{"id":"https://openalex.org/W1782407709","doi":"https://doi.org/10.1137/s1064827502406403","title":"Alternatives to the Rayleigh Quotient for the Quadratic Eigenvalue Problem","display_name":"Alternatives to the Rayleigh Quotient for the Quadratic Eigenvalue Problem","publication_year":2003,"publication_date":"2003-01-01","ids":{"openalex":"https://openalex.org/W1782407709","doi":"https://doi.org/10.1137/s1064827502406403","mag":"1782407709"},"language":"en","primary_location":{"id":"doi:10.1137/s1064827502406403","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s1064827502406403","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"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 Scientific Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://research.tue.nl/en/publications/6805deee-5a84-4934-b674-e3ed238c31c4","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5064293224","display_name":"Michiel E. Hochstenbach","orcid":"https://orcid.org/0000-0002-9196-8257"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Michiel E. Hochstenbach","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5109884380","display_name":"H.A. van der Vorst","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Henk A. van der Vorst","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5064293224"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.5781,"has_fulltext":true,"cited_by_count":20,"citation_normalized_percentile":{"value":0.89052541,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"25","issue":"2","first_page":"591","last_page":"603"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9462000131607056,"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":0.9462000131607056,"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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9247999787330627,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/rayleigh-quotient","display_name":"Rayleigh quotient","score":0.6769582629203796},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.581108808517456},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.5687821507453918},{"id":"https://openalex.org/keywords/quotient","display_name":"Quotient","score":0.5641622543334961},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.4732309579849243},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.40574878454208374},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.23171386122703552},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.12897053360939026},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.0976649820804596}],"concepts":[{"id":"https://openalex.org/C2778158742","wikidata":"https://www.wikidata.org/wiki/Q1665115","display_name":"Rayleigh quotient","level":3,"score":0.6769582629203796},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.581108808517456},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.5687821507453918},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.5641622543334961},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.4732309579849243},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.40574878454208374},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.23171386122703552},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.12897053360939026},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0976649820804596},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":11,"locations":[{"id":"doi:10.1137/s1064827502406403","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s1064827502406403","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"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 Scientific Computing","raw_type":"journal-article"},{"id":"pmh:oai:pure.tue.nl:openaire/6805deee-5a84-4934-b674-e3ed238c31c4","is_oa":true,"landing_page_url":"https://research.tue.nl/en/publications/6805deee-5a84-4934-b674-e3ed238c31c4","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":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Hochstenbach, M E & Vorst, van der, H A 2003, 'Alternatives to the Rayleigh quotient for the quadratic eigenvalue problem', SIAM Journal on Scientific Computing, vol. 25, no. 2, pp. 591-603. https://doi.org/10.1137/S1064827502406403","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:tue:oai:pure.tue.nl:publications/6805deee-5a84-4934-b674-e3ed238c31c4","is_oa":true,"landing_page_url":"https://research.tue.nl/nl/publications/6805deee-5a84-4934-b674-e3ed238c31c4","pdf_url":"https://research.tue.nl/nl/publications/6805deee-5a84-4934-b674-e3ed238c31c4","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":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Scientific Computing, 25(2), 591 - 603. Society for Industrial and Applied Mathematics (SIAM)","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:553433","is_oa":false,"landing_page_url":"http://library.tue.nl/csp/dare/LinkToRepository.csp?recordnumber=553433","pdf_url":null,"source":{"id":"https://openalex.org/S4306400553","display_name":"Munich Personal RePEc Archive (Ludwig Maximilian University of Munich)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I8204097","host_organization_name":"Ludwig-Maximilians-Universit\u00e4t M\u00fcnchen","host_organization_lineage":["https://openalex.org/I8204097"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":null},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.15.9049","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.15.9049","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.uu.nl/publications/preprints/1212.ps.gz","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.76.6676","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.76.6676","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.uu.nl/people/vorst/QEP2004.pdf","raw_type":"text"},{"id":"pmh:oai:dspace.library.uu.nl:1874/2479","is_oa":true,"landing_page_url":"http://dspace.library.uu.nl:8080/handle/1874/2479","pdf_url":null,"source":{"id":"https://openalex.org/S4306401649","display_name":"Utrecht University Repository (Utrecht University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I193662353","host_organization_name":"Utrecht University","host_organization_lineage":["https://openalex.org/I193662353"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Research paper"},{"id":"pmh:oai:library.tue.nl:553433","is_oa":false,"landing_page_url":"http://repository.tue.nl/553433","pdf_url":null,"source":{"id":"https://openalex.org/S4406923046","display_name":"TU/e Research Portal (Eindhoven University of Technology)","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"External research report"},{"id":"pmh:oai:library.tue.nl:646720","is_oa":false,"landing_page_url":"http://repository.tue.nl/646720","pdf_url":null,"source":{"id":"https://openalex.org/S4406923046","display_name":"TU/e Research Portal (Eindhoven University of Technology)","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ISSN:1064-8275","raw_type":"Article / Letter to the editor"},{"id":"pmh:uu:oai:dspace.library.uu.nl:1874/2479","is_oa":true,"landing_page_url":"https://dspace.library.uu.nl/handle/1874/2479","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":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/workingpaper"},{"id":"mag:1782407709","is_oa":false,"landing_page_url":"https://www.narcis.nl/publication/RecordID/oai%3Apure.tue.nl%3Apublications%2F734b1533-6301-4f8f-8a82-3ddf53bb6707","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"pmh:oai:pure.tue.nl:openaire/6805deee-5a84-4934-b674-e3ed238c31c4","is_oa":true,"landing_page_url":"https://research.tue.nl/en/publications/6805deee-5a84-4934-b674-e3ed238c31c4","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":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Hochstenbach, M E & Vorst, van der, H A 2003, 'Alternatives to the Rayleigh quotient for the quadratic eigenvalue problem', SIAM Journal on Scientific Computing, vol. 25, no. 2, pp. 591-603. https://doi.org/10.1137/S1064827502406403","raw_type":"info:eu-repo/semantics/publishedVersion"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":8,"referenced_works":["https://openalex.org/W1970606629","https://openalex.org/W2004464554","https://openalex.org/W2028576930","https://openalex.org/W2065014532","https://openalex.org/W2087549321","https://openalex.org/W2117686912","https://openalex.org/W3022873428","https://openalex.org/W4230094279"],"related_works":["https://openalex.org/W2163336887","https://openalex.org/W2027389940","https://openalex.org/W2179887283","https://openalex.org/W2099514750","https://openalex.org/W1970606629","https://openalex.org/W2051096875","https://openalex.org/W2978804717","https://openalex.org/W2185877463","https://openalex.org/W2023968097","https://openalex.org/W1981159057","https://openalex.org/W2811029936","https://openalex.org/W2102586601","https://openalex.org/W2606750429","https://openalex.org/W1964263339","https://openalex.org/W2501837644","https://openalex.org/W2489659966","https://openalex.org/W2267949825","https://openalex.org/W2487184502","https://openalex.org/W3162229724","https://openalex.org/W2283383937"],"abstract_inverted_index":{"We":[0,152],"consider":[1],"the":[2,41,51,76,80,99,121,125,128,132,154,171],"quadratic":[3,81],"eigenvalue":[4,43,173],"problem":[5],"$\\lambda^2":[6],"Ax":[7],"+":[8,11,59,62,85,88],"\\lambda":[9],"Bx":[10],"Cx":[12],"=":[13,56,90],"0.$":[14],"Suppose":[15],"that":[16,33,70,96],"u":[17,108],"is":[18,48,95,109,120,130],"an":[19,22,38],"approximation":[20,39],"to":[21,36,40,49,79,138,150,165,170],"eigenvector":[23],"x":[24],"(for":[25],"instance,":[26],"obtained":[27],"by":[28,158],"a":[29],"subspace":[30],"method)":[31],"and":[32,157,169],"we":[34,145],"want":[35],"determine":[37],"corresponding":[42],"$\\lambda$.":[44,151],"The":[45,161],"usual":[46],"approach":[47],"impose":[50],"Galerkin":[52],"condition":[53],"$r(\\theta,":[54],"u)":[55],"(\\theta^2":[57],"A":[58],"\\theta":[60,87],"B":[61],"C)u":[63],"\\perp":[64],"u$,":[65],"from":[66,167],"which":[67,118],"it":[68,113],"follows":[69],"$\\theta$":[71],"must":[72],"be":[73,116,135],"one":[74],"of":[75,127],"two":[77],"solutions":[78,133],"equation":[82,129],"$(u^*Au)":[83],"\\theta^2":[84],"(u^*Bu)":[86],"(u^*Cu)":[89],"0$.":[91],"An":[92],"unnatural":[93],"aspect":[94],"if":[97],"$u=x$,":[98],"second":[100],"solution":[101,119],"has":[102],"in":[103,140],"general":[104],"no":[105],"meaning.":[106],"When":[107],"not":[110,115],"very":[111,136],"accurate,":[112],"may":[114,134],"clear":[117],"best.":[122],"Moreover,":[123],"when":[124],"discriminant":[126],"small,":[131],"sensitive":[137],"perturbations":[139],"u.":[141],"In":[142],"this":[143],"paper":[144],"therefore":[146],"examine":[147],"alternative":[148],"approximations":[149,166],"compare":[153],"approaches":[155],"theoretically":[156],"numerical":[159],"experiments.":[160],"methods":[162],"are":[163],"extended":[164],"subspaces":[168],"polynomial":[172],"problem.":[174]},"counts_by_year":[{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-04T08:04:53.788161","created_date":"2025-10-10T00:00:00"}