On extension to "other" iteration roots

[Leo.W]

Does the P approach generalize to other functions? I will admit I'm still a little confused by it, but it seems to be working, lol. I ask because I don't see anything too specific to tetration, so I wonder if it works in more elaborate scenarios

I constructed another such superfunction, but not tetration case.
The two branches of superfunction is respectively,
\( f(z)=\frac{1}{2}z^3-\frac{2}{3}+\frac{7}{6},f(1)=1,s=f'(1)=\frac{5}{6} \)
(The principle value of the inverse of f
\( f^{-1}(z)=\frac{\sqrt[3]{2}\left(54z+\sqrt{(54z-79)(54z-47)}-63\right)^{2/3}+8}{3\2^{2/3}\sqrt[3]{54z+\sqrt{(54z-79)(54z-47)}-63}} \)
\( T_1(z+1)=f(T_1(z))=\sum_{n=0}^\infty{a_ns^{nz}},T_1(0)=\frac{9}{10} \)
\( T_2(z+1)=f(T_2(z))=\sum_{n=0}^\infty{b_ns^{nz}},T_2(0)=\frac{11}{10} \)
And by desire, the merged W has properties:
\( W(0)=\frac{9}{10},W\(\frac{1}{2}\)=\frac{11}{10},W(z+1)=f(W(z)) \) for almost all z (exempt for branch cuts)
After 300 iterations, we have precision at least of 16 decimal places.
The code to re-generate the function W (Exactly WW[z] in the code, and in the code T_1 and T_2 are denoted as Q2R[z] and Q1R[z]):

Code:

Clear[F, IF, Q1, Q1R, Q2, Q2R, P, W, WW]
F[z_] := 1/2 z^3 - 2/3 z + 7/6
IF[z_] := (
8 + 2^(1/3) (-63 + 54 z + Sqrt[(-79 + 54 z) (-47 + 54 z)])^(2/3))/(
3 2^(2/3) (-63 + 54 z + Sqrt[(-79 + 54 z) (-47 + 54 z)])^(1/3))
Q1[z_] :=
1 + 0.968621246264228916828157107449816182810930152921598222456445184\
6317052858231756541025631069776587540140548247332695498395833122330823\
4272992372578870632056986044418673843192629365959691789962876636013911\
9672626746884626283235792846770330683663907460652076768415141073833379\
5221864146698732518680321691976482108880400733467086459194066503147266\
9404376219000713887471047320499280878494772061225536595158481845766136\
7449503962585748616060147318870465771699347138608271493736016436874902\
0204204496407952806463212456744218066`498.23825778806594 (5/6)^z -
 10.13285288211625441473613269579116819828718117483820559857168873012\
7880442709475572161528278148972584181843784004829542195211470795067419\
0845128181484275681896818128460975068666636426412402676052897000769869\
1713511238080702292603616582402640165414468242772068530324128021226217\
7414790261162056980559076872426969238020735751516190590549062660276766\
4830756105234567804698480101863139134208585445441752018847752948328289\
8141171038015657244654178145656937734806854169891724477419061149923670\
64370579190055860612015600770554814`497.937227792402 (5/6)^(2 z) +
 113.8528004078953504299141926584404613245787115564098799459102726327\
3433554416881134638844152927838726924671490515780515670488529668951104\
3288590807237635075890793379149533948617878026170193350534537346898516\
4706534499773185212539505819111403353787382136707571225382442638350280\
2433165483823880417197100170366824341574028302880726451990317061429424\
7584918245667256808294163038404409880047520878271792011948045658798239\
5742089634493526508730509267526848349738411089337832586905873707666852\
630497535452060391334596748013957358`497.7611365333463 (5/6)^(3 z) -
 1340.412665344688793625764210212287659913617374109791585062488308588\
0109959185908642827932629644112671766501447494455210361860599112319911\
8292328980784571699071876618701085472499425314887201332284232448571217\
0290164495011956989003072381948436967916516377262314087722378425734480\
3140890146339433795028894816291511815576893823329510029451958845814145\
9420363242584019355576085928847806420312558521102738070546547816202932\
6641991392128740789880448570630041820654569293519304563832507727902130\
7929622258736793103983421817052668021`497.636197796738 (5/6)^(4 z) +
 16332.16934201566219201445325015112444889143632069558140596988507307\
8360354412614433004716011226188266673564133950065607795088193294101862\
9560418148870289893054102258819430628723682318219063268381565991273153\
9746433465569765254890669021430594822119351110593599704506195127855324\
4885378854847135943245597726229940408969206674465028290118981619991442\
0313161377277757840462936554687889360855742232779257823318976435201598\
6368103450696127401004676898436060749373424952905065254623751661896886\
6590447027613444167982651573365734839`497.53928778372995 (5/6)^(
  5 z) - 204420.\
4710939121023052257209070680893474864297422634522436870603425920672701\
5333460947938617710467446891693415431864647623977196168131601429706154\
5946302324801154199135605751218250315370385659806608798270273340826726\
8563714677765538477613562493354645222038546680284845587732429134582930\
3690183637762541566942225188058364939924801648082051823183235764223545\
5089738618617137265237399258871224740860998690509989399538456545474696\
9332317910571511279351675497188572477912202110526772037859840311385348\
883473810720477715015933813381`497.46010653768235 (5/6)^(6 z) +
 2.615238614486723195902289949592057843881232086577313063220924906579\
2976880146916542384480275442573493568316832777232926085968221904690047\
7722514831597460864974136167117732387784296196997387741662127640910081\
4473622346717755679842492471274135973280177946287768569521047059546394\
4671356468649928072176675226529591422584174607545972170766670486722801\
9701703877778415452238431259074904507782020216976170750147869365273736\
2209412394897460301915902091071675204521048367216538021616766622901618\
293512656981792596196`497.3931597480517*^6 (5/6)^(7 z) -
 3.407579630616294442646269433398524273803119461819435576104452408383\
3629885505767177150797464592798639336017767042113713041063785508118345\
1888227510228239376837669769042419342015181302880542314145403182855485\
2121784788792286841713907983496457920237979156884202265691731201289984\
9849745092432097481062763694478058502513309490262003077422378237322969\
8002146498332959347758322070096851677948960905074312636632536743406921\
1897599492792942412157901787477305771519676763191278380082009415667191\
5288811675207430176045`497.335167801074*^7 (5/6)^(8 z) +
 4.509747031285701927330969165474744943895526451685751874702732831246\
2985928401296984357557460817105504785072384898214555472439700327720516\
3113183920549686038609331448268987082434987180842624180096244478267707\
0411884713011812023256467750455105551357870461098767117255160284867732\
9282728944743349720272798460164112408055126713121988637994358129225746\
9999055870679251961767029930409453692694054519890475005232751267460383\
5469098688169487971374743709488201084641395153161831716275838406115974\
4855801151095753638`497.2840152786266*^8 (5/6)^(9 z) -
 6.049363002435427953429614004535592243219903703038967774131641195535\
7121498661723994635812644429289783379102246955515671996055712034658287\
0603970682195747271483031236546843616665972597163826109185629666344956\
0042818403148194228908836960641752236398099859456606579021133705677533\
4194748006524549404273635450876743844476475471152014577464888711590724\
7963162605614568517566454046775379799274563227631357098475209468087454\
7571506522686389367872409180863067125740579685406551700888564845731645\
192390608050614669638484`497.23825778806594*^9 (5/6)^(10 z)
Q1R[z_] :=
Nest[IF, Q1[SetPrecision[z + 100 - 4.52002161879124792`200, 500]],
 100]
Q2[z_] :=
1 - 0.007483468519172668962097768785918606108243474403709991202236146\
7254242278676264275564197726302889421453903847629129285141959982049878\
7855990740119804196278716913220182718671355968066791941829314144577645\
0810417239173036048094854113602569355189707273265643770260744695846705\
7012856011146023991794702575844599697174777932292110766853168033446325\
8832750361816769005115572955762356306819928015670728768266409743195697\
4423545471530569700934081393306787718218891143600913337275143788557386\
732874898750646181411834709792088254`496.2133569089641 (5/6)^z -
 0.000604824851636442491527770426607651912746428235587134749769286904\
2174195267006775598526981745050098198523579706758028523876456833914186\
6049856598113317555711556827623489476018541548369963990189840238196202\
0644484166200962523250088711831170467093725153148897701782586871059158\
2349342181788752077233585938014313274786048828411029360120977365147814\
5300699203872713242724709130380347382289912853739067391608103827802011\
8640638275932489196172730485220790573337544145946328123955729833336267\
525773691383768358250043248877094`495.91232691330015 (5/6)^(2 z) -
 0.000052503777747281330975465950347362934968599076126088282766389371\
6738939648769522138330055584534190388328512102110561670389931553639182\
5890738450517461323286029974234338650959482743561372037586911960527577\
7613711294571810361750880026951947344087893602355234365352467710173594\
5646582428968404370420612988982157258741616096612860947431171560571518\
2921499957525549214941117907705961663165561571692849170142493064508353\
2940456098414039961929844478896256159725743704162229476817793852376784\
9927454467552954203938764747042054`495.73623565424447 (5/6)^(3 z) -
 4.775669935984275493259952865444098290642055129268539207522335188974\
2584017079496660903392603462894199727880146061149503307254169624855572\
8792162564016265783400695841878119452040423464726213053494255628752007\
2228435525533875828185542095225490723794110013630847688487501081194205\
7380598676323225914995472085836918122795203261753649681742500300032934\
5442616339234878233640972614432065838809554518567180505388290177102975\
0289908231129213391563814673676082958715888737996203072816352384494298\
7992388607929521525162376451`495.61129691763614*^-6 (5/6)^(4 z) -
 4.495609706206277316817800311926282613579963937903270958637727214178\
5895135263254842261491428310527539534730016914702011484317110713453255\
5962408703344693757117197713601909767376108361461200695206504763520797\
4146803474022325766247941490507250461892727997975296840847231712425626\
9857239104647759211445055161901273981426015520674207312178251202957649\
7363462388981177135783347697260223584445548084051961072923192847782479\
9582530324792994136730125651678363324039631563945979035389667876093675\
12233896504978345189938783`495.51438690462805*^-7 (5/6)^(5 z) -
 4.347284431468142125814499950384196763606925619050572243722937658457\
2924562230091273699791621381819470269301888254924936680067426458886568\
4392200549833856275807291231842675435757565533843865875942855276124748\
1858750446957945757513109596356338381139214497132221394482016038300181\
1612787284437882399935667431115327725712539188183806496739789012597749\
2982593850670251878340245674343511594721371232157918785809454088789926\
3919656677029968803334952412128688972034794844950233298809604072938781\
40451781323317345889609799`495.4352056585805*^-8 (5/6)^(6 z) -
 4.296887035643382544229366555340055818601484438320641412945083276936\
0227443975078354064645764953063345469169060947449092391579795832783203\
9977911091951164215554965293313333534401548216320308650384318172552748\
0871504574761166108219983541424890002443584170843892874117938750703281\
1317630463904591623058034825612778197820800891047193509671489360071315\
4152791987144320367131744273274642726481556919085294838115210183973751\
5204333042501012356344051755446254358083068279665444316094532968758661\
8268372127093912196464477`495.36825886894985*^-9 (5/6)^(7 z) -
 4.325512535378080240484266943590981056163380317867063488202058432830\
0341039773837821197861902690444982888937883225989308514607463046271293\
2862291552856061380471006838253712049232099483862829359304855578029336\
7998875869054828134147320688212803626637352057006223513022661718482659\
6687789915312246956282263463914524094208862587056347421161518699096340\
0105753885070804099505732312878143132823485601678700645056383377954749\
7375816083919676014149353678610401173108408019057221483107860982112775\
465667134231912724662952`495.3102669219722*^-10 (5/6)^(8 z) -
 4.422753022434820172820944147923376485475726722248784130479977546392\
0321148340659456770829354393271014965067539186043381568708262494250369\
8376455530857151483054481192531137880178961382321206096711852957124125\
3975003813329791829680807528001172101024488683318660472723983094822543\
5920768077615932418323062720324689336163684355635338418256294064904969\
3202577800566035217198225615471719484620207542019071330623140317252564\
1185834248393142980099185794271013972694119297171932054849136030294005\
42045032528306947331551`495.2591143995248*^-11 (5/6)^(9 z) -
 4.583519651221088821794164273590697295912606900207761220905566863907\
3032316124867768794353818389461602079867760129872257010581052424324598\
1052354388233149864525780729542438471965761136549239821690378624610847\
0451061230801872858459033472483160376860427232943572212378044665945037\
9928604882020118783599242373575505958353043828331739800324423581928573\
5782366550644827078885989855100181633597130371096878984793976213361969\
5404725601981471603858073263030615413804515823829206124986168921194235\
2580785734206250556283`495.2133569089641*^-12 (5/6)^(10 z)
Q2R[z_] :=
Nest[IF, Q2[SetPrecision[z + 100 - 10.00736825289550022`500, 500]],
 100]
P[z_] := 1/2 + Cos[Pi z]/2
W[z_] := P[2 z] Q2R[z] + P[2 z + 1] Q1R[z - 1/2]
WW[z_] := Nest[IF, W[SetPrecision[z + 300, 500]], 300]

A value table, and a plot

Code:

{0.0000000000, 0.900000000},
 {0.0200000000, 0.937797663},
 {0.0400000000, 0.991699085},
 {0.0600000000, 1.028002075},
 {0.0800000000, 1.050320192},
 {0.1000000000, 1.064409767},
 {0.1200000000, 1.073715150},
 {0.1400000000, 1.080130624},
 {0.1600000000, 1.084719694},
 {0.1800000000, 1.088104600},
 {0.2000000000, 1.090665299},
 {0.2200000000, 1.092643100},
 {0.2400000000, 1.094196660},
 {0.2600000000, 1.095433493},
 {0.2800000000, 1.096428377},
 {0.3000000000, 1.097234500},
 {0.3200000000, 1.097890401},
 {0.3400000000, 1.098424443},
 {0.3600000000, 1.098857738},
 {0.3800000000, 1.099206120},
 {0.4000000000, 1.099481488},
 {0.4200000000, 1.099692733},
 {0.4400000000, 1.099846383},
 {0.4600000000, 1.099947051},
 {0.4800000000, 1.099997733},
 {0.5000000000, 1.100000000},
 {0.5200000000, 1.099954096},
 {0.5400000000, 1.099858963},
 {0.5600000000, 1.099712184},
 {0.5800000000, 1.099509845},
 {0.6000000000, 1.099246311},
 {0.6200000000, 1.098913871},
 {0.6400000000, 1.098502237},
 {0.6600000000, 1.097997820},
 {0.6800000000, 1.097382703},
 {0.7000000000, 1.096633169},
 {0.7200000000, 1.095717553},
 {0.7400000000, 1.094593085},
 {0.7600000000, 1.093201126},
 {0.7800000000, 1.091459852},
 {0.8000000000, 1.089252721},
 {0.8200000000, 1.086409831},
 {0.8400000000, 1.082676944},
 {0.8600000000, 1.077662642},
 {0.8800000000, 1.070746332},
 {0.9000000000, 1.060917891},
 {0.9200000000, 1.046515219},
 {0.9400000000, 1.024924076},
 {0.9600000000, 0.993077953},
 {0.9800000000, 0.953325670},
 {1.0000000000, 0.931166667},
 {1.0200000000, 0.953848081},
 {1.0400000000, 0.993185642},
 {1.0600000000, 1.024522215},
 {1.0800000000, 1.045795384},
 {1.1000000000, 1.060031339},
 {1.1200000000, 1.069780458},
 {1.1400000000, 1.076664151},
 {1.1600000000, 1.081669920},
 {1.1800000000, 1.085406083},
 {1.2000000000, 1.088257354},
 {1.2200000000, 1.090474266},
 {1.2400000000, 1.092224637},
 {1.2600000000, 1.093623820},
 {1.2800000000, 1.094752946},
 {1.3000000000, 1.095670226},
 {1.3200000000, 1.096418149},
 {1.3400000000, 1.097028162},
 {1.3600000000, 1.097523788},
 {1.3800000000, 1.097922735},
 {1.4000000000, 1.098238353},
 {1.4200000000, 1.098480644},
 {1.4400000000, 1.098656969},
 {1.4600000000, 1.098772535},
 {1.4800000000, 1.098830730},
 {1.5000000000, 1.098833333},
 {1.5200000000, 1.098780624},
 {1.5400000000, 1.098671409},
 {1.5600000000, 1.098502961},
 {1.5800000000, 1.098270868},
 {1.6000000000, 1.097968784},
 {1.6200000000, 1.097588041},
 {1.6400000000, 1.097117102},
 {1.6600000000, 1.096540773},
 {1.6800000000, 1.095839098},
 {1.7000000000, 1.094985773},
 {1.7200000000, 1.093945877},
 {1.7400000000, 1.092672551},
 {1.7600000000, 1.091102073},
 {1.7800000000, 1.089146427},
 {1.8000000000, 1.086681836},
 {1.8200000000, 1.083530778},
 {1.8400000000, 1.079433237},
 {1.8600000000, 1.074000308},
 {1.8800000000, 1.066639886},
 {1.9000000000, 1.056444426},
 {1.9200000000, 1.042058503},
 {1.9400000000, 1.021709619},
 {1.9600000000, 0.994303334},
 {1.9800000000, 0.964321625},
 {2.0000000000, 0.949582863}

Regards
Leo