{"id":"https://openalex.org/W2018586823","doi":"https://doi.org/10.1137/120869845","title":"On the Convergence Rates of Gauss and Clenshaw--Curtis Quadrature for Functions of Limited Regularity","display_name":"On the Convergence Rates of Gauss and Clenshaw--Curtis Quadrature for Functions of Limited Regularity","publication_year":2012,"publication_date":"2012-01-01","ids":{"openalex":"https://openalex.org/W2018586823","doi":"https://doi.org/10.1137/120869845","mag":"2018586823"},"language":"en","primary_location":{"id":"doi:10.1137/120869845","is_oa":false,"landing_page_url":"https://doi.org/10.1137/120869845","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/A5002179101","display_name":"Shuhuang Xiang","orcid":"https://orcid.org/0000-0002-6727-6170"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Shuhuang Xiang","raw_affiliation_strings":["central south university"],"affiliations":[{"raw_affiliation_string":"central south university","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5087948154","display_name":"Folkmar Bornemann","orcid":"https://orcid.org/0000-0001-9524-4259"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Folkmar Bornemann","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5002179101"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.9715,"has_fulltext":false,"cited_by_count":51,"citation_normalized_percentile":{"value":0.9531653,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"50","issue":"5","first_page":"2581","last_page":"2587"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.9998000264167786,"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":0.9998000264167786,"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/T12404","display_name":"Mathematical Approximation and Integration","score":0.9991000294685364,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9944999814033508,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8667558431625366},{"id":"https://openalex.org/keywords/gaussian-quadrature","display_name":"Gaussian quadrature","score":0.7292053699493408},{"id":"https://openalex.org/keywords/clenshaw\u2013curtis-quadrature","display_name":"Clenshaw\u2013Curtis quadrature","score":0.6833173036575317},{"id":"https://openalex.org/keywords/gauss\u2013jacobi-quadrature","display_name":"Gauss\u2013Jacobi quadrature","score":0.614611804485321},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.6111199855804443},{"id":"https://openalex.org/keywords/gauss\u2013kronrod-quadrature-formula","display_name":"Gauss\u2013Kronrod quadrature formula","score":0.6017064452171326},{"id":"https://openalex.org/keywords/quadrature","display_name":"Quadrature (astronomy)","score":0.5807148218154907},{"id":"https://openalex.org/keywords/tanh-sinh-quadrature","display_name":"Tanh-sinh quadrature","score":0.5550951361656189},{"id":"https://openalex.org/keywords/chebyshev-polynomials","display_name":"Chebyshev polynomials","score":0.5439995527267456},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5291151404380798},{"id":"https://openalex.org/keywords/numerical-integration","display_name":"Numerical integration","score":0.46587109565734863},{"id":"https://openalex.org/keywords/gauss","display_name":"Gauss","score":0.45940887928009033},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4289501905441284},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4204097390174866},{"id":"https://openalex.org/keywords/nystr\u00f6m-method","display_name":"Nystr\u00f6m method","score":0.16922533512115479},{"id":"https://openalex.org/keywords/integral-equation","display_name":"Integral equation","score":0.12111324071884155}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8667558431625366},{"id":"https://openalex.org/C167196314","wikidata":"https://www.wikidata.org/wiki/Q767680","display_name":"Gaussian quadrature","level":4,"score":0.7292053699493408},{"id":"https://openalex.org/C172657837","wikidata":"https://www.wikidata.org/wiki/Q5131618","display_name":"Clenshaw\u2013Curtis quadrature","level":5,"score":0.6833173036575317},{"id":"https://openalex.org/C54691486","wikidata":"https://www.wikidata.org/wiki/Q5527851","display_name":"Gauss\u2013Jacobi quadrature","level":5,"score":0.614611804485321},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.6111199855804443},{"id":"https://openalex.org/C14103991","wikidata":"https://www.wikidata.org/wiki/Q5527852","display_name":"Gauss\u2013Kronrod quadrature formula","level":4,"score":0.6017064452171326},{"id":"https://openalex.org/C62869609","wikidata":"https://www.wikidata.org/wiki/Q28137","display_name":"Quadrature (astronomy)","level":2,"score":0.5807148218154907},{"id":"https://openalex.org/C162046254","wikidata":"https://www.wikidata.org/wiki/Q7683194","display_name":"Tanh-sinh quadrature","level":5,"score":0.5550951361656189},{"id":"https://openalex.org/C129785596","wikidata":"https://www.wikidata.org/wiki/Q619511","display_name":"Chebyshev polynomials","level":2,"score":0.5439995527267456},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5291151404380798},{"id":"https://openalex.org/C127349201","wikidata":"https://www.wikidata.org/wiki/Q753445","display_name":"Numerical integration","level":2,"score":0.46587109565734863},{"id":"https://openalex.org/C161794534","wikidata":"https://www.wikidata.org/wiki/Q177493","display_name":"Gauss","level":2,"score":0.45940887928009033},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4289501905441284},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4204097390174866},{"id":"https://openalex.org/C48265008","wikidata":"https://www.wikidata.org/wiki/Q7071295","display_name":"Nystr\u00f6m method","level":3,"score":0.16922533512115479},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.12111324071884155},{"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/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","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},{"id":"https://openalex.org/C119599485","wikidata":"https://www.wikidata.org/wiki/Q43035","display_name":"Electrical engineering","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/120869845","is_oa":false,"landing_page_url":"https://doi.org/10.1137/120869845","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:CiteSeerX.psu:10.1.1.252.3935","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.252.3935","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www-m3.mathematik.tu-muenchen.de/foswiki/pub/M3/Allgemeines/Publications/86984.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.755.9691","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.755.9691","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1203.2445.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":15,"referenced_works":["https://openalex.org/W253307310","https://openalex.org/W1967564355","https://openalex.org/W1967779414","https://openalex.org/W1968437435","https://openalex.org/W1985305079","https://openalex.org/W1996160600","https://openalex.org/W2033022456","https://openalex.org/W2057817591","https://openalex.org/W2059998776","https://openalex.org/W2061816104","https://openalex.org/W2073732659","https://openalex.org/W2074573054","https://openalex.org/W2801179766","https://openalex.org/W2951868104","https://openalex.org/W3098914594"],"related_works":["https://openalex.org/W4387847482","https://openalex.org/W4390128132","https://openalex.org/W2148120208","https://openalex.org/W2049479394","https://openalex.org/W3115048450","https://openalex.org/W4317584040","https://openalex.org/W2765175526","https://openalex.org/W1490898760","https://openalex.org/W1968235172","https://openalex.org/W3154913590"],"abstract_inverted_index":{"We":[0],"study":[1],"the":[2,9,25,66,105,111,130],"optimal":[3,131],"general":[4],"rate":[5,31,87],"of":[6,8,13,21,59,88,97,113],"convergence":[7,86],"$n$-point":[10],"quadrature":[11,39,76,90,115],"rules":[12,91],"Gauss":[14,38,75],"and":[15,37,63,69],"Clenshaw--Curtis":[16,36],"when":[17],"applied":[18,77],"to":[19,78,82,94,126],"functions":[20],"limited":[22],"regularity:":[23],"if":[24,48],"Chebyshev":[26,79],"coefficients":[27],"decay":[28],"at":[29],"a":[30,71,114],"$O(n^{-s-1})$":[32],"for":[33,74],"some":[34],"$s>0$,":[35],"inherit":[40],"exactly":[41],"this":[42],"rate.":[43,132],"The":[44,85],"proof":[45,107],"(for":[46],"Gauss,":[47],"$0