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 Applying the iteration methods to simpler functions bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 11/06/2007, 10:36 AM Gottfried Wrote:You may also have a look at my operators-article... I discussed the three operations in connection with the matrix-method in few lines at the second last page. See here:http://math.eretrandre.org/tetrationforu...hp?aid=124 hmm. Don't know, seems, there is something broken with the pdf-attachments. Here is an external linkThe link works well. You write in this article: Quote:The eigensystem of P is degenerate; but it has an exceptional simple matrix-logarithm, by which then a general power can be easily computed when just multiplied with the h-parameter. There is even a general method that works with every matrix via the Jordan normal form. However it suffices to use a more relaxed form where the blocks of the Jordan normal form consist of upper triangular matrices with the eigenvalue $\lambda$ on the diagonal (instead having the eigenvalue on the diagonal and 1 on the diagonal above them, see wikipedia). We can take the power of such a block $B$ by the formula $B^t = \lambda^t(\frac{1}{\lambda}B-I+I)^t =\lambda^t \sum_{k=0}^\infty \left(t\\k\right)(\frac{1}{\lambda}B-I)^k$. Which however involves no limits for the entries. And the $t$th power of the whole matrix, which consists of several such blocks on the diagonal, is simply the $t$th power of each block on the diagonal. The matrix in the considered case above consisted of just one such block with eigenvalue $\lambda=1$. andydude Wrote:Do you consider parabolic/hyperbolic iteration (and Daniel Geisler's methods) to be part of regular iteration?Yes I consider both as regular iteration (though if directly handled they require different treatment) but both coincide with the matrix operator method if developed at the fixed point. « Next Oldest | Next Newest »

 Messages In This Thread Applying the iteration methods to simpler functions - by bo198214 - 11/05/2007, 11:39 PM RE: Applying the iteration methods to simpler functions - by Gottfried - 11/06/2007, 12:30 AM RE: Applying the iteration methods to simpler functions - by andydude - 11/06/2007, 03:32 AM RE: Applying the iteration methods to simpler functions - by bo198214 - 11/06/2007, 10:36 AM RE: Applying the iteration methods to simpler functions - by jaydfox - 11/09/2007, 05:34 AM RE: Applying the iteration methods to simpler functions - by bo198214 - 11/09/2007, 08:36 AM RE: Applying the iteration methods to simpler functions - by andydude - 11/12/2007, 09:40 AM RE: Applying the iteration methods to simpler functions - by Gottfried - 11/12/2007, 10:40 AM RE: Applying the iteration methods to simpler functions - by andydude - 11/13/2007, 12:39 AM RE: Applying the iteration methods to simpler functions - by bo198214 - 11/12/2007, 11:23 AM RE: Applying the iteration methods to simpler functions - by jaydfox - 11/12/2007, 04:41 PM

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