Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
regular sexp:different fixpoints
#1
When I was reading Dmitrie's & Henryk's (newest(?)) paper on superfunctions I tried to get an own impression about the differences of tetration when regular iteration is applied with different fixpoints. (see picture 4 at page 22)
I took the base b=sqrt(2) as used in the article and developed at fixpoints a2=2 and a4=4 and considered the range 2<x<4 which can be handled by both functions and (this is the tiny curve in pic 4 on that page)
However I changed the x-axis: instead of equal intervals of x I used equal intervals of h. Now the limits 2 and 4 are at the height-infinities, and there is no inherent center for h=0.
Derived from the graph of differences I used the point x=2.93462 as center-point assigning h=0 to it.
Then I defined the two functions tet2(h) and tet4(h), reflecting the two different fixpoints and plot the differences diff(h) = tet2(h)-tet4(h) for the interval -10 <=h <= 10, which is about 2.04<x<3.96
Here is the picture; the magenta line gives the value of tet2(h) which is between 4 and 2 for -inf<h<+inf, see the y-scale at the right border.

   

There are two aspects which make me headscratching.
(1) I could naively easier accept, if one of the functions proceeds faster and the other one slower; maybe with some modification, for instance a turning point at the center or something like that. But we have permanently changing signs - contradicting the assumtion of a somehow smoothely increasing function. But ok, the behave of the difference can be caused by one of the involved, say by the high (repelling)-fixpoint-version tet4.
(2) But this seems also not to hold. If I assume that at least the tet2-function is smoothely increasing, then a first guess may be, that all differences of all orders should have monotonuous behave. But that's also not true: looking at differences of high order (>24) we find sinusoidal behave in the magnitude of <1e-24. Consequence: very likely also the tet2-function, although based on the attracting fixpoint, has a sinusoidal component in that interval 2<x<4.
   
Gottfried Helms, Kassel
Reply


Messages In This Thread
regular sexp:different fixpoints - by Gottfried - 08/06/2009, 08:56 PM
RE: regular sexp:different fixpoints - by jaydfox - 08/11/2009, 06:47 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  fast accurate Kneser sexp algorithm sheldonison 38 84,488 01/14/2016, 05:05 AM
Last Post: sheldonison
  regular vs intuitive (formerly: natural) bo198214 7 12,373 06/24/2010, 11:37 AM
Last Post: Gottfried
  regular sexp: curve near h=-2 (h=-2 + eps*I) Gottfried 2 6,647 03/10/2010, 07:52 AM
Last Post: Gottfried
  sexp(strip) is winding around the fixed points Kouznetsov 8 14,720 06/29/2009, 10:05 AM
Last Post: bo198214
  small base b=0.04 via regular iteration and repelling fixpoint Gottfried 0 2,718 06/26/2009, 09:59 AM
Last Post: Gottfried
  sexp and slog at a microcalculator Kouznetsov 0 3,497 01/08/2009, 08:51 AM
Last Post: Kouznetsov
  diagonal vs regular bo198214 16 19,574 05/09/2008, 10:12 AM
Last Post: Gottfried
  Matrix-method: compare use of different fixpoints Gottfried 23 27,817 11/30/2007, 05:24 PM
Last Post: andydude



Users browsing this thread: 1 Guest(s)