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 matrix function like iteration without power series expansion Gottfried Ultimate Fellow Posts: 764 Threads: 118 Joined: Aug 2007 06/30/2008, 09:23 PM Using a matrix-expression this would be t°h(x) = W^-1 sum k=0..inf sum j=0..k (-1)^j * binomial(k,j) *diag(1,u^j,u^2j,...) * W sum j=0..k (-1)^j * binomial(k,j) *dV(u^j) = diag(u^j-1) *PPow( sum k=0..inf bo198214 Wrote:$f^{\circ t} = \sum_{n=0}^\infty \left(t\\n\right) \sum_{k=0}^n \left(n\\k\right) (-1)^{n-k} f^{\circ k}$ Notes: Perhaps Gottfried can jump in to provide summability in the divergent case $b>e^{1/e}$. Hmm, let me try (maybe I didn't get this right yet). $f^{\circ t} = \sum_{n=0}^\infty \left(t\\n\right) \sum_{k=0}^n \left(n\\k\right) (-1)^{n-k} f^{\circ k}$ is $f^{\circ t} = \sum_{n=0}^\infty \left(t\\n\right) c_n$ which is just a binomial weighting of the coefficients c_n. In my analyses I got the coefficients $f^{\circ t}(x) = \sum_{n=0} a_n x^n$ so, for instance, the c_n for the half-iteration are $c_n = a_n / \left(t\\n\right)$ The rate of growth of the a_n-coefficients for t=0.5 was asymptotically $a_n \sim~ u^{0.5} * \frac{u^{\frac{n^2-n}{2}}}{n!} * m_n$ where m_n are also growing coefficients, if only the leading coefficient of the polynomials at x^n are taken into account. Now the quotient of two consecutive binomials $\frac{ \left(0.5\\n\right) }{\left( 0.5\\n+1\right) }$ seem to approach -1, so the strong growth of about u^n^2/n!, or the quotient of two consecutive coefficients of ~ u^2n/n seems to dominate the characteristic of the c_n-coefficients. A series with quotient of increasing absolute value u^(2n)/n, u>1 cannot regularly be Euler-summed; maybe it can be summed with Borel-summation of higher orders. To be "not regularly" Euler-summable does not mean, we cannot have an approximation of a certain degree; however the problem with this is, that the partial sums may converge up to a certain index n, from where it "begins to diverge" - and it is not yet known to me, to what extent we can use the intermediate approximated value - I'm investigating for verification of some experimental summation-methods of the required power. Hmm - i hope this is not more confusing than clarifying - I've my head not really free today (have to prepare the final lesson tomorrow) Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread matrix function like iteration without power series expansion - by bo198214 - 06/30/2008, 03:01 PM RE: matrix function like iteration without power series expansion - by Gottfried - 06/30/2008, 09:23 PM RE: matrix function like iteration without power series expansion - by bo198214 - 07/01/2008, 06:48 AM RE: matrix function like iteration without power series expansion - by Gottfried - 07/01/2008, 02:18 PM RE: matrix function like iteration without power series expansion - by Gottfried - 07/02/2008, 11:23 AM RE: matrix function like iteration without power series expansion - by Gottfried - 07/08/2008, 06:56 AM RE: matrix function like iteration without power series expansion - by andydude - 07/08/2008, 06:01 PM RE: matrix function like iteration without power series expansion - by andydude - 07/08/2008, 09:35 PM RE: matrix function like iteration without power series expansion - by Gottfried - 07/08/2008, 06:23 PM RE: matrix function like iteration without power series expansion - by Gottfried - 07/09/2008, 08:58 AM RE: matrix function like iteration without power series expansion - by andydude - 07/09/2008, 02:38 PM RE: matrix function like iteration without power series expansion - by andydude - 07/09/2008, 02:57 PM RE: matrix function like iteration without power series expansion - by bo198214 - 07/09/2008, 06:18 PM RE: matrix function like iteration without power series expansion - by andydude - 07/14/2008, 06:21 PM RE: matrix function like iteration without power series expansion - by bo198214 - 07/14/2008, 09:55 PM RE: matrix function like iteration without power series expansion - by Gottfried - 07/10/2008, 07:01 AM

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