{"id":"https://openalex.org/W1984512230","doi":"https://doi.org/10.1137/s0895479800366859","title":"Computing Probabilistic Bounds for Extreme Eigenvalues of Symmetric Matrices with the Lanczos Method","display_name":"Computing Probabilistic Bounds for Extreme Eigenvalues of Symmetric Matrices with the Lanczos Method","publication_year":2001,"publication_date":"2001-01-01","ids":{"openalex":"https://openalex.org/W1984512230","doi":"https://doi.org/10.1137/s0895479800366859","mag":"1984512230"},"language":"en","primary_location":{"id":"doi:10.1137/s0895479800366859","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479800366859","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/A5045396195","display_name":"Jos L. M. van Dorsselaer","orcid":null},"institutions":[{"id":"https://openalex.org/I4210120676","display_name":"Mathematical Institute of the Slovak Academy of Sciences","ror":"https://ror.org/02fhy7464","country_code":"SK","type":"facility","lineage":["https://openalex.org/I207624831","https://openalex.org/I4210120676"]},{"id":"https://openalex.org/I193662353","display_name":"Utrecht University","ror":"https://ror.org/04pp8hn57","country_code":"NL","type":"education","lineage":["https://openalex.org/I193662353"]}],"countries":["NL","SK"],"is_corresponding":true,"raw_author_name":"Jos L. M. van Dorsselaer","raw_affiliation_strings":["Mathematical Institute, Utrecht University, P.O. Box 80.010, NL-3508 TA Utrecht, The Nether-lands"],"affiliations":[{"raw_affiliation_string":"Mathematical Institute, Utrecht University, P.O. Box 80.010, NL-3508 TA Utrecht, The Nether-lands","institution_ids":["https://openalex.org/I193662353","https://openalex.org/I4210120676"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5064293224","display_name":"Michiel E. Hochstenbach","orcid":"https://orcid.org/0000-0002-9196-8257"},"institutions":[{"id":"https://openalex.org/I193662353","display_name":"Utrecht University","ror":"https://ror.org/04pp8hn57","country_code":"NL","type":"education","lineage":["https://openalex.org/I193662353"]},{"id":"https://openalex.org/I4210120676","display_name":"Mathematical Institute of the Slovak Academy of Sciences","ror":"https://ror.org/02fhy7464","country_code":"SK","type":"facility","lineage":["https://openalex.org/I207624831","https://openalex.org/I4210120676"]}],"countries":["NL","SK"],"is_corresponding":false,"raw_author_name":"Michiel E. Hochstenbach","raw_affiliation_strings":["Mathematical Institute, Utrecht University, P.O. Box 80.010, NL-3508 TA Utrecht, The Nether-lands"],"affiliations":[{"raw_affiliation_string":"Mathematical Institute, Utrecht University, P.O. Box 80.010, NL-3508 TA Utrecht, The Nether-lands","institution_ids":["https://openalex.org/I193662353","https://openalex.org/I4210120676"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5109884380","display_name":"H.A. van der Vorst","orcid":null},"institutions":[{"id":"https://openalex.org/I4210120676","display_name":"Mathematical Institute of the Slovak Academy of Sciences","ror":"https://ror.org/02fhy7464","country_code":"SK","type":"facility","lineage":["https://openalex.org/I207624831","https://openalex.org/I4210120676"]},{"id":"https://openalex.org/I193662353","display_name":"Utrecht University","ror":"https://ror.org/04pp8hn57","country_code":"NL","type":"education","lineage":["https://openalex.org/I193662353"]}],"countries":["NL","SK"],"is_corresponding":false,"raw_author_name":"Henk A. van der Vorst","raw_affiliation_strings":["Mathematical Institute, Utrecht University, P.O. Box 80.010, NL-3508 TA Utrecht, The Nether-lands"],"affiliations":[{"raw_affiliation_string":"Mathematical Institute, Utrecht University, P.O. Box 80.010, NL-3508 TA Utrecht, The Nether-lands","institution_ids":["https://openalex.org/I193662353","https://openalex.org/I4210120676"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5045396195"],"corresponding_institution_ids":["https://openalex.org/I193662353","https://openalex.org/I4210120676"],"apc_list":null,"apc_paid":null,"fwci":0.4484,"has_fulltext":false,"cited_by_count":11,"citation_normalized_percentile":{"value":0.62944518,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"22","issue":"3","first_page":"837","last_page":"852"},"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/T10591","display_name":"Theoretical and Computational Physics","score":0.9923999905586243,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9904000163078308,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/lanczos-resampling","display_name":"Lanczos resampling","score":0.9100140333175659},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.8156189918518066},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7858912944793701},{"id":"https://openalex.org/keywords/hermitian-matrix","display_name":"Hermitian matrix","score":0.6778583526611328},{"id":"https://openalex.org/keywords/lanczos-algorithm","display_name":"Lanczos algorithm","score":0.5860664248466492},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5401932001113892},{"id":"https://openalex.org/keywords/ritz-method","display_name":"Ritz method","score":0.4824965000152588},{"id":"https://openalex.org/keywords/tridiagonal-matrix","display_name":"Tridiagonal matrix","score":0.47960197925567627},{"id":"https://openalex.org/keywords/chebyshev-filter","display_name":"Chebyshev filter","score":0.4611113369464874},{"id":"https://openalex.org/keywords/chebyshev-polynomials","display_name":"Chebyshev polynomials","score":0.44234782457351685},{"id":"https://openalex.org/keywords/probabilistic-logic","display_name":"Probabilistic logic","score":0.41170021891593933},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.319480836391449},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.23296049237251282},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.07361873984336853}],"concepts":[{"id":"https://openalex.org/C119256216","wikidata":"https://www.wikidata.org/wiki/Q913012","display_name":"Lanczos resampling","level":3,"score":0.9100140333175659},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.8156189918518066},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7858912944793701},{"id":"https://openalex.org/C94940","wikidata":"https://www.wikidata.org/wiki/Q652941","display_name":"Hermitian matrix","level":2,"score":0.6778583526611328},{"id":"https://openalex.org/C20501136","wikidata":"https://www.wikidata.org/wiki/Q366640","display_name":"Lanczos algorithm","level":4,"score":0.5860664248466492},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5401932001113892},{"id":"https://openalex.org/C21575042","wikidata":"https://www.wikidata.org/wiki/Q905132","display_name":"Ritz method","level":3,"score":0.4824965000152588},{"id":"https://openalex.org/C51647924","wikidata":"https://www.wikidata.org/wiki/Q1755277","display_name":"Tridiagonal matrix","level":3,"score":0.47960197925567627},{"id":"https://openalex.org/C21424316","wikidata":"https://www.wikidata.org/wiki/Q718621","display_name":"Chebyshev filter","level":2,"score":0.4611113369464874},{"id":"https://openalex.org/C129785596","wikidata":"https://www.wikidata.org/wiki/Q619511","display_name":"Chebyshev polynomials","level":2,"score":0.44234782457351685},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.41170021891593933},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.319480836391449},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.23296049237251282},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.07361873984336853},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","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}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1137/s0895479800366859","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0895479800366859","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:pure.tue.nl:openaire_cris_publications/67bb218c-2124-4b6c-b665-ed6d4bba4067","is_oa":false,"landing_page_url":"https://research.tue.nl/en/publications/67bb218c-2124-4b6c-b665-ed6d4bba4067","pdf_url":null,"source":{"id":"https://openalex.org/S4406922641","display_name":"TU/e Research Portal","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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Dorsselaer, van, J L M, Hochstenbach, M E & Vorst, van der, H A 2000, 'Computing probabilistic bounds for extreme eigenvalues of symmetric matrices with the Lanczos method', SIAM Journal on Matrix Analysis and Applications, vol. 22, no. 3, pp. 837-852. https://doi.org/10.1137/S0895479800366859","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.1012.1262","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.1012.1262","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://alexandria.tue.nl/repository/books/532451.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.1033.1525","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.1033.1525","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://dspace.library.uu.nl/bitstream/handle/1874/2041/1124.pdf%3Bjsessionid%3D9C4B74BE57B270BBC3A6F373269E1B23?sequence%3D1","raw_type":"text"},{"id":"pmh:tue:oai:pure.tue.nl:publications/67bb218c-2124-4b6c-b665-ed6d4bba4067","is_oa":false,"landing_page_url":"https://research.tue.nl/nl/publications/67bb218c-2124-4b6c-b665-ed6d4bba4067","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications, 22(3), 837 - 852. Society for Industrial and Applied Mathematics (SIAM)","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":16,"referenced_works":["https://openalex.org/W1976798627","https://openalex.org/W1981745143","https://openalex.org/W1982573289","https://openalex.org/W1989507268","https://openalex.org/W2012728721","https://openalex.org/W2013737143","https://openalex.org/W2040466831","https://openalex.org/W2059586807","https://openalex.org/W2059998776","https://openalex.org/W2090358611","https://openalex.org/W2110003100","https://openalex.org/W2167193748","https://openalex.org/W2170486042","https://openalex.org/W3022136271","https://openalex.org/W3035235874","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2075330145","https://openalex.org/W183203207","https://openalex.org/W2515307085","https://openalex.org/W2049166915","https://openalex.org/W1676602110","https://openalex.org/W1964217357","https://openalex.org/W2095282848","https://openalex.org/W1986034826","https://openalex.org/W4248073715","https://openalex.org/W2016499269"],"abstract_inverted_index":{"We":[0],"study":[1],"the":[2,20,27,31,34,42,54,61,64,103,118],"Lanczos":[3,65,96],"method":[4],"for":[5,53,102],"computing":[6],"extreme":[7,21,28,55],"eigenvalues":[8,56],"of":[9,33,63,70,105],"a":[10,98,124],"symmetric":[11],"or":[12,120],"Hermitian":[13],"matrix.":[14],"It":[15],"is":[16],"not":[17],"guaranteed":[18],"that":[19,41],"Ritz":[22],"values":[23],"are":[24,38,72,82],"close":[25],"to":[26,113,117],"eigenvalues---even":[29],"when":[30],"norms":[32],"corresponding":[35],"residual":[36],"vectors":[37],"small.":[39],"Assuming":[40],"starting":[43],"vector":[44],"has":[45],"been":[46],"chosen":[47],"randomly,":[48],"we":[49,88],"compute":[50],"probabilistic":[51,99],"bounds":[52,71,81],"from":[57],"data":[58],"available":[59],"during":[60],"execution":[62],"process.":[66],"Four":[67],"different":[68],"types":[69],"obtained":[73],"using":[74],"Lanczos,":[75],"Ritz,":[76],"and":[77,85],"Chebyshev":[78],"polynomials.":[79],"These":[80],"compared":[83],"theoretically":[84],"numerically.":[86],"Furthermore":[87],"show":[89],"how":[90],"one":[91],"can":[92],"determine,":[93],"after":[94],"each":[95],"step,":[97],"upper":[100],"bound":[101],"number":[104],"steps":[106],"still":[107],"needed":[108],"(without":[109],"performing":[110],"these":[111],"steps)":[112],"obtain":[114],"an":[115],"approximation":[116],"largest":[119],"smallest":[121],"eigenvalue":[122],"within":[123],"prescribed":[125],"tolerance.":[126]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}