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[Update] Comparision of 5 methods of interpolation to continuous tetration
#11
(10/15/2013, 07:14 PM)sheldonison Wrote: Gottfried, I think your solution is interesting. I see from your paper, that
the solution you get is real valued. Presumably, there is some sort of Taylor
series representation for your solution. I would be interested in seeing a
Taylor series at sexp(0) where sexp(0)=1.

The Kneser solution is defined by two basic things; which your solution would
need to match to converge.
1) it is real valued (it sounds like your function is also real valued).
2) the sexp limiting behavior in the complex plane, as imag(z) increases, is
the same as the Schroeder function solution. At imag(z)=0.75i, it is visually
the same, and convergence gets better as imag(z) increases, as defined by a 1-
cyclic scaling function that goes to a constant as imag(z) increases.
- Sheldon

Hi Sheldon -
I've still no real answer to your questions. However one more
observation might lead me into that direction.

Using the 64x64-Carlemanmatrix I got the impression that the
profile of the eigenvalues tend to something known: possibly,
with the index k they approximate to some u^(k^2), and with the
index running from -infinty to +infinity; and this reminds me
then much of your introduction of a theta-series into the
implementation of the Kneser-solution.


The problem that I cannot provide a true implementation of a
Schröder-function and of its inverse, and thus no true sexp and
slog, should be better characterized in the following way:

In the regular tetration we generate a triangular Carleman-matrix,
say C. Diagonalizing is exact and more important, constant for arbitrary
size of the matrix involved. Then we have
C = M * D * W
where D contains in its diagonal the consecutive powers of the multiplier u.
M and W are automatically of the Carleman-type, thus a proper Schröderfunction
s(x) can be built from the coefficients of the second column of M.

The inverse of the Schröderfunction, say "si(x)" can be built by the second
column of W and the consecutive powers of (s(x)*u) - this is just the
implementation of an ordinary power series.

With the polynomial-method we do initially the same: we diagonalize the matrix
C = M * D * W
But now neither M,nor W nor D are of the Carleman type, so we must evaluate
each column of M, getting a set of coefficients (not consecutive powers), then
D has a set of coefficients which seem to be unrelated with each other, and
the same is true with W.
So we dont have a schröderfunction and also not its logarithm the slog() and
we must evaluate the full matrix-product over all columns of W,D and the
second column of W.
Perhaps there is some finetuning possible though.



To the other question: here is a quick&dirty-version of a powerseries in h, which should give the
h'th iterate starting from z0=0:
f(h) = 0.938382809828*h - 0.167828379563*h^2
+ 0.287374412061*h^3 - 0.151317834929*h^4 + 0.152072891321*h^5 - 0.115906539199*h^6
+ 0.104252971656*h^7 - 0.0896484652321*h^8 + 0.0804335682514*h^9 -
0.0722083698851*h^10 + 0.0657781099848*h^11 - 0.0603034448582*h^12 +
0.0557107445178*h^13 - 0.0517589491520*h^14 + 0.0483386436954*h^15 -
0.0453431498049*h^16 + 0.0426994261734*h^17 - 0.0403475539535*h^18 +
0.0382412646892*h^19 - 0.0363431568004*h^20 + 0.0346230698175*h^21 -
0.0330562751755*h^22 + 0.0316223524107*h^23 - 0.0303042715741*h^24 +
0.0290877202845*h^25 - 0.0279605769637*h^26 + 0.0269125034199*h^27 -
0.0259346231153*h^28 + 0.0250192655171*h^29 - 0.0241597604217*h^30 +
0.0233502707006*h^31 - 0.0225856544388*h^32 + 0.0218613494467*h^33 -
0.0211732744878*h^34 + 0.0205177425741*h^35 - 0.0198913824642*h^36 +
0.0192910652121*h^37 - 0.0187138333754*h^38 + 0.0181568314451*h^39 -
0.0176172372736*h^40 + 0.0170921958071*h^41 - 0.0165787582106*h^42 +
0.0160738313658*h^43 - 0.0155741444362*h^44 + 0.0150762403807*h^45 -
0.0145765005325*h^46 + 0.0140712092755*h^47 - 0.0135566632029*h^48 +
0.0130293249244*h^49 - 0.0124860162007*h^50 + 0.0119241389267*h^51 -
0.0113419065535*h^52 + 0.0107385638363*h^53 - 0.0101145703178*h^54 +
0.00947172342934*h^55 - 0.00881320086624*h^56 + 0.00814350875574*h^57 -
0.00746833136793*h^58 + 0.00679428854523*h^59 - 0.00612861721557*h^60 +
0.00547880187039*h^61 - 0.00485218452897*h^62 + 0.00425558671715*h^63 -
0.00369497416472*h^64 + O(h^65)

Here the same for start at z0=1:

f(h) = 1.00000000000 + 1.30087612467*h + 0.613490605591*h^2 + 0.462613730384*h^3 + 0.258026678269*h^4 + 0.163196550177*h^5 + 0.0896232963846*h^6 + 0.0521088351004*h^7 + 0.0278728768121*h^8 + 0.0153988549222*h^9 + 0.00803787843644*h^10 + 0.00428567564173*h^11 + 0.00218991589639*h^12 + 0.00113674868744*h^13 + 0.000570258372500*h^14 + 0.000289775759114*h^15 + 0.000143052447222*h^16 + 0.0000714303563040*h^17 + 0.0000347673520279*h^18 + 0.0000171062518351*h^19 + 0.00000822180736115*h^20 + 0.00000399449536057*h^21 + 0.00000189817814974*h^22 + 0.000000912157817172*h^23 + 0.000000428985593627*h^24 + 0.000000204179146341*h^25 + 0.0000000951111473740*h^26 + 0.0000000448892717431*h^27 + 0.0000000207246449571*h^28 + 0.00000000970923382052*h^29 + 0.00000000444495348205*h^30 + 0.00000000206897297403*h^31 + 0.000000000939569121455*h^32 + 0.000000000434896014592*h^33 + 1.95950745547 E-10*h^34 + 9.02701746055 E-11*h^35 + 4.03579535562 E-11*h^36 + 1.85202423681 E-11*h^37 + 8.21532589062 E-12*h^38 + 3.75895249924 E-12*h^39 + 1.65398790921 E-12*h^40 + 7.55348988152 E-13*h^41 + 3.29536565950 E-13*h^42 + 1.50386648676 E-13*h^43 + 6.50048867954 E-14*h^44 + 2.96864069036 E-14*h^45 + 1.27005340533 E-14*h^46 + 5.81424087541 E-15*h^47 + 2.45836244673 E-15*h^48 + 1.13062762175 E-15*h^49 + 4.71495264346 E-16*h^50 + 2.18452094165 E-16*h^51 + 8.96002942713 E-17*h^52 + 4.19710010287 E-17*h^53 + 1.68674972326 E-17*h^54 + 8.02580312930 E-18*h^55 + 3.14409639911 E-18*h^56 + 1.52906227353 E-18*h^57 + 5.79803950799 E-19*h^58 + 2.90599482712 E-19*h^59 + 1.05635889369 E-19*h^60 + 5.51749837772 E-20*h^61 + 1.89731889651 E-20*h^62 + 1.04844959488 E-20*h^63 + 3.34780766727 E-21*h^64 + O(h^65)

No guarantee that this is correct; I just let Pari/GP expand the symbolic expression of the indicated computation with fixed x=<number> and h indeterminate.

Gottfried
Gottfried Helms, Kassel
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Messages In This Thread
RE: [Update] Comparision of 5 methods of interpolation to continuous tetration - by Gottfried - 10/16/2013, 12:54 PM

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