{"id":"https://openalex.org/W1601910470","doi":"https://doi.org/10.1137/14098003x","title":"The Scaling, Splitting, and Squaring Method for the Exponential of Perturbed Matrices","display_name":"The Scaling, Splitting, and Squaring Method for the Exponential of Perturbed Matrices","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W1601910470","doi":"https://doi.org/10.1137/14098003x","mag":"1601910470"},"language":"en","primary_location":{"id":"doi:10.1137/14098003x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/14098003x","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"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 Journal on Matrix Analysis and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1407.7485","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5045707678","display_name":"Philipp Bader","orcid":"https://orcid.org/0000-0002-7090-7105"},"institutions":[{"id":"https://openalex.org/I60053951","display_name":"Universitat Polit\u00e8cnica de Val\u00e8ncia","ror":"https://ror.org/01460j859","country_code":"ES","type":"education","lineage":["https://openalex.org/I60053951"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Philipp Bader","raw_affiliation_strings":["INSTITUTO DE MATEM \u00c1TICA MULTIDISCIPLINAR, UNIVERSITAT POLIT \u00c8CNICA DE VAL \u00c8NCIA, E-46022 VALENCIA, SPAIN"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"INSTITUTO DE MATEM \u00c1TICA MULTIDISCIPLINAR, UNIVERSITAT POLIT \u00c8CNICA DE VAL \u00c8NCIA, E-46022 VALENCIA, SPAIN","institution_ids":["https://openalex.org/I60053951"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5075969053","display_name":"Sergio Blanes","orcid":"https://orcid.org/0000-0001-5819-8898"},"institutions":[{"id":"https://openalex.org/I60053951","display_name":"Universitat Polit\u00e8cnica de Val\u00e8ncia","ror":"https://ror.org/01460j859","country_code":"ES","type":"education","lineage":["https://openalex.org/I60053951"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Sergio Blanes","raw_affiliation_strings":["INSTITUTO DE MATEM \u00c1TICA MULTIDISCIPLINAR, UNIVERSITAT POLIT \u00c8CNICA DE VAL \u00c8NCIA, E-46022 VALENCIA, SPAIN"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"INSTITUTO DE MATEM \u00c1TICA MULTIDISCIPLINAR, UNIVERSITAT POLIT \u00c8CNICA DE VAL \u00c8NCIA, E-46022 VALENCIA, SPAIN","institution_ids":["https://openalex.org/I60053951"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5078492749","display_name":"Muaz Seydao\u011flu","orcid":"https://orcid.org/0000-0003-3211-0864"},"institutions":[{"id":"https://openalex.org/I140912994","display_name":"Mu\u015f Alparslan University","ror":"https://ror.org/009axq942","country_code":"TR","type":"education","lineage":["https://openalex.org/I140912994"]}],"countries":["TR"],"is_corresponding":false,"raw_author_name":"Muaz Seydao\u011flu","raw_affiliation_strings":["DEPARTMENT OF MATHEMATICS, FACULTY OF ART AND SCIENCE, MUS \u00b8AL-PARSLAN UNIVERSITY, 49100 MUS \u00b8, TURKEY"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"DEPARTMENT OF MATHEMATICS, FACULTY OF ART AND SCIENCE, MUS \u00b8AL-PARSLAN UNIVERSITY, 49100 MUS \u00b8, TURKEY","institution_ids":["https://openalex.org/I140912994"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.0029,"has_fulltext":true,"cited_by_count":8,"citation_normalized_percentile":{"value":0.77488978,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"36","issue":"2","first_page":"594","last_page":"614"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9998000264167786,"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9968000054359436,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/matrix-exponential","display_name":"Matrix exponential","score":0.8273905515670776},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.81202232837677},{"id":"https://openalex.org/keywords/scaling","display_name":"Scaling","score":0.7128127217292786},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.6828241348266602},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.6042071580886841},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.5681385397911072},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.520715057849884},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.5151554346084595},{"id":"https://openalex.org/keywords/sparse-matrix","display_name":"Sparse matrix","score":0.4592551589012146},{"id":"https://openalex.org/keywords/taylor-series","display_name":"Taylor series","score":0.44633352756500244},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.4126637578010559},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3699437379837036},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.32650190591812134},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.30150264501571655},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09388643503189087},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.06919953227043152},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.0613006055355072}],"concepts":[{"id":"https://openalex.org/C195906000","wikidata":"https://www.wikidata.org/wiki/Q1191722","display_name":"Matrix exponential","level":3,"score":0.8273905515670776},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.81202232837677},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.7128127217292786},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.6828241348266602},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.6042071580886841},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.5681385397911072},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.520715057849884},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.5151554346084595},{"id":"https://openalex.org/C56372850","wikidata":"https://www.wikidata.org/wiki/Q1050404","display_name":"Sparse matrix","level":3,"score":0.4592551589012146},{"id":"https://openalex.org/C158946198","wikidata":"https://www.wikidata.org/wiki/Q131187","display_name":"Taylor series","level":2,"score":0.44633352756500244},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4126637578010559},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3699437379837036},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.32650190591812134},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.30150264501571655},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09388643503189087},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.06919953227043152},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0613006055355072},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"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},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"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/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1137/14098003x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/14098003x","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"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 Journal on Matrix Analysis and Applications","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1407.7485","is_oa":true,"landing_page_url":"https://arxiv.org/abs/1407.7485","pdf_url":"https://arxiv.org/pdf/1407.7485","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.767.4587","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.767.4587","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1407.7485.pdf","raw_type":"text"},{"id":"pmh:oai:akademikarsiv.alparslan.edu.tr:20.500.12639/1165","is_oa":true,"landing_page_url":"https://hdl.handle.net/20.500.12639/1165","pdf_url":null,"source":{"id":"https://openalex.org/S7407055300","display_name":"DSpace@M\u015eU","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"pmh:oai:riunet.upv.es:10251/99868","is_oa":true,"landing_page_url":"http://hdl.handle.net/10251/99868","pdf_url":null,"source":{"id":"https://openalex.org/S4306400639","display_name":"RiuNet (Universitat Polit\u00e8cnica de Val\u00e8ncia)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I60053951","host_organization_name":"Universitat Polit\u00e8cnica de Val\u00e8ncia","host_organization_lineage":["https://openalex.org/I60053951"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1407.7485","is_oa":true,"landing_page_url":"https://arxiv.org/abs/1407.7485","pdf_url":"https://arxiv.org/pdf/1407.7485","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":23,"referenced_works":["https://openalex.org/W70648946","https://openalex.org/W1749808203","https://openalex.org/W1981041413","https://openalex.org/W1981550893","https://openalex.org/W1988745808","https://openalex.org/W1990647940","https://openalex.org/W2003082771","https://openalex.org/W2005812552","https://openalex.org/W2006409769","https://openalex.org/W2012692201","https://openalex.org/W2055580199","https://openalex.org/W2058763860","https://openalex.org/W2061356302","https://openalex.org/W2062878332","https://openalex.org/W2071874678","https://openalex.org/W2093181810","https://openalex.org/W2106565812","https://openalex.org/W2120516507","https://openalex.org/W2125993136","https://openalex.org/W2130727150","https://openalex.org/W2142597043","https://openalex.org/W2163257413","https://openalex.org/W2952098845"],"related_works":["https://openalex.org/W2395105737","https://openalex.org/W4313274502","https://openalex.org/W4317660424","https://openalex.org/W2021279739","https://openalex.org/W4382762952","https://openalex.org/W2067382432","https://openalex.org/W2167705757","https://openalex.org/W2015216402","https://openalex.org/W2589376966","https://openalex.org/W4308299335"],"abstract_inverted_index":{"We":[0],"propose":[1],"splitting":[2,137],"methods":[3,74,138,151],"for":[4,84,136],"the":[5,8,18,56,85,91,95,112,115,123],"computation":[6],"of":[7,10,22,42,94,111,114,125],"exponential":[9,32],"perturbed":[11,116],"matrices":[12],"which":[13,40,65,108],"can":[14],"be":[15],"written":[16],"as":[17],"sum":[19],"$A=D+\\varepsilon":[20],"B$":[21,39],"a":[23,35,61,105,146],"sparse":[24,31],"and":[25,34,75,102,133,144],"efficiently":[26],"exponentiable":[27],"matrix":[28,37,58,117],"$D$":[29],"with":[30,47,141],"$e^D$":[33],"dense":[36],"$\\varepsilon":[38],"is":[41,52,66,155],"small":[43,62],"norm":[44],"in":[45,78,119],"comparison":[46],"$D$.":[48],"The":[49],"predominant":[50],"algorithm":[51],"based":[53],"on":[54,130],"scaling":[55],"large":[57],"$A$":[59],"by":[60,69],"number":[63,124],"$2^{-s}$,":[64],"then":[67],"exponentiated":[68],"efficient":[70],"Pad\u00e9":[71],"or":[72],"Taylor":[73],"finally":[76],"squared":[77],"order":[79,120],"to":[80,121],"obtain":[81],"an":[82],"approximation":[83],"full":[86],"exponential.":[87],"In":[88],"this":[89],"setting,":[90],"main":[92],"portion":[93],"computational":[96],"cost":[97],"arises":[98],"from":[99],"dense-matrix":[100],"multiplications":[101],"we":[103],"present":[104],"modified":[106],"squaring":[107],"takes":[109],"advantage":[110],"smallness":[113],"$B$":[118],"reduce":[122],"squarings":[126],"necessary.":[127],"Theoretical":[128],"results":[129],"local":[131],"error":[132,134],"propagation":[135],"are":[139],"complemented":[140],"numerical":[142],"experiments":[143],"show":[145],"clear":[147],"improvement":[148],"over":[149],"existing":[150],"when":[152],"medium":[153],"precision":[154],"sought.":[156]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2}],"updated_date":"2026-08-05T07:39:15.569665","created_date":"2016-06-24T00:00:00"}
