Andrew Robbins' Tetration Extension

[Gottfried]

Hi -
I reread this thread yesterday.

Now I tried a matrix-version of the slog - I'd now like to see, if the two methods agree.

In short: I suppose most matrices as known, as a reminder I only recall that the matrix Ut performs the decremented iterated exponentiation Ut(x) to base t here:

Code:

´
V(x)~ * Ut   = V(y)~ where y=t^x-1 = Ut(x)
V(x)~ * Ut^h = V(y)~ where y= Ut°h(x)

-----------

Now to ask for the superlog is to ask for the height h, given y and x=1.

This can surprisingly easy be solved using the known eigenmatrices Z of Ut
let u=log(t) then

Code:

´  Ut   = Z * dV(u) * Z^-1

then also

Code:

´  Ut^h = Z * dV(u^h) * Z^-1

Then the equation

Code:

´ V(x)~ * Ut^h = V(y)~

can be rewritten as

Code:

´
   V(x)~ * (Z * dV(u^h) * Z^-1) = V(y)~
  (V(x)~ * Z) * dV(u^h) * Z^-1  = V(y)~

and it follows that

Code:

´ (V(x)~ * Z) * dV(u^h) = (V(y)~ * Z)

--------------------------------------------

Since we need only the second column of the evaluated parentheses, let's denote the entries of the r'th row of the second column of Z as z_r

Then define a function

Code:

´ g(x) = z0 + z1*x + z2*x^2 + z3*x^3 + ...

and we have

Code:

´ g(x)*u^h = g(y)

and

Code:

´
u^h = g(y)/g(x)
h = log(g(y)/g(x)) / log(u)

follows.

If g(x) diverges conditionally, it may still be Euler-summable (see **)

For base t=2 the sequence of z_k diverges with a small rate, and seem all to be positive. If x is negative, then this is surely Euler-summable, if |x|<1/2 then it is even convergent.

Example-terms z_r for t=2

Code:

´
z_r= [0, 1.0000000, 1.1294457, 1.1985847, 1.2474591, 1.2856301, 1.3170719, 1.3439053, ... ]

-----------------------------------------

By the definition b=t^(1/t) we may use this also for the tetration-function (or better "iterated exponential" in Andrew's wording) of base b, since

Code:

´ Tb(x) = Ut(x')"   where x'=x/t-1 and x"=(x+1)*t

and the eigenvalues of Tb^h are the same as of Ut^h (namely = dV(u^h)).
(but remember, that this use of fixpoint-shift gives varying results dependent of the choice of the fixpoint - however small the differences may be, according to our current state of discussion)

So

Code:

´ for Ut°h(x) = y

the height-function hghU() gives

Code:

´ h = hghU_t(y) =  log( (g(y)/g(1)) / log(log(t))

and for

Code:

´  Tb°h(x) = y

the height-function hghT() gives

Code:

´
h = hghT_b(y) = slog_b(y)
           =  log( (g(y')/g(1')) / log(u)
           =  log( (g(y/t-1)/g(1/t-1)) / log(log(t))

The other good news are, 1) that I have also an extremely simple recursive eigensystem-solver for Ut, which needs only about 3-7 seconds for 96x96-matrices (if the Stirling-numbers are precomputed) depending on float-precision and 2) we need only the second column for this computation and the algorithm can thus be much reduced.

Gottfried

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(**) the Euler-summation adds coefficients e_k of weight to each term in g(x), so the Euler-summed variant eg(x) of the dim-truncated powerseries is then
eg(x) = e0*z0 + e1*z1*x + e2*z2*x^2 + e3*z3*x^3 + ... + e_dim*z_dim*x^dim

where the e_k have to be determined by a given size of matrix-truncation and a given appropriate Euler-order.