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 Solving tetration for base 0 < b < e^-e bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 09/13/2009, 11:23 AM (09/13/2009, 09:57 AM)Gottfried Wrote: The problem with the real fixpoint is, that the triangular Bell-matrices have (alternating signed) units on its diagonal. This prevents the computation of a matrix-logarithm as well of the diagonalization - at least in my implementations. That should not pose a problem. You need the first row of $B^t$, where $B$ is the Bell matrix. You make a Jordan decomposition $B = S J S^{-1}$ and then $B^t = S J^t S^{-1}$ where for each Jordanblock $J_m$ for eigenvalue $\lambd_m$ with multiplicity $M_m$ one sets $J_m^t = \sum_{n=0}^{M_m} \left(t\\n\right) (J_m-\lambda_m I)^n$. The sum is finite because $J_m-\lambda_m I$ is nilpotent: $J_m^{M_m}=0$. Unfortunately the Jordan decompostion is flawed in Sage so I could not try it myself. Quote:But well, let's see. It's surely not the highest summit of wisdom... and we also have the Newton-binomial-formula and others... I dont think that the Newton formula helps. It is real-valued and hence can not return a suitable solution for a decreasing base function. « Next Oldest | Next Newest »

 Messages In This Thread Solving tetration for base 0 < b < e^-e - by mike3 - 09/12/2009, 02:00 AM RE: Solving tetration for base 0 < b < e^-e - by bo198214 - 09/12/2009, 06:56 AM RE: Solving tetration for base 0 < b < e^-e - by bo198214 - 09/12/2009, 07:20 AM RE: Solving tetration for base 0 < b < e^-e - by mike3 - 09/12/2009, 07:40 AM RE: Solving tetration for base 0 < b < e^-e - by bo198214 - 09/12/2009, 07:47 AM RE: Solving tetration for base 0 < b < e^-e - by mike3 - 09/12/2009, 08:07 AM RE: Solving tetration for base 0 < b < e^-e - by bo198214 - 09/12/2009, 08:35 AM RE: Solving tetration for base 0 < b < e^-e - by mike3 - 09/12/2009, 08:50 AM RE: Solving tetration for base 0 < b < e^-e - by bo198214 - 09/12/2009, 03:48 PM RE: Solving tetration for base 0 < b < e^-e - by mike3 - 09/12/2009, 09:04 PM RE: Solving tetration for base 0 < b < e^-e - by Gottfried - 09/13/2009, 06:14 AM RE: Solving tetration for base 0 < b < e^-e - by mike3 - 09/13/2009, 07:24 AM RE: Solving tetration for base 0 < b < e^-e - by Gottfried - 09/13/2009, 09:57 AM RE: Solving tetration for base 0 < b < e^-e - by bo198214 - 09/13/2009, 11:23 AM RE: Solving tetration for base 0 < b < e^-e - by Gottfried - 09/13/2009, 11:44 AM RE: Solving tetration for base 0 < b < e^-e - by mike3 - 09/14/2009, 09:43 PM RE: Solving tetration for base 0 < b < e^-e - by bo198214 - 09/17/2009, 08:01 AM RE: Solving tetration for base 0 < b < e^-e - by mike3 - 09/17/2009, 11:03 AM RE: Solving tetration for base 0 < b < e^-e - by Gottfried - 09/13/2009, 01:34 PM RE: Solving tetration for base 0 < b < e^-e - by Gottfried - 09/13/2009, 07:34 PM RE: Solving tetration for base 0 < b < e^-e - by Gottfried - 09/14/2009, 02:02 PM RE: Solving tetration for base 0 < b < e^-e - by bo198214 - 09/13/2009, 08:08 AM RE: Solving tetration for base 0 < b < e^-e - by mike3 - 09/13/2009, 09:49 AM RE: Solving tetration for base 0 < b < e^-e - by tommy1729 - 09/18/2009, 12:16 PM RE: Solving tetration for base 0 < b < e^-e - by bo198214 - 09/18/2009, 01:00 PM

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