(08/31/2009, 09:26 PM)tommy1729 Wrote: matrix power tetration ?? = carleman matrix method ??
Take the Carleman matrix of a function, then take the t-th matrix power, extract the first line, these are the coefficients of f^t.
This is previously called diagonalization method, which is not completely correct, because we can also take the matrix power of non-diagonalizable matrix (e.g. for parabolic functions are only trigonalizable).
This method is equivalent to consider the Jordan decomposition of the Carleman matrix C = P^{-1} J P; C^t = P^{-1} J^t P. If J is a diagonal matrix then J^t is the diagonal matrix with the coefficients taken to the t-th power. In the non-diagonal case you can compute the t-th power of a Jordan block by some finite sum.
Quote:cauchy tetration ?? = andrews slog ?? = julia ?? = kouznetsov ?? = ??
Cauchy tetration is the tetration described and calculated by Dmitrii Kouznetsov which involves the Cauchy-/Contour- integrals.
We call Andrew's slog the intuitive tetration, it was AFAIK first described by Walker. It can also be described with help of the Carleman matrix.
The Julia function is usually used in the regular iteration of parabolic functions.