tetration from alternative fixed point

[sheldonison]

I can make graphs of the real contour of the superfunction developed from the alternative fixed point, with singularities at 0, 1, e, e^e, e^e^e ....

I think we need to go into the details here.
If I compute the Abel function, I dont get singularities at 1,e,e^e,... for the secondary fixed point.
I use something similar to the formula:
where c is the derivative at the secondary fixed point e[2] and is the branch of the logarithm with imaginary part between and , which implies that .

Henryk, I'm using the same equation you are. Instead of "-n", I mulitply by c^n before taking the log_c. This is mathematically equivalent, but it seems to help identify a unique branch point independent of the size of "n". Perhaps I can explain what I was trying to say with a graph of the SuperFunction (developed from the secondary fixed point), showing some contour lines. To generate these contours lines in the SuperFunction, I used the inverse super function with img(z)=-3pi*i contour with real(z) varying from . Then I graph the exponent of that contour line, which is repeated one unit away, with img(z)=0 and real(z) varying from - to 0, and the exponent of that contour line with img(z)=0 and real(z) varying from 0..1. The three contours are disconnected from each other, which I think you were pointing out to begin with. The Reimann mapping will not be analytic at the boundary points.

The pattern repeats, as shown by the "alt 3pi*i contour line." Each contour line would have alternates repeating endlessly in the diagonal direction. If you follow a path on the Reimann surface of the inverse superfunction from to , the path you take will be along a diagonal. But yes, there will be singularities at 0,1,e,e^e, ... but only two of them will be of interest to the Riemann mapping. I don't think I'm explaining it very well, apologies.

There are many other paths that can be be graphed in the SuperFunction from the alternate fixed point. I included the 2pi*i contour, and the pi*i contour as well, though they aren't my primary concern. They can also be used to generate repeating contour lines. This graph is in stark contrast to the superfunction from the primary fixed point, where the consecutive exponentiations fit each other like a glove, allowing the Riemann mapping to cancel out the singularity.
- Sheldon

[bo198214]

Henryk, I'm using the same equation you are. Instead of "-n", I mulitply by c^n before taking the log_c. This is mathematically equivalent,

I think it is not "equivalent" in the sense of "equal", because complex logarithm does not satisfy log(zw)=log(z)+log(w). I guess my formula lands on a completely different branch of slog, hm.
Well I will try to reproduce your curves, with your formula. And dont think that my understanding is that much deeper, perhaps we indeed can develop a secondary fixed point real analytic slog, if we dont start with a "plain" initial region, but with an overlapping initial region/manifold. I am really interested in this unitude/multitude topic. (For example what happens with Kouznetsov's method if we just plug in the secondary fixed point pair?)

[sheldonison]

I think it is not "equivalent" in the sense of "equal", because complex logarithm does not satisfy log(zw)=log(z)+log(w). I guess my formula lands on a completely different branch of slog, hm.
Well I will try to reproduce your curves, with your formula. And dont think that my understanding is that much deeper, perhaps we indeed can develop a secondary fixed point real analytic slog, if we dont start with a "plain" initial region, but with an overlapping initial region/manifold. I am really interested in this unitude/multitude topic. (For example what happens with Kouznetsov's method if we just plug in the secondary fixed point pair?)

The path from the real axis to the secondary fixed point is much less direct. Kouznetsov's method requires three consecutive exponents, (he uses 0, 1, e), and their vertical (along increasing imaginary) paths to the fixed point. The path to the primary fixed point is fairly direct. But not the path to the secondary fixed point. For example, consider the path from real=ln(0.5), 0.5, and exp(0.5) to the secondary fixed point (via the i*3pi contour). These are the real values near the maximum of the i*3pi contour, (see graph in earlier post). Here is a graph of the first two. The third, exp(0.5) path to the secondary fixed point is already very chaotic, with img varying in the hundreds, so I left it off the graph.



Notice how much more direct the path is to the primary fixed point is. Using this path to the primary fixed point as a seed, I would imagine Kouznetsov's method would converge very nicely. But it is difficult to imagine Kouzenetsov's method converging to anything meaningful for the path to the secondary fixed point (especially considering the even more chaotic exp(0.5) path). I also looked at the path to the secondary fixed point from i*pi contour, but that appears to be even less well behaved than the path via the i*3pi contour.
- Sheldon

[bo198214]

But not the path to the secondary fixed point. For example, consider the path from real=ln(0.5), 0.5, and exp(0.5) to the secondary fixed point (via the i*3pi contour). These are the real values near the maximum of the i*3pi contour, (see graph in earlier post). Here is a graph of the first two.

*nods* This is what I mean that we dont have an initial "plain" region (and I guess regardless how we choose the first curve, the second will always overlap the first or itself). In the best case we have some initial manifold (self overlapping region). But thats in the moment out of my sight how to apply Kouznetsov's method to a manifold.

PS: Dont be shy to upload pictures to the forum. I consider it a long term archive (though a very unsorted yet), while I dont know how long referenced sites would live.

[bo198214]

Kouznetsov's method requires three consecutive exponents, (he uses 0, 1, e), and their vertical (along increasing imaginary) paths to the fixed point.

As I think a second time about it, why does he need paths to the fixed point?
He integrates along the paths Re(z)=1, Re(z)=-1.
He merely forces the value of the superfunction to be the fixed point for imaginary part going to infinity. How the path behaves while going to the fixed point is not essential for his computation, isnt it?

[sheldonison]

Kouznetsov's method requires three consecutive exponents, (he uses 0, 1, e), and their vertical (along increasing imaginary) paths to the fixed point.

As I think a second time about it, why does he need paths to the fixed point?
He integrates along the paths Re(z)=1, Re(z)=-1.
He merely forces the value of the superfunction to be the fixed point for imaginary part going to infinity. How the path behaves while going to the fixed point is not essential for his computation, isnt it?

After reading the cauchy computation thread, I think the Cauchy algorithm is sensitive to getting a reasonable initial guess that is close enough to get convergence, and is also sensitive to updating the nodes in in an order that helps guarantee convergence.

Are there any links on the forum with Riemann (Knesser's solution) mapping results? I remember Jay posted some results. I've been toying with very simple iterative Riemann mapping, and was able to get some semi-reasonable results.

[sheldonison]

....To generate these contours lines in the SuperFunction, I used the inverse super function with img(z)=-3pi*i contour with real(z) varying from . Then I graph the exponent of that contour line, which is repeated one unit away, with img(z)=0 and real(z) varying from - to 0, and the exponent of that contour line with img(z)=0 and real(z) varying from 0..1. The three contours are disconnected from each other, which I think you were pointing out to begin with. The Reimann mapping will not be analytic at the boundary points.
....
- Sheldon

The complex periodicity is incorrect in this graph, which effects the "alternative 3*pi*i contour" graph. For the period I incorrectly used:
period = 2Pi*i/L
The correct value is
period = 2Pi*i/(L-2*Pi*I)

This is because the correct equation for the periodiicty is 2*Pi*i/ln(L)
Because L>2Pi*I, the primary ln(L)=L-2*Pi*I. The correct complex period=1.3769+2.1751*I.

I haven't verified the rest of the graph, but otherwise, I'm still using the same equations as I used when I made this graph, and those equations should have given correct values for the other complex contours. By the way, the equations are posted here. I found the problem when I wrote a pari-gp script for the secondary fixed point. With the fix, the alternative contour no longer fits snugly against the primary i=0 contour. I'm still interested in seeing if there's any way to Riemann map the contours back to a well defined real axis, but I still assume that it is not possible.
- Sheldon

[sheldonison]

....To generate these contours lines in the SuperFunction, I used the inverse super function with img(z)=-3pi*i contour with real(z) varying from ...

.... With the fix, the alternative contours no longer fits snugly... I'm still interested in seeing if there's any way to Riemann map the contours back to a well defined real axis, but I still assume that it is not possible.
- Sheldon

I'm pretty sure it is possible to generate an analytic sexp(z) function from the alternative fixed point after all! Looking at my older posts, I was also very very very close to seeing the solution 15 months ago. The problem is that there is more than one way to unwrap the inverse Schroder function into the complex plane, to generate the complex superfunction. From the best graph I previously posted, the correction for how to to unwrap the inverse Schroder function is to rotate the graph, and shrink it, so that superf(z+1)=exp(super(z)). But it would be better to just start over! Anyway, I have much prettier pictures this time, because I'm using Mike/Andy's complex graph coloring scheme.

Let's start with the complex superfunction from the secondary fixed point, L=2.0623 + 7.5886i. Here is a color plot of the complex superfunction. The period of the complex superfunction from the secondary fixed point is approximately 1.3769+2.17514i.
   
If you notice the light grey contour, you're looking at where the complex superfunction traces out the real number line from roughly -infinity to 4,000,000, or roughly from sexp(-2) to sexp(3), or five periods of z+theta(z). The negative real numbers are graphed in cyan. One more unit to the left of the cyan/grey contour, would be the 3pi i imaginary contour. Notice, that its instead of , because for this alternative solution, . The grey contour needs to get z+theta(z) mapped, so that this grey line becomes the real axis of the alternative sexp(z). I don't know how to calculate the theta(z) mapping, or the mathematically equivalent Riemann mapping, because this alternative sexp(z) function is not nearly as well behaved as the sexp(z) from the primary fixed point. Notice how quickly the function starts misbehaving as real(z) increases and imag(z) increases, above the grey contour. But in theory, it should be possible to calculate a theta/Riemann mapping, which would generate a 1 to 1 bijection between the superfunction from the secondary fixed point, and the upper half of the complex plane. I have some ideas for how to calculate it, although the existing Kneser.gp algorithm will not converge.

For comparison, here is the sexp(z) from the primary fixed point. Notice how nicely it is behaved, especially as imag(z) increases away from the real axis, and the function quickly converges to the primary fixed point! I also included the equivalent grey contour, for the real number line from roughly -infinity to 4,000,000, or sexp(-2) to sexp(3).
   

Here is the path from log(0.5) to the secondary fixed point. I didn't even try to include all of the path from 0.5 vertical to the fixed point. The path is even more chaotic than in my earlier post, with the rotated graph superfunction graph.
   

Finally, here is what the alternative sexp(z) graph would probably look like. As I said, I haven't calculated it yet, but this would have the requisite sexp(-3)..sexp(-2)=3pi i contour, where sexp(z) at integer values of z has the "z" term coefficient equal to 0, and the z^2 term coefficient also equal to zero.
- Sheldon
   

[sheldonison]

I'm pretty sure it is possible to generate an analytic sexp(z) function from the alternative fixed point after all....

I generated a taylor series and theta mapping, from the secondary fixed point. The complex plot is very pretty, and shows the z^3 pattern around sexp(z=-1). At the real axis, visually the sexp_l2(z) function looks as predicted in the previous post (see approximation graph). This sexp_l2(z) function is also analytic everywhere in the upper and lower half of the complex plane, with singularities at the real axis for integer values of z<=-2. So, this is another different analytic tetration solution for base(e), which meets all of the same requirements as the preferred solution, but obviously looks very different at the real axis, and in the complex plane, since it converges to the secondary fixed point, , as imag(z) increases or as real(z) decreases. The period of the superfunction ~=1.3769+2.1751i . Here, for the log(log(L2)), we use . Initially, the mistake I made was to use , which is a correct alternative way to unwrap the inverse schroder function to the complex plane to generate the superfunction, but it does not allow the theta(z) mapping to generate sexp(z).

Like the solution using the primary fixed point, the limiting value for the lower half of the complex plane is the conjugate of the L2.
   
Here is the graph, at a radius of 0.5, around z=-1, showing the three loops around the unit circle.
   
Here is the taylor series, also generated around z=-1. I also generated a z+theta(z) for imag(z)>0.1i. To get the two series to converge towards each other, I had to get a very good seed value, and even then, convergence was very slow, requiring perhaps 50 iterations to get these results. The algorithm I used to generate a seed value was to start with sexp(z) from the primary fixed point, and use . I used that to generate an initial theta(z) mapping, which still required tweaking before I could get convergence. Also, I was only able to use a radius of about 0.7 for the sexp(z) function about z=-1, so I wasn't sure if that would converge or not. To help improve convergence, I needed a better initial seed. So I also had to generate a real valued Fourier transform, , where theta2(z) had about a half a dozen terms. Both of these functions were iterated against . The taylor series below is accurate to about 32 decimal digits, when compared to for imag(z)>0.1i. . After each iteration generating the sexp_l2 approximation at z=-1, from the superfunction approximation, I forced a0=0, a1=0, and a2=0, which was required for convergence.

Code:

sexp from 2nd fixed point taylor series, generated around sexp(-1)=0.
a0=   0
a1=   0
a2=   0
a3=   6.0525428091015848467396904114867
a4=  -0.60849523747536175511806501610675
a5=  -5.8574541962938452347169436100018
a6=  -5.8154757843204092671165810360382
a7=   2.5347219528124619394630198728768
a8=   6.4668821970795532719465378531827
a9=   4.4871156839279985057981689319166
a10= -2.3500517589865009654200487701366
a11= -4.8774610594145425557598915202919
a12= -3.2684402012509980035269413156892
a13=  1.3654919521689999892599664565609
a14=  2.9823665195110101683443466612379
a15=  2.2661831696822414547275937778816
a16= -0.52913207870690413983768829488530
a17= -1.5156285520432398053435680304692
a18= -1.4572517446695661697863665832605
a19=  0.094286830609106271622763115792468
a20=  0.61110636950519339942198876521363
a21=  0.86271978601510899914905441427631
a22=  0.038974421486596403801894699769519
a23= -0.14905777043601514522125408517461
a24= -0.47896585094427931499121014795675
a25= -0.030389143449596281145270404911561
a26= -0.042825998034817123208188951486868
a27=  0.26235902102501291355107613937971
a28= -0.014143244950329277386417867196816
a29=  0.10059827919484809141333725260695
a30= -0.15454513160875109801594287654923
a31=  0.050006138519215390765363753443150
a32= -0.10520795919021832977856144839921
a33=  0.10624377494623484634694597176958
a34= -0.068484597990327511436726680258115
a35=  0.094944460228304998109629565511086
a36= -0.085693727084796985310665688533151
a37=  0.074122213721890768173136590316211
a38= -0.083843984677348783907640163390188
a39=  0.076235966508003484127877597757711
a40= -0.073069389756946858076727994027333
a41=  0.075222762588118941471666671581810
a42= -0.070612019609800271404236550635959
a43=  0.069441350373969278684007195642263
a44= -0.068881006958076546205688133213681
a45=  0.066217164743985940967056670962088
a46= -0.065261808225731542796012464732002
a47=  0.064014547545150315863062824154253
a48= -0.062316195718567494332301728926411
a49=  0.061294332906226487840725743704704
a50= -0.060029039396985886662256396924140
a51=  0.058764049550308815974907383174684
a52= -0.057730167455123192197602214419251
a53=  0.056599488372003113915391804142194
a54= -0.055541057151639765193298395532515
a55=  0.054560157576525914731163221375545
a56= -0.053565536674136590005525205437557
a57=  0.052629716197772950980480888304674
a58= -0.051728553326544954734152425491973
a59=  0.050844493606782859378117101810769
a60= -0.050000501525925342307229440392680
a61=  0.049181281229267069678508024865434
a62= -0.048386033785257460166130157341700
a63=  0.047619527381412221049137805806908
a64= -0.046875076515792168847026835482228
a65=  0.046153557896303522805721778538128
a66= -0.045454760295123160291375736294242
a67=  0.044776065510160199491799413861579
a68= -0.044117595385215902555503712705649
a69=  0.043478330251192648438922293764535
a70= -0.042857106284648360469480988317676
a71=  0.042253521301300292210596172100789
a72= -0.041666683066062752366849813544716
a73=  0.041095876145594636984485302734057
a74= -0.040540545496565613945077962499150
a75=  0.040000002117742705825994214770159
a76= -0.039473680162673857504905441992248
a77=  0.038961041511940238193398136688527
a78= -0.038461538061213742686998410078651
a79=  0.037974682757072796139145638913056
a80= -0.037500000856492225977549714336736
a81=  0.037037036646204616366175962080078
a82= -0.036585365813093889607615969483647
a83=  0.036144578520061409527864163769823
a84= -0.035714285553913653372657407963149
a85=  0.035294117694663812812247714829208
a86= -0.034883720959274005451397406041107
a87=  0.034482758574956584392754574654857
a88= -0.034090909117630668844739474066681
a89=  0.033707865165393714267654270592895
a90= -0.033333333324472619467885396896417
a91=  0.032967032976007199844301758708358
a92= -0.032608695648281905690683339130091
a93=  0.032258064515612316681201977767890
a94= -0.031914893619135479632053443899399
a95=  0.031578947366827529439356158688524
a96= -0.031250000000463649984311504584008
a97=  0.030927835051828540795825673426539
a98= -0.030612244897519756304248381285685
a99=  0.030303030303287137214584066663769
a100= -0.029999999999966282031229404541557
a101=  0.029702970296949837283873331539097
a102= -0.029411764705965021657699041094067
a103=  0.029126213592195908278373798109699
a104= -0.028846153846150793259645857652719
a105=  0.028571428571446783989204643005906
a106= -0.028301886792438526629856969692978
a107=  0.028037383177574696251348668866060
a108= -0.027777777777779823427720230977851
a109=  0.027522935779812846337183948157237
a110= -0.027272727272729551533867006685424
a111=  0.027027027027026594871373241617209
a112= -0.026785714285713708587427338763126
a113=  0.026548672566372367355487980292010
a114= -0.026315789473683846658285369595985
a115=  0.026086956521739115588647915103517
a116= -0.025862068965517422422639852899428
a117=  0.025641025641025545847388991708705
a118= -0.025423728813559251424682876256784
a119=  0.025210084033613403390708306581287
a120= -0.025000000000000210634068453080186
a121=  0.024793388429752111676536615771926
a122= -0.024590163934425784207269337708914
a123=  0.024390243902438972919951753608139
a124= -0.024193548387097878008656721979599
a125=  0.023999999999999874830826012878455
a126= -0.023809523809522177198866059223151
a127=  0.023622047244095016547447625736730
a128= -0.023437500000004730528259615801748
a129=  0.023255813953486943095952593420534
a130= -0.023076923076917084925513343342322
a131=  0.022900763358783736467796311748204
a132= -0.022727272727291509099320846247737
a133=  0.022556390977433940347123100851247
a134= -0.022388059701470332273740288257628
a135=  0.022222222222254205500382914734533
a136= -0.022058823529479415282418061328820
a137=  0.021897810218922467761077074543046
a138= -0.021739130434698909152627982450815
a139=  0.021582733813118207956767051755867
a140= -0.021428571428785024559533003066287
a141=  0.021276595744385680497554700219427
a142= -0.021126760563064445336698085322633
a143=  0.020979020979823863900839059151042
a144= -0.020833333333871461238610179305475
a145=  0.020689655170932338970321143814915
a146= -0.020547945204338179052061514895723
a147=  0.020408163268868439831447630713248
a148= -0.020270270270944964825128555194145
a149=  0.020134228180832249842369921195487
a150= -0.019999999996382145768944457088909
a151=  0.019867549683848916030390073901517
a152= -0.019736842101884200464284895313679
a153=  0.019607843104935080898558359139368
a154= -0.019480519472747153686365539175781
a155=  0.019354838769987628622500042979499
a156= -0.019230769194904427325515899160343
a157=  0.019108280114821735333622468050053
a158= -0.018987341782410682797657178061368
a159=  0.018867924762827561642329485236602
a160= -0.018749999782550921947274006260191
a161=  0.018633539801386849975790975198512
a162= -0.018518518784548856555456574041866
a163=  0.018404908859539817637569621034230
a164= -0.018292681833246517878352101357825
a165=  0.018181816011532742375200430960923
a166= -0.018072291311787620165426068307671
a167=  0.017964075079476270315483234073891
a168= -0.017857137826157011051469312107843
a169=  0.017751471802095868856191038213042
a170= -0.017647072166987844489621007896549
a171=  0.017543870851218341724085525290953
a172= -0.017441838139502430481295643756101
a173=  0.017341018294974304424367295663764
a174= -0.017241450709916318715211642676403
a175=  0.017142892764765992718387963486856
a176= -0.017045355278105614225544052063673
a177=  0.016949106072439699278655991418933
a178= -0.016854276424282969507424937927475
a179=  0.016759868130023045438921363043654
a180= -0.016666214315116921424792262275298
a181=  0.016574604276500489383923106468323
a182= -0.016485031090073264715644393566068
a183=  0.016393526409814549014622248699611
a184= -0.016302232081226604371621173580053
a185=  0.016217205012887436111175962990360
a186= -0.016135166958570918826264050175108
a187=  0.016041578337246276076860565681463
a188= -0.015947465741735751400414051023663
a189=  0.015880811347614127000659372647187
a190= -0.015812268974115238817285732324294
a191=  0.015693934950496055261552846318831
a192= -0.015578762616167368077787740783347
a193=  0.015590629166910915436453027039959
a194= -0.015540807357621775424779779754165
a195=  0.015292819039848609300335564328156
a196= -0.015101732862617970858553288354058
a197=  0.015474986702580039507486900124995
a198= -0.015381046865367596879509632876324
a199=  0.014517304156196916116826663108209

[sheldonison]

I generated a taylor series and theta mapping, from the secondary fixed point....

Below, there are graphs of the sexp(z) from the secondary fixed point, at the real axis, from sexp(-1.5) to sexp(1.5). I also graphed the first and second derivatives, and the equivalent functions from the primary fixed point. Notice how the derivative goes to zero at integer values of z.
   
I was able to get fairly clean convergence using two different algorithms, both with identical results. The simplest algorithm, with the quickest convergence required an initialization, very similar to the initialization used in the my kneser.gp program, followed by an initial approximation
. The initial sexp(z) need only have three terms in its Taylor series. Then this initial approximation required an additional 42 iterations, generating a theta(z) approximation from the secondary fixed point, followed by an sexp(z) approximation, from both the theta(z) and the sexp(z) approximation around z=-1. This gave results accurate to ~32 decimal digits. At each iteration, I forced the first three terms in the Taylor series to zero.
- Sheldon