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

[tommy1729]

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

this is not my expertise ... yet.

but i think i have seen those before in some far past.

for starters , i related your sums to equations of type f(x) = f(g(x)).

also , ergodic theory studies averages of type

F(x) = lim n-> oo 1/n (f^[0](x) + f^[1](x) + ... f^[n](x).)

hidden telescoping can indeed occur.

and sometimes we can rewrite to an integral.

but again , this is not my expertise yet.

you gave me extra question instead of an answer :p

in particular i do not understand your matrix idea in this thread.

my guess is that when you start at 1.0 , you use carleman matrices to compute the sum and one carleman matrix will not converge ( lies outside the radius ) for 1.0 ; so one is wrong and the other is not.

talking about alternating series 1/2 -1/3 + 1/5 -1/7 + 1/11 - ...

i believe this has a closed form/name and if i recall correctly its called the first mertens constant ...

there was something else i wanted to say ... forgot :s

edit : i do not know how to rewrite an average as a sum or superfunction ( do know integral and perhaps infinite product )... i say that because it might be usefull to see the link with the " ergodic average " ( or whatever its called ).

it bothers me , i wanna get rid of this " lim **/n " term for averages. ( might also be of benefit for number theory and statistics )