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 Matrix-method: compare use of different fixpoints bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 11/07/2007, 07:44 PM (This post was last modified: 11/07/2007, 07:45 PM by bo198214.) 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? 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. 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 ${_dV}(\ln(a))$! 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. « 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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