{"id":"https://openalex.org/W2029685795","doi":"https://doi.org/10.1145/1073884.1073898","title":"Hybrid symbolic-numeric integration in multiple dimensions via tensor-product series","display_name":"Hybrid symbolic-numeric integration in multiple dimensions via tensor-product series","publication_year":2005,"publication_date":"2005-07-24","ids":{"openalex":"https://openalex.org/W2029685795","doi":"https://doi.org/10.1145/1073884.1073898","mag":"2029685795"},"language":"en","primary_location":{"id":"doi:10.1145/1073884.1073898","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1073884.1073898","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","raw_type":"proceedings-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/A5055768810","display_name":"Orlando Alberto Carvajal","orcid":null},"institutions":[{"id":"https://openalex.org/I151746483","display_name":"University of Waterloo","ror":"https://ror.org/01aff2v68","country_code":"CA","type":"education","lineage":["https://openalex.org/I151746483"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Orlando A. Carvajal","raw_affiliation_strings":["University of Waterloo, Waterloo, ON, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Waterloo, Waterloo, ON, Canada","institution_ids":["https://openalex.org/I151746483"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5039868475","display_name":"Frederick W. Chapman","orcid":null},"institutions":[{"id":"https://openalex.org/I151746483","display_name":"University of Waterloo","ror":"https://ror.org/01aff2v68","country_code":"CA","type":"education","lineage":["https://openalex.org/I151746483"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Frederick W. Chapman","raw_affiliation_strings":["University of Waterloo, Waterloo, ON, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Waterloo, Waterloo, ON, Canada","institution_ids":["https://openalex.org/I151746483"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5002316838","display_name":"K. O. Geddes","orcid":null},"institutions":[{"id":"https://openalex.org/I151746483","display_name":"University of Waterloo","ror":"https://ror.org/01aff2v68","country_code":"CA","type":"education","lineage":["https://openalex.org/I151746483"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Keith O. Geddes","raw_affiliation_strings":["University of Waterloo, Waterloo, ON, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Waterloo, Waterloo, ON, Canada","institution_ids":["https://openalex.org/I151746483"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I151746483"],"apc_list":null,"apc_paid":null,"fwci":0.3375,"has_fulltext":false,"cited_by_count":12,"citation_normalized_percentile":{"value":0.61289858,"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":"84","last_page":"91"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","score":0.9980999827384949,"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":0.9980999827384949,"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/T12404","display_name":"Mathematical Approximation and Integration","score":0.9922000169754028,"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/T11435","display_name":"Polynomial and algebraic computation","score":0.9753000140190125,"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/bilinear-interpolation","display_name":"Bilinear interpolation","score":0.6891382932662964},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.582947850227356},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5358174443244934},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5203194618225098},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5115317106246948},{"id":"https://openalex.org/keywords/interpolation","display_name":"Interpolation (computer graphics)","score":0.5094900727272034},{"id":"https://openalex.org/keywords/tensor-product","display_name":"Tensor product","score":0.503332793712616},{"id":"https://openalex.org/keywords/numerical-integration","display_name":"Numerical integration","score":0.4935981035232544},{"id":"https://openalex.org/keywords/symbolic-computation","display_name":"Symbolic computation","score":0.4777882993221283},{"id":"https://openalex.org/keywords/tensor","display_name":"Tensor (intrinsic definition)","score":0.43274545669555664},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4205487072467804},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.37826552987098694},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3464627265930176},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.17286601662635803},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.16486281156539917},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.12781444191932678}],"concepts":[{"id":"https://openalex.org/C205203396","wikidata":"https://www.wikidata.org/wiki/Q612143","display_name":"Bilinear interpolation","level":2,"score":0.6891382932662964},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.582947850227356},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5358174443244934},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5203194618225098},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5115317106246948},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.5094900727272034},{"id":"https://openalex.org/C51255310","wikidata":"https://www.wikidata.org/wiki/Q1163016","display_name":"Tensor product","level":2,"score":0.503332793712616},{"id":"https://openalex.org/C127349201","wikidata":"https://www.wikidata.org/wiki/Q753445","display_name":"Numerical integration","level":2,"score":0.4935981035232544},{"id":"https://openalex.org/C110812573","wikidata":"https://www.wikidata.org/wiki/Q175515","display_name":"Symbolic computation","level":2,"score":0.4777882993221283},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.43274545669555664},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4205487072467804},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.37826552987098694},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3464627265930176},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.17286601662635803},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.16486281156539917},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.12781444191932678},{"id":"https://openalex.org/C104114177","wikidata":"https://www.wikidata.org/wiki/Q79782","display_name":"Motion (physics)","level":2,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/1073884.1073898","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1073884.1073898","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 2005 international symposium on Symbolic and algebraic computation","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":17,"referenced_works":["https://openalex.org/W115390329","https://openalex.org/W1489395389","https://openalex.org/W1976544585","https://openalex.org/W2001375248","https://openalex.org/W2013811797","https://openalex.org/W2014208555","https://openalex.org/W2023109807","https://openalex.org/W2046767550","https://openalex.org/W2078197553","https://openalex.org/W2086128303","https://openalex.org/W2089405781","https://openalex.org/W2133678399","https://openalex.org/W2329488429","https://openalex.org/W4206038828","https://openalex.org/W4291334562","https://openalex.org/W6808970728","https://openalex.org/W7063782797"],"related_works":["https://openalex.org/W2165589608","https://openalex.org/W3093557575","https://openalex.org/W2354231916","https://openalex.org/W3182767856","https://openalex.org/W2162140574","https://openalex.org/W2354073393","https://openalex.org/W4386687603","https://openalex.org/W4307007469","https://openalex.org/W4288558775","https://openalex.org/W3099027509"],"abstract_inverted_index":{"We":[0,112,245],"present":[1,114],"a":[2,160,188,201,205,233],"new":[3,75,120],"hybrid":[4],"symbolic-numeric":[5],"method":[6,20],"for":[7,23,96],"the":[8,66,140,181,185,217,229,242,248],"fast":[9],"and":[10,37,103,130],"accurate":[11,213],"evaluation":[12],"of":[13,26,40,55,77,85,100,106,139,149,192,208,216,222,237,239],"definite":[14],"integrals":[15,146,238],"in":[16,34,50,65,83,127,155,176,225],"multiple":[17,110,164],"dimensions.":[18],"This":[19,89],"is":[21,124],"well-suited":[22],"two":[24,35],"classes":[25],"problems:":[27],"(1)":[28],"analytic":[29,41],"integrands":[30,42],"over":[31,47],"general":[32],"regions":[33,49],"dimensions,":[36,178],"(2)":[38],"families":[39],"with":[43,131,147,204],"special":[44],"algebraic":[45,53],"structure":[46],"hyperrectangular":[48],"higher":[51],"dimensions.The":[52],"theory":[54],"multivariate":[56],"interpolation":[57],"via":[58],"natural":[59],"tensor":[60],"product":[61],"series":[62,79,99,203],"was":[63],"developed":[64],"doctoral":[67],"thesis":[68,87],"by":[69,199,259],"Chapman,":[70],"who":[71],"named":[72],"this":[73,107,197,226],"broad":[74],"scheme":[76],"bilinear":[78,98,235],"expansions":[80],"\"Geddes":[81],"series\"":[82],"honour":[84],"his":[86],"supervisor.":[88],"paper":[90],"describes":[91],"an":[92,212],"efficient":[93],"adaptive":[94,121],"algorithm":[95,108,123,141],"generating":[97],"Geddes-Newton":[101,202],"type":[102],"explores":[104],"applications":[105],"to":[109,151,180,183,210,232],"integration.":[111],"will":[113],"test":[115],"results":[116],"demonstrating":[117],"that":[118],"our":[119,136],"integration":[122,165],"effective":[125],"both":[126],"high":[128,132,177],"dimensions":[129,150,249],"accuracy.":[133],"For":[134],"example,":[135],"Maple":[137],"implementation":[138],"has":[142],"successfully":[143],"computed":[144,258],"nontrivial":[145],"hundreds":[148],"10-digit":[152],"accuracy,":[153],"each":[154],"under":[156],"3":[157],"minutes":[158],"on":[159],"desktop":[161],"computer.Current":[162],"numerical":[163],"methods":[166],"either":[167],"become":[168],"very":[169,189],"slow":[170],"or":[171,262],"yield":[172],"only":[173],"low":[174],"accuracy":[175],"due":[179],"necessity":[182],"sample":[184],"integrand":[186],"at":[187],"large":[190],"number":[191,207],"points.":[193],"Our":[194],"approach":[195],"overcomes":[196],"difficulty":[198],"using":[200],"modest":[206],"terms":[209],"construct":[211],"tensor-product":[214],"approximation":[215],"integrand.":[218],"The":[219],"partial":[220],"separation":[221],"variables":[223],"achieved":[224],"way":[227],"reduces":[228],"original":[230,243],"integral":[231],"manageable":[234],"combination":[236],"essentially":[240],"half":[241],"dimension.":[244],"continue":[246],"halving":[247],"recursively":[250],"until":[251],"obtaining":[252],"one-dimensional":[253],"integrals,":[254],"which":[255],"are":[256],"then":[257],"standard":[260],"numeric":[261],"symbolic":[263],"techniques.":[264]},"counts_by_year":[{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":3}],"updated_date":"2026-06-26T08:34:08.712188","created_date":"2025-10-10T00:00:00"}