{"id":"https://openalex.org/W2546125840","doi":"https://doi.org/10.23638/dmtcs-21-3-14","title":"On-line algorithms for multiplication and division in real and complex numeration systems","display_name":"On-line algorithms for multiplication and division in real and complex numeration systems","publication_year":2019,"publication_date":"2019-06-20","ids":{"openalex":"https://openalex.org/W2546125840","doi":"https://doi.org/10.23638/dmtcs-21-3-14","mag":"2546125840"},"language":"en","primary_location":{"id":"doi:10.23638/dmtcs-21-3-14","is_oa":true,"landing_page_url":"https://doi.org/10.23638/dmtcs-21-3-14","pdf_url":"https://dmtcs.episciences.org/5569/pdf","source":{"id":"https://openalex.org/S200334879","display_name":"Discrete Mathematics & Theoretical Computer Science","issn_l":"1365-8050","issn":["1365-8050","1462-7264"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4328141799","host_organization_name":"French association","host_organization_lineage":["https://openalex.org/P4328141799"],"host_organization_lineage_names":["French association"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete Mathematics & Theoretical Computer Science","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://dmtcs.episciences.org/5569/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5113448170","display_name":"Christiane Frougny","orcid":null},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I169173203","display_name":"D\u00e9l\u00e9gation Paris 7","ror":"https://ror.org/00bw5n526","country_code":"FR","type":"government","lineage":["https://openalex.org/I154526488","https://openalex.org/I169173203"]},{"id":"https://openalex.org/I204730241","display_name":"Universit\u00e9 Paris Cit\u00e9","ror":"https://ror.org/05f82e368","country_code":"FR","type":"education","lineage":["https://openalex.org/I204730241"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Christiane Frougny","raw_affiliation_strings":["IRIF, CNRS and Universit\u00e9 Paris-Diderot"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"IRIF, CNRS and Universit\u00e9 Paris-Diderot","institution_ids":["https://openalex.org/I204730241","https://openalex.org/I169173203","https://openalex.org/I1294671590"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5045684303","display_name":"Marta Pavelka","orcid":"https://orcid.org/0009-0007-0827-4291"},"institutions":[{"id":"https://openalex.org/I44504214","display_name":"Czech Technical University in Prague","ror":"https://ror.org/03kqpb082","country_code":"CZ","type":"education","lineage":["https://openalex.org/I44504214"]}],"countries":["CZ"],"is_corresponding":false,"raw_author_name":"Marta Pavelka","raw_affiliation_strings":["Department of Mathematics, FNSPE, Czech Technical University in Prague"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, FNSPE, Czech Technical University in Prague","institution_ids":["https://openalex.org/I44504214"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5053945124","display_name":"Edita Pelantov\u00e1","orcid":"https://orcid.org/0000-0003-3817-2943"},"institutions":[{"id":"https://openalex.org/I44504214","display_name":"Czech Technical University in Prague","ror":"https://ror.org/03kqpb082","country_code":"CZ","type":"education","lineage":["https://openalex.org/I44504214"]}],"countries":["CZ"],"is_corresponding":false,"raw_author_name":"Edita Pelantova","raw_affiliation_strings":["Department of Mathematics, FNSPE, Czech Technical University in Prague"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, FNSPE, Czech Technical University in Prague","institution_ids":["https://openalex.org/I44504214"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5027050564","display_name":"Milena Svobodov\u00e1","orcid":"https://orcid.org/0000-0003-0337-7922"},"institutions":[{"id":"https://openalex.org/I44504214","display_name":"Czech Technical University in Prague","ror":"https://ror.org/03kqpb082","country_code":"CZ","type":"education","lineage":["https://openalex.org/I44504214"]}],"countries":["CZ"],"is_corresponding":false,"raw_author_name":"Milena Svobodova","raw_affiliation_strings":["Department of Mathematics, FNSPE, Czech Technical University in Prague"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, FNSPE, Czech Technical University in Prague","institution_ids":["https://openalex.org/I44504214"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5113448170"],"corresponding_institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I169173203","https://openalex.org/I204730241"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.00392738,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"Vol. 21 no. 3","issue":"Discrete Algorithms","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"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"}},"topics":[{"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/T11034","display_name":"Digital Filter Design and Implementation","score":0.994700014591217,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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":0.9883000254631042,"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/division","display_name":"Division (mathematics)","score":0.8181250095367432},{"id":"https://openalex.org/keywords/multiplication","display_name":"Multiplication (music)","score":0.694347620010376},{"id":"https://openalex.org/keywords/division-algorithm","display_name":"Division algorithm","score":0.6115761399269104},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.582869291305542},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5593212842941284},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.47241631150245667},{"id":"https://openalex.org/keywords/line","display_name":"Line (geometry)","score":0.45506948232650757},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3031885325908661},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.09230801463127136}],"concepts":[{"id":"https://openalex.org/C60798267","wikidata":"https://www.wikidata.org/wiki/Q1226939","display_name":"Division (mathematics)","level":2,"score":0.8181250095367432},{"id":"https://openalex.org/C2780595030","wikidata":"https://www.wikidata.org/wiki/Q3860309","display_name":"Multiplication (music)","level":2,"score":0.694347620010376},{"id":"https://openalex.org/C190715199","wikidata":"https://www.wikidata.org/wiki/Q3526714","display_name":"Division algorithm","level":3,"score":0.6115761399269104},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.582869291305542},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5593212842941284},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.47241631150245667},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.45506948232650757},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3031885325908661},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.09230801463127136},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.23638/dmtcs-21-3-14","is_oa":true,"landing_page_url":"https://doi.org/10.23638/dmtcs-21-3-14","pdf_url":"https://dmtcs.episciences.org/5569/pdf","source":{"id":"https://openalex.org/S200334879","display_name":"Discrete Mathematics & Theoretical Computer Science","issn_l":"1365-8050","issn":["1365-8050","1462-7264"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4328141799","host_organization_name":"French association","host_organization_lineage":["https://openalex.org/P4328141799"],"host_organization_lineage_names":["French association"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete Mathematics & Theoretical Computer Science","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1610.08309","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1610.08309","pdf_url":"https://arxiv.org/pdf/1610.08309","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:doaj.org/article:00d0380a62f54709a85b08b01de2e67f","is_oa":true,"landing_page_url":"https://doaj.org/article/00d0380a62f54709a85b08b01de2e67f","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":"Discrete Mathematics & Theoretical Computer Science, Vol Vol. 21 no. 3, Iss Discrete Algorithms (2019)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.23638/dmtcs-21-3-14","is_oa":true,"landing_page_url":"https://doi.org/10.23638/dmtcs-21-3-14","pdf_url":"https://dmtcs.episciences.org/5569/pdf","source":{"id":"https://openalex.org/S200334879","display_name":"Discrete Mathematics & Theoretical Computer Science","issn_l":"1365-8050","issn":["1365-8050","1462-7264"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4328141799","host_organization_name":"French association","host_organization_lineage":["https://openalex.org/P4328141799"],"host_organization_lineage_names":["French association"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete Mathematics & Theoretical Computer Science","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Quality Education","id":"https://metadata.un.org/sdg/4","score":0.7900000214576721}],"awards":[],"funders":[{"id":"https://openalex.org/F4320320883","display_name":"Agence Nationale de la Recherche","ror":"https://ror.org/00rbzpz17"},{"id":"https://openalex.org/F4320321181","display_name":"Austrian Science Fund","ror":"https://ror.org/013tf3c58"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W2546125840.pdf"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W1482357992","https://openalex.org/W1548974324","https://openalex.org/W1853473721","https://openalex.org/W1981034639","https://openalex.org/W1986554231","https://openalex.org/W2015700892","https://openalex.org/W2024581566","https://openalex.org/W2089164015","https://openalex.org/W2122339186","https://openalex.org/W2133264311","https://openalex.org/W2152310641","https://openalex.org/W2187226754","https://openalex.org/W2508600411","https://openalex.org/W2916844179","https://openalex.org/W2963672539","https://openalex.org/W4243758454"],"related_works":["https://openalex.org/W2038759377","https://openalex.org/W3093500249","https://openalex.org/W3094080891","https://openalex.org/W2506252583","https://openalex.org/W2036954759","https://openalex.org/W4406620725","https://openalex.org/W1601569507","https://openalex.org/W1559374008","https://openalex.org/W2048249848","https://openalex.org/W4389142747"],"abstract_inverted_index":{"A":[0,185,211],"positional":[1],"numeration":[2],"system":[3,176,202,229],"is":[4,17,32,72,122,238],"given":[5],"by":[6,10,68,158],"a":[7,11,18,33,41,47,62,73,86,102,163,168,189],"base":[8,16,165,191,257,262,271],"and":[9,27,70,79,111,115,128,141,154,167,193,221,232,245,270],"set":[12,30,35,170,196],"of":[13,36,52,75,105,113,172,198,235,301],"digits.":[14,53],"The":[15],"real":[19,124,131,164],"or":[20,49,125,132],"complex":[21,126,190],"number":[22,42],"$\\beta$":[23,121,166,192],"such":[24],"that":[25,120,143,219,233],"$|\\beta|>1$,":[26],"the":[28,58,92,106,118,137,148,159,175,180,201,206,228,236],"digit":[29,87,169,195],"$A$":[31,171,197],"finite":[34,48],"digits":[37,129,260,265,287],"including":[38],"$0$.":[39],"Thus":[40],"can":[43],"be":[44],"seen":[45],"as":[46],"infinite":[50],"string":[51],"An":[54],"on-line":[55,107,152,241],"algorithm":[56],"processes":[57],"input":[59],"piece-by-piece":[60],"in":[61,85,225,227,255],"serial":[63,88],"fashion.":[64],"On-line":[65],"arithmetic,":[66],"introduced":[67],"Trivedi":[69,114],"Ercegovac,":[71],"mode":[74],"computation":[76],"where":[77,282],"operands":[78],"results":[80],"flow":[81],"through":[82],"arithmetic":[83],"units":[84],"manner,":[89],"starting":[90],"with":[91,259,264,286],"most":[93],"significant":[94],"digit.":[95],"In":[96],"this":[97],"paper,":[98],"we":[99],"first":[100],"formulate":[101],"generalized":[103],"version":[104,300],"algorithms":[108,242],"for":[109,117,243],"multiplication":[110,153,244],"division":[112,155,246],"Ercegovac":[116],"cases":[119],"any":[123],"number,":[127],"are":[130,156,223,253],"complex.":[133],"We":[134],"then":[135,151],"define":[136],"so-called":[138],"OL":[139,149,181,207],"Property,":[140,150],"show":[142],"if":[144,183,209],"$(\\beta,":[145,177,203,230],"A)$":[146,178,204,231],"has":[147,179,205],"feasible":[157],"Trivedi-Ercegovac":[160],"algorithms.":[161],"For":[162,188],"contiguous":[173,199],"integers,":[174,200],"Property":[182,208],"$\\#":[184,210],">":[186,212],"|\\beta|$.":[187],"symmetric":[194],"\\beta\\overline{\\beta}":[213],"+":[214,216,275,280],"|\\beta":[215],"\\overline{\\beta}|$.":[217],"Provided":[218],"addition":[220],"subtraction":[222],"realizable":[224],"parallel":[226],"preprocessing":[234],"denominator":[237],"possible,":[239],"our":[240],"have":[247],"linear":[248],"time":[249],"complexity.":[250],"Three":[251],"examples":[252],"presented":[254],"detail:":[256],"$\\beta=\\frac{3+\\sqrt{5}}{2}$":[258],"$A=\\{-1,0,1\\}$;":[261],"$\\beta=2i$":[263],"$A":[266,288],"=":[267,273,278,284,289],"\\{-2,-1,":[268],"0,1,2\\}$;":[269],"$\\beta":[272],"-\\frac{3}{2}":[274],"i":[276],"\\frac{\\sqrt{3}}{2}":[277],"-1":[279],"\\omega$,":[281],"$\\omega":[283],"\\exp{\\frac{2i\\pi}{3}}$,":[285],"\\{0,":[290],"\\pm":[291,293,295],"1,":[292],"\\omega,":[294],"\\omega^2":[296],"\\}$.":[297],"Comment:":[298],"Extended":[299],"contribution":[302],"on":[303,307],"23rd":[304],"IEEE":[305],"Symposium":[306],"Computer":[308],"Arithmetic":[309],"ARITH23":[310]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2015,"cited_by_count":1}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}