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+--- Thread: regular slog (/showthread.php?tid=70)
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RE: regular slog - bo198214 - 03/25/2009
Ansus Wrote:What is \( a \) in this formula?
\( a \) is the fixed point: \( b^a=a \).
RE: regular slog - bo198214 - 03/25/2009
Ansus Wrote:Is this correct:
\(
\text{alpha}[b,x]=\text{Log}\left[\text{Log}\left[-\frac{\text{ProductLog}[-\text{Log}[b]]}{\text{Log}[b]}\right],-\frac{\text{ProductLog}[-\text{Log}[b]]}{\text{Log}[b]}-\text{PowerTower}[b,n,x]\right]-n
\)
I am only familiar with Maple and Sage, so I can not help you with this. However in Maple the formula \( \frac{W(-\ln(b))}{-\ln(b)} \) works.
RE: regular slog - bo198214 - 03/26/2009
Ansus Wrote:\(
a=-\frac{W(-\ln b)}{\ln b}
\)
Maybe you have to specify the proper branch. (But as I told I can not test because I dont have Mathematica available.)
RE: regular slog - bo198214 - 03/26/2009
Ansus Wrote:Anyway with any value of a I cannot get anything close to what expected.
What shall I say? It worked for me.
For base \( \sqrt{2} \) the fixed point is \( 2 \), thatswhy this base is so preferred, you dont need to compute the fixed point seperately.
RE: regular slog - bo198214 - 03/27/2009
Ansus Wrote:Great! Now it works, but only for a limited range of bases. Particularly it works for the base \( \sqrt{2} \). I used this formula:
\( \operatorname{slog}_b(x)=\lim_{n\to\infty} \frac{\ln \left(\frac{\frac{W(-\ln b )}{\ln b}+\exp_b^{[n]}(x)}{\frac{W(-\ln b)}{\ln b}+\exp_b^{[n]}(1)}\right)}{\ln \ln \left(\frac{W(-\ln b)}{-\ln b}\right)} \)
I've added this formula to our wiki page:
http://en.wikipedia.org/wiki/Talk:Tetration/Summary#Evaluation_methods
RE: powerseries of regular slog - andydude - 04/02/2009
bo198214 Wrote:The Abel function has also a singularity at 0.Just realized, this is only if the fixed point is 0.
bo198214 Wrote:\( FS=cS \)This should be \( SF=cS \), which means you can't simplify the matrix like you did. The formula you give is a matrix representation of \( \sigma(f(x)) = \sigma(cx) \) if those are Bell matrices, or \( f(\sigma(x)) = c\sigma(x) \) if those are Carleman matrices.
Andrew Robbins
RE: powerseries of regular slog - bo198214 - 04/02/2009
andydude Wrote:otherwise at the fixed point. The regular iteration theory always assumes the fixed point at 0. If not one just considers the function \( f(x+a)-a \) where \( a \) is the fixed point.bo198214 Wrote:The Abel function has also a singularity at 0.Just realized, this is only if the fixed point is 0.
Quote:actually thats also wrong. However it is only an intermediate error in my derivation.bo198214 Wrote:\( FS=cS \)This should be \( SF=cS \),
Lets show the correct equations:
\( \sigma(f(x))=c\sigma(x) \) or, with \( \mu_c(x)=cx \):
\( \sigma\circ f = \mu_c \circ \sigma \)
if we take the Bell matrices:
\( FS=SM \)
where \( M \) is the Bell matrix of \( \mu_c \). This is the diagonal matrix:
\( M=\begin{pmatrix}
c &0 & 0 &\dots &0\\
0 & c^2 & 0&\dots& 0\\
&&\vdots&\\
0 & &\dots& 0 & c^n\\
\end{pmatrix} \)
I think Gottfried calls this the Vandermonde matrix.
The right multiplication of this matrix multiplies each \( k \)-th column with \( c^k \). If we truncate \( S \) to its first column \( \vec{\sigma} \) we get hence:
\( F\vec{\sigma} = c\vec{\sigma} \)
and this can then be transformed to:
\( (F-cI)\vec{\sigma}=0 \)
which I used for my further derivations.
RE: regular slog - Gottfried - 07/29/2009
(10/07/2007, 10:30 PM)bo198214 Wrote: Now there is the the so called principal Schroeder function \( \sigma_f \) of a function \( f \) with fixed point 0 with slope \( s:=f'(0) \), \( 0<s<1 \) given by:
\( \sigma_f(x) = \lim_{n\to\infty} \frac{f^{\circ n}(x)}{s^n} \)
This function particularly yields the regular iteration at 0, via \( f^{\circ t}(x)=\sigma^{-1}(s^t\sigma(x)) \).
Sometimes a thing needs a whole life to be recognized...
In the matrix-method I dealt with the eigen-decomposition of the (triangular) dxp_t() -Bellmatrix U_t to satisfy the relation
\( \hspace{24} U_t = W * D * W^{-1 } \)
While the recursion to compute W and W^-1 efficiently is easy and is working well, I did not have a deeper idea about the structure of the columns in W. Now I found, it just agrees with the above formula:
\( \hspace{24} W = \lim_{h\to\infty} {U_t}^h * diag({U_t}^h)^{-1} \)
which is exactly the above formula; we even can write this, if we refer to the second column of U_t^h as F°h, the second column of W as S, and s = F[1] while F°h[1] = F[1]^h =s^h , then we have
\( \hspace{24} S = \lim_{h\to\infty} \frac {F^{\circ h}}{s^h} \)
Something *very* stupid ... <sigh>
But, well, now also this detail is explained for me.
<Hmmm I don't know why the forum software merges my two replies (to two previous posts of Henryk) into one So here is the second post>
(04/02/2009, 02:31 PM)bo198214 Wrote: This is the diagonal matrix:Not exactly. I call the Vandermonde-*matrix* VZ (and ZV=VZ~) the *collection* of consecutive vandermonde V(x)-vectors
\( M=\begin{pmatrix}
c &0 & 0 &\dots &0\\
0 & c^2 & 0&\dots& 0\\
&&\vdots&\\
0 & &\dots& 0 & c^n\\
\end{pmatrix} \)
I think Gottfried calls this the Vandermonde matrix.
´ VZ = [V(0), V(1) , V(2), V(3), ...] \\ Vandermondematrix
Your M is just c*dV( c) in my notation: the vandermondevector V( c) used as diagonalmatrix (and since the first entry is not c^0 I noted the additional factor c)
Gottfried
RE: regular slog - bo198214 - 07/31/2009
(07/29/2009, 11:07 AM)Gottfried Wrote: Hmmm I don't know why the forum software merges my two replies (to two previous posts of Henryk) into one
I hopefully switched this behaviour off now.