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Neat stuff. I'm curious: how did you get the method to work? That is, when we take the Abel function of the Taylor approximation at z = 0 for the Fourier integral to get the warping map (the $\theta(z)$ mapping), how do you make sure that the values going into the Abel function are within its range of convergence? That seems to be the trick part that makes it difficult to extend the method to various other complex bases.
(03/10/2012, 06:59 AM)mike3 Wrote: [ -> ]Neat stuff. I'm curious: how did you get the method to work? That is, when we take the Abel function of the Taylor approximation at z = 0 for the Fourier integral to get the warping map (the $\theta(z)$ mapping), how do you make sure that the values going into the Abel function are within its range of convergence? That seems to be the trick part that makes it difficult to extend the method to various other complex bases.
Hey Mike,

Thanks for commenting. As far as the Abel function, goes, I do that evaluation over a unit length from -0.5 to 0.5, and I extended the imaginary delta to 0.175i, for the upper superfunction theta, and -0.175i for the lower superfunction eta. This helps remove ambiguity on "which logarithm branch" to use, and I also have some code tweaks to make sure the inverse Superfunction (or Abel function) is mapping to the same period for all sample points. The other problem is deciding how accurate the Schroder function needs to be. The program iterates logarithms (for the repelling case), before evaluating the Schroder function, until the sample point is within some bounds of L, where the Schroder function is accurate. One difficulty is how to know what to use as the bounds, and I'm still tweaking that, with adjustments for bases near eta, and will put an update of the code online shortly. Finally, I initialize and renormalize the superfunction so that B^sexp(-0.5+/-0.175i)=sexp(0.5+/-0.175i), which at least gives continuity to the theta calculations. This dramatically improves the numer of bits precision improvement I get at each iteration.

Right now, I'm numerically investigate the branchpoint singularity at eta, which is incredibly mild, as you and others have noticed, and I will post some surprising results about that.
- Sheldon
(03/10/2012, 08:53 PM)sheldonison Wrote: [ -> ]
(03/10/2012, 06:59 AM)mike3 Wrote: [ -> ]Neat stuff. I'm curious: how did you get the method to work? That is, when we take the Abel function of the Taylor approximation at z = 0 for the Fourier integral to get the warping map (the $\theta(z)$ mapping), how do you make sure that the values going into the Abel function are within its range of convergence? That seems to be the trick part that makes it difficult to extend the method to various other complex bases.
Hey Mike,

Thanks for commenting. As far as the Abel function, goes, I do that evaluation over a unit length from -0.5 to 0.5, and I extended the imaginary delta to 0.175i, for the upper superfunction theta, and -0.175i for the lower superfunction eta. This helps remove ambiguity on "which logarithm branch" to use, and I also have some code tweaks to make sure the inverse Superfunction (or Abel function) is mapping to the same period for all sample points. The other problem is deciding how accurate the Schroder function needs to be. The program iterates logarithms (for the repelling case), before evaluating the Schroder function, until the sample point is within some bounds of L, where the Schroder function is accurate. One difficulty is how to know what to use as the bounds, and I'm still tweaking that, with adjustments for bases near eta, and will put an update of the code online shortly. Finally, I initialize and renormalize the superfunction so that B^sexp(-0.5+/-0.175i)=sexp(0.5+/-0.175i), which at least gives continuity to the theta calculations. This dramatically improves the numer of bits precision improvement I get at each iteration.

What do you mean by "mapping to the same period"?

(03/10/2012, 08:53 PM)sheldonison Wrote: [ -> ]Right now, I'm numerically investigate the branchpoint singularity at eta, which is incredibly mild, as you and others have noticed, and I will post some surprising results about that.
- Sheldon

What did you find?
(03/11/2012, 12:27 AM)mike3 Wrote: [ -> ]
(03/10/2012, 08:53 PM)sheldonison Wrote: [ -> ]Hey Mike,

Thanks for commenting. As far as the Abel function, goes, I do that evaluation over a unit length from -0.5 to 0.5, and I extended the imaginary delta to 0.175i, for the upper superfunction theta, and -0.175i for the lower superfunction eta. This helps remove ambiguity on "which logarithm branch" to use, and I also have some code tweaks to make sure the inverse Superfunction (or Abel function) is mapping to the same period for all sample points.....

What do you mean by "mapping to the same period"?
What I was refering to was the last step in evaluating the inverse superfunction (or Abel function) to generate theta, after evaluating the Schroder function, and adjusting for iterated logarithms. This last step is:
$\log(\text{Schroder}(z-L))/\log(L\times\log(\text{base}))$
This logairthm of the Schroder function can be ambiguous, especially if fixup is required due to the accuracy range of the Schroder function. And the easiest fix is to compare adjacent points for the inverse superfunction, and to add or subtract the period so that the adjacent points are near each other.

Quote:
(03/10/2012, 08:53 PM)sheldonison Wrote: [ -> ]Right now, I'm numerically investigate the branchpoint singularity at eta, which is incredibly mild, as you and others have noticed, and I will post some surprising results about that.
- Sheldon

What did you find?

I'm still investigating and learning, and will post more details later. The experiment, is to develop taylor series for all of the coefficients of sexp_b(z), developed around complex sexp(z) in the neighborhood of base=2. The conundrum, which I'm beginning to be able to explain, is that the numerical results are way too good, and its hard to see the effects of the branch point at eta. In fact, numerically convergence seems limited by the more distant singularity for sexp_b(z), at b=1, and not at all by the closer singularity at eta. So far, I have a taylor series for all the coefficients for the neighborhood of base=2, and the resulting sexp(z) is accurate to 33 decimal digits. At eta, it is accurate to 31 decimal digits. Theory says the extrapolated sexp_b(z) function shouldn't even converge if the radius is outside 2-eta, but for the truncated series, that is not at all the case. For example, results for base e are still accurate to 19 decimal digits, and for base 1.3, the results are accurate to about 20 decimal digits. Precision declines nearly linearly from the sample radius (2-eta)=~0.555, towards the more distant singularity at b=1, with radius=1. More details later. I also put a new version of the code in the first post, with improvements near eta, and convergence for ultra high precision results for >p 67.
- Sheldon
Quote:... The experiment, is to develop taylor series for all of the coefficients of sexp_b(z), developed around complex sexp(z) in the neighborhood of base=2. The conundrum, which I'm beginning to be able to explain, is that the numerical results are way too good, and its hard to see the effects of the branch point at eta. In fact, numerically convergence seems limited by the more distant singularity for sexp_b(z), at b=1, and not at all by the closer singularity at eta....
I have a good way of explaining why the branch point at $\eta=\exp(1/e)$ is so slight. The graph below is for base $\eta-0.25\approx1.195$ Start with this formula for the merged sexp(z), after rotating counterclockwise around eta. Of course, there is a corresponding formula in terms of the lower repelling superfunction too, but the upper attracting superfunction is of primary interest here. Note that the attracting superfunction is real valued at the real axis, however, the theta(z) means that the sexp(z) has a small negative imaginary component at the real axis, for the counterclockwise sexp(z), and a small positive imaginary component for the clockwise sexp(z).
$\text{sexp}_{+\pi}(z)=\text{superf_u(z+\theta_u(z))$ (counterclockwise)
$\text{sexp}_{-\pi}(z)=\text{superf_u(z+\overline{\theta_u(z)})$ (clockwise)
[attachment=968]

Here, the graph on the left is for superf from the attracting fixed point, and the two graphs in the middle show the merged sexp(z) from both fixed points, rotating counter clockwise around eta, vs rotating clockwise around eta. The sexp(z) function looks much more like the attracting fixed point superfunction, than the repelling fixed point superfunction on the right. In fact, for $b=\eta-0.25$ at the real axis, $\theta_u(z)$ is an analytic function, dominated almost entirely by the first overtone, and very nearly equal to zero, with a peak negative imaginary component of about 7.06*10^-13. The graph below shows the 1-cyclic $\theta_u(z)$ function at the real axis, with its negative imaginary component in magenta, where the attracting superfunction(0) has been centered so that superfunction(0)=1.
[attachment=969]
$\theta_u(z)$ is very different than $\theta_l(z)$ which is for the repelling superfunction, which has a singularity at the real axis. So the clockwise versus counterclockwise functions are both nearly identical at the real axis, since both are very nearly identical to the attracting superfunction, depending on how small $\theta_u(z)$ is. First of all, $\theta_u(z)$ by definition converges to a constant as z goes to imag(infinity).
$\theta_u(z)=\sum_{n=0}^{\infty}t_n\times\exp(z\times 2n\pi i)$
It seems logical to assume that somewhere between the Period of the attracting superfunction, and the period of the repelling superfunction, $\theta_u(z)$ will have its singularity, and this matches the computational results. Also, the period of the attracting and repelling superfunctions are both functions of L. $\text{period}=2\pi i / \log(\log(L))$. In the neighbordhood of $\eta$, L(b) itself is an analytic function (real valued for b<eta) of $\sqrt{\eta-b}$, and approaches e at eta itself, so the log(log(L)) will also be analytic function of $\sqrt{\eta-b}$ and will approach zero. So the period approaches infinity, in the neighborhood of eta. And since $\theta_u(z)\approx \exp(-2\pi |\text{period}|)$, then theta_u is very small. The imaginary component of theta hits its minimum at -0.5. Emperically, the following approximation holds for the difference between the clockwise and counterclockwise at sexp(z=-0.5).
$\text{sexp}_{+\pi}(-0.5)-\text{sexp}_{-\pi}(-0.5)\approx 2 \times theta_u(z)\approx (1/3)\times\exp(-2\pi |\text{period}|)$.

So to the left of the branchpoint, the clockwise and counterclockwise functions both approach very closely to attracting superfunction, and hence very close to each other, with the difference between them being proportional to theta, and as the base approaches eta from the left, the branchpoint becomes very slight.

This delta corresponds to the difference in values between the clockwise sexp(z) and the counterclockwise sexp(z), which limits how accurate the truncated taylor series for the derivatives can be.
Code:
base<eta   Period           sexp clockwise/counterclockwise delta approx etaB-0.01 -31.13270473209*I 3.710448516980 E-86 etaB-0.02 -21.55773407651*I 4.979130539705 E-60 etaB-0.03 -17.28601302448*I 2.257479939076 E-48 etaB-0.04 -14.72147299231*I 2.247084770877 E-41 etaB-0.05 -12.95861790681*I 1.452161920541 E-36 etaB-0.06 -11.64760163792*I 5.488507194111 E-33 etaB-0.07 -10.62082878037*I 3.477470488040 E-30 etaB-0.08 -9.786588904067*I 6.572003322472 E-28 etaB-0.09 -9.089897234611*I 5.233558147561 E-26 etaB-0.10 -8.495506226695*I 2.191508331413 E-24 etaB-0.11 -7.979631626302*I 5.603215666548 E-23 etaB-0.12 -7.525558560592*I 9.716065775874 E-22 etaB-0.13 -7.121145360407*I 1.233201548139 E-20 etaB-0.14 -6.757326554412*I 1.212845868742 E-19 etaB-0.15 -6.427174188790*I 9.654055136983 E-19 etaB-0.16 -6.125286843939*I 6.434068563060 E-18 etaB-0.17 -5.847379092921*I 3.688308272101 E-17 etaB-0.18 -5.589997954079*I 1.858474866275 E-16 etaB-0.19 -5.350322279507*I 8.378616990659 E-16 etaB-0.20 -5.126017750125*I 3.429603628559 E-15 etaB-0.21 -4.915130017959*I 1.290340712260 E-14 etaB-0.22 -4.716004544129*I 4.508878950952 E-14 etaB-0.23 -4.527225439464*I 1.476386512487 E-13 etaB-0.24 -4.347568022104*I 4.564999487628 E-13 etaB-0.25 -4.175961379371*I 1.341877355523 E-12 etaB-0.26 -4.011458264023*I 3.772255892170 E-12 etaB-0.27 -3.853210351166*I 1.019578259126 E-11 etaB-0.28 -3.700447342634*I 2.662398374314 E-11 etaB-0.29 -3.552458696786*I 6.746794115812 E-11 etaB-0.30 -3.408576919042*I 1.666154414388 E-10 etaB-0.31 -3.268161383232*I 0.0000000004026006962741 etaB-0.32 -3.130581551846*I 0.0000000009556433874425 etaB-0.33 -2.995198175449*I 0.000000002237296864441 etaB-0.34 -2.861340468660*I 0.000000005187859356797 etaB-0.35 -2.728276154970*I 0.00000001196982497438 etaB-0.36 -2.595169164234*I 0.00000002762510017059 etaB-0.37 -2.461015584389*I 0.00000006417646711815 etaB-0.38 -2.324539653737*I 0.0000001512811687354 etaB-0.39 -2.184011424414*I 0.0000003658067014186 etaB-0.40 -2.036896446157*I 0.0000009219174865173 etaB-0.41 -1.879097130088*I 0.000002484776686097 etaB-0.42 -1.703001149178*I 0.000007512937020938 etaB-0.43 -1.490811357536*I 0.00002849857972573 etaB-0.44 -1.171208001292*I 0.0002122959653234

Finally, here is the best data I have for the taylor series of the first derivative of sexp_b(z), developed around b=2. The samples used to generate this series were accurate to better than 10^-41. The a0 coefficient is the first derivative of $\text{sexp}_2(z)$. I have similar approximation taylor series for all of the other derivatives. Here, the taylor series gives the first derivative for sexp(z) for bases, in the neighborhood base=2. Notice how the behavior changes around the 115th taylor series term, as the function switches from appearing to be dominated by the singularity at b=1, and switches to being dominated more and more by the branch singularity at the at eta, which is closer. The more terms in the taylor series that are included, the more the taylor series is limited by the branch singularity at eta, which leads to the difference between the clockwise versus counterclockwise sexp(z) functions. For the example below, where I was attempting to calculate sexp(z) for bases in the neighborhood of b=2. Up to a certain point, which appears to be around 30 decimal digits accuracy, with a pseudo convergence limited by the singularity at b=1, and the branch singularity at eta has little effect. But after that the branch singularity at eta effects numerical results, and eventually limits the convergence radius to 2-eta.
Code:
a0=   0.88936495462097637278974352283112566597959842312251 a1=   0.36669108107597092173217976440672326640242409195815 a2=  -0.16362154674166553061445059158034337946514620745021 a3=   0.090640323696839666639402392489128364529129886437045 a4=  -0.056092413061581014125833041210073512340458819978678 a5=   0.037429879347431534759732842843739952749936334953121 a6=  -0.026447566540555046167348051422368324180671884326480 a7=   0.019555271787834906960129053618264164629133914637930 a8=  -0.014998700484629922332412713318859127829768835212408 a9=   0.011852277973708718952425319210039273396382167341591 a10= -0.0095979249064313156164975512661790404203658685707124 a11=  0.0079311430805752725977382595100373190194734752188124 a12= -0.0066652715199993946528984284449977437088182904642631 a13=  0.0056815522808208952892183976587971611539857168959937 a14= -0.0049018796481256649565458874468783019138120256803352 a15=  0.0042733338454417762073839306915934253717329238935478 a16= -0.0037590911321363305318925585652200472516503816017069 a17=  0.0033329094106928993363221640124127419335693082919321 a18= -0.0029756842758379003928167202745212175475434077571195 a19=  0.0026732398131445450105912560861216188029536111841172 a20= -0.0024148761018878752698634155299989480692847576495069 a21=  0.0021923923431733994482485995594897921044163356982521 a22= -0.0019994160594281043243601961112895638663701583429693 a23=  0.0018309336059454422799865924545271346546674115585991 a24= -0.0016829557995207397774518990169204303777386188211848 a25=  0.0015522759492132892907543343668242961956689622577548 a26= -0.0014362921795159532216208753242354807067225985475890 a27=  0.0013328752061942965517387403695720455629275551040151 a28= -0.0012402687213875580166779370161425436514008743256891 a29=  0.0011570134929575005536334159397260465591648925387553 a30= -0.0010818889266422991816215579207984835873821554164544 a31=  0.0010138676373011954591188552275431839057943151948645 a32= -0.00095207981595956269637889395018389024825631550243812 a33=  0.00089578504693664818081819928516114832343581140952444 a34= -0.00084434984385899237861310863529479347779740346437169 a35=  0.00079722961381271718763168612347015038956007100110681 a36= -0.00075395407807855574712926956308909437049664444383764 a37=  0.00071411541160938946428818588581728166263111470660659 a38= -0.00067735853620409902799105130725043134392492254960256 a39=  0.00064337313124965386596766468500875413789077145692186 a40= -0.00061188702291154229626658219590368328702648806612004 a41=  0.00058266068623889760305492770771989104509265151955222 a42= -0.00055548265089542461882814031136209155736743753351527 a43=  0.00053016564452680670265915571375425262634292211455586 a44= -0.00050654334133633926255750493060219365857765874740199 a45=  0.00048446760962149932878851783636208732581181430898690 a46= -0.00046380617257324082001099728701592339448563511616058 a47=  0.00044444061286260480258610350811798318611503966510673 a48= -0.00042626466441773227586704765732445285587339365362382 a49=  0.00040918274507213061528217858906171204492942041496984 a50= -0.00039310869200873789672697722601821079213584904742657 a51=  0.00037796466856845280507923856825638182921587428012227 a52= -0.00036368021637146762203794495066833295512962792300986 a53=  0.00035019143107487044304578735914580352890072611416895 a54= -0.00033744024366323726129224584643119592836937214435671 a55=  0.00032537379209915666329061376587202828204010586321943 a56= -0.00031394387057292649992164597833483436135555278040621 a57=  0.00030310644558403565381865326178256698702599710893599 a58= -0.00029282122974019597041473217876872956215806169323111 a59=  0.00028305130553545529888036650354266314295302961767803 a60= -0.00027376279251768186397373561219018313261115169007070 a61=  0.00026492455221801863264234775726324693503039617233253 a62= -0.00025650792602353866363667026901576516933066334253096 a63=  0.00024848650185587439102216316319323328557430704192470 a64= -0.00024083590609468854748617568876614572411707594724692 a65=  0.00023353361767318228230186310880656093327918204473787 a66= -0.00022655880168788762141619164200057398983085275582110 a67=  0.00021989216021868017265308570152327937187302449701118 a68= -0.00021351579835709674505735864675680944924867708901373 a69=  0.00020741310369977137250138959377732530935733595485491 a70= -0.00020156863778583889009335810725309920146940227564785 a71=  0.00019596803814807842526366830452897875866432229654358 a72= -0.00019059792981201160762436281891258414477823508358813 a73=  0.00018544584521897067028814541769394653635462892269968 a74= -0.00018050015167147757639163313600795338153819088879476 a75=  0.00017574998550471612913893255538192624366439692559225 a76= -0.00017118519227851192795878191050308533487756636150850 a77=  0.00016679627236166650729250634108259328753423653710167 a78= -0.00016257433134587360165868974405869466784152208187347 a79=  0.00015851103478046157051073283942874969208400738183333 a80= -0.00015459856676202830364669661657386441709548083608241 a81=  0.00015082959194423570314702918991562002142493203027731 a82= -0.00014719722055144234518107114118344164903302738911123 a83=  0.00014369497598335467624451283154961745257066737932331 a84= -0.00014031676458308491554247858366115354314991475723378 a85=  0.00013705684710282338020723134032423602483573204119501 a86= -0.00013390981133232509659735342171329589620246096944970 a87=  0.00013087054524489410749850049900282861153695363681700 a88= -0.00012793420984833547352024557075709732171361649192285 a89=  0.00012509621068295869630008142469647164720784144646336 a90= -0.00012235216655488969008254181850562113073907649013970 a91=  0.00011969787358809869569954515710617029947418944225306 a92= -0.00011712926196282222536212792733308204118753827071088 a93=  0.00011464234169734474160271340456238026283744029908002 a94= -0.00011223313240623520511592382152797025290078408035316 a95=  0.00010989756996499343516887268260549947233769372698957 a96= -0.00010763138019507478088599419446017686268120495567600 a97=  0.00010542990572586879711082154570799636102261607883451 a98= -0.00010328786662887001547423189340462077942151910893610 a99=  0.00010119902760229155117987116555086751087119364128970 a100= -0.000099155733493257748427303874679264039448145645931645 a101=  0.000097148259484173663756910593709511569636217970447482 a102= -0.000095163900510316899459802740745143482696233109353337 a103=  0.000093185693831568627599711194886901547720375833413101 a104= -0.000091190625500058431170267531037152681582973453844691 a105=  0.000089147110577614111949313564259125356991960517880999 a106= -0.000087011451028699232260108246054448874076247235190155 a107=  0.000084722853861864699274074494454048190258669970406755 a108= -0.000082196420611059641593752759759312766864240276251934 a109=  0.000079313276695378040668769536450594708426817125548315 a110= -0.000075906665764395678727615632516854804533978379280764 a111=  0.000071742347662329608232959664625953008704720168616383 a112= -0.000066490949045316251665604983560227863859233441297859 a113=  0.000059688936607385319216806625661237510929842694777841 a114= -0.000050683491929795864387957494211684268403632531307108 a115=  0.000038554591215961131426091090024773805331362076252978 a116= -0.000022004781441186651525068565051593310584156771787147 a117= -0.00000079686479023490918352188515938718745623403314985007 a118=  0.000032435842306938425829042459347263929450929934064059 a119= -0.000076565714434638346050009566704962759286291581683884 a120=  0.00013835302514462329492715415906499072685291312101984 a121= -0.00022510927831603938204827769309448456479602850573677 a122=  0.00034718935704696551901128517239052916023360875271213 a123= -0.00051926976296097164502837945920672213845180105416167 a124=  0.00076216909449877172132451567849646712120932779759742 a125= -0.0011054431038213348013186478548646420559935081323403 a126=  0.0015910864138726589034799899476315237114635147747212 a127= -0.0022788182166267627311416287260322042878665023998693 a128=  0.0032536386376808833038924592897870534732898468104546 a129= -0.0046366371764573246242753419398304760841850786180881 a130=  0.0066004633054367071522484717818903138659172698774928 a131= -0.0093915057896219313853898700172838382932045079218239 a132=  0.013361717489740579769813199969075407115547914608975 a133= -0.019014272635751394692164096827789761140526602012705 a134=  0.027069173327981442637779312611596655121745248124818 a135= -0.038557729361091680133518568269684104356086857563680 a136=  0.054958493574037114777614806285063289961453277787629 a137= -0.078392826224250633795544680085007781810885500100699 a138=  0.11190745728458165537414831366818621760006682122102 a139= -0.15988295078616910465080607193584426374973435542011 a140=  0.22862172022946783097429914191082638526466150798879 a141= -0.32719818744634441833244810545444965620643808966197 a142=  0.46869628425465044206256240974234238301732522286161 a143= -0.67199708611904647090683577037755878112489281073301 a144=  0.96435215054958924907879854627278256335820106152765 a145= -1.3851485996958500780035723006344994178594599048135 a146=  1.9914129820358020140172084486011272635335577852069 a147= -2.8656866668959487638622965129796753023935425258549 a148=  4.1274679150084385370303387083856390745324722721731 a149= -5.9502821897220866692140379324763876062552002958396 a150=  8.5862986600847764918022749625888076525534010202031

This set of series for each of the taylor series coefficients developed in the neighborhood of base=2 is more accurate than the earlier one I posted. It would give improved results, with better accuracy around 1E-35 for bases inside the radius of convergence, and accuracy of 3E-33 for base eta.

I also generated the Taylor series for sexp(-0.5), as the base is varied, centered around base=2. The pattern changes near the 94th taylor series coefficient, when the closer singularity at base=eta begins contributing more to the taylor series than the more distant singularity at base=1. Here, a0=0.544764121459556733980121885825724470, which is sexp(-0.5) for base=2, and the taylor series is accurate to about 35 decimal digits.
- Sheldon
Code:
a0=   0.5447641214595567339801218858257244685854 a1=  -0.09026490293475114180982800726025252487179 a2=   0.05334642698935378617403396491528890594804 a3=  -0.03638190492562309183765608353362070821840 a4=   0.02665589484943122254265742189263438424835 a5=  -0.02047608577133435850738520805893632252252 a6=   0.01628939391559684527389871185757624228228 a7=  -0.01331802035638468229849633176805710250959 a8=   0.01113080347039454404398917618932270486539 a9=  -0.009471945601741301301799666960159500493414 a10=  0.008181870472918983418952481797363865140773 a11= -0.007156971109633091475785436209879906176635 a12=  0.006327698270413005651257016844418549882893 a13= -0.005646005057506155565996841622134687059648 a14=  0.005077852297548008377590502397935756807242 a15= -0.004598579564709383679003395147264261288216 a16=  0.004189960720720897899813797031566114828076 a17= -0.003838281581489355718364575037825080832826 a18=  0.003533055202798846826754080155559369484869 a19= -0.003266144873505376098605478730250127897586 a20=  0.003031153706186763516973507099991793317656 a21= -0.002822992160610217843850530938287773588695 a22=  0.002637566583362648226391359841543536218954 a23= -0.002471551499838161939220088380143251371568 a24=  0.002322220812129558330686955182616565785365 a25= -0.002187321052201244713039707812416923885954 a26=  0.002064975080105974232286030081964966694861 a27= -0.001953608108640999822645409586956590179027 a28=  0.001851890298539576237491912671987439471748 a29= -0.001758691790434811807511081574014212915353 a30=  0.001673047168806168273884584564550319035826 a31= -0.001594127148975512837191984637091772279620 a32=  0.001521215846034519175841930450931000214014 a33= -0.001453692394304569109742037483974061901089 a34=  0.001391015984731342378663288972983370893753 a35= -0.001332713607717317327426366538502721958881 a36=  0.001278369952559800466677607874323180950794 a37= -0.001227619037446880124253911053592339307366 a38=  0.001180137236855264191223682764928143626889 a39= -0.001135637444029102801867724419804092337925 a40=  0.001093864160641107689052187995385697434389 a41= -0.001054589347847405322195327651270883669674 a42=  0.001017608905751088252280461602251417505390 a43= -0.0009827396740070429313553656082778187553405 a44=  0.0009498168665858747234148639733725614346048 a45= -0.0009186918698079533934930814875314534697736 a46=  0.0008892303455966866573662840017051026789547 a47= -0.0008613105921955727892242590292733293823957 a48=  0.0008348221228916550711398042195621478636400 a49= -0.0008096644300085595816921514966352297536765 a50=  0.0007857459069005338594524734325749675459648 a51= -0.0007629829051481069438849922812669102687654 a52=  0.0007412989078245228555308404776565708380003 a53= -0.0007206238027261275601524070871984237792707 a54=  0.0007008932419630012770449395923516096144246 a55= -0.0006820480763866325438818551382445843370545 a56=  0.0006640338550679383137667663988829019940730 a57= -0.0006468003814946490331294868382872885969872 a58=  0.0006303013193831178111678644623963748370560 a59= -0.0006144938420376038744680011450272121474617 a60=  0.0005993383200741946673108758684076009577129 a61= -0.0005847980430851115584178527733490186488280 a62=  0.0005708389714760195065962019338923580583096 a63= -0.0005574295152845303003325133006371836953863 a64=  0.0005445403373002463827629604348689289744143 a65= -0.0005321441782716469402918017155794455019026 a66=  0.0005202157024181592772390603384936993523783 a67= -0.0005087313618820156701812678182987572351923 a68=  0.0004976692791697413648392248040346423784153 a69= -0.0004870091470646722871348673605764087543724 a70=  0.0004767321459596961918548061350965022366495 a71= -0.0004668208790873150944172435667760750155470 a72=  0.0004572593267416065803546985077871002284578 a73= -0.0004480328213309398865880156803821028719749 a74=  0.0004391280460191844112860195774985849996584 a75= -0.0004305330608687577109307488887504560310821 a76=  0.0004222373618726151852083171686555860996115 a77= -0.0004142319801615488633254732943499557223787 a78=  0.0004065096311401751693759190311085411363661 a79= -0.0003990649265288728503867805273227150792807 a80=  0.0003918946665218884447060007230972331199284 a81= -0.0003849982348510417760694970639142865634935 a82=  0.0003783781269217428875539830317674205408416 a83= -0.0003720406509700054967447945428476416910439 a84=  0.0003659968551918890293717641839010994444982 a85= -0.0003602637511201750326035146298122448667960 a86=  0.0003548659266520138706765657962715630833711 a87= -0.0003498376730740476659615080146624803134097 a88=  0.0003452257919088022517001222379805220095853 a89= -0.0003410933031098378510324434621473247433045 a90=  0.0003375243510812609314744505648626776451409 a91= -0.0003346307060211903759381304723940593677783 a92=  0.0003325603945037557882292885310966766766416 a93= -0.0003315091777419398624384857779237117250571 a94=  0.0003317358460153907833479388101334190239424 a95= -0.0003335826371421790840947235619676003126446 a96=  0.0003375025483293688889372157683210232309158 a97= -0.0003440959391986930633575807516000693209207 a98=  0.0003541596811015852018702302245010936694298 a99= -0.0003687532792553722994412436565871821184048 a100=  0.0003892879974033369355506985563538084697749 a101= -0.0004176472122482091920832394022681079546693 a102=  0.0004563492419325118494433069912923916819317 a103= -0.0005087680413650084606119114211931555096770 a104=  0.0005794328703416784956152178590887291116773 a105= -0.0006744359202315312364732927380874961661109 a106=  0.0008019877695049482375578463387186199880001 a107= -0.0009731755956181394060092634776613016580966 a108=  0.001202999930966446829426139160934525077288 a109= -0.001511794692123179861160636690582499120690 a110=  0.001927175422690043740636959464774028693374 a111= -0.002486716638299575521444734874764590083602 a112=  0.003241637115621436229181368952656278856890 a113= -0.004261880730986434210647558463981630316890 a114=  0.005643132413944208166578890706551114196855 a115= -0.007516521379651240969156586191270607737992 a116=  0.01006206163735266482138789454244816545884 a117= -0.01352729756512357159032473435823294438533 a118=  0.01825320917728106084640001141407135014463 a119= -0.02471025705364750094828467307169149418527 a120=  0.03354860917065776303709779278149423244864 a121= -0.04566823068915574646795726336973581269876 a122=  0.06231683139161434741416351784301382413064 a123= -0.08522693583797479716332856332368442457569 a124=  0.1168079697188183445926413360961134224671 a125= -0.1604158141872473246671531628893015031994 a126=  0.2207315839610322539821107999865013181935 a127= -0.3042945998433380627917194164914032453560 a128=  0.4202533179050196135850984244232386711566 a129= -0.5814247303292241247446316577711695736830 a130=  0.8057908832077465638002237284863080618377 a131= -1.118615564350797221844687119564496999668 a132=  1.555441935527913324823119191631251168563 a133= -2.166343033000847467475898770333491396571 a134=  3.021956187494805723428091061670418388443 a135= -4.222060471273729919015929034763285805797 a136=  5.907783486640862219163610751023466051097 a137= -8.278993788941744624065220674384056537610 a138=  11.61911094227013680329706281281624602869 a139= -16.33053766929996258807625048927333995682 a140=  22.98531974919989325688421392691882638831 a141= -32.39766033022431437194340100803250018929 a142=  45.72783261862920895891344502067529674014 a143= -64.63125037132339003279765074797075722636 a144=  91.47255488228885159381088483245314278778 a145= -129.6334095048019025160129531044130526933 a146=  183.9554899644497666098851981405887199496 a147= -261.3787224160197438528735946730321986942 a148=  371.8617901668770815153842203841958000090 a149= -529.7111364131130428052370084930064042101 a150=  755.5016929935324645636711119283331834056

Hmm, today I tried tetcomplex.gp with init(I). Unfortunately the program hangs/loops infinitely after the message

generating Schroder2 taylor series for isuperf2 function, scnt2 27

Using a real base, say init(1.44) works immediately.

What to do?

update: I found that it hangs in the routine "loop" when called from the "init(I)"-procedure. (Surprisingly "init()" doesn't provide an argument to the routine loop while that has a formal parameter "t")
update2: it enters "thetaup" and does not come back...

Gottfried
(02/06/2016, 01:37 AM)Gottfried Wrote: [ -> ]Hmm, today I tried tetcomplex.gp with init(I). Unfortunately the program hangs/loops infinitely after the message

generating Schroder2 taylor series for isuperf2 function, scnt2 27

Using a real base, say init(1.44) works immediately.

What to do?

update: I found that it hangs in the routine "loop" when called from the "init(I)"-procedure. (Surprisingly "init()" doesn't provide an argument to the routine loop while that has a formal parameter "t")
update2: it enters "thetaup" and does not come back...

Gottfried

hmmm, I haven't used tetcomplex in awhile. It is much much more limited in terms of what bases it will converge for than the newer fatou.gp program. For example, tetcomplex has no hope of converging anywhere near base(i), whereas fatou.gp works just fine. Also, the old program has no "fallback" algorithm to use for rationally indifferent fixed points, that are on the ShellThron boundary, whereas as the newer algorithm can work (although with less precision), without any Schroeder function whatsoever. The newer algorithm is much better, except for bases<eta, where someday I may allow for rotation angles >180 degrees for fatou.gp, but not yet. By the way, I finally had some time to post my answer to sexp base(i) on mathstack; http://math.stackexchange.com/questions/...35#1643235. I wojuld like to post more about fatou.gp here on the tetration forum as well.

Code:
\r fatou.gp setmaxconvergence(); /* base i is hard to compute */ sexpinit(i); sexp(0.5) 1.07571355731392 + 0.873217399108003*I
[update] Ah, got it working with Pari/GP v. 2.7 in a winxp-32bit virtual machine. Great, so I can recompute my example in MSE. I'll look for the incompatibility reasons and shall tell them later.[/update]

Here is the comparision of the regular/Schröder-solution and the (extended/generalized) Kneser-mechanism (see bottom of posting).
(This is the link to the discussion in MSE : http://math.stackexchange.com/questions/...ing-us-all )

-------------------------------------------------------

Hi Sheldon -

this is what I've done and what I've got with the just-downloaded fatou.gp: (this is the old Pari/GP 2.2.11-version)
Code:
\r f:\download\fatou.gp     seriesprecision = 21 significant terms    format = g0.15 help(); help2(); andrewjay(); for other functions \p 38  /* precis=38; 32-35 digits.  default \p 28 ~=24 decimal digits; */ /* generates Abel function for iterating z <= exp(z)-1+k; f(z) */ loop(k,nlim,nskip,looplim); sexpinit(b); /* b=exp(exp(k-1)); */ loop(1);  sexpinit(exp(1));  /* two examples for tetration for base e */ slog(z); sexp(z); abel(z); invabel(z,est); sexptaylor(center,radius,samples); slogtaylor(c,r,s); invabeltaylor(c,r,s); abeltaylor(c,r,s); fmode=0:abel  1:invabel  2:slog  3:sexp MakeGraph(width,height,x0,y0,x1,y1,filename, n); /* f(z); fmode */ debugprint=0; quietmode=0; x2mode=0; /* x2mode=1; iterate z^2+z+k */ prtpoly(wtaylor,t,name); setmaxconvergence(); /* base i is hard to compute */ thlogk=1; ctr=19/20; ir=57/64; ctfactor=85/100; disabautoctfactor=1; staylorstop=40; sexpinit(I);     seriesprecision = 21 significant terms    format = g0.15 1 0.474349095548301 0.0458093729068993 23 4 20 2 0.236926728615837 0.409974883179566 41 6 40 3 3.91014217666629 E144 4.33918410729562 E143 61 7 60   *** vector: negative number of components in vector. sexp(0.5)    *** if: incorrect type in comparison.

Discussing the (extended) Kneser-method the question of fixpoints is relevant. Here I have produced a picture of the fixpoints of tetration to base(î) , I found 2 simple fixpoint (attracting for exp, attracting for log, both used for the Kneser-method), and three periodic points - making things a bit more complicated. The fixpoints were sought using the Newton-algorithm for the joint threefold exponentiation $f(z)= \log_i(\log_i(\log_i(z)))$ and the iteration $z_{k+1} = f(z_k)$ .

This means for example, that for the point in the top-left edge with the blue color having the z-value z_0=-5 - 5i the Newton-algorithm using iteration $z_{k+1} = f(z_k)$ arrives at the fixpoint 3.0 (having the complex value of about -1.14+0.71I in a moderate number of iterations. The blue point at coordinate z_0=-2.5+1I needs less iterations and the a bit lighter blue points near the periodic point 3.0 need even less iterations.

Perhaps this post should be moved into a discussion of the Kneser-method or of the general problem of fixpoints.

Gottfried

Here is the picture:
(02/06/2016, 05:36 PM)Gottfried Wrote: [ -> ]Discussing the (extended) Kneser-method the question of fixpoints is relevant. Here I have produced a picture of the fixpoints of tetration to base(î) , I found 2 simple fixpoint (attracting for exp, attracting for log, both used for the Kneser-method), and three periodic points - making things a bit more complicated. The fixpoints were sought using the Newton-algorithm for the joint threefold exponentiation $f(z)= \log_i(\log_i(\log_i(z)))$ and the iteration $z_{k+1} = f(z_k)$ .

This means for example, that for the point in the top-left edge with the blue color having the z-value z_0=-5 - 5i the Newton-algorithm using iteration $z_{k+1} = f(z_k)$ arrives at the fixpoint 3.0 (having the complex value of about -1.14+0.71I in a moderate number of iterations. The blue point at coordinate z_0=-2.5+1I needs less iterations and the a bit lighter blue points near the periodic point 3.0 need even less iterations.

Perhaps this post should be moved into a discussion of the Kneser-method or of the general problem of fixpoints.

Gottfried

Here is the picture:

Yes, this thread should be moved; it has to do with fatou.gp, and an MSE question for tetration base(i), and the two primary fixed points for Henryk Trapmann's uniqueness sickle.

In the case at hand, the other fixed point, for the lower half of the complex plane for sexp(z), is -1.862-0.411i, which Gottfried has listed strangely as 3-periodic, where as it is a simple repelling fixed point for $i^z$. The Abel/Slog uniqueness sickle connects the two primary fixed points. For exp(z), both fixed points are repelling for exp(z). But if you move slowly from base(e), to base(i), you see the lower fixed point becomes -1.862-0.411i, which is still repelling, but the upper fixed point becomes attracting; 0.4383+0.3606i. The solution I posted on MSE is based on generated the slog(z) exactly between the two primary fixed points, which is what the fatou.gp program does. My answer on MSE includes the taylor series for p(z), which turns out to have a remarkably mild singularity at the two fixed points; finding that analytic Taylor series is the basis for the fatou.gp program solution, which leads directly to Henryk's uniqueness sickle.

$\alpha(z)=\frac{\ln(z-l_1)}{\ln(\lambda_1)} + \frac{\ln(z-l_2)}{\ln(\lambda_2)} + p(z)\;\;$ Abel function
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