02/13/2015, 12:23 PM

(02/13/2015, 10:09 AM)Gottfried Wrote: ..Good idea. Try to plot at least one complex map with your algorithm.

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.

Quote:If they match formally, they should give the same resultsQuote:I think, first we should compare our results for some fixed value of base, for example, for natural tetration , or for b=sqrt(2).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.

Then we may consider the derivatives with respect to base.

and the same maps. Confirm this and use the simplest one.

Quote: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.I think, the Carleman is complicated and slow compared to the asymptotic expansions.

Quote: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.I have declared, that with methods, described in my book, we can construct superfunction of ANY growing holomorphic function.

Including the exponential to the base you like.

Quote: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.Gottfried, I appreciate your interest, but:

1. The abilities of participants of this Forum to bring the new algorithms for evaluation of superfunctons greatly exceed by abilities to test them.

2. There is no reason to drill, analysing somebody's algorithm, until it is implemented and tested.

These follow from the TORI axioms.

Quote:I think, first, you may download my codes and test, that they run as they are supposed to. I can help you in this.

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.

Then you may implement any other algorithm and compare the efficiency.

Quote:I shall be glad, if you confirm my code (or if you indicate an error there). If it gives you a hint, how to construct an algorithm faster and/or more accurate – even better! Go ahead!

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.

Sincerely, Dmitrii.