Tetration Gottfried Helms - Univ Kassel            07' 2007 - 2008

# A provisorical computation for newsgroup-discussion using Pari/GP

Version 1.2

Gottfried Helms, 28.06.2009

## 1.       Provisorical computation of values for tetration with a base 0<b<e-e ; example b=0.04

The following is a very provisorical discussion of the regular iteration of tetration with an unusual base 0<b<e–e. There are some facts known and discussed, see for instance [1,2,3] in the "tetration-forum", but here I consider some specific matters using an example base.

The numerical approximations may be of much improvable quality (although I used matrix-size of 64x64 which usually gives at least 10 digits of accuracy), but is not of too much concern: the article may serve as a general impression of basic behaviour when such bases are involved in tetration.

### 1.1.     Function-graphs of integer iterates of bx

First, let us see a graph for some integer iterates of Tb°h(x); here we look at heights h=0 (the linear function f(x) = x) , at h=1 , the base-function Tb(x) = bx, then at h=2 where Tb°2(x) = bbx  , then at h=3 and up to h=7. Here is such a plot for the range x = 0..1 .

Graph 1: functions y=x, y=b^x, y=b^b^x,… up to h=7 for x=0..1

The diagonal black line is the graph for height h=0, the neighboured lines are for heights h=2,4,6 ; for h=1 the graph is the thin blue antidiagonal curve from top left (x,y)=(0,1) to bottom right (x,y) = (1,0.04).

### 1.2.     Basic properties,

All curves cross the point (x,y) = (0.33747…,0.33747…) where x=y.

So the tetration with b=0.04 has a fixpoint at t0 ~ 0.33747075; this fixpoint is repelling. (We can determine its coordinates, if we iterate x=log(x)/log(b) with initial x=0.5, until required precision)

Also we find a pair of points t1,low~ 0.08960084 and t1,high~ 0.749451269594 , which serve as oscillating fixpoints ("oscillation-points"). This pair of points is attracting (in the sense, that the trajectory of iterates converges to oscillation between t1,low  and t1,high) (We can determine their coordinates, if we iterate x=bx with initial x=0.5, until required precision, and store the two oscillating values)

### 1.3.     regular iteration using repelling fixpoint

However, having no single attracting fixpoint, regular iteration for the tetration at base b=0.04 must be based on the repelling fixpoint. If we use this fixpoint t0 and build the powerseries for a schröder-function s(x') implying fixpoint shift x'=x/t0 – 1 then the range of convergence of this powerseries is of small order of about |x'|<0.74 which means also |x|<0.57

If we fix one value for x, say x=0.5, (x'~0.4816) then we can use the schröder-function and its inverse to construct a continuous height-dependent tetration-function by way of introducing the decremented exponential-function dxpt(x) or Ut(x) and fixpoint-shift of the coordinate x and reshift of the result:

f(h)      = T0.04°h(0.5)
= t0+t0*U
t0°h(0.5/t0-1)
~ 0.3375 + 0.3375* U
t0°h(0.4816).

Here the auxiliary function Ut0°h(0.4816) is expressible as composition of the schröder-function s(x'), the log u0 of the fixpoint (u0 = log(t0)), and the coefficients dk of the inverse schröder-function  s-1().

The value of s(x') at x'=0.5/t0–1 is well approximable; we get about

s(x') = 0.478492972731…

The log of the fixpoint is

log(t0) = u0 = -1.08627643894

And the first 64 coefficients d0 to d63  of the inverse schröder-function are approximately

0,                                                  1,                                                        0.260338567475,                 -0.478529124194,                 -0.255584774365,
0.322652629730,             0.240156138491,                 -0.232248120860,                 -0.219164051023,                   0.170233465505,
0.195697313050,           -0.125057579776,                 -0.171751444825,                   0.0913985445843,                0.148607214519,
-0.0661583846439,        -0.127045882554,                   0.0472595033240,                0.107494944609,                 -0.0331925996259,
-0.0901345543311,          0.0228148251384,                0.0749770660406,              -0.0152442536051,              -0.0619263505424,
0.00979601118649,       0.0508213475010,              -0.00593865411329,           -0.0414672888395,                0.00326175202614,
0.0336573864076,        -0.00145057430857,           -0.0271872771240,                0.000265840596318,          0.0218640877667,
0.000472612107592,  -0.0175116034438,              -0.000898993867300,          0.0139726973180,                0.00111211415385,
-0.0111099044922,        -0.00118377332145,             0.00880479802174,             0.00116537486625,           -0.00695664546222,
-0.00109308525310,       0.00548068441165,             0.000991820197259,        -0.00430624888757,           -0.000878292961240,
0.00337489894557,       0.000763321069661,        -0.00263864811245,           -0.000653552995827,          0.00205834213769,
0.000552745963978,  -0.00160221423009,           -0.000462699992760,          0.00124462308003,             0.000383931449349,
-0.000964967998292,  -0.000316151326996,          0.000746768441396,          0.000258598763652

### 1.4.     Series for continuous iteration from x=0.5

With this we get the series for the function f(h) with first 64 terms and v replacing the parameter uh

f(h) = 0.337470750051 + 0.161477382402*v^ + 0.0201152657952*v^2 - 0.0176917666971*v^3 - 0.00452140459481*v^4
+ 0.00273117271977*
v^5 + 0.000972709623563*v^6 - 0.000450108582019*v^7 - 0.000203240365883*v^8
+ 0.0000755372470733*
v^9 + 0.0000415505368607*v^10 - 0.0000127050787221*v^11 - 0.00000834916999943*v^12
+ 0.00000212597302083*
v^13 + 0.00000165399398802*v^14 - 0.000000352333947383*v^15
- 0.000000323747001898*
v^16 + 0.0000000576248592785*v^17 + 0.0000000627168632205*v^18
- 0.00000000926644431352*
v^19 - 0.0000000120403394897*v^20 + 0.00000000145827733290*v^21
+ 0.00000000229312107961*
v^22 - 2.23090087535 E-10*v^23 - 0.000000000433635830151*v^24
+ 3.28227155570 E-11*
v^25 + 8.14792413631 E-11*v^26 - 4.55579733321 E-12*v^27 - 1.52215041555 E-11*v^28
+ 5.72899487311 E-13*
v^29 + 2.82867756201 E-12*v^30 - 5.83335761950 E-14*v^31 - 5.23142373948 E-13*v^32
+ 2.44765977804 E-15*
v^33 + 9.63244803577 E-14*v^34 + 9.96289882864 E-16*v^35 - 1.76637274388 E-14*v^36
- 4.33899084576 E-16*
v^37 + 3.22691718632 E-15*v^38 + 1.22894515234 E-16*v^39 - 5.87447449063 E-16*v^40
- 2.99504125468 E-17*
v^41 + 1.06593131877 E-16*v^42 + 6.75073462785 E-18*v^43 - 1.92823882609 E-17*v^44
- 1.44974143627 E-18*
v^45 + 3.47813853503 E-18*v^46 + 3.01176160687 E-19*v^47 - 6.25694497686 E-19*v^48
- 6.10630053223 E-20*
v^49 + 1.12272943819 E-19*v^50 + 1.21505839672 E-20*v^51 - 2.00977269600 E-20*v^52
- 2.38189075379 E-21*
v^53 + 3.58950165287 E-21*v^54 + 4.61230201111 E-22*v^55 - 6.39717609047 E-22*v^56
- 8.83980688494 E-23*
v^57 + 1.13777581755 E-22*v^58 + 1.67937661392 E-23*v^59 - 2.01968150107 E-23*v^60
- 3.16621607739 E-24*
v^61 + 3.57854976477 E-24*v^62 + 5.92956880693 E-25*v^63
+ O(
v^64)

If h=0 then vh=uh=1, then this seems to converges to 0.5.

If h–> –oo then uh–>0 and only the first coefficient remains, which is the value of the (repelling) fixpoint. Since the series converges for some continuous range h+0 to h+1, (for instance –2<=h<=–1) we can compute all fractional iterates from the results of this range appending exact integer iteration only.

If h–> +oo and is even (odd) then we arrive at the high(low) value of the pair of fixpoints t1,high ~0.74945…  (t1,low ~0.08960…)

### 1.5.     Fractional real heights give complex values

Fractional heights give complex values because this requires fractional powers of the negative value of log of the fixpoint: log(t0) = u0 = –1.08627643894… , so the real continuous interpolation between the integer iterates (based on this method of "regular iteration"/ "matrixpower-iteration") produces a graph through the complex plane. Here we deal only with the principal branch of the logarithm.

Here is a plot of such a graph for h=0 … 4 in steps of 1/20. The symbol y is here identified with f(h).

Graph 2: fractional heights for f(h)=T0.04°h(0.5) by method of regular iteration (matrix-method, size=64x64) using the repelling fixpoint t0 for parameter shift

(data: see appendix)

We see counter-clockwise for the shown handful of integer iterates:

f(0) = 0.5                    = x
f(1) = 0.2                    = 0.04
0.5
f(2)= 0.525…              = 0.04
0.2           = 0.040.040.5
f(3)= 0.184…              = 0.04
0.525…       = …
f(4)= 0.552…              = 0.04
0.184…        = …

connected by trajectories which pass the complex plane. The shape is that of a distorted spiral. The continued spiral will be limited at left and right hand by the pair of attracting oscillation-points t1,low and t1,high;

### 1.6.     Very sneaky trajectory with increasing iteration-height

Wrong hypothese in first version: I expected, that, when h diverges to +oo, the spiralling distorts to the shape of a rectangle (dotted red lines) of infinite height, meaning divergence to +/–oo * I .The trajectory shows a much more chaotic behaviour. Far from approximating a rectangle when h->+oo the spiral sneakes around and even tends to encircle the oscillation-points. The plot shows extrema at some heights, so in the near of h=10.5 and h=11.5; I don't know yet, whether we even get infinities there. Also I did not consider the consequences of multivaluedness and/or indeterminacies of fractional powers of negative bases so far.

Graph 3: fractional heights for f(h) approaching heights up to h=14 showing tendency to extremely uneven behave

### 1.7.     Superlogarithm and trajectory-shortcuts using complex heights

Looking at the trajectory in graph 2 we could ask, whether there is a complex iteration, which allows to proceed directly along the real axis from 0.5 to, say, 0.6 .

Graph 4: directly traverse iterations using complex-valued "heights"? Instead of spiralling along the trajectory of real heights through the complex plane (deep blue spiral trajectory) walk directly on the real-axis(red arrows)

This introduces the need for a logarithm-like function, which determines the "height" of a value, or more precisely, the height-distance between two values. The term "slog" or "superlog" for such a function is fairly well introduced; but since I compute its values by the diagonalization/schröder-function I'll use a distinct name and call this "iteration-height" or simply "height" or "hgh"-function.

The hghb()-function is essentially the log of the schröder-function. The schröder-function expresses increase of iteration-height h by h'th powers of u, in the sense that for a certain height h we have some relation uh -> f(h) and for iteration of an additional height j we have uh*uj = uh+j -> f(h+j) . For the (fixpoint-)unshifted version of the schröder-function we introduce the symbol σ such that for the following we use

σ(x) = s(x') = s(x/t-1)

where s(x') is the function as described above.

The schröder-function has no inherent zero; so it can be/must be normed. For the current article we assign to the value x0=0.5 = f(0) the height h=0. The unnormed schröder-function assigns the value σ(0.5) = 0.478492972731, which is also  σ(0.5) = u-0.006176346… + 0.2344679…*I . To have hghb(0.5) = 0 we must normalize

hghb(x) = logu(σ(x)/σ(0.5)) = logu (σ(x)/0.478492972731).

Unfortunately the log is multivalued and especially the attempt to work with a log of a negative base (u~-1.08…) as we do it here cannot supply unambiguous results. However, some first results are quite meaningful.

For instance,

σ(b0.5) = σ(0.2) =   –0.519775642476

and

σ(b0.5)/σ(0.5) =   –1.08627643894 = u1

thus

hgh(b0.5) = 1

as expected. We find such reasonable results also for heights 0≤h≤1 .

It is more interesting to increase x0=0.5 to x2 = bb0.5~ 0.525305560881 and see, whether we get the expected height-difference 2 this way. But here we run into the ambiguity of logarithms. The computation gives σ(x2)= 0.564620033956 from what

σ(x2)/σ(x0)= 1.17999650180 = u0.001386836 - 0.0526474*I

and the height-value would be

hghb(x2) = log(σ(x2)/σ(x0)) / log(u) = 0.001386836 - 0.0526474*I

thus a complex iteration height to proceed from x0=0.5 to x2~0.5253 directly along the real axis.  However, respecting the fact, that x2 is additionally one "winding" around the fixpoint from x0, and inserting 2*π*I in the log-formula for the height we get

hghb(x2) = ( log(σ(x2)/σ(x0)) + 2*π*I ) / log(u) = 2.000 + 0.000*I

which is what we originally expected for the height function.

Gottfried Helms, 26.6.2009

## 2.       Appendix:

### 2.1.     Data for graph of fractional heights

 h=height real(f(h)) imag(f(h)) 0.00 0.500000 0.000000 0.05 0.499530 0.023067 0.10 0.496798 0.046554 0.15 0.491428 0.070735 0.20 0.482719 0.095754 0.25 0.469596 0.121448 0.30 0.450635 0.147015 0.35 0.424374 0.170533 0.40 0.390228 0.188593 0.45 0.349996 0.196871 0.50 0.308675 0.192424 0.55 0.272536 0.176313 0.60 0.245431 0.153238 0.65 0.227361 0.128258 0.70 0.216146 0.104455 0.75 0.209404 0.082918 0.80 0.205374 0.063638 0.85 0.202943 0.046198 0.90 0.201456 0.030094 0.95 0.200541 0.014843 1.00 0.200000 0.000000 1.05 0.199751 -0.014859 1.10 0.199807 -0.030168 1.15 0.200289 -0.046408 1.20 0.201476 -0.064143 1.25 0.203923 -0.084044 1.30 0.208680 -0.106850 1.35 0.217642 -0.133116 1.40 0.233885 -0.162445 1.45 0.261201 -0.191932 1.50 0.301468 -0.214942 1.55 0.350717 -0.223578 1.60 0.399740 -0.214890 1.65 0.440605 -0.192994 1.70 0.470777 -0.164536 1.75 0.491598 -0.134415 1.80 0.505503 -0.105022 1.85 0.514610 -0.077095 1.90 0.520398 -0.050569 1.95 0.523793 -0.025044 2.00 0.525306 0.000000 2.05 0.525125 0.025135 2.10 0.523155 0.050962 2.15 0.518972 0.078107 2.20 0.511711 0.107180 2.25 0.499848 0.138622 2.30 0.480915 0.172250 2.35 0.451438 0.206211 2.40 0.408086 0.235222 2.45 0.351638 0.249873 2.50 0.291801 0.241753 2.55 0.243192 0.213154 2.60 0.212700 0.176159 2.65 0.196898 0.140931 2.70 0.189623 0.111007 2.75 0.186548 0.086157 2.80 0.185368 0.065165 2.85 0.184967 0.046867 2.90 0.184814 0.030352 2.95 0.184653 0.014918 3.00 0.184355 0.000000 3.05 0.183858 -0.014908 3.10 0.183143 -0.030315 3.15 0.182236 -0.046808 3.20 0.181252 -0.065137 3.25 0.180506 -0.086351 3.30 0.180811 -0.111967 3.35 0.184190 -0.144066 3.40 0.195404 -0.184665 3.45 0.223638 -0.232261 3.50 0.278406 -0.274413 3.55 0.353681 -0.289604 3.60 0.425345 -0.270857 3.65 0.476914 -0.232520 3.70 0.508842 -0.189973 3.75 0.527591 -0.150187 3.80 0.538570 -0.114657 3.85 0.545088 -0.082862 3.90 0.548990 -0.053807 3.95 0.551272 -0.026492 4.00 0.552437 0.000000