Thank you for your consideration of my proposal!
:-) - just based upon this method... : this method is just designed to define a half-iterate at a first hand.
Consequently we have the "uniqueness criterion" that it is just compatible with the alternating iteration series (call it "\( asum(x) \)" in the following ...
... where the half iterates are defined to be at the extrema of the \( asum(x) \).
The idea to generalize it to continuous real height is to define any real height via the inverse sinus function: we measure the \( asum(x) \) at each x, and we find one \( x_k \) where \( asum(x_k)=0 \). Because \( asum(x) \) is periodic with two iterations of \( b^x-1 \) we can choose a convenient *x* to be \( x_0 \) for which we define the height *h* to be zero. Then in the near we find one extremum of \( asum(x) \), where the *y*-value is the general amplitude for that periodicity. Then we measure each \( asum(x) \) for \( x=x_0 \) to \( x=x_2=b^{b^{x_0}-1}-1 \) and define the height to be the arcsine of \( {asum(x) \over amplitude } \) thus
I hope that some advantage of this definition over some other "naive" methods and 2-periodic (but otherways arbitrary) functions shall come out from the fact, that the resulting curve of the relation from x to \( h_x \) is smooth and the relation from \( h_x \) to \( asum(x_h) \) is a perfect sine-curve where we could compute x from h using the \( \sin \) and\( asum^{-1}(x) \) if...
... yes, if we had also an explicite inverse of the \( asum(x) \)-function. Currently I'm replacing that missing inverse by a binary search/newton-interpolation in the same way as Andy Robbins does it with his own sexp()-method, but that Newton method is not safe near the half-iterates where the derivative is near zero. I'm just investigating to find a better interpolation there.
It is, btw., very near to the "regular decremented exponentiation" in the development at the attracting fixpoint 0 (I've not yet checked the difference to the regular dxp at the repelling fixpoint, may be the asum-method defined here is for instance in the middle of that two methods). At the example-base b=1.3 the difference to the regular dxp at fractional heights is smaller than, say 1e-6.
Yes, this is a OWN method, and thus are its interpolated fractional heights/function values. Surely the method can only then be more than a fancy game if it has some interesting relations of its fractional iteration-definition to other places in number theory.
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The picture shows the relation of x_h and h for the base b=1.3. The upper (repelling) fixpoint is at about x=8.1, asymptotic to infinite negative height h (or: beyond the left side of the picture), the lower (attacting) fixpoint is at x=0 asymptotic to infinite positive height (or: beyond the right side of the picture)
(12/08/2012, 06:27 PM)tommy1729 Wrote: Gottfried assumed that they - the extrema - must be half-iterates ... but half-iterates based upon which method ?
:-) - just based upon this method... : this method is just designed to define a half-iterate at a first hand.
Consequently we have the "uniqueness criterion" that it is just compatible with the alternating iteration series (call it "\( asum(x) \)" in the following ...
... where the half iterates are defined to be at the extrema of the \( asum(x) \).
The idea to generalize it to continuous real height is to define any real height via the inverse sinus function: we measure the \( asum(x) \) at each x, and we find one \( x_k \) where \( asum(x_k)=0 \). Because \( asum(x) \) is periodic with two iterations of \( b^x-1 \) we can choose a convenient *x* to be \( x_0 \) for which we define the height *h* to be zero. Then in the near we find one extremum of \( asum(x) \), where the *y*-value is the general amplitude for that periodicity. Then we measure each \( asum(x) \) for \( x=x_0 \) to \( x=x_2=b^{b^{x_0}-1}-1 \) and define the height to be the arcsine of \( {asum(x) \over amplitude } \) thus
\( h_x = {\sin^{-1}{asum(x) \over amplitude} \over \pi } \)
I hope that some advantage of this definition over some other "naive" methods and 2-periodic (but otherways arbitrary) functions shall come out from the fact, that the resulting curve of the relation from x to \( h_x \) is smooth and the relation from \( h_x \) to \( asum(x_h) \) is a perfect sine-curve where we could compute x from h using the \( \sin \) and\( asum^{-1}(x) \) if...
... yes, if we had also an explicite inverse of the \( asum(x) \)-function. Currently I'm replacing that missing inverse by a binary search/newton-interpolation in the same way as Andy Robbins does it with his own sexp()-method, but that Newton method is not safe near the half-iterates where the derivative is near zero. I'm just investigating to find a better interpolation there.
It is, btw., very near to the "regular decremented exponentiation" in the development at the attracting fixpoint 0 (I've not yet checked the difference to the regular dxp at the repelling fixpoint, may be the asum-method defined here is for instance in the middle of that two methods). At the example-base b=1.3 the difference to the regular dxp at fractional heights is smaller than, say 1e-6.
Quote: Afterall , all methods agree on what an integer iteration is , but most disagree about what a half-iterate is. Not to mention requirements such as analytic and properties such as multiple fixpoints.The comment on the half iterate is above; the analycity is a delicate point. I'll look at it if I'll have a full working implementation.
Quote:So without properties or uniqueness criteria , it is just a half iterate by its OWN method.
Yes, this is a OWN method, and thus are its interpolated fractional heights/function values. Surely the method can only then be more than a fancy game if it has some interesting relations of its fractional iteration-definition to other places in number theory.
Quote:... no potential.Well, thanks anyway.
[hline]
The picture shows the relation of x_h and h for the base b=1.3. The upper (repelling) fixpoint is at about x=8.1, asymptotic to infinite negative height h (or: beyond the left side of the picture), the lower (attacting) fixpoint is at x=0 asymptotic to infinite positive height (or: beyond the right side of the picture)
Gottfried Helms, Kassel
