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Regular iteration using matrix-Jordan-form
#1
(editing remark: I changed the title or the original posting because of the contents of my current post in which I establish the connection between Regular iteration and the matrix-Jordan-form and give also a couple of online references)



Hi -

in answering a question in mathoverflow concerning a sequence of numbers given in OEIS (http://oeis.org/A008826), which are somehow related to fractional iteration for the f(x) = exp(x)-1 I found a surprising coincidence :
http://mathoverflow.net/questions/133593...281#151281

I've no further explanation so far, but it is interesting that the Jordan-form gives access to a variant of the Schröder-function to implement fractional iteration which was only possible with the matrix-logarithm (in that whole framework of the matrix-concept).
Perhaps I can say something more soon.

Gottfried
Gottfried Helms, Kassel
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#2
(12/11/2013, 01:04 AM)Gottfried Wrote: in answering a question in mathoverflow concerning a sequence of numbers given in OEIS (http://oeis.org/A008826), which are somehow related to fractional iteration for the f(x) = exp(x)-1 I found a surprising coincidence :
http://mathoverflow.net/questions/133593...281#151281
......
Any idea how this is related to the fractional iteration of exp(x)-1? I couldn't figure anything out from the thread, or the link.
- Sheldon
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#3
(12/12/2013, 09:44 AM)sheldonison Wrote: Any idea how this is related to the fractional iteration of exp(x)-1? I couldn't figure anything out from the thread, or the link.
- Sheldon
I've looked into the book of L. Comtet, from whom this whole sequence originated. Hmm, either me or the text seemed a bit obfuscated, but I've not yet understood, how he used this matrix of values; he talks about Faa di Bruno-Formula, but the whole argumentation/path to the fractional iterate is somehow complicated.(It's on the pages 144-148 in the"Advanced combinatorics" (Reidel, Dordrecht)) - I could provide a page scan if needed.

But after I see, that this set of polynomials perfectly agrees with the matrix of the Jordan-decomposition, which in the case of the diagonalizable Carlemanmatrix is the eigenmatrix and gives the coefficients of the Schröder-function, I think, it is a simple analogy here: it might simply implement a Schröder-function for the non-diagonalizable case for tha base b=exp(1) and the function f(x) = b^x - 1 .

Why I draw attention to this here is, because the method to use Jordandecomposition for nondiagonalizeble Bell/Carleman-matrices might be a very basic systematic extension for many related iteration-problems. I think, for instance, E Schröder could have come up with that generalization already, but Jordan decomposition might have seen to a very difficult procedure to be calculated and one does not find many references in the mass-literature. Even contemporary examples: while the for Pascal-matrix and their powers etc, which is frequently discussed even in matrix-contexts I could only find one article which studies Jordan-decomposition of the Pascal-matrix; but none for the same using the Stirling matrix - although one would arrive at the Comtet-numbers just completely natural...

Gottfried
Gottfried Helms, Kassel
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#4
I've found the key properties for the connection of the (matrix-) Jordanform of the Carleman-matrices and the "regular iteration".

Consider the (transposed, triangular) Carlemanmatrix U for the function f(x)= b^x-1 using some base b, where the infinite iteration converges for x in some convenient interval, so for instance b=2 .
The Carlemanmatrix-concept allows now to write

V(x) * U = V(f(x))
V(x) * U^2 = V(f(f(x)))
...
V(x) * U^h = V(f°^{h}(x))
...
V(x) * U^{-1} = V(f°^{-1}(x))
...

and so on, and thus allows to translate the function-iteration into matrix-powers.

If b<> exp(1) then "diagonalization" gives the connection with the concept of the schröder-function; by diagonalizing we get three matrices:
U = M * D * M^{-1}
where if the (triangular) M and M^-1 (let us write W for this now) are normed to have a unit-diagonal, they are the Carleman-matrices of the schröder-function \sigma(x) resp. its inverse such that we have
V(x) * M = V(\sigma(x))
V(\sigma(x)) * W = V(x)

and the diagonalmatrix D provides the possibility to express powers of U by powers of D such that
U^h = M * D^h * W
The coolest feature here is, that powers of D are made by powers of its scalar diagonal elements, and so we have by this the tool for arbitrary fractional and even complex powers of D, and thus of U and thus for the iteration-height of f(x).

This was impossible for base b=exp(1) because the diagonal in U is then the unit-vector and diagonalization cannot be done, and until now I just take the matrix-logarithm in this case to express fractional powers of U by the Log/Exp-expression
U^h = Exp ( h * Log( U))

While that matrices (having unit-diagonal) cannot be diagonalized they can be Jordan-decomposed, which is the best possible approximation to the diagonalization-concept. We get again
U = M * J * W
only that J is now not diagonal but has one additional unit-subdiagonal.
Powers of J cannot be computed simply like powers of D - again one has to introduce either the matrix-logarithm/-exponential for this matrix (but which has now a maximally simplified standard form), or the decomposition by the binomial-theorem for integer and/or fractional powers.

The really new information is now, that we do not get the Schröder-function, but we get an implementation of the Newton-series (or what I called the binomial-method for iteration), the product
V(x) * M = Y
gives now in the result vector Y the set of consecutive iterates of f(x) (in a slighty modified form). Then that iterates become just composed by the binomial-coefficients to allow integer and fractional iterations by compositions of the integer-iterations alone; it is the perfect implementation of that Newton-series-method, as for instance described in L.Comtet's "advanced combinatorics" written in the mid of the previous century (but Comtet seemingly didn't notice that relation to the Jordanform although he was also working with matrices).


Well, we do not get a new/better method for the computation of the fractional iterates; we are still left with the formal power series and their zero-radius of convergence for fractional iterates, but I think it is worth to record this additional connections between the Carlemanmatrix-/Schröderfunction concept and the Jordandecomposition and iteration-series - they give always valid (and the same) formal powerseries for the fractional iterates.


(The forum has still not MathJax activated; I'll convert this statement to Forum-TeX later)

Gottfried


Using google, found this earlier references:

http://www.sciencedirect.com/science/art...7X89902333
http://math.univ-lille1.fr/~gchen/articl...c-99.ps.gz
https://www.mittag-leffler.se/preprints/...03s-09.pdf

Gottfried Helms, Kassel
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#5
Maybe I read Gottfried's post too fast , but what is the difference/benefit between using Jordan form and the matrix-logarithm ?

Maybe its in the links , I did not read them yet.

Another thing :

I think fake function theory is suitable for formal power series / parabolic fixpoints.

That would really put Taylor series on the map.

One could then study for instance fake(exp^[1/2]) vs fake((exp(x)-1)^[1/2]).

I also wonder about the difference of their position of zero's.

Im not trying to claim superiority of fake function theory over matrices but I wanted to show a possible alternative way.

Need some time ...

regards

tommy1729
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#6
PS : Gottfried , do you know Vincenzo Librandi ?

regards

tommy1729
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#7
(09/29/2014, 10:35 PM)tommy1729 Wrote: Maybe I read Gottfried's post too fast , but what is the difference/benefit between using Jordan form and the matrix-logarithm ?

Maybe its in the links , I did not read them yet.
(...)

Well, not in the "links" but in the "connections" ;-) . If one wants to describe the Carleman-approach for the iterated exponential (and other iterations) then it is of interest to see as many connections to other parts in mathematics as possible to widen the view on the problem (remember there is still a lot of things missing, let us only remember the uniqueness problem). It is like collecting related information for some encyclopedic resource like wikipedia or the tetration-wiki, helpful for any newly interested collaborators.
The new information by the Jordan-decomposition is, that we introduce - based on the same logic as in the diagonalization - the use of iteration-series: one thing I expect/I hope to lead (in some future) to a better evaluable series than the otorious "zero-convergence-radius power series" for the fractional iterates, and also have a well known formalism which also connects the widely known binomial composition via Newton-series with the diagonalization-approach.
I think it's just worth to have that connections explicite.

Gottfried
Gottfried Helms, Kassel
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#8
(09/29/2014, 10:52 PM)tommy1729 Wrote: PS : Gottfried , do you know Vincenzo Librandi ?

regards

tommy1729

I had email contact with him, longer ago. Why?
Gottfried Helms, Kassel
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