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Matrix-method: compare use of different fixpoints
bo198214 Wrote:But this is than another method. The matrix operator method is to use truncated Matrices with unique Eigensystem decomposition and hence with a unique limit. That was the absolute good thing about this method as we dont have to take fixed points (with the attached question of which to choose) into consideration.

For the infinite matrices there is not even only one solution for each fixed point but also all the other possible solutions come into play, i.e. the non-regular Schroeder functions. So how should this help? If we know there are some solutions out there which we also want to have and than modify our method (that was designed to be unique) also to include the other solutions???
Hmm, I still don't see the problem here - may be, simply due to limitations of my horizon of knowledge.
First - I have no problem, if it comes out, that each entry of Bs can be computed in two or more different ways: this wouldn't then affect the validitiy of the use of Bs for the computation of tetration. We would still have the same coefficients for our powerseries.

Second - If two vectors
V(y0)*Bs = 1*V(y0) ~
V(y1)*Bs = 1* V(y1) ~
both satisfy these equation, then the two vectors satisfy the requirement of the definition being an eigenvector (see note 1). For the tetration-operator we know (simply by the occurence of multiple fixpoints), even multiple solutions for this must exist.
Now they have the same eigenvalue - but the eigenvalue occurs only once in the set of eigenvalues: so the special form of Bs suggests, that uniqueness is *not* given (in this sense -without affecting Bs itself) by the very problem of tetration and its operator (or: by its powerseries).
The matrix-method reflects this ambiguity perfectly - so this is even an argument *pro* appropriateness of the method.

Third - this ambiguity without consequences in the simple case (power of Bs is 1, so the above seems valid to define the same matrix Bs^1 expressed by identity of their entries) may introduce consequences in the question of powers of Bs. But for integer powers we may insert our truncated Eigensystems and thus use approximated and truncated Bs and still get identical (? let the unresolved "Bummer"-caveats aside for the moment) matrices for powers of Bs (I've done few numerical examples)

Forth - even if the matrices for integer powers should indeed be identical, then this does not mean they are identical for fractional powers. This is one of my current investigations, in fact, for fractional powers my method of construction of eigenmatrices based on different values for u (using the different branches) gives strange results, if I do not use the principal branches. So, if there should be no error in my composer-procedures, then we had indeed different versions of Bs^h (where h is non-integer) and we had a problem.
But since the construction of the eigenmatrices is complicated I assume an error of the type:
t^(1/t) =/= exp(log(t)/t) if for log(t) is a non-principal branch is required, but the model for the composition is developed based on the assumptions of principal branch of log(t) only. I did not consider this problem at all, when I developed the eigensystem-constructor, so by a careful reconsideration I may resolve this problem. Or not. If it persists - only then the/my matrix-method has a severe fundamental shortcoming. It may still be valid if only the first fixpoint is used, but the aesthetics of it were spoiled (at least in my view).


(1) more precisely of Bs~, but let this aside here
Gottfried Helms, Kassel

Messages In This Thread
RE: Matrix-method: compare use of different fixpoints - by Gottfried - 11/07/2007, 02:59 PM

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