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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` While 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: `...` 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)` RE: 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` 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)) ]` post 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. |