Reconsideration: a false interpolation-paradigm in tetration ?
- From: Gottfried Helms <helms@xxxxxxxxxxxxx>
- Date: Wed, 16 Sep 2009 12:27:45 +0200
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 [*1] 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 attention (see note below
and [*2]) 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 8 512 4096
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 orderpoylnomials 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 [*3], 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 (ref. by E.Sandifer):
Eneström-index E190. "Consideratio quarumdam serierum quae singularibus proprietatibus sunt
praeditae"
(“Consideration of some series which are distinguished by special properties”).
[*1] http://math.eretrandre.org/tetrationforum/showthread.php?tid=190&pid=2336#pid2336
pdf: http://go.helms-net.de/math/tetdocs/ExponentialPolynomialInterpolation.pdf
[*2] http://math.eretrandre.org/tetrationforum/showthread.php?tid=115
[*3] http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2050%20false%20log%20series.pdf
.
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