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.
And then possibly for some adaption/cure, so that the results can be made matching. For instance, Ramanujan-summation for divergent series includes one integral term to correct for the change-of-order-of-summation which is an internal detail in that summation method, possibly we should find something analoguous here.

also note that the 2 matrix-method number must sum to 1 !!

- which also shows the importance of the determinant !! -

Thank you for the double exclamation. They don't introduce a determinant of an infinite sized matrix but make much noise, which I do not like as you know from earlier conversations of mine in sci.math. So I'll stop that small conversation on your postings here as I don't have to say much more relevant at the moment for the other occasional and interested reader.

Gottfried