{"id":"https://openalex.org/W2006353862","doi":"https://doi.org/10.1090/s0025-5718-2013-02717-2","title":"Automatic evaluations of cross-derivatives","display_name":"Automatic evaluations of cross-derivatives","publication_year":2013,"publication_date":"2013-05-22","ids":{"openalex":"https://openalex.org/W2006353862","doi":"https://doi.org/10.1090/s0025-5718-2013-02717-2","mag":"2006353862"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-2013-02717-2","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2013-02717-2","pdf_url":"https://www.ams.org/mcom/2014-83-285/S0025-5718-2013-02717-2/S0025-5718-2013-02717-2.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/2014-83-285/S0025-5718-2013-02717-2/S0025-5718-2013-02717-2.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5079600875","display_name":"Andreas Griewank","orcid":"https://orcid.org/0000-0001-9839-1473"},"institutions":[{"id":"https://openalex.org/I39343248","display_name":"Humboldt-Universit\u00e4t zu Berlin","ror":"https://ror.org/01hcx6992","country_code":"DE","type":"education","lineage":["https://openalex.org/I39343248"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Andreas Griewank","raw_affiliation_strings":["Department of Mathematics, Humboldt-Universit\u00c3\u00a4t zu Berlin, Unter den Linden, 610099 Berlin","Department of Mathematics, Humboldt-Universit\u00e4t zu Berlin, Unter den Linden, 610099 Berlin"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Humboldt-Universit\u00c3\u00a4t zu Berlin, Unter den Linden, 610099 Berlin","institution_ids":[]},{"raw_affiliation_string":"Department of Mathematics, Humboldt-Universit\u00e4t zu Berlin, Unter den Linden, 610099 Berlin","institution_ids":["https://openalex.org/I39343248"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5065075869","display_name":"Lutz Lehmann","orcid":null},"institutions":[{"id":"https://openalex.org/I39343248","display_name":"Humboldt-Universit\u00e4t zu Berlin","ror":"https://ror.org/01hcx6992","country_code":"DE","type":"education","lineage":["https://openalex.org/I39343248"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Lutz Lehmann","raw_affiliation_strings":["Department of Mathematics, Humboldt-Universit\u00c3\u00a4t zu Berlin, Unter den Linden, 610099 Berlin","Department of Mathematics, Humboldt-Universit\u00e4t zu Berlin, Unter den Linden, 610099 Berlin"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Humboldt-Universit\u00c3\u00a4t zu Berlin, Unter den Linden, 610099 Berlin","institution_ids":[]},{"raw_affiliation_string":"Department of Mathematics, Humboldt-Universit\u00e4t zu Berlin, Unter den Linden, 610099 Berlin","institution_ids":["https://openalex.org/I39343248"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5042104231","display_name":"Hernan Le\u00f6vey","orcid":null},"institutions":[{"id":"https://openalex.org/I39343248","display_name":"Humboldt-Universit\u00e4t zu Berlin","ror":"https://ror.org/01hcx6992","country_code":"DE","type":"education","lineage":["https://openalex.org/I39343248"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Hernan Leovey","raw_affiliation_strings":["Department of Mathematics, Humboldt-Universit\u00c3\u00a4t zu Berlin, Unter den Linden, 610099 Berlin","Department of Mathematics, Humboldt-Universit\u00e4t zu Berlin, Unter den Linden, 610099 Berlin"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Humboldt-Universit\u00c3\u00a4t zu Berlin, Unter den Linden, 610099 Berlin","institution_ids":[]},{"raw_affiliation_string":"Department of Mathematics, Humboldt-Universit\u00e4t zu Berlin, Unter den Linden, 610099 Berlin","institution_ids":["https://openalex.org/I39343248"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5035908255","display_name":"Marat Zilberman","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Marat Zilberman","raw_affiliation_strings":["Haifa"],"affiliations":[{"raw_affiliation_string":"Haifa","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5079600875"],"corresponding_institution_ids":["https://openalex.org/I39343248"],"apc_list":null,"apc_paid":null,"fwci":3.2912,"has_fulltext":true,"cited_by_count":12,"citation_normalized_percentile":{"value":0.91188896,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"83","issue":"285","first_page":"251","last_page":"274"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12404","display_name":"Mathematical Approximation and Integration","score":0.9993000030517578,"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/T12404","display_name":"Mathematical Approximation and Integration","score":0.9993000030517578,"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":0.998199999332428,"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/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9965000152587891,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"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":0.817711591720581},{"id":"https://openalex.org/keywords/interpolation","display_name":"Interpolation (computer graphics)","score":0.5873355269432068},{"id":"https://openalex.org/keywords/computational-complexity-theory","display_name":"Computational complexity theory","score":0.5778676271438599},{"id":"https://openalex.org/keywords/fast-fourier-transform","display_name":"Fast Fourier transform","score":0.5304386615753174},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4958425462245941},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4574398994445801},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3493785858154297},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.30522310733795166},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.14768308401107788},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10184767842292786}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.817711591720581},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.5873355269432068},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.5778676271438599},{"id":"https://openalex.org/C75172450","wikidata":"https://www.wikidata.org/wiki/Q623950","display_name":"Fast Fourier transform","level":2,"score":0.5304386615753174},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4958425462245941},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4574398994445801},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3493785858154297},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.30522310733795166},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.14768308401107788},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10184767842292786},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"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/C502989409","wikidata":"https://www.wikidata.org/wiki/Q11425","display_name":"Animation","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1090/s0025-5718-2013-02717-2","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2013-02717-2","pdf_url":"https://www.ams.org/mcom/2014-83-285/S0025-5718-2013-02717-2/S0025-5718-2013-02717-2.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"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-2013-02717-2","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2013-02717-2","pdf_url":"https://www.ams.org/mcom/2014-83-285/S0025-5718-2013-02717-2/S0025-5718-2013-02717-2.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":[{"id":"https://openalex.org/F4320320875","display_name":"Deutscher Akademischer Austauschdienst","ror":"https://ror.org/039djdh30"},{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2006353862.pdf","grobid_xml":"https://content.openalex.org/works/W2006353862.grobid-xml"},"referenced_works_count":40,"referenced_works":["https://openalex.org/W30819010","https://openalex.org/W170075325","https://openalex.org/W598051755","https://openalex.org/W1484494266","https://openalex.org/W1512171754","https://openalex.org/W1543280314","https://openalex.org/W1585773866","https://openalex.org/W1602773783","https://openalex.org/W1608019378","https://openalex.org/W1820164671","https://openalex.org/W1938947360","https://openalex.org/W1968165559","https://openalex.org/W1972504036","https://openalex.org/W1981663184","https://openalex.org/W1981743146","https://openalex.org/W1992493944","https://openalex.org/W2001861853","https://openalex.org/W2004971474","https://openalex.org/W2011039300","https://openalex.org/W2011200438","https://openalex.org/W2012149337","https://openalex.org/W2022681869","https://openalex.org/W2042083287","https://openalex.org/W2043419861","https://openalex.org/W2045326192","https://openalex.org/W2065021724","https://openalex.org/W2083253722","https://openalex.org/W2095595785","https://openalex.org/W2101589741","https://openalex.org/W2141306842","https://openalex.org/W2183868163","https://openalex.org/W2478429860","https://openalex.org/W2482768955","https://openalex.org/W2497677836","https://openalex.org/W2752853835","https://openalex.org/W3119524009","https://openalex.org/W4234900207","https://openalex.org/W4238816768","https://openalex.org/W4285719527","https://openalex.org/W4300580367"],"related_works":["https://openalex.org/W4327521644","https://openalex.org/W2978884468","https://openalex.org/W3132558499","https://openalex.org/W2168413811","https://openalex.org/W2005846134","https://openalex.org/W2369237035","https://openalex.org/W2948041274","https://openalex.org/W2165986314","https://openalex.org/W4226515754","https://openalex.org/W2087878472"],"abstract_inverted_index":{"Cross-derivatives":[0],"are":[1,29],"mixed":[2],"partial":[3],"derivatives":[4],"involving":[5],"at":[6,68],"most":[7],"one":[8,12],"differentiation":[9],"in":[10,33,39],"each":[11],"of":[13,36,123,149],"<inline-formula":[14,50,77,95,157],"content-type=\"math/mathml\">":[15,51,78,96,158],"<mml:math":[16,52,79,97,159],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[17,53,80,98,160],"alttext=\"n\">":[18,161],"<mml:semantics>":[19,57,84,104,162],"<mml:mi>n</mml:mi>":[20,60,87,107,112,163],"<mml:annotation":[21,62,89,115,164],"encoding=\"application/x-tex\">n</mml:annotation>":[22,165],"</mml:semantics>":[23,64,91,118,166],"</mml:math>":[24,65,92,119,167],"</inline-formula>":[25,66,93,120,168],"coordinate":[26],"directions.":[27],"They":[28],"a":[30,69,132],"computational":[31,73],"tool":[32],"combinatorics":[34],"and":[35,94],"potential":[37],"use":[38],"high-dimensional":[40],"integration.":[41],"Here":[42],"we":[43],"present":[44],"two":[45],"methods":[46],"that":[47,122],"evaluate":[48],"all":[49],"alttext=\"2":[54],"Superscript":[55,82,102],"n\">":[56,83,103],"<mml:msup>":[58,85,106,110],"<mml:mn>2</mml:mn>":[59,108,111],"</mml:msup>":[61,88,109,113],"encoding=\"application/x-tex\">2^n</mml:annotation>":[63],"cross-derivatives":[67],"given":[70],"point.":[71],"The":[72,127],"complexity":[74,153],"is,":[75],"respectively,":[76],"alttext=\"3":[81],"<mml:mn>3</mml:mn>":[86],"encoding=\"application/x-tex\">3^n</mml:annotation>":[90],"alttext=\"n":[99],"squared":[100],"2":[101],"<mml:mrow>":[105],"</mml:mrow>":[114],"encoding=\"application/x-tex\">n^2":[116],"2^n</mml:annotation>":[117],"times":[121],"the":[124,147],"underlying":[125],"function.":[126],"asymptotically":[128],"faster":[129,173],"method":[130,178],"involves":[131],"final":[133],"interpolation":[134],"step,":[135],"which":[136],"can":[137,169],"easily":[138],"be":[139,170],"carried":[140],"out":[141],"using":[142],"extra-accurate":[143],"subtractions":[144],"to":[145],"reduce":[146],"effect":[148],"numerical":[150],"round-off.":[151],"Further":[152],"reductions":[154],"for":[155],"large":[156],"obtained":[171],"through":[172],"polynomial":[174],"multiplications,":[175],"e.g.,":[176],"Karatsuba\u2019s":[177],"or":[179],"FFT.":[180]},"counts_by_year":[{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":2}],"updated_date":"2026-04-22T06:01:30.510260","created_date":"2025-10-10T00:00:00"}
