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Condition numbers of the classical test matrices
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Numbers
$A_n$
$n$ 
$\kappa_2(A_n)$
Hilbert matrix
2:
19.28147006790397144831792337992132252333679086358732323389454148327244519544802785387898421871597525
Hilbert matrix
3:
524.0567775860608178707828459277826389282027210537370700223818857983732045183786337245287538826047870
Hilbert matrix
4:
15513.73873893258822257658341228279054988393607737835414587443915566453026377654442465109069584358851
Hilbert matrix
5:
476607.2502425608093811447795196152046928663241812939419678400825203841930782325253793674456254152353
Hilbert matrix
6:
14951058.64013121677509821491654659705160980645238264929146441574650661616442759402790751975685102803
Hilbert matrix
7:
475367354.9881789925590488183566092181350674661594968973815837982788531612385102004842230501459776533
Hilbert matrix
8:
15257575741.64694283908827407345546448476337599788053574235891522953676636832392494752399299092789594
Hilbert matrix
9:
493154926971.5421050802684068170249721507839040720391289493552650171045384154560600597709981371539292
Hilbert matrix
10:
16026286870216.88278839417329028495639716788348237454482282917214116409822195021863359019840277203763
Hilbert matrix
11:
523067739242940.8570898910256382193161941650902566939789594130802610211571153611189962209839725396722
Hilbert matrix
12:
17132289046970049.62598819655949944095614882499099353380517683765875754487618910536312944689840544339
Hilbert matrix
13:
562794237376007650.2808898912295100827994507734827710854721485220334119118104719616268915218838151092
Hilbert matrix
14:
18533817023471499356.52976037090170347118764228646171065993324992516346629101474678457198471892573047
Hilbert matrix
15:
611656579161984069371.4244380118813713528135237099774165225115993893038810659459986637999737756932714
Hilbert matrix
16:
20223459176745299156148.81207860942152569998226060304318415097379339430355031460065683817640525101331
Hilbert matrix
17:
669743898056063144272390.5843953050436852400267189529756420823454026871054524504674392443387409513118
Hilbert matrix
18:
22211900394338649181923816.21527593125881923241606218924089235134408638787048683346884644438240146473
Hilbert matrix
19:
737595118050884232273250307.3851311679024486774606353671447940820988545670800503731320900860983032039
Hilbert matrix
20:
24521565858153031724608315432.50872180495421955957950090666796337000772068104190589768545987901216161
Lehmer matrix
2:
3
Lehmer matrix
3:
6.663295412110753264930062243601386564923630514100470689440696158671979386513135440359035849019914270
Lehmer matrix
4:
12.20626419560859337378462862409422476996728211839713334322903604569936152775302659345356398077738131
Lehmer matrix
5:
19.65548972424304570040140813436145323896712967095556561325466951514871291761554183341629612722473277
Lehmer matrix
6:
29.04042865087580101666542960654054728601071366222142492480651603387339139503347117792268496544606133
Lehmer matrix
7:
40.38734100191432069163664264295304924130105341274898609573443246746480339772746422300298314557804731
Lehmer matrix
8:
53.71716211525641584576922732714458190672717377179247650506391966089723303114839078577965128570923195
Lehmer matrix
9:
69.04661228743515105024177181021835060066073351167677186243300773857673132236480035181070867687753301
Lehmer matrix
10:
86.38931620332281201794838455607982291201314158472442872330719092499410011351620813676231072314055736
Lehmer matrix
11:
105.7566022536090187091725511467158673216548353886121733930645136670933850636018518923865610206856862
Lehmer matrix
12:
127.1580559835941863070650031085290400743529849743044347790542391038016110774339870757617401623785854
Lehmer matrix
13:
150.6019094456142942742955658782243201592027369356422220938090524609560600453141388911783339796597187
Lehmer matrix
14:
176.0953217241984599298996800763093076381137360872592495026176532817956255975410280487701255005408026
Lehmer matrix
15:
203.6445859031697082529001143942779070635805086294962565850829113254331254807670349297152931919684261
Lehmer matrix
16:
233.2552851101576869014250654545516521548632980052667552742464728875857552901666391848952276158595276
Lehmer matrix
17:
264.9324124657716440688995987635987252378641151576122298177585705142112879043102063896294692048738730
Lehmer matrix
18:
298.6804648813727479370872194168866284125645655002349301732511464026171829622587010839632391402374404
Lehmer matrix
19:
334.5035175305438351002941810690652842746387916474533774769656669248906852383281253741350020196191364
Lehmer matrix
20:
372.4052837811381162666514512760857797900722077011827975271159877084107440783436421213383745772075730
symmetric Pascal matrix
2:
6.854101966249684544613760503096914353160927539417288586406345868115781388456707349121621612568173412
symmetric Pascal matrix
3:
61.98386676965933508143412319825919688666337364233272661270059012890786473549583226815429901486938814
symmetric Pascal matrix
4:
691.9374139700694235650943925628303171729052578959284478328599211329245942468004058763099644834458766
symmetric Pascal matrix
5:
8517.524361138743219408631818276628939545188504983117159877108090989062581343593373038558262072030479
symmetric Pascal matrix
6:
110786.6696800526695358665617707212545474560052169782237523724479605372119521913357228820476788659849
symmetric Pascal matrix
7:
1493442.321688535266800175722787648564791334480448901396704319615836795584933243408674144789740193786
symmetric Pascal matrix
8:
20645173.41542575383630090778199470919102636120598376002098207212681036308418102497819496433014472246
symmetric Pascal matrix
9:
290782729.8229740223850036683458400975910146895185995823035268464732988630108119090894300850943209753
symmetric Pascal matrix
10:
4155205697.177144144772224481514125845789146111877660153558559797636890720384045814587031339639804006
symmetric Pascal matrix
11:
60063526311.91008408749385416869463264668846881295416313020367665341002432536052926811135159434330087
symmetric Pascal matrix
12:
876394910921.1799472393900552499071327276675892908697606073945125568114883374808469327127833021353472
symmetric Pascal matrix
13:
12887562997963.22733666492360704862783595102106871502421935255050255997377540184364035711195389112168
symmetric Pascal matrix
14:
190763930554343.0713958383112947022009489147896702285460268683787842708652842453799820541404836970767
symmetric Pascal matrix
15:
2839640520004331.016641618391992093065938500628775710438510467073256975304756259664420817571061545747
symmetric Pascal matrix
16:
42475924456025123.49735150901097065498056568163705879605881354276120242809961091016534737577338708087
symmetric Pascal matrix
17:
638069311608929429.4590897190712099216213396249413655453482176102291979403064865565048559452852063730
symmetric Pascal matrix
18:
9620976839220254700.345104759189398995751462740303344856945419968058379744200709716984122691134324927
symmetric Pascal matrix
19:
145550741961148233334.7480764081133601365520591035155170064135484361547983379108100724598395293957830
symmetric Pascal matrix
20:
2208514849570099570432.231987827226918624791443066077271187583230278224985453401914851826841164365108
Redheffer matrix
3:
5.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
Redheffer matrix
4:
8.739681318220424342718522135449032485720572154205657521662710230692617096685643031362602103605742537
Redheffer matrix
5:
6.587955135104506409670230948082962337377541676060793370761442772795272436222427156005920487456573455
Redheffer matrix
6:
21.23359070667102751002581724704290998084971228338216453687697648495298717482545248866840930915402935
Redheffer matrix
7:
14.61624139652969868840148959403209372779406573285309453877094958101548935740630778055540794620419750
Redheffer matrix
8:
15.79768260992819023053776105950888672700950298660351705048456896005458458290695380803452980712611207
Redheffer matrix
9:
19.47892289335031680572020936572214567353221791131271069138596591089634137386019543990667775600575741
Redheffer matrix
10:
46.77181691292616098302947363478582763048763357849313162480447087916635517519833716545004409882629981
Redheffer matrix
11:
29.31031695963080268009636006380665230649368428992257470065177464948110661855637468885583277369293022
Redheffer matrix
12:
28.10272177036897067146187956155485334757317474973451621368254976404541471360104244817461076131752657
Redheffer matrix
13:
22.64722389148319717174159409105268253767494999288748755427427133429363000845696311231347547456728809
Redheffer matrix
14:
39.36403397991338424937817829424386468347511128437541347393579784214490720258640391715339350964304372
Redheffer matrix
15:
89.93936983645532453039289811042266797399008992415086192825134372786829091215592724540053114090179374
Redheffer matrix
16:
93.88532669983631584507197601234891160036168724761610490981388845715807841336666270030442199288729557
Redheffer matrix
17:
54.88054874310807231387313441162976052333047781517397101905123026871791225104961185468295328002151053
Redheffer matrix
18:
53.78887441969165746607774133311088351920922404395423282756814376757921962178538000684719732729776604
Redheffer matrix
19:
41.10609262857468230873227009298009543639279345523795118313121308189126083346243851382427981588518938
Redheffer matrix
20:
40.70623084849549939429500380335100652967263155761637244188745928326178671207344577405087607402978060
Vandermonde matrix
2:
6.854101966249684544613760503096914353160927539417288586406345868115781388456707349121621612568173412
Vandermonde matrix
3:
70.92313380667867256089820487444055931990886209341930294398562237143793142905167302184302445269581366
Vandermonde matrix
4:
1171.012685914953612137486622664292837084867697580887648993189524462500783283548617637433752400240399
Vandermonde matrix
5:
26169.68797063337548457166113227247274542076444186101847473102402036844178331355416349410511669986102
Vandermonde matrix
6:
731200.9387890256169016066182762837814360343075176807714561241393642615009743865545247961176402206120
Vandermonde matrix
7:
24459097.87264096851683491423416307737998120627135999852718767921290491257460483311020310773941975596
Vandermonde matrix
8:
952111739.1758977043101523232338048455554096366401942554808131038615863896418584843626843507097994615
Vandermonde matrix
9:
42261319068.20560937708425186402485386574751706873086392586816967425836276803426743423391941605470016
Vandermonde matrix
10:
2106257537026.937205488462681369668533204572029918424046056553441150154702417464236114231570159157625
Vandermonde matrix
11:
116446946417397.5817472391598488132715301919187574284254888064831552858776212188326933925235910190229
Vandermonde matrix
12:
7071899386995588.383482671871976846916818265471454238731764088831225311722112998119238450636069106546
Vandermonde matrix
13:
467971137039451674.9444274618094267217730627400774861581092270693868743413524911150717262642203563350
Vandermonde matrix
14:
33513665168473939633.44747044882576254025041375548515069318671551821270840767017013252203299476326576
Vandermonde matrix
15:
2582411198843621994707.210178004192594426180250807985466842229653401304345362223826864737436199013215
Vandermonde matrix
16:
213037400558373579654540.8595469933570059172562483535154784096033380989417633524046366174903642534091
Vandermonde matrix
17:
18733480865632011372601351.27490478127934727012688404119958869139736330773992423471675637788197926216
Vandermonde matrix
18:
1749229114545498222826797184.356563210744807537485428018077418288216694968745269232085526722418725530
Vandermonde matrix
19:
172847526852023587322227072095.3741540248670466764462340075682648395813152302390532376918507125756685
Vandermonde matrix
20:
18019770221051047870852974504651.13073830887111697609831574356873203631243265454160013304408855208834
Cauchy matrix
2:
38.47400842696444512771120873403933947783752449144988504429446532098620984348642135743218706774077122
Cauchy matrix
3:
1353.287005448660734629770287384826555435992879982690764225855905788402595030728360928182294248899124
Cauchy matrix
4:
45880.47554815595603057429276234424113540085589472395294773243019279790614744129317319370479535347449
Cauchy matrix
5:
1535043.895311232156596958668132165783432503503564817810907006261632739922111189397245950889365313341
Cauchy matrix
6:
51098162.99776819109150266739969077853207062479287911766556935369710763693972590695776651147859362492
Cauchy matrix
7:
1697836321.545401144739904106092685016465608419783981236152585813796422517883089452247428119087277331
Cauchy matrix
8:
56391849565.60453509934946866277337087504006782842579038462753957116230733210425569120591311520351753
Cauchy matrix
9:
1873512681139.627656702093909740411269960586183929344165585974508378650445097550694917264511368158966
Cauchy matrix
10:
62280515397220.94912624801163120999977914465468132284969670528353542859869897174442422381351463284537
Cauchy matrix
11:
2071871422302131.061215147065125339928472088034597542494315998842723698094683963805401549125496987710
Cauchy matrix
12:
68977846388277109.45042456955040937739778354487270116200594692305098254851691278620074023169085266205
Cauchy matrix
13:
2298231929403395058.414903067757533511202719354497195518016601165026495984073370874554375774682096675
Cauchy matrix
14:
76631320756842216140.21566878833715963606744137504187931003098502878501289836708924187733777574552102
Cauchy matrix
15:
2557011863610200390073.827241217787923645980528771930058735677963768810063947661504069895223748421743
Cauchy matrix
16:
85380100517811098148615.36431185078960183359188551967191062191449302352029262693659494931610726053585
Cauchy matrix
17:
2852731730720670364556154.292316885880371332898203321026609231266091101502849678007730380071205434558
Cauchy matrix
18:
95373703104866629524051499.07561887809030530368309196858101388927844467491447976956990754630890781722
Cauchy matrix
19:
3190388524127364236273746738.660505986686594981619186728630988477496002027783126924681584430933241276
Cauchy matrix
20:
106780120547379841085262149708.0622346974471890159406103564929957999075264022417502868133666683471355
Fiedler matrix
2:
1
Fiedler matrix
3:
3.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576
Fiedler matrix
4:
8.812559200041264077209755937373692847729845374313690624305081836193847842739150159609050051085724143
Fiedler matrix
5:
14.85876050659517617095823025943615761159193105655697804118623015782869389381721705403868532402742469
Fiedler matrix
6:
22.59628288864077342668253235274443537586845111174915667655743823210274811687965402932127940761963201
Fiedler matrix
7:
31.53500535924395345221159084827630633486440264760156779605319139996182217427535285963331018267353478
Fiedler matrix
8:
42.01051326895792351760270199001656902828649326953788618505321634298909182795348074124434434234050282
Fiedler matrix
9:
53.76922924439043127088899107541216513687807951126009194836124915207103513501471411822223577653376340
Fiedler matrix
10:
67.00488372234284933562868234105767000641507375004611925480903919809595888575593214755513473677645030
Fiedler matrix
11:
81.56190434290620466988176850129600231907454981889800816005049787131298888608535486191392421363689725
Fiedler matrix
12:
97.56641872691832851034171379628753731088043744418800201246722971172966597558344277000389310798661208
Fiedler matrix
13:
114.9130960048901581835379705222536733526910671983294774734564036623655658942534046055412710601043747
Fiedler matrix
14:
133.6905924048793388509222608929654773280632349631417172279048758792575474193986876734291812433310714
Fiedler matrix
15:
153.8228171531919334826464364780434214677591591642982492917113725167048355523092703031307049429978668
Fiedler matrix
16:
175.3754966306701851791790748712068909289993516994181323622351834242589704135265591961884211704632182
Fiedler matrix
17:
198.2910707243865022750937121936583297308714794813004780819366861553761523580360839433246018348367804
Fiedler matrix
18:
222.6202148135125405228909678109011297774346246297310384874235884243427622949098909159450841422799249
Fiedler matrix
19:
248.3178573277716960759715809512211739337697588510069475438044669109564878074104805922513897888175981
Fiedler matrix
20:
275.4242624796924577020346332157112849525190888044012717624415365620800413299626314454400553587276190
Kac-Murdock-Szegő matrix
2:
3
Kac-Murdock-Szegő matrix
3:
4.529210992451760747443979300582098494332599171743547137887454051462212770546526735135172012098795546
Kac-Murdock-Szegő matrix
4:
5.561552812808830274910704927987038512573599612686810217199316786547477173168810796793931825405342148
Kac-Murdock-Szegő matrix
5:
6.279135464188213446125318238126817350105078204343138107133894405151915601021045512938249063481401699
Kac-Murdock-Szegő matrix
6:
6.796394516776846759102708118557064470268722776616619646187543017006096272344643782644396301683557994
Kac-Murdock-Szegő matrix
7:
7.180972808321037694168272016971284665642045481196476483184922162172666630246827355335048557633535043
Kac-Murdock-Szegő matrix
8:
7.474333610189508473702734215857280014256303090529519994289347061806063478008669681692954822482634701
Kac-Murdock-Szegő matrix
9:
7.702977255563536677734056382318894874425963371730287251571621623625384587781772772283705693657706319
Kac-Murdock-Szegő matrix
10:
7.884472867533711658199141317541883064995115785973970707269570299915055161193526716379295427847314344
Kac-Murdock-Szegő matrix
11:
8.030836195110353210605100642072449077218409172481217578261441327973067500295521774122092177530493221
Kac-Murdock-Szegő matrix
12:
8.150505767501285436879004810853991048941924982617696193476313654593211699269431329305551672634703158
Kac-Murdock-Szegő matrix
13:
8.249545057777994174119098415422256820055189958224087394430745527790255752318386830585672325576494695
Kac-Murdock-Szegő matrix
14:
8.332398921577084525876618725160430771554183218653070671313226459407968986047732440922677246543112785
Kac-Murdock-Szegő matrix
15:
8.402383682615712610952888737124154517411644488306152245565707486266765371684399861155699285732886667
Kac-Murdock-Szegő matrix
16:
8.462012849559203191899266841677889302880741141948423320599374352454993051818362028461857015647436689
Kac-Murdock-Szegő matrix
17:
8.513218509348765862375789353303640960743816284123400558288617822865466151335252340360313082534002796
Kac-Murdock-Szegő matrix
18:
8.557504848050570881997852509474796492514408207149068285489064426731980825214664518285588301291249174
Kac-Murdock-Szegő matrix
19:
8.596056532114153145660330122422724064193796139032710445091667299873450282669131811849846168342495282
Kac-Murdock-Szegő matrix
20:
8.629816471241395444598406449584550411309827874543487179475708445822939761757125842024455223386568173
Minij matrix
2:
6.854101966249684544613760503096914353160927539417288586406345868115781388456707349121621612568173412
Minij matrix
3:
16.39373162228438300161665298907858879205371281996235974324577005197142374221953806961570878692375747
Minij matrix
4:
29.28405223595454165837667108698957397059357353634566683009637555597634006479745021352227667117566089
Minij matrix
5:
45.45516413147923407694373295520580532074941673105059566916548420391376099372811276398478060178254112
Minij matrix
6:
64.88554543475165193786527163974684798952701630717035890609195531051508245508059263115262173519540724
Minij matrix
7:
87.56683576630661512615494548415494697493072448106417062964254062272647868030237495518246880547861863
Minij matrix
8:
113.4952453781199494385669442164401540910419995802733567659399546007742877458758216969327894793918652
Minij matrix
9:
142.6688560612920010744363207238134360029292357161810516920411721898991810015536045484707753015956942
Minij matrix
10:
175.0866129584099208524596536608102550395413482821453899182311772960810993893042417356344976778296736
Minij matrix
11:
210.7478972104337118115083507958235835244882272516363997359627985039014710662589093074458260621427491
Minij matrix
12:
249.6523263910161478305303832793528447777281537478207797061189565998861204384683280991526652880881701
Minij matrix
13:
291.7996538827221575911796650307149169607412945646888109403570111336316138526577217357938589906818070
Minij matrix
14:
337.1897148756089327828611929082589498804533319500219589695245666031144970045121062659620576336072085
Minij matrix
15:
385.8223958464777198503218935499176151087346947318831933341765335163964793401701281319474100677998573
Minij matrix
16:
437.6976165385119832290589185482684097385102410478550824455001570960080803294078398700096689236345673
Minij matrix
17:
492.8153189160734788781123702070775010691149214232910783771151678158766482954742815636876111647920010
Minij matrix
18:
551.1754601719585293641430311456204881606410295459984340867356478756556343945628256139906221081829427
Minij matrix
19:
612.7780081730983108031947694381263643820533140432473411500901384070946954854890292807689807365436618
Minij matrix
20:
677.6229384193438793023516971087863918600312997974327413794551628066262058554993160341213234240246924
Grcar matrix
2:
1
Grcar matrix
3:
1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
Grcar matrix
4:
1.811291364304598879918960400438820288772767552565730006906671620714635635114704916628906965195169243
Grcar matrix
5:
2.509860164575649675974930459314745521651997079962301435599139648759801621460757023511668150973843563
Grcar matrix
6:
2.571939508333339773682582524902032150169242450107671309971003944566919391371868213845712512044337931
Grcar matrix
7:
2.608854065743297288145967507502161932660252379407934262315263438832664408923425599297558251186853695
Grcar matrix
8:
2.631019962157926437935079526028927044992436415034296820088522152087406812092255623324341878578830780
Grcar matrix
9:
2.840098127993314154273616056473907940647327790964903200088318657886120437209466612516617374367039734
Grcar matrix
10:
2.885940149695911756178659813412834559759757593748915368411271777443378570631676659833356149984621968
Grcar matrix
11:
2.860180908943119000555399774464937705416194404594350932063808248330724523441670505228260162189805148
Grcar matrix
12:
2.935755711662161167335744057114215577371763605784691369390191757828334374601655140054820666325124207
Grcar matrix
13:
2.995534727624118326430665013313800908997521825524856743069688086752302096536598264033353169956422920
Grcar matrix
14:
3.080043469861791130702953581212012211919605882431428770484367120678485503890511094622643554464555886
Grcar matrix
15:
3.028209773831895563010694332978661597425849148289310649249255188222182047468334381811430982878021547
Grcar matrix
16:
3.111265682500184123668242502744823331220989151815833731263093546867806339380359691748521677054690496
Grcar matrix
17:
3.098831684643462202205710197621710132523926768604829480187362907349531291250096113461447960943591385
Grcar matrix
18:
3.197755348641882027704830119906691445464763730588175562794031398674854415554649482630762799880348519
Grcar matrix
19:
3.164223128073759903830797317865593583574652932313102046015081173981936882328987068794305526264243391
Grcar matrix
20:
3.229391285410348671225822154862347642370252606160763857801571395448616536191140914755879070706551712
Frank matrix
2:
6.854101966249684544613760503096914353160927539417288586406345868115781388456707349121621612568173412
Frank matrix
3:
27.57958662941752713850603579022088623787382168255897347473262331325633956187869275229074662626629344
Frank matrix
4:
120.4488383722463782347637504718321340403786642241717174421088407252689478532055584538293656158454823
Frank matrix
5:
647.4683032841819018626546865658376811118272286835989937726121159357566664046778301496285137125825818
Frank matrix
6:
4208.256633444049386393300027537257699074134053735985673890784928668813926570666294777443813533973025
Frank matrix
7:
31913.23031925268771523719888151122630937467789954330220565986712916917389402435346950752671047525197
Frank matrix
8:
275624.6160507659493338936906106906019352897037344533035979095786681022610286802466724238317948087399
Frank matrix
9:
2666577.678340248802384673859159524286784490270109148829147782269837114184830426422107668176293363813
Frank matrix
10:
28543218.32614487950738675223149333283848936834274150673905622573765041055176547378004762901326592853
Frank matrix
11:
334752087.3332892175838047286543768183370119773246138063310147863948289663574037829851124111290426478
Frank matrix
12:
4267362283.919363129577046053532338712951766565975159403397337836669131852016373097680899843640616274
Frank matrix
13:
58739552870.27224524109805249191698563680897694249974066772363094482368011036788752651480352120003425
Frank matrix
14:
868146341460.7624641911394113454897188070167940762748645438474356322773467963797867212191078984724604
Frank matrix
15:
13710227934803.78743714318454442294890969278169686772918208000054503986763549785302219241644990880716
Frank matrix
16:
230386695272882.8394082183468808865171918511642737230660214283347337175545753227182403926730353770172
Frank matrix
17:
4104166129764008.008541632481174488384878322206748731469737172959565850577448655886425714163331920440
Frank matrix
18:
77254638917026346.74186042802058108839284349738631517594486046249995840749624064191001418834858224362
Frank matrix
19:
1532099194535720771.581749492809293282638979720191090873150832555891603151269979288365106183845673664
Frank matrix
20:
31928018806474977532.00921220126396747153803912356969250369512583600492818362906980330586052319327127
Wilkinson matrix
2:
3
Wilkinson matrix
3:
2
Wilkinson matrix
4:
5
Wilkinson matrix
5:
7.489686958661078597094809954483623639027331902763526383323485422772959346036558264622568309644304830
Wilkinson matrix
6:
13.05008522758678837722733299314065300671810635177637278549415401145556016736868856014340072905225635
Wilkinson matrix
7:
14.03832251817221747463366073761163423598215507970582454261294335955338854224498477969683269561550258
Wilkinson matrix
8:
19.67943753543910977903841148865064499247314366914205400283823615136543791346789457292850178941955788
Wilkinson matrix
9:
18.63685653822492800847889525124812386252819145057058529708057865713658513449558972216882752900050304
Wilkinson matrix
10:
24.65100234099671331371239633944664151005186094617466521075076940126704371349089649422693854394034474
Wilkinson matrix
11:
22.63700076077607784676291163403474298232980840522432769085104361010319316782266956209384976735538157
Wilkinson matrix
12:
29.37064487087760383118037789932176855050699778435399484219835357537750334228115240853017922337659690
Wilkinson matrix
13:
26.58003909679284829409797368443145125339791870875428024169501752823090810007068695577898212643680273
Wilkinson matrix
14:
34.07360730819533949532655475904165359554120525377041241541249573387299972092151968869105364327043964
Wilkinson matrix
15:
30.52015781453952051413743616212876919782834852386604518318759288864407122279484947959521521350810966
Wilkinson matrix
16:
38.77590465255302656733897335327888060281148281556901009130528499829947682026884455686489213918064446
Wilkinson matrix
17:
34.46018012017107388731055354494212427626311301993992256053481409855789493509397383945523757252046454
Wilkinson matrix
18:
43.47818392935143945430752420921624304052346920580838850340929457362202246000398074620132175852934388
Wilkinson matrix
19:
38.40020017766206895777934532676600000778526621953486692226745326807750695744819803137545807463410297
Wilkinson matrix
20:
48.18046284817080509740175689658623283648424256749460870923109983729831706979164340178991775611023265
Clement matrix
2:
1
Clement matrix
4:
4.441518440112252887999262362955145689146370092883477884571669901861729625759492147562832072252866863
Clement matrix
6:
10.27872566435090911565941130405932619643333279950775142674581484409985771544030385527008765345120313
Clement matrix
8:
22.13033926402618021202178344289552573322005588311143765185151833243119205651741062719305850071994303
Clement matrix
10:
46.57790899379434338685318499035284795922779898600005415460587418592945675179453311121641028244428476
Clement matrix
12:
97.03568528810451752509424276971124739773068804871780409627838306076496358484311176596024623296066186
Clement matrix
14:
200.9294391139554725924628780399028779543501705409357458565745080415255952706945817761861549080157169
Clement matrix
16:
414.2771036922582694548522758759288136456321429497401697341574284447189628239071542486387728171689090
Clement matrix
18:
851.3399369472781445043831107357275257201303972048794892081398559853109693487301086505213035828874157
Clement matrix
20:
1744.869575330080117753361531098782576281397932189307455057219535548959617649086813028752722697054950
Moler matrix
2:
6.854101966249684544613760503096914353160927539417288586406345868115781388456707349121621612568173412
Moler matrix
3:
29.28405223595454165837667108698957397059357353634566683009637555597634006479745021352227667117566089
Moler matrix
4:
153.5275998653947953814910085026890414770697110744936548983145695947735413726038166136065372523909346
Moler matrix
5:
865.9801223928554180034813680221692329644007206933790338045715648729656280740639930456615549717161549
Moler matrix
6:
4878.406423207143057192164319888538626400169545627791571455887817706858916494224335773260146926378736
Moler matrix
7:
26740.47593251011695237821766261589529728004118575165866738407670937776945804270088207313929728093950
Moler matrix
8:
141982.9469117203590236158999336277406620458413613308940868761544655695304217056041302087837173515652
Moler matrix
9:
732097.3543265305574633877429248710041685449025425288849855560054288489087560805753294641639109150461
Moler matrix
10:
3680591.911270500471327124342983507685292839110286622829974834251285330902060493888149042789553587483
Moler matrix
11:
18111714.97404205604536894704051441479370676573390646817851147849812847135859138952591631385256121890
Moler matrix
12:
87522898.09438388227965655929703523820923407256493761134188329483370816922956476412214263153795201976
Moler matrix
13:
416462155.9788828027786101548579783421100871274997824427151028157090796427507238970737322026143031035
Moler matrix
14:
1955575578.026996093578224140292098029650264476773938454546775021686222335439177531417924950232182624
Moler matrix
15:
9078131136.340535704858779888644696686139805101809581367219706564115488833000669366849305239788535733
Moler matrix
16:
41723355677.36085612218190382284069730731751542851777536237119990691462029263797739389558862171052013
Moler matrix
17:
190086303493.0561307667836701825165537946228472863091918634970050776598630607020987806826338999132248
Moler matrix
18:
859313396444.6587808162877794834606044081427880590732402389305776989831089848321722316975426850034817
Moler matrix
19:
3857908033778.770181036276660508525191157363166201427994004466036522992102891408856937517485419812380
Moler matrix
20:
17213361127608.77683284354851414595741824762635480885011892113645996521623473930448774772219981988747
Parter matrix
2:
1.767591879243998215519870211245082657708549428974207702118408842704527824715501740867436513669748453
Parter matrix
3:
1.955174075503658161478216107051173544131501903553808976466397341376030640820833809011572593304236413
Parter matrix
4:
2.064529164127144721953157564664523534544721355560299892108474895698918763472190607443629933649432795
Parter matrix
5:
2.149029032914014497729879877395767265673377198804670322021354041062554911048660318260541529040949502
Parter matrix
6:
2.218494420047682588313620795430870614006162585404661620020079528517041047779458596417973589748501629
Parter matrix
7:
2.277499717492104445613690052296525832832517466557475257106614856293648574680266870308077136292840537
Parter matrix
8:
2.328787976351587203831995560922456857916379918973059529777619970778622538056562082029664291427146252
Parter matrix
9:
2.374147571446823278457382064029967713187802278699819650471040507164807038433917671878951975452589096
Parter matrix
10:
2.414809641491287366849584331689982049531997683678862804728281224330331493365646694209683246707060703
Parter matrix
11:
2.451657790231373584271089059317047727260246438569781308366041099792240347050797104489772951737324177
Parter matrix
12:
2.485347634460338465735732796987268910340727271305338891871651890227053627888511251266433429951759777
Parter matrix
13:
2.516379027956037721189123107900995256987262036877516789444795946269841501585330260088276044836904968
Parter matrix
14:
2.545141787851143812482018694721175673532009003620172892645630670378765867605079110119290762060742199
Parter matrix
15:
2.571945789713183394003604959482914576992515228013764148886562710956328020124415551037737352149018449
Parter matrix
16:
2.597041432373484222826871480420769543140488069504883371739454709419680403414632860675452080797563582
Parter matrix
17:
2.620633947730724613685417549278561654241714307565342072564565066894774450903316686195803791497718344
Parter matrix
18:
2.642893650820354597641170667170625775909581749917424864693226516362068646037066243659380035586214988
Parter matrix
19:
2.663963438030878154875820448787819630686421593341229989143264497580614373589034009793532895900682153
Parter matrix
20:
2.683964374854636869014613042691734395925250787969011453203441695540058481554121629136437430275867487
Ris matrix
2:
1.767591879243998215519870211245082657708549428974207702118408842704527824715501740867436513669748453
Ris matrix
3:
1.955174075503658161478216107051173544131501903553808976466397341376030640820833809011572593304236413
Ris matrix
4:
2.064529164127144721953157564664523534544721355560299892108474895698918763472190607443629933649432795
Ris matrix
5:
2.149029032914014497729879877395767265673377198804670322021354041062554911048660318260541529040949502
Ris matrix
6:
2.218494420047682588313620795430870614006162585404661620020079528517041047779458596417973589748501629
Ris matrix
7:
2.277499717492104445613690052296525832832517466557475257106614856293648574680266870308077136292840537
Ris matrix
8:
2.328787976351587203831995560922456857916379918973059529777619970778622538056562082029664291427146252
Ris matrix
9:
2.374147571446823278457382064029967713187802278699819650471040507164807038433917671878951975452589096
Ris matrix
10:
2.414809641491287366849584331689982049531997683678862804728281224330331493365646694209683246707060703
Ris matrix
11:
2.451657790231373584271089059317047727260246438569781308366041099792240347050797104489772951737324177
Ris matrix
12:
2.485347634460338465735732796987268910340727271305338891871651890227053627888511251266433429951759777
Ris matrix
13:
2.516379027956037721189123107900995256987262036877516789444795946269841501585330260088276044836904968
Ris matrix
14:
2.545141787851143812482018694721175673532009003620172892645630670378765867605079110119290762060742199
Ris matrix
15:
2.571945789713183394003604959482914576992515228013764148886562710956328020124415551037737352149018449
Ris matrix
16:
2.597041432373484222826871480420769543140488069504883371739454709419680403414632860675452080797563582
Ris matrix
17:
2.620633947730724613685417549278561654241714307565342072564565066894774450903316686195803791497718344
Ris matrix
18:
2.642893650820354597641170667170625775909581749917424864693226516362068646037066243659380035586214988
Ris matrix
19:
2.663963438030878154875820448787819630686421593341229989143264497580614373589034009793532895900682153
Ris matrix
20:
2.683964374854636869014613042691734395925250787969011453203441695540058481554121629136437430275867487
Lotkin matrix
2:
14.09572316213714214908482712627305552408750807893088703681747432210231269014744760823548092273681177
Lotkin matrix
3:
482.9221408634444690880006992538854004828497029315067134960030461241458053665060612710295939339128556
Lotkin matrix
4:
17032.90060335015486820711298307520646737288124954963732899755144302748849119275475571767571165347242
Lotkin matrix
5:
591877.1218377181767175645387521386319648501140807366121786592635813056790344771839216477943111266732
Lotkin matrix
6:
20378509.97628039827540427608599163485093432195157366531251158133348760323083480251904807542251055442
Lotkin matrix
7:
698046150.3902913123797674355346343676446503076420519946604279546344977995633400619563580951501320043
Lotkin matrix
8:
23839682200.15309347860841037307875291829510059784092087515478716406501867555657866657340000687654842
Lotkin matrix
9:
812688986477.1179770684010892559405270035953703593582433130065714151162678565020293187627236102595879
Lotkin matrix
10:
27672097002232.43293353455268584331747647892251108030778198251957179357671278011124178641037194415072
Lotkin matrix
11:
941509988026486.8461767035322897680628404994397953851885147261036113208412458569188128677597720315648
Lotkin matrix
12:
32016959018788227.51511830914698084752362075826288996018747187647241926303178143833869273074452255039
Lotkin matrix
13:
1088370865409735141.926166069659390958253979710177933132813854231824795587537899977815967058540868584
Lotkin matrix
14:
36988073941626099269.29365903741830506040076654694844994332994929755935704466398118360575373137477173
Lotkin matrix
15:
1256800039936171800601.167580706218786486025889734230853990413340805005779884430062559896996257403733
Lotkin matrix
16:
42698483123659176306548.87190291224902455213973604986561788735398185627278060883833414513719439470409
Lotkin matrix
17:
1450495028176581931951705.788611430784657206676318293660698429367492943389982453914643069670419546389
Lotkin matrix
18:
49270724782201132389675425.13984265130457162213768018557575716693135644815058101561093530307201281297
Lotkin matrix
19:
1673550753322943677139780910.237643720199207213789559132538086159832074279182591685761176934181252968
Lotkin matrix
20:
56842374559081748337408125578.97626726799269312736318646435874417402030870389556802212489225545485619
Riemann matrix
2:
6.854101966249684544613760503096914353160927539417288586406345868115781388456707349121621612568173412
Riemann matrix
3:
8.193612579820439139807729500024362877098644371793435053236054807024325590557962720727064914442237099
Riemann matrix
4:
47.04981244690093290016297746997961008669616320848046841422464383771177638023018958803941545926647835
Riemann matrix
5:
13.99989583278952778367360787204153948013823406631685559148738588012780787869351835462012197172884057
Riemann matrix
6:
213.3164398824469905397714374169198248373700702273819896150815527190623137322291710693675758540621122
Riemann matrix
7:
239.1350675423491626031958913700039017738564448469936197750746804923882379942578846307653538912977565
Riemann matrix
8:
251.4628943544636710984252705892063190536363136808141204918262627435236313651093305196215299634336071
Riemann matrix
9:
28.84467489662542473641169530835524831424709116135886278548758591071114594142272163353378071412579624
Riemann matrix
10:
6276.127812156850708652974554271586995482304460420252150726688521894621443665988980120164633894573954
Riemann matrix
11:
6952.796605233813862494246229409710519206315672647295632530554770961485953119167046562978190436533463
Riemann matrix
12:
39.57707065017695543596613965577754128233339809855590683835965350176594704052056347053611492549439829
Riemann matrix
13:
553.7854975353958496641717130916005494387863059039340043642441975044619291634672774133355518356972311
Riemann matrix
14:
54.14527885651393485614374342497171148037620015059620771756993348097802464751582467265831527593446663
Riemann matrix
15:
58.76195850960064092277732523176480728811472065947904548179811722922500595226906109780221682166986761
Riemann matrix
16:
1906.284163671990698893164012100128863651956152455603255723660568762116567678877949849885602340411939
Riemann matrix
17:
2025.928458114988947571118773213284976265058081191653986284916143286543956672500725054864225831259818
Riemann matrix
18:
78.44346896192965337055801416654274439743547371230243013670091654361918495615939189588785009987929149
Riemann matrix
19:
81.54279457938095285749959052479858630058686898460053620730773175861077259464831368824224246760356410
Riemann matrix
20:
1343.922464578517622175568431172290201256726681453349869776328732144073320651620176215295914999585063
second-difference matrix
2:
3
second-difference matrix
3:
5.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
second-difference matrix
4:
9.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550
second-difference matrix
5:
13.92820323027550917410978536602348946777122101524152251222322791780773206763520014832458474702899430
second-difference matrix
6:
19.19566935808922125408885762809347889438555114422607404335289489837028858307541800061843169636078230
second-difference matrix
7:
25.27414236908818035565879310572184722319306526181401029182309426983953413970201522892768449688383563
second-difference matrix
8:
32.16343747752635842648488964163903691046598980487459609627694888152478248678922913142176845085155703
second-difference matrix
9:
39.86345818906140097284937164610780138503487108560327438481865913858276221721313128522637500795004189
second-difference matrix
10:
48.37415007870822885672446906933846071887432642789491957925447432190414161032068609792578916312975559
second-difference matrix
11:
57.69548054098103742542016242237930807303484566819254652680940861902855118463536084439220143596224574
second-difference matrix
12:
67.82742906960375599217923619233442644402674651864836608326724436346916587260060598501777094801217577
second-difference matrix
13:
78.76998224097115097975413473761310104295523962497076606550623735582317324474509748941972267207075533
second-difference matrix
14:
90.52313096777422640201207897989414603985020009878952484541289595000767234654256519918198585638818990
second-difference matrix
15:
103.0868689198174577967335656168509860726375039884518982947581815199148533077592782686114867789712344
second-difference matrix
16:
116.4611915774877529951308419061505515232394453304046225179347155191821034509803101317921584516505736
second-difference matrix
17:
130.6460956438598813605500036441875702343318517979129108034244296480747678641331773964883333667756477
second-difference matrix
18:
145.6415786680974458216413391134515702243861987225868043255285024600260891345422437517277312957439852
second-difference matrix
19:
161.4476387975884956973942691250209836576957984801531118741816699367686151736935011895700398051193012
second-difference matrix
20:
178.0642746108601779425991394266782722656718503092937653310301711504015264555016939341331840577660220
Definition
For a named classical test matrix $A_n$ of order $n$, this table gives the 2-norm condition number $\kappa_2(A_n)=\sigma_{\max}(A_n)/\sigma_{\min}(A_n)$ [1], the ratio of the largest to the smallest singular value.
Parameters
$A_n$
—   matrix family (one of the named matrix families listed in the comments)
$n$
—   order ($n\geq 1$)
Formulas
(1)
For the second-difference matrix, whose eigenvalues are $2-2\cos(k\pi/(n+1))$, $k=1,\ldots,n$, the condition number is $\kappa_2=\dfrac{2+2\cos(\pi/(n+1))}{2-2\cos(\pi/(n+1))}$. The cosine values in this expression are in $\cos(\pi x)$ for rational $x$.
Comments
(2)
Indices are $1$-based. The matrices are $A_{ij}=1/(i+j-1)$ for the Hilbert matrix [2]; $A_{ij}=\min(i,j)/\max(i,j)$ for the Lehmer matrix [3]; $A_{ij}=\binom{i+j-2}{i-1}$ for the symmetric Pascal matrix [4]; $A_{ij}=1$ if $i\mid j$ or $j=1$, and $0$ otherwise, for the Redheffer matrix [5]; and $A_{ij}=i^{j-1}$ for the Vandermonde matrix [6]. The Cauchy matrix is $A_{ij}=1/(i+j)$, the default scalar case of gallery("cauchy",n) [7]. The Fiedler matrix is $A_{ij}=|i-j|$. The Kac-Murdock-Szegő matrix is $A_{ij}=2^{-|i-j|}$. The Minij matrix is $A_{ij}=\min(i,j)$. The Grcar matrix has $-1$ on the subdiagonal and $1$ on the diagonal and on the first three superdiagonals [7] [8]. The second-difference matrix is tridiagonal, with $2$ on the diagonal and $-1$ on the first subdiagonal and first superdiagonal.
(3)
The Frank, Wilkinson, Clement, Moler, Parter, Ris, Lotkin and Riemann matrices use the definitions in the Test Matrix Toolbox source [8]. The Frank matrix is the unreflected upper Hessenberg form $A_{ij}=n-\max(i,j)+1$ for $j\geq i-1$, and $0$ otherwise. The Wilkinson matrix is symmetric tridiagonal with off-diagonal entries $1$ and diagonal entries $|(n-1)/2-(i-1)|$. The Clement matrix is the default nonsymmetric tridiagonal matrix, with zero diagonal, $A_{i,i-1}=n-i+1$ and $A_{i,i+1}=i$. The Moler matrix has $A_{ii}=i$ and $A_{ij}=\min(i,j)-2$ for $i\neq j$. The Parter matrix is $A_{ij}=1/(i-j+1/2)$. The Ris matrix, also called the Dingdong matrix, is $A_{ij}=(1/2)/(n-i-j+3/2)$. The Ris matrix is one half of the Parter matrix with its rows reversed, so the two have the same condition number at every order. The Lotkin matrix is the Hilbert matrix with first row replaced by ones. The Riemann matrix is the $n\times n$ matrix $A_{ij}=i$ if $(i+1)\mid(j+1)$ and $-1$ otherwise.
(4)
Only finite condition numbers are entries. If a default matrix is singular, $\sigma_{\min}(A_n)=0$ and $\kappa_2(A_n)$ is infinite rather than a real number, so there is no entry. This excludes the singular default Chow matrices for $n>1$ and singular orders of any other family.
(5)
The MathWorks gallery documentation [7] states the default parameter choices used here for the gallery families. The Test Matrix Toolbox source [8] is used where the entry-wise formula is not stated in the MathWorks page.
Programs
(P1)
Sage
from sage.matrix.constructor import matrix
from sage.rings.rational_field import QQ
from sage.rings.qqbar import AA

A = matrix(QQ, 3, 3, lambda i, j: QQ(1) / QQ((i + 1) + (j + 1) - 1))
B = A.transpose() * A
ev = sorted(B.charpoly().roots(AA, multiplicities=False))
(ev[-1] / ev[0]).sqrt().n(400)
Links
Similar tables
$\cos(\pi x)$ for rational $x$ —   the second-difference condition numbers are rational functions of the cosine values at $x=1/(n+1)$
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds finite condition numbers for the listed default matrix families with $2\leq n\leq20$; the default Chow matrix is omitted because it is singular for $n>1$, and singular Clement and Redheffer orders are omitted)
How they were obtained:

The generator builds each matrix over $\mathbb{Q}$, computes the eigenvalues of $A_n^{\mathsf T}A_n$ in Sage's algebraic real field, and returns the square root of the ratio of the largest positive eigenvalue to the smallest positive eigenvalue as a real ball with 96 guard bits beyond the 100 digits written.

more

When the algebraic value is proved to be a rational integer after a cheap numerical filter, that exact integer is returned instead. Singular matrices are detected by exact determinant computation and omitted, because their 2-norm condition numbers are infinite rather than finite real numbers. Before the entries were written, a dry run computed all 351 entries; the Frank determinants were checked to be $1$; the Redheffer determinants were checked against the Mertens function; the Clement characteristic polynomials were checked against the eigenvalues $\pm(n-1),\pm(n-3),\ldots$; the second-difference values were checked against (1); and every nonsingular entry with $n\leq6$ was compared with NumPy's independent singular-value decomposition.



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