{"id":"https://openalex.org/W2106565812","doi":"https://doi.org/10.1137/s00361445024180","title":"Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later","display_name":"Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later","publication_year":2003,"publication_date":"2003-01-01","ids":{"openalex":"https://openalex.org/W2106565812","doi":"https://doi.org/10.1137/s00361445024180","mag":"2106565812"},"language":"en","primary_location":{"id":"doi:10.1137/s00361445024180","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s00361445024180","pdf_url":null,"source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"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 Review","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/A5109088963","display_name":"C Moler","orcid":null},"institutions":[{"id":"https://openalex.org/I887054487","display_name":"MathWorks (United States)","ror":"https://ror.org/01n8qtk87","country_code":"US","type":"company","lineage":["https://openalex.org/I887054487"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Cleve Moler","raw_affiliation_strings":["MathWorks"],"affiliations":[{"raw_affiliation_string":"MathWorks","institution_ids":["https://openalex.org/I887054487"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5031635942","display_name":"Charles Van Loan","orcid":null},"institutions":[{"id":"https://openalex.org/I205783295","display_name":"Cornell University","ror":"https://ror.org/05bnh6r87","country_code":"US","type":"education","lineage":["https://openalex.org/I205783295"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Charles Van Loan","raw_affiliation_strings":["Cornell University"],"affiliations":[{"raw_affiliation_string":"Cornell University","institution_ids":["https://openalex.org/I205783295"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5109088963"],"corresponding_institution_ids":["https://openalex.org/I887054487"],"apc_list":null,"apc_paid":null,"fwci":45.0671,"has_fulltext":false,"cited_by_count":2348,"citation_normalized_percentile":{"value":0.99957329,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":99,"max":100},"biblio":{"volume":"45","issue":"1","first_page":"3","last_page":"49"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9997000098228455,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9997000098228455,"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":0.9854999780654907,"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/T13126","display_name":"Scientific Research and Discoveries","score":0.9751999974250793,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.7048910856246948},{"id":"https://openalex.org/keywords/matrix-exponential","display_name":"Matrix exponential","score":0.6112045049667358},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5947018265724182},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5932169556617737},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5683591961860657},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.5355287790298462},{"id":"https://openalex.org/keywords/matrix-differential-equation","display_name":"Matrix differential equation","score":0.5269687175750732},{"id":"https://openalex.org/keywords/polynomial-matrix","display_name":"Polynomial matrix","score":0.4429706335067749},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.4004051983356476},{"id":"https://openalex.org/keywords/matrix-polynomial","display_name":"Matrix polynomial","score":0.3992442488670349},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.39292266964912415},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.38554906845092773},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.38536471128463745},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.27073943614959717},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2551546096801758},{"id":"https://openalex.org/keywords/medicine","display_name":"Medicine","score":0.09059301018714905},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08988967537879944}],"concepts":[{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.7048910856246948},{"id":"https://openalex.org/C195906000","wikidata":"https://www.wikidata.org/wiki/Q1191722","display_name":"Matrix exponential","level":3,"score":0.6112045049667358},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5947018265724182},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5932169556617737},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5683591961860657},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.5355287790298462},{"id":"https://openalex.org/C105671362","wikidata":"https://www.wikidata.org/wiki/Q6787870","display_name":"Matrix differential equation","level":3,"score":0.5269687175750732},{"id":"https://openalex.org/C126352355","wikidata":"https://www.wikidata.org/wiki/Q2562273","display_name":"Polynomial matrix","level":4,"score":0.4429706335067749},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.4004051983356476},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.3992442488670349},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.39292266964912415},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.38554906845092773},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.38536471128463745},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.27073943614959717},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2551546096801758},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.09059301018714905},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08988967537879944},{"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/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/s00361445024180","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s00361445024180","pdf_url":null,"source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"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 Review","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.129.9283","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.129.9283","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://blackrim.org/resources/howtolike/41801.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.146.183","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.146.183","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cis.upenn.edu/~cis610/Mat-Exp-Dubious.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.92.5653","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.92.5653","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.cornell.edu/cv/ResearchPDF/19ways+.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":128,"referenced_works":["https://openalex.org/W6842531","https://openalex.org/W28980756","https://openalex.org/W165766857","https://openalex.org/W218580582","https://openalex.org/W1488435683","https://openalex.org/W1539593706","https://openalex.org/W1592378316","https://openalex.org/W1669869822","https://openalex.org/W1964262399","https://openalex.org/W1965854789","https://openalex.org/W1966159950","https://openalex.org/W1966728623","https://openalex.org/W1970592681","https://openalex.org/W1977350947","https://openalex.org/W1980819474","https://openalex.org/W1981041413","https://openalex.org/W1984347942","https://openalex.org/W1985909989","https://openalex.org/W1986034826","https://openalex.org/W1987664276","https://openalex.org/W1988427769","https://openalex.org/W1988843470","https://openalex.org/W1990753794","https://openalex.org/W1992992972","https://openalex.org/W1995516663","https://openalex.org/W1996747804","https://openalex.org/W1997126380","https://openalex.org/W1997180432","https://openalex.org/W1997643131","https://openalex.org/W2001958556","https://openalex.org/W2002057736","https://openalex.org/W2004584742","https://openalex.org/W2004837356","https://openalex.org/W2004961492","https://openalex.org/W2006409769","https://openalex.org/W2007746404","https://openalex.org/W2009743150","https://openalex.org/W2009794202","https://openalex.org/W2010352113","https://openalex.org/W2010652522","https://openalex.org/W2011286068","https://openalex.org/W2013339349","https://openalex.org/W2018148861","https://openalex.org/W2018979122","https://openalex.org/W2023288048","https://openalex.org/W2024671507","https://openalex.org/W2026225301","https://openalex.org/W2028789991","https://openalex.org/W2030569894","https://openalex.org/W2030947203","https://openalex.org/W2032735700","https://openalex.org/W2036734282","https://openalex.org/W2039047857","https://openalex.org/W2039700298","https://openalex.org/W2044263584","https://openalex.org/W2044543226","https://openalex.org/W2046795527","https://openalex.org/W2046982553","https://openalex.org/W2047859076","https://openalex.org/W2049764591","https://openalex.org/W2051058175","https://openalex.org/W2053509296","https://openalex.org/W2053851644","https://openalex.org/W2054078553","https://openalex.org/W2055226507","https://openalex.org/W2056636900","https://openalex.org/W2056795592","https://openalex.org/W2057602562","https://openalex.org/W2058994887","https://openalex.org/W2060769027","https://openalex.org/W2060884399","https://openalex.org/W2061440312","https://openalex.org/W2063776967","https://openalex.org/W2066683261","https://openalex.org/W2066825258","https://openalex.org/W2069532389","https://openalex.org/W2069665540","https://openalex.org/W2079477617","https://openalex.org/W2079699527","https://openalex.org/W2079780979","https://openalex.org/W2080475399","https://openalex.org/W2082463306","https://openalex.org/W2082923923","https://openalex.org/W2083408955","https://openalex.org/W2087883828","https://openalex.org/W2087911376","https://openalex.org/W2089770346","https://openalex.org/W2091782510","https://openalex.org/W2092205578","https://openalex.org/W2094114862","https://openalex.org/W2094133613","https://openalex.org/W2095357428","https://openalex.org/W2096104369","https://openalex.org/W2104756076","https://openalex.org/W2118430238","https://openalex.org/W2123259750","https://openalex.org/W2130727150","https://openalex.org/W2139120673","https://openalex.org/W2139481801","https://openalex.org/W2149614645","https://openalex.org/W2149733240","https://openalex.org/W2154062528","https://openalex.org/W2156464932","https://openalex.org/W2157342852","https://openalex.org/W2159454619","https://openalex.org/W2159671249","https://openalex.org/W2164143288","https://openalex.org/W2164252179","https://openalex.org/W2167193748","https://openalex.org/W2194001072","https://openalex.org/W2298672746","https://openalex.org/W2317827848","https://openalex.org/W2325100936","https://openalex.org/W2521283195","https://openalex.org/W2531149944","https://openalex.org/W2616614818","https://openalex.org/W2617854063","https://openalex.org/W3022136271","https://openalex.org/W4213169573","https://openalex.org/W4213206185","https://openalex.org/W4232010858","https://openalex.org/W4233501930","https://openalex.org/W4237057649","https://openalex.org/W4238169414","https://openalex.org/W4242107349","https://openalex.org/W4251297110","https://openalex.org/W4256186703","https://openalex.org/W4299150230"],"related_works":["https://openalex.org/W1489986405","https://openalex.org/W1980519147","https://openalex.org/W4286858160","https://openalex.org/W3214488505","https://openalex.org/W2466390518","https://openalex.org/W2072116916","https://openalex.org/W4200197172","https://openalex.org/W2118189260","https://openalex.org/W2360459946","https://openalex.org/W2044846538"],"abstract_inverted_index":{"In":[0,30],"principle,":[1],"the":[2,19,23,42],"exponential":[3],"of":[4,33,41,55],"a":[5,66,70],"matrix":[6,20,24],"could":[7],"be":[8],"computed":[9],"in":[10,61],"many":[11],"ways.":[12],"Methods":[13],"involving":[14],"approximation":[15],"theory,":[16],"differential":[17],"equations,":[18],"eigenvalues,":[21],"and":[22,36],"characteristic":[25],"polynomial":[26],"have":[27],"been":[28],"proposed.":[29],"practice,":[31],"consideration":[32],"computational":[34],"stability":[35],"efficiency":[37],"indicates":[38],"that":[39,49],"some":[40],"methods":[43],"are":[44,51],"preferable":[45],"to":[46],"others":[47],"but":[48],"none":[50],"completely":[52],"satisfactory.":[53],"Most":[54],"this":[56],"paper":[57],"was":[58],"originally":[59],"published":[60],"1978.":[62],"An":[63],"update,":[64],"with":[65],"separate":[67],"bibliography,":[68],"describes":[69],"few":[71],"recent":[72],"developments.":[73]},"counts_by_year":[{"year":2026,"cited_by_count":15},{"year":2025,"cited_by_count":95},{"year":2024,"cited_by_count":120},{"year":2023,"cited_by_count":88},{"year":2022,"cited_by_count":106},{"year":2021,"cited_by_count":124},{"year":2020,"cited_by_count":141},{"year":2019,"cited_by_count":146},{"year":2018,"cited_by_count":139},{"year":2017,"cited_by_count":158},{"year":2016,"cited_by_count":132},{"year":2015,"cited_by_count":128},{"year":2014,"cited_by_count":163},{"year":2013,"cited_by_count":116},{"year":2012,"cited_by_count":109}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
