Just for an impression. With the matrix-logarithm-method based on the exponentials recentered at e; g(h) = f(1-e,h)+e depending on the iteration height h I get the following coefficients for g(h)
which means
gives the the h'th iteration to base ae=exp(exp(-1))~ 1.4446... beginning at x0=1 in the version of the "regular tetration". This is valid for 20 digits accuracy for at least 0<=h<=2, so two unit-intervals of the iteration. The restriction to, say, 20 digits is because I've seen that ae^g(0.5) - g(1.5) ~ 1e-15 and possibly I need a significant extension of my matrices/powerseries to achieve more accuracy.
@James: as promised I'll send the Pari/GP-procedures, so you may extend the precision to higher degree yourself, but give me some time to harvest the relevant script procedures from the the jungle of my Pari/GP-script files library...
Code:
1.0
0.61109545377165144382
-0.23170261447676551100
0.091781287662054951830
-0.037564921705011073397
0.015773722201726337117
-0.0067614637772198390363
0.0029477476181950503707
-0.0013032939674486044656
0.00058306881272407925943
-0.00026347551443499556924
0.00012007976722060208750
-0.000055130237268268005131
0.000025472518304042709786
-0.000011834707545807564518
0.0000055251010640897040596
-0.0000025902729494681159146
0.0000012187271454481672018
-0.00000057508317717804681085
0.00000027193027228176594270
-0.00000012870902697527732596
0.000000060889266619581435803
-0.000000028734742331292897512
0.000000013494371318392422045
-0.0000000062882996391919038758
0.0000000028984843383423711421
-0.0000000013171193093747536382
0.00000000058812247376504184749
-0.00000000025724643591308081238
1.0991226259673585977E-10
-4.5760148165255342291E-11
1.8525282102101486163E-11
-7.2799477529346619758E-12
2.7731492194588571934E-12
-1.0228781132287377086E-12
3.6501818149236149850E-13
-1.2594229104590033680E-13
4.1994992731976094776E-14
-1.3528699883474457412E-14
4.2097881296159269770E-15
-1.2652080888516218381E-15
3.6723460072987803371E-16
-1.0294636636067164354E-16
2.7873275063956835675E-17
-7.2897324850428282565E-18
1.8417406660994510113E-18
-4.4956448474692462035E-19
1.0603756293758013064E-19
-2.4170833917511111844E-20
5.3253529443700961287E-21
-1.1342016411529572217E-21
2.3354793597373322700E-22
-4.6500987042706443614E-23
8.9536936179097868826E-24
-1.6674248602720150332E-24
3.0035886177128642416E-25
-5.2339146859694508036E-26
8.8235255818014501117E-27
-1.4391944413647261141E-27
2.2713605538391230710E-28
-3.4686799592348898270E-29
5.1259082336317069982E-30
-7.3302241660374222986E-31
1.0144000556887333339E-31Code:
g(h)= 1.0 + 0.611... h - 0.2317... h^2 + 0.09178... h^3 + ...gives the the h'th iteration to base ae=exp(exp(-1))~ 1.4446... beginning at x0=1 in the version of the "regular tetration". This is valid for 20 digits accuracy for at least 0<=h<=2, so two unit-intervals of the iteration. The restriction to, say, 20 digits is because I've seen that ae^g(0.5) - g(1.5) ~ 1e-15 and possibly I need a significant extension of my matrices/powerseries to achieve more accuracy.
@James: as promised I'll send the Pari/GP-procedures, so you may extend the precision to higher degree yourself, but give me some time to harvest the relevant script procedures from the the jungle of my Pari/GP-script files library...
Gottfried Helms, Kassel
