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short compilation:fractional iteration-> eigendecomposition
#1
Hi -

I've a short article, still in drafts-mode (no references, no proofs, no equation-numbering) in which I try to collect my heuristics about fractional iteration, a bit more general. I'd also like to get collaboration to improve this beast...

fractional iteration

Gottfried

P.s. it's not attached, since I'll have much editing/enhancing to do... I'll attach it here, if it has a more serious base.
Gottfried Helms, Kassel
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#2
So far, I like it!

Can some of this content can go in the FAQ? or was that the intent? As you will notice, there is very little in the Iteration chapter of the FAQ. Maybe you could help with this. Smile

Andrew Robbins
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#3
andydude Wrote:So far, I like it!

Can some of this content can go in the FAQ?

Why not? Only I've got many times some harsh criticism because of my lack of formalism, so normally I don't try to push my tractats into more than my cabinet of "mathematical miniatures"... The text is in word-format, and I'm a bit tired of writing since weeks, so I wouldn't like to be owed to transfer this also into mimetex :-( Ideas?

Quote:As you will notice, there is very little in the Iteration chapter of the FAQ. Maybe you could help with this. Smile
... would be a nice feature... Smile

Well, I'm scanning my literature for (understandable) portions of related theorems, proofs and examples as well and also internet-sources for links to insert more into the article. Any help with this (which, however, shouldn't make it a experts-only article) I would much appreciate.


Gottfried
Gottfried Helms, Kassel
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#4
I just skimmed through and agree that some part should go into the FAQ.
I like the illustration by examples.
Though I find the V(x)~ very cumbersome ...
Perhaps also the "Identities with binomials,Bernoulli- and other numbertheoretical numbers" at the bottom of the pages should be updated Wink

What I miss however is a suitable discussion of finite versus infinite matrices. For example if you approximate an infinite matrix M by finite matrices M_n then the inverse of the infinite matrix is not always the limit of the inverses of M_n.

Which is discussed in this video (unfortunately it seems currently not to be available) which I mentioned already here.
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#5
bo198214 Wrote:I just skimmed through and agree that some part should go into the FAQ.
I like the illustration by examples.
Though I find the V(x)~ very cumbersome ...
... this is Pari/GP... In my own matrix-calculator I implemented ' for transpose - but this is again misleading in a context of discussion of derivatives. What I really *hate* is the T-superscript Sad . So any good proposal is welcome...
Quote:Perhaps also the "Identities with binomials,Bernoulli- and other numbertheoretical numbers" at the bottom of the pages should be updated Wink
done
Quote:What I miss however is a suitable discussion of finite versus infinite matrices. For example if you approximate an infinite matrix M by finite matrices M_n then the inverse of the infinite matrix is not always the limit of the inverses of M_n.

Which is discussed in this video (unfortunately it seems currently not to be available) which I mentioned already here.

Well, it's a bit time ago, that I found small books concerning that matter and going into it in at least some detail. In general, I felt lost with that subject - may be I missed better treatises because I didn't recognize them - if the surrounding jungle of abstract terms/formalisms etc is too wild.
I really liked it, if I found something straightforward... In this view, the video was not helpful for me - not only was it very generally discussed beginning with infinite matrices to both sides, also I didn't find relations to my real problems with them (maybe I only was unable to see this): beginning with triangular real matrices, proceeding to complex-valued matrices and then, possibly, extension to fourway-infinite matrices. And including the concept of divergent series: I think it is needed to be familiar with this, since in our context we have to deal with such matrices/powerseries daily.

Gottfried
Gottfried Helms, Kassel
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#6
For considerations of the characteristic of the powerseries of U-tetration I recently uploaded an empirical table, but this link may have been overlooked in our threads. Here it is again:
coefficients for fractional iteration
I think, I've to correct my estimation there about growth of absolute values of terms. Better estimation (instead of hypergeometric) seems to be |term_k| ~ const*exp(k^2) asymptotically by inspection of first 96 terms with fractional heights. There is a "bump" at one index k, from where the absolute values grow after they have initially decreased. This "bump" moves to higher k with |1/2-fractional(h)|-> 1/2, and maybe we can say, it moves out to infinity, if h is integer, and the beginning of growth of absolute values of terms does not occur anywhere.

An older and shorter treatize of height-dependent coefficients is in
coefficients depending on h (older) Unfortunately I choose the letter U for the matrix, which would be the matrix "POLY" in my article, so this should no more be confusing (I'll change this today or tomorrow)

Gottfried
Gottfried Helms, Kassel
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#7
bo198214 Wrote:What I miss however is a suitable discussion of finite versus infinite matrices. For example if you approximate an infinite matrix M by finite matrices M_n then the inverse of the infinite matrix is not always the limit of the inverses of M_n.

Henryk -

I've one example loosely related to this: non-uniqueness of reciprocal.

Let base t=e, then the formal powerseries for f_t(x)=log(1+x)/log(t) is that of f(x) = log(1+x) and has the coefficients C_0=[0,1,-1/2,+1/3,-1/4,...]
Using them to construct the matrix-operator S1, we get the well known, infinite sized triangular matrix of Stirling-numbers 1'st kind (with factorial similarity scaling) S1; whose reciprocal is that of Stirling-numbers 2'nd kind, analoguously scaled.

I tried to include the property of multivaluedness of general logarithms, which is log(1+x) = y + k*2*Pi*i =y + w_k by replacing the leading zero in the above set of coefficients to obtain C_k=[w_k,1,-1/2,+1/3,-1/4,...]
I generated the according matrix-operator S1_k based on this formal powerseries.

Although we discuss theoretically infinite matrices the finite truncation of this made sense for k=1 and 2, so my approximations for S1_1 and S1_2 "worked" as expected (using sizes up to 64x64) :

I got, with good approximation to about 12 visible digits, the expected complex-valued logarithms, and even the reciprocity/inverse-conditions S1_1*S2 = I and even S1_2*S2 = I held (this was surely expected but was still somehow surprising Smile )

Anyway - I'd like to see more examples for problems with the infinite-size-inverse of triangular matrix-operators to get an idea about the basic characteristics of those problems. Do you know some?

Gottfried
Gottfried Helms, Kassel
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#8
I was considering remark of Dan Asimov, which Henryk cited in the thread "Bummer". Using my conversion-formula from T-tetration to U-tetration, which involves the fixpoints of the tetration-base b and application of the powerseries formula in my iteration-article, which I've announced here, I tend to conclude, that Asimov is right - because I get different powerseries, if I take different fixpoints, and where the related terms do not cancel pairwise.

Let b = t1^(1/t1) = t2^(1/t2) , u1=log(t1) and u2 = log(t2).

Then with the example b=sqrt(2), t1=2, u=u1=log(2), t2=4, u2=2*u and the conversion

Tb°h(x) = (Ut°h(x/t-1)+1)*t

I get for fixpoints t1 and t2 (and x=1 which is the assumption for tetration) the equation in question

(U2°h(-1/2)+1)*2 =?= (U4°h(-3/4)+1)*4

or rewritten

U2°h(-1/2) =?= U4°h(-3/4) + 1

The powerseries in u1 and u1^h resp u2 and u2^h seem so different, that one may possibly derive an argument from that, although I have not yet a convincing idea. Also possibly any general property of function-theory may be applicable then (for instance U2 is convergent and U4 is divergent, if I have it right).

See a better formatted short statement of this speculation here
[update] In a second view,and recalling the discussion and computations that we (and also myself) already had about the topic, the question is, why then is the difference so small? U_2 provides a convergent, U_4 a divergent series: the differences should be huge, if there are some, but: they are small. Hmmm...

Gottfried
Gottfried Helms, Kassel
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#9
I'm coming back to the proposal, to use some of the content of the article for the faq.
a) What part(s) would you like to include?
b) The text is in word-format, including bitmaps for matrix-display. Is there - optimally - a routine to transfer this to the source-format for the faq (or are other options available)?

Gottfried
Gottfried Helms, Kassel
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