{"id":"https://openalex.org/W1994629846","doi":"https://doi.org/10.1007/s11075-014-9925-x","title":"The three-term recursion for Chebyshev polynomials is mixed forward-backward stable","display_name":"The three-term recursion for Chebyshev polynomials is mixed forward-backward stable","publication_year":2014,"publication_date":"2014-11-12","ids":{"openalex":"https://openalex.org/W1994629846","doi":"https://doi.org/10.1007/s11075-014-9925-x","mag":"1994629846"},"language":"en","primary_location":{"id":"doi:10.1007/s11075-014-9925-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11075-014-9925-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11075-014-9925-x.pdf","source":{"id":"https://openalex.org/S117494069","display_name":"Numerical Algorithms","issn_l":"1017-1398","issn":["1017-1398","1572-9265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Algorithms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s11075-014-9925-x.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5110120959","display_name":"Alicja Smoktunowicz","orcid":null},"institutions":[{"id":"https://openalex.org/I98677209","display_name":"University of Edinburgh","ror":"https://ror.org/01nrxwf90","country_code":"GB","type":"education","lineage":["https://openalex.org/I98677209"]},{"id":"https://openalex.org/I108403487","display_name":"Warsaw University of Technology","ror":"https://ror.org/00y0xnp53","country_code":"PL","type":"education","lineage":["https://openalex.org/I108403487"]}],"countries":["GB","PL"],"is_corresponding":true,"raw_author_name":"Alicja Smoktunowicz","raw_affiliation_strings":["Faculty of Mathematics and Information Science, University of Technology, Koszykowa 75, Warsaw, 00-662, Poland","School of Mathematics, University of Edinburgh, Edinburgh, Scotland, EH9 3JZ, UK","Faculty of Mathematics and Information Science, University of Technology, Warsaw, Poland"],"affiliations":[{"raw_affiliation_string":"Faculty of Mathematics and Information Science, University of Technology, Koszykowa 75, Warsaw, 00-662, Poland","institution_ids":["https://openalex.org/I108403487"]},{"raw_affiliation_string":"School of Mathematics, University of Edinburgh, Edinburgh, Scotland, EH9 3JZ, UK","institution_ids":["https://openalex.org/I98677209"]},{"raw_affiliation_string":"Faculty of Mathematics and Information Science, University of Technology, Warsaw, Poland","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5008941125","display_name":"Agata Smoktunowicz","orcid":"https://orcid.org/0000-0001-9838-0432"},"institutions":[{"id":"https://openalex.org/I108403487","display_name":"Warsaw University of Technology","ror":"https://ror.org/00y0xnp53","country_code":"PL","type":"education","lineage":["https://openalex.org/I108403487"]},{"id":"https://openalex.org/I98677209","display_name":"University of Edinburgh","ror":"https://ror.org/01nrxwf90","country_code":"GB","type":"education","lineage":["https://openalex.org/I98677209"]}],"countries":["GB","PL"],"is_corresponding":false,"raw_author_name":"Agata Smoktunowicz","raw_affiliation_strings":["Faculty of Mathematics and Information Science, University of Technology, Koszykowa 75, Warsaw, 00-662, Poland","School of Mathematics, University of Edinburgh, Edinburgh, Scotland, EH9 3JZ, UK","University of Edinburgh"],"affiliations":[{"raw_affiliation_string":"Faculty of Mathematics and Information Science, University of Technology, Koszykowa 75, Warsaw, 00-662, Poland","institution_ids":["https://openalex.org/I108403487"]},{"raw_affiliation_string":"School of Mathematics, University of Edinburgh, Edinburgh, Scotland, EH9 3JZ, UK","institution_ids":["https://openalex.org/I98677209"]},{"raw_affiliation_string":"University of Edinburgh","institution_ids":["https://openalex.org/I98677209"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5040338996","display_name":"Ewa Pawelec","orcid":null},"institutions":[{"id":"https://openalex.org/I108403487","display_name":"Warsaw University of Technology","ror":"https://ror.org/00y0xnp53","country_code":"PL","type":"education","lineage":["https://openalex.org/I108403487"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Ewa Pawelec","raw_affiliation_strings":["Faculty of Mathematics and Information Science, University of Technology, Koszykowa 75, Warsaw, 00-662, Poland","Faculty of Mathematics and Information Science, University of Technology, Warsaw, Poland"],"affiliations":[{"raw_affiliation_string":"Faculty of Mathematics and Information Science, University of Technology, Koszykowa 75, Warsaw, 00-662, Poland","institution_ids":["https://openalex.org/I108403487"]},{"raw_affiliation_string":"Faculty of Mathematics and Information Science, University of Technology, Warsaw, Poland","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5110120959"],"corresponding_institution_ids":["https://openalex.org/I108403487","https://openalex.org/I98677209"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":0.4825,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.61301444,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"69","issue":"4","first_page":"785","last_page":"794"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9997000098228455,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9997000098228455,"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":0.9986000061035156,"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/T11191","display_name":"Mathematical functions and polynomials","score":0.9972000122070312,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8702788352966309},{"id":"https://openalex.org/keywords/chebyshev-polynomials","display_name":"Chebyshev polynomials","score":0.7158970832824707},{"id":"https://openalex.org/keywords/recurrence-relation","display_name":"Recurrence relation","score":0.5726702809333801},{"id":"https://openalex.org/keywords/term","display_name":"Term (time)","score":0.5704376697540283},{"id":"https://openalex.org/keywords/theory-of-computation","display_name":"Theory of computation","score":0.556904673576355},{"id":"https://openalex.org/keywords/chebyshev-nodes","display_name":"Chebyshev nodes","score":0.5502423048019409},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.5193315744400024},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5134740471839905},{"id":"https://openalex.org/keywords/chebyshev-filter","display_name":"Chebyshev filter","score":0.5092190504074097},{"id":"https://openalex.org/keywords/recursion","display_name":"Recursion (computer science)","score":0.5020196437835693},{"id":"https://openalex.org/keywords/chebyshev-equation","display_name":"Chebyshev equation","score":0.49730780720710754},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.48644018173217773},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.44892024993896484},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4066520929336548},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.38977742195129395},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3222978711128235},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.1588554084300995},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.10148048400878906},{"id":"https://openalex.org/keywords/classical-orthogonal-polynomials","display_name":"Classical orthogonal polynomials","score":0.09897240996360779},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06181648373603821}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8702788352966309},{"id":"https://openalex.org/C129785596","wikidata":"https://www.wikidata.org/wiki/Q619511","display_name":"Chebyshev polynomials","level":2,"score":0.7158970832824707},{"id":"https://openalex.org/C201292218","wikidata":"https://www.wikidata.org/wiki/Q740970","display_name":"Recurrence relation","level":2,"score":0.5726702809333801},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.5704376697540283},{"id":"https://openalex.org/C24858836","wikidata":"https://www.wikidata.org/wiki/Q844718","display_name":"Theory of computation","level":2,"score":0.556904673576355},{"id":"https://openalex.org/C171326582","wikidata":"https://www.wikidata.org/wiki/Q1052508","display_name":"Chebyshev nodes","level":3,"score":0.5502423048019409},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.5193315744400024},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5134740471839905},{"id":"https://openalex.org/C21424316","wikidata":"https://www.wikidata.org/wiki/Q718621","display_name":"Chebyshev filter","level":2,"score":0.5092190504074097},{"id":"https://openalex.org/C168773036","wikidata":"https://www.wikidata.org/wiki/Q264164","display_name":"Recursion (computer science)","level":2,"score":0.5020196437835693},{"id":"https://openalex.org/C142484921","wikidata":"https://www.wikidata.org/wiki/Q3890369","display_name":"Chebyshev equation","level":4,"score":0.49730780720710754},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.48644018173217773},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.44892024993896484},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4066520929336548},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.38977742195129395},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3222978711128235},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.1588554084300995},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.10148048400878906},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.09897240996360779},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06181648373603821},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s11075-014-9925-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11075-014-9925-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11075-014-9925-x.pdf","source":{"id":"https://openalex.org/S117494069","display_name":"Numerical Algorithms","issn_l":"1017-1398","issn":["1017-1398","1572-9265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Algorithms","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s11075-014-9925-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11075-014-9925-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11075-014-9925-x.pdf","source":{"id":"https://openalex.org/S117494069","display_name":"Numerical Algorithms","issn_l":"1017-1398","issn":["1017-1398","1572-9265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Algorithms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1994629846.pdf","grobid_xml":"https://content.openalex.org/works/W1994629846.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W31142903","https://openalex.org/W93996766","https://openalex.org/W154103404","https://openalex.org/W1593505867","https://openalex.org/W1981115560","https://openalex.org/W2013737143","https://openalex.org/W2020804487","https://openalex.org/W2041307747","https://openalex.org/W2080361077","https://openalex.org/W2081410910","https://openalex.org/W2084420415","https://openalex.org/W2113606817","https://openalex.org/W2124507810","https://openalex.org/W2126619277","https://openalex.org/W3022909171","https://openalex.org/W3038165843"],"related_works":["https://openalex.org/W2054912819","https://openalex.org/W2103708249","https://openalex.org/W2626380860","https://openalex.org/W3136409910","https://openalex.org/W2025186172","https://openalex.org/W4226174306","https://openalex.org/W2061078256","https://openalex.org/W2003704367","https://openalex.org/W2041163946","https://openalex.org/W2052749413"],"abstract_inverted_index":{"This":[0],"paper":[1],"provides":[2],"an":[3],"error":[4],"analysis":[5],"of":[6,21,25,42,56,62,67],"the":[7,19,22,26,33,39,53,57,75],"three-term":[8],"recurrence":[9,48],"relation":[10],"(TTRR)":[11],"T":[12,14,29,43,60],"n+1(x)=2x":[13],"n":[15],"(x)\u2212T":[16],"n\u22121(x)":[17],"for":[18,74],"evaluation":[20],"Chebyshev":[23,58],"polynomial":[24,59],"first":[27],"kind":[28],"N":[30,44,61],"(x)":[31,45],"in":[32],"interval":[34],"[\u22121,1].":[35],"We":[36],"prove":[37],"that":[38,88],"computed":[40],"value":[41,55,66],"from":[46],"this":[47],"is":[49],"very":[50],"close":[51],"to":[52],"exact":[54],"a":[63],"slightly":[64],"perturbed":[65],"x.":[68],"The":[69],"lower":[70],"and":[71],"upper":[72],"bounds":[73],"function":[76],"$C_{N}(x)=":[77],"|T_{N}(x)|":[78],"+":[79],"|x":[80],"T_{N}^{\\prime":[81],"}(x)|$":[82],"are":[83,93],"also":[84],"derived.":[85],"Numerical":[86],"examples":[87],"illustrate":[89],"our":[90],"theoretical":[91],"results":[92],"given.":[94]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2017,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
