Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
regular slog
#11
Ansus Wrote:What is in this formula?

is the fixed point: .
Reply
#12
Ansus Wrote:Is this correct:


I am only familiar with Maple and Sage, so I can not help you with this. However in Maple the formula works.
Reply
#13
Ansus Wrote:

Maybe you have to specify the proper branch. (But as I told I can not test because I dont have Mathematica available.)
Reply
#14
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 the fixed point is , thatswhy this base is so preferred, you dont need to compute the fixed point seperately.
Reply
#15
Ansus Wrote:Great! Now it works, but only for a limited range of bases. Particularly it works for the base . I used this formula:




I've added this formula to our wiki page:
http://en.wikipedia.org/wiki/Talk:Tetrat...on_methods

Smile
Reply
#16
bo198214 Wrote:The Abel function has also a singularity at 0.
Just realized, this is only if the fixed point is 0.

bo198214 Wrote:
This should be , which means you can't simplify the matrix like you did. The formula you give is a matrix representation of if those are Bell matrices, or if those are Carleman matrices.

Andrew Robbins
Reply
#17
andydude Wrote:
bo198214 Wrote:The Abel function has also a singularity at 0.
Just realized, this is only if the fixed point is 0.
otherwise at the fixed point. The regular iteration theory always assumes the fixed point at 0. If not one just considers the function where is the fixed point.

Quote:
bo198214 Wrote:
This should be ,
actually thats also wrong. However it is only an intermediate error in my derivation.
Lets show the correct equations:
or, with :

if we take the Bell matrices:

where is the Bell matrix of . This is the diagonal matrix:

I think Gottfried calls this the Vandermonde matrix.

The right multiplication of this matrix multiplies each -th column with . If we truncate to its first column we get hence:

and this can then be transformed to:

which I used for my further derivations.
Reply
#18
(10/07/2007, 10:30 PM)bo198214 Wrote: Now there is the the so called principal Schroeder function of a function with fixed point 0 with slope , given by:



This function particularly yields the regular iteration at 0, via .

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



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:



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



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:

I think Gottfried calls this the Vandermonde matrix.
Not exactly. I call the Vandermonde-*matrix* VZ (and ZV=VZ~) the *collection* of consecutive vandermonde V(x)-vectors

´ 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
Gottfried Helms, Kassel
Reply
#19
(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.
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  Some slog stuff tommy1729 15 11,327 05/14/2015, 09:25 PM
Last Post: tommy1729
  Regular iteration using matrix-Jordan-form Gottfried 7 8,080 09/29/2014, 11:39 PM
Last Post: Gottfried
  A limit exercise with Ei and slog. tommy1729 0 1,826 09/09/2014, 08:00 PM
Last Post: tommy1729
  A system of functional equations for slog(x) ? tommy1729 3 4,170 07/28/2014, 09:16 PM
Last Post: tommy1729
  slog(superfactorial(x)) = ? tommy1729 3 4,738 06/02/2014, 11:29 PM
Last Post: tommy1729
  [stuck] On the functional equation of the slog : slog(e^z) = slog(z)+1 tommy1729 1 2,325 04/28/2014, 09:23 PM
Last Post: tommy1729
  A simple yet unsolved equation for slog(z) ? tommy1729 0 1,783 04/27/2014, 08:02 PM
Last Post: tommy1729
  regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 Gottfried 7 8,586 06/25/2013, 01:37 PM
Last Post: sheldonison
  A kind of slog ? C + SUM f_n(x) ln^[n](x) ? tommy1729 1 2,281 03/09/2013, 02:46 PM
Last Post: tommy1729
  tetration base conversion, and sexp/slog limit equations sheldonison 44 51,833 02/27/2013, 07:05 PM
Last Post: sheldonison



Users browsing this thread: 1 Guest(s)