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

[Gottfried]

dear gottfried , im confused about your last post ...

plz clarify.

there is no secret at the alternating sums of iterates of consecutive (integer) heights: just approximate it using your favorite software which is capable of Cesaro-/Abel-/Eulersummation. In Pari/GP you use "sumalt":

Code:

\\ define function f(x) for forward iteration and g(x) for backward iteration (=negative height)
\\(additional parameter h for positive integer heights is possible)
f(x,h=1) = for(k=1,h,x = x^2 - 0.5 ); return (x) ;
g(x,h=1) = for(k=1,h,x = sqrt(0.5 + x) ); return (x) ;

\\ do analysis at central value for alternating sums x0=1
x0 = 1.0
sp = sumalt(h=0,(-1)^h * f(x0 , h))
sn = sumalt(h=0,(-1)^h * g(x0 , h))
y = sp + sn - x0

to reproduce the "serial" sums. (clearly you would optimize this by exploiting the fact, that the function-calls in sumalt use strictly consecutive integer heights)

Then y is in general (for real x0 in a unit-interval of iteration) not zero.

Using the matrix-method to compute the alternating sums I get systematically y=0 due to the rules of matrix-algebra. The interesting point is, that always one of the alternating sums is correct, either sp or sn - and I did not yet see, which one and when and why that one.

I've just done some more discussion of the iteration of this function due to the review of this question this days on math SE (But note, that that discussion has nothing to do with the problem of the relation to the matrix-based method as mentioned in my earlier post)