{"id":"https://openalex.org/W1988591471","doi":"https://doi.org/10.1137/120884468","title":"Bounds for the Rayleigh Quotient and the Spectrum of Self-Adjoint Operators","display_name":"Bounds for the Rayleigh Quotient and the Spectrum of Self-Adjoint Operators","publication_year":2013,"publication_date":"2013-01-01","ids":{"openalex":"https://openalex.org/W1988591471","doi":"https://doi.org/10.1137/120884468","mag":"1988591471"},"language":"en","primary_location":{"id":"doi:10.1137/120884468","is_oa":false,"landing_page_url":"https://doi.org/10.1137/120884468","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/1207.3240","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5082531182","display_name":"Peizhen Zhu","orcid":null},"institutions":[{"id":"https://openalex.org/I921990950","display_name":"University of Colorado Denver","ror":"https://ror.org/02hh7en24","country_code":"US","type":"education","lineage":["https://openalex.org/I921990950"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Peizhen Zhu","raw_affiliation_strings":["Department of Mathematical and Statistical Sciences; University of Colorado Denver, P.O. Box 173364, Campus Box 170, Denver, CO 80217-3364, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematical and Statistical Sciences; University of Colorado Denver, P.O. Box 173364, Campus Box 170, Denver, CO 80217-3364, USA","institution_ids":["https://openalex.org/I921990950"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5064504892","display_name":"Merico E. Argentati","orcid":null},"institutions":[{"id":"https://openalex.org/I921990950","display_name":"University of Colorado Denver","ror":"https://ror.org/02hh7en24","country_code":"US","type":"education","lineage":["https://openalex.org/I921990950"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Merico E. Argentati","raw_affiliation_strings":["Department of Mathematical and Statistical Sciences; University of Colorado Denver, P.O. Box 173364, Campus Box 170, Denver, CO 80217-3364, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematical and Statistical Sciences; University of Colorado Denver, P.O. Box 173364, Campus Box 170, Denver, CO 80217-3364, USA","institution_ids":["https://openalex.org/I921990950"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5064449686","display_name":"Andrew Knyazev","orcid":"https://orcid.org/0000-0002-1635-3711"},"institutions":[{"id":"https://openalex.org/I4210159266","display_name":"Mitsubishi Electric (United States)","ror":"https://ror.org/053jnhe44","country_code":"US","type":"company","lineage":["https://openalex.org/I1306287861","https://openalex.org/I4210133125","https://openalex.org/I4210159266"]},{"id":"https://openalex.org/I921990950","display_name":"University of Colorado Denver","ror":"https://ror.org/02hh7en24","country_code":"US","type":"education","lineage":["https://openalex.org/I921990950"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Andrew V. Knyazev","raw_affiliation_strings":["Department of Mathematical and Statistical Sciences; University of Colorado Denver, P.O. Box 173364, Campus Box 170, Denver, CO 80217-3364, USA","Mitsubishi Electric Research Laboratories; 201 Broadway Cambridge, MA 02139"],"affiliations":[{"raw_affiliation_string":"Department of Mathematical and Statistical Sciences; University of Colorado Denver, P.O. Box 173364, Campus Box 170, Denver, CO 80217-3364, USA","institution_ids":["https://openalex.org/I921990950"]},{"raw_affiliation_string":"Mitsubishi Electric Research Laboratories; 201 Broadway Cambridge, MA 02139","institution_ids":["https://openalex.org/I4210159266"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5082531182"],"corresponding_institution_ids":["https://openalex.org/I921990950"],"apc_list":null,"apc_paid":null,"fwci":0.3731,"has_fulltext":false,"cited_by_count":11,"citation_normalized_percentile":{"value":0.62893343,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"34","issue":"1","first_page":"244","last_page":"256"},"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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":0.9952999949455261,"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":0.8999389410018921},{"id":"https://openalex.org/keywords/rayleigh-quotient","display_name":"Rayleigh quotient","score":0.7931464910507202},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.6793689727783203},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.6339349150657654},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.620774507522583},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.561212420463562},{"id":"https://openalex.org/keywords/quotient","display_name":"Quotient","score":0.5240117311477661},{"id":"https://openalex.org/keywords/operator-norm","display_name":"Operator norm","score":0.5188727974891663},{"id":"https://openalex.org/keywords/self-adjoint-operator","display_name":"Self-adjoint operator","score":0.5052474141120911},{"id":"https://openalex.org/keywords/bounded-operator","display_name":"Bounded operator","score":0.4900771379470825},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4684935510158539},{"id":"https://openalex.org/keywords/residual","display_name":"Residual","score":0.46131396293640137},{"id":"https://openalex.org/keywords/normed-vector-space","display_name":"Normed vector space","score":0.4118732511997223},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.374062716960907},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.33622270822525024}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8999389410018921},{"id":"https://openalex.org/C2778158742","wikidata":"https://www.wikidata.org/wiki/Q1665115","display_name":"Rayleigh quotient","level":3,"score":0.7931464910507202},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6793689727783203},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6339349150657654},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.620774507522583},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.561212420463562},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.5240117311477661},{"id":"https://openalex.org/C68386474","wikidata":"https://www.wikidata.org/wiki/Q980557","display_name":"Operator norm","level":3,"score":0.5188727974891663},{"id":"https://openalex.org/C171972299","wikidata":"https://www.wikidata.org/wiki/Q6500908","display_name":"Self-adjoint operator","level":3,"score":0.5052474141120911},{"id":"https://openalex.org/C175454919","wikidata":"https://www.wikidata.org/wiki/Q2342396","display_name":"Bounded operator","level":3,"score":0.4900771379470825},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4684935510158539},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.46131396293640137},{"id":"https://openalex.org/C171739935","wikidata":"https://www.wikidata.org/wiki/Q726210","display_name":"Normed vector space","level":2,"score":0.4118732511997223},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.374062716960907},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.33622270822525024},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"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}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/120884468","is_oa":false,"landing_page_url":"https://doi.org/10.1137/120884468","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:1207.3240","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1207.3240","pdf_url":"https://arxiv.org/pdf/1207.3240","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1207.3240","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1207.3240","pdf_url":"https://arxiv.org/pdf/1207.3240","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":[{"id":"https://openalex.org/G4658120941","display_name":"Parallel Preconditioned Eigenvalue and Singular Value Solvers","funder_award_id":"1115734","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G848032724","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320328455","display_name":"Ipsen","ror":"https://ror.org/00d801g55"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W83404773","https://openalex.org/W658559791","https://openalex.org/W1485916122","https://openalex.org/W1563653928","https://openalex.org/W1592958377","https://openalex.org/W1970377488","https://openalex.org/W2004464554","https://openalex.org/W2014414743","https://openalex.org/W2026083721","https://openalex.org/W2048700868","https://openalex.org/W2053395975","https://openalex.org/W2059586807","https://openalex.org/W2075118289","https://openalex.org/W2080330289","https://openalex.org/W2100185573","https://openalex.org/W2102867755","https://openalex.org/W2116948188","https://openalex.org/W2138543857","https://openalex.org/W2480854438"],"related_works":["https://openalex.org/W3123914979","https://openalex.org/W2794086404","https://openalex.org/W4300098584","https://openalex.org/W2789172848","https://openalex.org/W3011352665","https://openalex.org/W2009893324","https://openalex.org/W2035999820","https://openalex.org/W4381805055","https://openalex.org/W2118722273","https://openalex.org/W2028111557"],"abstract_inverted_index":{"The":[0,155],"absolute":[1,62,70],"change":[2,23,63],"in":[3,10,13,24,38],"the":[4,16,19,22,25,43,46,61,65,69,80,102,113,116,127,130,133,145],"Rayleigh":[5],"quotient":[6],"(RQ)":[7],"is":[8,29,52,158],"bounded":[9,35],"this":[11,59],"paper":[12],"terms":[14],"of":[15,18,32,45,56,64,76,93,115,119,132,144],"norm":[17,114,131],"residual":[20,117],"and":[21,107,129,150],"vector.":[26],"If":[27],"$x$":[28,106],"an":[30,53,73],"eigenvector":[31],"a":[33,39,84,98,109,137],"self-adjoint":[34],"operator":[36],"$A$":[37,77],"Hilbert":[40],"space,":[41],"then":[42],"RQ":[44,66,81,162],"vector":[47,86,120,163],"$x$,":[48],"denoted":[49],"by":[50,79],"$\\rho(x)$,":[51],"exact":[54],"eigenvalue":[55,74,96],"$A$.":[57],"In":[58],"case,":[60],"$|\\rho(x)-\\rho(y)|$":[67],"becomes":[68],"error":[71],"for":[72,95],"$\\rho(x)$":[75],"approximated":[78],"$\\rho(y)$":[82],"on":[83,160],"given":[85],"$y.$":[87],"There":[88],"are":[89],"three":[90],"traditional":[91],"kinds":[92],"bounds":[94,100,111,124,143],"errors:":[97],"priori":[99],"via":[101,112],"angle":[103,128],"between":[104],"vectors":[105],"$y$;":[108,121],"posteriori":[110],"$Ay-\\rho(y)y$":[118],"mixed":[122],"type":[123],"using":[125],"both":[126],"residual.":[134],"We":[135],"propose":[136],"unifying":[138],"approach":[139,157],"to":[140],"prove":[141],"known":[142],"spectrum,":[146],"analyze":[147],"their":[148],"sharpness,":[149],"derive":[151],"new":[152],"sharper":[153],"bounds.":[154],"proof":[156],"based":[159],"novel":[161],"perturbation":[164],"identities.":[165]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2026-03-15T09:29:46.208133","created_date":"2016-06-24T00:00:00"}
