Feigenbaum constants to 1018 decimal places From: David Broadhurst 22-Mar-1999 To: Simon Plouffe CC: Keith Briggs, David Bailey, Steven Finch Subj: Feigenbaum constants to 1018 decimal places Keith Briggs' thesis gives values of alpha and delta to 576 decimal places. I found that his values are good to 346 and 344 places, respectively. The 1018 places below took 3 days, using 400MB on a 433MHz DECalpha, with a collocation of N=700 points, checked in 3.5 days at N=720. The Feigenbaum zero nearest to the origin was located to 1018 places. The nearest complex singularity was located to 868 places. Let g(x) and f(x) be even functions, with g(0)=f(0)=1 g(alpha*x)/alpha = g(g(x)) delta*f(alpha*x)/alpha = g'(g(x))*f(x) + f(g(x)) and delta as large as possible. Let {b,c,d} be positive numbers with g(b) = 0 = 1/g(c+i*d) and {b,c^2+d^2} as small as possible. Then alpha = -2. 5029078750958928222839028732182157863812713767271499773361920567 7923546317959020670329964974643383412959523186999585472394218237 7785445179272863314993372578112163594879503744781260997380598671 2397117373289276654044010306698313834600094139322364490657889951 2205843172507873377463087853424285351988587500042358246918740820 4281700901714823051821621619413199856066129382742649709844084470 1008054549677936760888126446406885181552709324007542506497157047 0475419932831783645332562415378693957125097066387979492654623137 6745918909813116752434221110130913127837160951158341230841503716 4997020224681219644081216686527458043026245782561067150138521821 6449532543349873487413352795815351016583605455763513276501810781 1948369459574850237398235452625632779475397269902012891516645793 9420198920248803394051699686551494477396533876979741232354061781 9896112494095990353128997733611849847377946108428833293833903950 9008914086351525626803381414669279913310743349705143545201344643 4264752001621384610729922641994332772918977769053802596851... delta = 4. 6692016091029906718532038204662016172581855774757686327456513430 0413433021131473713868974402394801381716598485518981513440862714 2027932522312442988890890859944935463236713411532481714219947455 6443658237932020095610583305754586176522220703854106467494942849 8145339172620056875566595233987560382563722564800409510712838906 1184470277585428541980111344017500242858538249833571552205223608 7250291678860362674527213399057131606875345083433934446103706309 4520191158769724322735898389037949462572512890979489867683346116 2688911656312347446057517953912204556247280709520219819909455858 1946136877445617396074115614074243754435499204869180982648652368 4387027996490173977934251347238087371362116018601281861020563818 1835409759847796417390032893617143215987824078977661439139576403 7760537119096932066998361984288981837003229412030210655743295550 3888458497370347275321219257069584140746618419819610061296401614 8771294441590140546794180019813325337859249336588307045999993837 5411726563553016862529032210862320550634510679399023341675... b = 0. 8323672369053164248490954462761044873784441510122511788018155377 9065114906172129921057496732507284896621794930858102189657909680 4959852291885861031738567513022623838312267142068518371187536349 1362035488204184582129489452800819510186963028258245294829711561 6292824705463696832007591526438387085870831936282420525992402697 3384608796419471351612698550341184443568618522876935047119761407 0499032433715378097841540314810087444324620581291322418748660093 2314842875181083628857143398269714251643290364438311172364921155 3432680016994298334304882138907259077911901592542340558036027211 4331844081840097934443345396018870714295750813278704117786464474 7099973387704004547129784444130409583579353429202642466627851163 3627652670725874364374403158535682677574519438277301740193741430 7912783836782859060956573909121951885375872115688323172785750683 7793827442294917455211250531331055106769424359756699521206009907 1908153946422594798482709706553583230746693176473981658506006584 7908039033441468866332261531378185904579907945405208836635... c = 1. 8312589849371314853421733978737816274412980903762379654774116464 7029628921518197466402695607431535931144675465676436776429991400 4666839100227569854447901401797416335863091512720997578390349533 2439795089089784388681012689908386206416711155926761527416018736 0898611713937017657422694088420338223695726664110997002430630745 7841960320446859147463063222130803898292617627948782384046850325 1944825401523679859659042762573329854239726864266992525400752698 0124640694256379140667623449103151146911291523321219944431825153 5493739570363881422570037221976986086042711311153795151520363497 1729459021044810405216758657973411661272874798290488341630337687 3341600697692943707912423921767074922481749634761867923871322423 8340981818145589131843412282926067965686630680002582442594767840 8371051279370262976429742726741473713533085787371144882449379816 288696672329596345137011738421245934... d = 2. 6831509004740718014499367586451482510386109205544354930936808684 2363044614290970217463629831538000035030339422550939048584211734 1138285347697276541097506616635246416832706746987691173068249055 6475241058846697132122168540788487314014805843580000406240806547 6191136626537334807666964715712440298704793780799957391521213073 3906516425221391841437351786903110743008572892901968458948759369 5673030577920734878406376454729255128864988186843811804643753211 3493050906206882192881702053594325185737303628120652857311749388 7076639229215868147954999487228917424186884812129648281543112579 4183720113976530733956992276836976718845735691313392850631440703 5017310790186846528690429234645420586978036796023986009971346119 5809839747144807686565166826236673574813851453412295336540638189 2334481762687754607251768929730673131412568009781912415169247189 487376763709719447715817358677632441... From: David Broadhurst 23-Mar-1999 To: Simon Plouffe CC: Keith Briggs, David Bailey, Steven Finch Subj: a 6th number PS: Let kappa be the order of the nearest singularity, with 1/g(c+i*d+z) = O(z^kappa) as z tends to zero. Then kappa = 1. 3554618047064087438634415359109709760367902710479931781617936681 3305881904302037128542124779889011575139648204842431398255153070 4598291257581907259042933385835455760492828646543341223734761311 3165734968192554683667914426935420848360809901275260696101461009 7604161091390036711370883350056179171708950441993592615164877487 2115920378795443873022005751165194076762612333789043750809684157 1580344173874824201726443759630518658616830558599958236051647773 6143754100887759157095778135218446563090645557927053564771153813 9030795677803655297311753938553555312440939529735326008123690374 1193550335118053205154820062292401345695295087659103653855843543 6736041821349634238108888145341385003155415317397509362822987280 4081543618458462897786717141793460171099257829045823147578244452 4715640398842764642211990339064314200753889117228668205544605239 291441546074631085975481673700466507... # This is the electronic signature for Plouffe's Inverter # # Ceci est la signature électronique pour l'Inverseur de Plouffe # # Copyright : Simon Plouffe/Plouffe's Inverter (c) 1986. # # http://www.lacim.uqam.ca/pi #