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Science Forum Index » Math - Numerical Analysis Forum » How do I compute e^e to thounands of decimal places?
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| john |
 Posted: Sat Dec 16, 2006 11:47 pm |
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Guest
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I know that I can compute e quickly by using the Taylor series with the
binary splitting.
I there any fast algorithm for computing e^e to high precision?
Thanks |
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| Guest |
| Posted: Sun Dec 17, 2006 1:11 am |
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john wrote:
Quote: I know that I can compute e quickly by using the Taylor series with the
binary splitting.
I there any fast algorithm for computing e^e to high precision?
Thanks
With Mathematica 5 on a mac G5
N[E^E,1000]
takes 56 microseconds. N[E^E,10000] takes .132 seconds. Why
do you want to do it yourself? |
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| Dave Seaman |
| Posted: Sun Dec 17, 2006 7:35 am |
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Guest
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On 17 Dec 2006 07:11:36 -0800, john wrote:
Quote: carlos@colorado.edu wrote:
john wrote:
I know that I can compute e quickly by using the Taylor series with the
binary splitting.
I there any fast algorithm for computing e^e to high precision?
Thanks
With Mathematica 5 on a mac G5
N[E^E,1000]
takes 56 microseconds. N[E^E,10000] takes .132 seconds. Why
do you want to do it yourself?
Mathematica is not accurate. The first thousand digits of N[E^E,1000]
is different than the first thounsand digits of N[E^E,10000]
The results are consistent, given that N[E^E,1000] gives a result that is
rounded to 1000 significant digits. If you want the result truncated to
1000 significant digits instead of rounded, you can take the first 1000
digits of N[E^E,1010].
--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/> |
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| john |
| Posted: Sun Dec 17, 2006 11:11 am |
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Guest
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carlos@colorado.edu wrote:
Quote: john wrote:
I know that I can compute e quickly by using the Taylor series with the
binary splitting.
I there any fast algorithm for computing e^e to high precision?
Thanks
With Mathematica 5 on a mac G5
N[E^E,1000]
takes 56 microseconds. N[E^E,10000] takes .132 seconds. Why
do you want to do it yourself?
Mathematica is not accurate. The first thousand digits of N[E^E,1000]
is different than the first thounsand digits of N[E^E,10000] |
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| Gottfried Helms |
| Posted: Sun Dec 17, 2006 11:30 am |
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Guest
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Am 17.12.2006 16:11 schrieb john:
Quote: carlos@colorado.edu wrote:
john wrote:
I know that I can compute e quickly by using the Taylor series with the
binary splitting.
I there any fast algorithm for computing e^e to high precision?
Thanks
With Mathematica 5 on a mac G5
N[E^E,1000]
takes 56 microseconds. N[E^E,10000] takes .132 seconds. Why
do you want to do it yourself?
Mathematica is not accurate. The first thousand digits of N[E^E,1000]
is different than the first thounsand digits of N[E^E,10000]
Pari/GP computes it in a blink.
I don't know, how many digits are accurate in the \precision 10000-example.
Gottfried Helms
---------------------------------------------
exp(exp(1))
%432 = 15.15426224147926418976043027
\p 1000
realprecision = 1001 significant digits (1000 digits displayed)
exp(exp(1))
%433 =
15.1542622414792641897604302726299119055285485368561397691407464059148309737309
3443260845696835787346051158726885285229584108349266426657649118779479704154810
4617616229388368454821943265188236980675811312322990354613338335185965954216525
0720487113169484124883702829810163094049574779199137245321728538732191068097791
4733658187699967694174778649038163390505612049776125348054466629607940201952987
7275185530879677728180527535931123975906005188808804151764154263227653969369419
2816814180488110501622857131251257368608417050247537255162547284751410457996493
3464925837773299779952674620708856662577940458954490095164618850324515554327610
2551379333718085468414791771323547050692212614636013851810485295066335920575541
4000937288132756611779760418697301696724871653429209936701021504088294995975066
1761671984261557111132629477412381031515161306636024230107996709808256269204109
4465730378984833988726011897854428408453570101750798918965140392429845444888600
38406071057436135092697254359616092967488612097910671
\p 10000
realprecision = 10008 significant digits (10000 digits displayed)
exp(exp(1))
%434 =
15.1542622414792641897604302726299119055285485368561397691407464059148309737309
3443260845696835787346051158726885285229584108349266426657649118779479704154810
4617616229388368454821943265188236980675811312322990354613338335185965954216525
0720487113169484124883702829810163094049574779199137245321728538732191068097791
4733658187699967694174778649038163390505612049776125348054466629607940201952987
7275185530879677728180527535931123975906005188808804151764154263227653969369419
2816814180488110501622857131251257368608417050247537255162547284751410457996493
3464925837773299779952674620708856662577940458954490095164618850324515554327610
2551379333718085468414791771323547050692212614636013851810485295066335920575541
4000937288132756611779760418697301696724871653429209936701021504088294995975066
1761671984261557111132629477412381031515161306636024230107996709808256269204109
4465730378984833988726011897854428408453570101750798918965140392429845444888600
3840607105743613509269725435961609296748861209791067089438400285351354408713918
8185236789558583228418697655027343664295059301685904163496485609333642048395482
0650384924580193949950779821150964872368373040063870083168095363027825895373270
2310307837847065593236711057022651887207411283498881856780573901619185020164160
2965145319622806997576325541488722314119730826099368099555364087698755915274173
7571115875654758656802173784981639101880003202229444914999791121833614814370224
6617292997391953208531924312212066217021547415290675137337431052819671080779942
5901987988955668469995950647941834081914265548283223496800308197268957546605820
6320833033553171990434965352214108774173325396334580400786128927180361486215530
6183896771477468145517390398881222962640793421552584774230508945485314307532737
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1135198832505594830473508310601555491417396748201669626466472692816532277039854
3417383872572338553723534609653269953199535443604157566414013521114991810159856
2866411214250152405529647896191413052812129961336151804153574833597099192198950
7267111810205564549566375194914643961437125489866537024131567362471454409935688
4966160936874318079196813692354128703012482855191590272470319112024833782249565
8746958393932826271796516391042999791759950682817945705010877252055249481298285
5489300252772768501931745621422213965593246683600380849978738590326834363049501
8500428227687154239048158012107122417751728406147114588081913493574014220174545
6018252949470567744935225268376925303977835265794403486044171894933825392225727
6060280689266927616439587320787034644985414588900486074924544863649001410532788
0977417728342518343323661516190343166133053023232190916296246312261587679333841
8128683703313656110756785714362533380344155735038075737094139606771127999463629
9773064048361492751822269677752090722432180113205610542937432496811150484238588
9348253629046490808709100506524404810787697290293400692434949201156168456689152
5306919296489225719430373453459784234205734061804455652163311734386767284367240
8343438435969806255924969881125591662929033399683994252425817994542044188669779
7002827286674261716138315282812846299380975679569516261540504625821056334650408
3597041156455606100143977115423103463033662486929805646359497997332809737823297
0850126064254372106043874398438310773607340468363257034333297014517998917905943
6075711672217137746786462455398688321671514797293371376965306257232395425821140
9767582212783112251466700945515279072003798479296836637853564843486728999442845
1734113206366060972279618756449519050792792333001388259838757983699293745013234
1660125754583303168476884368884783776901333348576560998479748343479424999109141
4576760283020321085917921483676306289226043294403972140781089371721151043959646
5330168085890043944090299377259947262258711285058435321708708389041603879946707
8773633844529619232245980830808370140512843601013885796938484928634964732028179
0945755371438694591107949627453702261473627885349799958569070926904769632677811
9301896549813521636695864061550369672075709028327004400720174003260300135057857
6031298916932994335547940766550017726047995411289533657285788255448203971892211
6901705771439261084706731865796851627908363481719788082194861444253539597245895
5611448835328694854331308627413229782333872979762544745619720082820044489589556
8467986103473639835452482285685721467997074208054392453888237037465760482895504
3173434580773594339800500439704660706416691956740859374496133170865588087283960
1094551135525880905896391587785956162412135233880959286645316011716385789225503
3670050844300365091444041769661254680177571496476199899105909095843679793129289
4165977577277554033195883574337488042018216365885539423179360841553520316262293
5685956648762956207797208329912848388494989439337425915675430930729703424168598
7180552814358009965022448742687043596308585106222419582650254943440160323664491
3899028165178005038912766704356593794112712214875952944344733046070680681512427
6260120208880081634867945300717685132337663564957234837179171590235557484185076
9645980735796620839722827628230362029619653830518898723550223680670434324493820
6167840763841773446925308262972280929377881185174780795798792586178581027583605
2989651991106783824254457337164151521529713100704620985518338280994059505048693
0764747651106554568321140741566243534775391678952733133033447135872914901595653
7479229719823329044630603029553971649957452761931691336815304074037234340058434
4275711332719297511477795066175511256877347408028952842898688383798237272836384
2571717926066249218158643701345909437883833236033600335015007518204373658184815
1750092777997798095755473768524693850758185957890692130436999137664467418703788
1381219879442586177403209325131340043258241298703327251563779269085783111032584
2774741248758567247152140727023826685093242441218855692556088621902432863210045
6956683175375231972388804735988949015569468005146968259496578426360490896026561
0254947296336080513930827310296461238509734240538769135378474340556221168562680
6514421610423332851534773324910285550843298301218367667846867970639025172767459
0091149137443761021269650014962550644036306120853925244729021936200156948727112
7446811973182309918530954984466900369991831746945753140761909163276748169845507
2648364572509395200098819848629629677233636299475917640892618352137731863432544
5028960466366391327440917029604016544109770103685562012325593434036555700710593
8139215159120935981460038930898046672040715224469524575920301804675154639643613
7254994369249986353529078666160453670668970376970701866635144363585370409224325
7967623744190705310396274885579259809553846357215187180773252212355695711615110
0331964474429200319056069653671054729407099912455764792097611470709923047064514
4185781140660360284248032350462498853106111289905515470209115472784921322521341
5581410212435950962077520282711095344439310055024461607709767521495851936971479
2368658945637722501481119788934335481832006419707699933662776694512590395546210
3249276273401459369776695813824971280416132352215253061561879801473550570159414
8092014290817859089553987325664847567138889443568182703206964339297644813686230
4239611018520909182631760611648526833654913633399044746263608388930554128799832
5493435808287788533218436148110015809534142167929797553661540995970568473807560
7920484997265517126017206377635876645113085143863237785286965886864523390490236
4849969810790280999301110144355333809105776115954820074468572804031792145733883
1830530017685493705682320581275073454455938583446726851659479909004259869372700
8528459486996809339491136524565933257842820316662858943418478580917039670914359
3015847823385389737374876850851013805967360850529417689159603092079433065458339
7021707131216142193980892484985219017307057101378688513466215880446431191086674
0832391553590102217475313144663215233483029451963957653827807362604188690218688
0122444445170512389669217015386850377298003316988124178301626228200342123014927
1312284825161307878406926011935895693164044569370208473402674357977405294989258
3208295181873235977371643485427277771833865617093581958084350046243803726006576
8677151784134213165501527585191297587111200959854888573648658029313459851776967
2006111636999603331679406048427717673482943509946912664391028642951615005684358
5834177023910410915844491777583488951665403757545648936604054929121730607106067
1458173854712796601155533176269739276616259137725384468113776971535028834997219
9040638502965061907698106049577155718101746299079463821412460185143967865046979
6719545080728552897388338062232238505484800277414896067038156224130762818001233
8763169907673977482366128427650747182545215798618536971812436307458543917047948
3144795713945550767268321578072625670126050447450692242298664074401264835476635
2251343096673315016119541107881153032275541009008965537682213694698954766652700
6295664675856741126761137705010190683065516028501826960474177629985771620816310
7595301049295556939671880155363143014342443073189736342368962279996028624619627
6944613586179692759538317078099561045665970567794014463313146913676028118384664
9718537494126028023416533656090198383897466755314991222794800443872830197900719
6701536704798195489254774630035389316725638043510826032194782076435960010407685
9988035059560852794503023411541560225124478020124251633255043486786504679606907
7885748906304603456364014565242494368884341980290529750830401985891808783592642
0261863624451283653847887180365947549116631224069000718264043299438109792776026
5494403419542949485357051314498407759243103197176779548276502958949855859529366
9717033468529148367578581258194081235286545628820633982432837002943466748186734
2813989726155810499789597976659602908972191132045933166656866970677124609574056
5669490660484534697534714914634137737316685987736430435145571210473762551934229
8934426919416389444046202207394640624043472972940424647276640289351408025338624
3932783174337365031421483588148165730213528967047850816634399926232352460424794
8223663055673516515718162454759294585233853635073905569664688672535990258287306
5531780169825283029888839864353571836314013751471414071720118690699258030491260
07496156648049114802685236104418088396819786811
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| john |
| Posted: Sun Dec 17, 2006 12:06 pm |
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Guest
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However, one cannot tell how accurate the result is computed from these
methods above.
Are there any fast methods for computing constants like e^e, pi^e,
sin(e), etc. to arbitrary precision to any accuracy?
Thanks |
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| Guest |
| Posted: Sun Dec 17, 2006 5:42 pm |
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Mathematica result for N[E^E,1000] is
15.154262241479264189760430272629911905528548536856139769140746405914830973730\
934432608456968357873460511587268852852295841083492664266576491187794797041548\
104617616229388368454821943265188236980675811312322990354613338335185965954216\
525072048711316948412488370282981016309404957477919913724532172853873219106809\
779147336581876999676941747786490381633905056120497761253480544666296079402019\
529877275185530879677728180527535931123975906005188808804151764154263227653969\
369419281681418048811050162285713125125736860841705024753725516254728475141045\
799649334649258377732997799526746207088566625779404589544900951646188503245155\
543276102551379333718085468414791771323547050692212614636013851810485295066335\
920575541400093728813275661177976041869730169672487165342920993670102150408829\
499597506617616719842615571111326294774123810315151613066360242301079967098082\
562692041094465730378984833988726011897854428408453570101750798918965140392429\
84544488860038406071057436135092697254359616092967488612097910671
This seems to agree to all 1000 places with the result from Pari/GP.
For 10000 digits it gives
15.154262241479264189760430272629911905528548536856139769140746405914830973730\
934432608456968357873460511587268852852295841083492664266576491187794797041548\
104617616229388368454821943265188236980675811312322990354613338335185965954216\
525072048711316948412488370282981016309404957477919913724532172853873219106809\
779147336581876999676941747786490381633905056120497761253480544666296079402019\
529877275185530879677728180527535931123975906005188808804151764154263227653969\
369419281681418048811050162285713125125736860841705024753725516254728475141045\
799649334649258377732997799526746207088566625779404589544900951646188503245155\
543276102551379333718085468414791771323547050692212614636013851810485295066335\
920575541400093728813275661177976041869730169672487165342920993670102150408829\
499597506617616719842615571111326294774123810315151613066360242301079967098082\
562692041094465730378984833988726011897854428408453570101750798918965140392429\
845444888600384060710574361350926972543596160929674886120979106708943840028535\
135440871391881852367895585832284186976550273436642950593016859041634964856093\
336420483954820650384924580193949950779821150964872368373040063870083168095363\
027825895373270231030783784706559323671105702265188720741128349888185678057390\
161918502016416029651453196228069975763255414887223141197308260993680995553640\
876987559152741737571115875654758656802173784981639101880003202229444914999791\
121833614814370224661729299739195320853192431221206621702154741529067513733743\
105281967108077994259019879889556684699959506479418340819142655482832234968003\
081972689575466058206320833033553171990434965352214108774173325396334580400786\
128927180361486215530618389677147746814551739039888122296264079342155258477423\
050894548531430753273706246888147420871112695796318805808940094545849750800038\
454605680063772534810291135198832505594830473508310601555491417396748201669626\
466472692816532277039854341738387257233855372353460965326995319953544360415756\
641401352111499181015985628664112142501524055296478961914130528121299613361518\
041535748335970991921989507267111810205564549566375194914643961437125489866537\
024131567362471454409935688496616093687431807919681369235412870301248285519159\
027247031911202483378224956587469583939328262717965163910429997917599506828179\
457050108772520552494812982855489300252772768501931745621422213965593246683600\
380849978738590326834363049501850042822768715423904815801210712241775172840614\
711458808191349357401422017454560182529494705677449352252683769253039778352657\
944034860441718949338253922257276060280689266927616439587320787034644985414588\
900486074924544863649001410532788097741772834251834332366151619034316613305302\
323219091629624631226158767933384181286837033136561107567857143625333803441557\
350380757370941396067711279994636299773064048361492751822269677752090722432180\
113205610542937432496811150484238588934825362904649080870910050652440481078769\
729029340069243494920115616845668915253069192964892257194303734534597842342057\
340618044556521633117343867672843672408343438435969806255924969881125591662929\
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607340468363257034333297014517998917905943607571167221713774678646245539868832\
167151479729337137696530625723239542582114097675822127831122514667009455152790\
720037984792968366378535648434867289994428451734113206366060972279618756449519\
050792792333001388259838757983699293745013234166012575458330316847688436888478\
377690133334857656099847974834347942499910914145767602830203210859179214836763\
062892260432944039721407810893717211510439596465330168085890043944090299377259\
947262258711285058435321708708389041603879946707877363384452961923224598083080\
837014051284360101388579693848492863496473202817909457553714386945911079496274\
537022614736278853497999585690709269047696326778119301896549813521636695864061\
550369672075709028327004400720174003260300135057857603129891693299433554794076\
655001772604799541128953365728578825544820397189221169017057714392610847067318\
657968516279083634817197880821948614442535395972458955611448835328694854331308\
627413229782333872979762544745619720082820044489589556846798610347363983545248\
228568572146799707420805439245388823703746576048289550431734345807735943398005\
004397046607064166919567408593744961331708655880872839601094551135525880905896\
391587785956162412135233880959286645316011716385789225503367005084430036509144\
404176966125468017757149647619989910590909584367979312928941659775772775540331\
958835743374880420182163658855394231793608415535203162622935685956648762956207\
797208329912848388494989439337425915675430930729703424168598718055281435800996\
502244874268704359630858510622241958265025494344016032366449138990281651780050\
389127667043565937941127122148759529443447330460706806815124276260120208880081\
634867945300717685132337663564957234837179171590235557484185076964598073579662\
083972282762823036202961965383051889872355022368067043432449382061678407638417\
734469253082629722809293778811851747807957987925861785810275836052989651991106\
783824254457337164151521529713100704620985518338280994059505048693076474765110\
655456832114074156624353477539167895273313303344713587291490159565374792297198\
233290446306030295539716499574527619316913368153040740372343400584344275711332\
719297511477795066175511256877347408028952842898688383798237272836384257171792\
606624921815864370134590943788383323603360033501500751820437365818481517500927\
779977980957554737685246938507581859578906921304369991376644674187037881381219\
879442586177403209325131340043258241298703327251563779269085783111032584277474\
124875856724715214072702382668509324244121885569255608862190243286321004569566\
831753752319723888047359889490155694680051469682594965784263604908960265610254\
947296336080513930827310296461238509734240538769135378474340556221168562680651\
442161042333285153477332491028555084329830121836766784686797063902517276745900\
911491374437610212696500149625506440363061208539252447290219362001569487271127\
446811973182309918530954984466900369991831746945753140761909163276748169845507\
264836457250939520009881984862962967723363629947591764089261835213773186343254\
450289604663663913274409170296040165441097701036855620123255934340365557007105\
938139215159120935981460038930898046672040715224469524575920301804675154639643\
613725499436924998635352907866616045367066897037697070186663514436358537040922\
432579676237441907053103962748855792598095538463572151871807732522123556957116\
151100331964474429200319056069653671054729407099912455764792097611470709923047\
064514418578114066036028424803235046249885310611128990551547020911547278492132\
252134155814102124359509620775202827110953444393100550244616077097675214958519\
369714792368658945637722501481119788934335481832006419707699933662776694512590\
395546210324927627340145936977669581382497128041613235221525306156187980147355\
057015941480920142908178590895539873256648475671388894435681827032069643392976\
448136862304239611018520909182631760611648526833654913633399044746263608388930\
554128799832549343580828778853321843614811001580953414216792979755366154099597\
056847380756079204849972655171260172063776358766451130851438632377852869658868\
645233904902364849969810790280999301110144355333809105776115954820074468572804\
031792145733883183053001768549370568232058127507345445593858344672685165947990\
900425986937270085284594869968093394911365245659332578428203166628589434184785\
809170396709143593015847823385389737374876850851013805967360850529417689159603\
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588044643119108667408323915535901022174753131446632152334830294519639576538278\
073626041886902186880122444445170512389669217015386850377298003316988124178301\
626228200342123014927131228482516130787840692601193589569316404456937020847340\
267435797740529498925832082951818732359773716434854272777718338656170935819580\
843500462438037260065768677151784134213165501527585191297587111200959854888573\
648658029313459851776967200611163699960333167940604842771767348294350994691266\
439102864295161500568435858341770239104109158444917775834889516654037575456489\
366040549291217306071060671458173854712796601155533176269739276616259137725384\
468113776971535028834997219904063850296506190769810604957715571810174629907946\
382141246018514396786504697967195450807285528973883380622322385054848002774148\
960670381562241307628180012338763169907673977482366128427650747182545215798618\
536971812436307458543917047948314479571394555076726832157807262567012605044745\
069224229866407440126483547663522513430966733150161195411078811530322755410090\
089655376822136946989547666527006295664675856741126761137705010190683065516028\
501826960474177629985771620816310759530104929555693967188015536314301434244307\
318973634236896227999602862461962769446135861796927595383170780995610456659705\
677940144633131469136760281183846649718537494126028023416533656090198383897466\
755314991222794800443872830197900719670153670479819548925477463003538931672563\
804351082603219478207643596001040768599880350595608527945030234115415602251244\
780201242516332550434867865046796069077885748906304603456364014565242494368884\
341980290529750830401985891808783592642026186362445128365384788718036594754911\
663122406900071826404329943810979277602654944034195429494853570513144984077592\
431031971767795482765029589498558595293669717033468529148367578581258194081235\
286545628820633982432837002943466748186734281398972615581049978959797665960290\
897219113204593316665686697067712460957405656694906604845346975347149146341377\
373166859877364304351455712104737625519342298934426919416389444046202207394640\
624043472972940424647276640289351408025338624393278317433736503142148358814816\
573021352896704785081663439992623235246042479482236630556735165157181624547592\
945852338536350739055696646886725359902582873065531780169825283029888839864353\
571836314013751471414071720118690699258030491260074961566480491148026852361044\
18088396819786811
which agrees to 9998 places. |
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| Dana DeLouis |
| Posted: Sun Dec 17, 2006 8:44 pm |
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Guest
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Some thoughts. Mathematica appears consistent in the last digit due to
rounding.
Here are the last 10 digits when rounded to 1000 digits.
Take[First[RealDigits[N[E^E, 1000]]],{991, 1000}]
{2, 0, 9, 7, 9, 1, 0, 6, 7, 1}
Here is 10000 digits, but we look at the same digits. It appears
consistent.
Take[First[RealDigits[N[E^E, 10000]]],{991, 1001}]
{2, 0, 9, 7, 9, 1, 0, 6, 7, 0, 8}
You say you can calculate E to a large value via Taylor.
I'm not sure how accurate this method is when we know the series isn't too
close to zero.
But...let's try...
s = Normal[Series[x^x, {x, 0, 5}]]
1 + x*Log[x] + (1/2)*x^2*
Log[x]^2 + (1/6)*x^3*
Log[x]^3 + (1/24)*x^4*
Log[x]^4 + (1/120)*x^5*
Log[x]^5
Change x to E
s /. x -> E
1 + E + E^2/2 + E^3/6 + E^4/24 + E^5/120
Another neat way to calculate E is to note the pattern in the continued
Fraction of E.
ContinuedFraction[E, 15]
{2, 1, 2, 1, 1, 4, 1, 1, 6,
1, 1, 8, 1, 1, 10}
--
HTH :>)
Dana DeLouis
"john" <xyz91234@yahoo.com> wrote in message
news:1166371617.054044.216360@n67g2000cwd.googlegroups.com...
Quote: However, one cannot tell how accurate the result is computed from these
methods above.
Are there any fast methods for computing constants like e^e, pi^e,
sin(e), etc. to arbitrary precision to any accuracy?
Thanks
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| Robert Israel |
| Posted: Sun Dec 17, 2006 9:56 pm |
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In article <1166327233.074973.319690@16g2000cwy.googlegroups.com>,
john <xyz91234@yahoo.com> wrote:
Quote: I know that I can compute e quickly by using the Taylor series with the
binary splitting.
I there any fast algorithm for computing e^e to high precision?
e^e = e sum_{k=0}^infty b_k /k!
where is the k'th Bell number. These are integers that can be
computed e.g. using the recurrence
b_{k+1} = sum_{j=0}^k (k choose j) b_j.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada |
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| Alex. Lupas |
| Posted: Sun Dec 17, 2006 11:50 pm |
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john wrote:
Quote: I know that I can compute e quickly by using the Taylor series with the
binary splitting.
I there any fast algorithm for computing e^e to high precision?
Thanks
===========
Hmm ! Nice...
[1] Construct the sequences (A_n) ,(B_n) in following manner
A_{n+1}=(4n+2)*A_{n} + A_{n-1} , n in {1,2,...}, A_0:=1, A_1:=1.
B_{n+1}=(4n+2)*B_{n} + B_{n-1} , n in {1,2,...}, B_0:=0, B_1:=1.
Suppose
C_n:=(A_n +B_n)/(A_n-B_n)
and define C:= C_{1000} .
[2] If the sequences (X_n) ,(Y_n) are generated by
X_{n+1}=(4n+2)*X_{n} +(C^2)* X_{n-1} , n in {1,2,...}, X_0:=1, X_1:=1.
Y_{n+1}=(4n+2)*Y_{n} + (C^2)*B_{n-1} , n in {1,2,...}, Y_0:=0, Y_1:=1.
[C being given at step [1] ] then
e^e =approx= Z_n:=(A_n +C*B_n)/(A_n-C*B_n) |
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| Han de Bruijn |
| Posted: Mon Dec 18, 2006 4:07 am |
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carlos@colorado.edu wrote:
Quote: john wrote:
I know that I can compute e quickly by using the Taylor series with the
binary splitting.
I there any fast algorithm for computing e^e to high precision?
Thanks
With Mathematica 5 on a mac G5
N[E^E,1000]
takes 56 microseconds. N[E^E,10000] takes .132 seconds. Why
do you want to do it yourself?
Yeah. Why should someone want to do mathematics all by himself?
Why play the piano if it's much easier to listen to a CD?
Because it's more fun, perhaps?
Han de Bruijn |
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| Guest |
| Posted: Mon Dec 18, 2006 1:12 pm |
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Robert Israel wrote:
Quote: In article <1166327233.074973.319690@16g2000cwy.googlegroups.com>,
john <xyz91234@yahoo.com> wrote:
I know that I can compute e quickly by using the Taylor series with the
binary splitting.
I there any fast algorithm for computing e^e to high precision?
e^e = e sum_{k=0}^infty b_k /k!
where is the k'th Bell number. These are integers that can be
computed e.g. using the recurrence
b_{k+1} = sum_{j=0}^k (k choose j) b_j.
Maybe a better way:
e^e = product_{j=0}^infty e^{1/j!}
= product_{j=0}^infty sum_{k=0}^infty 1/((j!)^k k!)
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada |
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| Alex. Lupas |
| Posted: Tue Dec 19, 2006 1:21 am |
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Guest
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john wrote:
Quote: I know that I can compute e quickly by using the Taylor series with the
binary splitting.
I there any fast algorithm for computing e^e to high precision?
Thanks
===========
Hmm ! Nice...
[1] Construct the sequences (A_n) ,(B_n) in following manner
A_{n+1}=(4n+2)*A_{n} + A_{n-1} , n in {1,2,...}, A_0:=1, A_1:=1.
B_{n+1}=(4n+2)*B_{n} + B_{n-1} , n in {1,2,...}, B_0:=0, B_1:=1.
Suppose
C_n:=(A_n +B_n)/(A_n-B_n)
and define C:= C_{1000} .
[2] If the sequences (X_n) ,(Y_n) are generated by
X_{n+1}=(4n+2)*X_{n} +(C^2)* X_{n-1} , n in {1,2,...}, X_0:=1, X_1:=1.
Y_{n+1}=(4n+2)*Y_{n} + (C^2)*Y_{n-1} , n in {1,2,...}, Y_0:=0, Y_1:=1.
[C being given at step [1] ] then
e^e =approx= Z_n:=(X_n +C*Y_n)/(X_n-C*Y_n) |
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