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 Matrix-method: compare use of different fixpoints Gottfried Ultimate Fellow Posts: 787 Threads: 121 Joined: Aug 2007 11/07/2007, 01:33 PM (This post was last modified: 11/07/2007, 01:55 PM by Gottfried.) bo198214 Wrote:I simply dont see how you get fixed points involved. You have Carleman matrix $B_b$ of $b^x$ and you uniquely decompose the finite truncations ${B_b}_{|n}$ into ${B_b}_{|n}=W_{|n} D_{|n} {W_{|n}}^{-1}$ and then you define ${B_b}^t = \lim_{n\to\infty} W_{|n} {D_{|n}}^t {W_{|n}}^{-1}$. 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) ~ thus 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) Gottfried --------------- (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 « Next Oldest | Next Newest »

 Messages In This Thread Matrix-method: compare use of different fixpoints - by Gottfried - 11/04/2007, 12:38 PM RE: Matrix-method: compare use of different fixpoints - by bo198214 - 11/04/2007, 12:59 PM RE: Matrix-method: compare use of different fixpoints - by Gottfried - 11/04/2007, 01:28 PM RE: Matrix-method: compare use of different fixpoints - by bo198214 - 11/04/2007, 01:31 PM RE: Matrix-method: compare use of different fixpoints - by Gottfried - 11/04/2007, 01:40 PM RE: Matrix-method: compare use of different fixpoints - by bo198214 - 11/07/2007, 10:52 AM RE: Matrix-method: compare use of different fixpoints - by Gottfried - 11/07/2007, 01:33 PM RE: Matrix-method: compare use of different fixpoints - by bo198214 - 11/07/2007, 01:57 PM RE: Matrix-method: compare use of different fixpoints - by Gottfried - 11/07/2007, 02:10 PM RE: Matrix-method: compare use of different fixpoints - by bo198214 - 11/07/2007, 02:21 PM RE: Matrix-method: compare use of different fixpoints - by Gottfried - 11/07/2007, 02:59 PM RE: Matrix-method: compare use of different fixpoints - by bo198214 - 11/07/2007, 03:35 PM RE: Matrix-method: compare use of different fixpoints - by Gottfried - 11/07/2007, 04:31 PM RE: Matrix-method: compare use of different fixpoints - by bo198214 - 11/07/2007, 07:44 PM RE: Matrix-method: compare use of different fixpoints - by Gottfried - 11/07/2007, 08:41 PM RE: Matrix-method: compare use of different fixpoints - by bo198214 - 11/07/2007, 09:32 PM RE: Matrix-method: compare use of different fixpoints - by Gottfried - 11/11/2007, 06:05 PM RE: Matrix-method: compare use of different fixpoints - by bo198214 - 11/11/2007, 10:05 PM RE: Matrix-method: compare use of different fixpoints - by Gottfried - 11/12/2007, 01:53 AM RE: Matrix-method: compare use of different fixpoints - by Gottfried - 11/13/2007, 05:48 PM RE: Matrix-method: compare use of different fixpoints - by Gottfried - 11/12/2007, 07:48 AM RE: Matrix-method: compare use of different fixpoints - by bo198214 - 11/12/2007, 11:52 AM RE: Matrix-method: compare use of different fixpoints - by Gottfried - 11/12/2007, 03:13 PM RE: Matrix-method: compare use of different fixpoints - by andydude - 11/30/2007, 05:24 PM

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