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A false interpolation paradigm (?); a reconsideration
Note: the following is a pretty long statement which I also want to post in sci.math, so please forgive that I did not put it into Tex-format

For the computation of tetration to fractional heights(iterates) I employ the diagonalization of operator matrices. This implements well-known manipulation of the coefficients of formal powerseries; in fact if the base b for tetration is b=exp(exp(-1)) this can be done by matrix-logarithm and if 1<b<exp(exp(-1)) we can directly apply diagonalization.

But because of the notorious difference of solutions, when the series are developed around different fixpoints, I'm still not confident, that this method is the final/the best solution.

In an earlier article I discussed a simple interpolation-approach, intended as a replacement for the diagonalization for difficult (for instance: complex) bases and instead found, that this agrees with the diagonalization down to the level of identity of the coefficients of the occuring powerseries.

This interpolation follows the common idea of polynomial interpolation resp. its generalization to the case of infinite order/ of powerseries, or of the use of (finite case) vandermonde-matrix. Here factors like (x-1),(x^2-1),(x^3-1) etc occur typically and essentially in numerators and denominators.

I had earlier brought the "false logarithmic series" to the readers' attention (see note below and this link) and this time I tried that interpolation-technique to the problem of re-engineering the series for the logarithm, and see, whether we get the correct series.

Now let's say better: "a" logarithm, because we find, that this interpolation gets correct results at integer arguments but systematically wrong results at fractional arguments - thus reflecting the observation in tetration, where the different fixpoints give identical results at integer and differing results at fractional heights, and so the fractional height are not reflected optimally with *any* such series developed around some fixpoint.

Let's look at a simple example for "a false logarithmic series".

We want to find a powerseries for the logarithm to base 2; such that with this series at parameter x we find the base-2-logarithm of x. Propose this with the initially unknown coefficients a,b,c,d,...
´ log_2(x) = a + bx + cx^2 + dx^3 + ...    // unknown coefficients a,b,c to be determined
and let's approach this problem stepwise from finite polynomials to a final generalization to a powerseries.

First we may set up a set of equations to find the unknown coefficients a,b,c,d for a cubic polynomial.
We write
´ x=2^0:   a + b 2^0 + c (2^0)^2 + d (2^0)^3  = 0
  x=2^1:   a + b 2^1 + c (2^1)^2 + d (2^1)^3  = 1
  x=2^2:   a + b 2^2 + c (2^2)^2 + d (2^2)^3  = 2
  x=2^3:   a + b 2^3 + c (2^3)^2 + d (2^3)^3  = 3
and solve by the vandermonde-method. Let's write this as matrix-equation

First we write the matrix of coefficients VV_3 (index 3 for dimension)
´  VV_3 = [1  1   1    1]
           1  2   4    8
           1  4  16   64
           1  8  64  512

    C  = columnvector[a,b,c,d]. the C-oefficients          
    L  = columnvector[0,1,2,3], the L-og values

and VV_3 * C = L
and solve C = VV_3^-1 * L

We get a polynomial in x
´  f_3(x) = -31/21 + 7/4*x - 7/24*x^2 + 1/56*x^3
For the integer exponents we get
´ f_3(2^0) = 0   = log_2(1)
  f_3(2^1) = 1   = log_2(2)
  f_3(2^2) = 2   = log_2(4)
  f_3(2^3) = 3   = log_2(8)
and for the interpolation to some fractional exponent we get, for instance
´  f_3(2^0.5) = 0.465857551857   =/= 0.5 = log_2(2^0.5)

From the computation-scheme it is obvious, how this can be generalized to higher order poylnomials and higher order approximates. However, we do not have an approximating procedure to the true logarithms at fractional exponents, however high the dimension (and thus the order of the polynomials) are.
´ f_12(2^0.5) = 0.473784748806
  f_24(2^0.5) = 0.473811031008
  f_48(2^0.5) = 0.473811037422
  f_96(2^0.5) = 0.473811037422
with a deviation from the correct value of about

f(2^0.5) = 0.5 - 0.0261889625777

The series, as dimension/order goes to infinity, approximates to
´ lim n->oo  f_n(x) = -1.60669515242 +  2*x - 4/9*x^2 + 8/147*x^3 - 16/4725*x^4 + 32/302715*x^5 + O(x^6)
which will give correct results for natural exponents but will be false with fractional exponents.

The method of interpolation is using the paradigm of polynomial interpolation, which even if generalized to infinite order of polynomials will remain to give false results for fractional exponents.
The matrix-method for tetration employs either directly the same interpolation-method (see my discussion on "exponential polynomial interpolation", an ugly term, but I did not find a better one) or in an obscured way (we can express an identity between diagonalization and this interpolation-method).

So -possibly- the same way as we need a move from this interpolation-paradigm to arrive at a meaningful series for logarithm, we need a move to arrive at a more meaningful interpolation for fractional tetration.

What do you think?

Gottfried Helms

Note: The original idea of the "false logarithm" was triggered by an article "How Euler did it - a false logarithm series" of Ed Sandifer in MAA-online, where he introduced to a similar analysis discussed by L.Euler
I didn't check the actual relation between the interpolation-method here and the Euler-series, but I note that the first coefficient -1.60669... occurs also in that article.
For the Euler-paper see:
Eneström-index E190. "Consideratio quarumdam serierum quae singularibus proprietatibus sunt praeditae"
(“Consideration of some series which are distinguished by special properties”).
Gottfried Helms, Kassel

Messages In This Thread
A false interpolation paradigm (?); a reconsideration - by Gottfried - 09/16/2009, 11:14 AM

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