Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Solving tetration for base 0 < b < e^-e
(09/12/2009, 09:04 PM)mike3 Wrote: So then since neither of those worked, it seems all we're left with is the Ansus formula and the Cauchy integral (but determining the correct contours and asymptotic behavior, now that's the rub...).

I'm also curious: What about exactly? You said it converges slowly, but how do you iterate it at all? What is the asymptotic as the tower ?

I used a complex fixpoint and could generate the matrices for regular iteration (in context of diagonalization). The fixpoint I used is
t0 = -0.1957457524880764 - 1.691199920910569*I
one of its logarithms is
u0 = 0.5320921219863799 + 4.597158013302573*I
where u0 = log(t0) + 2*Pi*I // log giving the principal branch
With this I create the triangular Bell-matrix and diagonalize.

The series has complex terms and is very difficult to evaluate - I accelerate slow converging series usually with Euler-summation, but the series has complex terms and it seems I need also complex order for Euler-summation. With 128 terms I could at least get results which reproduced the integer iteration to such an approximate that I'm confident that the series can be used in principle.
However, the fractional iterates behave even worse, and two half-iterates reproduce the integer iterate just to two decimals... Sad

The schröder-term s for schr(x') and x'=x/t0 - 1 at x=1 is, according to the last three partial sums of the series (128 terms):
[126]  -0.4119542792176348+1.439754774257274*I
[127]  -0.4119542792176264+1.439754774257268*I
[128]  -0.4119542792176181+1.439754774257268*I
where I assume s~ -0.411954279217... +1.439754774257...*I as correct decimals.
From here we can compute y' = schr°-1(s * u0^h ) and with h=1 I reproduce exp_b°1(1) = b =exp(-exp(1)) to 15 digits exact.
The last three partial sums of the series:
[126]   -1.004456441437337+0.03850266639539727*I
[127]   -1.004456441437337+0.03850266639539727*I
[128]   -1.004456441437337+0.03850266639539727*I
From here y' = -1.004456441437337+0.03850266639539727*I, y=(y'+1)*t0 , y-b = -7.405812 E-17 + 2.57894 E-15*I is an acceptable approximation. However - it is only the integer-iteration. I've not yet found en Euler-order or another convergence acceleration which stabilizes the partial sums for the half-iterate to an acceptable degree.

Gottfried Helms, Kassel

Messages In This Thread
RE: Solving tetration for base 0 < b < e^-e - by Gottfried - 09/13/2009, 06:14 AM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Complex Tetration, to base exp(1/e) Ember Edison 7 3,031 08/14/2019, 09:15 AM
Last Post: sheldonison
  Is bounded tetration is analytic in the base argument? JmsNxn 0 1,566 01/02/2017, 06:38 AM
Last Post: JmsNxn
  Solving tetration using differintegrals and super-roots JmsNxn 0 2,141 08/22/2016, 10:07 PM
Last Post: JmsNxn
  tetration base sqrt(e) tommy1729 2 3,673 02/14/2015, 12:36 AM
Last Post: tommy1729
  Explicit formula for the tetration to base [tex]e^{1/e}[/tex]? mike3 1 3,281 02/13/2015, 02:26 PM
Last Post: Gottfried
  tetration base > exp(2/5) tommy1729 2 3,448 02/11/2015, 12:29 AM
Last Post: tommy1729
  regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 Gottfried 7 10,132 06/25/2013, 01:37 PM
Last Post: sheldonison
  tetration base conversion, and sexp/slog limit equations sheldonison 44 60,379 02/27/2013, 07:05 PM
Last Post: sheldonison
  simple base conversion formula for tetration JmsNxn 0 3,491 09/22/2011, 07:41 PM
Last Post: JmsNxn
  Does anyone have taylor series approximations for tetration and slog base e^(1/e)? JmsNxn 18 22,930 06/05/2011, 08:47 PM
Last Post: sheldonison

Users browsing this thread: 1 Guest(s)