Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Tetra-series
#27
(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 be a matrix defined by , and let

then

so I thought, if we know G (1, -1, 1, -1, ...), then

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
Reply


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

Possibly Related Threads...
Thread Author Replies Views Last Post
Question Taylor series of i[x] Xorter 12 10,518 02/20/2018, 09:55 PM
Last Post: Xorter
  Taylor series of cheta Xorter 13 11,310 08/28/2016, 08:52 PM
Last Post: sheldonison
  Derivative of E tetra x Forehead 7 8,770 12/25/2015, 03:59 AM
Last Post: andydude
  [integral] How to integrate a fourier series ? tommy1729 1 2,353 05/04/2014, 03:19 PM
Last Post: tommy1729
  Iteration series: Series of powertowers - "T- geometric series" Gottfried 10 15,829 02/04/2012, 05:02 AM
Last Post: Kouznetsov
  Iteration series: Different fixpoints and iteration series (of an example polynomial) Gottfried 0 2,710 09/04/2011, 05:59 AM
Last Post: Gottfried
  What is the convergence radius of this power series? JmsNxn 9 15,034 07/04/2011, 09:08 PM
Last Post: JmsNxn
  An alternate power series representation for ln(x) JmsNxn 7 13,029 05/09/2011, 01:02 AM
Last Post: JmsNxn
  weird series expansion tommy1729 2 4,074 07/05/2010, 07:59 PM
Last Post: tommy1729
  Something interesting about Taylor series Ztolk 3 6,175 06/29/2010, 06:32 AM
Last Post: bo198214



Users browsing this thread: 1 Guest(s)