Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Tetra-series
#13
I've made a little progress with the problem of deviance of the serial summed alternaing series of powertowers of increasing heights ("Tetra-series"). Since T- and U-tetration can be mutually converted by shift of their parameter x, I can concentrate on the U-tetration here, which has the advantage, that its operator is a triangular matrix, whose integer-powers and eigensystem are easily and exact (within the unavoidable size-truncation for the actual computation) computable.
Code:
Denote the fixed base for

lb(x) = log(1+x)/log(b)
ub(x) = b^x - 1

The iterates

ltb(x,h) = ltb(lb(x),h-1)     ltb(x,0)= x
utb(x,h) = utb(ub(x),h-1)     utb(x,0)= x

The infinite alternating sums

ASLb(x) = x - ltb(x,1) + ltb(x,2) - ltb(x,3) + ... - ...
ASUb(x) = x - utb(x,1) + utb(x,2) - utb(x,3) + ... - ...
The matrix-approach suggests to compute the AS-series using the geometric-series of the U-tetration-matrices S1b and S2b, which I shall call MLb and MUb here. From here the conjecture was derived, that
Code:
ASLb(x) + ASUb(x) = x    // matrix-computation
since
Code:
MLb + MUb = I            // matrices

However, the computation of ASLb(x) along its partial sums and summation using Cesaro- or Euler-summation gives a different result:
Code:
ASLb(x) + ASUb(x) = x + db(x)  // serial-computation

The difference of the matrix- and serial-computation may then be expressed by db(x) only.

---------

In my previous posts I already mentioned, that the deviance between the two methods seem to be somehow periodic, wrt x as the variable parameter.

The first useful result was, that I found periodicity for db(x)
Code:
db(x) = - db(ltb(x,1)) = db(ltb(x,2)) = - db(ltb(x,3)) ...

for few bases and numerical accessible range for x. The plot showed also a sort of sinus-curve, but where the frequency was somehow distorted.

Today I could produce a very good approximation to a sinus-curve, using fractional tetrates for x.
I computed the k=0..32 fractional U-tetrates of x for height 1/16
Code:
x_k = ltb(1,k/16)
for base b=sqrt(2), and instead of x I used the index k as x-axis for the plot.
This provided a very good approximation to a sinus-curve for db(x_k)

In the plot I display the two near-lines for ASLb(x_k), ASUb(x_k) and db(x_k) and overlaid a sin-curve, whose parameters I set manually by inspection.
The curve for db(x_k) and sin() match very good; I added also a plot for their difference.

If I can manage to make my procedures more handy, I'll check the same for more parameters. The near match of the curves give apparently good hope that this line of investigation may be profitable...

Gottfried
[updated image]

   


Error-curve( deviation of db(x) from overlaid sinus-curve)

   
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
  Perhaps a new series for log^0.5(x) Gottfried 3 1,399 03/21/2020, 08:28 AM
Last Post: Daniel
Question Taylor series of i[x] Xorter 12 15,026 02/20/2018, 09:55 PM
Last Post: Xorter
  An explicit series for the tetration of a complex height Vladimir Reshetnikov 13 15,450 01/14/2017, 09:09 PM
Last Post: Vladimir Reshetnikov
  Complaining about MSE ; attitude against tetration and iteration series ! tommy1729 0 2,048 12/26/2016, 03:01 AM
Last Post: tommy1729
  2 fixpoints , 1 period --> method of iteration series tommy1729 0 2,096 12/21/2016, 01:27 PM
Last Post: tommy1729
  Taylor series of cheta Xorter 13 16,328 08/28/2016, 08:52 PM
Last Post: sheldonison
  Derivative of E tetra x Forehead 7 10,843 12/25/2015, 03:59 AM
Last Post: andydude
  [AIS] (alternating) Iteration series: Half-iterate using the AIS? Gottfried 33 46,685 03/27/2015, 11:28 PM
Last Post: tommy1729
  [integral] How to integrate a fourier series ? tommy1729 1 3,107 05/04/2014, 03:19 PM
Last Post: tommy1729
  Iteration series: Series of powertowers - "T- geometric series" Gottfried 10 19,115 02/04/2012, 05:02 AM
Last Post: Kouznetsov



Users browsing this thread: 1 Guest(s)