10/14/2013, 10:07 PM
(10/13/2013, 02:20 PM)Gottfried Wrote: I've just updated my discussion from 2010 where I provided pictures and short commentars for the basic introduction into different interpolation proposals for the tetration. I included now also the comaprision with the Kneser-method, where I used one of the Pari/GP-scripts which Sheldon has kindly provided here.
Here is the link:
http://go.helms-net.de/math/tetdocs/Comp...ations.pdf
Gottfried,
I was able to reproduce your graph on page 9, by plotting sexp(z+k), where \( k=1+\text{slog}(0.1i) \approx -0.002+0.107i \), and z varied from -7 to 1. You might try other values of k. As imag(k) increases, the Kneser technique behaves more and more like the Schroeder function. For example, try k=0.5i, graphing sexp(z+k) from z=-7 to z=1, and you see a very nice well defined spiral towards to fixed point, just like the Schroeder function solution. Here interpolation works very well, since interpolation naturally approaches the Schroeder function solution.
Closer to the real axis, the singularities at integer values <=-2 become more and more pronounced, which causes problems for interpolation.
- Sheldon
