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Matrix-method: compare use of different fixpoints
bo198214 Wrote:I simply dont see how you get fixed points involved.
You have Carleman matrix of and you uniquely decompose the finite truncations into
and then you define


Where are the fixed points used?

Assume the eigensystem-decomposition Bs = W0 * D0 * W0^-1
where the small s indicates the baseparameter s=b=base

Now this means also
W0^-1 * Bs = D0 * W0^-1

Now look at the first row of W0^-1, call this vector Y0. Then, since I assume the first eigenvalue d0_0 =1 we have
Y0 * Bs = 1 * Y0 = Y0
Now I observed, that Y0 has the form of a powerseries of the scalar parameter y0, so I may note,

V(y0)~ * Bs = V(y0)~

But on the other hand I know by construction of Bs that
V(x)~ * Bs = V(s^x)~

so the previous means
V(y0)~ * Bs = V(y0)~ = V(s^y0) ~
y0 = s^y0

and y0 is obviously a fixpoint.

The observation actually was: the first row in W^-1 is the powerseries in the first fixpoint y0.

Now consider this backways. Using another fixpoint y1 this relation holds again,
y1 = s^y1
V(y1)~ * Bs = V(s^y1)~ = V(y1)~
First row of W1^-1 is V(y1)~
and W0^-1 <> W1^-1 in their first rows and supposedly the same for the whole matrices W0 <> W1 and

W1^-1 * Bs = 1 * W1^-1

is another solution, provided that the form of the first row in W1^-1 is still that of a powerseries.

And in fact; if I introduce the coefficients u1 = alpha + beta*I and t1=exp(u1) in my analytical eigensystem-constructor as parameters, I get valid eigen-matrices W1 and W1^-1 based on these other fixpoints for the cases I've checked where it holds that
Bs = W1*D1*W1^-1 (but see note 1)

These had in fact the powerseries of the second fixpoint in the first row in W1^-1 - and I suppose, this will be the same with all other fixpoints (however the numerical computations become too difficult to do a general conjecture based on and backed by reliable approximations)


(1) This is directly related to your introductory posting in the "Bummer"-thread, and we must solve the discrepancy between these two statements!
Gottfried Helms, Kassel

Messages In This Thread
RE: Matrix-method: compare use of different fixpoints - by Gottfried - 11/07/2007, 01:33 PM

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