Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation...

[Gottfried]

0.580243966210
+
0.41975603379

=0.9999999999 = 1

simply because 1/(1+x) + 1/(1+(1/x)) = 1.

     

Yes, that observation was exactly what I was discussing when I presented these considerations here since 2007; especially I had a conversation with Andy on this. The next step which is obviously to do, is to search for the reason why powerseries-based methods disagree with the serial summation - and always only one of the results.    
(...)

It should be mentioned also in this thread, that the reason for this problem of matching the Carleman-based and the simple serial summation based results is simple and simple correctable.

1) The Carleman-matrix is always based on the power series of a function f(x) and more specifically of a function g(x+t_0)-t_0 where t_0 is the attracting fixpoint for the function f(x). For that option the Carleman-matrix-based and the serial summation approach evaluate to the same value.             

2) But for the other direction of the iteration series, with iterates of the inverse function f^[-1] () we need the Carleman matrix developed at that fixpoint t_1 which is attracting for f^[-1](x) and do the Neumann-series then of this Carlemanmatrix. This evaluates then again correctly and in concordance with the series summation. (Of course, "serial summation" means always to possibly include Cesaro or Euler summation or the like).       

So with the correct adapation of the required two Carleman-matrices and their Neumann-series we reproduce correctly the iteration-series in question in both directions.         

Gottfried