Hello Dmitrii -

That simple-to-formulate Carlemanmatrix-approach ( for the base 1<b<exp(1/e) using diagonalization to implement the regular tetration) can be simply extended for the limit-case base b=exp(1/e) using matrix-logarithm instead of the (well understood) diagonalization.

Only I did not know whether your approach to that specific base (let alone to the larger bases) could be reproduced by that matrix-log algebra . With that matrix-log-algebra the well known discussion concerning the iteration of f(x) = exp(x)-1 using its formal power series is properly reproduced and the -again well known- divergence of its powerseries at fractional height iterations is (hypothetically) overcome by the method of Noerlund-summation.

What I hoped/hope for was/is, that your method might be stronger (making the computation of actual values simpler than that divergent summation process), so my hope was, that I can adapt that quasi-standard procedures formally by some tweak to arrive at your results.

So I'll see, whether I can reproduce the values using your procedure of "primitive fit" and then how that can then possibly be expressed by building new elements on the conventional/known processes.

But that might now take some time, sigh...

Thanks first so far for your hints -

Gottfried

update

By the time I've computed approximate estimates for the base b=exp(1/e) tetration (always starting at z0=1) with iteration heights h=0 to 0.5. I've got this table of results

where the quote-symbol ' indicates the achieved trusted precision using summation of series with 128 terms to arrive at the coefficients c_k and 16 coefficients c_k are computed to form the polynomial which estimates the true tet_b(h) for the iteration-height argument h using the leading 16 coefficients

(02/13/2015, 02:04 AM)Kouznetsov Wrote:Well, here I see no problem; as you write that you use for that the regular tetration (which employs power series centered around the lower fixpoint (which is real)) - and that is nicely expressible by the Carlemanmatrix-algebra. So here we have automatically identity because the methods match even formally.Quote:If I find it I'd try this using Pari/GP to see myself, trying to use greater precision, whether my approach can be finetuned to arrive at your values at all.I think, first we should compare our results for some fixed value of base,

for example, for natural tetration , or for b=sqrt(2).

Then we may consider the derivatives with respect to base.

That simple-to-formulate Carlemanmatrix-approach ( for the base 1<b<exp(1/e) using diagonalization to implement the regular tetration) can be simply extended for the limit-case base b=exp(1/e) using matrix-logarithm instead of the (well understood) diagonalization.

Only I did not know whether your approach to that specific base (let alone to the larger bases) could be reproduced by that matrix-log algebra . With that matrix-log-algebra the well known discussion concerning the iteration of f(x) = exp(x)-1 using its formal power series is properly reproduced and the -again well known- divergence of its powerseries at fractional height iterations is (hypothetically) overcome by the method of Noerlund-summation.

What I hoped/hope for was/is, that your method might be stronger (making the computation of actual values simpler than that divergent summation process), so my hope was, that I can adapt that quasi-standard procedures formally by some tweak to arrive at your results.

So I'll see, whether I can reproduce the values using your procedure of "primitive fit" and then how that can then possibly be expressed by building new elements on the conventional/known processes.

But that might now take some time, sigh...

Thanks first so far for your hints -

Gottfried

update

By the time I've computed approximate estimates for the base b=exp(1/e) tetration (always starting at z0=1) with iteration heights h=0 to 0.5. I've got this table of results

Code:

`------------------------------------`

h tet_b(h)

------------------------------------

0.00 1

0.05 1.0299867588584812146

0.10 1.058880695285946978'0

0.15 1.08674287936747424'48

0.20 1.1136297821277916'69

0.25 1.139593709988212'70

0.30 1.16468319003852'55

0.35 1.1889433125893'41

0.40 1.212416036506'10

0.45 1.23514046202'20

0.50 1.2571530750'54

Gottfried Helms, Kassel