{"id":"https://openalex.org/W2043613796","doi":"https://doi.org/10.1137/s1064827595290450","title":"Error Analysis of Krylov Methods In a Nutshell","display_name":"Error Analysis of Krylov Methods In a Nutshell","publication_year":1998,"publication_date":"1998-01-01","ids":{"openalex":"https://openalex.org/W2043613796","doi":"https://doi.org/10.1137/s1064827595290450","mag":"2043613796"},"language":"en","primary_location":{"id":"doi:10.1137/s1064827595290450","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s1064827595290450","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":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5013995544","display_name":"Marlis Hochbruck","orcid":"https://orcid.org/0000-0002-5968-0480"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Marlis Hochbruck","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5065792988","display_name":"Christian Lubich","orcid":"https://orcid.org/0000-0003-0105-4927"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Christian Lubich","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.5504,"has_fulltext":false,"cited_by_count":31,"citation_normalized_percentile":{"value":0.89176962,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"19","issue":"2","first_page":"695","last_page":"701"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":1.0,"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":1.0,"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/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9951000213623047,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9889000058174133,"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/generalized-minimal-residual-method","display_name":"Generalized minimal residual method","score":0.8434435725212097},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7631129622459412},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5326067209243774},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.5293394327163696},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.48112884163856506},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.46522969007492065},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.4492890536785126},{"id":"https://openalex.org/keywords/preconditioner","display_name":"Preconditioner","score":0.44348084926605225},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.43612149357795715},{"id":"https://openalex.org/keywords/krylov-subspace","display_name":"Krylov subspace","score":0.4301567077636719},{"id":"https://openalex.org/keywords/arnoldi-iteration","display_name":"Arnoldi iteration","score":0.4207106828689575},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.29150018095970154},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.17912369966506958}],"concepts":[{"id":"https://openalex.org/C155332342","wikidata":"https://www.wikidata.org/wiki/Q1432976","display_name":"Generalized minimal residual method","level":3,"score":0.8434435725212097},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7631129622459412},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5326067209243774},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.5293394327163696},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.48112884163856506},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.46522969007492065},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.4492890536785126},{"id":"https://openalex.org/C167431342","wikidata":"https://www.wikidata.org/wiki/Q1754327","display_name":"Preconditioner","level":3,"score":0.44348084926605225},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.43612149357795715},{"id":"https://openalex.org/C147060835","wikidata":"https://www.wikidata.org/wiki/Q1757151","display_name":"Krylov subspace","level":3,"score":0.4301567077636719},{"id":"https://openalex.org/C43143463","wikidata":"https://www.wikidata.org/wiki/Q696822","display_name":"Arnoldi iteration","level":4,"score":0.4207106828689575},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.29150018095970154},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.17912369966506958},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","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},{"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/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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/s1064827595290450","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s1064827595290450","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":14,"referenced_works":["https://openalex.org/W101236918","https://openalex.org/W1965366611","https://openalex.org/W1998366068","https://openalex.org/W2049627148","https://openalex.org/W2051160435","https://openalex.org/W2068395392","https://openalex.org/W2085259167","https://openalex.org/W2090501010","https://openalex.org/W2100384730","https://openalex.org/W2117366131","https://openalex.org/W2117686912","https://openalex.org/W2137029533","https://openalex.org/W2139875628","https://openalex.org/W2140153041"],"related_works":["https://openalex.org/W1751018809","https://openalex.org/W4226335490","https://openalex.org/W2561347830","https://openalex.org/W2024149451","https://openalex.org/W2521868002","https://openalex.org/W1751968827","https://openalex.org/W2103271089","https://openalex.org/W2060439759","https://openalex.org/W2034236710","https://openalex.org/W2057217612"],"abstract_inverted_index":{"Previous":[0,556],"article":[1,557],"Error":[2,720],"Analysis":[3],"of":[4,50,56,86,114,184,223,254,321,325,353,355,598,616,645,650,695,715,718,812,833,839,856],"Krylov":[5,426,530,635,790,840,857],"Methods":[6,608,626,638,676],"In":[7],"a":[8,40,499,646],"NutshellMarlis":[9],"Hochbruck":[10,14],"and":[11,15,24,35,42,79,80,117,172,174,215,217,246,248,283,284,328,389,391,450,451,492,493,526,600,618,652,697,727,876,909,932],"Christian":[12,16],"LubichMarlis":[13],"Lubichhttps://doi.org/10.1137/S1064827595290450PDFBibTexSections":[17],"ToolsAdd":[18],"to":[19],"favoritesExport":[20],"CitationTrack":[21],"CitationsEmail":[22],"SectionsAboutAbstractError":[23],"residual":[25,291,459],"bounds":[26,555],"for":[27,181,259,293,318,429,461,529,579,639,670,747,751,770,793,817,872,907,930],"the":[28,54,57,87,115,255,319,322,671,678,692,716,719,837],"matrix":[29,58,399],"iteration":[30,316],"methods":[31,147,180,428,578,669,792,871],"BiCG,":[32],"QMR,":[33],"FOM,":[34],"GMRES":[36,118],"are":[37,397],"derived":[38],"in":[39,53,673,677,721,789],"simple":[41],"unified":[43],"way.[1]":[44],"W.":[45,209,240,281],"E.":[46,77],"Arnoldi,":[47],"The":[48,831],"principle":[49],"minimized":[51,359],"iterations":[52],"solution":[55,222,320],"eigenvalue":[59,323],"problem,":[60],"Quart.":[61],"Appl.":[62,72,135,201,417,481,503,513],"Math.,":[63,94,153,299,505],"9":[64,686],"(1951),":[65],"pp.":[66,97,127,156,193,230,268,302,338,368,408,439,473,508,539],"17\u201329.":[67],"qam":[68],"QAMAAY":[69],"0033-569X":[70],"Q.":[71],"Math.":[73,103,162,308,435,444,515],"CrossrefISIGoogle":[74,104,163,309,446,516],"Scholar[2]":[75],"R.":[76,208,239,280,494,522],"Bank":[78],"T.":[81],"F.":[82],"Chan,":[83],"An":[84,252,315],"analysis":[85,855],"composite":[88],"step":[89],"biconjugate":[90],"gradient":[91],"method,":[92],"Numer.":[93,102,152,161,189,204,298,307,504,514,535,546],"66":[95],"(1993),":[96,267],"295\u2013319.":[98],"num":[99,158,304],"NUMMA7":[100,159,305],"0029-599X":[101,160,306],"Scholar[3]":[105],"Google":[106,311,380,420,422,518,520,548],"Scholar[4]":[107],"P.":[108],"N.":[109,218,249,285,382,393],"Brown,":[110],"A":[111,289,456,633,766,810],"theoretical":[112],"comparison":[113],"Arnoldi":[116,501],"algorithms,":[119],"SIAM":[120,132,187,198,262,273,401,413,466,478],"J.":[121,137,188,203,263,274,331,343,361,373,402,414,467,483],"Sci.":[122,138,264,275,468,484],"Statist.":[123,469],"Comput.,":[124,265,470],"12":[125],"(1991),":[126,301],"58\u201378.":[128],"sss":[129,475],"SIJCD4":[130,476],"0196-5204":[131,477],"(Soc.":[133,199,479],"Ind.":[134,200,480],"Math.)":[136,202,482],"Stat.":[139,485],"Comput.":[140,276,445,486],"LinkISIGoogle":[141,206,278,418,487],"Scholar[5]":[142],"M.":[143,175,219,243,250,286,383,452],"Eiermann,":[144],"On":[145,691],"semiiterative":[146],"generated":[148],"by":[149,358,643],"Faber":[150],"polynomials,":[151],"56":[154],"(1989),":[155],"139\u2013156.":[157],"Scholar[6]":[164],"S.":[165,386],"C.":[166,170,313,350,387],"Eisenstat,":[167],",":[168,173,211,216,242,247,385,390],"H.":[169,176,213,244,453],"Elman":[171],"Schultz,":[177,454],"Variational":[178],"iterative":[179],"nonsymmetric":[182,398,463],"systems":[183,354],"linear":[185,224,296,326,356,433,464,794],"equations,":[186],"Anal.,":[190,536],"20":[191,570],"(1983),":[192],"345\u2013357.":[194],"sja":[195],"SJNAEQ":[196],"0036-1429":[197],"Anal.":[205,404,416,547],"Scholar[7]":[207],"Freund,":[210,241],"G.":[212],"Golub":[214],"Nachtigal,":[220,251,287,384],"Iterative":[221,637],"systems,":[225,297,434,465],"Acta":[226,235],"Numerica,":[227],"1":[228,610,628,781,803,805,884],"(1992),":[229,407],"57\u2013100.":[231],"ax2":[232],"ZZZZZZ":[233],"0962-4929":[234],"Numerica":[236],"CrossrefGoogle":[237,348,378],"Scholar[8]":[238],"Gutknecht":[245],"implementation":[253],"lookahead":[256],"Lanczos":[257],"algorithm":[258,460],"non\u2010Hermitian":[260],"matrices,":[261],"14":[266],"137\u2013158.":[269],"8sm":[270],"SJOCE3":[271],"1064-8275":[272],"(USA)":[277],"Scholar[9]":[279],"Freund":[282],"QMR:":[288],"quasi\u2010minimal":[290],"method":[292,317,746],"non\u2010":[294],"Hermitian":[295],"60":[300],"315\u2013339.":[303],"Scholar[10]":[310],"Scholar[11]":[312],"Lanczos,":[314,351],"problem":[324],"differential":[327],"integral":[329],"operators,":[330],"Res.":[332,344,362,374],"Natl.":[333,345,363,375],"Bur.":[334,346,364,376],"Stand.,":[335,365],"45":[336],"(1950),":[337],"255\u2013282.":[339],"jrn":[340,370],"JRNBAG":[341,371],"0160-1741":[342,372],"Stand.":[347,377],"Scholar[12]":[349],"Solution":[352],"equations":[357],"iterations,":[360],"49":[366],"(1952),":[367],"33\u201353.":[369],"Scholar[13]":[379],"Scholar[14]":[381],"Reddy":[388],"L.":[392],"Trefethen,":[394],"How":[395],"fast":[396],"iterations?,":[400],"Matrix":[403,415],"Appl.,":[405],"13":[406],"778\u2013795.":[409],"sjm":[410],"SJMAEL":[411],"0895-4798":[412],"Scholar[15]":[419],"Scholar[16]":[421],"Scholar[17]":[423],"Y.":[424,448],"Saad,":[425],"subspace":[427,531,550,791,841,858,912],"solving":[430,462],"large":[431],"unsymmetric":[432],"Comp.,":[436],"37":[437],"(1981),":[438],"105\u2013126.":[440],"mcm":[441],"MCMPAF":[442],"0025-5718":[443],"Scholar[18]":[447],"Saad":[449],"GMRES:":[455],"generalized":[457],"minimal":[458],"7":[471],"(1986),":[472],"856\u2013869.":[474],"Scholar[19]":[488],"B.":[489],"A.":[490],"Schmitt":[491],"Weiner,":[495],"Matrix\u2010free":[496],"W\u2010methods":[497],"using":[498],"multiple":[500],"iteration,":[502],"18":[506],"(1995),":[507],"307\u2013320.":[509],"aln":[510],"ANMAEL":[511],"0168-9274":[512],"Scholar[20]":[517],"Scholar[21]":[519],"Scholar[22]":[521],"Weiss,":[523],"Minimization":[524],"properties":[525],"short":[527],"recurrences":[528],"methods,":[532],"Electr.":[533],"Trans.":[534,545],"2":[537,863],"(1994),":[538],"57\u201375.":[540],"c27":[541],"ETNAB7":[542],"1097-4067":[543],"Electron.":[544],"ScholarKeywordsKrylov":[549],"methodsconjugate--gradient-type":[551,913],"methodsArnoldi":[552,914],"methodLanczos":[553,915],"methoderror":[554,916],"FiguresRelatedReferencesCited":[558],"ByDetails":[559],"Real":[560],"time":[561],"evolution":[562],"with":[563,725,798],"neural-network":[564],"quantum":[565],"statesQuantum,":[566],"Vol.":[567,586,655,681,705,734,756,776,800,821,844,860,879],"6":[568,779],"|":[569,590,609,627,659,685,709,738,760,780,804,825,848,864,883],"January":[571,806],"2022":[572],"Cross":[573,594,613,631,663,689,764,784,808,829,852,868,887],"Ref":[574,595,614,632,664,690,765,785,809,830,853,869,888],"Quotient-space":[575],"boundary":[576,667],"element":[577,668],"scattering":[580],"at":[581],"complex":[582],"screensBIT":[583],"Numerical":[584,754],"Mathematics,":[585,654,755],"61,":[587],"No.":[588,657,683,707,736,758,778,802,823,846,862,881],"4":[589,658,847],"29":[591],"March":[592,687],"2021":[593,612,630],"Linear":[596,723,796],"solversModeling":[597],"Resistivity":[599,617],"Acoustic":[601,619],"Borehole":[602,620],"Logging":[603,621],"Measurements":[604,622],"Using":[605,623],"Finite":[606,624],"Element":[607,625],"Jan":[611,629],"BibliographyModeling":[615],"Preconditioned":[634],"Subspace":[636],"Inverse":[640],"Source":[641],"Problem":[642],"Virtue":[644],"Regularizing":[647],"LM-DRBEMInternational":[648],"Journal":[649,701,730,894],"Applied":[651,679,910,933],"Computational":[653],"6,":[656],"11":[660],"June":[661,762],"2020":[662,688],"First\u2010kind":[665],"Galerkin":[666],"Hodge\u2010Laplacian":[672],"three":[674],"dimensionsMathematical":[675],"Sciences,":[680],"43,":[682],"8":[684,865],"Convergence":[693,854],"Behavior":[694],"IDR($s$)":[696],"Related":[698],"MethodsPeter":[699],"SonneveldSIAM":[700],"on":[702,731,774,836,895],"Scientific":[703,732,896],"Computing,":[704,733],"34,":[706],"5":[708,737,826],"25":[710,849],"September":[711],"2012AbstractPDF":[712],"(733":[713],"KB)Estimates":[714],"Norm":[717],"Solving":[722],"Systems":[724],"FOM":[726],"GMRESG\u00e9rard":[728],"MeurantSIAM":[729],"33,":[735],"27":[739],"October":[740],"2011AbstractPDF":[741],"(318":[742],"KB)A":[743],"quadrature":[744],"based":[745],"evaluating":[748],"exponential-type":[749],"functions":[750],"exponential":[752,815],"methodsBIT":[753],"50,":[757],"3":[759,761,824],"2010":[763],"Robust":[767],"Maxwell":[768],"Formulation":[769],"All":[771],"FrequenciesIEEE":[772],"Transactions":[773],"Magnetics,":[775],"44,":[777],"Jun":[782],"2008":[783],"Recent":[786],"computational":[787],"developments":[788],"systemsNumerical":[795],"Algebra":[797,875],"Applications,":[799,878],"14,":[801],"2007":[807],"class":[811],"explicit":[813],"multistep":[814],"integrators":[816],"semilinear":[818],"problemsNumerische":[819],"Mathematik,":[820,843],"102,":[822],"December":[827],"2005":[828,851],"effect":[832],"non-optimal":[834],"bases":[835],"convergence":[838],"methodsNumerische":[842],"100,":[845],"May":[850,866],"methodsGAMM-Mitteilungen,":[859],"27,":[861],"2014":[867],"GMRES-type":[870],"inconsistent":[873],"systemsLinear":[874],"its":[877],"316,":[880],"1-3":[882],"Sep":[885],"2000":[886],"Volume":[889],"19,":[890],"Issue":[891],"2|":[892],"1998SIAM":[893],"Computing319-701":[897],"History":[898],"Published":[899],"online:25":[900],"July":[901],"2006":[902],"InformationCopyright":[903],"\u00a9":[904],"1998":[905],"Society":[906],"Industrial":[908,931],"MathematicsKeywordsKrylov":[911],"boundsMSC":[917],"codes65F10PDF":[918],"Download":[919],"Article":[920],"&":[921],"Publication":[922],"DataArticle":[923],"DOI:10.1137/S1064827595290450Article":[924],"page":[925],"range:pp.":[926],"695-701ISSN":[927],"(print):1064-8275ISSN":[928],"(online):1095-7197Publisher:Society":[929],"Mathematics":[934]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}