{"id":"https://openalex.org/W2079513523","doi":"https://doi.org/10.1137/130920137","title":"The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential","display_name":"The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential","publication_year":2014,"publication_date":"2014-01-01","ids":{"openalex":"https://openalex.org/W2079513523","doi":"https://doi.org/10.1137/130920137","mag":"2079513523"},"language":"en","primary_location":{"id":"doi:10.1137/130920137","is_oa":true,"landing_page_url":"https://doi.org/10.1137/130920137","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://doi.org/10.1137/130920137","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5027032180","display_name":"Mary Aprahamian","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Mary Aprahamian","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5069339254","display_name":"Nicholas J. Higham","orcid":"https://orcid.org/0000-0001-5956-4976"},"institutions":[{"id":"https://openalex.org/I4210131439","display_name":"Applied Mathematics (United States)","ror":"https://ror.org/03seew607","country_code":"US","type":"company","lineage":["https://openalex.org/I4210131439"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Nicholas J. Higham","raw_affiliation_strings":["Applied Mathematics"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Applied Mathematics","institution_ids":["https://openalex.org/I4210131439"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.8053,"has_fulltext":true,"cited_by_count":21,"citation_normalized_percentile":{"value":0.9075405,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"35","issue":"1","first_page":"88","last_page":"109"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9993000030517578,"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":0.9993000030517578,"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":0.9990000128746033,"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/T11191","display_name":"Mathematical functions and polynomials","score":0.9966999888420105,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/mathematics","display_name":"Mathematics","score":0.7594665884971619},{"id":"https://openalex.org/keywords/matrix-function","display_name":"Matrix function","score":0.6810958981513977},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.6416870355606079},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5926502346992493},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.5857839584350586},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.5192953944206238},{"id":"https://openalex.org/keywords/square-root-of-a-2-by-2-matrix","display_name":"Square root of a 2 by 2 matrix","score":0.5105305910110474},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.5075269937515259},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4644562900066376},{"id":"https://openalex.org/keywords/scaling","display_name":"Scaling","score":0.4576503336429596},{"id":"https://openalex.org/keywords/matrix-exponential","display_name":"Matrix exponential","score":0.4461911618709564},{"id":"https://openalex.org/keywords/sign-function","display_name":"Sign function","score":0.4213048815727234},{"id":"https://openalex.org/keywords/symmetric-matrix","display_name":"Symmetric matrix","score":0.360178679227829},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2709839344024658},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.12375938892364502},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09344014525413513},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.08421209454536438}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7594665884971619},{"id":"https://openalex.org/C4263655","wikidata":"https://www.wikidata.org/wiki/Q2699958","display_name":"Matrix function","level":4,"score":0.6810958981513977},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.6416870355606079},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5926502346992493},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.5857839584350586},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.5192953944206238},{"id":"https://openalex.org/C14964150","wikidata":"https://www.wikidata.org/wiki/Q7582079","display_name":"Square root of a 2 by 2 matrix","level":5,"score":0.5105305910110474},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.5075269937515259},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4644562900066376},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.4576503336429596},{"id":"https://openalex.org/C195906000","wikidata":"https://www.wikidata.org/wiki/Q1191722","display_name":"Matrix exponential","level":3,"score":0.4461911618709564},{"id":"https://openalex.org/C4250","wikidata":"https://www.wikidata.org/wiki/Q236813","display_name":"Sign function","level":2,"score":0.4213048815727234},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.360178679227829},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2709839344024658},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.12375938892364502},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09344014525413513},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.08421209454536438},{"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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"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":8,"locations":[{"id":"doi:10.1137/130920137","is_oa":true,"landing_page_url":"https://doi.org/10.1137/130920137","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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:pure.atira.dk:openaire_cris_publications/4594aae7-978c-474f-8e62-5bf62bf9025f","is_oa":false,"landing_page_url":"https://research.manchester.ac.uk/en/publications/4594aae7-978c-474f-8e62-5bf62bf9025f","pdf_url":null,"source":{"id":"https://openalex.org/S4306400662","display_name":"Research Explorer (The University of Manchester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28407311","host_organization_name":"University of Manchester","host_organization_lineage":["https://openalex.org/I28407311"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Aprahamian, M & HIgham, N J 2014, 'The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential', S I A M Journal on Matrix Analysis and Applications, vol. 35, no. 1, pp. 88-109. https://doi.org/10.1137/130920137","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1974","is_oa":false,"landing_page_url":"http://eprints.maths.manchester.ac.uk/1974/1/paper.pdf","pdf_url":"http://eprints.maths.manchester.ac.uk/1974/1/paper.pdf","source":{"id":"https://openalex.org/S4306400452","display_name":"MIMS EPrints (University of Southampton)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I43439940","host_organization_name":"University of Southampton","host_organization_lineage":["https://openalex.org/I43439940"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"MIMS Preprint"},{"id":"pmh:oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:1980","is_oa":false,"landing_page_url":"http://eprints.maths.manchester.ac.uk/1980/1/paper.pdf","pdf_url":"http://eprints.maths.manchester.ac.uk/1980/1/paper.pdf","source":{"id":"https://openalex.org/S4306400452","display_name":"MIMS EPrints (University of Southampton)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I43439940","host_organization_name":"University of Southampton","host_organization_lineage":["https://openalex.org/I43439940"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"MIMS Preprint"},{"id":"pmh:oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:2023","is_oa":false,"landing_page_url":"http://eprints.maths.manchester.ac.uk/2023/1/paper.pdf","pdf_url":"http://eprints.maths.manchester.ac.uk/2023/1/paper.pdf","source":{"id":"https://openalex.org/S4306400452","display_name":"MIMS EPrints (University of Southampton)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I43439940","host_organization_name":"University of Southampton","host_organization_lineage":["https://openalex.org/I43439940"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"MIMS Preprint"},{"id":"pmh:oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:2032","is_oa":false,"landing_page_url":"http://eprints.maths.manchester.ac.uk/2032/1/paper.pdf","pdf_url":"http://eprints.maths.manchester.ac.uk/2032/1/paper.pdf","source":{"id":"https://openalex.org/S4306400452","display_name":"MIMS EPrints (University of Southampton)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I43439940","host_organization_name":"University of Southampton","host_organization_lineage":["https://openalex.org/I43439940"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"MIMS Preprint"},{"id":"pmh:oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:2094","is_oa":false,"landing_page_url":"http://eprints.maths.manchester.ac.uk/2094/2/unwinding.zip","pdf_url":null,"source":{"id":"https://openalex.org/S4306400452","display_name":"MIMS EPrints (University of Southampton)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I43439940","host_organization_name":"University of Southampton","host_organization_lineage":["https://openalex.org/I43439940"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:pure.atira.dk:publications/4594aae7-978c-474f-8e62-5bf62bf9025f","is_oa":false,"landing_page_url":"https://www.research.manchester.ac.uk/portal/en/publications/the-matrix-unwinding-function-with-an-application-to-computing-the-matrix-exponential(4594aae7-978c-474f-8e62-5bf62bf9025f).html","pdf_url":null,"source":{"id":"https://openalex.org/S4306400662","display_name":"Research Explorer (The University of Manchester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28407311","host_organization_name":"University of Manchester","host_organization_lineage":["https://openalex.org/I28407311"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Aprahamian, M & HIgham, N J 2014, 'The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential', S I A M Journal on Matrix Analysis and Applications, vol. 35, no. 1, pp. 88-109. https://doi.org/10.1137/130920137","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.1137/130920137","is_oa":true,"landing_page_url":"https://doi.org/10.1137/130920137","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4073111929","display_name":"The Manchester Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA)","funder_award_id":"EP/E050441/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G4916285367","display_name":null,"funder_award_id":"267526","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G4926088696","display_name":null,"funder_award_id":"EP/E050441/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W1483804921","https://openalex.org/W1578322733","https://openalex.org/W1585277169","https://openalex.org/W1898845316","https://openalex.org/W1964855068","https://openalex.org/W1976955562","https://openalex.org/W1981550893","https://openalex.org/W1990040771","https://openalex.org/W1990647940","https://openalex.org/W2000537011","https://openalex.org/W2056405080","https://openalex.org/W2056602674","https://openalex.org/W2061281381","https://openalex.org/W2067136705","https://openalex.org/W2068213203","https://openalex.org/W2082465767","https://openalex.org/W2086619045","https://openalex.org/W2090379260","https://openalex.org/W2091491180","https://openalex.org/W2096104369","https://openalex.org/W2106565812","https://openalex.org/W2112145895","https://openalex.org/W2120516507","https://openalex.org/W2160829115","https://openalex.org/W2798735774","https://openalex.org/W3022909171","https://openalex.org/W4252642284","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W1995516663","https://openalex.org/W1982019426","https://openalex.org/W2905395987","https://openalex.org/W3188579483","https://openalex.org/W2334948012","https://openalex.org/W2022992047","https://openalex.org/W144947161","https://openalex.org/W2953299013","https://openalex.org/W2972456653","https://openalex.org/W2079513523"],"abstract_inverted_index":{"A":[0],"new":[1],"matrix":[2,18,34,38,65,90],"function":[3,55,74,95],"corresponding":[4],"to":[5,24,59],"the":[6,33,43,64,72,77,94,108,123,126],"scalar":[7],"unwinding":[8,19,54,73],"number":[9],"of":[10,125],"Corless,":[11],"Hare,":[12],"and":[13,36,49,111,130],"Jeffrey":[14],"is":[15,22,56,84,87],"introduced.":[16],"This":[17],"function,":[20],"$\\mathcal{U}$,":[21],"shown":[23,58,88],"be":[25,60],"a":[26,81],"valuable":[27],"tool":[28],"for":[29,41,70,112],"deriving":[30],"identities":[31],"involving":[32],"logarithm":[35],"fractional":[37],"powers,":[39],"revealing,":[40],"example,":[42],"precise":[44],"relation":[45],"between":[46],"$\\log":[47],"A^\\alpha$":[48],"$\\alpha":[50],"\\log":[51],"A$.":[52],"The":[53],"also":[57],"closely":[61],"connected":[62],"with":[63,80,104],"sign":[66],"function.":[67],"An":[68],"algorithm":[69],"computing":[71],"based":[75],"on":[76],"Schur--Parlett":[78],"method":[79],"special":[82],"reordering":[83],"proposed.":[85],"It":[86],"that":[89],"argument":[91],"reduction":[92],"using":[93],"$\\mathrm{mod}(A)":[96],"=":[97,115],"A-2\\pi":[98],"i\\,":[99],"\\mathcal{U}(A)$,":[100],"which":[101,113],"has":[102],"eigenvalues":[103],"imaginary":[105],"parts":[106],"in":[107,122],"interval":[109],"$(-\\pi,\\pi]$":[110],"${\\mathrm{e}}^A":[114],"{\\mathrm{e}}^{\\mathrm{mod}(A)}$,":[116],"can":[117],"give":[118],"significant":[119],"computational":[120],"savings":[121],"evaluation":[124],"exponential":[127],"by":[128],"scaling":[129],"squaring":[131],"algorithms.":[132]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":2}],"updated_date":"2026-08-18T07:49:30.821534","created_date":"2025-10-10T00:00:00"}
