(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