Condition Numbers of the Hilbert Matrix

The Hilbert matrix is known as an ill-conditioned matrix. Eigenvalues are computed by exflib (multiple-precision arithmetic) and are checked by several methods.
ncond_2(H_n)cond_infty(H_n)required decimal digits
 
101.6026286870216882788394173290284956397168e+133.5357439251992000000000000000000000000000e+1396.3
202.4521565858153031724608315432508721804954e+286.2835796843178877070341940978021978021978e+28115.6
304.2278063881875260360579541415575268985885e+431.1776795727930895742328368602546927949175e+44134.9
407.6529138983103173844002731867959613647554e+582.2450971798796238086626608727098360607956e+59154.1
501.4229418415520910942696000390674280701492e+744.3303437485735977814705520100349208336627e+74192.7
602.6912885316802585330663748024990894657513e+898.4223743925963117400309833317150596684797e+89211.2
705.1508352289097289897115009310685699839710e+1041.6483514908240870683010835024762698065292e+105250.5
809.9444154094053210629478323721101159729963e+1193.2415706593731280050856814232712085524404e+120289.0
901.9327870549650482007900167549700251815822e+1356.3991977497632351598663037906600439168255e+135308.3
1003.7764862742167324533379490789926228890219e+1501.2672207874923736919350953935390153058393e+151346.8
1107.4106884736255844281145586377279938350130e+1652.5159759518360213261447151869020207428134e+166366.1
1201.4594041657382396418032454052003138353501e+1815.0062520182673256934143216099972816319427e+181404.6
1302.8826700000751482019238968345930581591461e+1969.9800738542222646708493190939753561997574e+196423.9
1405.7085569438220705384580370766850690618615e+2111.9927842183158400604694110948093756808018e+212462.4
1501.1329704359003462782208702079799862398853e+2273.9847739952983161490702345314675591941044e+227500.9
1602.2529389709477410338187734440730625439353e+2427.9812550121455854968916156131818340120828e+242520.2
1704.4876469861112979026029548955400781013955e+2571.6034563046309504924668446275806135608223e+258558.7
1808.9524761059720844199392747323873534340217e+2723.2237601348248275731563464636007029155409e+273578.0
1901.7883522161917942805133137210055765339372e+2886.4857871123372775127412045184255851409448e+288616.5
2003.5767547533691335911189646870629015820278e+3031.3056728687629900790043399887670037722036e+304655.0
2107.1614486375984647929647003876852237399053e+3182.6300104094957492114399756896512564040697e+319674.3
2201.4353063258980711785756801248089444099965e+3345.3004616277280819583950388419567050049504e+334712.8
2302.8792704483407947512774988485811693181374e+3491.0687758441003248007597151175149479319903e+350732.1
2405.7807128093685684793174108678267046646264e+3642.1560630112745518654491179038250495356869e+365770.6
2501.1614816156174711319879760342260859133063e+3804.3513573528276248399406043146827328061475e+380789.9
2602.3353373374829637248618761090977147518148e+3958.7854576089659498829506869471362120319644e+395828.4
2704.6986213195158315934048026144728739762716e+4101.7744726950698703503294121877059682827145e+411867.0
2809.4592013718831953252708754823937709190178e+4253.5853329905667858868958182541839630526956e+426886.2
2901.9053882628181199391718820142915309183955e+4417.2466241504686665921803707356581905337358e+441924.8
3003.8400873859981256908192877121364067142760e+4561.4653308181453854989480195120054515764501e+457944.0
3107.7430586779853674502241599912305264769122e+4712.9673611310213444647080073226232818444129e+472982.6
3201.5620122059241826324318012428317157088272e+4876.0102537230469244253584914313338153557801e+4871001.8
3303.1524235151253888910022849926197445136131e+5021.2175858091505739358982455918120636548703e+5031040.4
3406.3647575888122666231734715536083958097989e+5172.4671044017072464554076154712142301348954e+5181078.9
3501.2855418432399295946293411546945284046610e+5334.9998110786491821101242797041629590358988e+5331098.2
3602.5974567300992231127178219622944444108374e+5481.0134325375049029593142036183974114227543e+5491136.7
3705.2500042052106453708151599778616475260218e+5632.0545110021374619410250827630715717920818e+5641156.0
3801.0614813115294022054753612968810598651873e+5794.1657365342019295099329572773049073137610e+5791194.5
3902.1468375216369004455459382261818487540028e+5948.4477737135662273046003576581707091119676e+5941233.0
4004.3432359457727837381095944422368366263999e+6091.7133948368876302349554392190474528783285e+6101252.3
4108.7891905324624624895412172299975132099828e+6243.4756418841109683325894542864598182920173e+6251290.8
4201.7790976243110925381999328249661739131447e+6407.0513579898603618221809932169822758095542e+6401310.1
4303.6021414404233033089217195108102820786725e+6551.4307658647497730247241498397585089129712e+6561348.6
4407.2950257377181012382884617776829104952510e+6702.9035599628217864268217625086024264225346e+6711387.2
4501.4777239676055206213507020826722334840485e+6865.8982652564092182450922581815224468613629e+6861406.1
4602.9940280855415406825113904416472968631570e+7011.1982658815478665970730545811062156641522e+7021425.7
4706.0675086844129555814976635295026895414249e+7162.4345394948159287776558373282116653128499e+7171464.2
4801.2298527113093919249859376842970914712836e+7324.9466880554378468832308953435696721516478e+7321502.7
4902.4933334804146810091959904894166438022784e+7471.0051841217932622133371346794581611094184e+7481541.3
5005.0557877082092834325576745378871838194787e+7622.0427228025112561166646006806873747755664e+7631560.54

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