Re: Bringing back an old tetration curiosity?

On Sep 17, 3:31 pm, theronr...@xxxxxxxxx wrote:
On Sep 17, 9:45 pm, mike3 <mike4...@xxxxxxxxx> wrote:



On Sep 17, 11:22 am, theronr...@xxxxxxxxx wrote:

On Sep 17, 6:12 pm, theronr...@xxxxxxxxx wrote:
<snip>
After rethinking I think that the above statement is a bit
speculative. However I'm sure that there is no continuous f: (-2; +oo)
-> R that satisfies (1), (2) and (3). Here is why:

http://img403.imageshack.us/img403/1466/sqrttetrxjx1.png

This is a 5000x5000 plot of f(x) for -1 <= x <= 0 <= y <= 1.

Theron

This is interesting. There appears to be a gentle undulation
in it, oddly enough. It's intriguing to examine that in light
of a graph of ^x (0.1) on the integers, which oscillates quite
a bit.

I'd be curious to see your program, to root out various sources
of approximation error, for example, though.

The branching doesn't look to me as a side effect of possible rounding
errors. I can send you the C source code of the program, but I would
like an independent verification of the result. If your program
produces the same result (the same branching) then this will reaffirm
the result. It is not difficult to write such a program.

Theron

How did you write the program, exactly? That's why I'd like to see it.
My program just mirrors and recurses starting with an integer and
then plots all the iterates on the graph. This gives a more-or-less
sparse
graph, though. However on closer inspection it seems you might be
right -- there are points on the graph that make it seem sort of
"fuzzy" (in the way that would suggest the behavior you observed).
Decreasing precision of the numbers used did not appear to change
the graph. Of course I didn't change the _format_ of the number to
something (that might, say, handle overflows, for example,
differently) else, however.

This would suggest then that the tetrational function is _close_ to
symmetrical but not exactly so. I'd be curious to know if the
difference
between the value of ^0.5 sqrt(2) obtained using this method and that
obtained by assuming it is the square tetraroot of sqrt(2) is small
enough that perhaps the _correct_ curve for sqrt(2)'s tetrational
function would be able to pass through it at 0.5...

Regardless, the symmetry is obviously _close_, but no _cigar_.
I'd be curious to hear some more opinions from others here on
this group. It seems though like the guy's original idea, at least
in that form, is likely another dead end...


.

Relevant Pages

  • Re: Bringing back an old tetration curiosity?
    ... of a graph of ^x on the integers, ... of approximation error, for example, though. ... The branching doesn't look to me as a side effect of possible rounding ... I can send you the C source code of the program, ...
    (sci.math)
  • Re: Bringing back an old tetration curiosity?
    ... a graph of the tetrational function to the base 2, ... that it must converge to the _infinite_ tetraroot of two, ... were to use Nelson's superlog algorithm to ...
    (sci.math)