03/07/2012, 06:08 PM
(This post was last modified: 03/07/2012, 10:37 PM by sheldonison.)

(03/07/2012, 12:08 AM)sheldonison Wrote: I am fascinated by the concept of merged solutions on the Shell Thron boundary itself. .... the numerical method my algorithm uses cannot work on the boundary itself, where the period for one of the fixed points is real.So the algorithm, inspired by Dimitrii Kouznetsov, is to sample a bunch of different bases in the complex plane, and generate a Cauchy circle integral. Here, I sampled 52 bases, using the programs high precision mode, \p 67, and generated results accurate to ~26 decimal digits, or better, for each of those bases, for the taylor series around z=0. My goal, in picking the center, and the number of sample points, was to get the two samples closest to the Shell Thron boundary to both converge equally well, or equally poorly, and to have enough samples to get good convergance with the Cauchy circle integral. Here, the 52 samples are in magenta, equally spaced in the upper half of the complex plane. I also use the complex conjugates of those 52 points, reflected into the lower half of the complex plane. Then then 104 sample points are used to generate sexp(z) at other bases. The algorithm is to generate a Cauchy integral for each of the a_n taylor series coefficients. The algorithm generated the Cauchy integral for approximately 108 taylor series terms, which are then used to generate an extrapolated 108 taylor series polynomial sexp(z) series for some other base.

Here, I extrapolated results for five bases.

, accurate to >25 decimal digits

, accurate to ~22 decimal digits

, accurate to >25 decimal digits

, on the boundary, with "period"=5.309017.... Accurate to >25 decimal digits

, on the boundary with "period"=5. Accurate to >25 decimal digits

The accuracy was estimated by comparing B^sexp(-0.5+i) to sexp(0.5+i), for i varying from -0.85i to 0.85i.

Here is the picture for the extrapolated base with an irrationally indifferent period=5.309017.... For this base, my algorithm cannot generate results, since the base is on the Shell Thron boundary. But the extrapolation works fine. Then, from the extrapolated base, I generated a pari of exrapolated theta(z) function. This was tricky for the upper half of the complex plane, since the period is real, and it is numerically difficult to generate an accurate enough inverse Schroder and Schroder functions. I used Mike's Bell coefficient algorithm, which can generate about 40 terms, before pari-gp gets a memory failure, which is not really enough. I would need at least 200-400 hundred accurate terms to generate a theta mapping. So, I had to relax the upper theta(z) function to imaginary(z)=0.8i, due to numerical limitations. At 0.8i the extrapolated upper superfunction(z+theta(z)) and is consistent with sexp(z) to ~19 decimal digits. The extrapolated lower superfunction(z+theta(z)) is numerically straightforward, and is consistent with sexp(z) to ~27 decimal digits. I used the superfunctions, along with sexp(z) to generate this complex plane chart.

Here is the sexp(z) at the real axis, near z=0, where it is well behaved, with a radius of convergence=2.

As real z increases, the function at the real axis becomes more and more chaotic, with a smaller and smaller radius of nice behavior. Interestingly though, there are also places where graphs visually identical to the graph above can be generated, for example from z=218, to 228, because sexp(223) is nearly an even multiple of the irrational period. But, no matter how smooth they look, in these regions, sexp(z) has a poorly behaved taylor series, since in the complex plane, the function becomes very chaotic as imag(z) grows even very slightly negative. But as imag(z) grows positive in the complex plane, the function again becomes more and more well behaved, as would match the appearance of the complex plane graph.

So, at this point, we have numerical evidence that a Cauchy integral can be used to very accurately extrapolate sexp(z) around z=0, for arbitrary bases within a region in the complex plane. This Cauchy integral was used to extrapolate sexp(z) for a base on the Shell Thron boundary itself, with an irrationally indifferent fixed point, which would have otherwise been impossible to calculate using my algorithm. This base has a siegel disc, with a fractal at the analytic boundary. If the boundary of the siegel disc is mapped to the real axis, than this base's super function has a zero radius of converance at the real axis, which would include the points with f(z)=0, and f(z)=1, and would be a fractal at the real axis, but would be analytic for imag(z)>0. The merged fixed point mapping has cancelled out all of the fractal properties near sexp(z=0), but the fractal properties of the upper superfunction show up elsewhere at the real axis, and at the real axis the Taylor series is only well behaved near the origin. As imag(z) increases, both the upper superfunction and the sexp(z) behave similarly, and both are well behaved, and analytic. The merged sexp(z) solution approaches closer and closer to the upper fixed point as imag(z) increases.

The next post will be about the the extrapolated results to the base with a period=5, which is a rationally indifferent period, whose behavior is only nice in the region around z=0, and becomes chaotic as real(z) increases or decreases no matter what imaginary(z) is. For this base, with a rationally indifferent period=5, it can be proven that there is no upper superfunction.

For referance, these are the taylor series for B=1.9107783507843347325322538 + 0.33705042370189293106181810*I, which has an irrationally indifferent period=5.3090169943749474241022934 with a well behaved continued fraction, , which is actually not a golden ratio base, which I originally desired, but the base still has a well behaved continued fraction.

Code:

`a0= 1`

a1= 0.87603021541365056116984664 + 0.13008032471692635365087803*I

a2= -0.021250946094544245994982979 + 0.14342643007587819971477968*I

a3= 0.070834255634293596296394826 + 0.035166155746567142193324243*I

a4= -0.017915862554963081602439323 + 0.022299417770827164459165025*I

a5= 0.0070869354742820077704754204 + 0.0033829830028820083075630988*I

a6= -0.0044143080119538372115317067 + 0.0025005906192380288619092571*I

a7= 0.0011152454085614885854976346 - 0.0000098264079138854017724671814*I

a8= -0.00083840864042509099295429448 + 0.00025662849558257466543978876*I

a9= 0.00024819355658581378460725956 - 0.000070987964400799443047963972*I

a10= -0.00015360831642611174382166072 + 0.000033698324233098215071290265*I

a11= 0.000057668205457701602618977422 - 0.000018550219221721496316718389*I

a12= -0.000029783983385360276430406515 + 0.0000065169753160850850752501960*I

a13= 0.000012975170098880581392903812 - 0.0000038731100268853777824141925*I

a14= -0.0000061805361348595640247094473 + 0.0000014857796611434984539602050*I

a15= 0.0000028682842307374767356663523 - 0.00000079592471107782921627179016*I

a16= -0.0000013412951061644765905252752 + 0.00000034279889982566895782662590*I

a17= 0.00000063481720197760577494545003 - 0.00000016963091873306597426448420*I

a18= -0.00000029815413315353316219123371 + 0.000000078075132168311955995811340*I

a19= 0.00000014186882793188222399540980 - 0.000000037422284926647281998857067*I

a20= -0.000000067181914621764122409978112 + 0.000000017705418362137190352782436*I

a21= 0.000000032056063988141059971868987 - 0.0000000084345842970736111406898939*I

a22= -0.000000015282025774865765776734415 + 0.0000000040298473619878370249026015*I

a23= 0.0000000073134694655411576282600721 - 0.0000000019248149223017792394054566*I

a24= -0.0000000035033495624729484052267396 + 0.00000000092336690213977526569680704*I

a25= 0.0000000016818252688465622021641994 - 0.00000000044285086139748374983229721*I

a26= -0.00000000080854518972108134244989829 + 2.1302627058705047798069927 E-10*I

a27= 0.00000000038929931586616763319838796 - 1.0253481817373489889113409 E-10*I

a28= -1.8770113710921730316120804 E-10 + 4.9445555899780883904890435 E-11*I

a29= 9.0612979769664695025880120 E-11 - 2.3868083144216509371578144 E-11*I

a30= -4.3796871344870979197019744 E-11 + 1.1536736540610877080440934 E-11*I

a31= 2.1191857722171265658652164 E-11 - 5.5822067248374999829889584 E-12*I

a32= -1.0264866586085205879636856 E-11 + 2.7038920653394572051063904 E-12*I

a33= 4.9768901912645651959864977 E-12 - 1.3109799618841163798056260 E-12*I

a34= -2.4152600334154819899587276 E-12 + 6.3620934399764172245657077 E-13*I

a35= 1.1731254377963032928386288 E-12 - 3.0901673981596221923410891 E-13*I

a36= -5.7026955416048283278774795 E-13 + 1.5021621317370808428624807 E-13*I

a37= 2.7742840881808251364064066 E-13 - 7.3078247682148758003129920 E-14*I

a38= -1.3506383749208063842248715 E-13 + 3.5577543651338624863551890 E-14*I

a39= 6.5800333416637758624409081 E-14 - 1.7332656778874696808933119 E-14*I

a40= -3.2077662162397508230456995 E-14 + 8.4496685447635291097362079 E-15*I

a41= 1.5647640545028116788131633 E-14 - 4.1217899773528856929399879 E-15*I

a42= -7.6375387630196316444244105 E-15 + 2.0118259903882388226799739 E-15*I

a43= 3.7299608417133064316781009 E-15 - 9.8251968693830224786096369 E-16*I

a44= -1.8225944961126171173779948 E-15 + 4.8009484749758473516848550 E-16*I

a45= 8.9104620237224749162074654 E-16 - 2.3471303643608856504374437 E-16*I

a46= -4.3583781614834944396572130 E-16 + 1.1480529004004690210752221 E-16*I

a47= 2.1328233587266270758186165 E-16 - 5.6181312052304511507447599 E-17*I

a48= -1.0441947695604730717827668 E-16 + 2.7505434087624103059301397 E-17*I

a49= 5.1144233632166090477248687 E-17 - 1.3472049339218024442595126 E-17*I

a50= -2.5060674484791031257939137 E-17 + 6.6013041818838107097707949 E-18*I

a51= 1.2284644356695927024778266 E-17 - 3.2359334218356336445427255 E-18*I

a52= -6.0242005986901095436856816 E-18 + 1.5868519669356280136979224 E-18*I

a53= 2.9552682184979011137118537 E-18 - 7.7845568196921167647870737 E-19*I

a54= -1.4502705147564008138508763 E-18 + 3.8201991809472432025286617 E-19*I

a55= 7.1195097988706197181770603 E-19 - 1.8753705060557620763625272 E-19*I

a56= -3.4961878451526686392994130 E-19 + 9.2094087202457755611975861 E-20*I

a57= 1.7174256072962786736719248 E-19 - 4.5239200654622455635727298 E-20*I

a58= -8.4390741097412192449934994 E-20 + 2.2229607290936158578844648 E-20*I

a59= 4.1480194938531423909435060 E-20 - 1.0926417245085846939111156 E-20*I

a60= -2.0394429169198777996597870 E-20 + 5.3721551603694317187903587 E-21*I

a61= 1.0030047086524016478434906 E-20 - 2.6420434812801584928253664 E-21*I

a62= -4.9341360539169686675496014 E-21 + 1.2997149442860420441457280 E-21*I

a63= 2.4279083789699590633684417 E-21 - 6.3954239388348474607950009 E-22*I

a64= -1.1949862795435827725550778 E-21 + 3.1477485550985733507780444 E-22*I

a65= 5.8830089138250637261708164 E-22 - 1.5496605250150056265649114 E-22*I

a66= -2.8969341756568249718903724 E-22 + 7.6308876599098472168054220 E-23*I

a67= 1.4268466517197596200202064 E-22 - 3.7584879646841266046507384 E-23*I

a68= -7.0293234206338996742023346 E-23 + 1.8516097116573383893275390 E-23*I

a69= 3.4637355383618981390675866 E-23 - 9.1239746200782705618305516 E-24*I

a70= -1.7071321708154957787456481 E-23 + 4.4968176884870150380531194 E-24*I

a71= 8.4153413406814376001667834 E-24 - 2.2166916392693332423370017 E-24*I

a72= -4.1492461807449940841829943 E-24 + 1.0929298337518721812848341 E-24*I

a73= 2.0463246549496174943395562 E-24 - 5.3908967863665571554123910 E-25*I

a74= -1.0095560829869479255852945 E-24 + 2.6602414826634015954198073 E-25*I

a75= 4.9803751930402496382264439 E-25 - 1.3123964464199691856741067 E-25*I

a76= -2.4559415252898829007647481 E-25 + 6.4619324886054124930275233 E-26*I

a77= 1.2099980575626201021595812 E-25 - 3.1754840284004279640500558 E-26*I

a78= -5.9739907765311307592761373 E-26 + 1.5676010727997093759383397 E-26*I

a79= 2.9576909721142243802271205 E-26 - 7.8018084804795765364447938 E-27*I