Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Q: Exponentiation of a carleman-matrix
A couple of years ago I've fiddled with the question of the iterated exponentiation of the most simple(?) Carleman-matrix, that for the function f(x)=a+x, namely the Pascal-matrix (or the a'th power of it).

I have then found some interesting properties (see for the interested reader) , but besides of a still relatively vague idea it is difficult for me to say precisiely, what such an operation means and what it means, if then the tetrated pascal-matrix is used as provider of coefficients of some new associated operation.

Just playing around I've done this with the Carlemanmatrix for the , which is lower triangular, has units in the diagonal and for which a (matrix-) logarithm can be computed which can then be taken also as valid truncation of the infinite-sized-matrix.

Let's call the Carlemanmatrix for f(x) as "U", and its (matrix-)logarithm as "L" then the infinite exponentiation
(the identity-matrix) and then . Here the exponentiation is done by such that finally . This occurs at the n'th step of iteration because the matrices L*A are nilpotent to the chosen matrix size nxn.

This gives then coefficients for some function which I would like to characterize. Maybe it is Pentation, but I'm not sure about this.

Anyway, the coefficients in the second column of A are strongly diverging, more than the factorial and it seems they are all positive, so that to be able to evaluate it at all in terms of a function requires at least a negative x, and also sophisticated methods for the divergent summation.

Before I begin to invest much time and energy in this I would like to have an idea, what such function g(x) would do, how it could be characterized, at least qualitatively...


p.s.: if someone interested in this needs the Pari/GP-code I can provide this. Please consider, that it will be work to flesh out the relevant procedures from my (slightly unstructured ;-)) collection of Pari/GP code samples, so I'd like to do this if there is seriously interest only...
Gottfried Helms, Kassel

Possibly Related Threads...
Thread Author Replies Views Last Post
  A support for Andy's (P.Walker's) slog-matrix-method Gottfried 4 3,541 03/08/2021, 07:13 PM
Last Post: JmsNxn
  New Quantum Algorithms (Carleman linearization) Finally Crack Nonlinear Equations Daniel 2 630 01/10/2021, 12:33 AM
Last Post: marraco
  Tommy's matrix method for superlogarithm. tommy1729 0 2,788 05/07/2016, 12:28 PM
Last Post: tommy1729
  [2015] New zeration and matrix log ? tommy1729 1 4,945 03/24/2015, 07:07 AM
Last Post: marraco
  Regular iteration using matrix-Jordan-form Gottfried 7 12,434 09/29/2014, 11:39 PM
Last Post: Gottfried
  "Natural boundary", regular tetration, and Abel matrix mike3 9 20,296 06/24/2010, 07:19 AM
Last Post: Gottfried
  sum of log of eigenvalues of Carleman matrix bo198214 4 9,251 08/28/2009, 09:34 PM
Last Post: Gottfried
  spectrum of Carleman matrix bo198214 3 6,391 02/23/2009, 03:52 AM
Last Post: Gottfried
  Matrix Operator Method Gottfried 38 60,850 09/26/2008, 09:56 AM
Last Post: Gottfried
  Left associative exponentiation- an iteration exercise Gottfried 0 3,130 09/12/2008, 08:17 PM
Last Post: Gottfried

Users browsing this thread: 1 Guest(s)