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Attempt to find a limit point but each step needs doubling the precision...
#3
(11/03/2014, 09:41 PM)tommy1729 Wrote: Dear Gottfried,

(... [1] ...)
Maybe a few links will help.
( such as where this digital (freely available ?) paper occured , and which posts you are referring too )

(... [2] ...)
Also perhaps express the problem in notation we all use here ( " standard notation ... at this forum " ) such as slog and sexp ?

If I find the time to read carefully and understand the OP I guess I will remove this post ... on the other hand these suggestions might improve readability.

(... [3] ...)
EDIT : in particular your picture shows smooth curves , where is this jump point you talk about ??
I cant see it !

regards

tommy1729

Tommy, thanks for your consideration.

Answering your three questions:

[1]: the paper is in our database (but requires user/password for access); also at http://www.ams.org/journals/mcom/2010-79.../home.html Hmm. Seems to require also a password here; but I've the access to our database and can send the preprint to you. If you open the article you find pictures on the "physical" page 13 of the pdf.

[2]: It's also on my side, the difficulties with the notion of something (in this case with "superfunction", and how to be applied...)
My approach to tetration as iterated exponention is always: take a initial value assign a base b as an exponential expression to it and evaluate: so you have tetration of "height 1" from : . Assign b as base again, so you have tetration of height 2. Generalize this to negative, then fractional and then complex heights. Notation (For whatever reason we discuss superfunctions only using and reduce by this default the notation to b^^h ...) The generalization to fractional and complex heights is possible for bases between 1 and exp(exp(-1)) because that bases allow convergent power series in the computation and namely allow the Schröder-function-mechanism on real numbers with power series with real coefficients.

[3]: In the curve at the right hand there is the smooth thicker red curve. That is the trajectory beginning from in the near of 3.1 . Tetrating with increasing purely imaginary height it first ascends and then descends further to the right side, but does not arrive at the real axis (which is however expected to happen when the height is exactly . But the proceeding of the trajectory becomes radically stuffed: you are at the final height h_1 by about 99.9 % and still one cm away from the real axis. And if you go to 100.1 % you find the same value but below the real axis. So you conclude, the value at the final hight might be just in the middle. But that's not true. If you go nearer to to 100%-1e-6% there is another remarkable step with a horizontal component. After that, the default internal precision of the software (I do with 200 digits by default) does not suffice to improve the computation. In fact, one can proceed when 400 digits are used to come nearer to the 100%-level for the height, then 800 digits, then 1600 digits precision and the height can then approach - and one can see the tiny black circles with which I've marked that results. But obviously this type of computation cannot be stretched far more to see only a shadow of the true limit ... - conclusion: one needs an analytical approach (but I don't know how...)

If this explanations do not suffice, please ask for more.

Gottfried
Gottfried Helms, Kassel
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Messages In This Thread
RE: Attempt to find a limit point but each step needs doubling the precision... - by Gottfried - 11/03/2014, 10:46 PM

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