bo198214 Wrote:edit: Ah now I see, what puzzled me was U_t°h(x) and T_b°h(x). Gottfried, this is very prone to misinterpretation, my first reading was \( U_t\circ h (x) \), you know \( h(x) \) is used for the inversion of \( x^{1/x} \), but what you meant was \( U_t^{\circ h}(x) \)! If you dont use tex you need another presentation of iteration. The non-superscripted \( \circ \) has the completely different meaning of composition!
Yes, I'm getting a bit sloppy with this, sorry. In Tex it seems impossible to get this tiny circle, well I got it using \( U^{oh} \) (alas - I had it already - shame... )
I'd prefer a more serial notation for text, like {b,x}^^h and even better
x {[4],b} h , because this would then be concatenable:
x {[4],b} h1 {[4],b} h2 = x {[4],b} (h1+h2) and had adapted our current notation a [4] b, if... if the start-parameter (x in my notation) would be existent and the base parameter would not occupy the place, where only a concatenation is possible. So all easy ascii-available notations that I can think of are somehow exotic ... If we would have
x [4,b] h instead of b[4]h, or - well again I used h for height, so x [4,b]t instead of b[4]t, which seems to evolve as a standard currently, I would immediately adapt this notation
---
Numerics:
What surpries me a bit is, that acceleration by Euler-summation in this case does not yield much, even if the pre-transformation by Stirling matrix makes things much better. Many times I sat down to get a better insight in the characteristics of these powerseries and a good adaption by summation-methods, but this was not yet a fundamental enhancement. There is still something waiting to be discovered/characterized here.
Gottfried Helms, Kassel
