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

[Gottfried]

Sometimes we find easter-eggs even after easter...

For the alternating iteration-series
\( \hspace{48} sn(x,p)=\sum_{h=0}^{\infty} (-1)^h g(x,h) ^p \\
\hspace{48} \text{where } g(x)=\sqrt{0.5+x} \hspace{48} g(x,h)=g(g(x,h-1)) \hspace{48} g(x,1)=g(x) \)

(definitions as copied and extended from previous post, see below)

we find a rational polynomial for p=4. That means

\( \hspace{48} x^4 - g(x)^4 + g(x,2)^4-g(x,3)^4+ ... - ... \\
\hspace{48}= sn(x,4) = 1/8 - x^2 + x^4 \)


(maybe this is trivial and a telescoping sum only, didn't check this thorough)
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Another one:
\( \hspace{48} sn(x,1)+sn(x,2) = -0.25 + x^2
\)
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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
x = 1.0
sp(x) = sumalt(h=0,(-1)^h * f(x , h))
sn(x) = sumalt(h=0,(-1)^h * g(x , h))
y(x) = sp(x) + sn(x) - x