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+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
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):
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
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 monotonicIn 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
