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 Comments on examples on Daniel's "tetration.org" Gottfried Ultimate Fellow Posts: 764 Threads: 118 Joined: Aug 2007 03/08/2016, 12:24 PM (This post was last modified: 03/08/2016, 06:17 PM by Gottfried.) 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 $f^{\circ 0.5}(x-t)+t$ 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 Daniel Fellow Posts: 69 Threads: 25 Joined: Aug 2007 03/11/2016, 09:09 AM (This post was last modified: 03/11/2016, 07:24 PM by Daniel.) (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 $f^{\circ 0.5}(x-t)+t$ 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 « Next Oldest | Next Newest »

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