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
Assume s in the range 1/e^e<s<1 or 1<s<e^(1/e) and for convenience
simply take
Then, for nonnegative integer y
or
Denote the eigensystem-composition of B(s)
Assume assumption b) true, then for the set of eigenvalues of B(s) is
and also
thus
also for noninteger y
----------------------------------------------------------
Now the problem is how to compute
We take an admissible
With the above apparatus we can write the following:
Decomposing 2) into its eigensystem Determine the first part
so
Determine the second part
Combine
Since diagonal-matrices commute we may reorder diag(R1):
and since these are all diagonal matrices we can omit the V(1)~
summing-vectors and it must be for all diagonal-entries
Now the assumption is, that the entries of Ds2 are powers of L2,
so from the second entries alone we have the scalar equation:
from where
Using Ioannis' notation of the h()-function, where
============================================
I tried this numerically, with a surprising result. ;-)
I got
Having V(z) I can compute according to the previous formulae
from where finally y2 can be determined:
So, theoretically it should be
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
I get
and actually, using this value, empirically
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
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
simply take
Code:
: s = t^(1/t) 1<t<e
Code:
: V(1)~ * B(s)^y [,1]= s^^y
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, .... ])
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
Code:
: 2.1) V(1)~ * Qs2 * Ds2^y2 * Qs2^-1 = V(z) ~
--> V(1)~ * Qs2 * Ds2^y2 = V(z) ~ * Qs
Code:
: 2.2 V(1)~ * Qs2 = R1 ~ = V(1) * diag(R1)
so
Code:
: 2.3 V(1) * diag(R1) * Ds2^y2 = V(z) ~ * Qs
Code:
: 2.4 V(z)~ * Qs = R2~ = V(1)~ * diag(R2)
Code:
: 2.5 V(1)~ * diag(R1) * Ds2^y2 = V(1) ~ * diag(R2)
Code:
: 2.6 V(1)~ * Ds2^y2 = V(1) ~ * diag(R2)*diag(R1)^-1
summing-vectors and it must be for all diagonal-entries
Code:
: 2.7 Ds2[r,r]^y2 = R2[r]/R1[r]
so from the second entries alone we have the scalar equation:
Code:
: 2.8 L2^y2 = R2[1]/R1[1]
Code:
: 2.9 y2 = log(R2[1]/R1[1])/log(L2)
= log(R2[1]/R1[1])/log(log(t2))
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
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
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