{"id":"https://openalex.org/W2072524194","doi":"https://doi.org/10.1090/s0025-5718-99-01033-9","title":"For numerical differentiation, dimensionality can be a blessing!","display_name":"For numerical differentiation, dimensionality can be a blessing!","publication_year":1999,"publication_date":"1999-02-10","ids":{"openalex":"https://openalex.org/W2072524194","doi":"https://doi.org/10.1090/s0025-5718-99-01033-9","mag":"2072524194"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-99-01033-9","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-99-01033-9","pdf_url":"https://www.ams.org/mcom/1999-68-227/S0025-5718-99-01033-9/S0025-5718-99-01033-9.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/1999-68-227/S0025-5718-99-01033-9/S0025-5718-99-01033-9.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5065950971","display_name":"Robert Anderssen","orcid":"https://orcid.org/0000-0001-8344-2346"},"institutions":[{"id":"https://openalex.org/I1292875679","display_name":"Commonwealth Scientific and Industrial Research Organisation","ror":"https://ror.org/03qn8fb07","country_code":"AU","type":"government","lineage":["https://openalex.org/I1292875679","https://openalex.org/I2801453606","https://openalex.org/I4387156119"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Robert Anderssen","raw_affiliation_strings":["CSIRO Mathematical and Information Sciences, GPO Box 1965, Canberra ACT 2601, Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"CSIRO Mathematical and Information Sciences, GPO Box 1965, Canberra ACT 2601, Australia","institution_ids":["https://openalex.org/I1292875679"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5009262102","display_name":"Markus Hegland","orcid":"https://orcid.org/0000-0002-5136-2883"},"institutions":[{"id":"https://openalex.org/I118347636","display_name":"Australian National University","ror":"https://ror.org/019wvm592","country_code":"AU","type":"education","lineage":["https://openalex.org/I118347636"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Markus Hegland","raw_affiliation_strings":["Computer Sciences Laboratory, Australian National University, Canberra ACT 0200, Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Computer Sciences Laboratory, Australian National University, Canberra ACT 0200, Australia","institution_ids":["https://openalex.org/I118347636"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.9799,"has_fulltext":true,"cited_by_count":89,"citation_normalized_percentile":{"value":0.76607642,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"68","issue":"227","first_page":"1121","last_page":"1141"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","score":0.9987999796867371,"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":0.9987999796867371,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9983000159263611,"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":0.9973000288009644,"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/numerical-differentiation","display_name":"Numerical differentiation","score":0.8380632400512695},{"id":"https://openalex.org/keywords/automatic-differentiation","display_name":"Automatic differentiation","score":0.7036433219909668},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6200612187385559},{"id":"https://openalex.org/keywords/rounding","display_name":"Rounding","score":0.6119073033332825},{"id":"https://openalex.org/keywords/curse-of-dimensionality","display_name":"Curse of dimensionality","score":0.5730429291725159},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5485022664070129},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4475230276584625},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.33382147550582886},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3189024031162262},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.2684231400489807},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1941913366317749}],"concepts":[{"id":"https://openalex.org/C154181440","wikidata":"https://www.wikidata.org/wiki/Q1683292","display_name":"Numerical differentiation","level":2,"score":0.8380632400512695},{"id":"https://openalex.org/C133512626","wikidata":"https://www.wikidata.org/wiki/Q787371","display_name":"Automatic differentiation","level":3,"score":0.7036433219909668},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6200612187385559},{"id":"https://openalex.org/C136625980","wikidata":"https://www.wikidata.org/wiki/Q663208","display_name":"Rounding","level":2,"score":0.6119073033332825},{"id":"https://openalex.org/C111030470","wikidata":"https://www.wikidata.org/wiki/Q1430460","display_name":"Curse of dimensionality","level":2,"score":0.5730429291725159},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5485022664070129},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4475230276584625},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.33382147550582886},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3189024031162262},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.2684231400489807},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1941913366317749},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1090/s0025-5718-99-01033-9","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-99-01033-9","pdf_url":"https://www.ams.org/mcom/1999-68-227/S0025-5718-99-01033-9/S0025-5718-99-01033-9.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.47.8292","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.8292","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ftp://nimbus.anu.edu.au/pub/Hegland/andh97.ps.gz","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.97.1230","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.97.1230","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ams.org/mcom/1999-68-227/S0025-5718-99-01033-9/S0025-5718-99-01033-9.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-99-01033-9","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-99-01033-9","pdf_url":"https://www.ams.org/mcom/1999-68-227/S0025-5718-99-01033-9/S0025-5718-99-01033-9.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2072524194.pdf","grobid_xml":"https://content.openalex.org/works/W2072524194.grobid-xml"},"referenced_works_count":14,"referenced_works":["https://openalex.org/W2019203901","https://openalex.org/W2043419861","https://openalex.org/W2047168631","https://openalex.org/W2075667049","https://openalex.org/W2142968624","https://openalex.org/W2143022286","https://openalex.org/W2163591144","https://openalex.org/W2167973446","https://openalex.org/W2398667566","https://openalex.org/W3028195145","https://openalex.org/W4238717354","https://openalex.org/W4293257133","https://openalex.org/W4300580367","https://openalex.org/W4386571814"],"related_works":["https://openalex.org/W2381569690","https://openalex.org/W2179317017","https://openalex.org/W4253305913","https://openalex.org/W2281162465","https://openalex.org/W2382989034","https://openalex.org/W4251548445","https://openalex.org/W2072524194","https://openalex.org/W2768512587","https://openalex.org/W3047338125","https://openalex.org/W2011673959"],"abstract_inverted_index":{"Finite":[0],"difference":[1],"methods,":[2],"such":[3,23],"as":[4,114],"the":[5,13,35,59,79,88,101,118,166,178,188,198],"mid-point":[6],"rule,":[7],"have":[8,77],"been":[9],"applied":[10,26],"successfully":[11],"to":[12,27,32,111,177],"numerical":[14,147],"solution":[15],"of":[16,75,81,84,91,100,117,149,161,169,181,187,200],"ordinary":[17],"and":[18,48,68,109,192,197],"partial":[19],"differential":[20],"equations.":[21],"If":[22],"formulas":[24],"are":[25,55,66,71],"observational":[28,96,152],"data,":[29],"in":[30,52,58],"order":[31,93,145],"determine":[33],"derivatives,":[34],"results":[36],"can":[37,154],"be":[38,112,155],"disastrous.":[39],"The":[40,98,172,185],"reason":[41],"for":[42,165],"this":[43,135],"is":[44,138,174,194,203],"that":[45],"measurement":[46],"errors,":[47],"even":[49],"rounding":[50],"errors":[51],"computer":[53],"approximations,":[54],"strongly":[56],"amplified":[57],"differentiation":[60,148,168,180],"process,":[61],"especially":[62],"if":[63],"small":[64],"step-sizes":[65],"chosen":[67,113],"higher":[69,150,182],"derivatives":[70,94],"required.":[72],"A":[73],"number":[74],"authors":[76],"examined":[78],"use":[80],"various":[82,201],"forms":[83],"averaging":[85,102],"which":[86],"allows":[87],"stable":[89],"computation":[90],"low":[92],"from":[95],"data.":[97,171,184],"size":[99,120],"set":[103],"acts":[104],"like":[105],"a":[106,115,158],"regularization":[107],"parameter":[108],"has":[110],"function":[116],"grid":[119],"<inline-formula":[121],"content-type=\"math/mathml\">":[122],"<mml:math":[123],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[124],"alttext=\"h\">":[125],"<mml:semantics>":[126],"<mml:mi>h</mml:mi>":[127],"<mml:annotation":[128],"encoding=\"application/x-tex\">h</mml:annotation>":[129],"</mml:semantics>":[130],"</mml:math>":[131],"</inline-formula>":[132],".":[133],"In":[134],"paper,":[136],"it":[137],"initially":[139],"shown":[140],"how":[141],"first":[142],"(and":[143],"higher)":[144],"single-variate":[146],"dimensional":[151,183],"data":[153],"stabilized":[156],"with":[157],"reduced":[159],"loss":[160],"accuracy":[162],"than":[163],"occurs":[164],"corresponding":[167],"one-dimensional":[170],"result":[173],"then":[175],"extended":[176],"multivariate":[179],"nature":[186],"trade-off":[189],"between":[190],"convergence":[191],"stability":[193],"explicitly":[195],"characterized,":[196],"complexity":[199],"implementations":[202],"examined.":[204]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":7},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":5},{"year":2013,"cited_by_count":8},{"year":2012,"cited_by_count":3}],"updated_date":"2026-06-12T08:23:45.883708","created_date":"2025-10-10T00:00:00"}
