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 Matrix-method: compare use of different fixpoints Gottfried Ultimate Fellow Posts: 787 Threads: 121 Joined: Aug 2007 11/07/2007, 08:41 PM bo198214 Wrote:Gottfried Wrote:bo198214 Wrote:I want your acknowledgement on this. Well, you say it in the next sentence: let truncation aside. In all my considerations ...Is this a confirmation?I can only restate: in no situation I assumed the matrices as truncated by principle - all my considerations assume the unavoidable truncations in praxi as giving approximations in numerical evaluation as plcaeholder for the basically infinite matrices. I never discussed a finite (truncated) matrix being more than such an approximation for determination of any intermediate result. For instance: if the carleman-matrix is thought of finite size, then I don't see any relation between carleman matrices and any of my matrices. I don't know, whether this satisfies your request (if not, then please give even more explanation of what is the core of your question, I must then being unable to understand the relevant implications correctly) Quote:Gottfried though I am not 100% clear about your my-oh-my method, it seems as if it does nothing more than to find the regular iteration at a fixed point, we discussed something similar already here. That may all be - and whether it is "nothing more" or not: if it is the regular iteration, then fine; if not, then fine again. Quote:The regular iteration of a function $f$ with fixed point at 0 is given by $f^{\circ t} = \sigma^{-1}\circ \mu_{c^t} \circ \sigma$ with $c=f'(0)$, $\mu_d(x)=d\cdot x$ and $\sigma$ being the principal Schroeder function. If the fixed point is not at 0 but is at $a$ then $f=\tau_{-a}\circ \exp_b \circ \tau_a$ is a function with fixed point at 0 and $f'(0)=c=\ln(a)$, where $\tau_d(x)=d+x$. Putting this into the first equation we get: $\exp_b^{\circ t} = \tau_a\circ\sigma^{-1}\circ\mu_{{\ln(a)}^t}\circ\sigma\circ \tau_{-a}$. But if we translate this into matrix notation by simply replacing $\circ$ by the matrix multiplication and replacing each function by the corresponding Bell matrix (which is the transposed Carleman matrix) then we see a diagonalization of $B_b$ because the Bell matrix (and also the Carleman matrix) of $\mu_{\ln(a)}(x)=\ln(a)x$ is just your diagonal matrix ${_dV}(\ln(a))$! Yes, this seems so - but did we not already state the identity of the matrix B (or Bs) with the Bell/Carleman-transposes? I thought, that this had settled the question already? I was very happy, when you pointed out the relation in one of your previous posts - I couldn't have done it due to my lack of understanding of those concepts (described in elaborated articles, more than I could follow in detail). Quote:So it is nothing new, that we can diagonalize the untruncated $B_b$ for each fixed point $a$ with the diagonal matrix ${_dV}(\ln(a))$. As result we only get the plain old regular iteration at a fixed point. Hmm, for me this is an *achievement*, so my bottom up approach just out of the sandbox is then decoded into the terminology of "Schroeder-equation" and "regular iteration" - so: good! If from there some shortcomings of the method are already *known* then I would like to know them, too. Hmm, I've no more idea at this moment. I'll reread your post later this evening, perhaps I'm missing some point. Gottfried 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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