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Gottfried Helms\' change of base formula
#1
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) * B
Assume s in the range 1/e^e<s<1 or 1<s<e^(1/e) and for convenience
simply take

Code:
:   s = t^(1/t)    1<t<e
Then, for nonnegative integer y

Code:
:  V(1)~ * B(s)^y [,1]= s^^y
or
Code:
:   sum(r=0..inf) B(s)^y[r,1]  = s^^y

Denote the eigensystem-composition of B(s)
Code:
:    B(s) = Qs * Ds * Qs^-1



Assume assumption b) true, then for the set of eigenvalues of B(s) is
Code:
:  Ds = diag([1,log(t),log(t)^2,log(t)^3, .... ])
and also

Code:
:  B(s)^y = Qs * Ds^y * Qs^-1

thus

Code:
:   V(1)~ Qs * Ds^y * Qs^-1 [,1] = s^^y

also for noninteger y
----------------------------------------------------------

Now the problem is how to compute

Code:
:     s1^^y1  = z
and  s2^^y2  = z
    where s1,y1,s2 is given, z is computed and y2 is sought.

We take an admissible
Code:
:     t1, then s1 = t1^(1/t1), L1 = log(t1)
      t2, then s2 = t2^(1/t2), L2 = log(t2)

With the above apparatus we can write the following:

Code:
: 1)  V(1)~ * B(s1)^y1  = V(z)~   // where in the second column of the result is z  
  2)  V(1)~ * B(s2)^y2  = V(z)~   // where V(z) is computed by the previous
                                      and y2 is sought
Decomposing 2) into its eigensystem
Code:
: 2.1)  V(1)~ * Qs2 * Ds2^y2 * Qs2^-1 = V(z) ~
  -->   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) ~ * Qs
Determine 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)^-1
and 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)    ~   0.916290731874
I 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)= -0.102366635258
I 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
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