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 Tetra-series Gottfried Ultimate Fellow Posts: 767 Threads: 119 Joined: Aug 2007 11/01/2009, 07:45 AM (This post was last modified: 11/01/2009, 08:00 AM by Gottfried.) (10/31/2009, 09:37 PM)andydude Wrote: The way that I got the coefficients is slightly different than your method. I did this: Let $\mathbf{B}$ be a matrix defined by $B_{jk} = \frac{1}{j!} \text{spow}_k^{(j)}(1)$, and let $ \begin{tabular}{rl} f(x) & = \sum_{k=1}^\infty f_k (x - 1)^k \\ & = \sum_{k=1}^\infty g_k ({}^{k}x) \\ F &= (f_0, f_1, f_2, ...)^T \\ G &= (g_0, g_1, g_2, ...)^T \\ \end{tabular}$ then $\mathbf{B}.F = G$ so I thought, if we know G (1, -1, 1, -1, ...), then $F = \mathbf{B}^{-1}G$ and when the matrix size is even I get the first series, and when the matrix size is odd, I get the second series. Ah, now I understand, B is the matrix which transforms the f- into g - coefficients, G is given and F is sought... B is not triangular here: how do you get the correct entries for its inverse, btw? But whatever: I use this idea too, frequently. However in many instances I found in our context of exponentiation and especially iterated exponentiation, that the inverse of some matrix X represents highly divergent series, such that systematically results which are correct using the non-inverted matrix are not correct for the inverse problem using the (naive) inverse of X. This is especially the case for some matrix X, whose triangular LR-factors have the form of a q-binomial matrix. Such LR-factors occur by a square matrix X = x_{r,c} = base^(r*c) or X = x_{r,c} = base^(r*c)/r! or the like, and if X shall be inverted by inversion of its triangular factors. Such matrices X occur for example in the interpolation which I called "exponential polynomial interpolation" for the T-tetration (or sexp)-Bell-matrices. I used that matrix X also in the example for the "false interpolation for logarithm"-discussion. (But I could not yet find a workaround for the occuring inconsistencies with the inverse) Now I don't see the precise characteristics of your B-matrix so far; I've just to actually construct one and to look into it to be able to say more. Let's see... Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread Tetra-series - by Gottfried - 11/20/2007, 12:47 PM RE: Tetra-series - by andydude - 11/21/2007, 07:14 AM RE: Tetra-series - by Gottfried - 11/22/2007, 07:04 AM RE: Tetra-series - by andydude - 11/21/2007, 07:51 AM RE: Tetra-series - by Gottfried - 11/21/2007, 09:41 AM RE: Tetra-series - by Ivars - 11/21/2007, 03:58 PM RE: Tetra-series - by Gottfried - 11/21/2007, 04:37 PM RE: Tetra-series - by Gottfried - 11/21/2007, 06:59 PM RE: Tetra-series - by andydude - 11/21/2007, 07:24 PM RE: Tetra-series - by Gottfried - 11/21/2007, 07:49 PM RE: Tetra-series - by andydude - 11/21/2007, 08:39 PM RE: Tetra-series - by Gottfried - 11/23/2007, 10:47 AM RE: Tetra-series - by Gottfried - 12/26/2007, 07:39 PM RE: Tetra-series - by Gottfried - 02/18/2008, 07:19 PM RE: Tetra-series - by Gottfried - 06/13/2008, 07:15 AM RE: Tetra-series - by Gottfried - 06/22/2008, 05:25 PM Tetra-series / Inverse - by Gottfried - 06/29/2008, 09:41 PM RE: Tetra-series / Inverse - by Gottfried - 06/30/2008, 12:11 PM RE: Tetra-series / Inverse - by Gottfried - 07/02/2008, 11:01 AM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 10:38 AM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 11:01 AM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 01:25 PM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 02:40 PM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 09:37 PM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 10:33 PM RE: Tetra-series / Inverse - by Gottfried - 11/01/2009, 07:45 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 03:56 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 04:12 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 05:04 AM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 12:58 PM

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