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Comments on examples on Daniel's "tetration.org"
#1
Hi,

after Daniel showed up again (very nice! Hi Daniel!) I just took again a look at tetration.org to see whether there is something new.
I looked at the page concerned with convergence issues of the power series for the half-iterate of bases in the Euler-range.
The pictures show the logarithms of the absolute values of the coefficients c_k for where t is the (attractive) fixpoint - at least that is what I extracted/extrapolated from the description.
The logs of the c_k were shown up to k=200 or k=500, where for instance with base 1.2 around index k=260 a strange variation and non-monotonicity of the size of the coefficients begins to appear.
I tried to reproduce the curves using Pari/GP and the above mentioned logic. When I used internal float precision of 400 digits I got a very similar result, where also the variance appears in the region of k=250. But when I incresed the numerical precision to 800 internal digits that variance disappears.
Could it be that the variance in the pictures is indeed due to numerical errors? Or have I misinterpreted the computation of the series/the coefficients c_k ?

See below a shortened list using 400 and 800 digits internal for base b=1.2 .

After I've found this list I computed also the coefficients with b=1.414 with 800 internal digits. The following plot is what I've got for the coefficients of the half-iterate by the regular iteration

Also I'd recently made a picture for MO where I show a surely very good estimate for the bounding of coefficients of the half-iterative of exp(x)-1. The coefficients seem to grow not more than hypergeometric, see the (very nicely finetuned!) formula in the legend of the second picture.


Gottfried



Code:
ooo
  Nr  precision 400 digits     precision 800 digits
------------------------------------------------
    0          0.E-404          0.E-809
    1  -0.736335617832  -0.736335617832
    2   -3.29342924031   -3.29342924031
    3   -6.74952942230   -6.74952942230
    4   -11.1640223592   -11.1640223592
    5   -13.8772406918   -13.8772406918
    6   -15.7357887397   -15.7357887397
    7   -18.7009335095   -18.7009335095
    8   -21.1402569012   -21.1402569012
    9   -23.4318834230   -23.4318834230
   10   -29.3970096392   -29.3970096392
   11   -28.9730091697   -28.9730091697
   12   -32.3692301405   -32.3692301405
   13   -36.0190285675   -36.0190285675
   14   -38.0603347018   -38.0603347018
...
  100   -247.340861153   -247.340861153
  101   -249.920373417   -249.920373417
  102   -252.547923241   -252.547923241
  103   -255.253612894   -255.253612894
  104   -258.111034539   -258.111034539
  105   -261.410201491   -261.410201491
...
  260   -628.883874967   -628.888811285
  261   -631.315274265   -631.315274265
  262   -586.833986500   -633.744377999  **************** 400 digits begins chaotize 800 digits monotonic
  263   -635.519910541   -636.176390425
  264   -638.611625299   -638.611625299
  265   -620.443251435   -641.050453260 ****************
  266   -605.019659386   -643.493316449
  267   -621.637748997   -645.940748161
  268   -648.121466163   -648.393399759
  269   -650.852078313   -650.852078316
  270   -580.032796156   -653.317800588
  271   -629.631090938   -655.791872601
  272   -620.006579663   -658.276011021
  273   -578.024123139   -660.772535697
  274   -589.075864182   -663.284690255
  275   -611.485647503   -665.817208784
  276   -607.205440566   -668.377397161
  277   -604.407015388   -670.977417011
  278      -559.289958   -673.639849955
  279   -598.049647958   -676.414574090
  280      -558.131647   -679.455102067
  281   -600.440137766   -684.557775936
  282      -567.672679   -684.358223813
  283      -563.431384   -686.009204389
  284   -593.747632011   -687.973100731
  285   -576.820240502   -690.061036154
  286   -605.499569816   -692.216081207
...
  337      -526.072153   -811.463233363
  338      -538.319262   -813.835276331
  339     -510.7556936   -816.207780825
  340      -535.873848   -818.580736729
  341      -551.058889   -820.954134509
  342      -513.343578   -823.327965174
  343      -543.146408   -825.702220251
  344     -511.6512834   -828.076891757
  345      -532.230305   -830.451972167
  346      -544.900762   -832.827454397
  347      -532.166459   -835.203331779
  348     -506.8672636   -837.579598044
  349      -517.780157   -839.956247300
  350      -546.544068   -842.333274017
  351      -512.023096   -844.710673016
  352      -521.706680   -847.088439448
  353      -543.177928   -849.466568786
  354      -526.262523   -851.845056813
  355     -502.4597448   -854.223899610 *** 800 digits still monotonic


   

In the following picture I separated the sequence of coefficients into 4 partial sequences to get smoother curves (each of the four partial sequences becomes rather smooth, even sinusoidal, while if we tried to draw the curve from the original sequence it looks ugly/disinformative jittery):

   


Gottfried Helms, Kassel
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#2
(03/08/2016, 12:24 PM)Gottfried Wrote: Hi,

after Daniel showed up again (very nice! Hi Daniel!) I just took again a look at tetration.org to see whether there is something new.
I looked at the page concerned with convergence issues of the power series for the half-iterate of bases in the Euler-range.
The pictures show the logarithms of the absolute values of the coefficients c_k for where t is the (attractive) fixpoint - at least that is what I extracted/extrapolated from the description. ...

Hello Gottfried,
Thanks for the correction. I was looking for something elusive, but with your higher resolution the phenomena disappeared. I'll just pull the outdated information from my site. Good to be working with you again.
I've updated my website with a link to your work.
Daniel
Reply


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