared":[139],"their":[141],"implementation.":[142],"also":[144],"show":[145],"properly":[149],"handles":[150],"some":[151],"ill-conditioned":[152],"problems":[153],"containing":[156],"or":[158],"large":[159],"leading":[160],"coefficient.":[161]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":3}],"updated_date":"2026-08-14T07:06:38.338062","created_date":"2025-10-10T00:00:00"}
