02/25/2012, 06:35 AM
(This post was last modified: 02/25/2012, 04:35 PM by sheldonison.)

I'm in the process of updating Mike's merged fixed point code to work for a generic base in the complex plane. But first I needed a way to figure out both primary fixed points for a generic complex tetration base, and then to figure out if the base was inside or outside the Shell Thron region, which tells whether or not one of the fixed points is attracting. The formula to generate the base from the fixed point, is pretty well known, but what about the function's inverse, if you don't have access to the Lambert W function? What if you want to get the fixed point from the base, using something other than brute force? For this Taylor series formula, for a base b, z is the input to the Taylor series.

. In this formula, z=0 corresponds to base . Then the two primary fixed points are calculated by putting both the positive and negative square roots in the taylor series, which is accurate to >32 decimal digits for 1.05<=b<=25 or so, and gives the equation for the two fixed points.

Since I don't have access to the Lambert W function, (it is not a part of pari-gp), I think this Taylor series is very helpful. This Taylor series equation was inspired by the observation that you need to loop around , twice to return to your starting point, where the fixed point is slowly modified as you loop around . The log(log(b)) part of the equation was to maximize the radius of convergence, since there is a singularity the fixed point for b=1. The log(log(b)) moves the singularity at b=1 to infinity, so that helps this taylor series converge over a very large range of bases, and probably for any real base>1. I haven't had time to figure out the total range of convergence yet, but it is surprisingly large, and includes negative bases. For any base b, with abs(b)<4000, except in the immediate vicinity of b=1, and b=0, the Taylor series gives double precision accurate results for the fixed point.

- Sheldon

. In this formula, z=0 corresponds to base . Then the two primary fixed points are calculated by putting both the positive and negative square roots in the taylor series, which is accurate to >32 decimal digits for 1.05<=b<=25 or so, and gives the equation for the two fixed points.

Since I don't have access to the Lambert W function, (it is not a part of pari-gp), I think this Taylor series is very helpful. This Taylor series equation was inspired by the observation that you need to loop around , twice to return to your starting point, where the fixed point is slowly modified as you loop around . The log(log(b)) part of the equation was to maximize the radius of convergence, since there is a singularity the fixed point for b=1. The log(log(b)) moves the singularity at b=1 to infinity, so that helps this taylor series converge over a very large range of bases, and probably for any real base>1. I haven't had time to figure out the total range of convergence yet, but it is surprisingly large, and includes negative bases. For any base b, with abs(b)<4000, except in the immediate vicinity of b=1, and b=0, the Taylor series gives double precision accurate results for the fixed point.

- Sheldon

Code:

`a0= 2.7182818284590452353602874713526624977572470937000`

a1= 3.8442310281591168248636716374262768779881984009975

a2= 4.5304697140984087256004791189211041629287451561666

a3= 4.0577994186124010929116533950610700378764316454973

a4= 3.1310579579657891414705533466321408770463105412618

a5= 2.1392433777070640849472839389751781515471363324069

a6= 1.3201485117519207930224182716545764193987429888921

a7= 0.75026349567941987403067580894396345224122700450743

a8= 0.39664914883721664871255223561747527187781503373434

a9= 0.19675048132092445683854734270973946155931655263755

a10= 0.092284535541707553042271049603653939199684877326343

a11= 0.041128857617644107022183948771661806665481904754276

a12= 0.017499926566873175768104436013779089511797807188588

a13= 0.0071383045088424740927613278903873113685388260534036

a14= 0.0027990895495591130046356892963041542988072136991061

a15= 0.0010584397247571801837924827314069442927340627561410

a16= 0.00038690511378324604701554539024015099607150307841871

a17= 0.00013695921477906978645151135090639314281168212780761

a18= 0.000047060925749355490378155571865461380815063050655184

a19= 0.000015718459720078638139982218287502395265847645167202

a20= 0.0000051098833189178642859504435914542633052729886905741

a21= 0.0000016200756562979959213494113078792872671206842607515

a22= 0.00000050114395897125416867359285070262528766885907119212

a23= 0.00000015147122619645160295342350422242820377511300274821

a24= 0.000000044807348486894284474811959719674830325499646189215

a25= 0.000000012962878987582433364248157574392737421319847483067

a26= 0.0000000036777490474599150634033635884535264575305342994411

a27= 0.0000000010235317972261857886939462803228104812154269796464

a28= 0.00000000027892975104614435108979535736539187977244285193364

a29= 7.4913196693434785382673670840314908951403127366391 E-11

a30= 1.9717541191970651018708628895408641932956867632982 E-11

a31= 5.0915368080421762446029435219708682084557437930980 E-12

a32= 1.3066486276084097356358660994233585807052473233683 E-12

a33= 3.2467635216891226343648988598478613134554037958645 E-13

a34= 8.0328830300969462955204663241426866159060111914746 E-14

a35= 1.9884030816356147519332706021476848341643270874558 E-14

a36= 4.5604212355453915576497473836641686140815097764966 E-15

a37= 1.1375475027302582619728186577908772014830927317863 E-15

a38= 2.6388661035640136079598667240173987557818441629236 E-16

a39= 5.2684658313198763292443793305904997392068250988020 E-17

a40= 1.6456895069992339719420263709083815254122876574657 E-17

a41= 2.4709091962079383454024177450388535492756309345447 E-18

a42= 5.3707283652335457424977440370257556179749838766151 E-19

a43= 3.0349615536019343269887649835177688859124766896609 E-19

a44= -3.2787954527686945286108924002948599290667213646190 E-20

a45= 1.8345013123066661922492136389266941966903253270420 E-20

a46= 4.8481796161714217259736691210135868007908205669597 E-21

a47= -3.2824665245176595846850267392135417222393135087407 E-21

a48= 1.4956645179800024112898138808465573719639979782662 E-21

a49= -2.2810205766236867792442627645380203056636207166072 E-22

a50= -8.2025371766299993417377211085132693281802664744834 E-23

a51= 8.2820898235652527449736419417013483792769309534803 E-23

a52= -3.1987649031517273110151312741635882659324832087531 E-23

a53= 5.1028759500570603817123734231141374241956872771981 E-24

a54= 2.0745436935733458976804542834312653952415531741435 E-24

a55= -1.8974009421475881728609023489783737853747016759452 E-24

a56= 7.2399705291174268127829280413850875632584902836527 E-25

a57= -1.0843673887979929422153305212584762102272054654005 E-25

a58= -5.0277375699737374239824380947818496605390245813200 E-26

a59= 4.3954005288785835431790225413101260484756443385679 E-26

a60= -1.6441622742940879676195309996736139807832614792602 E-26

a61= 2.3377978549045549610283334095071378953325737820424 E-27

a62= 1.2153823729634690431272005530504305362371379491894 E-27

a63= -1.0217831791605172358442907077539785681280031170764 E-27

a64= 3.7583342703760100931146730323471004826516979556089 E-28

a65= -5.0842172513431152493662523617793691855778116872791 E-29

a66= -2.9298747678164593371749001842727339027440198115827 E-29

a67= 2.3843736042453943922739674466720939816392076010523 E-29

a68= -8.6404128743033872572631375344081227432768174762623 E-30

a69= 1.1147745631909792790933165843951598884838146816889 E-30

a70= 7.0532998709238197459676161152886849409067455445289 E-31

a71= -5.5840347009750660670277203195942046868244697211916 E-31

a72= 1.9968202962515529653768275377958257984435847587123 E-31

a73= -2.4624153585967391507506022808723761030685594008346 E-32

a74= -1.6971043513573951058377996551655952956127911075482 E-32

a75= 1.3121636252191637725547321104357974814481747681435 E-32

a76= -4.6366491360531875428354175436972903880388681629591 E-33

a77= 5.4759261055169523922054804602053975292206196019485 E-34

a78= 4.0835907618102687341427597190193171384178018179741 E-34

a79= -3.0931489399194580627859122578595983870151745833582 E-34

a80= 1.0813047412434062333080300695597773394885783093044 E-34

a81= -1.2252362103483261754655465170167431865554218135280 E-35

a82= -9.8300622452978664179128765338170742574844836034023 E-36

a83= 7.3130149232712161576150411896903892349851012529721 E-36

a84= -2.5316784946273245634403427410920911852929637185860 E-36

a85= 2.7569030159398000068528639302431024861323649994624 E-37

a86= 2.3678752475291946660830153577567179420330740924808 E-37

a87= -1.7337617902358690948312875490328268568214283564291 E-37

a88= 5.9489649580056122715011899267403080309530471951160 E-38

a89= -6.2353624456438702836611492343755661445986256939609 E-39

a90= -5.7084806755390469129435066349248018249886305456145 E-39

a91= 4.1209734225367270686731246993915008297939802940460 E-39

a92= -1.4025498788164094982161672209129834192788136568375 E-39

a93= 1.4169741817716908448099015810155102528257145161939 E-40

a94= 1.3774829427232759282536454610089933691524288647861 E-40

a95= -9.8186882484920920105275699610740814969572189953598 E-41

a96= 3.3168418862324804650808696747831173926792793354185 E-41

a97= -3.2341718090336638957257547112586099667893533476259 E-42

a98= -3.3272289382182763758362704190111486565549876399064 E-42

a99= 2.3446712211671867041330301250221977724670989098562 E-42

a100= -7.8660813856791455405739446872012737404587511037516 E-43

a101= 7.4117874092723636167895499876920034206925805479953 E-44

a102= 8.0449630071992383377823253016230949067068370627592 E-44

a103= -5.6107673309024071413165294113175917244659515740451 E-44

a104= 1.8703704290344439531751733113502708390740673057030 E-44

a105= -1.7049595279518816756078819029211146676550591619926 E-45

a106= -1.9472226441848835239296603130698560902214718601605 E-45

a107= 1.3452918010782273565438586849286267873298215727367 E-45

a108= -4.4581019582286464605966679709280753360940853885077 E-46

a109= 3.9356988280727884495539422587080135663067916265244 E-47

a110= 4.7179835992286895246829227989155589972139121713110 E-47

a111= -3.2315545147777352407953961572361820900127456511476 E-47

a112= 1.0650043716082824895276880145990182899014721900301 E-47

a113= -9.1147127786496282723800711785279226816331317383552 E-49

a114= -1.1443035082690186137554650391441932946474634683831 E-48

a115= 7.7760383631901205864928578474869898677279380698392 E-49

a116= -2.5495578122471133824914480360868584888184556180062 E-49

a117= 2.1173080587506716879335848811562106778592441836394 E-50

a118= 2.7781889012354101797726076421387624610649014703232 E-50

a119= -1.8741844847909908017849435440876945806324719777844 E-50

a120= 6.1154697933929833224163136341333661094234392463681 E-51

a121= -4.9324234160781771043738905360427488911054147505257 E-52

a122= -6.7516170974169804169671782560329059979919324533539 E-52

a123= 4.5240983802350759934936686549613478193076549996979 E-52

a124= -1.4695734330098409037700447100436584191843150894889 E-52

a125= 1.1521125191958365706864738454301151565496232238249 E-53

a126= 1.6423578046256356048202813386114874500427364236761 E-53

a127= -1.0936528703475943968388872126503033575039972975248 E-53

a128= 3.5375228294839848475909167953331987436497172454718 E-54

a129= -2.6978482304943505310735652292701489001943829219215 E-55

a130= -3.9987970359659040999742140405232720136297398130454 E-55

a131= 2.6474006812280438194473769206724269657102547304092 E-55

a132= -8.5291961019295574135442472443611745856312183189474 E-56

a133= 6.3323381046099492231654979714328684659496610032336 E-57

a134= 9.7449542393413625948030755094469137335603889989868 E-57

a135= -6.4168257129444529858674454277838539098605318969812 E-57

a136= 2.0595681078339135332218823360988545914331945409098 E-57

a137= -1.4896215397166610321145528593568864824078893656240 E-58

a138= -2.3768743298521482002068230707322255247633783728679 E-58

a139= 1.5572247840096992668430577433971873564773219712346 E-58