Re: Bringing back an old tetration curiosity?

Am 20.09.2007 11:12 schrieb I.N. Galidakis:
[snip]

In general I don't expect ANY calculable tetration function to pass "through"
the tetraroots. Any tetration extension to the reals is like a mathematical
"abomination" if you will. The best "approximation" of this function so far (by
Andrew Robbins) doesn't pass through the tetraroots, so all the rest are
expected to fail in this regard.

Hi Ioannis -

this statement surprises me, since I'm just conjecturing to having
solved this problem. But, true, there is still something slippery in
the mind, when I try to verbalize.

So plase try to translate your critique into the following formalism,
which works. I've expressed it in matrix-algebra.

Assume a type of vector V(x), containing the consecutive powers of x.
V(x) = [1,x,x^2,x^3,..]

Assume also a matrix Bs, which translates
V(x) * Bs = V(s^x)

This is an elementary problem, and the matrix Bs is well defined.

Now
(1) V(1) * Bs = V(s)
(2) V(s) * Bs = V(s^s)

The form (2) is equivalent to
(3) V(1) * Bs*Bs = V(1) * Bs^2 = V(s^s) = V(s^^2)
and generally
(4) V(1) * Bs^m = V(s^^m)

for integer m.

I think, there is no discussion needed about this.

Now I've solved the eigensystem-decomposition of Bs, so that
I can define arbitrary powers of Bs, say Bs^(1/2)

It is obvious (and I showed it several times with numerical
examples) that

(5) V(1) * Bs^(1/2) * Bs^(1/2) = V(1) * Bs = V(s^^1)

and generally we have from this

(6) V(1) * Bs^a * Bs^b = V(1)* Bs^(a+b) = V(s^^(a+b))
(7) V(1) * (Bs^(1/a))^ka = V(1)* Bs^(1/a*ka) = V(1) * Bs = V(s^^k)


Since we have these operations, completely plain matrix-algebra,
I'd say, this is
a) a working model for continuous tetration
b) the most useful one.

On the other hand I know you have done a lot about this subject,
so there may be some sophisticated problem either in my
reception of tetration or of which I've read about, or in your
pointing out, which I overlook here.

Where begin your critique and my steps (1)..(7) to separate their
path?

Regards -

Gottfried

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Gottfried Helms, Kassel
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