Maybe this was already adressed elsewhere, so if this is so, some kind reader may please link to that entry.
Playing again with the structures of Andy's slog I came to the following observation (so far with base e only, but I think it's trivial to extend).
Consider the power series for slog(z) as given by Andy's descriptions, and define slog0(z) by inserting zero as constant term instead of -1 as in slog(z). slog0(e^^h) gives now h+1 for the argument e^^h.
But moreover, now the series for slog0(z) can formally be inverted.
One can observe, that its coefficients are near that of the series for log(1+z), so let's define the inverse to the slog-function as tetration-function
taylorseries(T0) = serreverse(slog0) - taylorseries(log(1+x))
Then the coefficients of T0() decrease nicely, and we can compute e^^h (best for fractional h in the range -1<h<0 )
e^^h = T0(1+h) + log(2+h)
The series for T0() look much nicer than that of the slog(), but of course the coefficients are directly depending on the accuracy of the coefficients of the slog()-function, so I still used slog with, say, matrix-size of 96 or 128 for a handful of correct digits.
I've never worked with Jay D. Fox's extremely precise solutions for the slog-matrix so I cannot say anything how the coefficients of T0() would change. Would somebody like to check this?
Gottfried
Appendix: 32 Terms of T0(h) taken from slog0(z) with matrixsize of 64
Appendix 2:
Jay D. Fox has provided very accurate coefficients for the slog-function. Using the first 128 of that leading coefficients to recompute T0(h) I arrive at the remarkable solution for e^^pi where 20 digits match Jay's best estimate:
thread see at http://math.eretrandre.org/tetrationforu...php?tid=63 "Improving convergence of Andrew's slog"
post see at http://math.eretrandre.org/tetrationforu...920#pid920
The data of the taylor-series for slog with 700 terms are also in that thread.
Playing again with the structures of Andy's slog I came to the following observation (so far with base e only, but I think it's trivial to extend).
Consider the power series for slog(z) as given by Andy's descriptions, and define slog0(z) by inserting zero as constant term instead of -1 as in slog(z). slog0(e^^h) gives now h+1 for the argument e^^h.
But moreover, now the series for slog0(z) can formally be inverted.
One can observe, that its coefficients are near that of the series for log(1+z), so let's define the inverse to the slog-function as tetration-function
taylorseries(T0) = serreverse(slog0) - taylorseries(log(1+x))
Then the coefficients of T0() decrease nicely, and we can compute e^^h (best for fractional h in the range -1<h<0 )
e^^h = T0(1+h) + log(2+h)
The series for T0() look much nicer than that of the slog(), but of course the coefficients are directly depending on the accuracy of the coefficients of the slog()-function, so I still used slog with, say, matrix-size of 96 or 128 for a handful of correct digits.
I've never worked with Jay D. Fox's extremely precise solutions for the slog-matrix so I cannot say anything how the coefficients of T0() would change. Would somebody like to check this?
Gottfried
Appendix: 32 Terms of T0(h) taken from slog0(z) with matrixsize of 64
Code:
T0(h) = 0
+ 0.0917678575394 *h
+ 0.175505903737 *h^2
+ 0.0164995173026 *h^3
+ 0.0191448752458 *h^4
+ 0.00133512590560 *h^5
+ 0.00231496855708 *h^6
- 0.0000239484773943 *h^7
+ 0.000304699490128 *h^8
- 0.0000364374120911 *h^9
+ 0.0000455639411655 *h^10
- 0.0000114433561149 *h^11
+ 0.00000769463425171 *h^12
- 0.00000262533621718 *h^13
+ 0.00000145886258700 *h^14
- 0.000000604261454214 *h^15
+ 0.000000301260921078 *h^16
- 0.000000129633563113 *h^17
+ 0.0000000606527811916 *h^18
- 0.0000000293680769725 *h^19
+ 0.0000000147234175905 *h^20
- 0.00000000637812232351 *h^21
+ 0.00000000263672711471 *h^22
- 0.00000000137069796843 *h^23
+ 0.000000000858320261831 *h^24
- 0.000000000396925930057 *h^25
+ 7.48739318521E-11 *h^26
- 1.71265782834E-11 *h^27
+ 7.18443441580E-11 *h^28
- 5.88967800348E-11 *h^29
- 7.64569615630E-12 *h^30
+ 2.78278848579E-11 *h^31
+O(h^32)Appendix 2:
Jay D. Fox has provided very accurate coefficients for the slog-function. Using the first 128 of that leading coefficients to recompute T0(h) I arrive at the remarkable solution for e^^pi where 20 digits match Jay's best estimate:
Code:
37149801960.55698549 914478420500428635881 \\ using T0() with n=160 terms: e^^Pi = e^^(3+frac(Pi)) = e^e^e^[T0 (1+ frac(Pi) ) + log( 2 + frac(Pi)) ]
37149801960.55698549 914478420500428635881 \\ Using T0() with n=128 terms
37149801960.55698549 872339920573987 \\ J.d.Fox to about 25 digits precision; difference at 20. dec digitpost see at http://math.eretrandre.org/tetrationforu...920#pid920
The data of the taylor-series for slog with 700 terms are also in that thread.
Gottfried Helms, Kassel