{"id":"https://openalex.org/W2131125270","doi":"https://doi.org/10.1137/s0895479893256347","title":"Matrix Powers in Finite Precision Arithmetic","display_name":"Matrix Powers in Finite Precision Arithmetic","publication_year":1995,"publication_date":"1995-04-01","ids":{"openalex":"https://openalex.org/W2131125270","doi":"https://doi.org/10.1137/s0895479893256347","mag":"2131125270"},"language":"en","primary_location":{"id":"doi:10.1137/s0895479893256347","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479893256347","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":["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/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"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5002885020","display_name":"Philip A. Knight","orcid":"https://orcid.org/0000-0001-9511-5692"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Philip A. Knight","raw_affiliation_strings":["Mathematics And Statistics"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematics And Statistics","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.0682,"has_fulltext":false,"cited_by_count":22,"citation_normalized_percentile":{"value":0.78497409,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"16","issue":"2","first_page":"343","last_page":"358"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9998000264167786,"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":0.9998000264167786,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9955999851226807,"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/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.989799976348877,"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/mathematics","display_name":"Mathematics","score":0.8658905029296875},{"id":"https://openalex.org/keywords/spectral-radius","display_name":"Spectral radius","score":0.6957114934921265},{"id":"https://openalex.org/keywords/diagonalizable-matrix","display_name":"Diagonalizable matrix","score":0.6851462125778198},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.583450198173523},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5304426550865173},{"id":"https://openalex.org/keywords/condition-number","display_name":"Condition number","score":0.5091199278831482},{"id":"https://openalex.org/keywords/divergence","display_name":"Divergence (linguistics)","score":0.4767012298107147},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.46321067214012146},{"id":"https://openalex.org/keywords/radius","display_name":"RADIUS","score":0.4219553470611572},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4147064685821533},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3290334939956665},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.26169687509536743},{"id":"https://openalex.org/keywords/symmetric-matrix","display_name":"Symmetric matrix","score":0.14191222190856934}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8658905029296875},{"id":"https://openalex.org/C140532419","wikidata":"https://www.wikidata.org/wiki/Q249748","display_name":"Spectral radius","level":3,"score":0.6957114934921265},{"id":"https://openalex.org/C30072841","wikidata":"https://www.wikidata.org/wiki/Q1767080","display_name":"Diagonalizable matrix","level":4,"score":0.6851462125778198},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.583450198173523},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5304426550865173},{"id":"https://openalex.org/C84545080","wikidata":"https://www.wikidata.org/wiki/Q1147936","display_name":"Condition number","level":3,"score":0.5091199278831482},{"id":"https://openalex.org/C207390915","wikidata":"https://www.wikidata.org/wiki/Q1230525","display_name":"Divergence (linguistics)","level":2,"score":0.4767012298107147},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.46321067214012146},{"id":"https://openalex.org/C178635117","wikidata":"https://www.wikidata.org/wiki/Q747499","display_name":"RADIUS","level":2,"score":0.4219553470611572},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4147064685821533},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3290334939956665},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.26169687509536743},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.14191222190856934},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer 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/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.1137/s0895479893256347","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479893256347","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:eprints.maths.manchester.ac.uk.MIMS.EPrints:345","is_oa":false,"landing_page_url":null,"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:hdl:10068/651829","is_oa":false,"landing_page_url":"http://hdl.handle.net/10068/651829","pdf_url":null,"source":{"id":"https://openalex.org/S4377196900","display_name":"OpenGrey (Institut de l'Information Scientifique et Technique)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1295345811","host_organization_name":"Institut de l'Information Scientifique et Technique","host_organization_lineage":["https://openalex.org/I1295345811"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"I - Miscellaneous"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.230.837","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.230.837","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://eprints.ma.man.ac.uk/345/01/covered/MIMS_ep2006_166.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.34.7493","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.7493","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.maths.man.ac.uk/~nareports/narep235.ps.gz","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.52.7727","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.7727","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ftp://vtx.ma.man.ac.uk/pub/higham/narep235.ps.Z","raw_type":"text"},{"id":"pmh:oai:pure.atira.dk:openaire_cris_publications/4c44aa34-6915-4f4d-b87f-bec010479765","is_oa":false,"landing_page_url":"https://research.manchester.ac.uk/en/publications/4c44aa34-6915-4f4d-b87f-bec010479765","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":"Higham, N J & Knight, P A 1995, 'MATRIX POWERS IN FINITE PRECISION ARITHMETIC', SIAM Journal on Matrix Analysis and Applications, vol. 16, no. 2, pp. 343-358.","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:pure.atira.dk:publications/4c44aa34-6915-4f4d-b87f-bec010479765","is_oa":false,"landing_page_url":"https://www.research.manchester.ac.uk/portal/en/publications/matrix-powers-in-finite-precision-arithmetic(4c44aa34-6915-4f4d-b87f-bec010479765).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":"Higham, N J & Knight, P A 1995, 'MATRIX POWERS IN FINITE PRECISION ARITHMETIC', SIAM Journal on Matrix Analysis and Applications, vol. 16, no. 2, pp. 343-358.","raw_type":"info:eu-repo/semantics/publishedVersion"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W83404773","https://openalex.org/W391578156","https://openalex.org/W658559791","https://openalex.org/W1484752611","https://openalex.org/W1500709119","https://openalex.org/W1550563201","https://openalex.org/W1990695571","https://openalex.org/W2000917451","https://openalex.org/W2001802261","https://openalex.org/W2006411943","https://openalex.org/W2046305855","https://openalex.org/W2055903751","https://openalex.org/W2060092412","https://openalex.org/W2067136705","https://openalex.org/W2078675418","https://openalex.org/W2087843362","https://openalex.org/W2091332375","https://openalex.org/W2097898402","https://openalex.org/W2098533238","https://openalex.org/W2117919754","https://openalex.org/W2128978199","https://openalex.org/W2150287034","https://openalex.org/W2610857016","https://openalex.org/W2617854063","https://openalex.org/W2798909945","https://openalex.org/W4240942209"],"related_works":["https://openalex.org/W3149741176","https://openalex.org/W2584030696","https://openalex.org/W2977088962","https://openalex.org/W4302613585","https://openalex.org/W2369247774","https://openalex.org/W3148147474","https://openalex.org/W2801902682","https://openalex.org/W2038812594","https://openalex.org/W1655476355","https://openalex.org/W2131125270"],"abstract_inverted_index":{"If":[0],"A":[1,90,92,115],"is":[2,93,130],"a":[3,34,47,87],"square":[4],"matrix":[5],"with":[6],"spectral":[7,122],"radius":[8,123],"less":[9,131],"than":[10,132,135],"1":[11],"then":[12],"$A^k":[13],"\\to":[14,16,41,43],"0\\,{\\text{as}}\\,k":[15,42],"\\infty":[17,44],"$,":[18],"but":[19],"the":[20,61,70,78,102,111,121],"powers":[21,113],"computed":[22,112,124],"in":[23,58,84],"finite":[24],"precision":[25],"arithmetic":[26],"may":[27,29],"or":[28,108,133],"not":[30],"converge.":[31],"We":[32,75],"derive":[33],"sufficient":[35,79],"condition":[36,80],"for":[37],"$fl(":[38],"A^k":[39,53],")":[40,54],"$":[45],"and":[46],"bound":[48],"on":[49],"$\\|":[50],"fl":[51],"(":[52],"\\|$,":[55],"both":[56],"expressed":[57],"terms":[59,85],"of":[60,65,86,89,104,110,114],"Jordan":[62],"canonical":[63],"form":[64],"A.":[66],"Examples":[67],"show":[68,76],"that":[69,77,106],"results":[71],"can":[72,81,116],"be":[73,82,117],"sharp.":[74],"rephrased":[83],"pseudospectrum":[88],"when":[91],"diagonalizable,":[94],"under":[95],"certain":[96],"assumptions.":[97],"Our":[98],"analysis":[99],"leads":[100],"to":[101],"rule":[103],"thumb":[105],"convergence":[107],"divergence":[109],"expected":[118],"according":[119],"as":[120],"by":[125],"any":[126],"backward":[127],"stable":[128],"algorithm":[129],"greater":[134],"1.":[136]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}