tetration limit ??

[sheldonison]

Oh, you mean we have an upper fixed point of the tetrational for and the fixed point can then be computed by or . Ya, interesting. I dont know whether we even have a thread on the forum that dealt with the topic of the fixed point of tetrationals.
Of course there maybe always the dependency of the values from the chosen method of tetration.

slog_b (x) = x <=> b ^^ x = x => b ^^^ oo = x
yeah!

to remember:
one - 1^oo = 1
Euler - (e^(1/e)) ^^ oo = e

now new fixed point
(1.63532...) ^^^ oo ~= (3.08855...) what is this new result? it's a Super-Euler?

So how big is the difference between both methods, with respect to the computed fixed point?

to remember - you follow "regular slog" - http://en.wikipedia.org/wiki/Talk:Tetrat...on_methods lol

I calculated Nuninho's constant, to 32 decimal digits of precision using my latest kneser.gp program. For bases around 1.6, it takes about 70 seconds to generate sexp accurate to 32 decimal digits. Then we generate the Taylor series, centered around 3.0. Then we generate the Taylor series for sexp'(x), centered around 3.0. Calculate when sexp'(x)=1. Now we have the 'x' value for the local minimum of sexp(x)-x. For that x, calculate sexp(x)-x. If sexp(x)-x>0, then the current base is bigger than Nuinho's constant; if sexp(x)-x<0, then the current base is smaller than Nuinho's constant. Do a binary search .... Here's the result, accurate to 32 decimal digits.

Code:

base                    =  1.6353244967152763993453446183062
upfixed                 =  3.0885322718067176544821807826411
sexp'(upfixed)          =  1.0000000000000000000000000000000
sexp(upfixed)-upfixed   = -1.1371391135491644200632572659231 E-32
lowfixed                = -1.6408725757165933485612321510790
sexp(lowfixed)-lowfixed =  0.0000000000000000000000000000000
sexp'(lowfixed)         =  4.8060057543017516963843938970331

Here is a graph, showing sexp(x), and the line f(x)=x. The two graphs intersect each other at the lower fixed point, and at the upper fixed point. At the lower fixed point, the slope>1, so regular iteration is well defined. The slope at the upper fixed point=1, so this is a parabolic fixed point, much like eta.
- Sheldon
   

sexp taylor series, centered at 0, accurate to 32 digits

Code:

base 1.6353244967152763993453446183062
1.0000000000000000000000000000000
0.72816708264487902100067489564252
-0.14877354513094993489726432504500
0.074737145040443592555524996635139
-0.027303989699078802016765079880084
0.011995470121761344894107671889772
-0.0050360379041442966393132619914175
0.0022119303838134475052348019378829
-0.00097450334130330465540560312442488
0.00043642402109641675815655290883749
-0.00019709011821224654361547677039639
0.000089833086159903740469398544720969
-0.000041238706945915151891612184659684
0.000019053968532911172713720919436722
-0.0000088524992039690172128294516235260
0.0000041330272150278532917019791147867
-0.0000019379184778773538437457974949642
0.00000091213532301790212743743497422467
-0.00000043078402804966343713131475292944
0.00000020407220630397996832775388483792
-0.000000096939468490135916393282597246732
0.000000046163269001536943339028294028001
-0.000000022032976477566128111162386532553
0.000000010537670071263114939762057653181
-0.0000000050493505784723879328428158902764
0.0000000024237041954057779998388895184587
-0.0000000011652474462755328727057729290140
0.00000000056104666879689709891062841049440
-0.00000000027050515362351951023983514685701
1.3058885732387283938099128380771 E-10
-6.3117999575327010732122436268733 E-11
3.0540984120511190577663377256805 E-11
-1.4793294493258211341650723010964 E-11
7.1725081233265123525479755443758 E-12
-3.4807765474727949055646060967838 E-12
1.6906630700960019560537073128474 E-12
-8.2185016001496131023322444609729 E-13
3.9981901495825875858709893398879 E-13
-1.9464873683152398084034510942555 E-13
9.4828873682058897543982090883773 E-14
-4.6229076531095863895666660688233 E-14
2.2550769237060314009782522430640 E-14
-1.1006923143874751666653232160994 E-14
5.3754741141976101679195739564343 E-15
-2.6266521306806237834040544026443 E-15
1.2841410438370241361457279081570 E-15
-6.2811246779832092490540915955525 E-16
3.0737418659790733359751550952943 E-16
-1.5048527892943157286941274104004 E-16
7.3707075418117801037767115505475 E-17
-3.6116466962715746721560625538143 E-17
1.7704150474468843953046468580986 E-17
-8.6818430219633647701453137275718 E-18
4.2590173317992935945560952526965 E-18
-2.0900733203228146820652946545818 E-18
1.0260359936416863217848296514932 E-18
-5.0385696117267440739831494270566 E-19
2.4750868268436181589501842395506 E-19
-1.2162064580279337897647503015742 E-19
5.9779639462714596392285635124411 E-20
-2.9391656069274429139845002823088 E-20
1.4454912820989308608137149928767 E-20
-7.1108845329177578827105729098124 E-21
3.4990066749315417229455607454474 E-21
-1.7221673478194437291719490418385 E-21
8.4783623277310517244543582933675 E-22
-4.1749511462354160169741608691211 E-22
2.0563192212811097335696380185410 E-22
-1.0130396163685589250273727252998 E-22
4.9917894139953788654946337959986 E-23
-2.4602390683359020178003138967311 E-23
1.2127939069301191661639031402060 E-23
-5.9797477355407677288805838636317 E-24
2.9489166915266580573932329293086 E-24
-1.4545332328547903080488562466017 E-24
7.1756972822106364319766739978423 E-25
-3.5406401040453205932800972866763 E-25
1.7473288825024335263723440548007 E-25
-8.6246361187303828745205603559264 E-26
4.2577317531925345448445978953806 E-26
-2.1022550142841515972779800768394 E-26
1.0381506214729796161812587806332 E-26
-5.1274508064085962038074071753188 E-27
2.5328371055666086404133791761389 E-27
-1.2513416925459592867495070698346 E-27
6.1830996304612768192687699598035 E-28
-3.0555971956420774211650856970448 E-28
1.5102371365537669772158590708494 E-28
-7.4653373220930952905580064654050 E-29
3.6907228478957141273598480589907 E-29
-1.8248250536380089745541606438671 E-29
9.0238039349458111979273090686006 E-30
-4.4626321100200830762883445552940 E-30
2.2072716543167375806950968205447 E-30
-1.0917818862777843416562786095551 E-30
5.4010102698819202873494951185154 E-31
-2.6725235359162127092694499491243 E-31
1.3221556501875044764187601672257 E-31
-6.5583798017323607033639050931179 E-32
3.2442551084741376457891294676456 E-32

regular pentation generated from the lower fixed point, via pentation.gp code, pentation taylor series, centered at 0, accurate to ~21 digits

Code:

base 1.6353244967152763993453446183062
1.0000000000000000000005192248729
0.81779936973045395720071293214265
-0.20288561975535027441384945617561
0.0088786047456589776694546837262676
0.019219350732330865713239336968590
-0.0096966552676861733857155784547169
0.0019514051125590343992422677111750
0.00046437760665497425458895770723555
-0.00066778754659256411362708055845759
0.00028617189543862800787206737355819
0.000033665580602278323048951676875883
-0.000079454007870098296770230740659414
0.000014692767344986060783974244415048
0.000010474055892921335060914517323353
-0.0000038212173806832668738543563355660
-0.00000062027483834957308089542513290423
0.00000032318089329501075067509587120004
-0.000000019840408991749849622699357079378
0.000000067745810965836343131600904116424
-0.0000000058251455196449879916926009357848
-0.000000036482729520758717533689961006608
0.0000000079325719962896266186472362692267
0.000000010065411908157671788716163940183
-0.0000000030175551558297183590143318054742
-0.0000000021761785937770771152654327159320
0.00000000073633472536622621197919931082184
0.00000000041672032311134229621174565224437
-1.2285074992273508685395579548365 E-10
-7.9518263282218380475858744550042 E-11
9.7514169088525432105036375063450 E-12
1.6997610080669118896188802919875 E-11
2.2377960363057901950721056894192 E-12
-4.1046921572945523533720262850443 E-12
-1.3440219438117173904988603015592 E-12
1.0162692177419430576029255087264 E-12
4.3490698532209929828066693595609 E-13
-2.3609655934025589161994798534342 E-13
-1.1591299932858676127971072482002 E-13
4.8835256899010474072746185189629 E-14
2.8208435349728648547600118064044 E-14
-8.5015918464895476441605506462895 E-15
-6.5455218512546145350077565966573 E-15
1.0586549941943315125293352810983 E-15
1.4799324783739173257534631379136 E-15
-3.5648642871089965652275206214142 E-18
-3.2899826059741680560349674399479 E-16
-5.4568752539881806017312574688462 E-17
7.1622413809298965347483312234643 E-17
2.3703528551345290231430115929264 E-17
-1.4967240115919668278098012252538 E-17
-7.3758767032881589736768555996658 E-18
2.8831523110629674706610210115875 E-18
1.9635837518801332808940827485057 E-18
-4.7131822341175072827483676403593 E-19
-4.7478844794120580339842160065419 E-19
4.9730849306353870744775718808789 E-20
1.0752385621120588513176015629038 E-19
4.4349054243188470473077019531603 E-21
-2.3207449741059068344885938312759 E-20
-4.8107012333615483486510550415132 E-21
4.7981518512995167340705076629676 E-21
1.9009245566320189670069175442907 E-21
-9.3913645999478807261230373741907 E-22
-5.8049321491044260759969472389200 E-22
1.6698977178128083745531739054 E-22
1.5471939617682124872654801496 E-22
-2.3711240240715382390040881966 E-23
-3.8230531306677313062190683870 E-23
1.1307570367174951165397624466 E-24
9.6860551417077212769401805422 E-24
6.2578554176447631116043850053 E-25
-3.0700834538096470676346145367 E-24
3.4209410749157917380893492990148 E-25
1.1606628720265556441455230554 E-24
-1.0836394904688614450987914987 E-24
-4.1068959031308040748622684284 E-26
1.1631242740449412539214268446 E-24
-8.454186445453502413912267596 E-25
-6.2526608803890595387694651538 E-25
1.3123138393956501678978565230 E-24
-2.5218865114423677749272843280551 E-25
-1.1722018792055174047394751512 E-24
1.0570768224356190649006502317 E-24
4.7217362291599339847898474461 E-25
-1.4043538312117386249617760808 E-24
4.8577833663635964403036734425 E-25
1.1063437204958150575910381509 E-24
-1.2650944581344852978745299383 E-24
-2.6733069486489219154746185107 E-25
1.4881840360747858870759691092 E-24
-7.4870314510452372161187607061165 E-25
-1.0150709222160250770174606064 E-24
1.4712316462119861103644707796 E-24
2.4144503740560439830659577039 E-26
-1.5354822446101247032997950712 E-24
1.0483819898331090549069707551 E-24
8.612742065171026391328194093 E-25
-1.6930059010324648452317830381 E-24
2.8579292408098419907076502134 E-25
1.5611586068656302104116424319 E-24

sexp taylor series, centered at the upper fixed point. Parabolic regular iteration, since sexp(upfixed)-upfixed=0, and the derivative=1. However, the pentation series above was developed from the lower fixed point. It might also be interesting to develop the pentation from the upper fixed point.

Code:

base 1.6353244967152763993453446183062
upper fixed point 3.0885322718067176544821807826411
3.0885322718067176544821807826411
1.0000000000000000000000000000000
0.29348332594662679156185554416695
0.12006943677526115961845339690056
0.042289726008581658757917692352047
0.015406681466051593705278419486083
0.0053649969961165772207616839018923
0.0018630655977932198916899772571769
0.00063407537228816784082533370359873
0.00021400606080778703684108316364163
0.000071350368686559236499836370338975
0.000023591182312113468409361763152200
0.0000077307357866599052916288379173725
0.0000025148014797529359261692389144205
0.00000081222410528536302525231046897101
0.00000026067080724584451491376816826933
0.000000083156587277307998293961303674167
0.000000026382062425889945293743274742732
0.0000000083265988463026330560171882094710
0.0000000026153442829909609773032040022395
0.00000000081773434750878326642769707607752
0.00000000025458721090899662275665667332696
7.8940942014863353398515697803273 E-11
2.4383970740151010163418654302545 E-11
7.5045892057613467862823411677246 E-12
2.3016959657817279736641051418417 E-12
7.0362006799573268296411117493600 E-13
2.1441899019127249833543050355134 E-13
6.5145303264754241098603909417717 E-14
1.9735746229612285013998339658891 E-14
5.9624685317524456584119105248956 E-15
1.7965888918970083975596928948585 E-15
5.3996546628326549009825713515487 E-16
1.6188983497572185136799713979226 E-16
4.8422635593460558165533108592429 E-17
1.4450703785698551344484490685140 E-17
4.3030415164576701763666524081170 E-18
1.2786162600194922955158125010948 E-18
3.7915201105554629414895562907887 E-19
1.1220792748374648557975378688279 E-19
3.3143452564446513738023171333433 E-20
9.7715062076572343807657205107082 E-21
2.8756697774572856807994929489579 E-21
8.4479857850782485763764738145021 E-22
2.4775690067172222743409424497063 E-22
7.2539983667168612363130405456846 E-23
2.1204495177307628041026820799736 E-23
6.1886505172771185049788719771013 E-24
1.8034304113738667347358493697905 E-24
5.2475343265172758722223983303515 E-25
1.5246843780304616848992192317019 E-25
4.4237335335211977445129141410633 E-26
1.2817319966942693841808461450023 E-26
3.7086772208878018504657396083702 E-27
1.0716874126315835345680035791605 E-27
3.0928382435902888820021324131020 E-28
8.9145242970213826409341395287118 E-29
2.5662848881126763792030265558974 E-29
7.3788051820008966793254068896449 E-30
2.1193421380273394338847540585606 E-30
6.0824157348570199720495626504043 E-31
1.7421558940832369333368054728044 E-31
4.9765917498063797295934928999305 E-32
1.4062811115449320214491405056275 E-32
3.7934403719757943562626785237526 E-33
1.2305145019890227199662941927280 E-33
5.5711727230420028085149818418777 E-34
2.4186758292339716100055902047374 E-34
1.7732706656596300164917606853238 E-34
-1.3355809003646951549924724970099 E-34
-2.9072689285868805984220052919317 E-34
-6.9107402433856378705373186712384 E-35
4.4126731718368809510319147247328 E-36
1.7126322918948119164481144802202 E-34
2.7989257163342943183983118462271 E-34
-1.9985628783896615630523665162131 E-35
-1.4700179855300004173061277796181 E-34
-1.5536566643835194929354996925696 E-34
-1.9498134289485168975356191250626 E-34
9.9022922195740429874579196637929 E-35
2.4133016008594091094881247283662 E-34
9.9675730165405949191382441641302 E-35
6.6232586691919872798243258736776 E-35
-1.4692529755384798103039616882568 E-34
-2.6872630348380145791502305390272 E-34
-2.1981032501079913545456022870657 E-35
7.3537194481961900271945952344694 E-35
1.5483413632202028611513722984947 E-34
2.2373813096659179009012953828649 E-34
-5.8527500520290404167240680297540 E-35
-1.8596067914689981955501774376493 E-34
-1.2066043234322834168239487754860 E-34
-1.2374312463084364980127016870177 E-34
1.1860368344776018489712748078933 E-34
2.4573978020663713546965678757098 E-34
5.7985783818128634242359558552645 E-35
-4.7627589167074290469391891175827 E-36
-1.4677451984086100716918643719760 E-34
-2.3907044867916316647291674331143 E-34
1.8699789701736156540089113210717 E-35