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+--- Thread: Andrew Robbins' Tetration Extension (/showthread.php?tid=3)
RE: Andrew Robbins' Tetration Extension - Gottfried - 12/28/2009
There is one question open for me with this computation, maybe it is dealt elsewhere.
(In my matrix-notation) the coefficients for the slog-function to some base b are taken from the SLOGb-vector according to the idea
(I - Bb)*SLOGb = [0,1,0,0,...]
where I is the identity operator and Bb the operator which performs x->b^x
Because the matrix (I - Bb) is not invertible, Andrew's proposal is to remove the empty first column - and the last(!) row to make it invertible, let's call this "corrected(I - Bb)"
Then the coefficients for the slog-powerseries are taken from SLOGb, from a certain sufficiently approximate finite-dimension solution for
SLOGb = (corrected(I - Bb))^-1*[0,1,0,0,...]
and because the coefficients stabilize for higher dimension, the finite SLOGb is taken as a meaningful and also valid approximate to the true (infinite dimensional) SLOGb .
Btw, this approach resembles the problem of the iteration series for powertowers in a nice way: (I - Bb)^-1 would be a representation for I+Bb + BB^2 + BB^3 + ... which could then be used for the iteration-series of h^^b whith increasing heights h. Obviously such a discussion needed some more consideration because we deal with a nasty divergent series here, so let's leave this detail here.
The detail I want to point out is the following.
Consider the coefficients in the SLOGb vector. If we use a "nice" base, say b=sqrt(2), then for dimension=n the coefficients at k=0..n-1 decrease when k approaches n-2, but finally, at k=n-1, one relatively big coefficient follows, which supplies then the needed value for a good approximation of the order-n-polynomial for the slog - a suspicious effect!
This can also be seen with the partial sums; for the slog_b(b)-slog_b(1) we should get partial sums which approach 1. Here I document the deviation of the partial sums from the final value 1 at the last three terms of the n'th-order slog_b-polynomial
(For crosscheck see Pari/GP excerpt at end of msg)
Examples, always the ("partial sums" - 1) up to terms at k=n-3, k=n-2,k=n-1 are given, for some dimension n
Code:
dim n=4
...
-0.762957567623
-0.558724904310
-0.150078240781
dim n=8
...
-0.153309829172
-0.120439792559
-0.00882912480664
dim n=16
...
-0.00696424577339
-0.00629653092984
-0.0000322687018600
dim n=32
...
-0.0000228720888610
-0.0000223192966457
-0.000000000473007074189
dim n=64
...
-0.000000000331231525320
-0.000000000330433387110
-0.000000000000000000108While we see generally nice convergence with increasing dimension, there is a "step"-effect at the last partial sum (which also reflects an unusual relatively big last term)
Looking at some more of the last coefficients with dim n=64 we see the following
Code:
...
-0.000000000626200198250
-0.000000000492336933075
-0.000000000417440371765
-0.000000000376261655863
-0.000000000354008626669
-0.000000000342186109314
-0.000000000336009690814
-0.000000000332835946403
-0.000000000331231525320
-0.000000000330433387110
- - - - - - - - - - - - - - - -
-0.000000000000000000108 (= -1.08608090167E-19)What does this mean if dimension n->infinity: then, somehow, the correction term "is never reached" ?
Well, the deviation of the partial sums from 1 decreases too, so in a rigorous view we may find out, that this effect can indeed be neglected.
But I'd say, that this makes also a qualitative difference for the finite-dimension-based approximations for the superlog/iteration-height by the other known methods for tetration and its inverse.
What do you think?
Gottfried
Code:
b = sqrt(2)
N=64
\\ (...) computation of Bb
tmp = Bb-dV(1.0) ;
corrected = VE(tmp,N-1,1-N); \\ keep first n-1 rows and last n-1 columns
\\ computation for some dimension n<=N
n=64;
tmp=VE(corrected,n-1)^-1; \\ inverse of the dim-top/left segment of "corrected"
SLOGb = vectorv(n,r,if(r==1,-1,tmp[r-1,1])) \\ shift resulting coefficients; also set "-1" in SLOGb[1]
partsums = VE(DR,dim)*dV(b,dim)*SLOGb \\ DR provides partial summing
disp = partsums - V(1,dim) \\ partial sums - 1 : but document the last three entries for msgRE: Andrew Robbins' Tetration Extension - tommy1729 - 08/18/2016
I told this before but ...
Lets consider the system of equations from the OP.
We need to find v_n.
INSTEAD of truncating to n x n systems and letting n grow , i consider it differently.
We want the radius to be as large as possible.
So we minimize v_0 ^2 + v_1 ^2 + ...
And expect a radius Up to the fixpoints of exp.
We truncate to an n x (n+1) system and min the Sum of squares above for the relevant v_k.
So we take 9 equations with (v_1 ... v_10) and solve for the min of v_1 ^2 + ... v_10.
Then we proceed by adding 11 variables ( v_11,... v_22 ) and solve that system with plug-in the pevious values v_j and 10 equations , and again minimizing the Sum of squares.
Then repeat ...
So
V_1 .. V_10 then v_11 .. V_21 , v_22 ... V_33 etc
So eventually all v_j get solved.
And the equations hold almost at a triangulair number distance of iterztions.
Regards
Tommy1729
RE: Andrew Robbins' Tetration Extension - Gottfried - 08/22/2016
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
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/tetrationforum/showthread.php?tid=63&pid=920#pid920
The data of the taylor-series for slog with 700 terms are also in that thread.