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 Gottfried Helms\' change of base formula Gottfried Ultimate Fellow Posts: 789 Threads: 121 Joined: Aug 2007 08/16/2007, 08:28 PM Change of base, a view from the matrix approach There is already a lot of discussion about change of base here. Without having read it all, I thought I'd try my matrix-approach and post the results. With the a) eigensystem-approach, b) assumption about the set of eigenvalues (set of powers of parameter, see below) I can apparently approximate a solution for the base-change problem. Formulae: Assume the constant operator-matrix for tetration as usual B, the parametrized version Code::    B(s) = dV(s) * BAssume s in the range 1/e^e   V(1)~ * Qs2 * Ds2^y2          = V(z) ~ * Qs Determine the first part Code:: 2.2   V(1)~ * Qs2 = R1 ~ = V(1) * diag(R1) so Code:: 2.3   V(1) * diag(R1) * Ds2^y2      = V(z) ~ * QsDetermine the second part Code:: 2.4   V(z)~ * Qs  = R2~ = V(1)~ * diag(R2)Combine Code:: 2.5    V(1)~ * diag(R1) * Ds2^y2      = V(1) ~ * diag(R2)Since diagonal-matrices commute we may reorder diag(R1): Code:: 2.6    V(1)~ *  Ds2^y2      = V(1) ~ * diag(R2)*diag(R1)^-1and since these are all diagonal matrices we can omit the V(1)~ summing-vectors and it must be for all diagonal-entries Code:: 2.7    Ds2[r,r]^y2  = R2[r]/R1[r]Now the assumption is, that the entries of Ds2 are powers of L2, so from the second entries alone we have the scalar equation: Code:: 2.8   L2^y2  = R2[1]/R1[1]from where Code:: 2.9  y2 = log(R2[1]/R1[1])/log(L2)           = log(R2[1]/R1[1])/log(log(t2))Using Ioannis' notation of the h()-function, where Code::     from   t^(1/t) = s   ==>  t = h(s) Code:: 2.9.1 y2 = log(R2[1]/R1[1])/log(log(h(s2)) ============================================ I tried this numerically, with a surprising result. ;-) Code:: Using t1 = 2   ==> s1 = sqrt(2) ~ 1.414...         y1 = 3               ==> z = 1.414...^^3                t2 = 2.5     ==> s2 = 2.5^(2/5)  ~   1.44269990591         y2 = ?? (unknown, sought)                          L2 = log(t2)    ~   0.916290731874I got Code::   V(1)~ * B(s1)^3 = V(z)~     where     V(z) = [1.00000000000, 1.76083955588, 3.10055594155, 5.45958154710, ... ] and thus     z = V(z)[1] = 1.76083955588 Having V(z) I can compute according to the previous formulae Code::   R1~  = [1.00000000000   -675168330336.   107316405043.   -16669350847.3 ... ]     R2~  = [1.00000000000   -858097840871.   204571124661.   -60399411879.7 ... ]                        log(R2[1]/R1[1])= log(-858097840871./-675168330336.)= log(1.27093911594) ~ 0.239756088578 from where finally y2 can be determined: Code::   y2 = log(R2[1]/R1[1])/log(L2) =   -0.239756088578 / -0.0874215717908 = 2.74252777280 So, theoretically it should be Code:: a)  (2^(1/2))^^3               =  z = 1.76083955588   b)  (2.5^(2/5))^^2.74252777280 =  z = 1.76083955588 But with my small matrices of dim=24 (reduced because of eigenanalysis) the assumption about the eigenvalues (and thus of the diagonal-matrix Ds2 ) is aproximated only with high relative error. Errors occur also with the eigenvector-matrices (but in sum they seem to mutually cancel out for many cases) If I didn't use the theoretical eigenvalue L2, but the "empirical", L2emp, from the empirical diagonal-matrix Ds2 Code::   L2 =    0.916290731874     L2Emp = D2[1,1] = 0.902698529882  log(L2Emp)= -0.102366635258I get Code::   y2 = log(R2[1]/R1[1])/log(L2emp) = -0.239756088578 / -0.102366635258 = 2.34213118340 and actually, using this value, empirically Code::   V(1)~ * B(s2)^2.34213118340 = V(z) where     V(z)  = [   1.00000000000   1.76084065618   3.10055981646   5.45959178175 ... ]` which agrees with the version V(1)~ * B(s1)^3 = V(z) as shown above. ------------------------------ My eigensystem-analyses are unfortunately restricted to small dimensions, so deviations from theoretical expectations can be relatively high without obvious contradiction to the assumptions. The range of admissible parameters is unfortunately relatively small, it is in the range 1/e^e < s < e^(1/e), but additionally with relative wide epsilon areas at the limits and also around 1. So the methods should be improved. A bit of convenience allows the Euler-summation, which accelerates the convergence of oscillating series-terms down to such small number of accessible terms (which is needed in many sum-formulae) and gives acceptable approximates. Further interesting should be, what is, if one base s1<1 and the other base s2>1. Then I expect logarithms of negative numbers and tetration with complex exponent (but I didn't try this yet due to the numeric instability so far) Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

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