{"id":"https://openalex.org/W2072113904","doi":"https://doi.org/10.1137/120887278","title":"Approximate Calculation of Sums I: Bounds for the Zeros of Gram Polynomials","display_name":"Approximate Calculation of Sums I: Bounds for the Zeros of Gram Polynomials","publication_year":2014,"publication_date":"2014-01-01","ids":{"openalex":"https://openalex.org/W2072113904","doi":"https://doi.org/10.1137/120887278","mag":"2072113904"},"language":"en","primary_location":{"id":"doi:10.1137/120887278","is_oa":false,"landing_page_url":"https://doi.org/10.1137/120887278","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5024820291","display_name":"Iv\u00e1n Area","orcid":"https://orcid.org/0000-0003-0872-5017"},"institutions":[{"id":"https://openalex.org/I88273585","display_name":"Universidade Federal de S\u00e3o Paulo","ror":"https://ror.org/02k5swt12","country_code":"BR","type":"education","lineage":["https://openalex.org/I88273585"]}],"countries":["BR"],"is_corresponding":true,"raw_author_name":"Iv\u00e1n Area","raw_affiliation_strings":["Federal University of Sao Paulo#TAB#"],"affiliations":[{"raw_affiliation_string":"Federal University of Sao Paulo#TAB#","institution_ids":["https://openalex.org/I88273585"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5061055854","display_name":"Dimitar K. Dimitrov","orcid":"https://orcid.org/0000-0002-3078-2336"},"institutions":[{"id":"https://openalex.org/I315704651","display_name":"Leibniz Association","ror":"https://ror.org/01n6r0e97","country_code":"DE","type":"funder","lineage":["https://openalex.org/I315704651"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Dimitar K. Dimitrov","raw_affiliation_strings":["Leibniz Association, Berlin, Germany"],"affiliations":[{"raw_affiliation_string":"Leibniz Association, Berlin, Germany","institution_ids":["https://openalex.org/I315704651"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101921154","display_name":"E. Godoy","orcid":"https://orcid.org/0000-0003-1254-9407"},"institutions":[{"id":"https://openalex.org/I6289922","display_name":"Universidade de Vigo","ror":"https://ror.org/05rdf8595","country_code":"ES","type":"education","lineage":["https://openalex.org/I6289922"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Eduardo Godoy","raw_affiliation_strings":["Universidade de Vigo, Vigo, Spain"],"affiliations":[{"raw_affiliation_string":"Universidade de Vigo, Vigo, Spain","institution_ids":["https://openalex.org/I6289922"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5022188971","display_name":"Vanessa G. Paschoa","orcid":null},"institutions":[{"id":"https://openalex.org/I88273585","display_name":"Universidade Federal de S\u00e3o Paulo","ror":"https://ror.org/02k5swt12","country_code":"BR","type":"education","lineage":["https://openalex.org/I88273585"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"Vanessa Paschoa","raw_affiliation_strings":["Federal University of Sao Paulo#TAB#"],"affiliations":[{"raw_affiliation_string":"Federal University of Sao Paulo#TAB#","institution_ids":["https://openalex.org/I88273585"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5024820291"],"corresponding_institution_ids":["https://openalex.org/I88273585"],"apc_list":null,"apc_paid":null,"fwci":2.3376,"has_fulltext":false,"cited_by_count":8,"citation_normalized_percentile":{"value":0.87043679,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"52","issue":"4","first_page":"1867","last_page":"1886"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T11191","display_name":"Mathematical functions and polynomials","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T11428","display_name":"Advanced Mathematical Identities","score":0.9977999925613403,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11166","display_name":"Analytic Number Theory Research","score":0.996999979019165,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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":0.883352518081665},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7143891453742981},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.6104196310043335},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.49964380264282227},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.44224244356155396},{"id":"https://openalex.org/keywords/gram","display_name":"Gram","score":0.43454426527023315},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3974461555480957},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.20512604713439941}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.883352518081665},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7143891453742981},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.6104196310043335},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.49964380264282227},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.44224244356155396},{"id":"https://openalex.org/C161369605","wikidata":"https://www.wikidata.org/wiki/Q41803","display_name":"Gram","level":3,"score":0.43454426527023315},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3974461555480957},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.20512604713439941},{"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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C523546767","wikidata":"https://www.wikidata.org/wiki/Q10876","display_name":"Bacteria","level":2,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/120887278","is_oa":false,"landing_page_url":"https://doi.org/10.1137/120887278","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:acervodigital.unesp.br:11449/117189","is_oa":false,"landing_page_url":"http://acervodigital.unesp.br/handle/11449/117189","pdf_url":null,"source":{"id":"https://openalex.org/S4306400256","display_name":"Acervo Digital da Universidade Estadual Paulista (Universidade Estadual Paulista)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210115194","host_organization_name":"Funda\u00e7\u00e3o para o Desenvolvimento da UNESP","host_organization_lineage":["https://openalex.org/I4210115194"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"outro"},{"id":"pmh:oai:repositorio.unesp.br:11449/117189","is_oa":false,"landing_page_url":"http://hdl.handle.net/11449/117189","pdf_url":null,"source":{"id":"https://openalex.org/S4377196277","display_name":"UNESP Institutional Repository (S\u00e3o Paulo State University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I879563668","host_organization_name":"Universidade Estadual Paulista (Unesp)","host_organization_lineage":["https://openalex.org/I879563668"],"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:repositorio.unifesp.br:11600/37285","is_oa":false,"landing_page_url":"http://repositorio.unifesp.br/handle/11600/37285","pdf_url":null,"source":{"id":"https://openalex.org/S4377196380","display_name":"UNIFESP Institutional Repository (Universidade Federal de S\u00e3o Paulo)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I88273585","host_organization_name":"Universidade Federal de S\u00e3o Paulo","host_organization_lineage":["https://openalex.org/I88273585"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Artigo"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":32,"referenced_works":["https://openalex.org/W315472779","https://openalex.org/W634742382","https://openalex.org/W1244770280","https://openalex.org/W1528387448","https://openalex.org/W1966268321","https://openalex.org/W1989097818","https://openalex.org/W2003115626","https://openalex.org/W2009888798","https://openalex.org/W2026439994","https://openalex.org/W2029589275","https://openalex.org/W2030755808","https://openalex.org/W2031629670","https://openalex.org/W2037258180","https://openalex.org/W2039343706","https://openalex.org/W2042326540","https://openalex.org/W2045884230","https://openalex.org/W2058912430","https://openalex.org/W2069918448","https://openalex.org/W2082108265","https://openalex.org/W2087491571","https://openalex.org/W2087785740","https://openalex.org/W2103894063","https://openalex.org/W2107394963","https://openalex.org/W2118974146","https://openalex.org/W2125579366","https://openalex.org/W2502619932","https://openalex.org/W2597657582","https://openalex.org/W2798287718","https://openalex.org/W2980671223","https://openalex.org/W3038165843","https://openalex.org/W4248302793","https://openalex.org/W4401266791"],"related_works":["https://openalex.org/W1987781898","https://openalex.org/W2790582133","https://openalex.org/W2790867553","https://openalex.org/W1901380241","https://openalex.org/W2810359026","https://openalex.org/W2951327076","https://openalex.org/W1528146215","https://openalex.org/W2258523950","https://openalex.org/W4205141323","https://openalex.org/W3034390095"],"abstract_inverted_index":{"Let":[0],"$N$":[1,9,72],"be":[2,8,74],"a":[3,79,92],"positive":[4],"integer":[5],"and":[6,107,125],"$x_{j}$":[7],"equidistant":[10],"points.":[11],"We":[12],"propose":[13],"an":[14],"algorithmic":[15],"approach":[16],"for":[17,37,110,118,130],"approximate":[18],"calculation":[19,63,131],"of":[20,22,55,64,70,83,132],"sums":[21,65,77,133],"the":[23,32,53,56,62,116,119,127],"form":[24],"${\\sum_{j=1}^{N}":[25],"F(x_{j})}$.":[26],"The":[27,86],"method":[28,129],"is":[29,94,134],"based":[30],"on":[31],"Gaussian":[33],"type":[34],"quadrature":[35],"formula":[36,93],"sums,":[38],"\\[":[39],"\\sum_j=1^N":[40],"F(x_j)":[41],"\\approx":[42],"\\sum_k=1^n":[43],"B_n,k":[44],"F(g_n,k(N)),":[45],"n":[46],"\u0142l":[47],"N,":[48],"\\]":[49],"where":[50],"$g_{n,k}(N)$":[51,121],"are":[52,122],"zeros":[54,120],"so-called":[57],"Gram":[58],"polynomials.":[59],"This":[60],"allows":[61],"with":[66,78],"very":[67,123],"large":[68],"number":[69,82],"terms":[71],"to":[73,76,95],"reduced":[75],"much":[80],"smaller":[81],"summands":[84],"$n$.":[85],"first":[87],"task":[88],"in":[89],"constructing":[90],"such":[91],"calculate":[96],"its":[97],"nodes":[98],"$g_{n,k}(N)$.":[99,111],"In":[100],"this":[101],"paper":[102],"we":[103],"obtain":[104],"precise":[105],"lower":[106],"upper":[108],"bounds":[109],"Numerical":[112],"experiments":[113],"show":[114],"that":[115,126],"estimates":[117],"sharp":[124],"proposed":[128],"efficient.":[135]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}